A METEOROLOGICAL TREATISE ON THE o n . _ / c Circulation and Radiation IN THE ATMOSPHERES OF THE EARTH AND OF THE SUN BY FRANK H. BIGELOW, M.A., L.H.D. Professor of Meteorology in the U. S. Weather Bureau, 1891-1910, and in the Argentine Meteorological Office since 1910 FIRST EDITION FIRST THOUSAND NEW YORK JOHN WILEY & SONS, INC. LONDON: CHAPMAN & HALL, LIMITED 1915 Copyright, 1915, by FRANK H. BIGELOW PUBLISHERS PRINTING COMPANY 207-217 West Twenty-fifth Street, New York INTRODUCTION METEOROLOGY as a science has failed to make progress toward definite results for one fundamental reason. In a non- adiabatic atmosphere the terms of the general equations of motion, as computed from the ordinary prescribed formulas of thermodynamics, do not balance as required. There are two errors in the discussion: (1) There is a mixture of the non- adiabatic and the adiabatic systems, and (2) the important radiation terms have been omitted from the general equations. More specifically, for the Boyle-Gay Lussac Law, P = p T R y to be satisfied, at every point, it has been customary to borrow R = gas constant from the adiabatic system, and apply it in the non-adiabatic atmosphere. For example, three well known treatments follow: Bigelow. v. Bjerknes. Margules. W*T Pl=(Tl\T^ fi./ Tj P, \TJ P Q ~ \T P* \T PQ \TJ Po \T U ~ = Constant RI = R = Con. Margules' system is adiabatic, v. Bjerknes' is partly adiabatic and partly non-adiabatic, Bigelow's is strictly non-adiabatic. Ta-T ,_Cp. TX-TO " Cu Now it is true that each system satisfies P = p T R, but the individual values of PI, 7\, pi, RI, are very different in the three systems for the same initial values P , T , P O , RO, and on applying them to practical observations the systems that are not strictly non-adiabatic break down as regards computed and observed values which should be in agreement. We can easily see the iii 489536 IV INTRODUCTION separate consequences by the following equations that are readily demonstrated: Adiabatic: g fa - Zo ) = - Cp a (T a - To) = - Pa ~ - g fa ZQ) = ^1 Cp a (Ti TO) = - * Pad Non-adiabatic: g fa - z ) = - HI Cp lQ (7\ - T ) - HI (Cp a - g ( Zl -Z Q ) = - -^ - K (?i 2 - <?o 2 ) - (Qi - Co). Gravity = Pressure -f- Circulation + Radiation term. term. term. term. In the adiabatic system by definition R a and Cp a are constant, and there are no circulation and no radiation. In the non- adiabatic system there are both circulation and radiation, but these depend upon the departure of the specific heat from the adiabatic value ni (Cpa ~ Cpio) (7\ - To) = K (? 2 i - ? 2 o) + (Qi - <2o) Hence, if R is constant it is a contradiction in terms to discuss problems of circulation and radiation in the adiabatic or partially adiabatic systems, as has been universally the method. The facts of observation, furthermore, conform to r> p - ni Cpio (Ti - To) = -- > and this implies that circulation and radiation are required to make up the deficiency in respect to the gravity term between two strata z\ and Z Q . Finally, the radiation term (Qi Q ) is usually very much larger than the kinetic energy term ^ (qi 2 qf) y but it has never been incorporated in the meteorological equations. For these reasons the author has spent much time while in the United States Weather Bureau, and especially while in the INTRODUCTION V Argentine Meteorological Office, 1910 to date, in devising a simple adjustment of the thermodynamic adiabatic equations, found in all treatises, to an exact and practical form of computa- tion which will adapt them to the non-adiabatic system prevail- ing in the atmospheres of the earth and of the sun. The following Treatise sets forth this new method of discussing the meteorologi- cal problems, with sufficient detail to enable the reader to utilize the formulas in practical computations. It contains the solution of a number of problems that have heretofore been intractable along the old lines of procedure: 1. The diurnal convection and the semi-diurnal barometric waves, with the radiation. 2. The pressures and temperatures in cyclones and anti- cyclones, with the circulation and radiation. 3. The thermodynamics of the atmosphere from balloon ascensions to great altitudes. 4. The thermodynamics of the general circulation. 5. The distribution of the radiation in all latitudes and alti- tudes to 20,000 meters. 6. The "solar constant" of radiation and the conflicting results from pyrheliometers and bolometers. 7. The discrepancy in the absolute coefficient of electrical conduction as derived from the several apparatus for dissipation, and for the number and velocity of the ions. 8. The diurnal magnetic variations in the lower strata of the atmosphere. 9. The non-periodic magnetic variations in their relation to the solar radiation. 10. The magnetization and electrical terms in the sun at very high temperatures. I wish to express my appreciation and gratitude to my friend and colleague, Dr. Walter G. Davis, Director of the Argentine Meteorological Office, for his courteous co-operation in this work, and in memory of our good old Cordoba days. FRANK H. BIGELOW. CORDOBA, ARGENTINA. September, 1915. TABLE OF CONTENTS CHAPTER I PAGE METEOROLOGICAL CONSTANTS AND ELEMENTARY FORMULAS .... 1 The Status of Meteorology 1 The Constants and Formulas of Static Meteorology 9 Three Series of Constants in Three Systems of Units 12 Variations from the Standard P , Po, RO, T 0t for the Same Point and for Different Points on the Same Vertical Line 16 The Acceleration of Gravitation 18 The Density of the Atmosphere as a Mixture of Several Constituent Gases .20 The General Formulas for the Mixture of Gases 23 The Inner Kinetic Energies, Work and Heat 26 The Fundamental Laws of Physics 28 The Kinetic Theory of Gases for the Atmosphere 30 The Temperatures and Temperature Gradients Observed at Different Elevations in the Free Air 32 The Temperature Gradient in a Plateau 34 The Integral Mean Temperature and Gradient 36 The Virtual Temperature 37 /d T -jr 37 The General Barometric Formula 39 Corrections to the Barometer 41 Examples of the Barometric Reduction Tables . 46 CHAPTER II THERMODYNAMIC METEOROLOGY 50 General Formulas for the Computation of P, p, R, from the Ob- served Temperatures T in a Free Non-Adiabatic Atmosphere . 50 The Adiabatic Equations 52 The Working Non-Adiabatic Equations 53 The Variable Values of n = ^ 57 The Differentiation of (172) 58 The Two Laws of Thermodynamics 64 The First Law of Thermodynamics 65 Fundamental Equations and Definitions 68 The Second Law of Thermodynamics 70 The Reversible Process 71 The Irreversible Process 71 Carnot's Cyclic Process 72 vii Vlll CONTENTS PAGE THERMODYNAMIC METEOROLOGY Continued Cyclic Process for Vapors at Maximum Pressure 73 The Second Form of the Equations for Latent Heat . 75 Specific Heats 76 Examples of the Thermodynamic Data 77 Application of the Thermodynamic Formulas to the Non-Adiabatic Atmosphere 80 Application of the Thermodynamic Formulas to Various Meteoro- logical Problems 90 The Isothermal Region 91 The Diurnal Convection and the Semi-Diurnal Waves in the Lower Strata 98 The Thermodynamic Structure of Cyclones and Anticyclones . . . 104 The Planetary Circulation and Radiation 113 The Thermodynamic Tables of the Planetary Circulation and Radiation 118 CHAPTER III THE THERMODYNAMICS OF THE ATMOSPHERE 135 The Co-ordinate Axes 135 The Co-ordinate Velocities and Accelerations 136 The Constituents of the Force in any Direction 136 The Forces of Inertia, Expansion, and Contraction 137 The Forces of Rotation 139 The Pressure Gradients 143 The Potential Gradient 144 The Equations of Continuity 145 The Operator V 2 , and the Total Differential j- { 146 Summary of the Equations of Motion in Rectangular, Cylindrical, and Polar Co-ordinates 147 The Equations of Motion for the Rotating Earth in Cylindrical Co- ordinates 148 in Polar Co-ordinates 149 Connection between the peneral Equations of Motion and the Ther- mal Equations of Energy 151 The Equations for the Work of Circulation 153 CdP The Evaluation of the Term / 154 Numerical Check on the Two Systems of Formulas 157 Numerical Evaluations of the Pressure Gradient 157 Evaluation of the Ratios -j-^ and -j 159 dB dx To Find the Difference of Pressure (Bi B) at the Distance DI that will just Balance the Eastward Velocity v 159 The Angular Velocity of the Earth's Rotation, a> 3 160 The Linear Absolute and Relative Velocities 161 CONTENTS ix PAGE THE THERMODYNAMICS OF THE ATMOSPHERE Continued Evaluation of the Barometric Gradients 162 Application of the General Equations of Motion to the Local Circula- tions in the Earth's Atmosphere 163 Discussion of the Cylindrical Equations of Motion 164 Ferrel's Local Cyclone 164 The German Local Cyclone 166 The General Equation of Cylindrical Vortices 168 The Angular Velocity 171 The Total Pressure 171 Application of the Vortex Formulas to the Funnel-Shaped Tube . 172 to the Dumb-bell-Shaped Tube 172 The Total Pressure 173 The Relation Between Successive Vortex Tubes 173 The Second Form of the Cylindrical Equations of Motion in Terms of the Current Function ^ 175 The Funnel-Shaped Vortex 175 The Dumb-bell-Shaped Vortex 177 The Deflecting Force 178 The Force of Friction 179 The Transformation of Energy and the Circulation of the Atmosphere 180 Case I. The Change of Position of the Layers 181 The Evaluation of / T dm in Linear Vertical Temperature Changes 183 Case II. Effect of an Adiabatic Expansion or Contraction in a Non- Adiabatic Temperature Gradient 184 Case III. The Overturn of Deep Strata in the Column 185 Case IV. The Transformation of Two Masses of Different Tempera- tures on the Same Level into a State of Equilibrium .... 187 Case V. Local Changes between Two Strata of Different Tempera- tures 188 The General Circulation on a Hemisphere of the Earth's Atmosphere 189 Three Cases of the Slope of the Temperature Gradients and the Re- sulting Velocity of the East and West Circulations 191 CHAPTER IV EXAMPLES OF THE CONSTRUCTION OF VORTICES IN THE EARTH'S ATMOSPHERE 196 The Cottage City Water Spout, August 19, 1896 199 The St. Louis Tornado, May 27, 1896 202 The De Witte Typhoon, August 1-3, 1901 205 The Ocean and Land Cyclones 207 The Ocean Cyclone, October 11, 1905 209 The Composition of Vortices 212 The Reversed Dumb-bell Vortex 213 Historical Review of the Three Leading Theories Regarding the Physical Causes of Cyclones and Anticyclones in the Earth's Atmosphere 216 X CONTENTS PAGE EXAMPLES OF THE CONSTRUCTION OF VORTICES IN THE EARTH'S ATMOSPHERE Continued (1) Ferrel's Warm- and Cold-Center Cyclones 216 (2) Hann's 'Dynamic Cyclonic Whirls . . 216 (3) Bigelow's Asymmetric Cyclones and Anticyclones 216 The General and the Local Components 220 The Normal and the Local Velocities in Storms 221 The Normal and the Local Isobars in Cyclones and in Anticyclones . 229 The Graphic Construction of Resultants 232 The Normal and Local Isotherms in Cyclones and in Anticyclones . . 235 The Normal and Local Velocity Vectors in Cyclones and Anti- cyclones 240 The Land Cyclone 243 Recapitulation of the Formulas for the Dumb-bell-Shaped Vortex . 250 The Meaning of the Tangential Angle i 251 Example of the Evaluation of the Terms in the Equations of Motion for a Cyclone 254 CHAPTER V RADIATION, IONIZATION, AND MAGNETIC VECTORS IN THE EARTH'S ATMOSPHERE 261 The Determination of the Intensity of the Solar Radiation by Obser- vations with the Pyrheliometer and the Bolometer 262 The Pyrheliometer 263 Example of the Practical Observations with the Pyrheliometer at La Quiaca, Argentina, Sept. 22, 1912 264 The Bouguer Formula of Depletion 267 The Depletion of the Incoming Radiation from a Maximum Value on the Cirrus Levels, as Determined by Observations at Different Heights 274 The Relative Efficiency of 1 Gram of Aqueous Vapor .per Cubic Meter in Absorbing the Incoming Radiation 276 The Bolometer and Its Energy Spectrum of Radiation 277 Evaluation of the Wien-Planck Formula of Radiation 278 The Values of -r4f in the Radiation Formula 290 dz The Measures of the lonization of the Atmosphere 292 Notation and Elementary Relations 293 Electrostatic Relations per Unit Length 295 Conduction lonization Currents 296 Coefficient of Electrical Dissipation of the Atmosphere ^ . . . . 297 Elster and Geitel Dissipation Apparatus 297 Ebert Ion Counter 298 The Formulas for the Velocity of the Ions 299 Ebert Velocity Apparatus 300 Gerdian Apparatus for the Number and Velocity of the Ions . . . 302 CONTENTS XI PAGE RADIATION, IONIZATION, AND MAGNETIC VECTORS IN THE EARTH'S ATMOSPHERE Continued The Cause of the Discrepancy in the Values of the Conductivity of the Atmosphere as Determined by Two Methods 304 The Atmospheric Electric Potential 309 CHAPTER VI TERRESTRIAL AND SOLAR RELATIONS 312 The Five Types of the Diurnal Convection in the Earth's Atmosphere 312 The Diurnal Variations of the Meteorological, Electrical, and Magnetic Elements 317 The Diurnal Variations of the Terrestrial Magnetic Fields . . . 323 The Aperiodic Magnetic Vectors along the Meridians 329 The Synchronous Annual Variations of the Solar and the Terres- trial Elements . . 335 The Aqueous Vapor in the Atmosphere 342 The Laws of the Evaporation of Water from Lakes, Pans, and Soils with Plants 343 The Polarization of Sunlight in the Atmosphere 348 Solar Physics 352 The Spherical Astronomy of the Sun 356 The Magnetic Fields of the Earth and the Sun 360 Conclusion 363 CHAPTER VII EXTENSION OF THE THERMODYNAMIC COMPUTATIONS TO THE TOP OF THE ATMOSPHERE 364 Remarks on the Bouguer Formula 364 I. First Distribution of Temperature 365 Illustrations of the Use of Erroneous Densities . 372 The Thermodynamic Terms 374 The Constituents of the Solar and the Terrestrial Radiations in the Earth's Atmosphere 375 The Total, Transmitted and Absorbed Amounts of the Solar and the Terrestrial Radiations . . 377 II. Second Distribution of Temperature 380 Summary of the Computations for Twenty-one Balloon Ascensions . 384 The Effective Energy of Radiation and the Solar Constant .... 388 The Second Method of Discussing Pyrheliometric Data .... 395 The Annual Mean Variations 401 The General Summary 403 General Remarks 404 The Constants and Coefficients of Dry Air in the Kinetic Theory of Gases throughout the Atmosphere 406 Summary of the Dimensions with Special Reference to the Equiv- alents 414 General Problems in Atmospheric and Solar Physics 417 A METEOROLOGICAL TREATISE ON THE CIRCULATION AND RADIATION IN THE ATMOSPHERES OF THE EARTH AND OF THE SUN CHAPTER I Meteorological Constants and Elementary Formulae The Status of Meteorology MODERN meteorology may be said to have begun its "scientific development about the year 1870. Since that time an enormous mass of observations has been made, covering every branch of the subject, but the classification of the data in any systematic form has been singularly inadequate. This defective progress may be attributed chiefly to two causes, the first, the practical units in which the instrumental readings are commonly made, and the second, the usual misapplication of the Boyle-Gay Lussac Law in the atmosphere, whereby it is assumed that the gas coefficient is constant and the conditions strictly adiabatic. Thus, for the pressure P , thejiensity p Q , the temperature T , and the gas coefficient ^ , the law is, (1) Po = PO #o To, and it must be satisfied at every point, but in passing from one point to another, whereby PI p\ TI change values, the observa- tions made in balloon and kite ascensions are not verified without making R change with the height above the sea-level. This variation of R carries with it a variability of the specific heat by the formula, (2) Cp = -L- Ri where the ratio of the specific heats is k = Cp/Cv, and thence the changes from one level to another follow, for the free heat 02 1~ Co), the entropy (Si - 5 ), the work (Wi - W ), the inner [1] 2 * ' ' -'KLETEORdtbGlCAL CONSTANTS AND FORMULAS energy (Ui Uo), the radiation energy (Ki K ), and the coefficient c and exponent a in the radiation function, (3) K = c T. It can be proved, furthermore, that unless Cp varies from the adiabatic specific heat Cp a for R a constant, there can be no circulation and no radiation of heat in the atmosphere, such as actually exists. It is the purpose of this Treatise to develop the working formulae on a systematic plan, so that they shall be adapted to the computation of the necessary data throughout the atmosphere, as in the diurnal convection, the local and cyclonic circulation, and in the general planetary circulation, by means of which numerous problems may be studied without an undue amount of speculation. The difficulty regarding the type of the observations available is that the records are not suitable for use in the formulae without special reductions, and that a number of different sets of units are employed. There are three systems of units in more or less consistent use, (l) the meter-kilogram-second-Centigrade degree system (M. K. S. C), (2) the centimeter-gram-second-Centi- grade degree system (C. G. S. C), (3) the foot-pound-second- Fahrenheit degree system (F. P. S. F). The pressure, B n , for the fundamental formulae is recorded in barometric milli- meters of mercury, or in inches of mercury, instead of in force units, employing the acceleration of gravity, (4) g ^= Q.8060 (1 - 0.00260 cos 20) (l - y), where 9.8060 in meters is the acceleration per second in latitude y = 45 ? and at sea-level z = 0, for R = 6370191 meters, the radius of the earth. The pressure of one atmosphere in units of force is, (5) P = g OPm B , where p m = 13595.8 kilograms per cubic meter, the density of mercury. Similarly, (6) Po = go PO Jo, where p = 1.29305 density of air in the same units, and / = STATUS OF METEOROLOGY 3 7991.04 the height of the homogeneous atmosphere. For B n = 0.760 meter, the pressure P = 101323.5 kilograms per square meter. Again, in units of mass, (7) po = = p m Bo = Po h= 10332.8. Finally, in heat units, P n = 24.2106 where the mechanical equivalent of heat is (9) A = 426.837 kilogram meters, one large calorie. The pressure of the atmosphere is, therefore, Po = 101323.5 in units of force, po = 10332.8 in units of mass, p A = 24.2106 in units of heat, BQ = 0.760 in barometric units, so that the common barometer readings must be wholly trans- formed for practical computations in dynamic and thermody- namic problems which involve circulation and radiation. Similarly the (C. G. S. C) system and the (F. P. S. F) system each involves four other values for one atmospheric pressure, and it often happens that there occur crosses between these three systems, so that meteorological literature is confused and difficult to comprehend in any clear manner. The temperature is usually recorded in Centigrade degrees, C, or in Fahrenheit degrees, F, but the computations must be executed in degrees of absolute temperature. (10) T = 273 + / (C) Centigrade, (11) T = 459.4 + / (F) Fahrenheit, so that another series of transformations is required. It would improve matters greatly to mark barometers in terms of PO and thermometers in degrees T, and reconstruct the entire series of working tables. The relations between the barometric and the force pressures are as follows : 4 METEOROLOGICAL CONSTANTS AND FORMULAS TABLE 1 1 Scale division = 0.75 mm. A 5 = 100. AP A5 in millimeters AP units of force Unit Equivalents 100. 13332. 10. 1333.2 1.0 mm. A B = 133.32 A P 1. 133.32 0.1 13.332 0.75 mm. A B = 100. A P 75. 10000. 7.5 1000. 0.0075mm. A B = 1.0 A P 0.75 100. 0.075 10. If the divisions on the barometer scale are made 0.75 millimeter, the scale distance is 100 units of force in the (M. K. S. C) system, so that the readings of the barometric pressure are immediately available for all forms of dynamic and thermodynamic meteor- ology. This scale is equally valuable for public purposes and synoptic chart construction. If the absolute temperatures (J^, 7\) are observed on two levels of the atmosphere at the heights (z , 1), respectively, the formulas permit the corresponding values to be computed for (Po, Pi), (PO, PI), CRo, Ri), and (Cpo, Cpi). It can be shown that T, fl, FIG. 1. The temperature, velocity, and heat in the stratum (z l 2 ) if Cp a is the specific heat for the adiabatic R a , and Cpw is the mean specific heat for the stratum (z , Zi), the circulation and the radiation are given by the formula, (12) (Cp a - Cp lt ) (T a - - Qo), STATUS OF METEOROLOGY 5 where the terms are related to the points as indicated in Fig. 1 for the stratum (z , Zi). If (T a To) is the adiabatic temperature fall in the vertical distance (si z ), and (Ti T ) the actual observed tempera- ture fall, the ratio between them is, , - (T a ~ TQ) dp Qi - Zp) _0o = -(T,- To) "" a( Zl -z ) - a' Again, if (q , <?i) are the corresponding velocities without re- gard to direction, and (Qi - Q ) the loss of free heat in the stratum, these quantities are all connected together by the equation (12). This states that the kinetic energy of the circula- tion for the unit mass ^ (<?i 2 q 2 ), together with the heat exchange of the radiation energy of the free heat (Qi Q ), depends upon the divergence of the actual mean specific heat Cpio from the adiabatic value, (14) . Cp a = 993.5787 at constant pressure. The corresponding values for Cv a amd R a are (15) Cv a = 706.5453 at constant volume. Cp a - Cv a = R a = 287.0334 the gas coefficient. Hence. -~^ = k = 1.4062486 the ratio of the specific heats. Cv a (16) Cp a - Cv a = Cv a (k - 1) fto\ fl a (18) 777 - 7; = 5- = r -- 7 = 2.461545. Cp a - Cv a R a k - I The corresponding values in a non-adiabatic atmosphere become, Cp Cp k (19) (20) Cp -Cv ' R Cv Cv cp - Cv R k - r f T \n l where R = R a l^r/ as will be proved. 6 METEOROLOGICAL CONSTANTS AND FORMULAS These two systems will be more fully developed, it being the purpose here merely to point out the leading line of the construc- tion of this work. By the definition of the specific heat, (21) + Cp a (T. - T.) = - go (i - z), for the unit mass, so that = -~ = 0.0098695 C/meter, (23) a = - * 1 - jt = ^ =-12- = ^- C/meter. Zi Z Q tpio nCpa n From (17), (19), (22), and (23) k 1 R a R a . do R a a R a T\ T Q It will be shown from the data of observation that (25) C#,, (T a - To) = 2Z. PlO Subtracting (25) from (21), (26) (Cp a - Qio) (T a - To) = - ^ (21 - z ) ^^ Pio Equating (12) and (26), (27) - go ( Zl -**) = + - + Y 2 fe 2 - go 2 ) + (Qi - Co), Pio and this is the general equation of condition in the atmosphere, showing that for the unit mass the force of gravitation is balanced by the change of pressure, the kinetic energy of the circulation, and the radiating heat. Now, returning to the registration of the fundamental quan- tities, in addition to the temperature T, and the pressure P, there is also the velocity q. While the sum of the energy of the circula- tion and the radiation can be computed through the difference of the specific heats (Cp a Cpi Q ), there is no way to separate the circulation from the radiation except through the direct observation of the velocity. The radiation must be computed STATUS OF METEOROLOGY indirectly through the gravitation, pressure, and circulation terms taken together in an inequality. As these three terms seldom balance in the free atmosphere, which is continually ex- changing heat at every point, it is evident that the adiabatic conditions, in connection with the general equation of motion, are not capable of giving a complete solution of any of the im- portant thermodynamic problems of the atmosphere. The literature of meteorology is defective in this respect. It should be noted that the point of departure for this treatment of the problem consists in making R and Cp variable, as previously stated. The velocity vector q (, z>, w, a, 0) requires special con- sideration as to the axes of co-ordinates and the angular direc- Radius of the Earth + Z direction outward +W velocity upward (X.Y.Z.) (u.v.w.) West Parallel of latitude f y = direction East +V +V = velocity East FIG. 2. The rectangular co-ordinate axes with component velocities and angles tions. There is great confusion in meteorology in the manner of recording the motions of the atmosphere. The popular use of the compass points giving the directions from which the wind blows in the azimuth rotation N, E, S, W, is entirely inapplicable 8 METEOROLOGICAL CONSTANTS AND FORMULAS in meteorological computations. This system should be reversed in two respects, (1) The vector direction is that toward which the air moves, instead of that from which it blows, making an azimuth difference of 180; (2) the proper co-ordinate axes make the azimuth rotation (S, E, N, W). The result of these two changes is effected by the formula. (28) = 360 - A, where A is the azimuth in degrees from the north through the east, and the azimuth from the 'south through the east, the vector being changed to record the direction "towards" instead of the direction "from" which the wind moves. The rectangular co- ordinate axes are fixed by the common convention of a right- handed rotation about a radius of the earth with positive trans- lation outward. (Meridian) + x = axis positive southward; + u = velocity south. (Parallel) + y = axis positive eastward; + v = velocity east. (Radius) + z = axis positive outward; -f w = vertical velocity. (29) Horizontal Plane, s = (x 2 + y 2 )*. = (u 2 + *)*. (30) Vertical Plane. r = (x 2 + y* + zrf. q = (u 2 + v* + w 2 ) 1 . (31) Azimuth angle (S, E, N, W), tan = -- = . z w (32) Vertical angle (above horizon), tana = -- = . The same co-ordinate relations should be employed in terrestrial magnetism, atmospheric electricity, and vector physics generally. Besides recording the direction of motion very inconveniently, the velocities themselves usually require a series of transforma- tions to reduce than to practical dynamics. The anemometers are commonly graduated in kilometers per hour, or in miles per hour, but they should be graduated in meters per second for CONSTANTS AND FORMULAS 9 the (M. K. S.) system, and in feet per second for the (F. P. S.) system in order to conform with the other terms of equation (27). In the electrical self-registration of the wind direction it is common to limit the compass points to eight in number, N, NE, E, SE, S, SW, W, NW, but in all problems requiring accurate wind deflecting components, as in studies of the diurnal convec- tion, it is necessary to use at least 16 points of 22.5 each in order to compute the observed, resultant, and deflecting vectors. Finally, it would be much better to record the azimuth /3 (S, E, N, W) in degrees of arc, as can be readily done by a good me- chanical device. The vertical angle a must be computed instead of observed, because it is small except in tornadoes and cannot be ordinarily measured mechanically. It is very important to record the wind vector (q, 0, a) in the system thus described, in order to facilitate all studies in the higher problems. It may be noted that .there is evidence to show that the wind velocity recorded by the anemometer is on a scale about 20 per cent, greater in the United States than in Europe. This subject should be fully tested as soon as possible. It is also known that the ordinary anemometer registers excessive velocities as compared with a force dynamometer, such that the recorded value 40 means 33, 60 means 48, 80 means 62, thus introducing great errors in the dynamic equations unless corrected. The Constants and Formulas of Static Meteorology Meteorology distributes itself into three parts in accordance with the requirements of equation (27). Static meteorology develops, (33) - g(* - 2o ) = and it is that which is generally used in the construction of synoptic weather charts and the other elementary problems. Dynamic meteorology develops, (34) - g ( Zl - Zo ) = + ^^ + y, ( ?1 - <?'), Pio 10 METEOROLOGICAL CONSTANTS AND FORMULAS and is concerned with the several general equations of motion connecting circulation and pressure. Thermodynamic meteorology develops, (27) - g (ft - so) = + ^-=-^2 + 1 A (<?i 2 - <?o 2 ) + (Qi - Co), Pio and unites the radiation with the circulation and the pressure through the functions of work and inner energy. It follows that the term g (z\ z ) may be broken up into three parts : (36) - g (z - Z Q ) = - g (zi - Z Q ) - g (zz - zi) - g (* - 22) where we have, respectively, r> 7> (33) g (zi s ) = H L the pressure effect, Pio (37) - g (z 2 - zi) = + y 2 (qi 2 - g 2 ) the circulation effect, (38) - g fa - a) = + (Qi - Qo) the radiation effect. Each of these terms is effective in disturbing the normal pressure, temperature, and density levels, which would assume fixed positions when uninfluenced by the absorption and the emission of solar and terrestrial radiation, the entire process being the means of continually returning to normal equilibrium. In order to derive the constants and the formulas for static meteorology, the formulas (25) and (33) are united to form, (39) - Pl ~J = fto fe - z ) = - Qio (T, - To), where the mean gravity and the mean specific heat between the two vertical points, z\ and z 0l are to be used. Then', (40) - (Pi.- Po) = 10 Pio fe -Zo) = - pioQio (Ti - To). Since for a column on a base of unit square area pio (z\ Z Q ) = M, the mass that produces the pressure (Pi P ) when acted upon by the force of gravity gi , in the differential equation, is, (41) - J dP = J gdm = j gpdz = - JpCp d T. If the upper limit is at the top of the atmosphere, and the lower limit at the bottom of it, on the sea level in latitude 45, CONSTANTS AND FORMULAS 11 and for the temperature T = 273, (40) reduces to, (42) Po = go po Zi = PO Cpo (273 - 7\). When the temperature of reduction is (43) 3Ti - 273 the last form in (42) disappears. The constants of static meteor- ology conform to (42) for any substance whatsoever: water, mercury, dry air, aqueous vapor, or mixtures. If P^ is the pressure of one standard atmosphere the density must change Tn an inverse proportion with the height. In the following notation, Substance Density Height Column Water Pw HUJ Water column Mercury Pm Bo Barometer Dry air Po /o Homogeneous Aqueous vapor 'Pz h Vapor column (42) becomes, specifically, (44) Po = go P w h w = go P m BQ = go Po /O = go pZ /2- and, (water) (mercury) (dry air) (aqueous vapor) p (45) = p w h w = p m B Q = pp/p_ P2/2. Before evaluating equations (44) and (45), it is necessary to adopt the standard constants * of transformation between the three fundamental systems (M. K. S.), (C. G. S.), (F. P. S.). The equivalent units of length and volume are, (46) 1 meter = 100 centimeters = 3.2809 feet. 1 meter 3 = 1000000 cm 3 = 35.3166 cu. ft. The standard relation between volume and mass is, 1 cubic centimeter of water = 1 gram at the temperature 276.9. (47) 1 kilogram = 1000 grams = 2.20462 pounds. 1000 kilograms = 1000000 grams = 2204.62 pounds. *The subject of units and physical constants can be studied in Everett's " Units and Physical Constants," Gray's " Smithsonian Physical Tables," and in the text-books on physics generally. 12 METEOROLOGICAL CONSTANTS AND FORMULAS Hence, by division, the equivalents become, kilograms (48)1000 , (meter) 3 (centimeter) 3 = 62.4237 16 . 0198 ki!2SI^? = 0.0160198^= 1 m 3 cm 3 (foot) 3 ' pound Three Series of Constants in Three Systems of Units In Static Meteorology there are three series of constants for force units, mass units, and heat units in the three systems of units (M. K. S. C), (C. G. S. C), (F. P. S. F). These de- velop from fundamental principles or definitions. Thus, to illustrate by pressure: 1. Force pressure = mass X acceleration. (49) P = p m B go = po/ogo = M go = pogo = PA~- 2. Mass pressure = heat pressure X mechanical equivalent of heat. (50) u A PA M = = -r. go A 3. Heat pressure = force pressure X heat equivalent of gravity work. (51) P A = AM = Po- = Ap Q . o These transformations apply to the heat terms R, Cp, Cv, in the several systems. These factors become in the several unit systems : TABLE 2 GRAVITY AND MECHANICAL EQUIVALENTS OF HEAT Work and Heat Equivalents (M. K. S.) (C. G. S.) (F. P. S.) go Acceleration of gravity - Work equivalent of heat Gravity-work of heat. 9.8060 426.837 4185 1 980.60 42683.7 41851000 32.173 777.93 25028 2 A A Heat equivalent of work . . . A_ J Heat equivalent of gravity go ' work 0.002343 0.00023894 0.00002343 0.000000023894 0.0012855 0.000039954 A X (Heat in mechanical units) = Heat units of heat = calories. THREE SYSTEMS OF UNITS 13 In Tables 3, 4, and 5 have been collected together the con- stants in the three unit systems. They illustrate practically the formulas (1), (17), (18), (22), and (44) in Table 3, (50) in Table 4, and (51) in Table 5. By combining these constants and formulas a very large amount of static meteorology is derived. TABLE 3 THREE SERIES OF CONSTANTS IN THREE SYSTEMS OF UNITS (1) Gravitational Force Units Formulas S (M. K. S. C) Meter-Kg-Second (C. G. S. C) Cm.-Gram-Second* (F. P. S. F) Foot-Pound-Second go pw hw Po go pm Bo Po go PO It Po go P2 h Po ft k -1 ft k-l 1 -l Ro To po Po lo go a Ro Cp a Cv a ao Log. 9.8060 0.99149 1000 .0 3 . 00000 10.3329 1.01422 101323.5 5.00571 9 . 8060 . 99149 13595.8 4.13340 . 760 9 . 88081 101323 .5 5 . 00571 9.8060 0.99149 1.29305 0.11162 7991.04 3.90260 101323.5 5.00571 9.8060 0.99149 0.80427 9.90540 12847.6 4.10882 101323.5 5.00571 1.4062486 0.14806 0.4062486 9.60879 3.461545 0.53927 2.461545 0.39121 287.0334 2.45793 273. 2.43616 1.29305 0.11162 101323.5 5.00571 7991.04 3.90260 9.8060 0.99149 0.003663 7.56384 287.0334 2.45793 993.5787 2.99720 706.5453 2.84914 0.0098695 7.99429 Log. 980.60 2.99149 1.0000 0.00000 10332.9 3.01422 1013235. 6.00571 980.60 2.99149 13 . 5958 1 . 13340 76.0 1.88081 1013235. 6.00571 980.60 2.99149 .00129305 7.11162 799104. 5.90260 1013235. 6.00571 980.60 2.99149 .00080427 6.90540 1284760. 6.10882 1013235. 6.00571 1.4062486 0.14806 0.4062486 9.60879 3.461545 0.53927 2.461545 0.39121 2870334. 6.45793 273. 2.43616 0.00129305 7.11162 1013235. 6.00571 799104. 5.90260 980.60 2.99149 0.003663 7.56384 2870334 6.45793 9935787. 6.99720 7065453. 6.84914 .000098695 5.99429 Log. 32.173 1.50749 62.4237 1.79535 33.901 1.53021 68085 4.83305 32 .173 1 . 50749 848.70 2.92875 2.4935 0.39681 68085 4 . 83305 32.173 1.50749 0.080717 8.90696 26218.1 4.41860 68085. 4.83305 32 .173 1 . 50749 0.056009 8.69974 42249.1 4.62582 68085. 4.83305 1.4062486 0.14806 0.4062486 9.60879 3.461545 0.53927 2.461545 0.39121 1716.52 3.23465 491.4 2.69144 0.080717 8.90696 68085 4.83305 26218.1 4.41860 32.173 1.50749 0.002035 7.30856 1716.52 3.23465 5941.86 3.77392 4225.34 3.62586 0.0054146 7.73357 Density of water Height (44) Po = gopwhw.... Gravity Density of mercury . . . Barometer height (44) Po = gopmBo. . . . Height (44) Po - go polo Gravity Density aqueous vapor. Height (44) Po = gop 2 /2 Cp Specific heats -^ Ratio ^ - 1 . . C v (17) Const, press (18) Const, vol The Boyle-Gay Lussac Law (1) Po = P oRoTo lo go = Ro To (1) and (44) 1 a, = "Tfr To Ro ~ lo So a. (17) CP =Ko jp-j.... (18) Cv = Ro jri (22)- d /-^- ' d z Cpa 14 METEOROLOGICAL CONSTANTS AND FORMULAS (2) TABLE 4 Mass or Units of Weight Formulas 5 (M. K. S. C) Meter-Kilgm.-Sec. (C. G. S. C) Cm.-Gm.-Sec. (F. P. S. F) Ft.-Pound-Sec. (50) p = Pa/go R = p/poTo.... P R CP Cv Log. 10332.8 4.01422 29.2713 1.46644 101.3235 2.00571 72.0522 1.85765 Log. 1033.28 3.01422 2927.13 3.46644 10132.35 4.00571 7205.22 3.85765 Log. 2116.20 3.32556 53.353 1.72716 184.683 2.26643 131.330 2.11837 Ci>-R-^ Cv=Cp - R TABLE 5 (3) Heat Units Formulas S (M. K. S. C) (C. G. S. C) (F. P. S. F) Work equiv. heat 1 A A PA R A CP A CA 426.837 2.63022 0.002343 7.36978 24.2106 1.38400 0.068583 8.83622 0.237406 9.37549 0.168823 9.22743 42683.7 4.63022 0.00002343 5.36978 0.024106 8.38400 0.068583 8.83622 0.237406 9.37549 0.168823 9.22743 777.93 2.89094 0.0012855 7.10906 2.72025 0.43461 0.068583 8.83622 0.237406 9.37549 0.168823 9.22743 Heat equiv. work (5D P A =AP R A = AR C *A = ACp Cv A = ACv... . Work and Heat Units One large calorie is the heat required to raise 1 kilogram of water from to 1 C. One small calorie (therm.) is the heat required to raise 1 gram of water from to 1 C. One British thermal unit is the heat required to raise 1 pound of water from 32 to 33 F. One calorie = 1000 therms = 3.968 Br. th. u. = 426.837 kilogram meters. One therm = 0.003968 Br. th. u. (3.968 = 2.2046 X 1.8.) One dyne is the force which acting upon a gram for one second generates a velocity of one centimeter per second; it produces the C. G. S. unit of acceleration on one gram; it produces the C. G. S. unit of momentum on any mass per second. One erg is the amount of work done by one dyne working THREE SYSTEMS OF UNITS 15 through the distance of one centimeter; it is the C. G. S. unit of energy. One erg = 1 centimeter dyne = ^TTT;; = 0.0010198 gram cm. - 980.60 xUx 100 = 0-000000010198 kilogram meter. One large calorie = 1 kilogram-degree C water = 426.837 kilogram meters = 426.837 X 980.60 X 1000 X 100 = 4.1851 X 10 10 ergs, C. G. S. One small calorie = 1 gram-degree C water = 426.837 gram meters = 426.837 X 980.60 X 100 = 4.1851 X 10 7 ergs, C. G. S. One British thermal unit = 1 pound-degree F water o 280Q = 426.837 X - -= 777.93 foot-pounds. l.o Work to Heat T. The mechanical equivalent of heat is the work required by work-friction to produce the given heat. Log. (52) -j = 426.837 2.63022 kilogram meters (M. K. S.) = 42683.7 4.63022 gram centimeters (C. G. S.) = 777.93 2.89094 foot-pounds (F. P. S.) (53) il= 4185.1 3.62171 joules (M. K. S.) = 41851000. 7.62171 ergs (C. G. S.) = 25028.2 4.39843 absolute units (F. P. S.) Heat to Work A. The heat equivalent of work is the heat that is required to do a given amount of work. 16 METEOROLOGICAL CONSTANTS AND FORMULAS Log. (54) A = 0.002343 7.36978 kilogram calorie (M. K. S.) = 0.00002343 5.36978 gram therm (C. G. S.) = 0.0012855 7.10906 Br. th. units (F. P. S.) (55) = 0.00023894 6.37829 (M. K. S.) = 0.000000023894 2.37829 (C. G. S.) = 0.000039954 5.60157 (F. P. S.) I. FOR THE SAME POINT OR STATION Variations from the Standard P , p , R Q , T Q The several formulas derived from P = PO ^o TO apply only to the sea level on latitude 45, but the variations, PI, pi, RI, TI, are incessant in the earth's atmosphere, and the formulas must be derived for passing from one condition to another. Meteorology divides itself into two main branches according as R is taken constant or variable, and it is a principal part of this work to discuss the formulas when R is variable from one stratum to another. When the point of observation is on the sea level or on the lower land areas, it is proper to assume RI = RQ constant, which greatly simplifies the computations. If the variations oc- cur at the same place or station, go is also constant. For two variable conditions of dry air at a given place we may write the two in a ratio in several forms, using (44), (49), (50), (51) : Hence, . PAOJ ,__N P_ j>_ pA _ P T RQ _ Pm B go _ p I go PO pQ pAQ PoToRo Pm^ogo PO IQ go We shall confine our attention chiefly to the force pressures P, PO in the (M. K. S.) system, and to the first series of constants in Table 3. There are numerous equivalents which are easily derived from the formulas. VARIATIONS IN THE VERTICAL LINE 17 P T o I B (58) The pressure ratio, -^- = ^ = - = . LQ PO J- PO*0 -OQ The temperature ratio becomes, The pressure ratio may have several forms, from which are derived the pressure-density ratio, (60) = ~ = (1 + a t). Auxiliaries. p po 1 o po ^ Pm go / i A D D - (1 + a/). Po = #o P m go. po = PO ^o (1 + a 0- = VQ. Po = Ro T Q (1 + a t). P V = R T . T P D (61) The temperature ratio, -=.. 1 o " <y P (62) The density ratio, = ^ . ^ Po -TO ^ ^ -^ The temperature varies as the pressure and inversely as the density; the density varies as the pressure and inversely as the temperature. II. FOR DIFFERENT POINTS ON ANY VERTICAL LINE z The Variations of Gravity, Density, Temperature, and Pressure The practical problems in static meteorology consist to a considerable extent of the reduction of barometric pressures from one elevation to another along a radius of the earth extended, or inversely the determination of the difference of elevation be- tween two measured barometric pressures along the same vertical. ' The former process is applied in forming the synchronous charts of pressure reduced to the sea level, or to any other adopted plane which are used in public forecast charts of storm and weather conditions, and the latter to preliminary surveys in 18 METEOROLOGICAL CONSTANTS AND FORMULAS mountain and plateau regions. The entire process is, however, very complex in its application to particular cases, and it re- quires much experience in managing the details of the computa- tions. It will be necessary to describe somewhat fully the several terms that are to be integrated from one level to another, which enter the final barometric and hypsometric formulas. Those here developed are such as are found in the Standard ^Treatises, but another method of reduction will be introduced at a later section. The first problem to describe is the gravity value in any latitude,^, and at any height above the sea level, z; the second considers the density of the air as a mixture of gases and aqueous vapor in varying proportions; the third is to de- termine the temperature gradients in a vertical direction in the free air and within the land masses; and the fouxth is the use of the barometer as an instrument of precision, together with the discussion of the observed heights of the mercury column. All the details easily found in good works on meteorology will be very briefly mentioned. 7. The Acceleration of Gravitation J g$ z d z. From formula (4), which is of geodetic origin, (63) g$ = go (1 - 0.00260 cos 2 0), the latitude variation. *" (64) g z = gj ^1 -~rj, the elevation variation. Hence, (65) f^ d z = 9.8060 (1 - 0.00260 cos The force of gravity varies inversely as the square of the distance from the center of the earth. ( 66 >>l- Rz (bb) go " (R + *) - -Z ). (67) - dz = (z - * ) - ACCELERATION OF GRAVITATION 19 The radius of the earth may be taken in the mean from Bessel's spheroid, (68) JRp = 6370191 meters, 6.8041525 log. Rt = 20899600 feet, 7.3201380 log. The computed values of formula (4), without any integration in latitude and altitude, are given for a few selected points in Table 6. TABLE 6 EVALUATION OF FORMULA (4) gfa = 9.8060 (1 - 0.00260 cos 2 0) ( 1 - ^) z 90 80 70 60 50 45 40 30 20 10 Meters. 20000 15000 10000 5000 000 m. 9 . 7703 9.7857 9.8011 9.8165 m. 9.7688 9.7842 9.7996 9.8150 m. 9.7643 9.7797 9.7951 9.8105 m. 9.7574 9.7728 9.7882 9.8036 m. 9 . 7489 9 . 7643 9.7797 9.7951 m. 9 . 7444 9.7598 9.7752 9.7906 m. 9.7399 9.7553 9.7707 9.7861 m. 9.7314 9.7468 9.7622 9.7776 m. 9.7245 9.7399 9.7553 9.7707 m. 9.7200 9 . 7354 9.7508 9 . 7662 m. 9.7185 9.7339 9.7493 9 . 7647 9.8319 9 . 8304 9.8259 9.81909.8105 9.8060 9.8015 9 . 7930 9.7861 9.7816 9.7801 2z The variation in height g -^- = 0.00308 meters per 1,000 meters, and 0.00308 feet per 1000 feet for g = 32.173 feefT There has been a discussion as to the effect of the land masses upon the action of gravity, whether the coefficient in the formula should be 2.00, as developed in (66), or be modified. Ferrel claims that it should remain 2.00; the Smithsonian Meteorologi- cal Tables have adopted 1.96; and the International Meteoro- logical Tables have taken 1.25, which latter value is here adopted. The plateau regions of North and South America, Asia, rr and Africa will be best represented by 1.25 -~-, where z has a con- siderable value, reckoned T rom the sea level. In balloon ascen- sions from the ocean or from the low plains it may be better to increase the value to 2.00, but this can be determined from obser- vations by means of mercurial and aneroid barometers. In fact, the aneroid barometer, when perfectly adjusted as a mechanism, 20 METEOROLOGICAL CONSTANTS AND FORMULAS measures the local hydrostatic pressure without any gravity factor. Among the computations to be introduced in a later section there will be examples of this action. Admitting the coefficient 1.25, we have (69) g z = g (l 1-25^) = 9.806 (1 - 0.000000196s) metric. = 32.173 (1 - .0000000598 z) English. = 9.806 - 0.00000192 z metric. = 32.173 - 0.00000192 z English. Similarly, the correction for height can be applied to any other value of #0 , as found on the lower line of Table 6. 2. The Density of the Atmosphere as a Mixture of Several Con- stituent Gases For the practice of barometry it is sufficient to take account of the atmosphere as a mixture of dry air, aqueous vapor and carbon dioxide, commonly called carbonic acid.* In physical problems there are the gases oxygen, nitrogen, hydrogen, carbonic oxide, and traces of argon, helium, neon, krypton. We shall summarize the treatment of several gases in a mixture and the data of the kinetic theory of gases in this connection. Adopt the notation for standard conditions as expressed by Po = PO #o To, or Po v = Ro TV Mixture. Dry Air. A J ueous Car A b ic Vapor. Acid. Density Pm0 p^ ^ P30 Volume v m0 v w v 20 v 30 The general equation for mixture is, (70) p mQ V m(> = pio fllO + P20 ^20 + P30 ^30- The values to be assigned to the terms are: Dry air Plo = 1.29278. v 10 = v mo - v 2Q - v 30 . * In view of the possible existence of the real carbonic acid (H 2 C O 3 ) at the low temperature of the isothermal layer, the use of the word " acid " instead of anhydride can not be commended. DENSITY OF THE ATMOSPHERE 21 Aqueous vapor p2 = 0.622 pi . v 2Q = variable amounts. Carbonic acid pso = 1.529 pi . 30 = 0.0004 v m0 . Mixture p m0 to be computed. v m0 = v w + v 20 + v 30 . Introducing these values in (70) and dividing by v mQ (71) ^ = 1.29278(1-^ - ^ + 0.622^ + 1.529^), \ Z>mO VmO V m0 V m O/ = 1.29278 fl - 0.378 + 0.529 }. \ v m0 v m0 J Since the ratio of the volume of the constituent to the volume of the mixture is the same as that of the partial pressure of the constituent to the total pressure of the mixture, we have generally VnO Pn0 pn_ e_ v m0 " p m0 " p m " B' where p n0 and p m0 are for the normal data (P TO), p n and p m are for any (P T) , e and B are the barometric pressures. Hence, = , and ^ = 0.0004, so that, Z>mO >Q V m O (73) p m0 = 1.29278 (l - 0.378^ + 0.00021 V It is customary to unite the terms for the dry air and the carbonic acid in the normal density, (74) po = 1.29278 + 0.00027 = 1.29305 per cubic meter. In (73) e^is to be taken in meters of mercury in the (M. K. S.) system, in millimeters in the (C. G. S.) system, and in feet in the (F. P. S.) system. Since e varies incessantly in the atmosphere no fixed value can be assigned to it on any level. The reduction from the normal (P Q p m0 T ) to any other condition (P p m T) on the same level z is given by (60), substituting p m for /o, and Pmo for /o, p 'T' p 'T 1 / p (75) p m = - PmQ = ~ P Q 1 ~ 0.378 - p 22 METEOROLOGICAL CONSTANTS AND FORMULAS B 1 1 (76) p m = -5- p m0 6/) (l + 0.378) Since we retain the density p = 1.29305 for the value of p m0 , with dry air at normal pressure in this equation, the corresponding barometric pressure in the fraction expressing the partial pressure of aqueous vapor must be B -, butj^may be any pressure whatever, according to the dryness of the air. C z e The Integral Mean / 0.378 -=-. 'so >o The pressure of the aqueous vapor decreases from the ground upward in a geometric ratio, which is expressed approximately on the average by the formula, (77) e = e 10 6517 in meters, e = e 21381 in feet. It will be shown that the barometric pressure diminishes by a similar law, (78) B = Bo 10 1840 in meters, B = B 10 60367 in feet. p By combining these in the ratio 5- it becomes, (79) 4- = -^ 1Q~ 10091 in meters, ~- = -~ lO' 33 ^ in feet. > >Q & JL>O These can be reduced from the common base 10 to the Naperian base e by the modulus M = 0.43429. 10091 X M = 4383, 33108 X M = 14378. The expression for the integral mean from^Zpjto z is (80) j8 = - 0.378 ~ / V ^ssa j z f or t he metric system. Z ZQ >Q ^ZQ I en C z - (81) j8 = - 0.378 - / e 14378 d z for the English system. Z ZQ >Q 'ZQ These can be developed in a series, as shown in the "Report on the International Cloud Observations," UJ3.JSLJ3., 198 Z _ page 491,^" FORMULAS FOR MIXTURE OF GASES 23 metnc - i / z \ z ~] ~T\ \4383 / ^ * ' ' J (83) ft = 0.378-1 [l - 4l yb* ) + 4t (74^) 2 - JJQ ^i ^J-TcO/O / ol ^XTtOiO' 1 / V IT -4!Vl4378)+-"J En S llsh - That is to say, having the vapor pressure j^ and the barometric pressure B Q at the surface one can compute the average value & of the integral of the term 0.378 ^ up to the height z. It is commonly impractical to measure the values of e and B at several points in the atmosphere, and for many computations this method of mean integration upward from the surface is quite sufficientjpr_pjractical work. Also, it is a very expeditious method when using the humidity table 92, page 548, for metric measures and Table 19 of the "Barometry Report/' U. S. W. B., o 1900, page 108, for English measures. The mean values of -g for the air column is often taken as the arithmetical mean of the observed values at an upper statiqn_z and a lower station J;Q.____ In the case of balloon and kite ascension, the registered relative humidity, temperature, and pressure can be computed to the integral mean value required. When only temperature and pressure are registered, this correction to the density of the atmosphere is not available in the hypsometric formula. The General Formulas for the Mixture of Gases The general principles controlling the mixture of gases are so often useful in meteorology that it will be convenient to collect together the common formulas expressing the several processes. It will now be proper to pass from the system of M. K. S. units to the system of C. G. S. units, and to bring forward the terms 24 METEOROLOGICAL CONSTANTS AND FORMULAS applicable to the thermodynamics of the atmosphere and the kinetic theory of gases. We adopt the following notation: a = atomic weight. K m = molecular weight = -5-. K. f (84) K = the absolute gas constant = m R = ~ . * Compute K in the (C. G. S.) system, using the gas hydrogen, m H = 2, p H = 0.000089996. Log = 5.95422 - 10. 82482000 = 1-9708 small calories or therms. Using the values for air m = 28.736, p = 0.00129305, the same result is obtained. This formula applies to the three systems and to all gases. The density and molecular weight of hydrogen are related to those of other gases so that, (85) m p H = m H p, and for this reason hydrogen is the standard. n = the number of molecules in a unit volume. N = the number of molecules in a V- volume. M = N m = Mass. (86) Number. N = = V n = ^~ m K 1 P V m P V (87) Mass. M = N m = V nm = T = TT^ = P M_ PM R^T ' nm ~ RT p' M M MKT (88) Volume. V = M v = - -=- = p nm m P FORMULAS FOR MIXTURE OF GASES 25 (89) Pressure. jP^ = - -= = nK T = =- = ~RT = nmRT = p R T. (90) Density. p = = WW = TT= yr = n~^. , m v . 1 1 F A" T RT (91) Volume of unit mass, v = = - -= = = . p nm M m P P (92) Constant. R = = - = = - = -^-. m M T T nm T p Referring to the standard gas hydrogen there are some special values for the gas constant. Logs. n p ~v H ~ m gram cm. 2 (94) (95) K = ^^* = ^^ = 82482000 gram X PH PH J- o sec. 2 temp. If 7? r is the gas constant for the heat energy at T, R m is the gas constant for the molecular energy, R a is the gas constant for the atomic energy. Then, 3 r the mean kinetic energy ( Ub ) KI = K T + K m -\- K n = K = =- = - 2 T absolute temperature . (97) R z = ^ Ri R = Cv the specific heat at constant volume. o (98) R 3 = - RI = Cp the specific heat at constant pressure. o These are related to the potential and kinetic energies in the following relations, and thence to the specific heats at constant pressure and constant volume. 26 METEOROLOGICAL CONSTANTS AND FORMULAS Inner Potential Energies (99) / = J m -f J a * The inner potential energy of molecules and atoms. o o r> o (100) J m =-Pv=- = - RT. Inner molecular potential w & p & energy. / = Inner atomic potential energy. 2 /o = ^ 7^ (inner viriol). Initial inner potential energy. o (101) V = - I S (X x + Y y + Z.) the mean viriol for the force (X. Y. Z.). Inner Kinetic Energies (102) H = H m -f H a . The inner kinetic energy of molecules and atoms. H m = Inner molecular kinetic energy. H a = Inner atomic kinetic energy. Total Heat and Work Energies Q = The heat or total inner energy = Cv T. W = the work or total external potential energy. (103) Q = J + H + W = (| 1 - R) T + / . Total inner energy. p 2 (104) W = Pv = -- = RT = -V e (outer viriol). External p o potential energy. *J m and To relate to the trifling rearrangements of parts which are the only changes that can occur in ordinary chemical and physical reactions. We can not attack the enormous stores of energy shut up within the atoms. INNER KINETIC ENERGIES, WORK, AND HEAT 27 (105) U = J - /o + W = J m + J a - /o + W. Potential energy. (106) U W = J JQ. Accession of inner potential energy. (107) V = H = - l - S (X x + Y y + ZJ = | U = mean viriol or work done. The gas constants are again denned thus: TJ (108) R T = 7p (heat) = the ratio of the inner kinetic energy to T. (109) R m = j^-^mq 2 where q 2 = the mean square velocity. (110) R a = -^ -n m^i 2 , n = the number of atoms in a molecule. 1 2> The Specific Heats of Monatomic Gases 5 55 (111) Cp= -R l -R + R==R 2 + R = -R l = -R. 7? 2 7? 2 7? = 3* 1 = 3 R * / ~* - <-\\ /~*at 3? 7? Z? 3? . J2 - 3? 1? 3 T> T) T> 2 ^ _ Kl _ y^ 2 , ON Cp R Z + R 5 (113) pr- = ^ = r- = 1.67 = k for monatomic gases (mercury). c-c k - ' (20) ' (116) Cp - Cv = R = ^. Compare (15), (16). O 2 CD 28 METEOROLOGICAL CONSTANTS AND FORMULAS , 11ft v Ja Jm 3 Cp-Cv 5Cv-3Cp (118) -Q- -Q- - 2 oT ~2C~ V The above formulas are in mechanical units, and they can be transposed into heat units on multiplying them by A = the heat equivalent of work which is A = 0.00002343 in ,the C. G. S. system by Table 2. The Fundamental Laws of Physics (119) Boyle's (Mariotte) Law. P v Q = Pv= (-} = R T = \p/ T constant for constant T. CT) \ ~D =rj = = constant for constant v. Pv (121) Boyle-Gay Lussac Law. -=r = R = constant in per- fect gases. (122) Avogadro's Law. N = = T^ = constant for (P. m A i V. T.) constant. (123) Clausius' Law. J mi <?i 2 = f m z qf constant kinetic energy. (124) Dalton's Law. P = PI + P 2 + . . . = f (J mi qS + J mz qz* + ...). Pressures. In ordinary gases as distinguished from perfect and ideal gases, all these formulas are more or less defective on account of the internal action of the atoms and the molecules upon each other under the stresses of electrical and other mechanical forces. Many formulas, with constant coefficients and exponents, have been devised to take account of these physical variations, but they will not be further mentioned in this place. The Formulas for the Mixture of Several Gases (125) Pressure. P = P RT = ~- = nmR T = n K T = K . RTM FORMULAS FOR THE MIXTURE OF SEVERAL GASES 29 (126) P = P, + P 2 + P 3 + . . . = (! + m + (127) Pi = . *' - P. n n (128) Pi : P 2 : P 3 . . . = V l : V 2 : V 3 . . . = Rt-Mi:RtM 9 :KiM^ (129) P V = (M l R l + M 2 R 2 + . . .) T = (130) Mass. = , , > ^^ m mi nh m$ ... M 1 +M 2 +M 3 +. n Mi M* Ma (1 32) n^,n,...,,... = P,,P,,P,... (133) Density. n m = HI mi + n 2 nh + n s m 3 + . . . = PI + P2 + PS + . . . = P = -. (134) 11 (135) Gas Constant. R - . . . K n 2 m 2 * RI R 2 R 3) etc., in these equations refer to the various gas coefficients of the several gases. 30 METEOROLOGICAL CONSTANTS AND FORMULAS (136) Inner Energy. M U = M l U l + M 2 / 2 + M 3 U 3 + . . . . (137) Entropy. M S = M^ Si -f M 2 S 2 + M z S 3 + . . . . (138) Concentration, c = ~ '- LL -. (139) c R = The Kinetic Theory of Gases for the Atmosphere The various formulas involving specific heat can all be de- duced from the kinetic theory of gases, and it is therefore desir- able to have at least approximate values of the constants of the principal gases which are the constituents of the atmosphere. These are arranged in Table 7, so that the formulas from which they are derived suggest by definition the exact meaning of the several terms. It is much better to depend upon formulas for defining constants than upon any extended verbal description for the sake of accuracy and brevity. On the other hand, it is not possible to study any advanced research problem in atmos- pheric physics without depending upon the several terms in the kinetic theory of gases. In the present status of physics, research is attempting to make out the connection between the theory of mechanical collisions in the molecules of a gas and the corresponding dynamic electric and magnetic forces, but this investigation is incomplete. It would be very desirable that some international commission should adopt a series of consistent constants for the terms of Table 7, in order that all computations may be made on the same basis. At present there are small variations in the values in consequence of adopting slightly different fundamental con- stants from which the others are derived. It is probable that sufficient agreement exists among chemists and physicists as to these elementary constants, in order to make this a practical proposition. KINETIC THEORY OF GASES 31 .a o> 118 .; O O 5.$ ir *H "O a=a H ':!TOiii4i ii : II 5^.y?- t SSSS-eo i S tec I I IS C 10 fl B .2 S 3 Hi n v P 32 METEOROLOGICAL CONSTANTS AND FORMULAS This table was computed for the constants, K = 82481110. Absolute gas constant, dynes/cm. 2 *- P = 1013235. One atmosphere in dynes/cm. 2 -j- = 41852800. Mech. equivalent heat in ergs. It is very desirable that the International Meteorological Committee should fix standard values throughout the table. The Temperature and the Temperature Gradients Observed at Different Elevations in the Free Air The actual temperature of the atmosphere at any point is the resultant of the force of gravitation as balanced by the TABLE 8 EXAMPLES OF TEMPERATURE AND TEMPERATURE GRADIENTS AT DIFFERENT ELEVATIONS Lindenburg Lindenburg Atlantic Ocean Victoria Nyanza Station Apr. 27, 1909 May 5, 1909 Sept. 25, 1907 Lat. Summer, 1908 Lat. + 52 Lat. + 52 3 + 35Long. + 36 J Lat. J Height z AT AT AT AT in meters T 1000 1000 1000 1000 18000 226 5 4-30 190 5 6 5 17000 223.5 + 1.0 222.9 + 2.7 197.1 -5.5 16000 222.5 4-18 220.2 + 0.1 202.6 - 4 2 15000 220.7 4-10 220.1 + 0.2 206 8 4 14000 219.7 + 28 219.9 - 0.2 210.8 -52 / 13000 202.7 - 0.8 216.9 + 5.6 220.1 - 0.2 216.0 -6.6 12000 203.5 - 6.4 211.3 - 1.7 220.3 - 7.8 222.6 -8.8 11000 209.9 - 8.8 213.0 - 8.4 228.1 - 7.9 231.4 -7.5 10000 218.7 - 9.9 221.4 - 9.1 236.0 -10.0 238.9 -7.2 9000 228.6 -11.9 230.5 - 8.8 246.0 - 7.7 246.1 -4.6 8000 240.5 - 8.8 239.3 - 8.4 253.7 - 7.4 250.7 -7.3 7000 249.3 - 7.0 247.7 - 7.9 261.1 - 6.3 258.0 -5.4 6000 256.3 - 6.6 255.6 - 7.3 267.4 - 6.6 263.4 -5.8 5000 262.9 - 6.2 262.9 - 4.2 274.0 - 4.5 269.2 -6.5 4000 269.1 - 6.6 267.1 - 5.8 278.5 - 3.4 274.7 -6.1 3000 , 275.7 - 5.3 272.9 - 3.9 281.9 - 5.2 280.8 -7.6 2000 281.0 - 5.5 276.8 - 2.9 287.1 - 3.4 288.4 -7.8 1000 286 5 7 9 279 7 1 6 290 5 6 296 2 000 294.4 281.3 296.5 hydrostatic pressure, the circulation, and the radiation. It is the most important element to be observed, and from it all the TEMPERATURE GRADIENTS IN FREE AIR 33 other terms can be computed, provided the velocity and the vapor pressure are also given by the observations. In order to have the data in concrete form so that the formulas may become practical, four examples are taken from the observations, at Lindenburg, Germany, in the Tropic North Atlantic Ocean, and at Victoria Nyanza, Africa. Table 8 records the height in metersj^ the absolute temperature T, the vertical temperature ^ T gradient per 1000 meters , and Table 9 the relative humidity -LUUU R. H., and the vapor pressure in millimeters e. The latter is computed by taking from the Tables of vapor pressure in saturat- ed air at given temperatures the dew-point vapor pressure, and multiplying by the relative humidity. The Smithsonian tables have been extended to include approximately the vapor pressure TABLE 9 EXAMPLES OF THE CORRESPONDING RELATIVE HUMIDITY AND VAPOR PRESSURE AT DIFFERENT ELEVATIONS Height z R.H. e R.H. e R.H. e R.H. e 19000 per cent. mm. per cent. mm. per cent. mm. per cent. mm* 18000 43 0.020 33 0.000 17000 43 .004 47 0.014 33 .000 16000 . 43 .012 47 .012 33 .001 15000 44 .011 47 .012 34 .002 14000 44 .010 47 .012 34 .003 13000 61 '6.002 44 .007 47 .012 34 .004 12000 61 .002 43 .004 47 .012 34 .010 11000 61 .043 43 .004 47 .027 34 .029 10000 61 .012 43 .012 47 .063 34 .058 9000 61 .039 43 .033 47 .183 34 .134 8000 62 .135 44 .084 48 .402 35 .218 7000 64 .284 45 .209 51 .843 36 .449 6000 69 .741 47 .473 54 1.544 41 .831 5000 72 1.400 50 .972 56 2.750 50 1.667 4000 67 2.370 51 1.420 58 3.910 73 3.770 3000 67 3.710 42 1.910 66 5.600 69 5.440 2000 70 5.590 41 2.460 65 7.770 60 7.800 1000 63 7.250 60 4.390 72 10.700 57 12.030 000 54 15.620 58 4.730 78 16.770 at very low temperatures. These values of T and e will be used in illustrating the barometric reduction formulas. By formula 34 METEOROLOGICAL CONSTANTS AND FORMULAS (22) the adiabatic temperature gradient per 1000 meters is -9.869 C., or per 1000 feet -5.415 F. In Table 8 it is seen how widely the actual temperature gradients differ from this value, and it is this circumstance that compels us to reconstruct the entire range of standard thermodynamic formulas, in order to adapt them to practical work in the earth's atmosphere. The slow progress of meteorological physics is due to this difference of gradient more than to any other cause. The temperature gradients are incessantly varying in the atmosphere from large temperature falls, A jT, to considerable temperature gains, -f A T. This change of sign gives rise to the subject of the inversion of temperature of which examples occur often at night near the ground, and usually in the isothermal region. The values of the vapor pressure below T = 273 are those of the Smithsonian Tables extended. The Temperature Gradient in a Plateau from the Sea Level to the Surface of the Ground An important part of barometry is the determination of the temperature gradient within the land mass forming a plateau region, as in the Rocky Mountain district of the United States, by means of which the pressure observed at a station on the surface may be reduced to the sea level, in order to be combined with those stations having low elevations, so as to make a synchronous map of storm conditions for the entire country. This problem is one of considerable difficulty, and it must be solved in accordance with the prevailing local conditions, so that no fixed rules can be given for its treatment. A very extensive reduction for the United States is found in the "Report on Barometry," already mentioned, but its leading principles can be briefly summarized. Having low-level stations in the eastern, central, and Pacific districts, the problem is to connect up the stations on the plateau at different elevations above the sea level, by means of the average temperature gradient within the land mass, which is very different from the gradient above the plateau in the free air. The first step is to construct from the available data approximate temperature gradients, which TEMPERATURE GRADIENT IN A PLATEAU 35 can be used for short distance reductions, in longitude, in latitude, and on the vertical. Then certain reference vertical lines are chosen, as the intersection of the parallels and meridians for each 5-degree interval, and for the planes 1000 feet apart. To these points are reduced the stations by numerous combinations, so that the same station is reduced to several selected reference points. These points now lie on vertical lines, and each line of tempera- tures may by plotting be extended downward to sea level. The second step is to draw the sea-level isotherms between the central and the Pacific districts, joining across the plateau region by the most probable curves. The third is to compare by interpolation these horizontal temperatures with those obtained from vertical extension, and by mutual adjustments the two sets may be made to agree harmoniously. This inter- locking of a horizontal system with a vertical system is able to produce by mutual checking a very exact agreement between the two sets. In this way the sea-level temperature was found beneath the plateau, and thence the temperature gradients in a vertical direction were computed. These gradients differ greatly from one another in different parts of the plateau, they differ from month to month at the same place, and there is no fixed gradient which can be used at any given station. The station gradients fall into distinct classes in the several parts of the plateau in respect to the yearly variations, and diagrams were constructed to serve for the individual stations. The monthly gradient may be roughly summarized for comparison with the free-air gradients. A T C. AT F. The reduction from 7^7- - to - _ , -- is effected by 100 meters 100 feet the factor 1 = 0.55. This is from 1 X ^~ = p^ ' 55 ' This plateau gradient is only 37 per cent, of the free air adiabatic gradient. Extensive reduction barometric tables were con- structed for many stations in the United States, which are used in compiling the weather forecast charts, of which further mention will be made. 36 METEOROLOGICAL CONSTANTS AND FORMULAS TABLE 10 THE MEAN MONTHLY TEMPERATURE GRADIENTS IN THE ROCKY MOUNTAIN PLATEAU OF THE UNITED STATES A 27100 Jan. Feb. March Apr. May June Per 100 ft. Per 100m. -0.191 -0.348 -0.174 -0.317 -0.145 -0.264 -0.202 -0.368 -0.194 -0.353 -0.215 -0.391 A 77100 July Aug. Sept. Oct. Nov. Dec. Year Per 100 ft. Per 100m. -0.175 -0.319 -0.179 -0.326 F -0.168 -0.306 -0.191 -0.348 -0.181 -0.329 -0.187 -0.340 op -0.202 -0.368 The Integral Mean Temperature and Gradient In reducing pressures from one level to another it is necessary to know the mean temperature of the actual air column in the free air, or in the hypothetical air column within a plateau, or between a mountain summit and the sea level, or other plane of reference. These are found from the summation of the tem- peratures at several levels as in Table 8, and dividing the sum by the number of the strata taken. (140) from (23). For two strata it is the arithmetical mean, (141) 7\ = J (7 1 ! + To). For example, in the balloon ascension, April 27, 1909, sum the T z through the several strata; also, only the top and bottom of the same thick layer, as indicated. INTEGRAL MEAN TEMPERATURE AND GRADIENT 37 Layer in meters -s r z i (TJ. + r ) 000 to 4000 281.34 281.75 000 to 9000 264.43 260.15 000 to 13000 248.51 243.55 It is not usually sufficient to take the mean value of the top and bottom temperature of a thick layer for the mean tempera- ture of the column, and the error of reduction is dependent upon the discrepancy between T m and T iQ . If the gradient is uniform between two strata T m = T w , and the difference vanishes. The length of vertical distance that permits 7\ to be used depends upon local temperature distributions, and each case must be carefully examined. The same rule applies to the vapor pressure e, and any other meteorological element. The determination of the integral mean with accuracy is one of the hardships of practi- cal meteorology, upon which a large amount of labor is necessarily expended. The Virtual Temperature T r . It is sometimes convenient to combine the actual mean temperature T m with the expression for the vapor pressure p term 0.378-^ to form the so-called virtual temperature T r , by the formula, (142) 2T- = T w (l + 0.378 J) The barometric reduction can then be carried forward as if the dry air and the aqueous vapor were compounded in one gas whose equivalent temperature is T r . /I rrt ^jT. The ratio of the change of temperature d T to the prevailing temperature T is related to the logarithm in the follov/ing useful auxiliary formulas, which are often needed in substitutions. 38 METEOROLOGICAL CONSTANTS AND FORMULAS They are applicable through such strata, thin or thick, as have uniform temperature gradients, whether the temperature in- creases or diminishes in a vertical direction. They are given for the adiabatic and the non-adiabatic temperature variations. /< ^o\ I a a 1 a /I T 1 1 T> \ (143) / -=- = T = M lo S r" = M ( lo S r ~ lo S r o). J 1 a J- aO M- 1 o M (144) f^ = r '~ r = i log p = ^ (log Tx - log To). J 1 lio M 1 Q M Ti-T 1 10 1 (n- 1) M g R ~ (n- 1) M The Temperature Variations and the Specific Heat It is convenient to make the transfer from the non-adiabatic temperature loss to the adiabatic temperature loss, in connection with the specific heat, by using, as in (13), T a -T (146) i Cp a (T, - To) = Cp a (T a -To) = - go(zi-z ). (147) m Cp 10 (T, - To) = Cp 10 (T a - To) = Pl ~ P Pi0 These formulas will be fully illustrated in a later chapter. It is evident that many combinations can be made by employing the formulas (143) to (147), and they are very practical in de- veloping the formulas. T f j- r^ Bo I 0r Zi Z \ Transformation of -5- = -^ f 1 + 1.25 ^ J. The introduction of the plateau effect upon gravity in (69) has its parallel in the effect upon the barometric pressure, which is similarly modified. We have for both cases, (i> . -i. (i + GENERAL BAROMETRIC FORMULA 39 Since 1 + 1.25 ~^~ is a small variation from unity, the K general formula is applicable, (150) Com. log (1 + x) = M (x - % x* + % x* - . . .). Passing to common logarithms (149) becomes on neglecting the powers above x, (151) log J = log | + log (l + 1.25 *-^) , . Bo , M 1.25 , = log -g + -- (2i - Z ). It will be found in the barometric formula that, P ' B (152) Zi z = K log -5- (approx.) = K log -_- (approx.). Hence, /ieo\ 1 ^ 1 -^ i 1 nt-M K BQ (153) log - = log - + 1.25 -- log - , logj (1 + 0.00157) = log ~ (1 + y). The General Barometric Formula The several auxiliary formulas now deduced make it very simple to derive the barometric reduction formula connecting together the heights (z\ . z ) and the mercurial pressures (5i, B ). From (41) the differential pressure is, (154) - dP = p.gdz. Divide by P, (155) - d -j-=^gdz. Divide by |, d P P (156) - -5- . - = g . d z. Substitute from (75), * P (157) - -- . - l + 0.378 = g d z. po 1 o 40 METEOROLOGICAL CONSTANTS AND FORMULAS Substitute P from (49) and g+ f from (63), (64), _ dP Bo Pm go d i Q 370 __\ = X Po J- \ -t>Q/ go (1 - 0.0026 cos2(j>) (l - -jj{} d z. Pass to common logarithms by the factor -^. and integrate, /IKON i -^ -^Q Pm T m / e\ (159) log -p- . -vp- . ( 1 + 0.378 -=- ) = r M po 1 o \ JDQ/ m e =- o DQ m - 0.0026 cos The last gravity form is from (67). The constant (160) K = 2~ = 18400 (Metric), and JRTi = 60367.7 (English). M po T m is the mean temperature of the column (140), and the integral mean of ( 1 + 0.378-^-) is accomplished by the observa- \ JJQ/ tions along the column, or by integrating from the surface by (82) and (83) . Substituting (153) (161) log ~ (1 + 0.00157) K(l+ 0.00367 9) (l + 0.378 } & \ -L>0/ m Barometer. " Const. Temperature. pSsure. (1 + 0.0026 cos 20) l + l = zi - Z . Gravity in Gravity in Latitude. Height. Substituting the numerical values and combining, (162) log So = log B + 18429+67 Y0 2 C.+ o.003 < - 0.378^-) (1 - 0.0026 cos 20). Metric. (163) log = logB ~ - 0.378^-) (1 - 0.0026 cos 20). English. CORRECTIONS TO THE BAROMETER 41 These can be expressed in the general form, (164) log B, = log B + m (1 - 0) (1 ~r) = log B + m m ft m ?. In view of the uncertainty attaching to our knowledge at any time of the distribution of the vapor pressure in the air column, it is desirable to keep the term m ft separate as a correc- tion to the difference between the logarithms. Similarly the gravity term is retained by itself because in many computations it is small and can be neglected. Complete reduction tables are given in the " Report on the International Cloud Observa- tions" for the metric system, Tables 91, 92, 93, and in the " Report on the Barometry of the United States, Canada, and the West Indies" for the English system, Tables 13 to 21. From these logarithm tables many forms of numerical tables without logarithms can be constructed for special purposes. Corrections to the Barometer The mercurial barometer requires several corrections before the pressure can be used in practice. 1. Correction to the Standard Temperature. The instrument is constructed of parts whose coefficients of expansion with changes of temperature are not the same, as for the mercury and the brass scale. Adopting the notation, t = the temperature of the attached thermometer. t m = the standard temperature of mercury, C., 32 F. t s = the standard temperature of the brass scale, C., 62 F. m = the coefficient of expansion of mercury, 0.0001818 per degree Centigrade, 0.0001010 per degree Fahrenheit. n = the coefficient of expansion of brass, 0.0000184 per degree Centigrade, 0.0000102 per degree Fahrenheit. The accepted formulas are as follows : (165) B n - B = - B ( m ~ n \ t f or B n and B in millimeters. 1 + m t inches. 42 METEOROLOGICAL CONSTANTS AND FORMULAS The English form reduces to, (167) B n - B -B 10978 + 1.112 /' The necessary reduction tables are found in nearly all compilations of Meteorological Tables. 2. Correction to the Standard Gravity, g&. This is the gravity variation in latitude as given in (63), from which is obtained, (168) 45 - B+ = - B+ (l - ~ } = - B+ 0.00260 cos 2<. V 45 / The temperature and gravity corrections are applied as instrumental corrections to the actual barometric reading at a given hour. 3. Correction to a Standard Barometer or Patron. Each barometer as an instrument has certain minor deficiencies which cannot be readily analyzed, and in order to make a number of barometers homogeneous, so as to give strictly comparable pressures, it is necessary that they be severally standardized by comparison with an adopted normal or patron barometer. The Kew barometer is used for many standard comparisons, and each weather service keeps its own standard which has been carefully compared with the Kew instrument. Within each national service the barometers are compared, and an instru- mental correction is given for each barometer before sending to a station. Sometimes these corrections hold steadily for long intervals, and sometimes they change suddenly and erratically. Whenever there are local removals, or whenever a barometer is cleaned, its correction must again be determined. Frequent inspections and comparisons with a portable secondary standard are necessary if a homogeneous series of pressures is to be secured . It is not possible to be too painstaking in respect of the inter- barometric corrections. 4. The Station or Removal Correction. If it happens that at a given station there are any removals of the barometer from one office to another, as so frequently happens in large cities, and the elevation is thereby changed from time to time, it is necessary to adopt a standard elevation for the station and reduce the CORRECTIONS TO THE BAROMETER 43 series of readings taken at any other height to this level, which will persist as the adopted station elevation from the beginning to the end of the service. When the change in height is con- siderable these corrections depend upon the temperature in the course of the year. A correction card for instrumental and station removal errors should accompany each barometer, and be carefully recorded as part of the history of the instrument. In preparing homogeneous tables of pressure for use in solar physics and other cosmical problems, it is indispensable that all barometer readings should be carefully treated in this manner. The homogeneous system for the United States has thus been prepared by the author to cover the barometric pressure, the temperature, the vapor pressure, and the precipitation from the year 1871 to date, and the published data of the Weather Bureau are all on that basis. Similar homogeneous data are being pre- pared for Argentina and for other countries. 5. The Correction from the Surface Temperature (t) to the Mean Temperature of the Air Column (8) in Barometric Reductions. It is obviously so difficult to determine the relation of the surface temperature / to the mean air column temperature within a land mass 0, as from a station on a plateau to the sea level, that a special study was made of this subject in order to facilitate a prompt reduction of the observed pressure to the corresponding sea-level pressure. These are needed for transmission by telegraph to a central office where the daily weather forecast charts are constructed. Unfortunately there is no simple rule connecting / and 0, and in many cases the difference 6 t is very variable. Reduction tables are first computed with the adopted elevation H, and a series of assumed values of 6 for several barometric pressures in steps of 0.10 inch. Then the relation between / and having been found, the surface tem- perature is used as the argument for the table in place of 6. The practical value of / taken in the United States, where the observa- tions are made at 8 A.M. and 8 P.M. daily, is the mean of the cur- rent dry-bulb temperature and that taken twelve hours before. This gives a fair temperature average for the day, and it tends to eliminate some of the local effects of passing storms. It has 44 METEOROLOGICAL CONSTANTS AND FORMULAS been found to work well in the practice of ten years. In order to illustrate the differences between / and 6 in the course of the year, as the temperatures change from summer to winter, a few examples are extracted from Table 53 of the Barometry Report, where the heights are in feet, and temperatures are Fahrenheit. TABLE 11 RELATION BETWEEN THE SURFACE TEMPERATURE / AND THE MEAN Boise, Salt Lake Independence, Helena, Pike's Peak, Battleford, Idaho City, Utah Cal. Mont. Colo. Canada 2739 4366 3910 4110 14111 1608 1 e t e t 9 / e t e t e -42 -40 -48 -40 -61 -40 -32 -30 -37 -30 -49 -30 -40 -30 -23 -20 -24 -20 -26 -20 -35 -20 -29 -20 -13 -10 -14 -10 -15 -10 -23 -10 -17 -10 - 3 - - 5 - 3 '6 - 4 -12 - 6 7 10 5 10 6 10 7 10 - 2 10 5 10 17 20 14 20 15 20 20 20 5 20 16 20 27 30 24 30 24 30 29 30 10 30 27 30 37 40 34 40 35 40 38 40 16 40 38 40 47 50 44 50 48 50 46 50 22 50 49 50 57 60 55 60 60 60 55 60 30 60 59 60 67 70 66 70 71 70 66 70 38 70 69 70 77 80 76 80 81 80 76 80 47 80 79 80 87 90 86 90 92 90 86 90 56 90 89 90 97 100 96 100 102 100 96 100 99 100 Similarly, the relations between the surface / and the mean free-air temperature, or the mean plateau temperature, 6, have been prepared for reductions to the sea-level plane, the 3,500- foot level, and the 10,000-foot planes for over 200 stations, so that synchronous charts can be constructed on each of these three planes simultaneously. Such charts were prepared for one year, in part by telegram and in part by card reports, so that the pressure charts could be studied on the sea level, on the mean-plateau level, and in the two-mile level. These com- parisons were so suggestive and instructive in respect of the progress of storms and the areas of precipitation as to make them of great value in practical forecasts of weather conditions. The trend of the upper-level isobars shows clearly the course of CORRECTIONS TO THE BAROMETER 45 the storm track for 24 to 36 hours, whereas the sea-level isobars have very little evidence of this kind. This is because the closed isobars on the sea level have generally opened up into sweeping curves on the two-mile level. Similarly, the rain areas are indicated by the region of most oblique crossing of the lower with the upper isobars. There is a great future for meteorology in the use of these upper level charts. 6. The Plateau Correction C A0 H. An extensive discussion of the reduced pressures on the sea-level plane showed a series of plateau differences depending upon a station constant C = 0.001 usually, A = the departure of the monthly from the annual 6, and H = the height of the station in feet in units of 1,000 feet, so that A B = C. A 6. H. This plateau correction was computed and applied to all the plateau stations of the United States. It seems to take account of the effect of the land mass in the course of the year upon the temperature distribution, which is very complex in its action. 7. The Local Correction A A. After the corrections above mentioned have been applied, there are still a few stations which require a small correction A A to make them harmonize with the pressure system surrounding them. The cause is still obscure and is very local, possibly due to the wind action near the office. 8. The Local Vapor Pressure Correction. The prevailing relative humidity and the corresponding vapor pressure are approximate functions of the temperature in each locality, so that an approximate value of the correction to the barometer due to the presence of the aqueous vapor can be found for each station and applied along with the other corrections. 9. The Station Pressure Reduction Charts. It should be noted that all the barometric corrections have been made in terms of the surface temperature so that this t and the barometer reading B. when corrected for the several instrumental errors, become the arguments for the reduction to any plane. In practical work, instead of corrections from the station to the sea level or other plane of reference being furnished to the several stations, there have been prepared, for the arguments (/, B), the reduced sea- level pressure at once in a sufficiently expanded form of tables to 46 METEOROLOGICAL CONSTANTS AND FORMULAS permit of quick and accurate interpolation; similar tables were provided for the 3,500-foot and the 10,000-foot planes. With these auxiliary station tables the reduced pressures are promptly obtained, and transformed into the telegraph cipher code for transmission to other offices. In this way a large number of stations in the United States receive in several telegraph circuits, by interchange of messages, the necessary data for all map con- struction. Making the observations at 8 A.M., 75th meridian time in all districts, for example, the data are received, recorded, interpreted as forecasts, and usually retransmitted to all parts of the country within two hours, or by 10 o'clock. Examples of the Barometric Reduction Tables "Barometry Report," U. S. W. B., 1900-01, Tables 13-21. " International Cloud Report/' U. S. W. B., 1898-99, Tables 91-93. In order to illustrate the barometric formulas in practice, the example of Santa Fe, N. M., is here given. TABLE 12 REDUCTION TO THE SEA LEVEL BY THE W-TABLES IN LOGARITHMS SANTA FE, NEW MEXICO Height = 7013 feet. Longitude, 105 57' Latitude, 35 41' Arguments Jan. April July Oct. Year Station B Temperature . . .6 Vapor Pressure CQ 23.180 in. 32.0F. 0.145 in. 23.177 in. 54. 3 F. 0.199 in. 23.362 in. 79.0F. 0.574 in. 23.294 in. 58.0F. 0.294 in. 23. 248 in. 56.0F. 0.274 in. Logarithm log B. Table 17 m Table 19C, - $m Table 20 ym Sum. log BO 1.36511 + .11594 -.00016 -.00011 1 48078 1.36506 + .11090 -.00022 -.00010 1.47564 1.36851 + .10581 -.00061 -.00010 1 . 47361 1.36725 + .11011 -.00033 -.00010 1 . 47693 1.36638 + .11054 -.00031 -.00010 1 47651 Sea level .... B 30.254 29.898 29.759 29.987 29.958 Table 19A Arg I . Tablel9BArg.II .0018 .0014 .0025 .0020 .0073 .0058 .0037 .0030 .0035 .0028 Reduction B B 7.074 6.721 6.397 6.693 6.710 EXAMPLES OF BAROMETRIC REDUCTION SECOND FORM OF TABLE, NUMERICAL 47 Table 21, Sec. I. . 7.028 6.762 6.487 6.719 6.742 Table 21, Sec. II. -f. 064 -.018 -.041 + .004 -.003 /?w Ym -.018 --.022 . -.049 -.030 -.028 Reduction 7.074 6.722 6.397 6 693 6.711 The station pressure B is already corrected for the tempera- ture, gravity, instrumental and removal errors. For the argument ^ (t Sp + O or y 2 (t 8a + t 8p ), take 0. For the arguments (B. 0), in Table 17, take m. For the arguments (B . e ) in Table 19 A, take Arg. I (below). For the arguments (Arg. I, H), in Table 19 J5, take Arg. II (below) . For the arguments (Arg. II, m), Table 19 C, take - m. For the arguments ($, m), in Table 20, take f m. Numerical Form For the arguments (H, B = 30 inches), in Table 21, 1, take first reduction. For the arguments H, B^ (approx.), in Table 21, II, proceed by trials. Interpolate the correction for humidity from Table 21, III. Interpolate the correction for gravity from Table 21, IV. These two methods work very rapidly after a little practice, and the reductions are valid to the 0.001 inch of pressure. In order to illustrate the method of reduction for the plateau the following example is given in Table 13. The assumed station pressure B has the four station correc- tions applied. The body of the reduction table was computed for assumed values of 6, which correspond with certain surface temperatures /, computed from the .consecutive 8 o'clock pairs as observed. This t becomes then the practical argument for station reductions. The mean annual 6 for Santa Fe was taken 63 F., and A B at any time is the variation (0 63). The 48 METEOROLOGICAL CONSTANTS AND FORMULAS TABLE 13 SANTA FE, NEW MEXICO REDUCTION OF PRESSURE TO THE SEA LEVEL, THE 3,500- AND 10,000-FooT PLANES I. Reduction to sea level Elevation, 7,013 feet. Longitude, 105 57'. Latitude, 35 41' Temp. Correction for 22.40 22.60 22.80 23.00 23.20 23.40 23.60 23.80 24.00 24.20 I e C.A 0.H e LA Reduction Bo B from w-Table. -27 -16 - 5 5 16 26 35 43 50 58 67 77 88 -20 -10 10 20 30 40 50 60 70 80 90 100 -.50 -.44 -.38 -.32 -.26 20 +7.79 .59 .40 .22 7.05 6 88 7.86 . 66 .46 .28 7.11 6 94 7.93 .73 .53 .34 .17 7 00 8.00 7.80 .60 .41 .24 7 07 8.07 7.86 .66 .47 .30 7 13 8.14 7.93 .73 .54 .36 19 8.21 7.99 .79 .60 .42 25 8.28 8.06 7.86 .67 .49 31 8.35 8.3 7.93 .73 .55 37 8.42 .20 8.00 7.80 .61 43 .00 -.01 01 .00 .00 00 -.14 -.01 00 .72 .78 6.84 6.90 6.96 7.02 7.08 7.14 .20 .26 -.08 -.02 + 04 -.01 -.01 - 02 .00 .00 00 .57 .43 .29 .63 .49 35 .69 .55 41 .75 .60 46 .81 .66 52 6.87 .72 58 6.93 .77 63 6.99 .83 68 7.05 6.89 74 7.11 6.95 ,80 + .10 + .16 + .22 -.03 -.03 -.03 .00 .00 .00 .16 6.04 5.91 .22 6.09 5.97 .28 .15 6.02 .33 .20 6.07 .39 .26 6.13 .44 .31 6.18 .49 .36 6.23 .54 .41 6.28 .60 .47 6.34 .66 .52 6.39 Date Jan. Feb. Mch. Apr. May June July Aug. Sept. Oct. Nov. Dec. 5 28.9 30.6 37.0 44.2 52.7 62.0 67.4 67.0 62.0 53.2 41.7 33.1 15 27.9 32.0 39.5 46.6 55.7 65.1 68.5 66.3 59.9 49.8 37.6 30.8 25 29.3 34.5 41.9 49.6 58.8 66.2 67.8 64.2 56.5 45.7 35.3 29.8 Note. A A and C.&O.H have been united. II. Reduction to the 3,500-foot plane Temp. Correction for 22.40 22.60 22.80 23.00 23.20 23.40 23.60 23.80 24.00 24.20 I 0i C.A 0.H e AA Reduction Bi B from w-Table -24 -20 -.27 +3.61 3.65 3.68 3.71 3.75 3.78 3.81 3.84 3.87 3.90 14 10 24 .52 56 59 62 66 69 72 75 78 81 - 4 -.21 .44 .47 .50 .53 .57 .60 .63 .66 .69 .72 fi 10 18 36 39 42 45 48 51 54 57 60 63 16 ?0 -.15 .29 .32 35 38 41 44 46 49 52 55 ?6 30 -.12 22 25 28 30 33 36 39 42 45 47 36 40 -.09 . . . .15 .18 .21 .23 .26 .29 .32 .35 .38 .40 45 50 -.06 .08 .11 .14 .16 .19 .22 .25 .28 .31 .33 55 60 -.03 .66 .00 3.02 3.05 .08 .10 .13 .16 .18 .21 .24 .26 64 70 .00 -.01 .00 2.96 2.99 3.02 3.04 .07 .10 .12 .15 .17 .19 74 80 + .03 -.01 .00 .90 .93 2.96 2.98 3.01 3.04 .06 .09 .11 .13 84 90 + .06 -.02 .00 .84 .87 .90 .92 2.95 2.98 3.00 3.03 3.05 .07 94 100 + .09 -.02 .00 2.79 2.82 2.84 2.86 2.89 2.92 2.94 2.97 2.99 3.01 EXAMPLES OF BAROMETRIC REDUCTION III. Reduction to the 10,000-foot plane 49 Temp. Correction for 22.40 22.60 22.80 23.00 23.20! 23. 40 23.60 23.80 24.00 24.20 t 02 C.A e.H e AA ' Reduction Bt - B from m-Table. -20 -?0 -.21 -2.67 2.70 2.73 2.75 2.78 2.80 2.82 2.85 2.87 2.89 -10 -10 -.18 - .61 .64 .67 .69 .72 .74 .76 .79 .81 .83 1 -.15 ... - .56 .59 .61 .63 .66 .68 .70 .73 .75 .77 12 10 - 12! ... .51 .54 .56 .58 .61 .63 .65 .67 .69 .71 23 20 09 46 .49 .51 .53 .56 .58 .60 .62 .64 .66 34 30 -.06 - .41 .44 .46 .48 .51 .53 .55 .57 .59 .61 45 40 -.03 .00 .66 - .37 .39 .41 .43 .46 .48 .50 .52 .54 .56 56 50 -.01 + .01 .00 - .32 .34 .36 .38 .41 .43 .45 .47 .49 .51 67 60 + .01 + .01 .00 - .28 .30 .32 .34 .37 .39 .41 .43 .45 .47 78 89 70 +.03 80 +.06 + .01 + .02 .00 .00 - .24 - .20 .26 .22 .28 .24 .30 ' .26 .32 .28 .34 .30 .36 .32 .38 .34 .40 .36 .42 .38 100 90 + .09 + .02 .00 -2.16 2.18 2.20 2.22 2.24 2.26 2.28 2.30 2.32 2.34 value of C for Santa Fe happens to be 0.00086 and H is taken in units of 1,000 feet, 7.01. Hence, for the sea level, for = - 20, C.A0.H = 0.00086 X (-83) X 7.01 (-63) 20 40 60 80 100 (-43) (-23) (- 3) (+17) (+37) - 0.50 -0.38 -0.26 - 0.14 -0.02 + 0.10 + 0.22 The plateau stations always seem to require such a correction in order to make a harmonious network of pressures with the surrounding low-level stations. It is easier to make this correc- tion in the form given above, rather than attempt to trace out its effect upon the mean temperature 6, as related to the surface temperature t. The entire subject needs a fuller theoretical discussion if possible. The vapor pressure correction e is the mean value as for the argument surface /, and suffices for these station reduction tables up to the 0.01 inch. The final station reductions to the sea level were made for the arguments (t. B}, and applied to the assumed values of B, so that for the same arguments (t. B), the value of B Q is immediately read by an easy interpolation. Similar reductions were made for the 3,500-foot plane, and the 10,000-foot plane. They were checked by reduction from the station to 3,500 feet, to sea level, from sea level to 10,000 feet, and thence in a circuit back to the station pressure. This was done for all the numerous plateau stations in the United States. CHAPTER II Thermodynamic Meteorology General Formulas for the Computation of P, p, R,from the Observed Temperatures T in a Free Non-Adiabatic Atmosphere IT is easily seen from the discussion of the barometer how many complexities this instrument introduces in practice, on account of the series of corrections, and by reason of the system of units employed, which separates the data from all other thermodynamic terms occurring in meteorology. There is need, then, of developing another system of reduction, by which it may be possible to pass from the temperatures T t observed in the free air up to great heights, to the corresponding pressures P, densities p, and gas coefficients R, so that the general law P = p R T shall continuously be satisfied throughout the atmosphere. If the mercurial barometer is needed on the surface to give a base for vertical reductions, it is not practical to carry it to heights on kites and balloons. The aneroid may be used to check the resulting computed pressures, but not to give the actual pressure for the dependent formulas. Fortunately, there is a simple and comprehensive set of formulas for this purpose, which will now be developed. For any temperature vertical gradient a, the temperature T at the height z above T is, J rr* (169) T = T Q - a z, so that, d T = -adz, and d z = --- . The differential equation for pressure variations with the height is from (41), (170) dP = pg dz= -- p godT, by substituting d z. From the Boyle-Gay Lussac Law, P = p R Q T, by division, we obtain, since by (24) ^- , a /CQ K 1 dT nk dT aR T ~ k - 1 T 50 GENERAL FORMULAS FOR NON-ADIABATIC ATMOSPHERE 51 Passing to logarithms and to limits, this gives, - 1 - (^\ PO \T O ) nk Having observed 7\ and T on two levels, at the vertical distance apart z\ z , the pressure PI can be computed from the pressure P on the lower level. We proceed to determine the density pi, and gas coefficient RI, which correspond with PI = piRiTi on the Zi-level, when P = p R Q To is given on the_2g-leyel. By successive stages from the surface the same formulas will arrive at any altitude where the temperature TV- is known. From two successive levels, we have the ratio, (173) 75- = 5~^F , and by transforming, /^ r'o p KQ 1 o ( . * ; At this point the entire treatment of thermodynamic meteorology diverges. If the gas coefficient is taken constant, RI = RQ, and (175) fl. For example, V. Bjerknejs in equation J jageJ>l, " Dynamic Meteorology and Hydrography," Carnegie Institution of Wash- ington, 1910, uses this form for the ratio pi/p , since in his system of units go = 1. This is the common way of treating the matter, but it is easy to see that this derivation of the non- adiabatic densities from the well-known adiabatic equation is inconsistent with the analogue of the pressures in (172), which k simply multiplies the exponent T - - by n, so that for the K 1 densities the exponent should be , _ . Proceeding in the second way it is obvious that, preserving the same treatment for density as for pressure, we should take, 52 THERMODYNAMIC METEOROLOGY / 17A v (176) = ( jT po VI o Ri /TV*- 1 ' Ro =: \T ) In order to check these results by (172) and (173), - ^ +(n-l) + l > which is correct. This process makes R a variable in the existing non-adiabatic atmosphere, so that the air is not distributed by gravitation like an adiabatically expanding gas, in which there is no circulation and no change of heat contents by radiation and absorption from level to level. On the contrary, the observations prove that usually there is circulation and radiation going on to preserve the gravitation equilibrium with the existing pressure variations or gradients. As stated, the entire system of ther- modynamics takes on a new form through the fact that the specific heat must also be a variable along with the gas coefficient. (2) Cp-^R. We shall return to explain the consequences of this funda- mental property of the atmosphere, which is in reality a gaseous mixture of rapidly varying thermodynamic capacities, in con- sequence of the effect of the absorption of solar radiation and the emission of atmospheric radiation in various ways. The Adiabatic Equations The correlative adiabatic equations follow at once by putting n = 1, and a = a , (179) Pressure = -TO \J- o (180) Density^ = (^ Po W o> (181) Gas constant RI = R . WORKING NON-ADIABATIC EQUATIONS 53 The Working N on- Adiabatic Equations (182) Pressure, log P, - log P = ^ (log 7\ - log r ). (183) Density, log Pl - log Po = ^~ (log 7\ - log T ). (184) Gas coefficient, log RI - log R Q = (-!) (log Z\ - log T ). These equations were published in the Monthly Weather Review, March, 1906, (38), (39), (40), and they have been illustrated by numerous applications to balloon and kite ascen- sions with excellent results, up to great altitudes, as 20,000 meters. The following example shows the method of arranging the computation so as to proceed from level to level, the com- puted PI, pi, RI of one becoming P , p , R for that next above it. The example is taken at random from our computations, some of the results being compiled in Bulletin No. 3, Argentine Meteorological Office, 1913. The constants are taken from Table 3 in the (M. K. S.) system. The surface values of.P , PO, R , TQ are assumed to conform to the adiabatic system, while PI, pi, RI, TI above the surface are computed by the non-adiabatic system. At the height z = 116 meters the density is computed from the adiabatic formula, P Also, P = go p m Bo, where B Q is in meters. It will be noted that the check is complete. It may be stated that the ob- served values of B at Lindenburg are usually about 1 mm. higher than the computed values B c . This constitutes a cri- terion upon the adjustment of the aneroid, which, in ascending, lags in registration- and records a pressure corresponding to a lower level than that assumed for the temperature T at the height z. 54 THERMODYNAMIC METEOROLOGY 88-" 22 <N H M H CO S J O <M O t^ TjH |> t^ T i O CO 00 O CO <N t>- Tt< Q *O <M rt< O iO f^ l> T-H 3 3 8 ? r g^g^ "8 co i> oo t- TJH T^ 00 1> b- o o <N TjH T-H I> sg I o =>S 00 Ttl O "* i-H CO TtH O 10 Tt^ 8 05 CO O i I T-H O i> co co i> Tf IO N I *f o o -^ r* ^E 00 <N % 2g OS 00 O CO O TjH CO CO CO TfH " Tt< i- 1 CO CO CO CO o o CO CO ^ I i rH CO CO O T-H CD iO <N CO O T-H rt< 00 GO IO O CO CO O iO O5 C^l OS GO 00 ^^ ^* ^^ CO <N CO t^* t^- l>- CO CO 00 I I cc v~ Tf Tj T g '3 ^ O( ^ O H D ) iO (M CO (M !> OO 00 CO O5 Oi t^* t^ O5 O5 OO " s 4 5 r s b-H -t t i r^ ^ fcj c 8 1 M 4- '5 1 t" J rE C *. ; i 1 1 si ; c I i; S i t ^ ^_ ' J v X 1 3 ^ X ^ C| ? 1 5 j 1 -H , 8 ^ | H 1 Ot t. j 1 1 bj ^ o 1 1 ^_- D J j D *8 ' j 7^ b/ ^C ,1 "^ ) ] PQ == '. bfl CO . O WORKING NON-ADIABATIC EQUATIONS 55 *o CO 7 i t^ CM iO CMTt<O5 CO i I d '+ ^ ^ ^ ^ CM O5 b- i CM t>- *-* 00 i-H Tt< TjH CM 00 CO Tf< t>- O O5 i-H cO l> O5* t^. I O5 CM CO CM ^ CO CO O5 CO CO w S rH 00 CO O5 i-H CO O5 00 O O 00 O ^ CM II I d d _,. ^o O5 CO p CO p 07 V s CM i-H 00 i-H o i-H i-H 00 b- CM OO co O ^ t- , CM s l> O C^ CO CO ^D CO CO CO 05CO + co >O f (X, 1 =- a w> b % ' bo n bo 5 44 5 bO I S bo bO bo bo j? s 7 1 tf ^ ^ ^ . J o 6 ^^ S I * I * 8 I -H+ Q - L^ K - II *O bo -= ^ o bo & I o u 'Q, . cyo ^ I I KJTJ bO bO .2 ^ -2 ^ co" ^co" bf) 00 oo 00 t- ^ 11 i-H i-H i-H 00 (* 05 00 d d CO Oi d d CO O5 ^ d s CO s 8 II o v2 a; cs e 3 ff S"S I! "rt * (U O (51 * 56 THERMODYNAMIC METEOROLOGY TABLE 15 COMPARISON OF THE PRESSURES COMPUTED BY THE NON-ADIABATIC FORMULA AND THOSE OBSERVED AT GREAT HEIGHTS Heights Lindenburg Lindenburg Atlantic Ocean Atlantic Ocean in May 5, 1909 July 27, 1908 Sept. 9, 1907 June 19, 1906 Meters Lat. +52 Lat. +52 Lat. +26 Lat. -2 z Be Bo Bc-B, Be Bo Bc-Bo Be B Bc-Bo Be Bo Bc-Bt 19000 18000 .0589 .0600 -.0011 17000 .0679 .0670 + .0009 .0692 .0690 + 2 .0742 .0740 + !6662 16000 .0794 .0790 + 4 .0808 .0790 + 18 .0861 .0860 + 1 15000 .0927 .0930 3 .0945 .0950 - 5 .1004 .1010 - 6 .1021 .1020 + .0001 14000 .1082 .1090 - 8 .1101 .1110 - 9 .1177 .1170 + 7 .1203 .1200 + 3 13000 .1266 .1270 4 .1289 .1300 - 11 .1381 .1380 + 1 .1409 .1410 - 1 12000 .1484 .1490 6 .1512 .1520 - 8 .1615 .1610 + 5 .1641 .1640 + 1 11000 .1744 .1750 6 .1772 .1780 8 .1881 .1880 -1- 1 .1902 .1900 + 2 10000 .2041 .2050 - 9 .2067 .2080 - 13 .2179 .2180 - 1 .2191 .2190 + 1 9000 .2374 .2390 - 16 .2399 .2410 - 11 .2509 .2510 1 .2512 .2510 + 2 8000 .2746 .2760 - 14 .2769 .2780 - 11 .2875 .2880 5 .2870 .2870 7000 .3159 .3180 - 21 .3181 .3190 - 9 .3281 .3290 9 .3268 .3260 + 8 6000 .3619 .3630 - 11 .3639 .3650 - 11 .3732 .3740 8 .3709 .3710 - 1 5000 .4128 .4130 - 2 .4149 .4160 - 11 .4233 .4240 - 7 .4200 .4200 4000 .4696 .4700 4 .4716 .4720 - 4 .4789 .4800 - 11 .4747 .4750 3 3000 .5330 .5340 - 10 .5344 .5340 + 4 .5402 .5410 - 8 .5356 .5350 + 6 2000 .6035 .6040 - 5 .6040 .6040 .6077 .6060 + 17 .6031 .6030 + 1 1000 .6823 .6840 - 17 .6806 .6810 - 4 .6821 .6820 + 1 .6783 .6780 + 3 Surface .7599 .7599 .7551 .7551 .7640 .7640 .7610 .7610 The differences between B c and B Q are probably due to an assignment of the temperature to a slightly erroneous height,* owing to the movement of the balloon ahead of the record of the barograph and thermograph, which requires a correction for lag. The pressure recorded by the aneroid is for the mixture of dry air and aqueous vapor, so that by (75) for the same height, where T = T and p = p , (185) P = P (l - 0.378 J; Bo = B (l - 0.378 ^ where P or B is the dry air pressure, and P or B Q the pressure in the mixture, e being the vapor pressure in the * The heights have been read from the aneroid record and are not corrected for the supposed lag; but the error is less for T than for P because T changes more slowly than P. THE VARIABLE VALUES OF H = -~ 57 same system of units. The connection between (182) and (159) is such that they can easily be shown to be identical, after the action of the mercurial barometer has been made to equal that of an aneroid. The Variable Values of n = - The introduction of n into the adiabatic formulas converts them into the non-adiabatic formulas, and at the same time adds circulation and radiation to a static atmosphere. Hence, by (13), = _ _T l -T _ -(T a - To) a (186) n = = marks the natural transition from static to dynamic and ther- modynamic meteorology. It is important, therefore, to under- stand the full significance of the ratio between the adiabatic and the non-adiabatic temperature gradients. Since (T a T ) TABLE 16 EXAMPLES OF THE VALUE OF n Height z Lindenburg May 5, 1909 Lindenburg July 27, 1908 Atlantic Ocean Sept. 9, 1907 18000 - 3.6553 17000 +1.0966 -16.4489 - 2.5972 16000 -5.4830 - 6.1684 - 0.9399 15000 -9.8694 -49.3467 - 2.9027 14000 -3.5248 - 6.1684 + 4.9347 13000 -1.7624 - 2.5972 + 1.9739 12000 +5.8055 + 2.0999 + 1.3337 11000 + 1.1749 + 1.1611 + 0.9676 10000 +1.0846 + 1.2986 + .1611 9000 +1.1215 + 1.1344 + .1611 8000 +1.1749 + 1.1749 + .3901 6000 +1.3520 + 1.6179 + .3901 4000 + 1.7016 + 1.8980 + .2653 2000 +3.5248 + 1.2986 + 2.5972 1000 -4.9347 + 12.3366 -12.3367 and (Ti TO) are usually each negative with an increase in elevation, n is generally a positive quantity, but it becomes 58 THERMODYNAMIC METEOROLOGY negative whenever there is an inversion of temperature, or temperature increase with the height, as near the surface of the ground in the early morning, or in the isothermal layer at great heights. If TI = Ta, n = 1, and the gradient is adiabatic; if TI = T Q , n = and there is no temperature change with the height; if TI > T Q , n is negative, and if TI is only a little greater than T Q for the change in elevation z\ z , n will be a large negative quantity. Table 16 gives a few examples of the values of n. Table 16 indicates the wide range through which n passes in practical reductions, and it is easily seen how valueless the formulas become for meteorological discussions where n is assumed to be unity, as is commonly the procedure. Further- more, since the value of n must always be carried to the fourth decimal it has not seemed worth while to construct general reduction tables, because they would be very extensive or re- quire complex interpolation. The Differentiation of (172) Since n is a variable in equation (172), we proceed to differ- entiate it for P, T, n, variables. (187) log = log . Differentiate, nk TT T k dn 4-1 r. + log r k Substitute P = p R T and Cp = R , rv J. (190) = n Cp d T + CpTlQgTdn. p M. Take the integral between limits for the mean p, (191) Pl ~ P = % Cfco (3Ti - To) + ^ CA, r 10 log p PlO M 2 o DIFFERENTIATION OF (172) 59 By (144), (146), (147), using the mean values Cp w , (192) Pl ~ o P = * Cp w (7\ - To) + (m - n ) Cp a (Tt - To). = ! c# 10 (T\ - r ) + i c/> a (ri - r ) - WoC> (Ti-To) = cfco (r a - r ) + cp a (T a - TO) - cp a (T a - To) = Cp a (T a - To) - (Cp a - Cfco) (T a - T ). Since p and Cp are variable in the stratum (zi z ), the mean values are pio and Cp W) while HI continues constant within the stratum, which must not be taken too thick to allow this approximation to hold true. The result is twofold. First, (193) Pl ~ P = C# w (T a - To) = i C#i (r x - To). Pio The adiabatic system, on the other hand, gives, (194) Pra ~ P = C# B (r fl - To) = - g fe - so). PaO Hence, the difference between the two systems is, (195) - - = (Cp a - Cp) (T a - To). PaO PIO From the common dynamic equation for pressure and velocity, which will be proved in a later section, and adding a term for the dynamic energy of radiation heat, we obtain by substitution in (192) the working and fundamental equation, as the second result, (196) go fe - a.) = - fl ~ P - J (?, 2 - <?o 2 ) - 2i - Co). Pio These have already been quoted in (21), (25), (26), (27). From (192), using the last form, we find, (197) go (z, - Sb) * - ^ ^ - (Cp a - Cp 10 ) (T a - To), Pio and by comparison of the last terms (196), (197), (198) - (ft - Co) - i (?i 2 - <?o>) - - (Cp a -Cp 1Q ) (T a -T Q ). 60 THERMODYNAMIC METEOROLOGY (199) - (<2i - <2o) = - (Cpa ~ C# w ) (T a - To) + i (?i 2 -?o 2 ). That is to say, the variation from the true adiabatic system is due to the kinetic energy of heat (<2i Q ), and circulation J (qi z g 2 ) for the unit mass, which is equivalent to the .variation of the mean specific heat of the stratum from the adiabatic specific heat, Cp a = 993.5787 in Table 3, multiplied by the change in temperature between the bottom T and top Ta of the layer z\ ZQ. Unfortunately, there seems to be no way to separate (Qi Q Q ) from \ (qj <? 2 ) through the specific heat, except by using the direct observations of the velocity, and then computing (Qi Q Q ) by means of (199). Those observatories which record P, T, the pressure and the temperature, but not the R. H. and <?, the relative humidity and velocity of motion of the stratum, cannot enter upon any problem in circulation or radiation in the atmosphere. From equation (196) we obtain, (200) - = gdz + qdq + dQ, and, P (201) -dP = gpdz+pqdq+pdQ. From the equation (190) is derived, (202) - d P = p n Cp d T - p -^ Cp T log T. d n, and by means of (146), (144), (21), this becomes, (203) - d P = g p d z - p j Cp T log T. d n. The usual adiabatic pressure variation dP = gpdz, as in (170), is converted into the non-adiabatic form with circulation and radiation by subtracting p T> Cp T log T d n, which includes the differentials first for heat, and second for circulation. If equation (203) is treated in the same manner as (170) we shall obtain by substitutions, foriA\ ( T \ ( T \ \ Zj-Zp l dp Zi- (204) f^) -(~r) =(a -a) T = which is the difference between the temperature ratios in the adiabatic and the non-adiabatic systems. DIFFERENTIATION OF (172) 61 (M 1 * CO 00 iO l^ GO 00 <M "tf 1 <N O OS C^ ^* iO 00 GO .... t> IO rHCOrH OSrH (N rH rH O C5 OSCO TjH OS IO I O I>C^rH CO ^ os osos + I _j_ + , + b- o co to i> ^ 00 1> O CD t> - GO OS OS 8 28S GO CO l o TF CD OO 00 IO OS 00 00 Tf CO GO GO OS OS OS ^^ O OS co os co rH 5 g 8 O CO CO O O5 O^ O^ .. 1, s I V O O 00 OS d d co co + I ? ^ l> CO 2 $ co c^ I s " *o O O <N t^ CO CO OS CD iO O t>- co os co ^* CO (M O CO 1 > * OS OS Os o-. -.1 CO OS CO "*i CO 5 ^S^_ id d co co t>- 00 OS rH OS OS OS _|_ ^ GO O O s IO rt ^ 7 TjH rH- % OS OS CO O CO rH <M (M OO CD co r- t^ 10 GO 01 + I OS o o CO CO OS O O rH CD rH I d 8 Tt< O CO O CO <M CO O + + ^ CO ^ (N CO OS CO 00 7 + IO l> I JO OS iO 58 O CD CO d O 00 t>- OS CO r^ IO 00 os id '* co d OS CO l>- CO ? ' o o *o . ^^ Ttl IO l>- rH CO rH IO 3^ 00 rH ^ rH O rH O ^ CO co o co CC r" $ > If I 2 i C c I a eg > o a 1 x / N /^ C ' g CC X G- rH & . c E^ < 1 1 * rH | ^J ^ 1 <i ^_ X" 2 g ' s~~ I Q^ C & c > ^ c i ^J c ^ 3 S ^ ' --5 M ^fc I , - ' 4 I i 5 t sl H "d I '^ ^:c , T i 2 * OS + 7 c <5 i c a> bo E ^ 62 THERMODYNAMIC METEOROLOGY 00 O O CO os co co 88 I 10 o 10 8i + 8 z * C I 3 o!^ OQ eSU ^ SC- i s-s. 00 CO CO o o CO CO o^ o^ o o CO CO oo *-;-* co co rjJ rjJ O^ Oi GO OO o o CO CO CO T* CO CO S S O O 00 00 CO CO CO CO I I CO CO <N Cjl 00 00 CO CO DIFFERENTIATION OF (172) 63 There are a few critical remarks that can now be made to advantage, in view of the principles that have been assumed in many important meteorological papers. (1) By (201) and (203) it is not proper to assume that the variation of the pressure in a vertical direction is proportional to the mass variation, d P = g d m, because this excludes the circulation and the radiation. It is a contradiction in terms to seek for solutions of radiation problems under this limitation. (2) By (198), (199), if it is assumed that R and Cp are constants and the change of temperature is adiabatic, there can be no circulation and no radiation. It is a contradiction in terms to seek for solutions of radiation problems under this assumption. i*d P (3) By (193) the integral / depends upon a varying Cp. It follows that if P. T. are observed by the instruments, if RQ is taken constant in the Boyle-Gay Lussac Law, and if p is then computed directly from P = p R Q T, this value in the /dP will always lead to fictitious results, because the p circulation and the radiation are excluded from the dis- cussion. (4) In the integration of (190), in strata where n is constant, as they can be made by taking the layer thin enough, it is proper to use the mean values of the several terms, pio, Cpio, TIQ, jp p _ _ p 1 i / "\ PlO = 2 \Pl + POJ. /* J P f Pio nCpdT = n Cp 1Q (^ - T ), Cp w = J (Cpi + Cp ). I T I TI M TQ M TQ since Cp = Cp a and n = 1 in this term. (5) It has been customary to evaluate the equations of motion in the atmosphere while omitting the heat term (Qi Qo) in (196). The result is that it is impossible to balance the other three terms, so that the great problem of the relation of the circulation to the observed pressures in the general and the local circulations has been insoluble. In order to exhibit more fully 64 THERMODYNAMIC METEOROLOGY the relation of these four terms, two examples of balloon ascen- sions to great heights are added. TABLE 18 THE EVALUATION OF (Q l Q ) IN BALLOON ASCENSIONS (196) Station Lindenburg, May 5, 1909 Atlantic Ocean, Sept. 9, 1907 jj * _/_ ._ \ Pi-Po -J -(Qi- Pt-Po -J _ (Ql _ pio (v-^ Qo) pio- (V-tf Qo) 17000 9806.0 6437.6 + 3.6 3358.6 + 6.2 6755.8 -26.8 3065.9 +11.1 16000 9806.0 6629.2 + 5.6 3169.2 + 2.0 7301.2 + 6.4 2496.1 +2.3 15000 9806.0 6963.7 +11.2 2817.1 +14.0 7886.4 +38.9 1885.6 - 4.9 14000 9806.0 7357.0 + 9.2 2433 . 5 + 6.3 8288.6 +104.8 1424.2 -11.6 13000 9806.0 7851.0 + 5.6 1939.7 + 9.7 8508.6 +53.0 1228.5 +15.9 12000 9806.0 8300 . 4 + 1.7 1499.8 + 4.1 8666.2 -89.3 1227 . 9 + 1.2 11000 9806.0 8489 . - 0.9 1313.9 + 4.0 8706.6 -33.2 1130.2 + 2.4 10000 9806.0 8530.6 + 0.9 1268.8 + 5.7 8721.0 - 1.4 1079.9 + 6.5 9000 9806.0 8565.0 + 8.1 1227.5 + 5.4 8769.4 + 4.2 1025.7 + 6.7 8000 9806.0 8615.8 +13.2 1176.9 + 0.1 8845.2 +30.4 928.1 + 2.3 7000 9806.0 8668.8 +16.1 1114.1 + 7.0 8958.0 -54.3 895.0 + 7.3 6000 9806.0 8750.2 + 8.8 1044.0 + 3.0 9074.0 -53.8 777.6 + 8.2 5000 9806.0 8885.0 + 5.2 909.2 + 6.6 9222 . +18.3 568.1 - 2.4 4000 9806.0 9051 . 5 + 5.4 745.6 + 3.5 9340.5 -24.6 485.9 + 4.2 3000 4903.0 4589.3 + 7.0 312.3 - 5.6 4706.0 - 2.2 204.1 - 4.9 2500 4903.0 4636.4 +11.7 257.5 - 2.6 4725.8 - 6.2 179.2 + 4.2 2000 4903.0 4689.1 - 5.9 217.3 + 2.5 4764 . 9 - 4.5 135.7 + 6.9 1500 4903.0 4738.9 -22.1 174.4 +11.8 4815.6 - 0.8 87.6 + 0.6 1000 4903.0 4813.8 -32.5 104.4 +17.3 4846.6 +13.7 9.9 +32.8 500 3765.5 3753.8 -11.5 19.5 + 3.7 4865.7 -10.7 0.9 - 1.9 * More exactly g should diminish with height. It is easy to see that in the higher levels (Qi Qo) is a domi- nant term, and that there is no possibility of a balance between the other three taken by themselves.. There is a delicate interaction of the four terms, such that the circulation seeks to adjust the pressure and the radiation to the demands of gravity. The minor errors A are to be chiefly ascribed to the fact that the observed temperatures are not quite correct at the height z. In many cases there are pair values of -f A and A easily adjusted on this hypothesis. The Two Laws of Thermodynamics Before proceeding further it is necessary to summarize the two fundamental laws of thermodynamics, the first being that of the conservation of energy, and the second being that of the THE TWO LAWS OF THERMODYNAMICS 65 decrease of entropy or increase of expenditure in a non-conserva- tive system. The first law is defined as follows: In a conservative system, from which no heat escapes and into which no heat is re- ceived, the sum of all the changes in the energy, whether large or small, remains constant. In a system in communication with the outside world the amount of the energy gained or lost by it is equal to that delivered to or received from the outside world. The second law may be described in several ways. Heat cannot by itself pass out of a colder into a warmer body (Clausius). It is impossible to construct a periodically acting machine which does nothing else than raise a weight in expending work and cooling a reservoir of heat (Planck). There exists in nature a quantity which always changes itself only in the same direction in all the variations which take place in nature; this is the entropy. It is in nowise possible to diminish the entropy of a system of bodies without there being left changes in other bodies. If such changes do not remain, then the entropy of a system can continue the same. Every physical or chemical change takes place in such a way that entropy either decreases or remains the same, but the outside world tends continually toward maximum entropy. It is necessary and sufficient for the equilibrium of a separate structure that for all possible changes in the state of the structure, the changes of entropy be zero or negative (Weinstein). The First Law of Thermodynamics The following series of formulas are available in the adi- abatic and non-adiabatic systems using the proper values of P. p. R. T. P p ~'~ R f RT i) = . dT dP (205) (206) (207) (208) P Pv dT d v = P RT = RT. Pdv + vd P dP = P R R T dT P ' P 66 THERMODYNAMIC METEOROLOGY dv dT dP_ (209) : ~Y p Referring to the series of equations 84-118 for the definition of the terms, we have generally, (210) Inner Energy. U = H + J = Q-W. (211) U = Cv fT.pdv = - fp.dv. (212) dU (213) 7? 7" 1 (2H) d U = Cv d T + -- dv - Pdv. (215) dU = CpdT - Pdv. (216) d U = Cvd T = (Cp - R) d T. (217) External Work. W = (K) + V = J + H + V. (218) dW = dQ-dU. (219) dW = TdS -dU. (220) dW = d(TS- U} -SdT. (221) dW =* -dF -SdT. (222) d W = P d v. dP (223) dW = RdT - . (224) Heat Energy. Q = W + U. (225) () = TF + # + / = # + #- (226) Q = [(K) + V] external + [H + J] inter- nal + (R) friction. (227) (228) (229 dQ = CvdT-\-RdT v dP = Cp d T (228) JQ = Cz;jr+ ~ - dv = Cvd T + Pdv. THE FIRST LAW OF THERMODYNAMICS 67 (230) dQ-C, K (231) dQ = TdS. From (218) in heat units, we obtain, (232) First Law. A d W = d Q - A d U, and by (222), AdU (233 ) A dPd ..(AQ\ d [ dTdv ~ dT\dv) ~J~v dP dAP dCv (234) A IT = ^T ~ ^ *(->) V * <** (236) - Qi = - Ui - Wi, Heat expended outward (negative). + ft = + #2 + P^>, Heat received inward (positive). (237) A Q = Q 2 - Qi = (U 2 - Ui) + (W> - Wi), Resultant heat energy. Among the definitions we have, V = the external gravity potential acting inwards, to- gether with the centrifugal force of the earth's rotation at the angular velocity w = , at the perpendicular distance tcr from the axis, (238) V = - fg d z + y*f (vutfdm. (239) (K) = the kinetic energy of motion of the mass m with the velocity q. (K) = | m (f = H + /. (240) F = the free energy or the thermodynamic potential at a constant volume. F = U T S. (241) U = The bound energy. U - F = T S. (242) = the thermodynamic potential at a constant pressure, F + Pv. <f> = U - TS + Pv. = U-TS + RT. 68 THERMODYNAMIC METEOROLOGY Fundamental Equations and Definitions It is convenient to have for ready reference the fundamental equations and several definitions of thermodynamic processes, though they cannot be further developed in this connection. It is the purpose of this treatise to prepare such data for meteor- ology as can be admitted into the large group of well-known equations which have been heretofore inapplicable in the atmos- phere for lack of the necessary correct values of P. />. R. T. There are several variables : P.v.T. Q.W.U .S.R. (243) dU = A l dQ- dW = A l TdS - Pdv. = Pdv. (244) dF = - A^SdT - Pdv. fA*\ JT dF + Pdv dF (245) dT== ~~~A^S ' dv== ~~ d<f> = - Adidbatic Processes, d Q = and d S = 0. Adiabatic. d Q = signifies no gain of heat from the outside and no loss of heat to the outside from the system. Isentropic. dS = signifies that the entropy remains con- stant. Isodynamic Processes, d U = and d T = 0. Isodynamic. d U = 0. The inner energy remains constant. Isothermal, d T = 0. The temperature remains constant. Isoenergetic Processes. dW = 0, d P = Q, dv = 0, d p = Q. Isometric, d W = 0. The expenditure of external work is the same. Isobaric. d P = 0. The pressure remains unchanged. Isochoric. dv = 0. The volume is constant during the process. Isopyknic. d p = 0. The density is constant during the process. FUNDAMENTAL EQUATIONS AND DEFINITIONS 69 Isopiestic Processes, d P = 0. Isopiestic. dP = 0. The pressure is constant while the other variables change. Isoelastic Processes. dR = 0. Isoelastic. dR = 0. In ideal gases the gas coefficient is constant. Evaluation of dQ in Terms of P. v. T. through the Entropy S. Taking the following three pairs of variables, they lead to the definition of the specific, latent, and expansion heats. (246) Variables (v.T). d Q = T (||) dT + T (~) d v = \O JL ' v V O V ' T + C T dv. (247) Variables (P. T). dQ = T d T + TJ dP (248) Variables (P.v). d Q = T (|) dT+T (|) d v = Hence by comparison the definitions become, Specific Heats. Latent Heats. Expansion Heats. (249) C P = T ( . (250) C T = r( (251) r p = r(|f ) . p . T p p (252) c, = T . (253) rr = r. (254) r > r) These occur in These occur in These occur in radiation. evaporation. convection. The subscript indicates the term which remains constant. Evaluation ofdQin Terms of P. v. T. through the Inner Energy U. (255) Variables (r. T). dQ =A (|^) d T + A [ (|^) T +P~]d v. 70 THERM ODYNAMIC METEOROLOGY (256) Variables (P. T). d Q =A [ (|~) + P |~]d T + (257) Variables (P. v) . d Q = A (|^) d P + f (|^) + P~| * o Jr ' p UAO v * i Specific Heat d v. p Latent Heat Expansion Heat ^ - () Latent Heat Specific Heat - (H). -dfi Expansion Heat (263) r,= By inter comparisons and substitutions very numerous equa- tions can be constructed. Compare Weinstein's "Thermody- namik." Those for entropy are, (264) S vT = S + Cv log T + A R log v. (265) S TP = S + Cp log T - A R log P. The Second Law of Thermodynamics This is derived from equation (231). (266) Second Law. dS =-~. This gives rise to two processes in nature, the reversible, in which after a series of transformations the original state is reached, and the irreversible process, in which the original state is permanently lost. THE REVERSIBLE AND IRREVERSIBLE PROCESSES 71 For the reversible process, d S = -=r = 0. From (225) for d Ui = d U 2 = d W 2 = 0, we have, (267) Q 2 - Q l + W 1 = 0. (268) From the first law. d Q 2 - d Qi + d Wi = 0. (269) From the second law. d S 2 - d Si = -r - - = 0. L2 1\ T (270) Solving these equations, d Qi = ^ d Wl ' r,' a - Q 2 " TI - 2Y d - g 2 " TX - r 2 * (272) ^ = (Tl ~ ^ =1 - ^- 2 . The efficiency of the engine. ^i ^ i i ^ of 1 M ^ - n ( 1 -} - ft - r. -- ft - - (274) Carnot's Function = -^ in mechanical units for 7\ > T 2 1 1 = -T-TFT in heat units. A T! T! = the temperature of the source of heat energy. T 2 = the temperature of the sink of the energy. The energy runs down from the source to the sink. For the irreversible process, d S = -TJT > 0. (275) From the first law. d Q 2 - d Q l + d Wi = 0. (276) From the second law. d S 2 - d Si = -J^- - -Jr* > 0. lz J-i Solving these equations. (277) dW l 72 THERMODYNAMIC METEOROLOGY (278) Ford IF, = 0, (279) d Qi >~r . 1 1 d Q 2 Carnot's Cyclic Process As an example of a reversible process we may describe the Carnot Cycle, in which a unit mass^with the initial condition (Pi. i. jfi) passes by an isothermal change to a second condition O Volume v FIG. 3. Carnot's cycle (P 2 - 2. ^i), then by an adiabatic change to the third condition (Ps. z; 3 . T 2 ), thence reversing from this extreme point by another isothermal change to (P 4 . z> 4 . T 2 ), and finally by another adiabatic change to the initial (Pi.Vi. TI). This is illustrated in Fig. 3. Ti = temperature of the source RI. Tz temperature of the sink R 2 . 1 and 3 = Isothermal processes. 2 and 4 = Adiabatic processes. 1 and 2 = Work of expansion. 3 and 4 = Work of compression. + Q = Heat received. Q = Heat expended. + W = Work of expansion. W = Work of compression. CARNOT'S CYCLIC PROCESS 73 Summarizing by the first law. Isotherm. Adiab. Isotherm. Adiab. (280) Q = Q l -Q 2 = W = Wi + W 2 -W 3 -Wt nTi V3 T* vi Ts vtTi (281) = IPdv +/Pdv -IPdv -IPdv, Jv\ Ti Jv2 Ti Jv* T* Jvi Ti (282) Since P = ^ = - Cv j^- by (228) iordQ = 0. /Ti r T * r V4 T 2 r Tl = R f -dv - fCvdT +R/ ~dv - fCv d T, J VI v JTI Jvz v JT-L (283) Since R log v + Cv log T = Const, by (228) for d Q = 0. (284) = R (T, - r.) log ^ = R (T, - T 2 ) log = It follows that log = log , or = . (285) (k - 1) log Pl = log TV By (180). (286) +(k-i) log vi + log Tt = (k- 1) log v, + log TV (287) + (k - 1) log v 2 .+ log T! = (k - 1) log fl 3 + log T 2 . From the Second Law by (271) and (180), k- i , k- i LATENT HEAT Cyclic Process for Vapors at Maximum Pressure A second example of the reversible process is found in the cycle through which vapors pass in changing to liquids, by the latent heat which is required in effecting this transformation. 74 THERMODYNAMIC METEOROLOGY Vapor Liquid Solid Total Mass Mi M 2 M 3 M = Mi + M 2 = Constant. Specific Heat Ci C 2 C 3 Latent Heat ... r\ r 2 Volume Vi v 2 v 3 v = Vi + v%. Liquid \ Vapor -Q, \ O e h f V FIG. 4. Cyclic process for maximum vapor pressure (289) Product. M (vi + v 2 ) = Mi Vi+M 2 v 2 = Mi Vi + (M MI) v 2 . (291) dMv = M dv = (vi v 2 ) dMi. This is the mass which evaporates in the expansion while (z> 3 z; 4 ) d M 2 condenses during compression. Hence (292) ^ = -^- d v Vi v 2 The general equation of condition is, (293) A P d v = r 2 - , d v = r 2 - - d v in heat units. d v Vi v 2 For d v = we have by (209), P = T -r^. Hence, dP M (294) APdv = AT T ^dv = r 2 -dv,s (295) (296) r 2 A Ti ~ T latent heat of vaporization of liquid to vapor (^i v 2 ) in heat units. r 3 = A T 2 ^- 2 - 7-=-, latent heat of melting of i a 1 2 solid to liquid (v 2 v 3 ) in heat units. SECOND FORM OF THE EQUATIONS FOR LATENT HEAT 75 The Second Form of the Equations for Latent Heat For external equilibrium where there is no exchange with the surrounding medium, the conditions are : Total Vapor Liquid Solid (297) Masses. M = M i + M 2 + M s . (298) Specific volumes, v = v\ + % + Vs- (299) Volume. Mv = M i Vi + M 2 v 2 + M 3 %. (300) Energy. M Z7 = Mi C/i + Jlf 2 U 2 + M s U 3 . (301) Entropy. MS = M l Si + M 2 S 2 + M 3 5*3. For internal equilibrium the general equation is, (302) yl 1 M 5 5 = A 1 S M a 5 5 a + ^l 1 S S a 8 M a , Where a takes the values 1. 2. 3. in succession. Since U = Q - W, and TS = U + W = U + Pdv = Q, we have by differentiation and substitution, (303) SS = ^ + L^ (304) A*M SS = S ^-^ + 2 Ma ^ 3ga + A*S a SM a = 0. -^ a -'a The three independent conditions for interpreting this equation are, (305) For the masses, S 6 M a = 0. (306) For the volumes, S M a 5 v a + 2 fl 5 Af a = 0. (307) For the energies, S Af a 6 / a + S Z7 fl 6 M a = 0. After eliminating from the equations 6 Jl/2 . 5 V2 . 6 C/2- (308) A^ d S = (4- - 7^-} M, d Ui - (^ - ^-} M* d U 3 \1 1 1 2/ \^ 2 ^ 3/ = 0. 76 THERMODYNAMIC METEOROLOGY The conditions of equilibrium for the maximum entropy are uniform temperature, T = T\ = T 2 = jT 3 , and uniform pressure, Pi = P* = Pi. Hence we have by selection, (309) r 2 = Ti (Si - 5 2 ) = A (Ui - U 2 ) + PI (vi - v 2 ) for vapor and liquid. (310) r, = T 2 (S 2 - S t ) = A (U 2 - Z7 8 ) + P* (v* - %) for liquid and solid. SPECIFIC HEATS A third example of a reversible cyclic process is given in the specific heats, C\ for vapor, C 2 for fluid. From the first law by (234), we have, , dP d.AP d.Cv (234) A jf = -^ ~j^. Develop these terms successively by substitutions. Differentiate the last form of (293), dividing by d v, d ' AP M r 2 d (DI v 2 ) -i M Ldf ~ (vi - v 2 ) d T] vi-v* The specific heat of the mixture is, by (136), (312) Cv = C 2 (M-M 1 )+CiMi-r 2 j~. Hence, by (289), 'fQiA\ f^\^L - C*dMi CidMi r 2 \~dvi-d ^ 2 "1 L4j (2) dv " dv dv " (v! + v 2 ) I dT J ^ By (292), dCv Subtract (1) - (2). SPECIFIC HEATS 77 /i^ i ^ /- . . , (317) -7-=; -\- C 2 Ci = A TT -p= in heat units. # i M. u, L The Specific Heats in Terms of the Latent Heats It can be proved by differentiation of the first forms of (309) and (310) and the necessary substitutions that, (318) (Cfc - (Q) 2 - f* - + _*_ [ ( f| )p _ f ?) J, vapor-liquid. <> *> - > -I? - ? + ^[.(t?),- (I?) ,) liquid-solid. Compare Planck's "Thermodynamik." Examples of the Thermodynamic Data 1. Carnot's Cycle. In Fig. 3 the area enclosed between the isotherms (1.3) and the adiabats (2.4) represents the work done, W, in the cyclic process, and the figure is called the indicator diagram. This is used in studying the efficiency of engines, whether the process is natural or mechanical, and there is a very large literature on the subject. No applications have as yet been made in the atmosphere, for two reasons, the first because of the difficulty of tracing out the history of a given mass, and the second because the values of Cv in the formula are not constant, and the true values of it have not heretofore been computed. However, it is possible to take a standard mass, as one kilogram of air at an initial point, and trace out the conditions through which it must have passed in rising from the surface through changing P. v. T. till it arrives at the surface again in the original state, even though the path during its circulation may not be known. This work will be reserved for further studies. 2. Cyclic Process for Vapors. In Fig. 4, we have, (320) (z> 2 Vi) d Mi the mass which evaporates in expansion. (321) (z> 3 z> 4 ) d M 2 the mass which condenses in compression. 78 THERMODYNAMIC METEOROLOGY (322) b k = -TJ, d T the increase in the pressure. (323) W = (vi v 2 ) -T-, .d Mid T the work done in the area a, b, c, d. (324) Qi = r 2 dMi = C 2 d MidT the heat received in expansion. (325) Q 2 (326) Q 2 = CidMidT --.d Mi d T the heat expended in compression. (327) Q = Qi-Q 2 = (j^ + C 2 - Since A = ^, this becomes as in (317), (328) -~ + C 2 - Ci = A (vi - v 2 ) jj, in heat units. Values of the Latent Heat and Specific Heats (329) r 2 = 606.5 - 0.708 t for water to aqueous vapor. n = 80.066 for ice to water. Cpi = 0.4810 Cp 2 = 1.0000 Cp* = 0.5020. Example 3. Water to Vapor at 100 C., by Vaporization Ti = 273 + 100 = 373. z>i = 1658 the volume of 1 gram of aqueous vapor at 100 C. ^2 = 1 the volume of 1 gram of water at 0. vi-v 2 = 1658 - 1 = 1657. -r~ = 27.2 millimeters of mercury. - 2 - 3894 x 10 ~ 8 = H85TOOO- ^ = 2 ' 37829 - 10) - X = 3.1856 X 10 ~ 5 (log = 5.50319 - 10) go 7oU reduction from work units and mm. to C. G. S. units heat. VALUES OF LATENT HEAT AND SPECIFIC HEATS 79 (295) Latent Heat. r 2 = T : fa - %) ~ = 373 X 1657 X go a 1 27.2 X 3.1856 X 10 " 5 = 535.5. Example 4. Ice to Water at by Melting T 2 = 273 + = 273. v z = 1.00 volume of 1 gram of water at 0. v 3 = 1.09 volume of 1 gram of ice at 0. % - v 3 = - 0.09. d P -r= = 134.6 atmospheres. (296) Latent Heat. r 8 = . T 2 fa - v s ) ~ = 273 X (- 0.09) (-134.6) X 3.1856 X 10 ~ 5 X 760 = 80.066. Example 5. Aqueous Vapor in Contact with Water at 100. &-- 0.708 (||) = 4.931 (||) = 0.001 dT \oTJ p \o 1 J P Cp 2 = 1.0300 P = 760mm. T = 373 <3, 8 , , 0.001] = - 0.5504. Specific Heat. C#i = 1.0300 - 0.5490 = 0.4810. Example 6. Water in Contact with Ice at ^ = 0.6400 (||)=- 0.00006 (||) = + 0.00011. dT \oTJ P \ol / p Cp 2 = 1.0000 water at 0. 80 80 = 0.6400 -273 + ^^^t- 0.00006- 0.00011] = 0.6400 - 0.2930 + 0.1510 = 0.4980. Specific Heat. Cp z = 1.0000 - 0.4980 = 0.5020. THERMODYNAMIC METEOROLOGY Example 7. Pressure of Vapor in Contact with Water and Ice P 12 = 4.57 vapor-water. Vi = 205000. p 13 = 760 vapor-ice. v 2 = 1.00 p 23 = water-ice. v 3 = 1.09 T = 0.0074 C., the fundamental temperature. r 12 41851000 760 (295) Vapor on water, -j^ -- T ^_ ~ - 1013235- 606.5 3.1391 X 10 4 204999 = 0.3402mm. Vapor on ice. (606.5 + 80) 3.1391 X 10 4 273 " 204999 -0.3851mm. ^0 (296) Water on ice. - - 80.066 3.1391 X 10 4 ' 102294 ' mm - (-0.09) Compare Planck's "Thermodynamik." Application of the Thermodynamic Formulas to the Non-Adiabatic Atmosphere The foregoing formulas would apply to an adiabatic atmos- phere, using the constants of Table 3, wherein Cp, Cv, R are constants, but they do not apply to the existing non-adiabatic atmosphere, because it is not an ideal gas, rather a mixture of gases which are undergoing rapid changes of condition through variations in the heat contents by insolation and radiation. They can, however, be adapted to the earth's atmosphere by suitable modifications, which depend upon the formulas de- veloped under static meteorology. The following summary is sufficient for working purposes. APPLICATION TO NON-ADIABATIC ATMOSPHERE 81 Entropy T jr - R / QQ1 \ e e ft ~ Qo r>. TI T Q 1 PI (331) 61 - 6 = rp ~ = HI Cp a ^ -- TT Rio log -5- ^ dQ dT dP dT v (330) dS = -jr '= Cp-jr - R~p~ = Cp^r - j,dP. 10 JL 10 M Work Against External Forces (332) dW = Pdv = RdT - = Cp ^-^ d T - . p k p (333) Wi - Wo = Pio (vi - VQ) = R w (T a - T ) L Pio k - 1 , p, Pio (334) Pio Energy k-ldP = CpdT r -- . k p (335) A - U = Cvu (T a - To) = (Cp a - RIO) (T a - T ) = Cp a (T a ~ To) ~ Cp w ~^ (T a - T ) = (Qi - Go) - (W l - Wo) = Cp a (T a ~ To) ~ pio Heat Energy (336) dQ = CvdT + RT~ = Cv d T + Cp ^ d T - k - 82 THERMODYNAMIC METEOROLOGY (337) Q 1 -Q = T w (S, -So) = (Cp a - Cp w ] (T a - T ) = Cv w (T a - T ) + Pio (vi - vo). = (Ui - Uo) + (Wi - Wo) = Cpa (Ta - To) - Pi- Po Pio Radiation Function P - TdS p- P* ~ ~~ ~ f - (Krt r >- '- P "'- p (339) * = 3^ . = -^^ - P 10 = VI _ VQ - Pm (341) log K! - log ^o = A (log r, - log T ). (342) , _ log *i - log go (3 . 42) - log r, - log r.- The Radiation Coefficients and Exponents (343) #10 = C Ti Q A . log #10 = log C + ,4 log r 10 . (344) KM = c T 10 a . log #10 = log c +a log T 10 . (345) log c = log Co + (-4 - 4) log B. C = 9.12 X 10 ~ 5 . B = 1.66 X 10 ~ 2 . (346) logc = - 5.906 - (2.220) (A - 4). (347) c =C Q B A ~ 4 = 9.12 X 10~ 5 (1.66 X lQ-y~ 4 . These formulas will be fully explained and illustrated in the examples that follow (pages 8485). Working Equations -Co (331) Si - So (333) Wi - W = R'w (T a - To) - Pi - Po Pio WORKING EQUATIONS 83 (335) Ui-U = (Qi-Qo)-(Wi-W ). (339) K 10 = Ul ~ Uo . (340) =() Ao Wo/ log Zi log In order to illustrate the formulas of computation (331) to (342) the data of Table 17 are continued in Table 19. It must be especially noted that - (Cp a - Cpio) (T a - T ), which by (198) includes the kinetic energy of circulation and the kinetic energy of radiation - J (q? - q 2 ) - (Q l - Q ), is not carried forward, but only (Qi Q ), the energy of radiation. If the former were taken for the computations beyond this point the circulation would be treated as true radiation, which is improper. The sign - (Q l - Q ) in Table 17 is changed to -f (Qi Qo) in Table 19. In computing the mean entropy from one level to another, the mean temperature T iQ = f (T[ -f To) is taken from Table 14 in successive pairs. The entropy generally increases with the height, and always does so unless there is an inversion of temperature, or an excess of wind varia- tion in velocity between the levels, such as occurred in this case between 500 and 1,000 meters. In applying (333) for computing the work (Wi Wo) we must now compute R\o corresponding with (Qi Q Q ), which differs from R 1Q taken by pairs in Table 14, since R w implies the circulation as well as the radiation. The formulas now be- come (348) (Cp a - Cp' lo ) (T a - To) = (Qi - Qo). (349) Cp\ Q = Cpa - %-^ = 993.58 - ^=-|. a - (350) 84 THERMODYNAMIC METEOROLOGY M H 05 U O5 s ^ g s 1 1 <J w fl I M B H D CM ! l> CM CM CO I I CM rH s 1 I rH 1 1 O5 O GO O t>- CO O5 O5 CO J> CO O5 TH O O rH TH 00 CM <M TH TH TH !> T-H 10 1 1 1 1 O5 1 I sis s CO.H i i 00 O 1 1 CO >O (N CO CO O5 O5 1^ CO CO O O | + O5 >O IO O5 TH CO 05 CO TH S CO-H CO 1 I O (M s CM 00 CM O 00 O (M co 05 O5 CO CO O !> O5 >O O CO O5 TH CO O O II CM (M l> 10 rH 00 T( CO TH i 7 1 1 o>o> I I Co Co 1 I Co Co bfl 1 1 .<N + I CO + + + Tt< CO CO + + I + + I 00 l> 10 CO CO WORKING EQUATIONS oo o oo b- 00 CM ^H O rH CO" i-H IO Ob- T}^COO(MTft i ( CO T-H O OOCO O CO O CO CM OS O T-H OS + I I 00 00 O "# I I ! t^ CO GO T-H OS 1O Tt< 1C 00 rH (N l> 00 O OS O rH OS 8 < I I * ' 00 00 O CO I I I 00 CO b- (M O5 CO CO r^ t^ CO ^* CO CO t^* CO OS *O CO CO GO Tfrl CO iO OS O O O CO CO C5 O r-t '06 I + CO O O O CO CO CO t I 00 CO 00 iO I 7 D CO CO CO ' 00 l> O I I I OO CO TJH CM CO CO OS b- CM CO CO CO b- O3 -^f OS CM CM CO O 10 oo i i ^H ^ ^H oc^ T i co 10 oo co IO OS CO CO CO CO OS CO CO C^ *O CO OO CO OSOT-i'o6 TJH'O 'o6l>Or}J i + i T i i i ^^ T^ OS 1( ^ *O OS OO CO CO CO ^ lO^JiiOCl T-I COO1O(N OO T-HOOSC'MCO O COCMOSCO TJH OOs2?T^(N C<lcO T-t(NT^l>.ii >O O OS w . O CO O5I> OC^OCOI> CO OOs'oo' TJH' TJ< *o6l>OT}< i + i T i i i T-H OS ^ OS rf* TJH Olt^iOcOOS r^C<l ^o oososoo osi^ CO CO OS O ' 00 rji b- 00 I +11 Tf lO 1C OS O b- (M b- 00 00 CM ' 00 t>^ O rji I I I 1>* CO l>* !> CO CO co co T- i co t^ co o ^^ Oi Oi ^^ *-O O5 CO O OS O '06 Tj<' CN + I \ CO i- 1 O T-H 00 t O CO I I I OS S CO O OS O 00 I + OS lO OS IO ^ OS I \ ^ 2 "^r 00 O OS O i I I 85 86 THERMODYNAMIC METEOROLOGY COMPARISON OF Rio WITH R\ Height z 116 500 1000 1500 2000 2500 3000 4000 5000 RIO R'IO 287.28 282.91 286.87 287.35 285.34 285.65 283.14 281.68 280.50 277.95 278.21 275.46 275.59 274.09 272.06 268.89 R'IO is generally smaller than R w in these observations. P P The term is taken directly from Table 17, and PlO Wo is easily computed. Then Ui U Q follows from (335). In computing the radiation, Ui-Ui the values of v are the reciprocal of the density p in Table 14. Had p been computed by formula (175), which takes R constant, instead of by (176), with R variable, it is seen at once how erroneous would have been the derived radiations, because the values of KIQ depend upon the small differences fyi v ) in succession. These radiations are mean values for the strata concerned. It is important to study the relation of the radiation to the tem- perature, and to compare the exponents of formula (340) with the exponent of a full radiation in the Stephan Law, which is 4. This subject is complex in the earth's atmosphere as will be in- dicated. The problem is as follows: The values of K in re- lation to T by (340) are in the form of ratios, whereas in the Stephan Law (344) they stand related through a coefficient. If the constituent of the ratio is in the form (343), it is quite certain that the coefficients (C . c} are not equal, nor are the expo- nents (^4 .a). We proceed to develop the relations between C and c, A and a. The equation (343) gives three terms, K w , A, TIG, from which to compute C, and it is necessary to indicate what are the relative values of log C and A . With the data of Table 19 in the first section of Table 20, compute A log T 10 and subtract this from log K iQ to obtain log C. The negative sign before the logarithm affects only the characteristic. Thus, logarithm 11.944 gives the number 8.79 X 10~~". In this WORKING EQUATIONS 87 way the values of log C and A were computed for twelve balloon ascensions, of which two examples are given in Table 21. It is readily seen that log C is negative, as the temperature T de- creases with the height z, and positive in regions of inversion of temperature. The magnitude of log C depends upon the ratio rri If the temperature changes slowly with the height the TABLE 20 COMPUTATION OF LOG C AND ( a . LOG c} Section I. log C = log Kio - A log Tio (343) Lindenburg, April 27, 1909 z TlQ log Tio 116 500 292.25 1000 288.30 2 45984 1500 284.90 2 45469 2000 282 . 15 2 45048 2500 279.85 2 44693 3000 277.20 2 44279 4000 272 .40 2 43521 5000 266.00 2 42488 log.log Tio 39091 39000 38925 38862 38789 38654 38469 log A 0.78581 68298 67320 . 68362 0.67620 0.55160 62707 A 6 11 4 82 4 71 4 83 4 74 3 56 4 24 log (A log Tio) 1.17672* 1.07298 1.06245 1.07224 1.06409 0.93814 1.01176 A log Tio 15.021 11.830 11 547 11.810 11 . 590 8.672 10 275 log Kio 4 96534 4 94187 4 92519 4 90796 4 83566 4 84816 4 80528 Kio 92330 87472 84176 80902 76853 70495 63867 loeC 11 944 7 112 -7 378 7 098 7 246 4 176 6 530 C ll 8.79X10 7 1.29X10 7 2.39X10 7 1.25X10 7 1.76X10 1.50X10 3.39X10 Section II. log c = log Kio - a log Tio (344) Assumed 3 82 3 82 3 82 3 82 3.82 3.81 3.81 log ai 0.5821 0.5821 0.5821 0.5821 0.5821 0.5809 0.5809 log.log Tio 0.3909 0.3900 0.3893 0.3886 0.3879 0.3865 0.3847 log ai log Tio 9730 . 9721 0.9714 0.9707 0.9700 0.9674 0.9656 9 397 9 378 9 363 9.348 9.333 9.277 9.238 log Kio 4.965 4 942 4.925 4.908 4.836 4.848 4.805 log c 5 568 5 564 5 562 -5.560 -5.503 -5.571 -5.567 3 824 3 822 3 821 3.820 3.794 3.825 3.823 Second Assumed 3.807 3.817 3.817 0.5806 0.5817 0.5817 log log 7\o 0.3879 0.3865 0.3847 log . 02 log Tio 0.9685 0.9682 0.9664 9.300 9.294 9.256 log Kio 4.836 4.848 4.805 -5.536 -5.554 -5.549 a 3.809 3.817 3.815 88 THERMODYNAMIC METEOROLOGY Working Formulas (343) K 1Q = C7V log C = log KM A log TIG (344) K 10 = c 2V log KIQ = log c + a log JIG log C = log KIQ a log TIQ. Assume trial #1 and adjust by Table 22. The final pair log c and a should fall on the same line. TABLE 21 - EXAMPLES OF LOG C . A AND LOG c . a THE COEFFICIENTS AND EXPONENTS IN THE RADIATION FORMULAS (343), (344) z Lindenburg, April 27, 1909 Lindenburg, May 5, 1909 logC A logc a logC A logc a 17000 16000 +28 . 034 10 . 34 5.435 3.764 15000 +52.871 -21.11 -5.461 3.775 14000 + 16.828 5.78 -5.489 3.788 13000 - 4.356 3.45 -5.531 3.807 + 10.437 - 3.41 -5.537 3.810 12000 - 7.140 4.86 -5.563 3.822 -27.533 13.29 -5.563 3.821 11000 - 5.962 3.64 -5.571 3.825 - 6.181 4.40 -5.548 3.814 10000 - 1.410 3.44 -5.560 3.820 - 3.288 3.09 -5.558 3.819 9000 - 2.114 2.73 -5.549 3.815 - 4.021 3.62 -5.558 3.819 8000 - 3.211 3.11 -5.539 3.810 - 6.506 4.26 -5.560 3.820 7000 - 7.178 4.79 -5.538 3.810 - 5.249 3.95 -5.557 3.818 6000 - 9.719 5.40 -5.548 3.814 - 5.401 3.89 -5.560 3.820 5000 - 6.530 4.24 -5.549 3.815 -11.132 6.48 -5.547 3.814 4000 - 4.176 3.56 -5.554 3.817 - 4.050 3.63 -5.560 3.820 3000 - 7.246 4.74 -5.536 3.809 - 8.115 5.25 -5.567 3.823 2500 - 7.098 4.83 -5.560 3.820 -18.387 9.23 -5.577 3.828 2000 - 7.378 4.71 -5.562 3.821 -20.982 9.81 -5.590 3.833 1500 - 7.112 4.82 O . OvVx 3.822 -14.180 7.64 -5.601 3.838 1000 -11.944 6.11 -5.568 3.824 +24,708 -8.29 -5.593 3.834 500 ratio is small. At the same time the ratio h~ J , which does \Ar,/ not have a coefficient or exponent, registers a change in the WORKING FORMULAS 89 atmosphere which is closely connected with the variation of the pressure P. Hence there are large changes in log C and A which are opposite in sign, but both increasing or diminishing together. Under nearly normal conditions it is seen that A is approximately 4.00, which would be the value for a full radiating black body. The entire series of values A, log C, were collected in groups, and the mean values when plotted fall on a straight line, of which the equation was found to be, (347) C = C B A ~\ (345) log C = log Co + (A - 4) log B. (346) logC = - 5.960 + (A - 4) (- 2.220). (347) C = 9.12 X 10~ 5 (1.66 X 10 ~ 2 ) A ^ 4 . The development of a portion of this formula from A = 4.00 to A = 3.50 is given in Table 22. A = 4.00 corresponds with a full black radiator and A = 3.50 corresponds with the theoretical value for the atmospheric air. The mean value was found to be A = 3.82 near the surface. The notation (log c . a) indicates the values in the constituent formula, while (log C . A) are used for the ratio formula. TABLE 22 EVALUATION OF THE FORMULA (346) a logc a log c a log c a logc 4.00 -5.960 3.99 -5.938 3.98 -5.916 3.97 -5.893 3.96 -5.871 3.95 -5.849 3 94 5 827 3.85 -5.627 3.84 -5.605 3.83 -5.583 3.82 -5.560 3.81 -5.538 3.80 -5.516 3 79 _5 494 3.70 -5.294 3.69 -5.272 3.68 -5.250 3.67 -5.227 3.66 -5.205 3.65 -5.183 3 64 -5 161 3.55 -6.961 3.54 -6.939 3.53 -6.917 3.52 -6.894 3.51 -6.872 3.50 -6.850 3.93 5.805 3.78 -5.472 3.63 -5.139 3.92 -5.782 3.77 -5 449 3.62 5.116 3 91 5 760 3 76 5 427 3 61 5 094 3.90 -5.738 3 75 5 405 3 60 - 5 072 3.89 -5.716 3.88 -5 694 3.74 -5.383 3 73 5 361 3.59 -5.050 3 58 -5 028 3.87 -5.671 3.72 -5.338 3.57 5.005 3.86 -5 649 3 71 -5 316 3 56 -6 983 90 THERMODYNAMIC METEOROLOGY The negative sign applies only to the characteristic log Co = - 5.960 Co = 9.12 X 10~ 5 logB = - 2.220 B = 1.66 X 10~ 2 . By means of Table 22 we proceed in Section II of Table 20 to compute log c and a from log C and A . Assume an approximate value ai, as 3.82, and compute di log 7\ from the value in Section I. Subtract from log K\ Q for log c, and in Table 22 interpolate that value a , which is the pair value of log c. If this value a Q agrees with the assumed a\ the check is complete. If, on the other hand, these values of logc and a do not quite agree, as in the examples under z = 3000, 4000, 5000, take the mean value between the assumed a\ and computed a Q , and proceed again with a 2 to compute logc and a. The second trial is usually successful if a\ has been chosen with some practice. The corresponding values of (log c . a) are found in the examples of Table 21, and it is seen that the irregularities of (log C . A) have disappeared. Log c and a usually decrease slowly with the elevation and with the increase of latitude from the equator. These results check by log c + a log TIQ = log KIQ. Application of the Thermodynamic Formulas to Various Meteor- ological Problems It has seemed necessary to give an extended example of the method of computing the thermodynamic values in the non- adiabatic atmosphere, on account of the complexity of the computations, and because of the numerous valuable results dependent upon them. In Bulletin No. 3 of the Argentine Meteorological Office, 1913, will be found the results for many types of data in considerable detail. We can here summarize them briefly, depending upon diagrams to bring out the general ideas, in particular respecting the isothermal region, the diurnal convection, the circulation in cyclones and anti-cyclones, and the general circulation of the atmosphere. THE ISOTHERMAL REGION 91 The Isothermal Region It has been found by balloon ascensions to great elevations, up to 20,000 meters or more, that the temperature of the atmo- sphere diminishes at the rate of about 6.0 C. per 1,000 meters up to an elevation of 12,000 meters in Europe, or 15,000 meters in the tropics, or even to 20,000 meters over the equator, while above these elevations the temperature is nearly constant or increases a little to the highest levels explored. There have been many conjectures as to the cause of the permanence of the heat of this isothermal region, as overflow of the tropic heat to mid-latitudes, conductional transportation of heat from the lower to the higher levels, production of ozone by the incoming solar radiation in the upper atmosphere and absorption of the short waves of the solar radiation in the same region. There are objections to each of these hypotheses so obvious that we pro- ceed at once to examine the thermodynamic data for at least a statement of the case, if not a complete explanation of the facts. The computations were executed for the following balloon ascensions, as reported in the volumes: Europe Atlantic Tropics Lindenburg, April 27, 1909 (52). Sept. 25, 1907 (35). May 5, 1909 " Sept. 9, 1907 (25). May 6, 1909 " Aug. 29, 1907 (13). July 27, 1908 " July 29, 1907 (13). Sept. 2, 1909 " June 19, 1906 (-2). Mailand, Sept. 7, 1906 (45). Victoria Nyanza, 1908 (0). The mean values are compiled in Table 23, and illus- trated in Figs. 5 and 6. Since the data in the isothermal region are not so complete as below it, these results are to be considered as instructive rather than definitive. It will require the work of many years to accumulate and compute the data necessary for normal conditions. The temperatures show that there are as wide local fluctuations in the isothermal region as below it. Furthermore, the temperatures are lowest over the equator, 200, and gradually increase to 210 in the tropics, or 215 in 92 THERMODYNAMIC METEOROLOGY Europe. From these minimum values the temperatures in the isothermal region increase about 10 degrees. These facts appear clearly in the Table 23, and the diagrams, where 13000 10000 5000 000 P- K Pressure Radiation Energy Density 25000 50000 75000 100000 0.7500 1.0000 1.2500 1-5000 10000 5000 000 200 C Temperatu e \ Free Heat 250 275 -3000 Full .Line for Europe DottedJLine for Tropics FIG. 5 -2000 -1000 the other data are presented. The data on Fig. 5 (P, K, p, Qi QQ) may be considered primary, and those on Fig. 6 (Cp, Si - So, Wi - W Q , Ui - UQ) secondary, as being the machinery of the thermal engine. It is noted that (P,K, p) THE ISOTHERMAL REGION 93 have one configuration, and (Q, Cp, S, W, U) another configura- tion. The former is more immediately under the control of gravitation acting downward, and the latter is the result of 15000 5000 000 C p Specific Heat En tropy \ 600 700 800 900 1000 -10.00 -5.00 15000 10000 5000 Work - U Inner Energy 000 4000 5000 6000 7000 -8000 -7500 -7000 -6500 Full Line for Europe Dotted Line for Tropics FIG. 6 radiation. The circulation q is the balancing governor to the engine which keeps the other two parts in equilibrium. (<2i - Qo). There is a persistent supply of heat from four conditions: (l) That from the heated earth as the source; (2) that in the lower strata, due to convection within 2,000 94 THERMODYNAMIC METEOROLOGY TABLE 23 MEAN THERMODYNAMIC VALUES FOR EUROPE z T P p Cp g Qi-Qo Si -So Wi- W Ui-U KlQ 20000 19000 18000 17000 219.5 9141 0.2220 657.15 0.8 -3231 -14.747 4658 -7887 16704 16000 223.4 10983 .2476 689.35 2.5 -2791 -12.620 4962 -7753 18497 15000 221.2 12819 .2764 731.44 3.9 -2392 -11.158 5209 -7602 20083 14000 220.3 14972 .3086 769.99 8.6 -1955 - 8.641 5540 -7494 21850 13000 215.7 17017 .3422 799.60 14.3 -1702 - 7.487 5730 -743l!23883 12000 215.8 19942 .3830 836.65 16.4 -1381 - 6.326 5948 -7329J26548 11000 218.7 23341 .4283 863.20 17.8 -1190 - 5.345 6096 -7285 29953 10000 226.0 27198 .4780 873.16 19.3 -1089 - 4.722 6164 -7253 34502 9000 232.8 31606 .5315 884.82 20.7 -1131 - 4.768 6248 -737940630 8000 241.1 36509 .5890 890.49 18.4 -1034 - 4.229 6302 -733645275 7000 249.0 41972 .6504 897.75 16.3 - 920 - 3.647 6329 -7249 51345 6000 256.1 48053 .7162 907.35 12.6 - 793 - 3.083 6405 -7198 57729 5000 262.6 54819 .7865 919.62 14.1 - 650 - 2.578 6511 -7160 64352 4000 268.3 62353 .8620 934.08 13.0 - 530 - 1.957 6601 -7131 71739 3000 274.3 70722 .9429 947.41 12.6 - 380 - 1.375 6717 -7097 77430 2000 279.1 80030 1.0299 964.88 11.8 - 241 - 0.857 6825 -7066 86011 1000 284.8 90345 1.1224 979.50 9.2 - 76 - 0.241 6969 -7045 94582 100 289.4 100419 1.2101 993.58 6.1 meters of the ground, and involving a supply of latent heat by the condensation of aqueous vapor into water in cloud formation by (295); (3) that in the cirrus cloud region, 9,000 to 15,000 meters elevation, according to the latitude, due to ice formation from frozen water or vapor by (296). In this cirrus region there are other sources of heat supply, such as an accumulation of heat from absorption of radiation producing the new rate of loss of free heat per 1,000 meters. Take the differences in the ((?i (?o) columns, and the mean values fall into two groups, omitting those in the cirrus layers. Europe, 12000 to 17000 A (Q l - Q ) = - 370 per 1000 meters. Tropics, 14000 to 17000 = - 476 Europe, 1000 to 11000 =-133 Tropics, 1000 to 13000 " = - 140 " " THE ISOTHERMAL REGION 95 TABLE 23 MEAN THERMODYNAMIC VALUES FOR THE ATLANTIC TROPICS a 20000 19000 18000 r P p Cp <? Qi-Qo Si -So Wi- Wo Ui-Ut Kio 17000 227.0 9894 0.2256 668.93 8.0 -3012 -13.359 4911 -7922 17488 16000 217.3 11213 .2513 709.00 7.4 -2570 -10.292 5203 -7774 18647 15000 209.9 13331 .2788 788.87 8.8 -1898 - 8.567 5609 -7506 19328 14000 212.8 15665 .3128 814.91 8.5 -1584 - 7.361 5750 -7334 21544 13000 215.9 18398 .3502 835.84 14.3 -1470 - 6.697 5911 -7381 24731 12000 221.6 21442 .3911 855.58 13.8 -1345 - 5.949 6024 -7369 28286 11000 230.3 24942 .4355 861.08 13.3 -1302 - 5.545 6060 -7362 32588 10000 238.9 28853 .4830 865.62 12.4 -1250 - 5.135 6086 -7336 37288 9000 247.7 33204 .5338 869.26 12.0 -1182 - 4.699 6127 -7309 42938 8000 254.9 38038 .5879 878.51 11.8 -1100 - 4.252 6199 -7298 47822 7000 260.1 43413 .6459 887.63 10.7 -1000 - 3.769 6271 -7271 53935 6000 268.3 49379 .7079 899.98 9.9 - 881 - 3.247 6368 -7248 59582 5000 274.1 56008 .7742 913.55 6.6 - 700 - 2.534 6473 -7173 65980 4000 278.5 63380 .8453 931.88 6.5 - 532 - 1.907 6591 -7123 72304 3000 283.9 71569 .9217 946.73 6.5 - 397 - 1.388 6694 -7091 78606 2000 288.8 80625 1.0031 963.44 5.8 - 187 - 0.641 6846 -7033 86055 1000 292.4 90699 1.0908 984.35 4.4 - 24 - 0.081 6911 -6935 93656 000 299.5 101753 1.1837 993.58 5.8 In the Europe group 9,000 to 10,000 is omitted, and in the tropic group 10,000 to 12,000 is omitted, as being regions of special local supply. It appears that heat is lost at about three times greater rate in the isothermal region than in the lower levels. This occurs at the same time the temperature is rising in the isothermal region but falling in the lower levels. The Victoria Nyanza ascensions give A (Qi Q ) = 144 through- out the region 3,000 to 18,000, but in this case no inversion of T was found. (4) The principal fact to be explained is the slow rate of loss of heat in the convectional region, 140 per 1,000 meters, as compared with that in the isothermal region, about 400. This is easily accounted for by the following facts : The incoming solar radiation is of short wave lengths, and pen- etrates to the earth's surface, having only a small amount of 96 THERMODYNAMIC METEOROLOGY selective absorption of radiant energy. This is transformed at the earth into long waves, in changing the temperature energy from 7,000 to 300, and this heat escapes to space partly by radi- ation and partly by vertical convection, the latter extending to the isothermal layer, whose height varies with the latitude, and the heat contents of the air at the surface. In the general ver- tical convection of the atmosphere, as the temperature of a unit mass changes from T to 7\, in the vertical distance z\ z , there is an evolution of heat Qi Qo = Cp 1Q (T\ jT ), which is added to the atmosphere throughout the convectional region. If (Qi Qo) is the natural loss of heat by radiation without convection, and ((V QQ} the amount evolved by convective cooling of the temperature of the rising mass, then we have, 400 = (Qi Qo) the heat loss in the isothermal region, and 140 = (Qi ~~ Qo) ~~ (Qi ~ Qo) that in the convection region. Hence 260 = (Qi Qo) the heat evolved by vertical motion. This subject will require a fuller development than is at present available, and it is complicated by the fact that the vertical distance through which the mass moves is not well known. The air that has risen by convection with cooling and evolution of heat in one place falls again in other places with heating and ab- sorption of heat. Such places of descending air are, during the night, in the permanent high-pressure belts and in the wander- ing anti-cyclones. This subject will be sufficiently illustrated in the following sections. P, - K and p. Pressure, Radiation-Potential, and Density. The pressure is found to change continuously in a smooth curve from the surface upward, that in the tropics being some- what higher in value than in Europe. An entirely similar curve is developed by the potential radiation K, the values being always higher than P. The density is also given in a smooth curve, the tropics and Europe crossing at about 7,000 meters. It would seem, then, that the ultimate purpose of T. Q and the other dependent terms is to so regulate the pressure PRESSURE, RADIATION-POTENTIAL, AND DENSITY 97 and radiation that they shall change steadily from level to level, under the attraction of the earth's gravitation, and that all the other thermodynamic values mutually adjust themselves to produce this simple result. Hence, the problem depends upon the rate of loss of A (Qi Q ) in the lower and the isothermal regions, which is distinctly a physical question. A reason has been already indicated why the rate of transfer of heat should be greater in one region than in the other, and why the iso- thermal separates from the convectional region. By (339) we have the following equation: A<n K . (339) K 10 - This equation is immediately derived from (259), where C\ /^ C T = ( J is the latent heat while the temperature is constant; it also comes from (309), where the latent heat r 2 of vaporization is derived by primary analysis; or it may be taken from (338), (339); finally it is found by computation that the last form through the gas coefficients R a , RIO is satisfied, as can easily be verified by the data of Tables 14 and 19. The small dis- crepancies are due entirely to the velocity term which was eliminated by means of Table 17. KIQ is negative in sign because (U\ U ) is negative while fyi v ) is positive. It seems, then, that the latent heat, Hence, the escape of heat or radiation in the earth's atmosphere depends entirely upon the divergence of the gas coefficient ^10 from the adiabatic value R a , as was stated. The divergence of the lines P 10 and K^ in Fig. 5 measures this term. If the velocity is also considered it will be a line near the KIQ line, slightly adjusting it to make the pressure transitions gradual. This is the function of the horizontal cloud motions of flowing 98 THERMODYNAMIC METEOROLOGY strata so generally seen in the atmosphere. This confirms the principle of equations (36) to (38), which indicate the relations of pressure, circulation, and radiation to gravitation. It is easy to see that such data are capable of making all the general thermodynamic formulas (205) to (328), and many others, applicable to the earth's non-adiabatic atmosphere. It should be carefully noted that the density p, and the gas coefficient R, must be computed by (176), (177), and not by (175) for R a con- stant; that the effective specific heat Cp is variable, and that radiation depends upon this fact. The principal quantities to obtain by observation are the temperature T, and the velocity of circulation q at the height z, and hence the observations for temperature alone, omitting q, are not capable of giving correct radiation data. Finally, the variation of pressure d P is not proportional to the mass gpdz = gdm, but by (201) the terms p q d q + p d Q must be added for circulation and radiation, or else P = K, which is to exclude them from the problem, and reduce it to the unusual adiabatic case. One can now perceive that there is no possibility of solving the general equations of motion in cyclones and anti-cyclones, and in all the other types of circulation, without first eliminating the heat term d Q. Nearly all attempts of meteorologists to solve the circulation problems have been futile chiefly on this account, because of the assumed necessity of ascribing to friction, and to the deflecting force of the earth's rotation on a moving mass, values which they do not actually possess. We shall be able to explain this more fully in the chapter on Dynamic Meteorology, but now proceed to illustrate more at length the thermodynamic terms in other typical conditions of the atmosphere. The Diurnal Convection and the Semi-diurnal Waves in the Lower Strata There is a series of problems relating to the semi-diurnal waves observed at the surface, which have been much discussed without satisfactory results, as the semi-diurnal barometric waves and the several electrical and magnetic waves which are CONVECTION AND SEMI-DIURNAL WAVES 99 associated with the diurnal convection. At the surface the temperature has only a single diurnal wave, and it has not been possible to match these two series of data in a definite relation of cause and effect. The difficulty in studying the general problem has been the lack of observations in the free air above the surface, especially during the night. The only exception to this defect is the series of kite ascensions at the Blue Hill Observatory, 1897-1902, which were discussed in my papers, Monthly Weather Review, February to August, 1905. In these it was shown that the single diurnal temperature wave changes into a semi-diurnal wave at about 400 meters above the surface, and that the semi-diurnal waves die away within two or three thousand meters of the ground. It was also indicated that the other data, namely, vapor pressure, atmospheric electric potential, ionization, and magnetic fields, have diurnal variations closely matching the diurnal circulation. This section gives the result of another discussion of the subject, using the data of the Cordoba and Pilar stations, Argentina. There are two hypotheses regarding the origin of the semi- diurnal barometric waves: (1) The forced oscillation of the entire atmosphere, as proposed by Lord Kelvin, and developed by Margules, Hann, Jaerisch, Gold, and others; (2) the effect of the diurnal convection, proposed in general terms by Espy, and studied by Ferrel, Koppen, Sprung, Bigelow, and others. The important objections to the former hypothesis are, that these waves do not embrace the entire atmosphere and are usually confined within 2,000 meters of the surface; that the analytical equations and harmonic analyses merely represent, in other forms, the data assumed for their coefficients, and do not reach the origin of the physical causes; and that these equations have not the radiation and heat terms, which are more important than the friction and the deflecting force of rotation. The method of treating the Cordoba-Pilar data was to assume a uniform temperature T, and pressure P, on the 3,500-meter level, and then, by studying the observed temperatures and wind velocities on several levels, 000, 200, 400 ... 3,500 meters, at 100 THERMODYNAMIC METEOROLOGY Scale AT p.m. 2a.m. 6a.m. 10 a.m. 2p.m. 6p.m. 10 p.m. 2a.m. 2500 2000 1500 1000 400 200 000 + 3.0 + 2.0 + 1.0 -1.0 -2.0 -3.0 -40 280.9 284.7 NN. 291.5 293.3 295.0 296.3 296.4 295.8 FIG. 7. A T Temperature of the semi-diurnal waves above 400 meters and the diurnal wave near the surface T is the mean temperature on the given level. AP .10 p.mj 2 a.m. 6a.m. 10 a.m. 2p.m. 6p.m. 10 p.m. 2a.m. 2500 2000 1500 1000 800 600 400 200 000 75463 Units +160 -1-120 f 80 + 40 40 -80 -120 -160 \X 85087 90253 92387 94555 96771 99027 101338 FIG. 8. A P Pressure of the semi-diurnal waves in all these strata vanish- ing at about the level 2500 meters CONVECTION AND SEMI-DYURNAi; ^WAVES' J 101 the hours (2, 6, 10) A.M., (2, 6, 10) P.M., proceed by computations entirely similar to those of Tables 14, 17, 19, to derive a pressure P at the surface which would exactly match the mean AB 10p.m. 2a.m. 6a.m. 10a.m. 2p.m. 6p.m. 10p.m. 2a.m. + 1.00 0.00 -1.00 _~n ^^ ^ \ fr '^ \ \ / wn T. ne=Otse Line = Cc rved Data mputed D ^VJ ita / Dotted FIG. 9. Computed and observed pressure waves at the surface 10 p.m. 2 a.m. 6 a.m. 10 a.m. 2 p.m. 6 p.m. 10 p.m. 2a.m. AQ -70 -70 2500 2000 1500 -50 1000 800 -50 -40 600 400 200 000 -30 -20 -10 + 10 + 20 y/ 0+20 FIG. 10. The loss of heat for every 200 meters A Q = (Q l Q ) pressure as observed at Cordoba. It required several trial computations to accomplish this result, especially during the night hours where direct observations were lacking above the 102 TIIERVOD ^NAMIC METEOROLOGY surface, and the outcome is indicated in Tables 24 and 25, and in Figs. 7, 8, 9, 10. All the other data, as computed, will be found in Bulletin No. 3, O. M. A. Transferring the data of Table 24 to the diagrams, it is seen that the single diurnal temperature wave at the surface transforms into semi-diurnal waves at 400 meters, and that these die away at 2,500 meters. This result confirms the discussion of the Blue Hill data in all respects. The corresponding pressure waves are semi-diurnal throughout these strata from the surface to 2,500 meters, and there they vanish. The night wave is weaker than the day wave in consequence of the temperature inversion near the surface, both waves being nearly equal at 1,000 meters. Since the barometric variations are alone of interest, the base line has been taken as that near sea level. The discrepancy at 2 P.M. is due to an imperfect temperature distribution at that hour, which should be made cooler by a few tenths of a degree. By adjusting the vertical temperatures a little it would be possible to reproduce the observed curve with precision through the non-adiabatic computations. There are no other tempera- tures which could reproduce this pressure wave at the surface, and this fact is proof of the cause of the observed pressure system. In order to understand more fully the origin of the temperature system, the values of A (Qi Q Q ), the variations on the daily mean are plotted on Fig. 10 and the curves of equal heat losses are drawn. The data are somewhat imperfect, but the general result is not doubtful. It shows that there are two principal axes of heat exchange, that in the afternoon, 2 P.M., at the surface, to 8 P.M., at 2,500 meters, and that in the night from 1 A.M at 2,500, to 4 A.M., at the surface. That is to say, the air rises obliquely in the afternoon to the right and falls in the night, also to the right. The air rises and falls in such a zigzag path as gives a turning- point at about 10 P.M. above and 10 A.M. below, judging by the crests. The rising air cools by expansion, and the falling air heats by compression, the former producing the afternoon wave and the latter the night wave to within 400 meters of the surface on Fig. 7. At this level the more rapid cooling of the ground during the night makes itself felt, and there is radiation CONVECTION AND SEMI-DIURNAL WAVES 103 TABLE 24 THE SEMI-DIURNAL TEMPERATURE WAVES IN THE STRATA 400 TO 2,500 METERS, TOGETHER WITH THE SINGLE DIURNAL TEMPERATURE WAVE AT THE SURFACE z r P 2 -02H2o) T P 2 -02H2o) 2 A. VI. 6 A .M. 2500 281. 75464 3.0 + 151.8 280. 8 75466 6.0 + 140.2 2000 285. 80153 4.0 +131.9 284 .2 80162 7.0 +120.1 1500 288 .6 85080 5.0 + 111.9 287 .5 85104 7.0 + 97.6 1000 291 .9 90230 5.0 + 38.5 290 .6 90288 6.0 + 34.9 800 293 .6 92368 5.0 + 40.2 292 .3 92428 5.0 + 33.1 600 295 .0 94530 4.0 + 30.0 294 .3 94604 4.0 + 25.8 400 295 .8 96743 4.0 + 16.1 295 .4 96820 4.0 + 13.2 200 293 .7 99018 3.0 - 11.7 293 .4 99095 3.0 - 15.8 000 231 .7 101349 2.0 291 .0 101440 2.0 10 A. M. 2p M. 2500 280. 7 75462 9.4 + 144.2 280. 9 75462 9.7 + 146.7 2000 284 .0 80173 9.4 + 114.2 284 .8 80162 10.3 +133.0 1500 287 .0 85112 9.4 + 87.4 288 .8 85084 10.6 + 133.0 1000 289 .5 90313 8.3 + 23.0 292 .6 90228 9.2 + 49.2 800 291 .0 92460 8.1 + 19.7 294 .8 92355 8.7 + 49.2 600 292 .5 94656 7.9 + 18.9 296 .9 94510 8.3 + 49.9 400 294 .5 96886 7.6 + 16.2 299 .0 96703 7.9 + 52.2 200 296 .2 99154 6.0 + 23.4 300 .5 98930 7.0 + 52.2 000 297 .7 101459 4.0 301 .3 101214 5.0 6 P.I I. 10 F .M. 2500 281. 5 75458 8.0 +159.3 280. 8 75466 5.0 + 142.0 2000 286 .0 80142 9.0 + 156.5 284 .3 80170 6.0 +117.6 1500 290 .0 85038 9.0 + 145.6 287 .1 85102 6.0 + 88.5 1000 293 .5 90166 8.0 + 51.9 290 .7 90294 6.0 + 34.9 800 296 .0 92278 8.0 + 58.9 292 .2 92432 5.0 + 28.0 600 297 .3 94426 7.0 + 51.6 294 .2 94606 5.0 + 29.0 400 298 .0 96636 6.0 + 42.1 295 .1 96835 4.0 + 19.9 200 298 .6 98867 5.0 + 31.6 295 .7 99095 3.0 + 3.2 000 299 .0 101149 4.0 294 .3 101414 2.0 from the descending air to the ground, and an inversion of tem- perature under ordinary circumstances. The friction and the earth's deflection have very little influence on the temperature and pressure conditions, and the circulation cannot be studied by itself until the radiation or heating terms have been eliminated. 104 THERMODYNAMIC METEOROLOGY TABLE 25 THE CORRESPONDING VALUES OF B c =P/g Pm ON THE LEVEL 2 = 000 METERS ARE Now GIVEN, AND COMPARED WITH THE OBSERVED VALUES OF # BY MEANS OF THE DIFFERENCES Surface 2A.M. 6A.M. 10 A.M. 2 P.M. 6P.M. 10 P.M. Mean B c Bo A BC 760.20 760.16 +0 08 760.88 760.75 +0 76 761.03 760.98 +0 91 759.18 759.52 -0 94 758.70 758.71 1 42 760.70 760.59 +0 58 760.12 760.12 A BO +0 04 +0 63 +0 86 -0 60 1 41 +0 47 These curves can be reduced to the harmonics if desired. The value of the radiation exponent is a = 3.82 throughout the twenty-four hours. The Thermodynamic Structure of Cyclones and Anticyclones There has been much speculation regarding the forces that generate the powerful circulations in storms known by the name of cyclones and anticyclones, or low-pressure and high-pressure areas respectively. These will be more fully mentioned in the chapter on Dynamic Meteorology, but here we proceed to apply the principles just illustrated in the diurnal convection. From numerous kite and balloon ascensions in all parts of these local circulations, it has been learned what is the usual distribution of the temperature, and from the cloud observations what is the direction and velocity of the wind motion or " vector" in all areas, and all altitudes up to at least 10,000 meters. Compare the International Cloud Report, 1898, the Monthly Weather Review, January to July, 1902, April to June, 1904, January to August, 1906, October, 1907, to February, 1909, also the daily Weather Synoptic Charts, for numerous studies and details. From these data we have selected the temperatures T, and veloc- ities q, given in Tables 26, 25, and Figs. 11, 12. The computed values of the pressure P, and the free heat (Qi - Qo) are given in the same tables and figures, while the other thermodynamic data are summarized in Bulletin No. 3, 0. M. A. Only three diagrams are extracted from the tables, but these are enough to 6000 meterp High Area Low Area 266 2000 meters 260 284 288 292 Full lines = Temperatures T Dotted lines = Variations of Heat (Q, Q ) FIG. ii. The temperatures and heat variations in high and low areas High Area 5000 meter? FIG. 12. Pressures and wind velocities in high and low areas THERMODYNAMIC STRUCTURE OF CYCLONES TABLE 26 SOME VALUES OF T. P. q(Q l - Q ) IN CYCLONES TEMPERATURE T 107 000 500 1000 1500 2000 2500 3000 4000 5000 5. 760 S. 750 5. 740 C. 735 290.0 287.5 284.0 280.0 289.0 286.4 282.6 279.0 288.0 284.5 280.8 277.5 286.3 282.7 279.0 275.5 284.3 280.4 276.8 273.2 282.1 278.0 274.4 271.0 279.8 276.0 271.6 269.0 274.5 270.5 266.0 263.0 268.0 264.0 260.0 257.0 E. 760 E. 750 E. 740 278.0 281.0 283.5 276.6 279.3 281.4 274.8 277.0 279.1 272.4 274.6 276.6 269.7 272.0 274.0 267.0 269.5 271.0 264.5 267.0 268.0 259.5 261.7 263.0 255.0 256.0 257.0 N. 750 I N. 750 II N. 740 266.0 272.5 277.0 265.8 271.2 276.0 264.5 269.7 274.8 263.0 268.0 273.0 261.5 266.0 270.8 259.5 263.9 268.5 257.0 261.7 266.0 250.0 256.0 260.5 243.0 249.0 254.0 W. 760 W. 750 W. 740 268.0 273.0 277.0 267.5 272.6 276.0 266.8 272.0 275.0 266.0 271.0 273.7 265.0 269.8 272.0 264.0 268.0 270.0 262.0 266.0 268.0 258.0 261.0 263.0 253.0 255.0 256.5 PRESSURE P S. 760 101322 95518 90010 84814 79893 75208 70778 62566 55161 5. 750 99988 94194 88724 83548 78628 73953 69540 61367 54000 S. 740 98655 92873 87418 82230 77340 72683 68278 60127 52801 C. 735 97990 92158 86672 81482 76565 71908 67498 59366 52051 E. 760 101322 95275 89562 84140 78997 74124 69515 61016 53424 E. 750 99988 94064 88462 83150 78114 73338 68812 60467 52993 E. 740 98655 92864 87380 82166 77223 72532 68072 59861 52486 N. 750 I 99988 93706 87884 82372 77165 72273 67641 59114 51465 2V. 750 II 99988 93890 88142 82708 77587 72747 68163 59737 52179 N. 740 98655 92747 87176 81896 76915 72187 67720 59481 52080 W. 760 101322 95067 89220 83654 78460 73532 68911 60404 52860 W. 750 99988 93983 88252 82868 77790 72997 68477 60150 52690 W. 740 98655 92745 87156 81890 76930 72223 67779 59603 52256 VELOCITY q S. 760 4.1 6.5 9.0 13.9 21.0 26.5 27.0 24.0 23.0 5. 750 4.9 9.0 12.0 15.4 21.2 27.0 26.0 24.0 26.0 S. 740 6.5 11.0 14.0 16.9 22.8 30.0 34.0 34.0 33.0 C. 735 7.0 11.0 15.0 19.0 30.0 40.0 42.0 40.0 39.0 E. 760 5.2 8.6 12.4 16.0 19.8 23.0 24.0 25.0 26.0 E. 750 5.0 9.0 13.0 17.0 22.0 26.0 27.0 27.0 27.0 E. 740 5.0 7.5 11.0 15.0 22.0 29.0 29.0 29.0 28.0 TV. 750 I 3.5 5.0 5.8 6.0 7.0 7.5 8.0 8.0 7.5 N. 750 II 4.5 6.0 8.0 11.0 14.5 18.0 20.5 24.5 25.0 TV. 740 6.0 8.0 11.0 15.0 18.0 22.0 26.0 31.0 33.0 W. 760 5.0 8.0 12.0 17.0 22.0 26.0 27.0 28.0 31.0 W. 750 6.0 9.0 13.0 19.0 24.0 27.0 28.0 30.0 33.0 W. 740 6.0 10.0 14.0 20.0 25.0 29.0 32.0 34.0 37.0 1 108 THERMODYNAMIC METEOROLOGY FREE HEAT (Qi - Qo) 5. 760 -45.9 -118.4 -209.7 -333.0 -386.3 -313.4 -789.6 -872.3 5. 750 -61.2 -124.5 -189.3 -297.2 -374.2 -253.8 -631.9 -861.6 5. 740 -69.2 -125.2 -186.0 -316.2 -425.8 -403.3 -659.0 -764.7 C. 735 ... -70.5 -150.6 -221.4 -465.6 -597.1 -377.6 -625.6 -810.9 E. 760 -54.5 -129.3 -189.8 -248.3 -282.7 -283.5 -671.9 -851.9 E. 750 -55.7 -122.8 -184.1 -263.8 -303.7 -276.4 -621.4 -773.4 E. 740 -55.0 -104.1 -167.4 -286.0 -377.2 -232.2 -571.8 -695.6 N. 750 I -50.1 -124.3 -184.7 -251.3 -304.7 -352.3 -791.3 -890.2 N. 750 II -40.6 -109.8 -183.5 -254.0 -316.8 -356.8 -828.4 -874.3 2V. 740 -48.6 -130.3 -213.5 -261.7 -338.5 -398.1 -864.0 -919 6 W. 760 -59.7 -158.1 -260.1 -360.9 -427.8 -417.6 -960.4 -1-204 . W. 750 -62.5 -167.1 -286.6 -364.4 -392.4 -394.8 -925.4 -1110.3 JF. 740 -66.7 -151.7 -271.5 -341.0 -389.2 -417.7 -860.9 -1041.4 TABLE 27 SOME VALUES OF T. P. q (Q l <2 ) IN ANTICYCLONES TEMPERATURE T A C B 000 500 1000 1500 2000 2500 3000 4000 5000 14 15 16 17 S. 770 I S. 770 II S. 775 C. 777 278.0 273.5 272.0 272.0 277.4 273.0 271.0 271.0 276.0 272.0 269.5 269.2 274.6 270.4 267.8 267.0 273.0 268.7 266.0 264.5 271.0 267.0 264.0 262.0 268.6 265.0 262.0 259.5 264.0 260.0 256.5 254 . 5 259.0 254.0 250.0 248.0 11 = 18 19 20 E. 760 E. 770 E. 775 268.0 264.0 266.0 267.5 263.3 265.0 266.8 262.7 264.0 266.0 262.0 262.9 265.0 261.2 261.5 264.0 259.7 259.8 262.0 258.0 258.0 258.0 254.0 254.0 253.0 249.5 249.0 21 22 23 N. 760 N. 770 N. 775 271.0 272.0 272.0 269.8 271.4 271.0 268.2 270.3 269.5 266.3 269.1 267.6 264.2 267.6 265.5 262.0 265.4 263.1 259.4 262.6 260.4 254.0 257.0 255.0 248.0 251.0 249.5 24 25 26 W. 760 W. 770 W. 775 282.0 278.0 274.0 281.2 277.1 273.3 280.3 276.0 271.8 279.1 274.7 270.0 277.7 273.0 268.0 276.0 271.0 265.9 273.8 268.5 263.7 268.0 263.0 258.0 262.0 257.0 252.0 PRESSURE P 14 5. 770 I 102655 96583 90794 85330 80146 75292 70677 62160 54551 15 5. 770 II 102655 96594 90550 85028 79805 74865 70220 61644 53970 16 S. 775 103322 97020 91080 85466 80164 i 75157 70435 61741 53949 17 C. 777 103588 97270 91316 85670 80343 75289 70520 61740 53888 18 E. 760 101322 95067 89220 83654 78460 73532 68911 60404 52860 19 E. 770 102655 96232 90184 84492 79144 74122 69392 60711 53016 20 E. 775 103322 96906 90818 85126 79753 74698 69928 61187 53419 21 N. 760 101322 95110 89266 83736 78512 73577 68913 60321 52646 22 N. 770 102655 96394 90676 85088 79857 74893 70197 61549 53803 23 N. 775 103322 97020 91080 85472 80153 75142 70395 61656 53845 24 W. 760 101322 95376 89716 84410 79394 74638 70136 61824 54349 25 W. 770 102655 96563 90772 85312 80146 ! 75273 70648 62111 54476 26 W. 775 103322 97073 91176 85596 80327 75357 70647 61974 54203 THERMODYNAMIC STRUCTURE OF CYCLONES VELOCITY q 109 14 5. 770 I 5.0 7.0 10.0 12.0 16.0 19.0 20.0 21.0 22.0 15 5. 770 II 5.0 7.0 8.0 9.0 12.0 15.0 16.0 17.0 18.0 16 5. 775 3.0 5.0 7.0 8.0 9.0 11.0 12.0 14.0 16.0 17 C. 777 4.0 5.0 6.0 8.0 9.0 10.0 11.0 12.0 14.0 18 E. 760 5.0 8.0 12.0 17.0 22.0 26.0 27.0 28.0 31.0 19 E. 770 5.0 7.0 10.0 14.0 19.0 23.0 25.0 26.0 28.0 20 E. 775 4.0 6.0 9.0 12.0 16.0 19.0 20.0 22.0 24.0 21 N. 760 5.0 8.0 11.0 14.0 18.0 22.0 24.0 26.0 27.0 22 2V. 770 6.0 9.0 13.0 16.0 20 23.0 26.0 28.0 30.0 23 N. 775 4.0 7.0 10.0 13.0 17.0 21.0 23.0 24.0 26.0 24 W r . 760 5.0 8.0 10.0 13.0 14.0 15.0 16.0 17.0 18.0 25 PF. 770 5.0 7.0 9.0 11.0 12.0 13.0 14.0 15.0 16.0 26 W. 775 4.0 6.0 8.0 10.0 12.0 12.5 13.0 14.0 15.0 FREE HEAT (Qi - Qo) 14 5. 770 I 49 7 131 4 -189 282 -337 1 350 6 804 4 977 1 15 5. 770 II 51 8 121 5 -186 1 266 1 331 1 359 837 1 988 16 5. 775 -43.3 -113.2 -167.7 -224.8 -289.5 -332.2 -794.6 -934.0 17 C. 777 -39.8 -104.0 -164.6 -204.8 -248.9 -293.1 -706.1 -868.1 18 19 E. 760 E. 770 .**.* -59.7 51 -158.1 142 8 -260.1 -242 8 -360.9 352 -427.8 420 2 -417.6 442 4 -960.4 972 2 -1204.5 -1194 7 20 E. 775 -45.9 -130.2 -209.3 -299.6 -356.1 -379.1 -918.7 -1109.7 21 22 23 TV. 760 TV. 770 N. 775 -53.2 -61.3 51 8 -125.7 -154.6 126.7 -191.5 -219.9 192 9 -270.1 -311.0 270 2 -335.3 -356.9 333 5 -346.0 -408.2 343 4 -770.4 -834.3 740 4 -895.7 -980.2 923 1 24 W. 760 55 2 123 9 206 7 246 8 304 4 355 1 -808 8 -942 4 25 W. 770 -47.2 -119.8 -188.3 -238.6 -292.2 -338.8 -780.6 -921.9 26 W. 775 -47.7 -119.5 181.1 -238.1 -272.5 -321.0 -762.3 -903.0 indicate clearly the principles that are involved in their structure. Continuing the computations to higher levels, it is found that the temperature lines or isotherms, the pressure lines or isobars, and the velocity lines or vectors, coincide at every point in direction. In the lower levels these lines cross each other at various angles in the areas marked 1 to 26 on the sea level of Fig. 12, which shows the order of the computations: (1) The upper undisturbed circulation, where T. P. q coincide, belongs to the general circulation, and (2) the lower disturbed circulation, where T. P. q do not coincide, to the combined general and local circula- tion; (3) the purely local circulation, the cyclone and anti- cyclone proper, can be separated from the second by vector composition, since (2) is the resultant of (l) and (3). It is found that the disturbing circulation (3) is similar in configuration 110 THERMODYNAMIC METEOROLOGY to that at sea level up to 10,000 meters, if it penetrates the general eastward drift so high, that it usually increases in intensity to the 3,000-meter level, and then gradually dies out as the head is stripped away in the rapidly flowing upper currents. There is no evidence of a change in the type of the circulation, and therefore the physical origin of the structure of a cyclone is the same throughout. In the International Cloud Report it was shown that a series of warm currents from the south interlock alternately with another series of cold currents from the north, in the United States and adjacent regions, and that these local circulations are the mixing regions where the interchange of the temperature goes on toward a thermodynamic equilibrium under the force of gravitation. Fig. 11 shows the temperature distribution T and the free-heat distribution (Qi Qo). The temperature is deflected to the south on the east side of the high area, and to the north on the east side of the low area. The deflections of the isotherms diminish with the altitude and finally disappear as these disturbing currents diminish in strength. There are no cold-center anticyclones, and no warm-center cyclones, as has been assumed in many theoretical discussions. The distribution of (Qi Q ) is in elliptical figures whose centers are on the border of the high and low areas, and they show where the exchange of heat is going on most vigorously. The radiation heat increases with the height in consequence of the general radiation of the atmosphere increasing upward. It would be well to separate the purely local (Qi Q ) from the general as can be done by computation. The pressures of Fig. 12 depend entirely upon the temperature assigned to the several areas, and not upon the circulation, the deflecting force, the centrifugal force, or the friction, or any other minor condition. The air column, though temporary in position at a given instant, presses upon the level of computation, in consequence of the air masses which are determined only by the density, since this in turn is a function of the temperature. The given temperature structure must be continuously renewed by circulation of air from the warm and cold regions, or else the gravitation would soon flatten down all the disturbed temperature and pressure levels. The WARM, COLD, AND LEAKAGE CURRENTS 111 ultimate source of heat is the sun's insolation, chiefly on the tropics, and radiation in general from the atmosphere. The tropics are the boiler and the polar regions the condenser of the thermal engine, and the cyclones and anticyclones are the working machinery of motion. The general circulation depends upon the heat of the tropics, with westward drift to the south of 33 latitude, and eastward drift in middle latitudes from 33 to 66 latitude, that in the polar zone being irregular. There are, however, centers of general action along the high-pressure belt separating the westward from the eastward drifts, such that there are leakage currents from one of these zones to the other. There are such centers of action over the tropic north and south Atlantic and Pacific Oceans, those in the same hemi- sphere being broken through by the western and eastern conti- nents. Such leakage currents flow northward from the Gulf of Mexico over the United States, and from the north Pacific upon the northwestern States in a southward direction. The inter- flowing of these two series of warm and cold currents upon the United States and Canada is the immediate cause of the num- erous cyclones and anticyclones that wander eastward over this region. Forecasts are made of the probable detailed action of the weather conditions in all areas, as learned by experience with the types that these local circulations assume. The operation of the several elements, temperature, pressure, wind, and precipitation, is very complex and irregular, so that practical forecasts are difficult and uncertain, except in the cases of vigorously developed storms, which move along paths quite well determined by the pressure and temperature distributions. It should be noted in Fig. 12 that there is a saddle of higher pressure to the north of the center in low areas, and to the south of high areas there is a saddle of low pressure. These gradually diminish with the height and usually disappear above the 3,000- meter level. It can be seen that the winds on the sea level generally blow out of a high area into a low area, by curves having reversed spiral forms crossing the isobars at angles varying with the place. The winds flow more closely along the isobars at higher levels, as stated, and from 5,000 meters to 112 THERMODYNAMIC METEOROLOGY 10,000 meters it would be safe to draw the isobars and isotherms from the wind directions as observed in the high clouds. It follows that high-level pressure and temperature charts are the true indicators of the general movement of storms across the continent, because they show the direction of the eastward drift when the isobars on the sea level do not clearly indicate it. Such charts were prepared by the author for the sea level, the 3,500-foot plane, and the 10,000-foot plane for the United States, and they proved to be most instructive for the public forecast service. Attention is called to the fact that the isobars are all marked in the notation of Table 1, units of force B = P/100, and the isotherms in absolute temperatures T. Thus, we have for the pressure : D Tt Mercury f D mm. 103322 1033.2 775.0 102655 1026.5 770.0 101322 1013.2 760.0 99988 999.9 750.0 98655 986.6 740.0 97990 979.9 735.0 P is the dynamic pressure in the M. K. S. System; B is this pressure divided by 100 for practical use; mm. is the milli- meters of mercury of a barometer. B and mm. are related very closely in the ratio 4 to 3. Hence, by making the scale of a barometer in divisions each three-fourths of a millimeter, it would be only necessary to multiply the reading by 100 to obtain the dynamic pressures useful in all computations on the dynamics and thermodynamics of the atmosphere. This change in units is so simple, and so far-reaching in its beneficial results, that it is strongly recommended to meteorologists. Generally, a full set of Tables should be constructed to supersede the mercurial British and the Metric systems now in use. Further attention is called to the fact that these computa- tions fully satisfy equation (196) in a non-adiabatic atmosphere, and that, therefore, the author's theory of the non-asymmetric cyclone and anticyclone, due to interflowing currents at different temperatures subject to the attraction of gravitation at every PLANETARY CIRCULATION AND RADIATION 113 point, is fully verified, since the computed and the observed values are in agreement. The Planetary Circulation and Radiation. The Observations of Temperature and Velocity The greatest difficulty in discussing the problems of the planetary circulation and radiation consists in determining the proper temperatures and velocities of the circulation in all latitudes from the equator to the pole, and at all altitudes from the surface up to the practical limit of the balloon ascensions, as 30,000 meters. The number of available observations is very limited throughout the tropics, they are lacking entirely in the arctic zone, and above 14,000 meters in the isothermal region they are insufficient for our purposes. In spite of these difficulties it has been thought proper to execute the extensive computations, for the sake of the general instruction regarding various unsolved problems of meteorology, which depend upon such data. There are several accessible reports and compilations on the results of balloon ascensions, and we utilize them without further references: Rykachef for Russia, Dines for England, Teisserenc de Bort for France, Wegener for Germany, Rotch for St- Louis, Teisserenc de Bort and Rotch for the Atlantic Ocean, Berson for Victoria Nyanza and East Africa. Table 27 contains a summary of the original mean observations arranged according to the latitude, and Table 30 contains the adopted temperature system, which fairly represents this type of distribu- tion. An inspection of these original temperatures presents a great difficulty when they are compared with the wind velocities and directions in the tropics. It is seen that there is a decrease of temperature in the convectional region from the equator to the pole, except in the low levels of the tropics, as indicated in Fig. 13, Case II. When the temperature rises towards the pole there is westward wind, as in the trades of the tropics; when the temperature falls toward the pole there is eastward drift, as in the temperate zones. This was first developed by Bigelow, 1904, and confirmed by De Bort and Rotch in their report, 1909, thus establishing a fundamental property of all 114 THERMODYNAMIC METEOROLOGY atmospheric motions as indicated by Helmholtz. The trades blow steadily westward at a moderate velocity, while the east- ward drift reaches a mean velocity of about 35 meters per second z Height in Meters 18000 17000 16000 15000 14000 13000 12000 11000 10000 9000 8000 7000 6000 5000 4000 3000 2000 1000 000 Case II High Temperatures in the Isothermal Region 90 80 70 60 50 40 30 20 10 Case I Low Temperatures in the Isothermal Region 70 60 50 40 30 20 10 O c FIG. 13. Two typical cases of the observed temperatures in the earth's atmosphere up to 19,000 meters in middle latitudes, where the temperature falls rapidly toward the pole at high elevations. These cases illustrate the movement of a temperature maxi- mum from the tropics into the temperate zones of the isothermal region. The general questions of temperature are greatly complicated by the necessity of adapting them to the observed velocities, and for those the observations are too limited in number in the isothermal region and in the arctic zone to be decisive. Table 33 contains one system of velocities, which conform to the adopted temperatures of Table 30. The velocities are directed westward in the convection region of the tropics, with maximum on the 4,000-meter level, minimum at the 13,000- meter level, and a region of alternately westward and eastward velocities above that level, except immediately over the equator, where the wind is steadily westward. In the latitudes on the poleward side of the high-pressure belt, which is in latitude 30, PLANETARY CIRCULATION AND RADIATION 115 TABLE 27 THE MEAN OBSERVED TEMPERATURES IN THE ATMOSPHERE FROM BALLOON ASCENSIONS ARRANGED IN THE ORDER OF LATITUDE Russia Eng- land Ger- many France St. Louis Atlantic Ocean Tropics Viet. Nyanza Lati- tude 56 53 51 49 39 35 25 15 5 Num- ber 143 200 380 581 23 12 6 8 6 12 19000 18000 190 5 17000 197 1 16000 202 6 15000 213 3 206 8 14000 219.1 218.6 218.9 212.2 211.2 210.7 210.0 214.1 210.8 13000 .... 219.3 218.5 218.6 214.2 214.6 212.6 216.2 218.1 216.0 12000 218.3 219.6 218.8 217.8 216.7 219.4 217.4 223.6 224.7 222.6 11000 217.7 219.4 220.2 219.0 221.0 225.8 224.7 231.8 231.8 231.4 10000 218.9 223.1 223.4 223.7 226.2 233.7 233.2 240.2 239.3 238.9 9000 224.3 228.4 228.6 229 .5 232 . 9 242.0 241.8 248.2 247.2 246.1 8000 231.4 235.2 235.0 236.8 239.8 250.0 249.6 255.5 254.6 250.7 7000 239.1 241.8 242.2 244.0 248.5 257.6 256.6 261.8 261.4 258.0 6000 246.4 248.8 249.3 251.1 256.1 264.4 263.0 267.6 267.3 263.4 5000 253.1 255.4 256.1 257.6 262.7 270.5 268.8 272.8 272.1 269.2 4000 259.3 261.7 262.3 263.6 268.6 275.8 274.6 277.9 277.2 274.7 3000 265.3 268.8 268.0 269.0 273.5 280.9 279.3 282.8 282.0 280.8 2000 269.9 272.5 273.1 273.7 277.8 285.4 283.9 287.5 286.7 288.4 1000 274.3 2,77.1 277.6 278.3 281.1 290.9 288.7 292.2 292.3 296.2 000 277.1 281.3 282.9 282.5 285.9 298.9 296.2 298.4 300.8 as given in Table 29 for all elevations, the wind is eastward, increasing from the surface to a maximum on the 9,000-10,000- meter level, where it suddenly falls in velocity, and prevails eastward or variable throughout the isothermal region. These velocity conditions conform to Bigelow's observations at Wash- ington, D. C., 1896-97, International Cloud Report, Charts 11, 14, and Monthly Weather Review, April, May, June, 1904. The reports of the International Committee show that for the hemisphere at large, these westward and eastward circulations tend to concentrate about "centers of action," wherein the continuity of the high-pressure belt around the globe in longitude is broken up into sections, one over the Atlantic Ocean in each hemisphere, and another over the Pacific Ocean in each hemi- sphere. This subdivision is due to the mutual influence of 116 THERMODYNAMIC METEOROLOGY oceans and continents through the induced temperatures, and the configurations of the great currents of the general circulation depending upon them. The high-pressure belt is itself produced, in that latitude, by the downflow of air which has originally risen in the tropics; the segregation is accompanied by low-level " leakage currents " from the tropics to the temperate zones, which form the warm parts of cyclones and anticyclones. The cold streams from the polar zones meet these warm currents in mid- latitudes, and their interaction produces the local circulations of storms, under the force of gravitation acting on warm and cold masses in contact with each other. For our special purpose in this connection, there has been much discussion regarding the existence of the " antitrade" winds blowing eastward in the upper levels, as above 12,000 meters. The observational data are themselves conflicting, but this points to a very important feature in the theory of the planetary circulation. Collecting some of the data for very high altitudes, 9,000 to 17,000 meters, we have the following typical exhibit: TABLE 28 THE NUMBER OF WINDS FROM EIGHT COMPASS POINTS, IN DIFFERENT LATITUDES, AT 9,000 TO 17,000 METERS ALTITUDE Station N. N.E. E. S.E. [s. S.W. w. N.W. To'l Lat. Lindenburg 18 17 14 4 1 16 2 q 81 52 Ponta Delgada . 6 16 8 4 1 10 1? q 66 37.7 and Madeira. 32.6 Teneriffe 3 ?, ?, 3 ?,?, 1 g 38 28.5 St. Vincent ?, 13 4 5 2 26 17 Victoria Nyanza Ascension 5 1 10 o 36 5 10 2 4 4 10 8 81 14 -8 It is evident that there is an alternation of wind direction between N.E. and S.W. in all latitudes, but that the westward wind prevails over the equator, and to some extent predominates in all latitudes to 50 or 60. This can only mean that in the isothermal region the temperatures increase from the equator PLANETARY CIRCULATION AND RADIATION 117 to that latitude, as on Fig. 13, Case II, "High Temperatures in the Isothermal Region"; for all the wind directions blowing eastward, the opposite temperature gradient must occur, of fall from the equator toward the pole, as in Case I, "Low Temperatures in the Isothermal Region." We infer that in the general circulation there are heat maxima, or warm crests, which form near the equator, Case I, and move toward the pole to middle latitudes, Case II. The temperature gradients in the isothermal region are, therefore, very unsteady at any place, and there is a continuous mixture of the air currents, along with a vigorous radiation of heat from below, as will be further indicated. The computed data of Case II are here produced in Tables 29-42. While there are instances of "antitrade" winds at high elevations, the "trade" winds apparently penetrate to very high altitudes at other times. These observations were all made on the eastern side of the north Atlantic Tropic Ocean, and on the eastern edge of the high-pressure section of that region, where westward winds from N.E.- to S.W.-ward prevail as part of the forced circula- tion. On the western side of the ocean and western edge of this same section, it is probable that the " antitrade" wind will pre- dominate much more vigorously, but this is again a localized effect of temperature and pressure distribution. An inspection of the Victoria Nyanza temperatures, Table 27, indicate a very pronounced fall of temperature on the levels from 16,000 to 18,000 meters, such as would produce a violent westward circulation, which does not seem to exist. It would be proper to confirm these valuable observations at other points over the equator, whenever practical. Similarly the temperatures of St. Louis relative to Europe would demand a violent westward wind at the high levels, which likewise does not regularly exist. These facts show how difficult it is to construct a satisfactory system of temperatures and velocities for the planetary circula- tion. There is great need for high-level balloon ascensions recording temperature, humidity, and wind velocity and direc- tion, upon which to base the computations for the other terms in the problems of the atmosphere. 118 THERMODYNAMIC METEOROLOGY The Thermodynamic Tables of the Planetary Circulation and Radiation After the preceding explanations regarding the observational data and formulas, the reader can easily study the results of the computations for Case II, in Tables 29-42, so that only special points of interest will be indicated. Table 29. The pressure maximum is near latitude 30 at all elevations; the minimum near the pole is much lower than that at the equator. Table 30. The temperature maximum is in the high-press- ure maximum throughout the convectional region, and in the isothermal region it moves from near the equator toward the pole; there is a sharp drop in the temperature in passing from the convectional to the isothermal region; this boundary is located at 9,000-12,000 meters on the poleward side of 40 and it lies between 12,000 and 16,000 in the tropics; when the temperatures in the isothermal region are relatively cold the boundary is at high elevations, and when warm at low elevations respectively; over anticyclones the isothermal region is at high altitudes, and over cyclones at low latitudes; it is high in winter and low in summer; its elevation depends upon the temperature and gravitation conditions in the convection region and not on any inherent forces of its own; it is distinguished in its physical properties from the convectional by some properties which will be indicated under the topic of radiation. Table 31. The density has nearly the same value on the same level of the tropics as a broad minimum, and it in- creases toward the pole, much more in the convectional than in the isothermal region. Table 32. The gas coefficient and the dependent specific heat are variables, though assumed to be constant at the surface before radiation changes it with the elevation, but decreas- ing upward generally, much more near the pole than over the equator; there is an irregularity in passing to the isothermal region, accompanied by the change of temperature and velocity of circulation. The check P = p T R is confirmed at every point. THERMODYNAMIC TABLES 119 TABLE 29 THE PLANETARY CIRCULATION AND RADIATION The pressure P in the units of force (M. K. S.) 90 80 70 60 50 40 30 20 10 19000 5795 5854 5964 6184 6356 6587 j 6945 6832 6637 6445 18000 6768 6838 6980 7224 7409 7709 8128 8038 7848 7683 17000 7922 8003 8148 8433 8672 9023 i 9550 9457 9273 9132 16000 9247 9367 9536 9843 10110 10519 11177 11127 10955 10829 15000 10823 10935 11136 11521 11833 12311 13133 13084 12915 12807 14000 12667 12797 13028 13448 13794 14408 15407 15374 15215 15108 13000 14825 14978 15248 15740 16144 16837 18041 18028 17869 17754 12000 17351 17530 17846 18422 18821 19674 21076 21069 20899 20782 11000 20308 20513 20887 21560 22028 22972 24555 24523 24345 24224 10000 23768 24013 24405 25135 25781 26808 28481 28403 28220 28098 9000 27818 28105 28596 29367 30099 31177 32860 32726 32542 32418 8000 32509 32838 33367 34213 35012 36068 37721 37534 37344 37222 7000 i 37859 38226 38772 39666 40517 41530 43113 42877 42678 42558 6000 43930 44344 44876 45781 46666 47620 49111 48812 48605 48495 5000 50811 51251 51742 52631 53516 54414 55772 55427 55219 55110 4000 58606 59038 59464 60297 61156 61967 63164 62791 62576 62491 3000 67437 67816 68143 68867 69678 70378 71355 70963 70772 70707 2000 i 77432 77703 77877 78455 79164 79722 80428 80046 79862 79838 1000 ! 88730 88838 88836 89168 89704 90104 90462 90126 89958 89950 000 101521 101388 101135 101149 101414 101588 101548 101216 101042 101058 Formula: log P = log Po nk (log r - log To). (182) TABLE 30 THE TEMPERATURE T IN ABSOLUTE DEGREES CENTIGRADE z 90 80 70 60 50 40 30 20 10 19000 219.0 219.2 219.4 220 .4 220.8 218.6 214.8 210.9 203.0 193.0 18000 218.6 218.8 219.0 220.0 220.6 218.4 214.6 210.6 204.0 196.0 17000 218.3 218.5 218.7 219.7 220.4 218.2 214.4 210.3 205.0 199.0 16000 218.0 218.2 218.4 219.4 220.2 218.0 214.2 210.0 206.0 202.0 15000 217.7 217.9 218.1 219.1 220.0 217.8 214.0 211.0 208.0 205.0 14000 217.4 217.6 217.8 218.8 219.8 218 215.0 213.0 210.0 209.0 13000 217.2 217.4 217.6 218.6 219.6 219.0 218.0 216.2 215.0 214.0 12000 217.0 217.2 217.4 218.4 219.4 220.0 221.0 222.0 221.0 220.0 11000 216.8 217.0 217.2 218.2 219.2 221.0 226.4 228.3 227.0 226.0 10000 216.6 216.8 217.0 218.0 219.0 222.1 234.3 236.3 235.5 234.5 9000 216.4 217 218.5 220.3 222.2 230.4 243.6 245.4 244.2 243.2 8000 221.8 222.1 224.5 227.3 230.0 238.6 251.6 253.2 252.5 251.5 7000 227.0 227.5 230.5 234.8 238.0 246.2 259.2 260.5 259.5 258.5 6000 232.8 233.0 237.0 241.8 245.7 253.0 265.8 266.3 265.5 264.8 5000 237.3 239.0 243.0 248.3 253.0 259.5 271.7 271.5 270.5 269.5 4000 241.5 244.2 248.3 254.2 259.0 265.8 277.3 276 5 275.5 274.0 3000 245.3 248.7 253.3 259.7 264.8 271.3 282.9 281.4 280.0 279.0 2000 249.0 253.0 257.7 264.6 270.6 276.6 288.4 286.2 285.0 284.0 1000 252.5 257.0 261.9 269.0 276.0 281.6 292.0 290.0 289.5 289.0 000 255.0 260.0 265.0 273.0 281.0 288.0 299.5 298.5 298.7 298.0 The observations made in balloon and kite ascensions. 120 THERMODYNAMIC METEOROLOGY Table 33. The adopted wind velocity indicates an east- ward movement with the positive sign (+), westward with the negative sign ( ), and alternating with the (=F) signs; it is quite likely that further observations will enable us to improve this mean table of velocities; these must ultimately be so ad- justed to the air masses associated with them that the sum of the momenta of rotation about the earth's axis shall be equal to zero in order that the period of the earth's rotation may be constant, as indicated by astronomical observations, and this involves the corresponding pressure, temperature, density, and radiation of heat from point to point throughout the entire atmosphere. Table 34. The kinetic energy of circulation from one level to another acts as a balance in the action of gravitation against the pressure and heat terms in the general equation. This action is very pronounced in passing from the convectional to the isothermal region; it is strong in some parts of cyclones and anticyclones, and in tornadoes, being due to rapid changes in the temperatures for short distances. When the pressure and heat terms are deficient, the kinetic energy makes it up by an increase in the velocity; when in excess, it balances the same by decreasing the velocity; the numerous horizontal currents in the atmosphere, as seen by the cloud motions in different direc- tions, exhibit this process which is incessantly at work adjusting these delicate differences between pressure and radiation to the controlling force of gravitation ; these adjust themselves mutually at every point in the atmosphere, and are not propagated at long range from one distant point to another. Table 35. The hydrostatic pressure per unit density is computed from Tables 29, 31, for use in the general equation (196), which must always be satisfied. It should be carefully remembered that the density is to be computed by equation (176), and not by equation (175). The term decreases upward, and generally from the equator to the pole, though there is a small maximum in the convectional region just north of the high-pressure belt, in the levels 5,000 to 9,000 meters. Table 36. The free heat on which radiation depends gives the change in the heat contents of the unit mass per 1,000 THERMODYNAMIC TABLES 121 TABLE 31 THE PLANETARY CIRCULATION AND RADIATION The density p in kilograms per cubic meter z 90 80 70 60 50 40 30 20 10 O 8 19000 0.1810 0.1788 0.1776 0.1769 0.1754 0.1756 0.1753 0.1737 0.1700 0.1670 18000 .2022 .1997 .1986 .1976 .1956 .1964 .1961 .1950 .1915 .1892 17000 .2261 .2233 .2217 .2206 .2188 .2197 .2199 .2189 .2156 .2140 16000 .2524 .2498 .2480 .2463 .2440 .2456 .2459 .2458 .2428 .2416 15000 .2823 .2788 .2768 .2754 .2729 .2740 .2758 .2758 .2730 .2722 14000 0.3157 0.3118 0.3096 0.3074 0.3043 0.3064 0.3090 0.3093 0.3066 0.3061 13000 .3531 .3487 .3463 .3437 .3403 .3423 .3457 .3464 .3438 .3433 12000 .3949 .3900 .3872 .3835 .3796 .3824 .3861 .3870 .3843 .3840 11000 .4417 .4362 .4331 .4300 .4245 .4270 .4304 .4311 .4284 .4282 10000 .4939 .4878 .4844 .4796 .4748 .4765 .4783 .4786 .4758 .4759 9000 0.5524 0.5456 0.5415 0.5357 0.5300 0.5306 0.5295 0.5293 . 5265 0.5269 8000 .6172 .6094 .6043 .5972 .5902 .5885 .5841 .5835 .5806 .5813 7000 .6878 .6790 .6724 .6634 .6548 .6505 .6424 .6414 .6385 .6394 6000 .7645 .7546 .7660 .7346 .7240 .7170 .7047 .7033 .7003 .7016 5000 .8479 .8364 .8255 .8112 .7981 .7884 .7714 .7698 .7669 .7684 4000 0.9384 0.9249 0.9114 0.8935 0.8775 0.8647 . 8428 0.8413 0.8382 0.8402 3000 1.0370 1.0207 1.0041 0.9821 0.9628 0.9466 0.9191 0.9177 0.9148 0.9173 2000 1 . 1440 1 . 1244 1.1071 1 . 0775 1.0543 1 . 0344 1 . 0008 0.9998 0.9969 1 . 0001 1000 1.2604 1.2367 1.2125 1.1802 1 . 1523 1.1284 1 . 0880 1.0878 1.0850 1.0886 000 1.3870 1.3586 1.3296 1.2908 1.2574 1.2289 1.1812 1.1814 1.1785 1 1825 Formula: log p log Po + j- (log T - log To). (183) TABLE 32 THE GAS COEFFICIENT R, WHICH Is A VARIABLE z 90 80 70 60 50 40 30 20 10 19000 146.14 149.37 153.03 158.60 164.12 171.56 184.39 186.48 192 . 33 199.92 18000 153 . 12 156.51 160.45 166.18 171.70 179.71 193.14 195.72 200.88 207.12 17000 160.47 164.01 168.01 174.01 179 . 86 188.24 202 . 54 205.43 209.77 214.42 16000 168.03 171.88 176.06 182.21 188.17 196.94 217 . 10 215.62 219.05 221.92 15000 176.09 179.98 184.36 190.95 197.10 206.29 222.48 224 . 88 227.53 229.53 14000 184.53 188.61 193.20 199.95 206.21 215.68 231.91 233.39 236.28 236.14 13000 193.29 197.56 202.37 209.44 216.00 224.58 239.38 240.73 241.75 241.63 12000 202.46 206.94 211.93 219.37 225.99 233.85 246.98 245.24 246 . 08 245.98 11000 212.07 216.76 222 . 04 229.79 236.72 243.45 251.97 249.17 250.38 250.28 10000 222.14 227.05 232.58 246.01 247.95 253.29 254.13 250.64 251.86 251.77 9000 232 . 68 237.39 241.68 248.85 255.56 255.06 254.74 251.96 253 . 09 253.01 8000 237.48 242.61 245.94 252 . 06 257.92 256.88 256.66 254.07 254.69 254.62 7000 242.48 247.47 250.16 254.65 259 . 99 259.30 258.96 256.62 257.57 257.51 6900 247.35 252.22 253.79 257.74 262.33 262.51 262.21 260 . 62 261.39 261.05 5000 252 . 54 256.39 257.92 261.31 265.05 265.98 266.11 265.19 266.19 266.15 4000 258.59 261.39 262.76 265.47 269.08 269 . 62 270.28 269.94 270.97 271.46 3000 265.12 267 . 15 267.91 270.02 273.29 274.04 274.43 274.72 276.27 276.28 2000 271.82 273.15 273.71 275.18 277.48 278.65 278.66 279.74 281.07 281.11 1000 278.81 279 . 51 279.75 280.88 282.05 283.56 284.74 285.70 286.38 285.93 000 287.03 287.03 287 . 03 287.03 287 . 03 287 . 03 287.03 287.03 287.03 287.03 Formula: log R = log Ro + (n-l) (log T - log To). . . . (184) Check P = Tp Rat every point (173) 122 TIIERMODYNAMIC METEOROLOGY meters, and the table indicates how far the atmosphere has de- parted from the adiabatic state. The term increases upward, and generally from the equator to the pole, but there is a mini- mum on entering the isothermal region in middle latitudes and a region of marked irregular progression in the values. Un- fortunately there is no way to compute this term directly, as it depends upon the evaluation of the specific heat and the velocity of the circulation, by equation (199). It is therefore a great loss to science when an observatory measures the temperatures, but not the humidity and wind velocity, at different levels, because the entire subject of thermodynamic meteorology is thereby excluded from further discussions. The magnitude of the term (Qi Qo) can be seen in Table 36 to be very great in the upper levels, and that in comparison with it the kinetic energy of the circulation, as in Table 34, is very small. Hence, all those the- ories of atmospheric circulation which depend upon gravitation, pressure, and circulation alone, and omit heat changes through radiation, either absorption or emission, have no permanent value. The difficulty of determining the heat term has, no doubt, been the cause of this defect which prevails in meteorological literature, but it is none the less a fatal defect in this branch of science. Table 43, p. 131, gives the second differences of the heat contents, and it is the rate of change of the heat per 1,000 meters, or the radiation rate, which prevails on the average in all parts of the atmosphere, and it is fundamental to all studies of "solar constant " and solar insolation. This table will be dis- cussed more fully under the subject of Bolometry and Pyrhe- liometry, and at this place only its leading features will be noted. Taking the hemisphere as a whole, the mean rate of radiation is 157.1 in the convectional region, and 283.1 in the isother- mal region, almost twice as great in the latter. This changes greatly in latitude, as can be seen in the means for the isothermal region (/), and the convectional region (C), respectively. The means on the table are taken outside the layers of transition, and these give nearly a constant value in C from the equator to lati- tude 60, but an increasing value to the pole; in the / region there is a minimum in the tropics, maximum in the middle latitudes, THERMODYNAMIC TABLES 123 TABLE 33 THE PLANETARY CIRCULATION AND RADIATION The velocity q in meters per second z 90 80 70 60 50 40 30 20 10 19000 0.0 3.0 4.0 4.0 5.0 T5.0 T6.0 T7.0 + 9.0 -12.0 18000 0.0 3.0 4.0 5.0 6.0 6.0 T7.0 =F8.0 +~10.0 -12.0 17000 0.0 3.0 4.0 6.0 7.0 7.0 T8.0 +9.0 +10.0 -12.0 16000 0.0 4.0 5.0 7.0 8.0 9.0 T6.0 T6.0 T10.0 -11.0 15000 0.0 5.0 6.0 8.0 9.0 10.0 T4.0 +4.0 T 7.0 -10.0 14000 0.0 6.0 7.0 9.0 10.0 11.0 T2.0 T2.0 =F 4.0 - 9.0 13000 0.0 6.0 8.0 10.0 11.0 12.0 4.0 5.0 T 2.0 - 8.0 12000 0.0 7.0 9.0 11.0 12.0 13.0 9.0 5.0 - 3.0 - 7.0 11000 0.0 7.0 11.0 13.0 15.0 16.0 12.0 6.0 - 4.0 - 5.0 10000 0.0 8.0 15.0 17.0 20.0 20.0 15.0 6.0 - 5.0 - 3.0 9000 0.0 5.6 25.2 31.6 35.0 34.0 21.7 0.8 - 3.8 - 2.0 8000 0.0 4.8 22.8 28.9 31.5 29.0 16.0 .-2.6 - 5.0 - 2.5 7000 0.0 4.0 19.6 25.8 26.9 22.0 9.0 -4.9 - 6.2 - 2.6 6000 0.0 3.2 15.9 21.4 21.8 15.4 2.4 -6.8 - 7.0 - 2.7 5000 0.0 2.0 12.4 17.6 17.2 11.4 2.6 -8.6 - 7.4 - 2.8 4000 0.0 0.8 9.0 13.8 13.6 9.2 6.6 -9.4 - 7.6 - 2.8 3000 0.0 0.4 6.6 11.0 11.0 7.4 7.3 -9.4 - 7.2 - 2.8 2000 0.0 1.6 5.0 9.4 9.4 6.4 6.1 -8.4 - 6.2 - 2.4 1000 0.0 2.4 3.8 8.0 8.0 5.6 4.1 -6.7 - 4.6 - 1.4 000 0.0 3.4 3.4 7.4 7.1 5.1 2.0 -4.6 - 2.6 - 0.4 From the observations made in balloon and kite ascensions. TABLE 34 THE KINETIC ENERGY OF CIRCULATION X (2 2 i <Z 2 o) z 90 80 70 60 50 40 30 20 10 19000 0.0 0.0 0.0 + 4.5 + 5.5 + 5.5 + 6.5 + 7.5 + 9.5 0.0 18000 0.0 0.0 0.0 + 5.5 + 6.5 + 6.5 + 7.5 + 8.5 0.0 0.0 17000 0.0 + 3.5 + 8.5 + 6.5 + 7.5 + 16.0 - 14.0 -22.5 0.0 -11.5 16000 0.0 + 4.5 + 5.5 + 7.5 + 8.5 + 9.5 - 10.0 -10.0 -25.5 -10.5 15000 0.0 + 5.5 + 6.5 + 8.5 + 9.5 + 10.5 - 6.0 - 6.0 -16.5 - 9.5 14000 0.0 0.0 + 7.5 + 9.5 + 22.0 + 11.5 + 6.0 +10.5 - 6.0 - 8.5 13000 0.0 + 6.5 + 8.5 + 10.5 + 11.5 + 12.5 + 32.5 + 5.0 + 2.5 - 7.1 12000 0.0 0.0 +20.0 + 24.0 + 40.5 + 43.5 + 31.5 + 5.5 + 3.5 -12.5 1 1000 0.0 + 7.5 +52.0 + 60.0 + 87.5 + 72.0 + 40.5 - 0.0 + 4.5 - 8.0 10000 0.0 +16.3 -54.9 +354.8 +412.5 +378.0 +123.0 -17.7 - 5.3 - 2.5 9000 0.0 - 4.2 -57.6 -81.3 -116.4 -157.5 -107.4 + 3.1 + 5.3 + 1.1 8000 0.0 - 3.5 -67.8 - 84.8 -134.3 -178.5 - 87.5 + 8.6 + 6.7 + 0.3 7000 0.0 - 2.9 -65.7 -103.8 -124.2 -123.4 - 37.6 +11.1 + 5.3 + 0.2 6000 0.0 - 3.1 -49.5 - 74.1 - 89.7 - 53.6 + 0.5 +13.9 + 2.9 + 0.2 5000 0.0 - 1.7 -36.4 - 59.7 - 55.4 - 22.7 + 18.4 + 7.2 + 1.5 0.0 4000 0.0 - 0.2 -18.7 - 34.7 - 32.0 -14.9 + 4.8 0.0 - 3.0 0.0 3000 0.0 + 1.2 - 9.3 - 16.3 - 16.3 - 6.9 - 8.0 - 8.9 - 6.7 - 1.0 2000 0.0 + 1.6 - 5.3 - 12.2 - 12.2 - 4.8 - 10.2 -12.8 - 8.6 - 1.9 1000 0.0 + 2.9 - 1.4 - 4.6 - 6.8 - 2.7 - 6.4 -11.8 - 7.2 - 0.9 000 0.0 To obtain the kinetic energy these numbers must be multiplied by the appropriate values of p:o. 124 THERMODYNAMIC METEOROLOGY and minimum in the polar zone. The broad fact is clear that about twice as much heat radiates through the isothermal region as through the convection region. At the strata of transition the radiation is in some latitudes positive, as at the 10,000-meter level from 30 to 60 latitude. These special sources of heat will be further considered in discussing the value of the "solar- constant" of radiation. The importance of such data in radia- tion problems is very great, because it seems to explain several processes in the atmosphere that have been very obscure. Table 37. The entropy increases upward and from the equator to the pole, though the same variation occurs in the transition to the isothermal region as was noted for the heat contents. Table 38. The work against external forces , such as expansion when a mass of air is raised from one level to another in the circulation, is to be computed through a new value of R', which differs from the R of (184) in consequence of the elimination of the velocity of the circulation, as explained in (199), (333), on Table 19. It is the circulation term which continu- ally interferes with a perfect check to this series of equations, and on that account, therefore, must be most carefully observed. Table 39. The inner energy is the heat contents remaining after the work has been done externally, and it increases upward and from the equator to the pole, with the same variation be- tween the convectional and the isothermal regions. Table 40. The radiation function is the rate of change of the inner energy with the change of the volume, and it may be computed by several formulas. The K\Q differs from Pi by the rate of change of the heat contents with the change in volume, or it may be seen to depend upon the variation of the gas coeffi- cient R. The value of K i0 generally decreases with the altitude, and there is a small maximum in the high-pressure belt of the convectional region, but this shifts toward the pole in the iso- thermal region. The immediate problem is to determine the relation to the temperature through the approximate Stefan formula, and this has been done in the same way as that followed THERMODYNAMIC TABLES 125 TABLE 35 THE PLANETARY CIRCULATION AND RADIATION Pi - The Hydrostatic Pressure per Unit Density PlO z 90 80 70 60 50 40 30 20 10 19000 5078.2 5198.1 5401 . 4 5552.5 5676.6 6032.3 6370.4 6540.2 6697.9 6651 . 2 18000 5387.5 5508.6 5556.6 5782.0 6095.6 6314.3 6836.6 6855.0 6998.9 7187.5 17000 5537.1 5765.0 5909.0 6038.5 6214.4 6428.8 6985.8 7185.8 7338.7 7449.5 16000 5893 . 9 5932.7 6082.3 6431.6 6665.4 6897.7 7497.2 7503 . 8 7600.0 7699.7 15000 6167.3 6305.5 6466.7 6532.2 6794.8 7226.0 7777.0 7826.5 7936.5 7956.5 14000 6453 . 3 6603 . 1 6766.3 6962.3 7291.3 7490 . 8045.2 8094.0 8161.2 8149.2 13000 6754.0 6908.4 7082 . 8 7366.2 7436.2 7828.3 8294 . 7 8293.0 8321.8 8325.7 12000 7069.0 7220.8 7413.5 7704.4 7975.7 8149.3 8520.6 8445.0 8479.5 8475.8 11000 7396.3 7575.8 7668.0 7860.7 8345.6 8490.4 8640 . 8531.2 8571.2 8569.0 10000 7740.8 7919.5 8169.8 8335.6 8594.6 8675.4 8690 . 2 8577.4 8623.2 8615.8 9000 8021.6 8195.8 8327.8 8555.8 8771.6 8740.2 8730.4 8641.2 8674.0 8669.8 8000 8199.2 8364.0 8466.6 8643 . 6 8843.4 8816.8 8799.4 8724 . 6 8750.0 8742.0 7000 8359.8 8595.0 8606.8 8748.2 8919.2 8906.2 8897.0 8826.6 8854 . 8854 . 6 6000 8535.2 8702.6 8737.6 8862.8 9001.4 9026.2 9025.6 8980.5 9015.7 9000 . 6000 8727.2 8842 . 8 8892.2 8993 . 6 9119.0 9137.4 9158.6 9141.0 9166.4 9177.0 4000 8940 . 8 9023.6 9061.5 9138.4 9261 . 9287.8 9297.4 9291.6 9350.8 9349.0 3000 9165.4 9217.6 9234.4 9310.4 9405.0 9433.6 9451.0 9473.2 9509.4 9524.5 2000 9397.8 9432 . 9461.2 9490 . 8 9553.0 9600.4 9607.2 9657.0 9698.4 9682.0 1000 9663.0 9671.6 9676.5 9697.5 9719.4 9743.6 9770 . 8 9776.8 9793.3 9781.4 000 .... Computed from Tables 29, 31. TABLE 36 THE HEAT WHICH GIVES RISE TO RADIATION (Q\ Q ) z 90 80 70 60 50 40 30 20 10 19000 4694.2 4581.1 4451.2 4253.8 4064.2 3800.2 3350 . 6 3270.0 3079.6 2853.1 18000 4449.3 4331.0 4195.5 3982.7 3794.3 3514.3 2990.1 2944.8 2791.4 2605.0 17000 4204.4 4064.9 3920.2 3707.8 3512.0 3210.4 2736.1 2677 .4 2481.0 2363.8 16000 3927.8 3793.2 3643.8 3424 . 3 3216.4 2908.5 2391.6 2291.5 2203.3 2104.8 15000 3645.9 3504.4 3350.2 3140.1 2907.2 2587.5 2050.2 1934.7 1900.1 1861.0 14000 3352.2 3131.0 3041.4 2803.6 2571.7 2273.9 1749.7 1696.7 1646.3 1653.7 13000 3045.8 2811.2 2719.8 2470.7 2244 . 6 1962.7 1465.6 1499.7 1470.5 1483.8 12000 2724.1 2568.5 2372.2 2109.6 1866.1 1609.4 1233.1 1355.2 1322.5 1341.5 11000 2388.8 2217.5 1988.4 1714.0 1439.6 1248.7 1120.4 1268.5 1222.6 1238.1 10000 2036.9 1888.9 1759.7 1093.8 792.7 744.4 990.6 1238.5 1185.9 1185.9 9000 1774.7 1611.1 1534.0 1331.4 1151.3 1218.7 1177.8 1159.1 1126.8 1133.6 8000 1607.5 1438.2 1399.5 1235.4 1093.6 1167.3 1085.8 1074.0 1049.0 1057.6 7000 1439.1 1273.3 1263.3 1157.2 1008.1 1016.1 941.2 959.0 936.0 948.0 6000 1267.2 1121.2 1114.6 1013.9 887.3 832.2 781.0 810.4 791.2 800.4 5000 1075.2 958.1 948.3 867.6 736.2 697.8 625.3 658.0 628.9 622.9 4000 860.2 777.7 760.0 693.8 573.4 534.2 496.8 501.1 461.1 449.7 3000 634.2 575.6 563.7 559.4 414.2 372.3 366.3 342.5 292.4 285.9 2000 400.3 364.2 357.3 319.7 260.5 202 .4 192.4 160.1 121.7 122.1 1000 140.4 125.2 125.8 109.7 91.9 62.1 45.5 34.6 18.5 24.4 000 Formula: - (Qi - Qo) = - (Cp* - Check q (zi - Zo) = l - (T* - To) + \ (<Zi 2 - <Z0 2 ) - Q ) (199). (196). Comparing the numerical values of these terms, it is seen how impossible it is to solve any problems in dynamic meteorology without the term (Qi Qo). 126 THERMODYNAMIC METEOROLOGY on Table 20, the intermediate steps of computing log C and A for the ratios Ki/K Q and Ti/To being here omitted. Table 41. The log c in the formula diminishes slowly from the surface upward, it increases from the equator to the pole in the convectional region, but decreases in the isothermal region, the result being a nearly uniform value on the high levels of the isothermal region. The minus sign affects only the characteristic of the logarithm, and for a mean value log c = 5.500, c = 3.162X 10~ 5 . The Kurlbaum coefficient in the Stefan formula for a perfect radiator is taken at 7.68 X 10~ n (C. G. min. C) = 5.32 X 10~ 6 joules per square meter per sec., so that the air radiates at six times the rate of a perfect radiator in the ether, as when the sun transmits its radiant energy to the earth. The quantity c relates to a summation of the successive radia- tions from one air volume to another throughout a layer 1,000 m. deep. The number of repeated transfers in this depth can only be approximately inferred from the knowledge that the efficient radiating layer of the atmosphere, or that for which any further increase of depth does not add to the radiant in- tensity, amounts to several meters. The latter quantity is con- siderably smaller than the superficial radiation from a black solid. Table 42. The exponent a diminishes from the surface up- ward; it increases from the equator to the pole in the convec- tional region, but diminishes in the isothermal region. Its mean value is 3.82 instead of 4.00, as in the Stefan formula, at the surface. Comparing the data we have for the Perfect Radiator, K w =5.32 X 10~ 6 TV' 00 joule/sq. m. sec. (K. M. S. system) and for the Atmosphere, K 10 = 3.162 X 10~ 5 Tic 3 ' 82 per 1,000 m. The consequences of these data in theories of atmospheric radiation will require much careful investigation in several directions. THERMODYNAMIC TABLES 127 TABLE 37 THE PLANETARY CIRCULATION AND RADIATION The Entropy (5, - S ) 2 90 80 70 60 50 40 30 20 10 19000 -21.454 -20.919 -20.307 -19.318 -18.415 -17.392 -15.606 -15.516 -15.133 -14.669 18000 -20.363 -19.799 -19.166 -18.111 -17.208 -16.099 -13.938 -13.993 -13.650 -13.190 17000 -19.269 -18.612 -17.933 -16.885 -15.942 -14.720 -12.768 -12.740 -12.073 -11.790 16000 -18.026 -17.392 -16.692 -15.615 -14.613 -13.348 -11.171 -10.884 -10.644 -10.343 15000 -16.755 -16.090 -15.363 -14.338 -13.221 -11.875 - 9.558 - 9.126 - 9.091 - 8.990 14000 -15.427 -13.936 -13.971 -12.819 -11.705 -10.407 - 8.082 - 7.906 - 7.748 - 7.819 13000 -14.029 -12.937 -12.505 -11.308 -10.226 - 8.942 - 6.677 - 6.845 - 6.745 - 6.838 12000 -12.559 -11.831 -10.917 - 9.664 - 8.509 - 7.299 - 5.512 - 6.019 - 5.904 - 6.019 11000 -11.024 -10.224 - 9.159 - 7.859 - 6.570 - 5.635 - 4.864 - 5.330 - 5.287 - 5.377 10000 - 9.408 8.510 - 8.079 - 4.990 - 3.593 - 3.291 - 4.146 - 5.031 - 4.944 - 4.965 9000 - 8.100 - 7.337 - 6.926 - 5.949 - 5.092 - 5.197 - 4.757 - 4.649 - 4.537 - 4.583 8000 - 7.164 - 6.398 - 6.152 - 5.348 - 4.672 - 4.816 - 4.251 - 4.182 - 4.098 - 4.147 7000 - 6.268 - 5.531 - 5.403 - 4.836 - 4.169 - 4.071 - 3.728 - 3.643 - 3.566 - 3.623 6000 - 5.397 - 4.751 - 4.644 - 4.138 - 3.558 - 3.248 - 2.905 - 3.014 - 2.952 - 2.996 5000 - 4.491 - 3.966 - 3.861 - 3.454 - 2.876 - 2.589 - 2.278 - 2.402 - 2.304 - 2.992 4000 - 3.534 - 3.156 - 3.030 - 2.700 - 2.181 - 1.989 - 1.774 - 1.796 - 1.660 - 1.626 3000 - 2.566 - 2.295 - 2.206 - 2.134 - 1.547 - 1.359 - 1.283 - 1.207 - 1.035 - 1.016 2000 - 1.596 - 1.428 - 1.375 - 1.198 - 0.953 - 0.725 - 0.663 - 0.609 - 0.424 - 0.426 1000 - 0.553 - 0.484 - 0.478 - 0.405 - 0.330 - 0.218 - 0.154 - 0.118 - 0.063 - 0.083 000 Formula : (Si -So) - . - (331) . TABLE 38 THE EXTERNAL WORK (Wi - W ) z 90 80 70 60 50 40 30 20 10 19000 3601.4 3688.7 3854.4 3948 . 5 4017.9 4256.9 4344.3 4652.0 4754.7 4942.6 18000 3849.6 3926.9 3935.8 4099.8 4358 . 9 4496.7 4867.6 4872.0 4972 . 4 5107.2 17000 3918.8 4109.2 4208.6 4276.8 4396.1 4523.4 4943.4 5126.4 5222.6 5299.5 16000 4195.8 4195.7 4302.1 4588.0 4761.7 4905.1 5355.2 5333 . 5403 . 6 5474.9 15000 4387.8 4485.0 4601.7 4606.5 4801.8 5140.7 5536.4 5552.7 5652.6 5661.2 14000 4588.9 4645.9 4812.1 4939.4 5201.4 5314.0 5717.8 5751.3 5804 . 5794.1 13000 4801.0 4887.7 5035.7 5247.1 5251.8 5562.5 5885.3 5893.3 5913.8 5921.5 12000 5023.1 5129.9 5266.0 5581.0 5681.9 5781.4 6044.0 6003.6 6028.7 6030.5 11000 5253 . 5 5383 . 6 5409.6 5523 . 5928.7 6018.3 6130.8 6065.7 6091.6 6093.9 10000 5496.4 6532.3 5845.3 5818.7 6019.3 6056.5 6143.5 6102.3 6133.0 6125.6 9000 5701.4 5828.4 5938.1 6107.6 6271.4 6259.4 6237.8 6143.2 6166.6 6164.4 000 5830.8 5945.7 6038.0 6167.7 6326.5 6321.2 6280.2 6202.0 6220.2 6214.7 7000 5942.7 6130.0 6138.9 6249.8 6377.6 6366.8 6336.1 6271.0 6291.2 6295.6 6000 6068.4 6193.7 6226.8 6322.9 6424.9 6433.7 6418.4 6381.8 6411.4 6398.4 5000 6205.0 6286.7 6333.3 6411.4 6498.8 6501.0 6506.4 6498.2 6515.2 6524.1 4000 6356.4 6415.4 6448.2 6506.0 6593.8 6609.2 6608.1 6603.5 6651.2 6646.0 3000 6515.8 6551.0 6564.4 6639 . 2 6722 . 8 6708 . 4 6724.1 6739.3 6761.0 6774.3 2000 6680.6 6704 . 3 6731 . 6 6750.3 6795.4 6826 . 1 6830.0 6874 . 9 6900.7 6884.4 1000 6870.7 6874 . 9 6880.0 6896.4 6911.3 6928.7 6951.1 6953.9 6965.8 6955.6 000 Formula: (Wi - Wo) = R'IO (T* - To) - Plo .... (333). Cp'io . . . (349) and (350). 128 THERMODYNAMIC METEOROLOGY o <N CO 00 O O O O CO CD 00 O 00 (N CO 00 t^ 00 <N 1 l> O5 'f O l> CO TH Ol 1>- ^ Oi T}1 OT> CD O OU _iiOiO <* rfi CO CO CO <N <N <N TH TH OOOO5 t> !> t^* I s * t^ t^ t^* i"** * I s * JS t"* IS* t > * 1^ t > * 1 s * CD I I I I I .Mill I I I I I III! co op co p i> %%%E% t l> I I I I I I CO C^ C<l Ci CO CO TH ~fi r^ j^ LJ L^J ^i CQ CQ CQ CQ 77777 CO O5 t^ (N ^ O^ CO C^ C^ "^ (N (N CQ <N TH 77777 CO T*< Tt< CO (N CO (N Tjn TH 10 o^ oo i-H ^^ CO O^ l> 1> I> CD I I I I O l> CD O TH O^ 00 00 CO ^^ 77777 O O 00 (M 00 00 CO 00 ^ O Tt< CO CO CO CO 77777 COOO(NC<I CDCOO^O N CO O Cfl CO O t^ CO O5 iO ro <M c^ TH TH 77777 ^ TH iO 00 O 00 CO 00 rH O O O5 I I I I OS N. lO 00 CD ^ t>- Oi CD CD O5 O !> ^ OO CD 00 CO t^- 1O !>. t^ l>- l> l> I I I I I IO p TH CC| rH l> O t^- TH rt< CO IO !> 1C CO CD O CO O5 TtH TjH CD . t^ l> I I I I I IO CD t^- O5 T-H rH CO t~ O5 CO TfH CO <M i-l T-H -# O <N O O5 <M T I O O I 1 I I I I I I I TH O 00 CD O5 (M 00 O OS 1O O5 OS 00 TH CO CO 00 TH CO TH <N ' 00 1> O O 00 00 1> I I I I I 00 (N O5 CO O O iO CO <M 00 i> t>- 1> i> co I I I I I 00 00 <M O l> 00 OO CO 00 CO cq T-H CO Tt< u I O.J, ?S8 R^S2 ^g^2 rHO5O51>. I>TJHIOCOOO !> t^* t*^* t^* V s * 1^* r** CO X) 00 I I I I I I I I I I r^ CO CO (M l> t^ 1> 1> t- I I I I I O O5 co >o co co co <N <N OOOCOOO OTH C*5 CO CO I s * ^D t^* C^ TH TJH TfH TH Oi CO TH ^ ^^ Oi ^^ !> !> 1 s * CO C^ O^ ooot^oot^ t^t^t^-t^co I I I I I I I I I I CO t> CO O5 00 co o TH 01 OS t^- CO I I I I I I I I I COCOOOO5O5 lOTHOOiOTH COiOOOOOiO OCOIN-^iO OOCOO5O COTHTHO5O5 OOt^-COCOCO 0000001>1>- 1>1>1>1>I> I I I I I I I I I I CQ I> d TH TH i> co o "* oo Tt< -^ rfi CO <N 77777 (N TH p 00 i> F^ t^ i> I I I I 00 O5 TH p Tj< l> ^ 00 C5 00 00 t>- J> Mill p Oi Tf< TH CO 00 00 TH l>* O^ Oi ^^ l> CO CO CO 7 I I M <N IO O5 CO O5 00 8 O5 CO CO Tt< 00 O TH CO H/l CO (M > t^ 1> l> 1> I I I I I TH CO 1C TH CO CO GO O OJ <M CO O TH TH O O CD 05 (N CD b- TH 00 (N CO CO lOOOCOCOCO i-HCOl-^C^CO C^ C^ TH TH CO OS 00 t^ CO IO oooooooooo i>i>t^-r>i> I I I I I I I I I I TH CO CO CD <N CD 00 TH 10 O t^ CO 00 CO 00 T}H TfH CO CO (N CO O O5 TH CO O O TH TH IO 00 TH C<l TH O O I I I I I I I I I THERMODYNAMIC TABLES 129 88 TH C3 Tf t>* CQ CO TH cO CO T^H CO CO Tt^ CO Oi 00 TH Tj< 1>- TH IO -i TH TH TH TH <N <N CQ CO CO I I I I I I I I I I 00 iO TH CO <N CN O TH TH CO C^l 00 ^i^ rfri rj< iO iO CO I I I I I t^OOOO 00 t^ OO T- i oo <M i t co O OO CO iO l> 1> OO O5 I I I I ^^ O^ l> O^ 00 <M TH CO CO Ci 00 l> CO TH O Oi CO Tt< t^ <N CO TJH i> TH 00 lt TjH 1> TH CO HrHTHi-HTH <N<N<NCOCO I I I I I I I I I I (M >O (M TH rfi TH CO C^ 00 ^ 7 7 tTT ^Soi i i i I I I I I (N CO 00 iO Oi i i >O O5 CO C<1 O5 O5 i i Oi CO >O O5 CO CO I> CO CO O Oi (N t^ 00 TH _ . rj< I> (N <N O iO t^ Oi <N O 10 !> i I >O I I I I I I I I I I I I I I I t^ 00 O IO M O5 t^ CO I> t>- 00 OS I I I I Mil! OO CO C^ CO CO (N <M (N (N CO I I I I I OOOO^CO CONOCO I I I I I I I I I rJH 00 ?SS8 ^ TH TH TH TH C^ I I I I I <N (N <M CO CO I I I I I Oi IO l> 00 TH CO co^oioco t^t^ooo I I I I I I I I I TjH TH Oi &~~ CO 7 I I i I rt< 1> 00 CO CO 00 _ Tt^OOSTHCO f~-<McOOO C<l <M>OI^OTH OO-^Ot* 1 ^ THOi|>cO C^IMC^COCO COr^iOiOCO 1>-I>OOO5 I I I I I I I I I I I I I I 00 CO .._323 O5 iO O O> 00 CO iO l> OO O rH TH TH TH C^l I I I I I OTH OOI>COcOiO lOCOTtn t--O COI>C^COI> 00<N(N (MO5 O5CO-^OOOO OOcO OO500OTH OOI>COCO t^co-^t^o coi>coco O5 co O5 c^ Oi Oi co ^* oo c^i *o t^" co ^f oo ^^ co co co TH Oi t*** d(M<NCOCO COTtHiOiOCO l>t>-00 I I I I I I I I 1 I I I I I iO IO 1-1 O I I I I I (M O (N CO iO CO !> C^ CO 00 CO CO "* Tt< IO l> O5 t-H TH i-H ^H TH C<l I I I I I rH Tf t^ C^ CO Oi C^ ^^ C^l CO C^ CO C^l CO OilS-OOOiO COOOi^fb- THOO^f'^f ^COOOi-HO Oi-^Ot^CO THOit>-cO NC^C<JCOCO COrtiiOtOcO l>I>OOOi I I I I I I I I I I MM s I I 130 THERMODYNAMIC METEOROLOGY TABLE 41 THE PLANETARY CIRCULATION AND RADIATION The coefficient log c in log K 10 = log c + a log T 10 . . . (344) e 90 80 70 60 50 40 30 20 10 19000 -5.383 -5.416 -5.386 -5.380 -5.381 -5.378 -5.361 -5.387 -5.419 -5.433 18000 -5.404 -5.434 -5.408 -5.406 -5.404 -5.404 -5.387 -5.409 -5.426 -5.449 17000 -5.423 -5.445 -5.431 -5.426 -5.418 -5.424 -5.418 -5.439 -5.454 -5.465 16000 -5.436 -5.474 -5.454 -5.449 -5.444 -5.447 -5.435 -5.454 -5.469 -5.480 15000 -5.459 -5.486 -5.475 -5.468 -5.460 -5.466 -5.477 -5.463 -5.481 -5.491 14000 -5.481 -5.503 -5.496 -5.489 -5.484 -5.489 -5.490 -5.496 -5.504 -5.500 13000 -5.503 -5.522 -5.520 -5.513 -5.494 -5.504 -5.503 -5.518 -5.521 -5.510 12000 -5.526 -5.526 -5.542 -5.538 -5.510 -5.523 -5.507 -5.505 -5.507 -5.515 11000 -5.559 -5.571 -5.56'0 -5.544 -5.571 -5.540 -5.514 -5.510 -5.513 -5.518 10000 -5.578 -5.580 -5.589 -5.560 -5.547 -5.531 -5.512 -5.511 -5.513 -5.519 9000 -5.606 -5.601 -5.595 -5.589 -5.569 -5.552 -5.526 -5.511 -5.513 -5.519 8000 -5.614 -5.611 -5.596 -5.603 -5.563 -5.555 -5.524 -5.514 -5.517 -5.524 7000 -5.622 -5.618 -5.608 -5.608 -5.576 -5.556 -5.557 -5.524 -5.526 -5.531 6000 -5.628 -5.626 -5.610 -5.607 -5.577 -5.558 -5.536 -5.532 -5.537 -5.543 5000 -5.637 -5.627 -5.615 -5.615 -5.581 -5.564 -5.547 -5.531 -5.527 -5.536 4000 -5.638 -5.635 -5.622 -5.600 -5.589 -5.574 -5.534 -5.537 -5.542 -5.548 3000 -5.656 -5.646 -5.628 -5.616 -5.588 -5.561 -5.546 -5.554 -5.555 -5.561 2000 -5.664 -5.652 -5.634 -5.614 -5.597 -5.582 -5.561 -5.588 -5.546 -5.549 1000 000 The minus sign affects only the characteristic log c = 5.664. c = 4.61 X 10 ~ TABLE 42 THE EXPONENT a IN K w = c T i0 a . . . (344) z 90 80 70 60 50 40 30 20 10 19000 3.740 3.755 3.741 3.739 3.739 3.738 3.730 3.742 3.757 3.763 18000 3.750 3.763 3.752 3.750 3.750 3.749 3.742 3.752 3.764 3.770 17000 3.758 3.768 3.762 3.760 3.756 3.759 3.756 3.766 3.772 3.777 16000 3.764 3.781 3.772 3.770 3.768 3.769 3.764 3.772 3.779 3.783 15000 3.775 3.786 3.782 3.778 3.775 3.778 3.782 3.776 3.784 3.789 14000 3.784 3.794 3.791 3.788 3.785 3.788 3.788 3.791 3.796 3.793 13000 3.794 3.803 3.802 3.798 3.790 3.794 3.794 3.801 3.802 3.797 12000 3.804 3.804 3.812 3.810 3.797 3.803 3.796 3.795 3.796 3.799 11000 3.820 3.825 3.820 3.813 3.825 3.811 3.799 3.797 3.799 3.801 10000 3.828 3.829 3.833 3.820 3.814 3.807 3.798 3.798 3.799 3.801 9000 3.841 3.838 3.835 3.833 3.824 3.816 3.804 3.798 3.799 3.802 8000 3.844 3.843 3.836 3.833 3.822 3.818 3.804 3.799 3.800 3.804 7000 3.848 3.846 3.841 3.833 3.823 3.818 3.818 3.804 3.804 3.807 6000 3.850 3.850 3.842 3.831 3.828 3.819 3.809 3.807 3.810 3.812 5000 3.854 3.850 3.844 3.837 3.829 3.822 3.814 3.807 3.805 3.809 4000 3.855 3.854 3.848 3.838 3.833 3.826 3.808 3.810 3.812 3.814 3000 3.863 3.858 3.850 3.845 3.832 3.820 3.814 3 . 817 3.818 3.820 2000 3.867 3.862 3.853 3.844 3.836 3.830 3.820 3.832 3.814 3.815 1000 000 The exponent a - 3.50 in dry air; a = 4.00 in a perfect radiator. THERMODYNAMIC TABLES 131 V) d 1 S 00 00 TH t^. TH IO tO TH TH O CM CM CO CM O CO CO CO CO CO II II 1 O O TH o t> TH T-H 00 CO CO 777 i i t> CO CO CM CO 00 t^ <0 T) rH O Oi TH CO CM to i 1 CO TH TH CO CO CO CO CO CM II II 1 77 ii i TH CO t 00 b- CT. CO 00 TH O> T 1 tO TH CO TH CO TH TH CO CO CO CO CO CO CM II II 1 gggSS; 777 i i o 2 CM 00 TH CM O CO O5 Ci 1> to CM CO CO CO CM CM II II 1 TJH CO CO t* lO ggss^ 77 i i i o 2 O co w i -* 00 00 O5 to TjH CM CO CO CO CO CO CO CO CM II II 1 to 0> CD ^- CO 01 CO CO 10 OJ TH 0* 7 i ii i g CO t^ CO CM CM CM T^ TH IO Tf CO CO CO CO CO II II 1 to 0> CD i-l to O CD IO IO CO CM iH W * 1 II + 1 en W CO CM CO ^ to TH CO TH 00 CD * to til < i CO CO CO CO CO CO TH co O t- 777 i i 3 w T* to to to CO CM CO tO CO CD ^ *- 8 * 5 O CM CO CO CO CO CO CO CO CO TH CO to CO CO CO t>- !> O 00 7777 i 1. 10 CM 05 TH TH t^. o TH to oo fe w t, U CO to O5 Oi ^ co co co co co II II 1 OO CO 0^ TH CO tO to t TH CM CO CM TH TH TH Mill CO l> CM CM 00 CO CM CO TH O5 b 00 !> CO to TH CM CM CO CO CO II II 1 77777 CO CM O CM tO 00 CO CO CO CO o 8 CM CO O5 CM t^ 00 O^ ^^ CM CO CM CM CO CO CO Mill to O5 CO O5 CO TH CO O5 00 *> CO CM TH TH TH II II 1 N O5 00 t>- CO to TH CO CM TH O 132 THERMODYNAMIC METEOROLOGY CO r^ rH CO <M iO O 00 rH CO CO O5 <N T-H CD CO 00 rH T-H T-H GO !> O5 O O5 rH t>- iO CO O5 l>- CD CO O5 CO rH CO (M rH O5 O C^ to !> iO O O (M l>- CO CO t^- T-H 00 to M M 1 M M 1 Mill i M i : 1 1 1 rH (M O 00 CO GO T-H O5 CO t* Oi CO TH CN! CO s ^ g g O CO CO tO (M CO 0i 1 s ** 1 s * CO GO 00 t^ ^' co co i> e* ' 00 O 00 co oo ^ M M 1 M M 1 M M 1 MM: 1 1 1 <NTi^<N CO O O OS t- iH 00 O 00 CO 00 t-t-(NlO <N CO (N 82^SS? t2*oco8 t* CO r^ (M t^ t T-H rJH CO CO S^SS : rH CO t^ M 1 M M M 1 M M 1 MM: 1 1 1 <N rh C5 GO O t- * 1-1 tO rH Tfrl O5 CO rfl to tO <*1U3 O ^OGoScO o^ooco^ O rJH Oi (N CO 00 (M iO *tf tO 00 (N CO tO TjH 00 w & M M 1 M M 1 M M 1 MM: 1 1 1 d tO O to ^ ^O T-H XO t- 00 O* O CO (M 1> tO tOOOlO CO to 00 M M O Tt< TH T-H O CO >O ^t 1 Tf t>- oo co *-i oa oo O Tt< CO >0 (M iF2^^ : 23 1 M M 1 M M + M M 1 MM: 1 1 1 HH w 1 O5 O5 O5 CO CO to CO T-H T-H CO GO O O (N T-H <N CO l> CO CO 1- i CO O Tji Tjf T-H iO CD O t" r-i rH CO <N to IO to OO tO rfi O5 O5 CO iH rH O5 O C* CO CO TJ< <O CO O5 to CO 00 t^ Tf r-H CD g i 1 1 1 1 1111 + M M 1 MM: 1 1 1 H M O5 CO CO C^l *O ^H 10X00X0 t- IO 00 rH 00 (M l> CO O> 00 rH |>. 5 H <; CO CO O^ ^^ CO t^ 00 CO O 00 (N l> (M ^ O |> IO O T-H (N IO 00 (M to CO O5 CO OO TH to to CO O >O 00 (M CO CO 2 M i M M 1 11-11 + M M 1 MM: 1 1 1 a rH O5 IO (N 10 O5 ^^ CO C^ CO o e* co co GO - Tt< t^ l> O l> CO t 1 w i 1 Tt< CO -tf CO t^ !> GO GO CO M S 0> W CO 00 CO CO CO O> t* ^f Th CO T^ Oi ^^ O5 tO 00 CD (M T-H 10 O M M 1 1 1 1 1 + M M 1 MM: 1 1 1 1. b- co TJH co oo CO CO 00 l> t^ to <M i> co co CO ^t 1 to CO <Nt- to 1 w to to CO CO 00 c^^ooc^w rH CO 00 CO 00 -tf co TH co oo CD CD rH IO O5 rH CO C?fl 00 C<1 00 rH O5 t^ en 3 M M 1 M M 1 M M 1 MM: 1 1 1 T-H T 1 !> GO Tf 00 l> O CO 00 05 05 rH ^HT^ rH^ <N CO T-H Tj< co2o! O W T-H 00 t^ (M rf iO (M l> (N TJH (N CO O to to iO to OO o coS T^oJoo M M 1 M M 1 M M 1 MM: 1 1 1 O5 O5 CO O5 t^- rt< l> CO O5 (M <N TH 05 00505-* T-H CO TH Tji CO rH CO -# TjH t^ 00 Oi M M 1 ^^ C^l CO *-O CO CO CO CO CO (M M M i l> 00 rH (M tO CO CO l> O^ ^H 1 1 T 1 T 1 1 i C^ M M 1 ?ff!i 777 O5 00 l> CO >O ^ CO C^ rH O O5 00 l>- CD >O rj< CO <M rH O SEVERAL IMPORTANT CONCLUSIONS 133 The Case Ii of low temperatures in the isothermal region shows more rapid radiation than Case L of high temperatures, but both indicate about twice as much radiation as in the con- vectional region C. There are two special sources of heat, (1) in the upper cirrus region, and (2) at the surface of the earth, or in the lower cumulus region. The incoming radiation of the sun is divided into two nearly equal parts, the effective radiation penetrating to the surface, but diminishing in intensity in and below the cirrus region, 30 to 10 kilometers, and the returning radiation neutralizing an equal amount of the solar radiation. Pyrheliometer observations at the surface give 2.00 calories per square centimeter per minute, and the bolometer observations probably require the solar constant of 4.00 calories. Hence 2.00 calories are penetrating to the surface and 2.00 calories are returning to space. Several Important Conclusions From the formulas, g * - * = - ~ - <?0 - - , g Pio (2i -Z ) = - (Pi - Po) ~ J Pio (?i 2 - ?o 2 ) - Pio (Qi-Qo), we must lay down several propositions. (1) There is properly no such a thing as purely dynamic meteorology, which is defined as a balance between the three terms, g pio since these conditions are fulfilled in nature only under tem- porary circumstances. (2) In Mar gules' paper, "Ueber die Energie der Sturme," Jahrbuch der K. K. Central-Anstalt fur Meteorologie und Erdmagnetismus, Wien, 1903, the formulas are nearly all adiabatic, the gas coefficient R and the specific heat Cp are constant, so that the radiation term (Qi Qo) cannot be com- puted, and there is no balance possible among the other terms. (3) In V. Bjerknes' paper, " Dynamic Meteorology and 134 THERMODYNAMIC METEOROLOGY Hydrography," Carnegie Institution of Washington, D. C., 1910, the density is computed by formula (175) instead of by (176), and it is, therefore, a mixed system, since the pressure depends upon (172), and R, Cp, are taken constant, so that there is no theoretical circulation and radiation to be computed. (4) In Gold's paper, "The Isothermal Layer of the Atmo- sphere and Atmospheric Radiation," Proceedings of the Royal Society, A, Vol. 82, the assumption is made that the mass is proportional to the pressure. This omits the important terms I PIO (<?i 2 <?o 2 ), PIO (Qi Qo), involving the circulation and radiation, so that the dependent formulas are not properly sup- ported, because the adiabatic case is in reality assumed. (5) There is a very large literature in meteorology based upon attempts to make a balance of the equation, P - - P g (zi - * ) = - - - - i (?i 2 - ?o 2 ), Pio but it is in reality without substantial importance. In spite of the serious difficulty that exists in determining the (QiQo) term, it is necessary that this should be done. There are several large observatories for balloon and kite ascensions which record pressure and temperature, but not humidity and wind velocity, and these entirely fail of their purpose in advancing the interests of meteorology. (6) In spite of the fact that only thermodynamic meteor- ology can have any permanent value in science, there are yet many subordinate problems in the atmosphere which are to be studied without the heat term, such as the stream lines of the circulation, however their forces may have been developed. Ac- cordingly, we proceed, under dynamic meteorology, to deduce the general equations of motion, in the rectangular, the cylin- drical, and the polar co-ordinate systems, together with several minor terms, in order to study their application in local storms and circulations of various types. CHAPTER III The Hydrodynamics of the Atmosphere The Co-ordinate Axes THE general equations of motion will be assumed from hydrodynamics, because they are well known, and the proof is Zenith y / South x, y, z (Rectangular) accessible in many treatises, but the equations needed in meteor- ology will be deduced from them as briefly as possible. There are three systems of co-ordinates, the rectangular, the cylindrical, and the polar, as represented in the diagrams of Fig. 14. Starting at the point as the origin of co-ordinates, one can reach the point P directly along the line P, or indirectly, in rectangular co-ordinates along the distances x, y, z, in succession parallel to the axes; in cylindrical co-ordinates along the radius or, through an angle <f> counted 135 r, \ B (Polar) FIG. 14. Three systems of co- ordinate axes. Compare Fig. 2. 136 THE HYDRODYNAMICS OF THE ATMOSPHERE from an initial line of reference where = 0, and along z to P; in polar co-ordinates along a line r whose position in space is determined by the angle A in the plane x y counted from an initial line at the axis of x on that plane, and in the plane z v at the angular distance 6 from the axis of rotation z. These systems are convenient in different problems and must each be developed. The Co-ordinate Velocities If q is the velocity along the line O P with which a mass is moving, the co-ordinate velocities are as follows: Rectangular Cylindrical Polar (352) u = -jfi (353) u=^' (354) u - r^y dy d<t> d^ v = dT "*'*Tf V = rsm9 d-i dz dz dr The Co-ordinate Accelerations d t d t 2 d ' dw d*z . d z z . ' d z r The Constituents of the Force in Any Direction Force is measured by the acceleration of a mass in any direction, because it takes force to change the velocity of the THE CONSTITUENTS OF THE FORCE IN ANY DIRECTION 137 mass m at a given point. This will be taken as unity, m = 1, in the preliminary equations. A mass of gas or liquid can undergo changes in inertia, changes in volume by expansion or contrac- tion, and changes in figure by rotation, and these three types of forces must be placed in the general equations of motion. The causes that produce these forces in the atmosphere of the earth are external and internal. The external forces are due to gravitation or the potential changes with position; the internal forces are due to pressures which vary in different directions, and cause motion during the restoration to normal equilibrium. The primary source of these pressure forces is the distribution of the thermal energy derived from the solar radiation, or trans- ported in currents of circulation. The Force of Inertia This is a partial differential of the velocity in the direction of the co-ordinate axes, and is du dv dw in each of the co-ordinate systems. The Forces of Expansion or Contraction It is convenient to express the differentials of the linear dis- placements in each co-ordinate system, referred to the x, y, z system. Rectangular Cylindrical Polar (359) 3 # = 3 # (360) 3 x = 3 w. (361) 3 x = r 3 fr dy = d y> dy = ttr 30. 3 ? = r sin 03 A. 82 = 32. 32 = 32. 3^ = 3 r. Using these values of 3 x, 3 y, 3 2, we obtain for the accelera- tions due to expansion or contraction: 138 THE HYDRODYNAMICS OF THE ATMOSPHERE Rectangular , x 3# du 3w (362) u + v^- + w. dx dy 82 3fl dv dv u + v -r 1- w ^ . 80; dy 32 u Cylindrical dv 3 z> v - dw ^ dy . dz + V to- 90 dz' d~z dw d~z (364)^+,-, du 32; dw dw dw U h V ; ~ + W . rd& r sin 3 A 3 r In order to give some practical idea of the meaning of these terms, the following example is taken from the Cottage City water-spout, without explanation. TABLE 44 THE RADII IX AND VELOCITIES , v, w, IN (1) OUTER AND (2) INNER TUBES Height Radius W (1) (2) Velocity u (1) (2) Velocity v (1) (2) Velocity w (1) (2) az 90 83.3 51.9 11.52 18.49 0.77 1.98 80 84.0 52.3 - 2.00 - 3. 21 11.34 18.20 0.76 1.95 70 86.0 53.6 - 3.94 - 6. 32 10.82 17.37 0.72 1.86 60 89.6 55.8 - 5.76 - 9. 24 9.97 16.01 0.66 1.71 50 95.2 59.3 - 7.40 -11. 88 8.82 14.16 0.58 1.52 40 103.9 64.8 - 8.82 -14. 16 7.40 11.88 0.50 1.27 30 117.9 73.4 - 9.97 -16. 01 5.76 9.24 0.38 0.99 20 142.5 88.8 -10.82 -17. 37 3.94 6.32 0.26 0.68 10 200.0 124.6 -11.34 -18. 20 2.00 3.21 0.13 0.34 00 00 -11.52 -18. 49 It is seen that the velocities in meters per second in the lower half of the dumb-bell vortex of this water-spout undergo changes from the vortex tube (1) to the vortex tube (2), and that as the radius w and the height az change, these velocities change by certain laws. By taking the differences 3w, 3, 3w, 3 -or, THE FORCES OF ROTATION 139 or 3 0, 3 z in cylindrical co-ordinates, and using the mean values of u m , v m , w m , the forces that caused these changes from point to point in the vortex can be computed. The Forces of Rotation If a mass moves relatively to fixed axes without rotation there are no forces other than inertia and compression or expansion. If the mass also rotates, a new set of forces of rotation is intro- duced which may be analyzed as follows: Assume that there is a set of rectangular axes fixed in space, and that another set of 4-zo> 2 VU 1 +y<i FIG. 15. Angular velocities of motion about fixed axes. axes attached to the rotating body moves relatively to the fixed axes with the co-ordinate angular velocities wi about the axis x, o) 2 about the axis y, and ^ 3 about the axis z. The entire system of rotational velocities at the distances x, y, z, from the origin of rotation is shown on Fig. 15. The right-handed rotation is defined as that in which a right-handed screw is turned to advance while the axis x moves toward y, the axis y toward z, the axis z toward x, in transla- tion along z, x, y, respectively, and all co-ordinates, of fixed and moving axes, are so related. Thus, in rotation about z, with the angular velocity, ^ 3 , at the distance x there is instantaneous 140 THE HYDRODYNAMICS OF THE ATMOSPHERE velocity + x co 3 parallel to the axis y } and at the distance y there is instantaneous velocity y co 3 parallel to the axis x, that is, in the negative direction. The same considerations give the component linear velocities parallel to the axes. (365) Parallel to the axis x, y co 3 + z o> 2 , Parallel to the axis y, z coi + x co 3 , Parallel to the axis z, x 0)2 + y + y "i- These symbols are all arranged in the cyclic order, and are easily verified from Fig. 15. Similarly the component accelerations are found by substi- tuting u, v, Wj for x, y, z, in succession, and we have the accelera- tions parallel to the axes: (366) Parallel to the axis x, v co 3 + w co 2 , Parallel to the axis y, w wi + u co 3 , Parallel to the axis z, u o> 2 + v coi. Since these forms are entirely general, it is only necessary to substitute the special values of i, co 2 , co 3 , for given cases, to apply the formulas to particular problems. If only fixed axes are employed, we have i = 0, co 2 = 0, co 3 = 0. If cylindrical axes are employed, there is rotation about the axis z only, so u that coi = 0, co 2 = 0, co 3 = -f- . If polar co-ordinates are employed the angular velocities become i = -- , co 2 = + o> 3 = + - -. Placing these results in tabular form, Y tan u have, as can be seen by the definition of angular velocity, Rectangular Fixed Axes Cylindrical Co-ordinates Polar Co-ordinates (367) coi == (368) i = 0. (369) coi = - - |. C0 2 = 0. C0 3 = 0. co 2 = 0. V w <* = + 7- y l 0)8 + w' THE FORCES OF ROTATION 141 In cylindrical co-ordinates the angle increases with the angular velocity ^3 = ^* In polar co-ordinates the angular velocities may be illustrated by Fig. 16. Let a mass move from P to P 3 relative to the rotating earth, whose axis of rotation is Z. Lay down the fixed axes x, y, z, Pole FIG. 1 6. Angular velocities of moving axes relative to fixed at the center O, draw the radius r to P Q at the polar distance 0, in longitude A counted from an initial meridian. The angular velocity of the rotating earth is ^ 3 ; r sin 6 is the perpendicular distance of P Q from Z, and r tan 6 is the tangential distance from P Q to O Z. If the mass moves from P Q to PI it rotates about the u axis y with the angular velocity co 2 = -, since it moves in the positive direction of x; if it moves from P to P% it rotates about v z with the angular velocity o/ 3 = rzrr^J ^ ^ moves from rtan0 142 THE HYDRODYNAMICS OF THE ATMOSPHERE to P 2 it rotates about x with angular velocity coi = -, because in the motion from P to PZ the velocity is in the negative direction of rotation. If the mass moves from P to P 3 along the radius there is no rotation. These rotational relations can be rigorously proved by analytical demonstration, but as the analysis is rather complicated the reader is referred to the standard treatises by Basset, Lamb, W. Wien, and others. The formulas for the angular velocities in terms of the linear velocities are as follows : (370) Rectangular co-ordinates. 2 coi = TT . (372) Polar co-ordinates. 8 u 'dw o ^ 8 z 8#* 8 v du 2 COs 7T ~^ . 8* dy (371) Cylindrical co-ordinates. 2 co 2 8 co - 8 v iff 8</> 8 z du 8_co 8-or* 8 v du v 2 cos = ~ ^ + 2coi = 8 co 8 v r sin 6 8 A 8 r du dw 2 co 2 = ^- - -^-2 2 0) 3 = 8 v 8 u r'dB ~~ r sin 8 r tan If the linear velocities u, v, co, at the point x, y, z, are known, the angular velocities coi, co 2 , co 3 , can be computed in the three systems of co-ordinates. THE PRESSURE GRADIENTS 143 It should be observed that since, v . v z;cos0 " 3 = rlfo*' and " = TtoTo = ,- (373) 0/3 = o) 3 cos 0, so that at the pole for = 0, cos = 1, o/ 3 = co 3 , and at the equator for = 90, cos = 0, o/ 3 = 0. At the pole a point rotates about the axis of rotation with the angular velocity of the rotating earth, but at the equator it does not have angular velocity about z. The Pressure Gradients The earth in its rotation, acted upon by the gravitation of its own mass, has assumed the form of an oblate spheroid, being flattened at the poles, and if the atmosphere were not acted upon by thermal forces it would be arranged in layers of density parallel to those of the earth's solid mass. The heating of the atmosphere in the tropics by solar radiation, and secondarily in the other latitudes, disturbs these level surfaces of pressure, by lifting some areas and depressing other areas during the processes of heating, cooling, and circulation. The change in pressure from the normal pressure in a given linear distance is the pressure gradient -7-, and since the forces are directed from the higher to the lower pressure the minus sign is used, -7-. Finally, the forces are all to be reduced to the unit density so that the forms become, for the three co-ordinate systems, respectively, Rectangular Cylindrical Polar Co-ordinates Co-ordinates Co-ordinates (374) - (375) - (376) - - . 3 x p a-or pr 30 ap iap iap py p t*r< iap iap i ap p dz' p dz' p a r * 144 THE HYDRODYNAMICS OF THE ATMOSPHERE These expressions can be evaluated into many forms, which are convenient for practical computations, as will be shown in a later section. The Potential Gradient If the potential of the external forces of attraction of the earth's mass is V, then the forces due to such a potential are, 8 V 8 V In the case of the earth's mass the forces -r and -^ , dx dy on the meridians and on the parallels of latitude, respectively, are chiefly concerned with the determination of its existing figure. In the problems of meteorology these forces can be neglected, so that there remains only the vertical potential gradient, (378) -!--* There has been considerable confusion in the literature of this subject because the positive direction is taken upward by some authors, but downward by others. If the positive direction is upward, as where the positive motion along the radius is outward from the center of the earth, we have (379) ^ = + g, and V = + g z. If the positive direction is inward, as along the path of a falling body, we have, d V (380) -j^ = - g, and V = - g z. The positive direction upward is used by Ferrel, Sprung, Oberbeck, Basset in some sections, Helmholtz, Bigelow; the positive direction inward is used by Basset in some sections, Lamb, V. Bjerknes. The force of gravity is. however, modified on the rotating EQUATIONS OF CONTINUITY 145 earth by the centrifugal force, which acts only in planes per- pendicular to the axis of rotation. These forces can be resolved along and perpendicular to the axis, as follows, since the centrif- ugal force is J co 2 w 2 . The total potential is, (381) V = g r + i coo 2 w 2 , where co is the angular velocity, and co = n in the notation for the rotating earth. By (66) we have R 2 (382) g = go , and R 2 (383) 'gr = g , so that (384) V Taking the differential along the axis z, and perpendicular to it along -or, there results: dV R 2 dr -77 -+?-*;' dV R 2 dr (386) -^ = + so-^-" 2 o*r. We have: (387) r 2 = x 2 + y 2 + z 2 = -or 2 + z\ Differentiating, (388) 2rdr = 2wd'&+2zdz, and (389) Tz = 7* = 7' Hence > for * = >* (390) - -^ = g ' j = g cos e. (391) - -7 = g ' - co? 2 w = g sin - co 3 2 or. a w Y Equations of Continuity When a mass of air streams through a given space, as a cubic meter at a given place, as much air must pass out of it as enters 146 THE HYDRODYNAMICS OF THE ATMOSPHERE it, or else there will be congestion, and a change in the con- tinuity. The equations which finally control any solution of current functions must satisfy these equations of continuity. (392) Rectangular du 8_y dw = Co-ordinates, ox ~ (393) (394) Cylindrical Co-ordinates. (395) Polar f 9 ( sin 6) + f dv + ^ $ 3 (>>) = Q Co-ordinates. o 8 A 8 r Operator V 2 The sum of the second differentials in three co-ordinates is often used in analytical discussions, and the symbol V 2 has been adopted for this process. (396) Rectangular y2 = 8 2 _, 9 2 _, 8 2 Co-ordinates. ' 3*. 8/ ,8*" (397) Cylindrical y2= _?L_,J__3_ + 1 9 2 , S 2 Co-ordinates. 3^ 2 tcr 8-ar -or 2 8 ^ 2 8 2 2 ' (398) Polar y.'^J!. + ljL. + lI _*.-.,. 2* 4- + Co-ordinates. dr 2 r dr r 2 8 0* r 2 d 1 _^_ r 2 sin 2 8 A 2 ' d jTAe Toto/ Differential -r. at The symbol -7^ is often used to include the terms of the inertia and expansion or contraction. (399) A = 3- + u JL + v Ar w A <// 8 8ic 83; 82* There are a series of complicated terms used to express the internal forces caused by the stretches, the shears, the dilatation, the tractions due to elasticity and viscosity within the masses, but these will be omitted in this place. They are summarized on pages 499-501 of the Cloud Report. SUMMARY OF EQUATIONS OF MOTION 147 Summary of the Equations of Motion By putting together the terms that have now been explained, there result the general equations of motion which are at the basis of all the dynamic meteorology, omitting the heat term d Q. Rectangular Co-ordinates /,\ 18P du du du t du (400) -- = =-^T:+W +v + w a co 3 + w w 2 . p dx dt dx d y dz 1 8P dv dv dv dv - P dj=^i +u -d'x +v d-y + w d- z - w " l+ua3 - 1 8P dw dw dw dw Cylindrical Co-ordinates , . 18P8w 8w 8w du (401) -- - = + ^ + v r- + w ^ P 8zD" 3/ dv- ttdp 82 ^ "JJ 1 8P 8^ -, + o -- r- ^~ + w ^ ^ P -nr8^ 8^ 9w WO^P 80 ^"3 1 8P dw dw dw dw ^ .5 -- = - -U 11 - -\- 71 - -I- 7i -- ^ Co-ordinates 8P 8w 8w s r r g 1 8P dv dv dv dv uv t wv cot + . _ ^ _ __ _ " ''^ U ~* V dl^ W 3r ~ u 2 v 2 148 THE HYDRODYNAMICS OF THE ATMOSPHERE Equations of Motion for the Rotating Earth The equations of motion for moving axes are further modified when the axes are attached to the earth which is rotating with the constant angular velocity oj s . Cylindrical Co-ordinates on the Rotating Earth The linear velocities remain the same except that the linear velocity eastward is increased by the term w 3 .ttr cos 6. The angular velocity about z is changed by the addition of the term co 3 COS 6. (403) Linear velocities, u' = u. i) r = v+(*> 3 . IB cos e. w' = w. Angular velocities, a/i = 0. /2 = 0. 6 H . iff The partial differentials are also modified: 8 z/ 8 (v + o) 3 . ttr cos 0) dv , 97-- ^T ;-9 F + ..cos. 8 w' Substituting these terms in the general equations (400), and using (399), we have: (405) - ^ = -r! - (ojs.-ar cos0 + fl)(co 3 cos + ). p ow at \ TX J 1 dP dv / v\ ^T = -r* + u - ^ cos + u co 3 cos + ) . p wo$ at \ TX J _1_8^P dw ' p dz = dt +g ' POLAR CO-ORDINATES ON ROTATING EARTH 149 Performing the algebraic work, and substituting for the v relative angular velocity eastward v = , we find: 1 9P du v 2 du (406) -ZT- = -77 2 o> 3 cos B.v = -77 p Qfff a t tff a t f^ (2o? 3 cos B -f- v) v. i V- 1 9P dv uv dv z* 5^ = Ti + 2a> 3 cos O.u + = -T-, + p tcrc/0 d f *& a i (2 o> 3 cos -f v) w. _1_^P_^ dw 'p92 ~ dt S ' dt S ' Co-ordinates on the Rotating Earth The linear velocity eastward is increased by the term o> 3 r sin 6, and consequently the angular velocities o>i and co 3 are modified to conform with it. (407) Linear velocities, southward, u f = u. eastward, v' = v + co 3 . r sin 6. zenithward, w' = w. v + co 3 r sin (408) Angular velocities, about the axis re, a? i = - - . axis y, 0/2 = + . , v + co 3 r sin B axis 0, w i = r tan (409j The partial differentials ^ = ^--. u t o t *dv' 9 (v + o> 3 r sin 0) _ 9 F7 = 9/ = 97 " u . o) 3 cos + w . w 3 sin B. d~t = 97* 150 THE HYDRODYNAMICS OF THE ATMOSPHERE Substituting these values in the equations (400), we have, 1 8P du 1 8 P d v /v + co 3 r sin rtane w . or _av /v + cc 3 r sin 0\ 7rsin08A = ~57~ *V r J" /v + co 3 r sin 0\ M ( rung ) / -x = -77 w . (v + co 3 r sm 0) p 8 r dt r i) -f- o> 3 r sin /^ -T co 3 : \ r Performing the multiplications and reductions, 18P d u v 2 uw dt r r 2 co 3 cos . . . r co 3 2 sin cos 0. *S* a 1 8P dv UV WV 2 co 3 cos . w + 2 co 3 sin . w. . . . ^SP _ dw u* + v 2 _ p d r dt r 2 co 3 sin . v . . . r co 3 2 sin 2 0. The terms in 2 co 3 represent the deflecting forces due to the earth's rotation, which always act at right angles to the linear velocity q with which a mass is moving in any direction. They deflect a moving body to the right in the northern hemisphere, but to the left in the southern hemisphere. The deflecting force is a maximum and equal to 2 co 3 q on the horizontal plane at the poles; it is equal to zero at the equator, for all velocities in the horizontal plane. If the velocity is vertical the deflecting force due to this term is zero at the pole and equal to 2 cos w at the equator. The terms in r co 3 2 represent the forces which change the figure of the earth from a sphere to an oblate spheroid. CONNECTION BETWEEN THE GENERAL EQUATIONS 151 They need not be considered in practical meteorology, but are important in geodesy. The derivation of the general equations of motion given above is exceedingly simple and direct, showing immediately where all the terms come from which are concerned with gen- eral motions. They are equally true for the sun, the earth, and the planets, and all observed motions must conform to them. Connection Between the General Equations of Motion and the Thermal Equations of Energy We found that with changes of the temperature gradient the variation of the pressure is expressed by equation (190), when there is no change in heat Q not otherwise accounted for in this process of motion. But in case all the heat energy is not expended in motion, as where a part escapes in radiation or in internal molecular or atomic motions, as in ionization, a new term must be added to take this into the account, so that o -p (412) = -J + nCpdTt+CpTlogT.dn. In order to avoid confusion of symbols in this set of equations, take v = o>3 the angular velocity of the rotating earth, and combine the equations (411) and (412). It should be noted v that since V = : -, by Fig. 16, we have r sin 0' , ^ v v 2 cot uv cot 6 (413) = v sin 0, = v cos . v, = u cos . v, and we shall make use of them in the following transformations. In order to connect together the hydrodynamic and the ther- modynamic systems by the law of the conservation of energy, we have the same in both systems. Combining these terms P and making the reductions, substituting d x t dy, d z for the corresponding expressions of the linear displacements in polar co-ordinates, and adding d Q for the change in the heat contents, 152 THE HYDRODYNAMICS OF THE ATMOSPHERE and d J for the energy in the form of electric and magnetic forces, there result for the forces of acceleration, The General Hydrodynamic and Thermodynamic Equations of Motion , A1A . 1 dP du x uw (414) - = cos 6 (2 co 3 + v) v + o o x a t r ^ - g = -TT + cos 6 (2 w 3 + v) u + sin 9(2 w 3 + 1 QP dw Corresponding with these co-ordinate equations is the differ- ential equation, dP dq dQ dJ , dT (415) ~- = ^^ + + - = -^-- This equation has been already discussed in its vertical variations, but the more difficult task is to explain its meaning in the horizontal directions (#, y). The terms -77, -T-, -7 U / U / U / contain the inertia and the forces of expansion and contraction as expressed in equations (358)-(364), (396)-(398), and these must generally be employed in the study of tornadoes, water- spouts, hurricanes, ocean and land cyclones. The terms in / can hardly be discussed until the subjects of absorption in the spectrum, scattering in the atmosphere, electric and magnetic THE EQUATIONS FOR THE WORK OF CIRCULATION 153 forces can be more thoroughly worked out. The difficulties in determining the value of Q have already been stated. The Equations for the Work of Circulation If the equations for the force of acceleration (414) are multi- plied by d x, d y, d z, respectively, that is, if the force is multi- plied by the distance through which it acts, they give the work expended in transporting the mass from one point to another. Still retaining the unit mass, m = 1, the work-equations become, (415) -- = du-j- cos d (2 co 3 -f- v) v d x -f --- = dv -77 + cos (2 co 3 + v) udy + p at ! Q d y. 3P dz = d w -j-. sin 6 (2 oj 3 -f v) v d z p at uudz r dz We have the auxiliary equations, (416) v d x = u d y, w d x = u dz, w d y = v d z, and it is seen that by substitution all the terms cancel except the following, after using the total differential, - The integral equation is known as Bernoulli's, /cL P f* = 1 ( W 2 + Z ,2_|_ W 2)_|_ / gdz + (Q + J), one limit, (419) - = | (qi 2 - o 2 ) + g (zi - z ) + (Qi - Qo) + PIO (/i / ), two limits. 154 THE HYDRODYNAMICS OF THE ATMOSPHERE The other combination, in (414), gives, (420) - f = - fnCpdT + f Cp T log T d n. as (202). J p J J In case / J = 0, we have, in practice, for the static pressure P' and the dynamic pressure P", if P = P' + P". (421) / = g (ZIZQ) = Cpni(Ti T<?), static pressure, J p (422) - f = J (gi 2 - g 2 ) + (Qi - Qo), circulation and J p radiation, //*** .A. /"""* A \ /T* T 1 \ //"* A. /"^ t \ = w^a ~~ Lpio) (J- a J- o) n\ \Lpa ~ ^pio) (T, - r ). The equation (421) has already been discussed under Barom- etry (39) with others; also, under gradients (12) with others. The equation (422) remains to be considered under circulation and the variation of the gradients. /dP . p There are several methods of treating the computations involved in the thermodynamic and static equation (419), which considers the circulation and the vertical pressure along with the thermodynamic sources of the pressure differences. Since all the terms in (415), which represent the deflecting and the centrifugal forces, disappeared on the substitution of the values in (416), it follows that these forces are always acting at right angles to the direction of the motion of the mass moving with the velocity q in any direction relatively to the surface of the rotating earth. The deflecting and the centrifugal forces have, therefore, no effect upon the work of the circulation of the atmosphere, any more than the central forces do upon the work of the motion of a planet in its orbit. They change the direction of the motion by the composition of the forces, but it requires no additional pressure to overcome these forces at right angles to the path of motion of a current of air. rdP EVALUATION OF TERM - / - 155 J p /d P . in terms of p the temperature, eliminating the density. This can be done by substituting from (62), p ~ po P To po T Q P~ T P' Substituting and using the values of the constants in Table 3, omitting henceforward (Qi Qo), ^ ~f '--.*?<*-'> + ' $<>-* Integrating between limits for natural logarithms, (425) log Po - log P = ^^ (f-qf) + ( - so). (426) lo g Po-logP= I ^ 5 .^(f- ?0 2 ) 7991.04 T The same formulas in common logarithms become. 1 (427) log Po - log P = 1321837 (428) log Po - log P = 3^2 ' T 18400 T ' g Q (429) The constants are derived in succession: 574.067 = . 287.033 = ./ ^ 156720 = 2 g l Q . 7991.04 = 1 . 1321.837=^. 660.919 = ^ TO M 1 o 360862 = -. 18400 = . 156 THE HYDRODYNAMICS OF THE ATMOSPHERE The second term of (428) is the same as in formula (159), to which static barometry is usually confined, but the term in the velocity should be added for accurate computations. In the integrations between two points the mean temperature, JIG = T m , and the mean gravity, gi = g m , of the air column should be employed, as already explained. II. Since by the Boyle-Gay Lussac Law, we have: I _ RT P " P 1 this value can be introduced into the equation (419), so that (430) -/ -j- = ~ (f - <?o 2 ) + ^ (* - *o). This is correct because 2R = 574.067, by Table 3, and 79904' III. Since, by equations (176) and (178), we have, for n = 1, this value can be introduced into the equation (419). Then, /fi P i - = /P\AzI (T\ n Since (-5-) * = l^r) , we find again, for n = 1, V -t ' ^ ' -t 0' ^ (433) -^-= In case w is not equal to unity, and the non-adiabatic tem- perature distribution of the air is considered, we have: (434) - / ^ = j^ [i (? 2 -?o 2 ) + g (z - to) ]. IV. In case it is desired to reduce the equation of motion to a form where the standard density is unity, we shall have, p = 1, and /dP 157 P k-\ (435) = P 17 * . P~ l/k = (R, r ) v *. j " . Hence, 1-fe (436) - flj- = (1 g o r o )i/* [J (? 2 - go 2 ) + * (* - *o)l. This form eliminates the temperature and throws all the discussion upon P, being an impure form, since P can be found only by trials. CdP V. It is often desirable to integrate the term J -- with- out using the mean temperature of the column, and this can be done by determining the mean density of the column p m . Then, (437) -=- P Pm (Qi - Co), and, (438) - Pl ~ P = + Hl (Cp a - Cp 10 ) (7\ - To) - Pm i C#io (Ti - To). Numerical Check on the Two Systems of Formulas We have already found, if the humidity, gravity, and moun- tain terms are now omitted, that (439) log f = log f = 18400 2 7 67<6 ,, by (159), and that (440) log J = log |? = - n ^ (log T - log T ), by (182), so that the two methods of computation, through the mean temperature of the air column 6 and the gradient ratio n can be compared and checked. They prove by trial examples to be in agreement. Numerical Evaluations of the Pressure Gradient It is necessary to reduce the difference on the barometric pressure, at any distance apart on a horizontal level, to a 158 THE HYDRODYNAMICS OF THE ATMOSPHERE standard distance which is taken as 1 degree = 111 111 meters on the surface of the earth. G = the barometric difference for the distance 111111 meters. D Q = 1 degree or 111 111 meters on the surface of the earth. A = any distance on the surface. (441) = D 111111 DC dB dx A ?! ~ B A . Hence, (442) G = ^ (ft - 5). Lower pressure Higher pressure B dx FIG. 17. The derived gradient for the distance of I degree. The term can be evaluated in many ways. Since P = g Pm B = go p, we have: fAA\ (443) = goPm-^=S<>2- x - (444) = 1.200 G (meter) = 0.0012 G (mm.). dp dx 111 111 (445) 5f = Pm - = 0.12236 G (meter) 0.00012236 G (mm.). Hence, from these formulas and others preceding, (446) -7t^ = -7F P U *v Po - dP = _ gQ 'o dx go p m B n Td log P Po TO dx NUMERICAL EVALUATIONS OF PRESSURE GRADIENT 159 7 1 = - 7 ^" H = ~ 7 Lao G (meter) - - - 0.0012 G (mm.). p (448) -7H = ~7 p -lf = -7 - 12236 G (meters) = - - 0.000 12236 G (mm.). P T/fe Evaluation of the Ratios -j-^ and -7 . a r> ax We have, from Fig. 17, (450) -j = y^ = jr = jj (for the top of the homogeneous atmosphere). dh_ h _Do Pi I I 7991.04 d B ~ B Q - B ~ G Di ~ Bi - BD t ~ B t ~ 0.760 10514.5. The pressure B = at the top of the homogeneous atmo- sphere. BI is the pressure of the atmosphere 0.760 m., and / = R T = the height of the homogeneous atmosphere 7991.04. For the ratio -7 , we obtain, by (449) and (451), a x dh _d_h L Bo- B __ dh^ t G_ _ 10514.5 ' dx~dB' D ~dB'D<>~niin 0.09463 G. This makes it possible to compute approximately the height of the required isobar above the surface, at the horizontal distance from the place of 1 degree = 111 111 meters. To Find the Difference of Pressure (Bi B) at the Distance A that will just balance the Eastward Velocity v The eastward velocity of the general circulation v produces a pressure directed southward along the meridian, and it is required 160 THE HYDRODYNAMICS OF THE ATMOSPHERE to find the difference of pressure in the meridian, higher to the south in the northern hemisphere, that will keep the velocity of motion directed exactly eastward. By equation (414), for steady motion, -77 = 0, and neglecting the term in w, we have for v in meters per second, (453) ; = cos 6 (2 co 3 +v ) v. p d oo Since d P = g p d k, this becomes: (454) g -T- = cos d (2 co 3 + v) v. d oc By equations (452) and (442), f . 10514.5 D (Bi-B) , N (455) g . - -~ - = cose (2 co 3 + v) v. Hence, (456) Bi-B For any temperature and pressure other than the standard, T = 273 and B = 760 mm., if we take A = D Q = 111 111 111, so that BI = B Q , that is, DI = the pressure at a distance of 1 degree southward on the meridian, we have, for g = go, B " " B = Wo cos e (2 The angular velocity of the rotating earth is 2 co 3 = 0.0001458 and if v is neglected, we find, T> (458) B Q - B = 0.05644 ^ v cos (in millimeters). It should be learned from this example how to apply the general equations of motion on the rotating earth to special cases, by making the proper limitations in the use of the terms. The Angular Velocity of the Earth's Rotation, a> 3 Allowing for the sidereal time the angular velocity is, o 2 V ^ 141 'iQ {(23X60) +56} 60 660 LINEAR, ABSOLUTE AND RELATIVE VELOCITIES 161 (460) 2 co 3 = 0.000145846. The Linear, Absolute and Relative Velocities The absolute linear velocity at any latitude is (461) v' = r o> 3 sin at the distance r = R -f- h from the center of the earth, where R = 6370191 meters of 208996600 feet, and h is the height in the atmosphere above the surface. The relative linear velocity of a body moving eastward over the surface of the earth is (462) v = r v sin 0. Hence, , Aao . v' 2v f cos 2v'cot0 (463) co 3 = r - , and 2 co 3 cos = ' ,._.* (464) (465) , r sin ' r sin z> cos v cot It follows that formulas (414) can be written in another form, remembering that co = n for the angular velocity, 18P du cot , w (466) ---- = --- r (2 9*4- 9 ) v + -> r . 3^ f/ ^ r r rf z; cot , .w ay= 9P In some respects the fourth form of polar equations offers distinct advantages in practical computations, especially by the aid of a few auxiliary tables, such as Tables 104, 105, 106 of the Cloud Report. 162 THE HYDRODYNAMICS OF THE ATMOSPHERE Evaluation of the Barometric Gradients in the Fourth Form of the Polar Equations of Motion From equations (446), (5), and (449), we have: 1 8P 1 BoT G x ( ^ 7) 7 9^ == PO ~B fi ' * Pm ' Do" , , _!_ 8P _ 760 X 9.806 X 13595.8 T 1 ' P 9# " 1.29305 X 273 X 111 111 111 B x = 0.0025833 ^G x . > d u For steady motion, -7-7=0, and neglecting the term in w, a t in (466) (469) - 0.0025833 ^ G x = - ^^ (2 v' + ) Hence, r> K (470) G, = 387.1 1 ~ (2 ^ + ) . G z = 387.1 (2 '+)r + - *. The gradients are, in millimeters of mercury, G x = (Bi B) X) along the meridian, G y = (Bi - B) y , along the parallel of latitude, G z = (Bi - B\ = along the vertical. As an example of the computation take, The north polar distance, = 90 - = 30, The radius of the earth, R = 6370191, B = 700 mm., T = 260 C. The angular velocity, 2 co 3 = 0.00014584, - The eastward velocity of the earth at d, v r = Rn sin 0, The relative eastward velocity over the earth, v = 40 meters per sec. EVALUATION OF BAROMETRIC GRADIENTS 163 Logarithms By Tables 104 and 106 387.1 2.58782 v' ^ cot 0.2v 0.005052 700 2.84510 cot . v v 0.000435 cot 30 0.23856 0.005487 (2 i/ +v)v 4.30492 Logarithms 9.97640 cot0.v [-j 4- ) 7.73933 378.1 2.58782 r 2.41497 B 2.84510 R 6.80415 3.17225 0.75728 9.21912 T 2.41497 5.717 0.75728 Logarithms R = 6370191 6.80415 2 u>3 = 0.000145846.16388-10 sin = sin 30 9.69897 2v' = 464.5 2.66700 v = 40.0 2 v' + v = 504 . 5 (2v f + v)v = 2018.0 C x = (Bi - B) x = 5.717 Application of the General Equations of Motion to the Local Circula- tions in the Earth's Atmosphere The Local Circulations The circulations of the earth's atmosphere can be con- veniently analyzed under two classes: the first, or general circulation, including the large movements that are primarily related to the axis of the earth as the line of reference for the angular motion, and the second, or local circulation, including the minor movements that are referred to axes which are wander- ing over the surface of the earth. The general circulation takes account of the great polar whirls covering the entire hemisphere, one north and one south of the equator, including the trade winds in the tropics, the eastward drifts in the temperate zones, and the minor circulations near the poles. These great zonal currents break up into localized circulations, as determined by the ocean and land areas, which constitute the first disintegration of the general circulation into smaller circulations. The true local circulations are commonly known as cyclones, anticyclones, hurricanes, tornadoes, and waterspouts, and these are referred to axes which move over the earth in paths that are determined by their relation to the breaks in the normal general circulation. Finally, there are very numerous minor whirls, as eddies, small vortices of many types, which constitute the effectual internal friction through the operation of the law of inertia in the moving masses. The analysis of these motions, by the application of the laws of hydrodynamics and thermodynamics given in this chapter, determines the principal problems in theoretical meteor- 164 THE HYDRODYNAMICS OF THE ATMOSPHERE ology. It is proposed to set forth the main features of this subject with sufficient fulness to guide other students to the problems of the research which are pressing for solution. Discussion of the Cylindrical Equations of Motion The cylindrical equations of motion are most convenient for application to the discussion of the phenomena of the local circulations in cyclones, hurricanes, and tornadoes. There have been several attempts to adapt these equations to the observed data, and the two best-known systems, that of Ferrel and that of the German School of Meteorologists, will be briefly mentioned before taking up the form of vortex that I have been led to adopt in my researches. It will be desirable, at the outset, to assume that the motion is symmetrical about the z-axis in cylindrical co-ordinates, and that the isobars are centered as circles upon this axis, though this is not the case in nature, except for waterspouts, tornadoes, and hurricanes. The pure vortex law does not apply directly to cyclones and anticyclones, and the disturbing terms which make the transition between pure and impure vortices can be studied only by comparing the pure vortex underlying a cyclone with the data obtained by observa- tion. It has been difficult for meteorologists to do this, because the actual conditions in the free air above the surface are found only indirectly by computation, or in an inadequate manner by occasional ascensions with kites and balloons. In recent years, however, enough data have been accumulated to make it possible to advance these studies in the right direction. It will greatly assist those who are engaged in the study of the atmosphere above the ground, if, in planning and executing the observations, the fundamental principles upon which the actual motions must depend are clearly understood. Ferret's Local Cyclone If the vortex is assumed to be symmetrical about the z-axis, and the friction terms k u, k v can be neglected, the second FERREL'S LOCAL CYCLONE 165 equation in the second form of cylindrical equations (406) be- comes: (471) - + (2 co 3 cos 6 + v) u = 0. w -7- d t v dw Substituting, v = , u = -7-, and multiplying by tcr, this " becomes, ozo' d v (472) 2 co 3 cos 0. zrr -7- + tzr -y- + = 0. Integrating, we have for each particle in gyration: (473) ttr 2 co 3 cos 6 + ttr y = -or 2 (o> 3 cos + v) = c. fed initial value of v, m (474) Take C = - for the entire gyrating mass, if Vo is the f or 2 (w3 cos + v) dm (475) C = - - = I w 2 (co 3 cos + Vo). 7W If the initial gyration is zero, v = 0, and we have: (476) -or 2 ( co 3 cos + ~ = ^o 2 co 3 cos ^. This is equal to the moment of inertia of the whole mass at the distance \ tffo. We obtain: (477) = ~^- 2 o) 3 cos 6 co 3 cos 0, and / 7K 2 \ v = \^~~ 2 l) ^co 3 cos0, the tangential velocity at to- . If the tangential velocity vanishes, v = 0, for co = R, we have: (478) R 2 = ^, and R = 0.707 If a cylinder with the radius W is drawn around the gyrating mass, then at the distance 0.707 ttr the velocity is zero; inside this R the fluid rotates in one direction and outside of it the rotation is in the opposite direction. The general circulation of 166 THE HYDRODYNAMICS OF THE ATMOSPHERE the atmosphere is arranged to operate in this way, the air north of the latitude 35 16' rotating eastward, and the air south of this parallel rotating westward. If water be enclosed in a cylindrical vessel, and an upward current be formed in the center, by heating the lower surface, or by a wheel operating in the water to raise a central column, it will circulate in this way. The water will gyrate about the central axis as it rises, till at the distance 0.707 or from the center the gyration ceases, while beyond this radius the gyration reverses and the water descends in a ring bounded on the outside by the vessel. Ferrel supposed that this type of vortex would represent the local cyclone as well as the general circulation, but this is not the case. In the general circulation there is a fixed mass in gyration, as there is in the cylindrical vessel of the experiment, but observations show that the cyclone, hurricane, and tornado are constructed by means of another type of vortex in which the mass is continuously changing. Other mechanical difficulties are mentioned in the Cloud Report, 1898. The German Local Cyclone In the second form of cylindrical equations (406), the second equation becomes: (479) fi + ** + **.+-**--(*, where /I = 2 co 3 cos 9 and k is the coefficient of friction. This equation has two solutions, and the vortex has been divided into an inner and an outer part to correspond with them. Thus First Solution Second Solution (Inner Part) (Outer Part) (480) Radial velocity = - - or. u = . "ZET A c * c (481) Tangential velocity v = 7 ; -U . v = -r . K C A K> to (482) Vertical velocity w = c z. w = 0. THE GERMAN LOCAL CYCLONE 167 The constant c depends upon the dimensions of the vortex, and must be determined by observations. These values of u, v, w, can be readily verified by substituting in equation (479). If we take the following current functions, called the Stokes functions, and the vortex law, (484) w = + - (485) v W = $ there results for the two solutions, First Solution Second Solution (Inner Part) (Outer Part) (486) Current c i-i A '/'i = T* tCT^ 2 \l/\ = -f- C 2. Function. ! 2 (487) Radial 1 8^1 c 1 9^i c Velocity. u ~ " -or 3z ~ ~ ~2' w " ~ttr8lT = ~m" (488) Vertical iri v W = + 7^-=- = + C Z. Velocity. -or 9w (489) Vortex I c Law. ~~ k c 2 (490) V -c 2* A; 0* It is seen that two forms of the current function are required to satisfy the general vortex law which will be deduced later. Even if the constants k and c could be determined the solution is not consistent for the entire vortex in either the inner or the outer part. There is in nature no such division of the vortex, that is, there is no outward part without vertical velocity, as compared with an inward part having vertical velocity, w = + c z. It is for this reason that the application of these formulas to the cyclone has not been successful. The cyclone is constructed upon quite different principles. The solutions of the second equation of motion can be satisfied by yet other values, which 168 THE HYDRODYNAMICS OF THE ATMOSPHERE give a consistent current function for all values of the currents and the vortex law. The General Equation of Cylindrical Vortices If the definition of a cylinder be extended to include any figure of revolution formed by rotating any line as a generatrix about an axis, the vortices of meteorology can be designated as cylindrical or columnar vortices, the axis being approximately vertical in direction. The cylindrical equations of motion will therefore be adopted, and they are transformed in the following manner. In discussing problems in vortex motion, it is con- venient to use the current function ^, which is deduced from the equation of continuity (394) : This may be put into another form, (493) J- -- (w) + ^ - 0. ttr v This is satisfied by substituting the velocities, (494) =--p, ZD- 8s 1 8^ (495) *=+-^, which are known as Stokes's functions. In order that the equation of continuity may satisfy the second equation of motion, assuming steady motion and = 0, u t this becomes, from (40l) 2 , with no deflecting force and no friction, , . dv dv uv (496) w+w +--=0, dur 90 nr and it is sufficient to make (497) , = , GENERAL EQUATION OF CYLINDRICAL VORTICES 169 so that vor = ^ = constant is the usual vortex law, or in its most general form, v ttr = a . \J/, where the \l/ of the Stokes func- tions is made to cover the vortex law by the constant factor a. By differentiation, 9w vs w w (499) |?-^t*. dz & 3 z Substituting these values in (496), it becomes: .j ,t _ t w 2 aw 9 z """ w 82 y w aw a 3 w 3 a 3 ^ ' If the equation (496) is multiplied by zcr, since ^ = 0, it can be written, (501) uv -f- ztr- + wttr - -f- ^ y = 0. and, therefore, otir 02 oz . , in the form, p. c\ (502) u (ttrz;) + w (tcrz;) = 0. This shows that tffv = \f/ = constant is a solution of the equation of continuity. Any function of \l/ which satisfies this equation will be a solution of the second equation of motion. Inasmuch as there are several such values of ^ known, it is only necessary to choose the one which is in harmony with the observed phenomena in the earth's atmosphere in order to obtain the solution of the motions found in cyclones, hurricanes, and tornadoes, or waterspouts. Hence an arbitrary function of ^, (503) wz> =/(*), is a solution of the second equation of motion. The potential and the pressure terms in the first and third equations can be eliminated by the following process. From (401) in the case of symmetry about the 2-axis, 8V dP du . du du v* -- 8V 8P dw dw dw 170 THE HYDRODYNAMICS OF THE ATMOSPHERE Differentiate the first equation to 3 z, the second to 3 w, and subtract with the result, , v __ _ _ = at \d~z ~ dv) + d^ z 9 3 w\ 3 / fl 2 \ ary dm dz We derive the following auxiliary differentiations: _- _ - " " - _ _ JL 82 "9w" " w v z Since v w = / (\l/), v 2 w 2 = [f (^)] 2 , = L>/ vy/J ? we have, W "*" dz zcr w Making these substitutions in (505), we obtain, + ' T ^ _9 2 fU) 9 /^ ia^.a 2 ^\~i az vaw 2 "' Any function of ^ satisfying this equation is capable of giving a vortex motion. In the application to the atmosphere some simple forms will be considered and illustrated by examples. The first form, (511) t = Aw 2 z, gives a funnel-shaped vortex, and the second form, (512) ^ = A w 2 sin a z, gives a dumb-bell-shaped vortex. These are the common ones GENERAL EQUATION OF CYLINDRICAL VORTICES 171 in the atmosphere, as will be shown by the observations. Un- fortunately the motions under the complex local forces that generate storms do not often produce pure vortex motion, but it is the province of meteorology to consider the perturbations as observed and to give an account of their causes. The Angular Velocity By formula (371), the angular velocity is, and if an arbitrary function of ^ is taken, it follows that, By differentiations it follows that f ^ R \ 8 (516) - Pressure Since by formula (418), omitting Q and 7, (518) - - = i ( W 2 + ^2 + ^2) + g z> it follows, by using Stokes's functions, that, /KinN P X f/ 8 ^\ 2 , / r9 *\ 2 n , i a2 ^ 2 (519) 7 ".? Ks) + (a"J J + * -iT for one limit. The difference of pressure between two points, designated by n and n + 1, becomes, by using the mean density 172 THE HYDRODYNAMICS OF THE ATMOSPHERE (520) P P . -f- \ -1 fY^V + /^ (520) P n+1 -P n - 2 -( _, /i\ /a\ 2 UwJ. ~ U 2 J.+ The Application of the Vortex Formulas to the Funnel-shaped Tube Employing formulas (494), (495), (497), (503), we readily obtain the following group of relations, r w Current ^ ' ^ " "2 ' function. (w>\ r JL JL ^L _^ Vortex = Tff*z = vz ' iff 2z constant. 1 8i \L> ww v Radial ( 523 ) u = 4- = C tcr = = = . , -or 82; ttrz 222 velocity. . . \1/ \1/ w Tangential (524) v = =Ciff z = =- -& = uz. , .. -or -or 2 velocity. (525) 2 ttr Vertical velocity. The Application of the Vortex Formulas to the Dumb-Bell-Shaped Tube (526) ^ = A TB* sin az. /rr _ N 18^ A aw 2 cos #3 (527) w= ^ = - - = A a TX cos az. TX d z iff / C oo\ a $ A aw 2 sin az (528) v = = + - - = + A a iff sin az. TX Tff /__ N 1 8 ^ 2 A ttr sin (529) * = APPLICATION OF VORTEX FORMULAS 173 The Total Pressure For the funnel-shaped tube, omitting the expansion terms, (530) - - = J (u 2 + v 2 + w 2 ) + gz = | C 2 (ttr 2 + w 2 z 2 + 4s 2 ) + g 2 -f- constant. For the dumb-bell-shaped tube, p (531) -- = (A 2 a 2 m 2 cos 2 az + ,4 2 a 2 tcr2 s in 2 az + 4 yl 2 sin 2 az) + gz + constant. p (532) - = A 2 a 2 iff 2 +2A 2 sin 2 as + g z + constant. p (533) - = A 2 a 2 tcr 2 + 4 2 (1 '- cos 2az) + gz + constant. It should be noted that the signs of the Stokes functions have been taken opposite to one another in the funnel-shaped and the dumb-bell-shaped vortices. This is because it is more convenient to place the plane of reference for the funnel-shaped vortex at the base of the cloud from which it is developed, with the positive direction of the z-axis downward, while in the dumb-bell-shaped vortex the first plane of reference is taken at or below the surface of the sea or ground, and the positive direction of the z-axis is upward to the second plane of reference. These will be explained further by diagrams and examples. The Relations Between Successive Vortex Tubes A vortex is so constructed that a section through it perpendi- cular to the 2-axis at any height, z or az } cuts off a series of rings so regulated in size that the successive radii stand in a constant ratio to each other. Take this ratio, (534) P = ~, and (535) 174 THE HYDRODYNAMICS OF THE ATMOSPHERE If tff n is the radius of the outer ring Wi, and or 2j or 3 , zcr 4 ... of the successive rings inward, then, / KQA X 1 2 3 (536) p= _ = _ = _ This constant ratio p plays a very important part in the computation of these vortices, and it is found that we can pass from one value of the radius and the velocities to the w others in succession by employing the following formulas: (535) Ratio of the radii. log p = log -^-. Wn+l (536) Vortex constant, log C n = log d + 2 n log p. (537) Radii of rings, log w n = log^i n log p. (538) Radial velocity, log u n = log HI + n log p. (539) Tangential velocity, log v n = log Ui + n log p. (540) Vertical velocity, log w n = logwi + 2n log p. (541) Horizontal angle i, log tan i = constant. (542) Vertical angle >?, log tan y = log tan ^ + w log p. (543) Time of one rotation /, log / = log ti 2 n log p. (544) Volume through rings, V = TT ( 2 ttr n ttr 2 n+ i) w m = constant. (545) Centrifugal force, log (-) = log (-) -f 3 n log p. \w/.n Vw/i (546) Barometric pressure, logEj = log - + log D n -\ B n Wn+i (547) Total velocity g, # = (w 2 -f v* + ^ 2 ) J = z; sec i sec w. The relations shown by these formulas will be made clearer by a diagram giving the connection between the angles and the velocities. (548) tani-. (549) (550) tan y = a sin tfi - .. v sec^ RELATIONS BETWEEN SUCCESSIVE TUBES 175 ttr, = the cylindrical co-ordinates of a point on the x y plane q (u, v, w) = the co-ordinates of the velocity at the point (zEr,0,z), ff (u } v) = the component of q on the horizontal plane; i = the FIG. 1 8. The relations of the angles and velocities in the formulas. angle from v to <r, positive outward from the tangent; y = the angle from <r to q, positive upwards. The Second Form of the Cylindrical Equations of Motion (406) in Terms of the Current Function \f/ The equations of motion for cylindrical vortices can be readily transformed into terms depending upon the current function ^. Writing equations (406) in their full forms, they become, , fn . (552) du , du , du v 2 + u - h w^- --- 2 o> 3 cos 6 . v + k u. pottr ot out oz iff 1 dP dv 1 dP dw du> dv uv dw It is only necessary to develop the differential terms from the velocities given in (523)-(525) and (527)-529). (553) For the Funnel-Shaped Vortex u = C iff. v = C iff z. w - 2Cz. dv = C z. dz 176 THE HYDRODYNAMICS OF THE ATMOSPHERE 8w 8w w 2 u = C 2 Tff . O'Ztf w = 0. 82 - = C 2 W 2 2 . -or u 1) 8y z> u fa =0ws . w; ;r- = 2 C 2 ttr 82 z. = C 2 or 3. -or dw u = 0. 8w w = 4 C 2 2. (554) Hence the second form of the cylindrical equations of motion becomes, for the funnel-shaped vortex, (555) - - = + C 2 w- C 2 z 2 -os - 2 tcr 3 cos . v + k u. 8P ^~ ~~ ' == ~~^ ~i~ 4 O 2 "T* ^ r~ k IV . p u Z 9 / Multiply the equations (555), respectively, by 8 -or, -or 8$, and 82, and integrate for the total pressure, and we obtain, omitting the friction terms, (556) - f = = J (w 2 +z; 2 +w 2 )-f J C 2 ttr 2 - J C 2 2 2 ^ 2 + / p p Substituting the values of u\ v 2 i, r w i \ this becomes : p (557) = C 2 or, 2 + 4 C 2 2 2 + g 2, at the point (ttr, 2). The difference of pressure between two points (&i z) n and w+1 may be expressed, rt = (C 2 w 2 ) n +1 - (C 2 w 2 ) n + 4 [(C 2 2 2 ) n + i- (C 2 s 2 )J + Pm It has been customary in meteorology to use the formula (530) as an expression for the total pressure integral, but it is evident that (556) and (557) are the complete forms for the funnel-shaped vortex. If the inertia terms are omitted the formula becomes, without friction, (558) -- = |C 2 *r 2 -K 2 * 2 ^ 2 FOR THE FUNNEL-SHAPED VORTEX 177 Hence, we can summarize the result for : p (556) includes the inertia and the expansion terms; (519) contains the inertia, but omits the expansion; (558) omits the inertia but contains the expansion. For the Dumb-Bell-Shaped Vortex From the equations (527)-(529) we have by differentiation and substitution, using \J/ = A ttr 2 sin az, u = A a or cos az. (559) (560) = A a cos az. du = A a 2 w sin az. oz v = A a -or sin az. = A a sin az. oz cosaz. w = z A sin az. dw = 2 A a cos az. o z u ~ = A z a z tff cos 2 az. dv u = A dw ;^- = 2 A 2 a? iff sin 2 oz. O ^- oz dw -^- =4A 2 asm az cos az. dz = A 2 a 2 -or sin 2 or wz ., 2 = A a W sm az cos <zz. or Substitute these values in the general equations (552) : 1 dP (561) otcr = ^-. -f A 2 a 2 tff 2 -or 3 cos . v w. 1 9P 3w - = ^ + 2^4 2 . 2 sin a z cos a z. a + F + kw. p o z o t 178 THE HYDRODYNAMICS OF THE ATMOSPEEfcE Multiply by 3 ztr, iff 3 <, 3 z, respectively, omit the &-terms, and integrate for the total pressure, remembering that, (562) 2J sin az cos a 8 az = sin 2 az, (563) - - = \(tf + i; 2 + w 2 ) + ^ 2 a 2 to-2 + 2,4 2 sin 2 az + g z. p The term for the velocity square is, (564) i (w 2 + ^ + w 2 ) = | ^ 2 a 2 w 2 + 2 ^ 2 sin 2 az, so that the total pressure with the inertia and expansion becomes: p (565) = A 2 a 2 iff* + 4 A 2 sin 2 az + gz, for the point (of z). p The total pressure without the expansion is: (566) - - = A 2 a? w 2 + 2 A 2 sin 2 az + gz. p The total pressure without the inertia is : (567) -- = |^ 2 a 2 ^ 2 + 2A 2 sm 2 az + gz. It is apparent that in the dumb-bell-shaped vortex the pressure p p difference , required to overcome the inertia resist- Pm ance, is the same as that which is needed to overcome the resist- ance to expansion. The Deflecting Force The terms in 0, the polar distance, 2 w 3 cos . v and + 2 co 3 cos 6 . u, disappear from the equation of total pressure in the summation because we have, (568) vdiff = urn d(f>, just as in the rectangular co-ordinates, (569) v d x = u d y, which shows that the deflecting force is at right angles to the ifi uv direction of motion. The centrifugal force , , being at right angles to the direction of motion and induced by the velocities w, v y together with the deflecting force, has no influence upon THE DEFLECTING FORCE 179 the circulation except to change the direction without producing acceleration. In the same way a planet falls toward a body ex- erting central force, and thus moves in an orbit about it, but the velocity in the orbit is not changed by these forces acting at right angles to the direction of motion. The Force of Friction The viscous friction in the atmosphere is a very small quan- tity, and k would be a small coefficient were it not that in all large movements of the air there are numerous small vortices produced within and carried along in the great current. These minor whirls have a strong force of resistance and they are largely concerned in frittering down the energy of motion contained in the large current. It is customary to take the term expressing friction proportional to the velocity, k u, k v, k w. This is a subject that has not been satisfactorily cleared up, and it will require much careful research. There can be no doubt that k is a variable coefficient, and differs widely in tornadoes passing over a city or rough country from that in a cyclone over an ocean area. It is not certain that the velocity enters the equations as the simple first power, but that remains to be determined. The resistance due to the friction, whatever function may be found to express it, acts along the line of motion to retard the velocity, so that the pressure difference must increase to overcome this type of resistance. We may write the final equation for all the terms, when the pressure-difference between two points is required, using (v t z) n + i and (w, z ) n , p _n+i n+i n (570)-- =^ 2 a 2 ^ 2 + 4^ 2 sin 2 0z Pm J n J n _J n The mean density p m along the path between the two points must be used, and in general the mean conditions of all the terms along the path of the integration must be carefully con- sidered. 180 THE HYDRODYNAMICS OF THE ATMOSPHERE The Transformation of Energy in the Circulation of the Atmosphere The circulation of the atmosphere is the process of the trans- formation of energy, the transportation of warm and cold masses of air from one place to another being the evidence that a dis- turbed thermal condition is seeking its normal equilibrium. These currents are so complex that at present there is no possi- bility of working out a comprehensive system of equilibrium. The direction and velocity of the currents in all levels and in all latitudes and the temperatures of the masses, must be worked out by numerous observations before that can be undertaken. All the integrations heretofore proposed assume that a nearly perfect vortex law can be laid at the base of the discussion of the general and the local circulations, but as the vortices on the hemisphere and in the cyclones are very imperfect a more com- plicated treatment is necessary. At present it is possible to lay down only some isolated, detached propositions which con- tribute to the ultimate solution of the problems of atmospheric circulation. The following discussions merely introduce a subject of great value, which is capable of unlimited development. CASE I. The Change of Position of the Layers in a Column of Air When a layer of air in a column is not at the temperature which belongs to its elevation it must move upward or down- ward in order to gain a position of equilibrium, upward if too warm, and downward if too cold for its place. This occurs when a cold sheet overruns a warm layer, when there will be an inter- change of position in certain streams, which may have a vortical structure more or less fully developed. The following proposi- tions take no account of the form of the current lines, but they explain the amount of energy that can be transformed into a velocity q. The chief imperfection in these propositions consists in the omission of the powerful heat terms (Q\ QQ). From the equations (196) to (199), we find, (571) - - = g fe- Pio THE CHANGE OF POSITION OF LAYERS 181 so that the velocity equation becomes, for the mass M, (572) J (q? - ? 2 ) M = -g( Zl -Zo)M- (! - n ) Cpu (7\ - T ) M M-(Q 1 -Q )M. The evaluation of the term (HI n Q ) Cpi (Ti TO) m is difficult, because the moment a mass of air moves up or down, it at the first instant has an adiabatic gradient, w = 1, of ex- pansion or contraction, which sets up a minor circulation within the mass whose gradient is HI, so that this internal circulation cannot be followed, and it is necessary to treat it as a resultant mass whose general gradient is n\. We, therefore, omit this clM 2 pfc fnitial Final FIG. 19. Change of position of the layers in a column of air. term, also the initial velocity g 2 and the initial height z for convenience, and have for the kinetic energy, for several masses, (573) I m f 2 [ n\ Cpio (Ti T ) m g z m]. These terms must be evaluated and substituted in the general formula. (Compare Margules' "Energie der Stiirme.") Change of Position of the Layers in a Column of Air Suppose that the thin layer mi at the height Zi, pressure Pi, temperature Ti, is too warm for its place, but that it must rise to the height 2 to be in equilibrium, while a column M z of height h falls through a short distance. The mass Mo is not affected 182 THE HYDRODYNAMICS OF THE ATMOSPHERE while the mass M h above z 2 falls as in a piston without changing the pressure or temperature. The changes in the mass M 2 must be integrated through the layers, M 2 = / d M 2 . In the /o exchanges the pressure of mi changes from PI to PI, and the temperature from TI to 7Y; the pressure of the differential layer d M 2 changes from P 2 to P 2 ', and the temperature from T 2 to Tz, while the height h changes slightly as the large mass con- tracts in falling. Layer (574) Initial Pi' = Final g mi 2Y = To evaluate T 2 we have ,/ fc-l /r~r\ rp , rp /* 2 \ IT rp I + . 6_lllf A n k T (575J 1 2 /i(-"l*" = ^2 I 1 +^ WA! = ^ 2 _ " l2 R wyfe Substitute these values in equation (573). /Vi-r/^N 1 . ^2 * {~* J\ \oi\)j ~2 mi c[ = ni ^P] nCpP 2 ' (577) | wi g 2 = % Cp w since in the d M2-term, (578) m CAO (r, - r 2 -/g )"* J wi - g (2 - f dM 2 r l -- =- gmi) dz=-gmih. / p 2 * The two gravity terms in (576) nearly disappear by the summation. The available kinetic energy ^ mi q 2 caused by displacing a thin layer by a thick layer can be computed in this way, but there is no account given of the form of the currents produced by the transformation, nor of the energy lost in the EVALUATION IN TEMPERATURE CHANGES 183 small internal vortices with the accompanying inertia and friction, nor the energy lost by radiation. The Evaluation of JT dm in Linear Vertical Temperature Changes Since the integration of the term JT dm may frequently occur for a simple linear vertical gradient, it is proper to secure the general auxiliary theorem that will express this term when the temperature is defined by (579) T = To - a z. If a is not constant, as is seldom the case except for short vertical distances, then another solution will be required. We have to evaluate, for T = T Q a z, (580) $Tdm=$T P dz. It is convenient to have before us the equivalents, P /T\ nCp /T\ /r\ * /T\ 8/Ra By substitutions, we find, r z 1 f* 1 r z t T \ g/Ra (582) -g/Ra g/Ra T dz. o Change the limits of integration from z to T. Since, jrrt (583) T = To - a z, dT = - adz, - ~- = dz, we have: (584) T P dz = --Po T,- ga T 8a d T = - P T ~ g/Ra r jK# 1 F T g/Ra +1 Tg/Ra+l^\ a+ll 10 ]' g/Ra+l T (P T -PT). Hence, g + Ra (586) fldm = CT pdz = r : (P T - PT), /n Jo . K 1 t^TI 184 THE HYDRODYNAMICS OF THE ATMOSPHERE The difference of the products of the pressure and the tem- perature at two points, multiplied by the coefficient depending on the w-coefficient of the gradient of the temperature between them, is the integral of this term. It is, however, much simpler to integrate by means of T and p for the stratum (zi 2 ), (587) fT dm** fr p d z = Tio Pw Oi - * ), /0 /0 which gives close approximate values. CASE II. Effect of an Adiabatic Expansion or Contraction in a N on- Adiabatic Temperature Gradient Since a mass in moving from one level to another level in the atmosphere begins to change adiabatically, while the pre- vailing temperature gradient is non-adiabatic, it becomes desirable to define the relation of these facts to the velocities which are immediately set up in the mixing medium. The equation (573) is to be evaluated under adiabatic conditions, by which it becomes, (588) i fin g 2 = Cp From = ' we have ' f r r * = Tl " a h ' From the binomial theorem, we have the formula, 1 (590) (l- X ) n =l- nx* --, so that (591) i m, q* = Cp ^-T, (1 -m + n* . | - 2 EFFECT OF EXPANSION OR CONTRACTION 185 (592) 1 (593) The mass mi is driven from its position with a velocity energy inversely proportional to the temperature, so that warm air has less driving power than cold air. The drive depends upon the departure-ratio n and vanishes when n = 1, that is, for adiabatic expansion in an adiabatic gradient. When a > do the mass m : is in unstable equilibrium, that is, too cold for its position and tends to fall. Example, for n = 0.5, a = 19.747, a = 9.87. When a < a Q the mass mi is in stable equilibrium. Example, for n = 2, a = 4.94 <0 = 9.87. It is not possible to drive the small mass mi through any great height in the atmosphere, because the differential energy of the expanding mass sets up minor whirls which tend to interchange the heat energy by mechanical effects and internal friction and radiation. The result is to change the gradient from a Q to. If the displace- ment of the mass mi takes place in the medium of gradient a, then the drive may be expressed by terms of the form /r^,\ i i h* /a a Q \ i h* (594) zmif = $g Yi mi (- - -) = ^g^mi where HI is the effective temperature ratio of the moving mass, and n is that of the prevailing general temperature gradient. There are two primary type cases of the distribution of the masses of different temperatures: (1) That in which they are superposed, and (2) that in which the masses are located side by side on the same levels. CASE III. The Cher turn of Deep Strata in the Column Let the pressures, temperatures, and heights be arranged in the initial and final states as indicated in Fig. 20. 185 THE HYDRODYNAMICS OF THE ATMOSPHERE The Overturn of the Deep Strata in the Column The greatest entropy in the initial state in 1 is less than the least in 2, so that the cold mass 1 will fall beneath the warm T ftl hi ICold 2' Warm Pi Tfi p / T . Pi Ti 2 P< T'^ 2 Warm Cold Po To 2 Po Toi Initial Final FIG. 20. The overturn of the deep strata in the column. mass 2. The heights of the masses will change, as well as the pressures and temperatures. Assume that P , r 02 , h 2 , Ti\, hi are known in the initial states. We shall have the following formulas for computing the other required terms, in a non-adiabatic atmosphere. T T T ~ Tm ~ (595) Pi = F (596) P h = 1 Substitute mCp(jTdm jTi d nti) successively, using (715) . (597) Initial. (7+ U)a = i m q 2 = Cp JT d m = ^^ (Po T 02 - Pi T i2 + P, T a - P h T hl ) + const. (598) Final. (V + U) e = = ^p-j (PO r i'- P/ r a '+p/ T#-P H T W '. TRANSFORMATION OF MASSES 187 (599) Kinetic energy = (7 + U) a - (V + U) e = \ M q 2 = ' 2 (approx.). 6 (600) Heights, h' = ^ (TV - TV). o (601) hi = ^f (TV - 2V). O The approximate solution of this case (Margules) is (602) Velocity, i q 2 = & fr ? +~h T CASE IV. The Transformation of Two Masses of Different Temperatures on the Same Levels into a State of Equilibrium. Tft2 F M 1 Cold (Sj) POI T ol 2 Warm (S 2 ) P o2 T 02 Initial 2' Warm P/ T/ 2 l' Cool T Po To Final FIG. 21. Transformation of two masses on the same levels. Transformation of Two Masses on the Same Levels Given the initial data at the height h, Thi, TV, PH, the areas Bi, B 2 , and the entropies Si < S 2 . Hence by the formulas we shall have, .T N fife (603) Pm = Ph (^ = (604) P ^ Ph (^ = (605) P; = P A + \ (P M - (606) Po' - P + *(**- (607) r.- (608) 188 THE HYDRODYNAMICS OF THE ATMOSPHERE (609) Initial. (V+lT) a = ^r r f (Poi T 01 - P h T kL + O -j I . P 02 T Q2 - P h T h2 ) + const. (610) Final. (V + U) e = \_.B (P ; T ' - P/ T,' + i i _ _ P/TV - P A r w ) + const. (611) Kinetic energy = J M ? 2 = (F + Z7) a - (V + t/) e . (612) Mass. Jkf = - (Po ; - P h ). o (613) Height. V = ^ (r c ' - zy). /^ (614) k,'^(T ff '-T u ). An approximate solution is given (Margules). (615) Take r = 2 ~ \ T* = TtTt, M = B P h -^, = j. j\. B p h (approx.) (616) ^ M q z = ^ M-^^-ghr. These solutions must be handled cautiously in practice, because the internal motions of the atmosphere introduce ele- ments of pressure, temperature, and velocity which it is very difficult to follow, and take account of in forming the elements of the integrals, and there is no term for the radiation. CASE V. For local changes between two strata of different tem- peratures, where on the boundary the pressure P = PI' = P 2 ' and the temperature is discontinuous Take the following conditions: (617) w 2 , P 2 T 2 , P 2 ' = P 2 + g m 1} T 2 ' = T 2 {~JT^ (618) m lt Pi T\, Px' = Px - g. m,, TV = ^ I ^ \ ^ k-< J LOCAL CHANGES 189 The condition of equilibrium becomes, for PI = P 2 ' = P, (619) Kinetic energy = Cp [mi (7\ - TV) + m* (T 2 - T 2 ')] (.20) , c (621) = mi fl r> 'T 1 -| T> 'T 1 (622) Since ^ = and ^ = , therefore, fl Pl -T2 P2 /OO\ 1 1/f .-,2 L " (\)Z6) 2^ M ^ = Wl W 2 ~ . H pi P2 The kinetic energy inducing an interchange is proportional to the difference of the densities, and inversely proportional to the product of the densities. Hence, if strata of different den- sities are flowing over one another in the general circulation, which is temporarily stratified, these two strata tend to mix by interpenetration according to this law. There are numerous other cases which can be worked out in accordance with the law which may be assumed for the distribu- tion of the temperature in a vertical and in a horizontal direction. Compare " Ueber die Energie der Stiirme," von Max Mar- gules. Jahrbuch der k.k. Cent.-Anst. fur Meteo. u. Erdeng. Wien, 1903. '' The Thermodynamics of the Atmosphere," F. H. Bigelow. Monthly Weather Review, 1906, and Bulletin W. B. No. 372. The General Circulation on a Hemisphere of the Earth } s Atmosphere While it is impracticable to take up the problems of the general circulation with the purpose of forming integrals that will take account of the entire circulation, there are yet a few propositions which are of interest in the premises. Resume equation (414) 2 , and limit it by assuming symmetry about the axis of rotation, so that -~ = 0; also, by omitting 190 THE HYDRODYNAMICS OF THE ATMOSPHERE the small term in w, so that we shall have as a special case, wherein the vertical current and the friction are omitted from the general motion, (624) cos (2 co 3 + v) u + -T- = 2 cos ( 3 +v) w - v cos . w+ Multiply this equation by r sin 6, and substitute from (413), (625^ -7- V cos . u = r sin , so that, at ot (626) 2 r sin (w 3 + v) r cos + (r sin 0) 2 = 0. Integrating for each gyrating particle, (627) r 2 sin 2 (co 3 + v) = c. Let C = constant for the entire rotating mass, if Vo is the initial relative angular velocity, m fr 2 sin 2 (o> 3 + V ) d m ) (628) c This is equal to the moment of inertia of the entire mass at the distance f r. If the initial state is that of rest on the rotat- ing earth V = 0. Finally, we shall have: (629) r 2 sin 2 ( <o 3 + ^-J = f r 2 , and (630) rco 3 sin0 + z^ = f>or (631) v= This is the eastward velocity at the north polar distance 0. (632) If v = 0, sin 2 = f , sin = 0.8165, = 54 44'. At latitude = 90 - - 90 - 54 44'= 35 16' the east- ward velocity vanishes at the surface of the earth. Observations indicate that v = extends upward in a direction sloping to- wards the equator from latitude 35 north and south, its position being indefinite above 20,000 meters. TEMPERATURE GRADIENTS AND VELOCITY 191 The equation (624) can be obtained as follows: Assume the vortex principle of the conservation of the momenta of inertia, (633) or 2 (co 3 + v) = (r sin 0) 2 (o> 3 + v) = c a constant. This is not strictly true in the atmosphere, because it is not circulating in a perfect vortex, and this faulty assumption has been generally made in discussing this subject. How far it departs from a vortex law remains to be determined by the observations. Differentiate, divide by r sin d /, and we obtain (634) 2 cos 6 (co 3 + v) r-j-. + r sin B-r-. = 0. d t (It Since r-j-. = u, we find from (624), dv dv (635) v cos 6. Jj.+ f $m f jj - 0, and this is the same as (625). Ferrel discusses these equations, and gives some approxi- mately correct views regarding the general circulation. Ober- beck's treatment embraces the three equations of motion, and the solution approaches more closely to the flow of currents actually observed. The complete integration of the system is, however, more complex than has been admitted, and the problem awaits a better treatment. The actual velocities and direction of motion, together with the temperatures, must be so handled as to embrace the general and the local circulations in a single comprehensive solution. Three Cases of the Slope of the Temperature Gradients and the Resulting Velocity of the East and West Circulations In the earth's atmosphere there are three general cases of the distribution of the temperature gradients and the resulting circulation which can be distinguished, though the solution will not be complete until the radiation term has been accounted for in the equations of motion. These cases are: (l) for the eastward drift in the temperate zone where the velocity increases upwards, while the temperature decreases towards the pole in a 192 THE HYDRODYNAMICS OF THE ATMOSPHERE line parallel to the axis of the earth's rotation; (2) for the lower levels of the westward drift in the tropics, up to an altitude of about 5,000 meters, wherein the westward motion increases with the height, while the temperature increases towards the pole; (3) for the upper layers of the westward drift, above 5,000 meters into the isothermal region, wherein the velocity decreases with the height, while the temperature increases towards the pole. The case (3) seems to agree with the conditions observed in the atmosphere of the sun, which has decreasing velocity from the equator to the pole, and decreasing velocity from the surface upwards in all latitudes, accompanied by increasing tempera- tures towards the pole. The observations in the tropics on the cloud velocities give an increasing velocity westward to 5,000 ii in FIG. 22. The relative values of d r and dTJf in three cases. meters, then decreasing to 11,000, then increasing to the limit of balloon observations, so that cases (2) and (3) alternate to some extent. or = the distance from the axis of rotation of the earth. r the radius from the center of the earth. Draw a tangent to the circle at the initial point of the iso- therm. Draw d TX and d r to second points on the same isotherm, to show its slope relative to the horizon and axis of rotation. We have to determine the relation of the temperatures and the velocities of motion in space to each other at any point in the earth's atmosphere. Take the general integral of motion (417), omitting the TEMPERATURE GRADIENTS AND VELOCITY 193 (Q + J) term, and supply the centrifugal force in the gravitation term, -- | v Q 2 , where v is the linear velocity of the rotating surface at the given latitude, and we obtain, (636) - f = | (u* + v z + w*) - ! i>o 2 + g r + constant. */ p 1 7? T Substitute - = p-, and pass to logarithms, also put g r = from the general law of gravity. (637) - logP . R T = | (u*+w*} + i (z; 2 -^ 2 ) + + C. We give different values of the temperature (7\, T 2 ) to two adjacent strata flowing over each other at different velocities (fli, %), but since the pressures cannot be discontinuous at the bounding surface, we take PI = P 2 . Hence, by substitution in two strata, and transformation of the terms for differentiation, l l C, This is the general equation to be fulfilled at every point. Now differentiate (637) to r, the change along the radius, for two adjacent strata at pressures PI, P 2 , and we have: ^ (log Pi-log P 2 ) l l\ (r.-7\) omitting the small terms in u and w. Again differentiate (637) to -or, holding the angular momentum (v&) constant in each stratum. At the surface of the earth the velocity v<? = co 3 2 tcr 2 . Hence, d V(? 2 co 2 3 zcr 2 2 v 2 (640) -T =2co 3 2 ttr=- - = -- . Using this form, d iff tD" iff and differentiating for two adjacent strata, i -log A) _ 1 /Ei 2 -Eo 2 _ ^2 2 -^o 2 \ -or V Ti 2s/ i rfa 2 -v)r 2 - (^- orL ^r, 194 THE HYDRODYNAMICS OF THE ATMOSPHERE Divide (641) by (639) and the ratio -r becomes, /~,\ dv 1 r~u>i 2 ZJo 2 ) T% (vz 2 Vo 2 ) '. (642) ^=~^L Tv^rT This equation connects the velocities and temperatures with the slope of the isotherms, and it is capable of three solutions which are expressed as follows: Case I Case II Case III dr _ (+) drf (-) T 2 <T 1 +dr _ ( ) (+) T 2 >T 1 East West(lower) "West (upper) and Sun FIG. 23. The relative values of v , m, v 2 , Ti, T 2 in three Cases. If dr = 0, and the isotherm is parallel to the surface, it follows that fa - v 2 ) T 2 = (vf - vf) T! so that the crossed products of the square of the relative velocities at any point in the atmosphere by the alternating temperatures of the two adjacent strata are equal. The warm stratum assumes greater velocity than the cold stratum, in order to maintain a gradual change in the value of the vertical hydrostatic pressure, such as was developed in Chapter II. If d r changes from 0, in the three cases described, and typically illustrated, the temperature gradients take on slopes that respectively balance the velocities of the air movements, generally above the series of tangents to the horizon in the tropics, but below them in middle latitudes, as have been found from the direct observations in balloon ascensions. In Chapter II, it has been shown how powerfully the (Qi Qo), the change of the heat contents per unit mass TEMPERATURE GRADIENTS AND VELOCITY 195 from one level to another reacts upon the velocity system, so that this problem cannot be fully solved through velocity and temperature functions. These theorems can be extended to very useful inferences in the case of the sun where velocities can be measured, but where it is very difficult to determine the ab- solute temperatures prevailing in different strata. CHAPTER IV Examples of the Construction of Vortices in the Earth's Atmosphere AN extensive computation on vortices has been published in the Monthly Weather Review, October, 1907, and subse- quent numbers, giving in sufficient detail the method of handling the data. These comprise the funnel-shaped vortex of the Cottage City waterspout, August 19, 1896, the dumb-bell-shaped vortex of the same Cottage City waterspout, the truncated dumb-bell-shaped vortex of the St. Louis tornado, May 27, 1896, the De Witte typhoon, August 1-3, 1901, the impure dumb-bell vortex in the ocean cyclone, October 11, 1905, and the very imperfect vortices of the land cyclones of the United States. In Figs. 24, 25, 26, are given typical (w . z) lines in the Cottage City waterspout, the St. Louis tornado, and the De Witte typhoon, respectively, to which further references will be made. In the funnel-shaped vortex tube of the Cottage City water- spout, the plane of reference is at the base of the cloud, 1,100 meters above the sea level, and the axis extends downwards, this being the convenient form for tornadoes generally. In the dumb-bell-shaped vortex there are two planes of reference, and the lower one is placed below the sea level while the upper one is placed at the cloud level. The axis between these planes 180 is divided into 180 degrees so that, a = 1 2QQ = 0.15, this being the value in the current function \f/ for this case. It has been found that these vortices are generated at the cloud base by the thermodynamic action of strata of different temperatures, and that they are propagated downwards to the sea level or to the surface of the ground. These vortices seem to be cut off or truncated at some distance above the lower plane of reference, and on this supposition the vortex laws, when applied to the observed phenomena, appear to meet satisfactorily all the 196 EXAMPLES 197 requirements of the problem. A few details of the computations will be introduced in this connection. Funnel- shaped Water Spout Cloud Level looVN 200 * 300 400 500 600 700 800 900 1000 1100 Dumb-bell shaped Water Spout 0Cloud Level 160 140 120 100 80 60' Se&.Level 1200 FIG. 24. The Cottage City, Mass., waterspout, August 19, 1896. St. Louis Tornado, May 27, 1896. FIG. 25. Illustrating the truncated dumb-bell-shaped vortex. The vertical ordinate is magnified ten times. 100000 200000 300000 400000 FIG. 26. The De Witte typhoon, August 1-3, 1901, In constructing a vortex of either type it is necessary to know two facts from observations: (1) the tangential velocity v at a point whose radius is S5 in meters from the axis, on a plane defined by 2 in the funnel-shaped vortex, or by az in the dumb- 198 CONSTRUCTION OF VORTICES IN EARTH'S ATMOSPHERE bell-shaped vortex; (2) the ratio p, which is the ratio of the successive radii -or in the tubes. By a series of studies on the spacing of the isobars in the De Witte typhoon, the St. Louis tornado, and the ocean cyclone of October 11, 1905, it was found that log p = 0.20546 seems to comply with the positions of the tubes in these vortices as developed in the earth's atmos- phere. In the cyclones and anticyclones there is a wide de- parture from this simple constant ratio, which indicates that another source of energy is at work besides the one generating these simple vortices, but this will require a fuller explanation. In Tables 45-53, and on Figs. 24, 25, 26, are given the results of the computations in sufficient detail to illustrate the scope of the formulas, the dimensions of the vortices, and the velocities, together with the angles of the helices which they make in the tubes. The funnel-shaped tube of Fig. 24 is constructed from or z in Table 45, using the tube (l) ; the truncated dumb-bell- shaped vortex of the St. Louis tornado is constructed from Table 48. An examination of the tables of the velocities and the angles suggests numerous remarks on their relations, but as they can be very clearly perceived it is not necessary to write them down. The vortices differ from one another in their dimensions, the waterspout, tornado, and hurricane being illustrations of the dumb-bell vortex. The cyclone shows a close relationship to this type of vortex, but it is distinctly modified by a different distribution of the thermal energy. The meteorological data are so extensive as to make it impracticable to reproduce them in this Treatise. Table 45 contains the necessary initial data and the formulas for developing a funnel-shaped vortex in all its tubes from the outer to the inner in succession. Taking the assumed data z = 100, iff = 60, v = 6.67 m/sec, \l/ = vw, we proceed through the formula to construct for tube (1), which has the largest radius, the constant of that tube C, the velocities u, v, w, and the horizontal angle i and the vertical angle y as defined in Fig. 14 for cylindrical co-ordinates, and Fig. 18 for vortices. Then follows the successive application of the formulas (535) to (550) by which the data of tubes (2), (3), (4), (5), (6) are computed. EXAMPLES 199 Thus by subtracting 0.20546 from log or, in succession, the radii of the other tubes ttr 2 , ttr 3j ztr 4 , or 5) ttr 6 , are obtained; similarly, the log p as indicated is to be applied for C, u, v, w, i, y. Table 48 contains the initial assumed data for the dumb-bell- shaped vortex of the St. Louis tornado, truncated on the plane az = 60. Thus, for az = 60, i = - 30, v = 960, v = 13.1 m/sec, we find in succession, a = 0.100, A, u, v, w, ttr, i, y, on the tube (l), and on tubes (2), (3), (4), (5), (6), by applying log p or its multiples as indicated by the working group of formulas (535)-(550). In the same manner we proceed with the De Witte typhoon, the ocean cyclone, and similar highly developed vortices. The land cyclone and anticyclone are imperfect vortices, and they involve a system of hydrodynamic stream lines which are highly complex in their origin and develop- ment. It must be constantly remembered that the important radiation terms do not appear in these vortex formulas and examples, so that a fuller treatment would be much more com- plex than the one here briefly summarized. A further illustra- tion will be added in discussing the origin of the cyclone. TABLE 45 THE COTTAGE CITY WATERSPOUT, AUGUST 19, 1896 The Funnel-shaped Vortex Tubes, V = C tff z z Collection of the Constants and Working Formulas Assumed data, z = 100 meters, distance below cloud plane Iff 60 meters, radius of cloud sheath. v = 6.67 m/sec, tangential velocity at (z, tCC). $ Formulas. C = o~ the constant for each tube. Wi u C ttr the radial velocity. ^ v = the tangential velocity. ttr w = 2 C z the vertical velocity. V = v Iff constant, log V = 2.60206. The ratio of the successive radii, P = n ' log p = 0.20546. n + 1 The successive radii, \og'0f n+l = logttr n log p. The successive velocities, log w n + i = log u n + log P. logfn+i = logfl n + log/ , log w n +i = log w n + 2 log p. 200 CONSTRUCTION OF VORTICES IN EARTH'S ATMOSPHERE The successive angles, log tan i n = constant. u tan i = v log tan Jj n+l = log tan ?+ log p. tan y = TABLE 46 THE VALUES OF tcr, C, u, v, w, ON THE PLANE z = 100 w. V sec i Formula (1) (2) (3) (4) (5) (6) log P log or or 1.77815 60.0 1 . 57269 37.4 1.36723 23.3 1.16177 14.5 0.95631 9.0 0.75085 5.6 l 0.20546 log C C 7.04576 0.001111 7.45668 0.002862 7.86760 0.007372 8.27852 0.01899 8.68944 0.04891 9.10036 0.12600 I + 0.41092 log M U 8.82391 0.06667 9.02937 0.1070 9.23483 0.1717 9.44029 0.2756 9.6*575 0.4423 9.85121 0.7099 J + 0.20546 log i 1 V 0.82391 6.67 1 . 02937 10.70 1.23483 17.17 1.44029 27.56 1.64575 44.23 1.85121 70.99 1 + I 0.20546 loga> w -9.34679 -0.222 -9.75771 -0.572 -0.16863 -1.474 -0.57955 -3.798 -0.99047 -9.783 - 1.40139 -25.199 B 0.41092 log tan i i 8.00000 34' 8.00000 34' 8.00000 34' 8.00000 34 ' 8.00000 34' 8.00000 34' s o U log tan 77 8.52288 1 55 8.72834 3 4 8.93380 4 54 9 . 13926 7 51 9.34472 12 28 9.55018 19 33 + 0.20546 TABLE 47 THE VALUES OF ttr, u, v, w, i, f\ ON SEVERAL PLANES The Radii Iff z (1) (2) (3) (4) (5) (6) 00 CO CO oo oo 00 1 600.0 373.8 232.9 145.1 90.4 56.3 10 189.7 118.2 73.7 45.9 28.6 17.8 25 120.0 74.8 46.6 29.0 18.1 11.3 50 84.9 52.9 32.9 20.5 12.8 8.0 100 60.0 37.4 23.3 14.5 9.0 5.6 200 42.4 26.4 16.5 10.3 6.4 4.0 300 34.6 21.6 13.5 8.4 5.2 3.3 400 30.0 18.7 11.7 7.3 4.5 2.8 500 26.8 16.7 10.4 6.5 4.0 2.5 700 22.7 14.1 8.8 5.5 3.4 2.1 900 20.0 12.5 7.8 4.8 3.0 1.9 1100 18.1 11.3 7.0 4.4 2.7 1.7 EXAMPLES TABLE 47. CONTINUED 201 (1) (2) (3) (4) (5) (6) The Radial Velocity u 1 00 0.667 oo 1.070 00 1.717 00 2.756 00 4.423 00 7.099 10 0.211 0.339 0.544 0.874 1.402 2.250 25 0.133 0.214 0.343 0.551 0.885 1.420 50 0.094 0.151 0.243 0.390 0.626 1.004 100 0.067 0.107 0.172 0.276 0.442 0.710 200 0.047 0.076 0.121 0.195 0.313 0.502 300 0.039 0.062 0.099 0.159 0.255 0.410 400 0.033 0.054 0.086 0.138 0.221 0.355 500 0.030 0.048 0.077 0.123 0.198 0.318 700 0.025 0.040 0.065 0.104 0.167 0.268 900 0.022 0.036 0.057 0.092 0.147 0.237 1100 0.020 0.032 0.052 0.083 0.133 0.214 The Tangential Velocity v 1 0.7 1.1 1.7 2.8 4.4 7.1 10 2.1 3.4 5.4 8.7 14.0 22.5 25 3.3 5.4 8.6 13.8 22.1 35.5 50 4.7 7.6 12.1 19.5 31.3 50.2 100 6.7 10.7 17.2 27.6 44.2 71.0 200 9.4 15.1 24.3 39.0 62.6 100.4 300 11.6 18.5 29.7 47.7 76.6 122.9 400 13.3 21.4 34.3 55.1 88.5 142.0 500 14.9 23.9 38.4 61.6 98.9 158.7 700 17.6 28.3 45.4 72.9 117.0 187.8 900 20.0 32.1 51.5 82.7 132.7 213.0 1100 22.1 35.5 57.0 91.4 146.7 235.7 The Vertical Velocity w 1 -0.0022 -0.0057 - 0.0147 - 0.0380 - 0.0978 - 0.2520 10 -0.0222 -0.0572 - 0.1474 - 0.3798 - 0.9783 - 2.520 25 -0.0556 -0.1431 - 0.3686 - 0.9495 - 2.446 - 6.300 50 -0.1110 -0.2862 - 0.7372 - 1.889 - 4.891 - 12.60 100 -0.222 -0.572 - 1.474 - 3.798 - 9.785 - 25.20 200 -0.444 -1.145 - 2.949 - 7.596 - 19.57 - 50.40 300 -0.667 -1.717 - 4.423 -11.39 - 29.35 - 75.60 400 -0.889 -2.290 - 5.898 -15.19 - 39.13 -100.80 500 -1.111 -2.862 - 7.372 -18.99 - 48.91 -126.00 700 -1.556 -4.007 -10.32 -26.59 - 68.48 -176.40 900 -2.000 -5.152 -13.27 -34.18 - 88.05 -226.80 1100 -2.444 -6.297 -16.22 -41.78 -107.61 -277.20 202 CONSTRUCTION OF VORTICES IN EARTH'S ATMOSPHERE The Horizontal Angle i / / o 90 Constant . 90 o 10 5 43 5 43 50 1 9 11 1 9 100 34 11 34 300 11 11 500 7 tt 7 700 5 " 5 900 4 4 1100 3 3 The Vertical Angle rj / / / / / / 10 36 58 1 33 2 29 3 59 6 22 50 1 18 2 10 3 28 5 34 8 53 14 5 100 1 55 3 4 4 54 7 51 12 28 19 33 300 3 18 5 18 8 28 13 25 20 58 31 35 500 4 16 6 49 10 52 17 8 26 19 38 26 700 5 2 8 3 12 48 20 2 30 20 43 12 900 5 43 9 7 14 27 22 28 33 34 46 48 1100 6 19 10 4 15 54 24 34 36 16 49 39 All the data on the level z = 100 meters have been computed in Table 46. The St. Louis Tornado, May 27, 1896 The Truncated Dumb-bell-shaped Vortex Tubes, ip = A z Tff* sin az TABLE 48 COLLECTION OF THE CONSTANTS AND WORKING FORMULAS Assumed data a z = 60 i = 30 on the sea-level plane. ttT = 960 meters, radius of the outer tube. v =13.1 m/sec. tangential velocity at (ttr . az). a - I20ST600 * - 10 log P = 0.20546. log a sin az = 8.93753 log a cos az = 8.69897. A = : , constant for each tube. atCTsm az u = A a tcr cos az, the radial velocity. aV v = , the tangential velocity. w = + A a Iff sin az, the vertical velocity. Ztr = ( : = ) , the radius on different levels. \A a sm az) EXAMPLES 203 tan * = tan (90 + az) = cot az = . v tan ^ = :, vertical angle. vsect q = v sec i sec v, total velocity. TABLE 49 THE VALUES OF -or, A, u, v, w ON THE PLANE a 2 = 60 Formula (i) (2) (3) (4) (5) (6) logP log Iff 2.98227 2.77681 2.57135 2.36589 2.16043 1.95497 - 0.20546 TV 960.0 598.2 372.7 232.2 144.7 90.2 log A 9.19672 9.60764 0.01856 0.42948 0.84040 1.25132 + 0.41092 A 0.1573 0.4052 1.0437 2 . 6883 6.9247 17.8371 f log -0.87796 -1.08342 -1.28888 -1.49434 -1.69980 -1.90526 Y - 0.20546 u -7.6 -12.1 -19.5 -31.2 -50.1 -80.4 *G 8 log v 1.11652 1.32198 1.52744 1.73290 1.93836 2 . 14382 ~ + 0.20546 V 13.1 21.0 33.7 54.1 86.8 139.3 1 log w 9.43528 9.84620 0.25712 0.66804 1 . 07896 1.48988 -f 0.41092 w 0.27 0.70 1.81 4.66 12.00 30.89 M 8 log tan t 9.76144 Constant 9.76144 a 30 30 u ' log tan ?/ 8.25629 8.46175 8.66721 8.87267 9.07813 9.28359 + 0.20546 ? 1 2' 1 39' 2 40' 4 16' 6 50' 10 53' THE TABLE 50 VALUES OF iff, u, v, w, i, ~n ON SEVERAL PLANES a z (1) (2) (3) (4) (5) (6) The Radii 180 00 00 00 00 00 00 170 2143.9 1335.8 832.3 518.6 323.1 201.3 160 1527.6 951.8 593.0 369.5 230.2 143.5 150 1263.4 787.2 490.5 305.6 190.4 118.6 140 1114.3 694.3 432.6 269.5 167.9 104.6 130 1020.7 636.0 396.3 246.9 153.8 95.9 120 960.0 598.1 372.7 232.2 144.7 90.2 110 921.6 574.2 357.8 222.9 138.9 86.5 100 900.2 560.9 349.5 217.8 135.7 84.5 90 893.4 556.6 346.8 216.1 134.6 83.9 80 900.2 560.9 349.5 217.8 135.7 84.5 70 921.6 574.2 357.8 222.9 138.9 86.5 60 960.0 598.1 372.7 232.2 144.7 90.2 204 CONSTRUCTION OF VORTICES IN EARTH'S ATMOSPHERE TABLE 50. CONTINUED (1) (2) (3) (4) (5) (6) The Radial Velocity u 180 00 00 00 00 00 00 170 33.2 53.3 85.5 137.3 220.3 353.6 160 22.6 36.2 58.2 93.4 149.8 240.4 150 17.2 27.6 44.3 71.2 114.2 183.3 140 13.4 21.6 34.6 55.5 89.1 143.0 130 10.3 16.6 26.6 42.7 68.5 109.9 120 7.6 12.1 19.5 31.2 50.1 80.4 110 5.0 8.0 12.8 20.5 32.9 52.8 100 2.5 4.0 6.3 10.2 16.3 26.2 90 0.0 0.0 0.0 0.0 0.0 0.0 80 -2.5 - 4.0 - 6.3 -10.2 -16.3 -26.2 70 -5.0 - 8.0 -12.8 -20.5 -32.9 -52.8 60 -7.6 -12.1 -19.5 -31.2 -50.1 -80.4 The Tangential Velocity v 180 170 5.9 9.4 15.1 24.2 38.9 62.4 160 8.2 13.2 21.2 34.0 54.5 87.5 150 9.9 16.0 25.6 41.1 65.9 105.8 140 11.3 18.1 29.0 46.6 74.8 120.0 130 12.3 19.7 31.7 50.9 81.6 131.0 120 13.1 21.0 33.7 54.1 86.8 139.3 110 13.6 21.9 35.1 56.3 90.4 145.1 100 13.9 22.4 35.9 57.7 92.5 148.5 90 14.1 22.6 36.2 58.1 93.2 149.6 80 13.9 22.4 35.9 57.7 92.5 148.5 70 13.6 21.9 35.1 56.3 90.4 145.1 60 13.1 21.0 33.7 54.1 86.8 139.3 The Vertical Velocity -w 180 170 0.06 0.14 0.36 0.93 2.41 6.20 160 0.11 0.28 0.71 1.84 4.74 12.20 150 0.16 0.41 1.04 2.69 6.93 17.84 140 0.20 0.52 1.34 3.46 8.90 22.93 130 0.24 0.62 1.60 4.22 10.86 27.96 120 0.27 0.70 1.81 4.66 11.99 30.90 110 0.30 0.76 1.96 5.05 13.01 33.52 100 0.31 0.80 2.06 5.30 13.64 35.13 90 0.32 0.81 2.09 5.38 13.85 35.67 80 0.31 0.80 2.06 5.30 13.64 35.13 70 0.30 0.76 1.96 5.05 13.01 33.52 60 0.27 0.70 1.81 4.66 11.99 30.90 EXAMPLES The Horizontal Angle i 205 180 +90 Constant +90 160 +70 +70 140 +50 +50 120 +30 . +30 100 + 10 +10 90 a 80 -10 H -10 70 -20 II -20 60 -30 II -30 The Vertical Angle rj 180 0' 0' 0' 0' 0' 0' 170 6 9 14 23 37 59 160 15 25 40 1 4 1 42 2 44 150 27 44 1 10 1 52 3 4 49 140 40 1 4 1 42 2 44 4 23 7 130 52 1 23 2 13 3 38 5 49 9 17 120 2 1 39 2 40 4 16 6 50 10 53 110 10 1 52 3 4 49 7 42 12 15 100 15 2 1 3 14 5 10 8 16 13 7 90 17 2 3 3 18 5 17 8 27 13 25 80 15 2 1 3 14 5 10 8 16 13 7 70 10 1 52 3 4 49 7 42 12 15 60 1 2 1 39 2 40 4 16 6 50 10 53 The data on the level a z have been computed in Table 49. TABLE 51 THE DE WITTE TYPHOON, AUGUST 1-3, 1901, IN THE CHINA SEA Results from the plane a z = 60 Initial z = 12000 meters a = 180 C 12000 + 600 = 0.010 z (1) (2) (3) (4) (5) (6) A .002016 .005193 .013375 .034452 .088744 .248922 206 CONSTRUCTION OF VORTICES IN EARTH'S ATMOSPHERE The Radii W in Meters a z = 180 00 00 00 00 00 00 170 887600 534338 332938 207443 129253 80534 160 611071 380742 237232 147813 92098 57384 150 505389 314900 196205 122250 76172 47460 140 445740 277727 173044 107820 67180 41858 130 408310 254406 158515 98766 61539 38344 120 384018 239272 149083 92890 57878 36062 110 368655 229700 143120 89174 55563 34619 100 360117 224379 139803 87108 54275 33818 90 357367 222663 138735 86444 53860 33559 80 360117 224379 139803 87108 54275 33818 70 368655 229700 143120 89174 55563 34619 60 384018 239272 149083 92890 57878 36062 The Radial Velocity u in Meters per Second az = 180 00 00 00 00 00 00 170 17.03 27.32 43.86 70.38 112.96 181.30 160 11.58 18.58 29.82 47.85 76.80 123.27 150 8.82 14.16 22.72 36.47 58.53 93.94 140 6.88 11.05 17.73 28.46 45.67 73.30 130 5.29 8.49 13.63 21.87 35.10 56.34 120 3.87 6.21 9.97 16.00 25.68 41.22 110 2.54 4.08 6.55 10.51 16.86 27.07 100 1.28 2.02 3.25 5.21 8.36 13.42 90 0.00 0.00 0.00 0.00 0.00 0.00 80 -1.28 -2.02 -3.25 - 5.21 - 8.36 -13.42 70 -2.54 -4.08 -6.55 -10.51 -16.86 -27.07 60 -3.87 -6.21 -9.97 -16.00 -25.68 -41.22 The Tangential Velocity v in Meters per Second a z = 180 0.00 0.00 0.00 0.00 0.00 0.00 170 3.00 4.82 7.73 12.41 19.92 31.97 160 4.21 6.76 10.85 17.42 27.95 44.86 150 5.09 8.17 13.12 21.06 33.79 54.24 140 5.78 9.27 14.88 23.88 38.32 61.50 130 6.31 10.12 16.24 26.07 41.84 67.14 120 6.70 10.76 17.27 27.72 44.48 71.39 110 6.98 11.21 17.99 28.87 46.34 74.36 100 7.15 11.47 18.42 29.55 47.43 76.13 90 7.20 11.56 18.56 29.78 47.80 76.71 80 7.15 11.47 18.42 29.55 47.43 76.13 70 6.98 11.21 17.99 28.87 46.34 74.36 60 6.70 10.76 17.27 27.72 44.48 71.39 THE OCEAN AND THE LAND CYCLONES The Vertical Velocity iv in Meters per Second 207 a z = 180 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 170 0.0007 0.0018 0.0046 0.0120 0.0308 0.0794 160 0.0014 0.0036 0.0091 0.0236 0.0607 0.1564 150 0.0020 0.0052 0.0134 0.0345 0.0887 0.2286 140 0.0026 0.0067 0.0172 0.0443 0.1141 0.2939 130 0.0031 0.0080 0.0205 0.0528 0.1340 0.3502 120 0.0035 0.0090 0.0232 0.0597 0.1537 0.3959 110 0.0038 0.0098 0.0251 0.0648 0.1668 0.4296 100 0.0040 0.0102 0.0263 0.0679 0.1748 0.4502 90 0.0040 0.0104 0.0268 0.0689 0.1775 0.4572 80 0.0040 0.0102 0.0263 0.0679 0.1748 0.4502 70 0.0038 0.0098 0.0251 0.0648 0.1668 0.4296 60 0.0035 0.0090 0.0232 0.0597 0.1537 0.3959 The Ocean and the Land Cyclones The tornadoes and hurricanes always occur in strata of air which are practically quiescent in the vertical direction, the tornadoes in the lower levels of stagnant air during hot summer afternoons, and the hurricanes in the neighborhood of the latitudes 30 to 35, where the east and west movements in the general circulation practically disappear. Should hurricanes move into higher latitudes, where the eastward drift prevails with an increase of its velocity in proportion to the height above the ocean, the nearly perfect dumb-bell vortices which represent them are transformed into imperfect vortices of the same general type. The penetration of the head of the vortex into the midst of the eastward drift introduces components of resistance which deplete and even destroy the type in the upper levels, so that it is degraded to a cyclone, or imperfect dumb-bell vortex by the mere mechanical action. Furthermore, the temperature dis- tribution is distinctly different in hurricanes and in cyclones. In the former the temperature differences are separated by horizontal planes in the upper levels, while in the latter the temperatures are separated chiefly in a vertical direction. The hurricanes have a symmetrical horizontal distribution of tem- peratures, but in cyclones the temperature distribution is de- cidedly asymmetrical, as is well known from the weather maps on the surface. The same asymmetry of temperature continues 208 CONSTRUCTION OF VORTICES IN EARTH'S ATMOSPHERE to the levels as high as 10,000 meters, warm on the east and cold on the west side of cyclones in the United States. These broad, thin sheets of warm and cold air, under the action of gravity, tend to return to a horizontal symmetry by the production of stream lines, whereby the cold air underruns the warm stream to the east and to the west by dividing into two branches, while the warm air overruns the cold air to the east and to the west similarly in two branches. Complicated stream lines are thus produced, which are those observed in the free air, after entering into composition with the velocities of the general circulation of the locality. This complex subject will require much more study than has been possible up to the present time in order to secure a complete analysis of the data, but it is clear that the research must proceed along certain lines which can be briefly indicated. The first problem is to separate the imperfect from the perfect vortices, and to assign the components of resistance, that is, to construct a reverse vortex which is practically equiva- lent to the system of reactions that prevents the dumb-bell vortex from developing into a pure form. The second problem is to determine the stream lines by which the masses of air at different temperatures are drawn by the force of gravity into these im- perfect cyclonic vortices. The ocean cyclone, October 11, 1905, has been taken to illustrate the composition of vortices, and the land cyclones must be studied more at length from the data provided by balloon and kite ascensions in Europe and the United States. The ocean cyclone is more highly developed than the land cyclone, and affords a convenient transition from the hurricane to the ordinary cyclonic storm. The cyclone of October 11, 1905, has been reduced to an equivalent cylindrical vortex by taking the mean radii as measured in four directions at right angles to each other. This mechanical process need not be repeated here, but the result is that the radii are not spaced in the vortical geometrical ratio. They diverge from that model which belongs to the perfect vortex. The corresponding velocities tangential to the equivalent circular isobars were constructed from the observed values in different parts of the THE OCEAN AND THE LAND CYCLONES 209 cyclone as reported by the 110 vessels that made observations on that date. TABLE 52 THE OCEAN CYCLONE, OCTOBER 11, 1905 The Imperfect Dumb-bell-shaped Vortex V'l Results for the Plane a z = 60 180 Initial z = 8000 meters, a = QQQ + 4QQQ = 0.015, log p =0.10600. z (1) (2) (3) (4) (5) (6) (7) (8) A, .00089 .00127 .00184 .00265 .00382 .00551 .00795 .01146 The Radii Iff in Kilometers a z = 180 00 00 00 00 00 00 00 00 170 3079.9 2412.9 1890.3 1480.6 1160.2 909.0 712.1 557.9 160 2194.6 1719.3 1346.9 1055.2 826.7 647.7 507.4 397.5 150 1815.0 1422.0 1114.0 872.8 683.7 535.7 419.6 328.8 . 140 1600.8 1254.1 982.5 769.7 603.0 472.4 370.1 290.0 130 1466.4 1148.8 900.0 705.1 552.4 432.8 339.0 265.6 120 1379.2 1080.5 846.5 663.2 519.5 407.0 318.9 249.8 110 1324.0 1037.2 812.6 636.6 498.8 390.7 306.1 239.8 100 1293.3 1013.2 793.8 621.9 487.2 381.7 299.0 234.3 90 1283.4 1005 . 5 787.7 617.1 483.5 378.8 296.7 232.5 80 1293.3 1013.2 793.8 621.9 487.2 381.7 299.0 234.3 70 1324.0 1037.2 812.6 636.6 498.8 390.7 306.1 239.8 60 1379.2 1080.5 846.5 663.2 519.5 407.0 318.9 249.8 The Radial Velocity u\ in Meters per Second a z = 180 00 00 00 00 00 00 00 00 170 40.2 45.4 51.3 58.0 65.5 74.0 83.6 94.4 160 27.3 30.9 34.9 39.4 44.5 50.3 56.8 64.2 150 20.8 23.5 26.6 30.0 33.9 38.3 43.0 48.9 140 16.2 18.4 20.7 23.4 26.5 29.9 33.8 38.1 130 12.5 14.1 16.0 18.0 20.4 23.0 26.0 29.4 120 9.1 10.3 11.7 13.2 14.9 16.8 19.0 21.5 110 6.0 6.8 7.7 8.7 9.8 11.0 12.5 14.1 100 3.0 3.4 3.8 4.3 4.8 5.5 6.2 7.0 90 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 80 - 3.0 - 3.4 - 3.8 - 4.3 - 4.8 - 5.5 - 6.2 - 7.0 70 - 6.0 - 6.8 - 7.7 - 8.7 - 9.8 -11.0 -12.5 -14.1 60 - 9.1 -10.3 -11.7 -13.2 -14.9 -16.8 -19.0 -21.5 210 CONSTRUCTION OF VORTICES IN EARTH'S ATMOSPHERE The Tangential Velocity vi in Meters per Second az = 180 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 170 7.1 8.0 9.0 10.2 11.6 13.0 14.7 16.7 160 9.9 11.2 12.7 14.3 16.2 18.3 20.7 23.4 150 12.0 13.6 15.4 17.3 19.6 22.1 25.0 28.3 140 13.6 15.4 17.4 19.7 22.2 25.1 28.4 32.0 130 14.9 16.8 19.0 21.5 24.3 27.4 31.0 35.0 120 15.8 17.9 20.2 22.8 25.8 29.1 32.9 37.2 110 16.5 18.6 21.0 23.8 26.9 30.3 34.3 38.7 100 16.9 19.1 21.5 24.3 27.5 31.1 35.1 39.7 90 17.0 19.2 21.7 24.5 27.7 31.3 35.4 40.0 80 16.9 19.1 21.5 24.3 27.5 31.1 35.1 39.7 70 16.5 18.6 21.0 23.8 26.9 30.3 34.3 38.7 60 15.8 17.9 20.2 22.8 25.8 29.1 32.9 37.2 The Vertical Velocity w\ in Meters per Second o az = 180 .0000 .0000 .0000 .0000 .0000 .0000 .0000 .0000 170 .0003 .0004 .0006 .0009 .0013 .0019 .0028 .0040 160 .0006 .0009 .0013 .0018 .0026 .0038 .0054 .0078 150 .0009 .0013 .0018 .0026 .0038 .0055 .0079 .0115 140 .0011 .0016 .0024 .0034 .0049 .0071 .0102 .0147 130 .0014 .0020 .0028 .0041 .0059 .0084 .0122 .0176 120 .0015 .0022 .0032 .0046 .0066 .0095 .0138 .0198 110 .0017 .0024 .0035 .0050 .0072 .0104 .0149 .0215 100 .0017 .0025 .0036 .0052 .0075 .0109 .0157 .0226 90 .0018 .0025 .0037 .0053 .0076 .0110 .0159 .0229 80 .0017 .0025 .0036 .0052 .0075 .0109 .0157 .0226 70 .0017 .0024 .0035 .0050 .0072 .0104 .0149 .0215 60 .0015 .0022 .0032 .0046 .0066 .0095 .0138 .0198 TABLE 53 THE OCEAN CYCLONE, OCTOBER 11, 1905 The Perfect Dumb-bell-Shaped Vortex Vo The Radii ZCT remain the Same z A, (1) .00089 (2) .00144 (3) .00235 (4) .00382 (5) .00623 (6) .01014 (7) .01653 (8) .02699 The Radial Velocity a z = 180 00 00 00 00 oo 00 00 00 170 40.2 51.3 65.5 83.6 106.7 136.2 173.8 221.9 160 27.3 34.9 44.5 56.8 72.5 92.6 118.2 150.9 150 20.8 26.6 33.9 43.3 55.3 70.6 90.1 115.0 140 16.2 20.7 26.5 33.8 43.1 55.1 70.3 89.7 130 12.5 15.9 20.4 26.0 33.2 42.3 54.0 69.0 120 9.1 11.7 14.9 19.0 24.3 31.0 39.5 50.4 110 6.0 7.7 9.7 12.5 15.9 20.3 26.0 33.1 100 3.0 3.8 4.8 6.2 7.9 10.1 12.9 16.4 90 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 THE OCEAN AND THE LAND CYCLONES 211 The Tangential Velocity v az = 180 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 170 7.1 9.0 11.6 14.7 18.8 24.0 30.7 39.1 160 9.9 12.7 16.2 20.7 26.4 33.7 43.0 54.9 150 12.0 15.4 19.6 25.0 31.9 40.8 52.0 66.4 140 13.6 17.4 22.2 28.4 36.2 46.2 59.0 75.3 130 14.9 19.0 24.2 31.0 39.5 50.4 64.4 82.2 120 15.8 20.2 25.8 32.8 42.0 53.6 68.5 87.4 110 16.5 21.0 26.9 34.3 43.8 55.9 71.3 91.0 100 16.9 21.5 27.5 35.1 44.8 57.2 73.0 93.2 90 17.0 21.7 27.7 35.4 45.1 57.6 73.6 93.9 The Vertical Velocity a z = 180 .0000 .0000 .0000 .0000 .0000 .0000 .0000 .0000 170 .0003 .0005 .0008 .0013 .0022 .0035 .0057 .0094 160 .0006 .0010 .0016 .0026 .0043 .0069 .0113 .0184 150 .0009 .0014 .0024 .0038 .0062 .0101 .0165 .0269 140 .0011 .0018 .0030 .0049 .0080 .0130 .0212 .0346 130 .0014 .0022 .0036 .0058 .0095 .0155 .0253 .0412 120 .0015 .0025 .0041 .0066 .0108 .0176 .0286 .0466 110 .0017 .0027 .0044 .0072 .0117 .0191 .0311 .0506 100 .0017 .0028 .0046 .0075 .0123 .0202 .0326 .0530 90 .0018 .0029 .0047 .0076 .0124 .0203 .0331 .0538 TABLE 54 THE REVERSING OR COMPONENT VORTEX, Vi At. = Ai Ao .00000 -.00017 -.00051 -.00117 -.00241 -.00463 -.00858 -.01553 The radial velocity of the reverse vortex, uz az = 180 0.0 170 0.0 -5.9 -14.2 -25.6 -41.2 -62.2 -90.2 -127.5 160 0.0 -4.0 - 9.6 -17.4 -28.0 -42.3 -61.4 - 86.7 150 0.0 -3.1 - 7.3 -13.3 -21.4 -32.3 -47.1 - 66.1 140 0.0 -2.3 - 5.8 -10.4 -16.6 -25.2 -36.5 - 51.6 130 0.0 -1.8 - 4.4 - 8.0 -12.8 -19.3 -28.0 - 39.6 120 0.0 -1.4 - 3.2 - 5.8 - 9.4 -14.2 -20.5 - 28.9 110 0.0 -0.9 - 2.0 - 3.8 - 6.1 - 9.3 -13.5 - 19.0 100 0.0 -0.4 - 1.0 - 1.9 - 3.1 - 4.6 - 6.7 - 9.4 90 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 The tangential velocity of the reverse vortex, a z = 180 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 170 0.0 -1.0 -2.6 - 4.5 - 7.2 -11.0 -16.0 -22.4 160 0.0 -1.5 -3.5 - 6.4 -10.2 -15.4 -22.3 -31.5 150 0.0 -1.8 -4.2 - 7.7 -12.3 -18.7 -27.0 -38.1 140 0.0 -2.0 -4.8 - 8.7 -14.0 -21.1 -30.6 -43.3 130 0.0 -2.2 -5.2 - 9.5 -15.2 -23.0 -33.4 -47.2 120 0.0 -2.3 -5.6 -10.0 -16.2 -24.5 -35.6 -50.2 110 0.0 -2.4 -5.9 -10.5 -16.9 -25.6 -37.0 -52.3 100 0.0 -2.4 -6.0 -10.8 -17.3 -26.1 -37.9 -53.5 90 0.0 -2.5 -6.0 -10.9 -17.4 -26.3 -38.2 -53.9 212 CONSTRUCTION OF VORTICES IN EARTH'S ATMOSPHERE The vertical velocity of the reverse vortex, az = 180 .0000 .0000 .0000 .0000 .0000 .0000 ,0000 .0000 170 .0000 -.0001 - . 0002 - . 0004 - . 0009 -.0016 - . 0029 - . 0054 160 .0000 -.0001 - . 0003 - . 0008 -.0017 -.0031 -.0059 -.0106 150 .0000 -.0001 - . 0005 -.0012 - . 0024 -.0046 - . 0086 -.0154 140 .0000 -.0002 - . 0006 -.0015 - . 0031 -.0059 -.0110 -.0199 130 .0000 - . 0002 -.0008 -.0017 - . 0036 - . O0'< 1 -.0131 - . 0236 120 .0000 - . 0003 - . 0009 - . 0020 - . 0042 -.0081 -.0148 - . 0268 110 .0000 - . 0003 - . 0009 - . 0022 -.0045 - . 0087 -.0162 -.0291 100 .0000 -.0003 -.0010 - . 0023 - . 0047 - . 0091 -.0169 - . 0301 90 .0000 - . 0004 -.0010 -.0023 - . 0048 -.0093 - . 0172 -.0309 The Composition of Vortices By subtracting the computed velocities of the perfect vortex from those of the imperfect vortex, u\ U Q = u 2 , Vi V Q = v 2 , Wi WQ = w 2 , we have a component vortex which, added to the perfect vortex, will produce the observed imperfect vortex, u\ UQ -f ^2, v\ = VQ + v 2 , Wi = w -\- w 2 , the signs being added algebraically. The corresponding values of the constants A, AI AQ = A 2 , can be found by computing the values of A for the derived velocities by the formulas, (643) A 2 = U 2 -W 2 a w cos a z sin a z 2 sin a z' More simply, the algebraic values of A 2 are derived im- mediately from AI (imperfect vortex) A (perfect vortex), whence the corresponding velocities u 2 , v 2 , w 2 can be computed in the usual manner. Table 52 gives the values of AI, MI, Vi, w\ in the imperfect vortex; Table 53 those of A , u , V Q , w d in the perfect vortex, and Table 54 those of A 2j 2, v 2 , w 2 in the component reversing vortex. A comparison of the velocities in these tables shows that, by starting with the same radius and velocity on the outer isobar (1), the observed imperfect vortex departs more and more from the corresponding perfect vortex in proportion as the veloci- ties approach the axis. The component vortex which is equiva- lent to these differences is a vortex reversed in all respects to the original vortex, revolving in the opposite direction and directed downward from the clouds to the surface of the sea. This principle of the composition of vortices through the constants A THE REVERSED DUMB-BELL VORTEX 213 of the successive tubes is very important, and leads to many practical researches in the theory of cyclones, because it enables us to take account of the numerous departures from the pure vortex law, without giving up the advantages of the method of vortex computations. The Reversed Dumb-bell Vortex A very erroneous impression would be left if it were sup- posed that the imperfect dumb-bell vortex could be applied directly to the study of the common cyclones in the atmosphere. The dumb-bell vortex seems to be essentially reversed, turned Height in meters (6) (5) (4) (3) (2) (I) 10000 0000 8000 7000 6000 5000 1000 3000 2000 1000 000 11500 FIG. 27. The constant a A lines as derived from observations in the land cyclones. inside out, as can be seen by Table 55, and Fig. 27. The dis- cussion of the velocity components resulted as given in the Cloud Report, in Table 126. Taking the velocities in that table and plotting them on diagrams, a consistent system was deduced which conforms on the average to that there given. It 214 CONSTRUCTION OF VORTICES IN EARTH'S ATMOSPHERE is produced on Table 55, I, II, for every 1,000-meter level, and for the radial distances given in the normal land cyclone. Since tan a z = , by (527) and (528), we can compute a A for each level and tube. The result appears in Table 55, III. On Fig. 27, at the radial distances, 80, 300, 508, 733, 975, 1,250 kilometers, these values of a A were plotted down, and lines of equal a A were drawn and they are given on Fig. 27. They contain the surprising result that the radial distances W are arranged nearly on a geometrical ratio system, as can be readily seen* by making the tests, and that the old ratio value log p = 0.20546 is quite competent to satisfy the average conditions. The lines a A are, however, concave towards the axis, instead of convex as in the hurricane, and the lines are closed up on the outer circles rather than on the inner, this being a complete reversal of the configuration. The theoretical and the thermodynamical con- ditions which produce this circulating structure have been indicated in the series of papers on the " Thermodynamics of the Atmosphere," W. B. 372, 1907. The subject will require further study and investigation. TABLE 55 THE OBSERVED RADIAL AND TANGENTIAL VELOCITIES IN CYCLONES. INTERNATIONAL CLOUD REPORT I. u = the observed radial velocities Height in Meters (i) 1250000 (2) 975000 (3) 733000 (4) 508000 (5) 300000 (6) 80000 10000 -0.5 -0.7 -1.0 -1.3 -1.6 -2.0 9000 -1.5 -2.1 -2.7 -3.0 -3.7 -4.5 8000 -2.0 -3.0 -4.0 -4.5 -5.5 -6.0 7000 -3.0 -4.0 -5.0 -6.0 -7.0 -7.5 6000 -3.0 -4.0 -5.0 -6.0 -7.0 -8.0 5000 -2.5 -3.0 -4.0 -4.5 -6.0 -7.0 4000 -2.0 -2.5 -3.0 -4.0 -5.0 -6.0 3000 -2.0 -2.0 -2.0 -2.5 -3.5 -4.5 2000 -2.0 -2.0 -2.0 -2.0 -2.0 -2.0 1000 -3.5 -3.5 -4.0 -4.0 -4.0 -4.0 000 -4.5 -5.0 -6.0 -6.5 -6.5 -6.0 II. THE REVERSED DUMB-BELL VORTEX v = the observed tangential velocities 215 10000 +2.0 + 2.5 + 3.0 + 4.0 + 4.5 + 5.0 9000 +4.0 + 6.0 + 8.0 . + 8.5 + 8.5 + 9.0 8000 +4.0 + 7.0 + 10.0 + 12.0 + 12.0 +11.0 7000 +5.0 +10.0 + 13.0 +14.0 +14.0 +14.0 6000 +6.0 +11.0 + 14.0 +15.0 +15.0 +16.0 5000 +7.0 +12.0 + 15.0 +16.0 + 18.0 +19.0 4000 +7.0 +11.0 +14.0 +17.0 + 19.0 +21.0 3QOO +6.0 + 10.0 +13.0 +16.0 +19.0 +23.0 2000 +5.0 + 9.0 +12.0 +15.0 +18.0 +22.0 1000 +5.0 + 6.0 + 8.0 +10.0 + 12.0 +14.0 000 +5.0 + 5.5 + 6.0 + 6.0 + 6.5 + 7.0 III. The computed a A = u Iff sin az Iff cos a z 10000 165 266 431 828 1593 6732 9000 342 652 1152 1775 3091 12577 8000 358 781 1469 2523 4401 15659 7000 467 1105 1900 2998 5217 19852 6000 537 1201 2028 3180 5518 24801 5000 595 1269 2118 3270 6325 25306 4000 585 1157 1953 3439 6548 27298 3000 506 1046 1794 3187 6441 29300 2000 431 946 1659 2978 6040 27598 1000 488 713 1220 2120 4216 18198 000 538 762 1158 1741 3064 11524 The unit = .00000001 = 1 X 10 - 8. It is obvious that the velocities u, v can be computed from the formulas, knowing the values of the constants a A, or, on the other hand, the constants can be computed from the observed u, v velocities. The results of this computation, Table 55, Fig. 27, show that the dumb-bell vortex system has been entirely reversed. The a A lines are concave toward the axis, they are geometrically spaced but closed up on the outer rather than on the inner areas of the cyclone. The temperature distribution conforms to this arrangement, and the cause is probably due to the penetration of the vortex of the lower strata into the rapidly moving drift of the upper strata. 216 CONSTRUCTION OF VORTICES IN EARTH'S ATMOSPHERE Historical Review of the Three Leading Theories Regarding the Physical Causes of Cyclones and Anticyclones in the Earth's Atmosphere In the Astronomical and Astro physical Journal, January, 1894, the writer made a summary of the three most important general theories regarding the physical causes which generate the local storms, called cyclones, in the earth's atmosphere, and the following extracts from that review are sufficiently explicit for ordinary purposes. The three theories are: (l) Ferrel's warm-center and cold-center cyclones; (2) Hands dynamic production of temperatures as found by observation; (3) Bige- low's asymmetric cyclone with warm and cold currents arranged in ridges, or streams of different densities, and driven into local cyclonic and anticyclonic circulations by the force of gravitation acting upon them. Ferrel had in mind for his cyclone the type of the general circulation of the atmosphere, and conceived that the same principles dominate both of them. The general cir- culation is described as a cold-center cyclone, with eastward movement from the pole to the high-pressure belt in latitude 33, and a warm ring of westward movement in the tropics; the local cyclone is warm at the center and has right-handed rotation from the axis to a ridge of high pressure, outside of which a cold ring circulates in the anti-right-handed direction in the northern hemisphere, these directions of motion being reversed in the southern hemisphere. Ferrel's practical difficulty was to account for the originating heat energy in the central column of the cyclone, and this must precede any criticism of the circula- tion that depends upon it. He writes, "In the ordinary cyclonic disturbances of the atmosphere, the causes are similar to those in the general circulation but more local, and consist of a differ- ence of density arising mostly from a difference of temperature between some central area and the external surrounding parts of the atmosphere." This dominant idea proved fatal to Ferrel's successful development of sound fundamental principles, and has greatly influenced many students to travel a road whose HISTORICAL REVIEW OF LEADING THEORIES 217 end has never been found. He was evidently unable to account satisfactorily for the energy implied in the temperature difference required to do the work observed in the motions of the cyclones and anticyclones. In his Coast and Geodetic Survey Report, p. 183, he remarks: "If for any reason there is kept up a continued interchange of air between the central and exterior part"; p. 201, "The condensation of aqueous vapor plays an important part in cyclonic disturbances, but is by no means either a primary or a principal cause of cyclones"; p. 239, "Rainfall is not essential to the formation of areas of low barometer, and is not the principal cause of their formation or of their progressive motion"; in Waldo's edition of FerrePs "Hydrodynamics," p. 39, "The theory which attributes the whole of the barometrical oscillations to the rarefaction of the atmosphere produced by the condensation of vapor in the forma- tion of clouds and rain cannot be maintained." However, being hard pushed to find a cause for his central area of high tempera- ture in cyclones, he gradually weakened from this position, en- dorsed Espy's condensation theory of the development of latent heat by the formation of clouds and rain, and in the last year of his life could write in Science, December 19, 1890, "All this has been done in the condensation theory of cyclones, with results so satisfactory as scarcely to leave a doubt as to the truth of the whole theory." This was written in reply to Dr. Hann's revolt against 'the sufficiency of this cause to produce cyclones as observed, who took the ground that these local gyrations are only subordinate whirls in the general circulation, which depend upon the effects of the equatorial radiation only, and are independent of any local cause. Hann even went so far as to conclude, that "the actual motion of the atmosphere is not a product of the temperature (FerrePs idea), but is in spite of it; the temperature is a product of the motion," Science May 30, 1890. Ferrel was loyal to the theory that temperature differences cause the motion always and everywhere, and Hann, in adopting the inverse proposition, has surely erred against first principles. FerrePs mechanical theory urged him to adopt a ridge of 218 CONSTRUCTION OF VORTICES IN EARTH'S ATMOSPHERE high pressure surrounding every cyclone, as indicated in formulas (471) to (478), but this is opposed by several fundamental con- ditions which show that it does not conform with modern observa- tions. Another solution for the warm-center cyclone is given in equations (479) to (490), but this is equally opposed by the observed conditions. Both of these mathematical solutions have been practically abandoned, chiefly because there are, in fact, no warm-centered cyclones and no cold-centered anti- cyclones in existence. The hurricane probably has a warm- centered system of motion, but it is entirely different in structure. When a cold sheet of air overflows a warm sheet, the warm sheet flows outward radially from a central point in all directions, like the spokes on a wheel, and this outward movement in the high cloud levels drags behind it the vortex tube described in Fig. 26, and Tables 51. This, however, is entirely different from the temperature conditions of motion in cyclones and anticyclones. Bigelow writes in the same paper of the Astronomical and Astrophysical Journal, "I must admit freely that I am unable to see in the daily weather maps that formation as fundamental which Ferrel and meteorologists generally assume to be the pri- mary state. I propose to see in temperature differences, ar- ranged in waves or ridges, the true cause of the observed pressures and the antecedent of the precipitation. It is therefore necessary to account for cold and warm temperature waves passing over the United States." "The passage of winds past each other in opposite directions tends toward local gyrations, which all drift eastward with the prevailing component in middle latitudes. All this simply depends upon the difference between the polar and the equatorial temperature, and is fully in accordance with the views of Ferrel and the latest expressions by Dr. Hann." "The formation of these low- and high-pressure areas is the result of the existence of the warm or cold sections of waves lying athwart the maximum crest. From first principles the warm and cold masses will be impelled toward each other, because of the action of gravitation on media of differing density. They will tend to encounter along or near the ridge of greatest temperature variation. Along the line of greatest temperature HISTORICAL REVIEW OF LEADING THEORIES 219 change, with cold air to the west and warm air to the east of it, the gyrating cyclone is formed, the couple existing from the system of causes thus described. Likewise, along the next ridge, with cold air to the east and warm air to the west, and often to the south of the maximum crest, the anticyclone is produced. A corollary remark is that the storm track along the north United- States seems to be the effort of the general circulation to restore the permanent polar low-pressure belt which is interrupted by the continent. Another is that tornadoes and hurricanes are due to precisely the same cause, namely, the juxtaposition of masses of air having great temperature differences." The ideas were illustrated by the cyclone of November 16, 17, 1893. The origin of these cold and warm waves, or ridges of different densities, has been discussed at great length in the International Cloud Report, and the streams from the cold north and warm south were called "leakage" currents, because these are in fact sporadic offshoots from the general circulation into middle lati- tudes. The warm currents in the United States are thrown off by the Atlantic center of action, from the Gulf of Mexico to the north; those upon southeastern Asia from the Pacific center of action ; those upon northwestern United States from the same Pacific center of action, or else from the Arctic zone over British America; those upon northwest Europe, from the Atlantic center of action, or else from the Arctic circulation. The continents and the oceans react upon the general circulation in such a manner as greatly to disturb and distort its free operation, so that finally southerly currents prevail in certain regions and northerly currents are dominant in other regions. In the United States the southerly warm currents and the northerly cold currents encounter in long streams, flowing past each other in waves or ridges of density, and under the force of gravitation they are compelled to flow in cyclonic and anticyclonic circula- tions toward a thermal equilibrium. The exact mathematical conditions prevailing at every point have been indicated in Chapter II of this Treatise, and in that place, and in Bulletin No. 3, Argentine Meteorological Office, 1912, the practical details 220 CONSTRUCTION OF VORTICES IN EARTH'S ATMOSPHERE have been illustrated. Bigelow, in 1894, laid down the principle of the force of gravitation acting upon density masses in alternate juxtaposition, whether side by side as in cyclones, anticyclones, and tornadoes, or in vertical superposition as in hurricanes, with intermediate cases between these principal positions; and in 1912 he worked out a method of computing the very important terms of the radiation energy in the general equations of motion, which had heretofore been entirely omitted from the discussions. A very brief summary of some of the most important features of the general and the local circulations are added in this place, though the student must consult the weather maps of various countries for any complete knowledge of such a complex subject as the actual circulation. The General and Local Components of the Velocities, Pressures, and Temperatures in the Circulation of the Atmosphere The General and the Local Components There are certain distributions of the temperature, pressure, wind direction, and velocity, which are characteristic of the general motions of the atmosphere, and others which belong to the local circulations peculiar to the cyclones and anticyclones. It is necessary to separate them from the observed values that are the resultants of those two components. It is practicable to observe certain values of the velocities, pressures, and tem- peratures at a given station, from which the general or normal values are computed, so that by subtracting in the form of vectors the normal values from the observed values, the local or component terms may be found. This great labor has been performed for the United States and the West Indies, the results for the velocities being recorded in the Cloud Report, 1898, for the pressures in the Barometry Report, 1901, and for the tem- peratures in the Report on Homogeneous Normals, 1909, together with numerous papers in the Monthly Weather Review. This Treatise is concerned with the methods of com- putation and discussion appropriate to meteorology, rather than with statistical results, so that only a brief summary of NORMAL AND LOCAL VELOCITIES IN STORMS 221 these data can be presented in this connection. Unfortunately, the data of meteorology are so bulky that it becomes very difficult to do justice to the subject within the limits of a reasonable volume. The Normal and Local Velocities in Storms In the Cloud Report are to be found the resultant velocities and directions of the wind at the surface, in the cumulus levels (1,000-2,000 meters), and in the cirrus levels (8,000-10,000 meters), for all parts of cyclones and anticyclones, when the centers of these areas are located in different parts of the United States, as the Dakotas, the Lake region, New England, Colorado, Texas, and East Gulf States, respectively. They were obtained in the several areas by making a composite chart from about FIG. 28. Observed stream lines of air in the cumulus cloud level (2,000) over a cyclone whose center is in the Lake Region. fifty charts for each type of storms. For this purpose the United States was divided into small areas by the parallels of latitude and the meridians, the centers of the fifty storms were made to coin- cide, and the vectors or arrows were transferred to a common chart, from which the resultant vector was carefully computed. These charts are of great theoretical value for the student, as well as of practical value for the forecaster, and they should be 222 CONSTRUCTION OF VORTICES IN EARTH'S ATMOSPHERE thoroughly examined. Specimens of these charts are given on Fig. 28 for a cyclone centered in the lake region, transcribing the FIG. 29. Observed stream lines in the cumulus cloud level (2,000) over a cyclone whose center is in the West Gulf States. FIG. 30. Observed stream lines in the cirrus cloud level (10,000) over a hurricane whose center is in the South Atlantic States. These cur- rents all show that there is a U-shaped formation in the circulation gen- erally opening to the northeastward, though it is also found pointing westward and southward. lower cloud level; on Fig. 29 for a cyclone in the West Gulf States for the cumulus cloud level; and on Fig. 30 for a hurri- NORMAL AND LOCAL VELOCITIES IN STORMS 223 cane in the East Gulf States, upon the upper or cirrus cloud level. It is seen in all these cases that the currents of air form U-shaped figures like the isobars and isotherms, in entering a cyclonic vortex, and that the eastward drift is locally diverted into this configuration. There is a bridge across the top of the U-shaped vortex, and in the isobars a well-defined saddle is always constructed, where the high-pressure areas are tem- porarily broken through in the construction of a vortex circu- lation. Fig. 31 gives a representation of a typical circulation of air in connection with the isobars in three levels: sea level, 3,500-foot level, and 10,000-foot level, showing the relation of the currents to the isobars. The high-pressure cusps tend to approach over a bridge or saddle at C S C, the pressure being lower to the north and to the south of it. The number of- the closed isobars decreases with the height, and it is usual for them to disappear at the level of 3,000-4,000 meters, and sometimes even lower. This is a proof that the dumb-bell vortex which dominates in hurricanes has almost entirely vanished in cyclones except in the lowest levels, the top being entirely depleted in the higher levels. This throws back the theory of cyclones into quite a different category of imperfect vortices and, considering the asymmetrical distribution of the temperature, it is evident that the currents are due to pressure gradients in the thin sheets of air of different temperatures in the process of mixing in the middle latitudes. The tendency for the currents to divide into two streams mutually underrunning and overrunning each other should be carefully noted. In the Cloud Report are contained the data from which the average heights were computed where the several clouds are formed, and the average velocities were deduced at which they move nearly eastward, after the cyclonic and the anticyclonic components have been eliminated. Table 56 contains a summary of the velocities in high and low areas, the northward and south- ward components over high and low areas, the northward and southward components between the centers in the warm and cold streams, and the seasonal velocities. See Fig. 32. 224 CONSTRUCTION OF VORTICES IN EARTH'S ATMOSPHERE The cloud forms, stratus, cumulus, strato-cumulus, alto- cumulus, alto-stratus, cirro-cumulus, cirro-stratus, cirrus, occur at certain well-defined heights on the average, and it is found by observation that they drift over the earth's surface at certain average velocities as shown on the diagram. This increase of velocity upward, from 7 meters per second at the surface to 40 Typical abnormal Isobars, (Sea Level) FIG. 3 1 a. FIG. 3 1 a, b, c. Showing the relation of the local circulation to the typical isobars in high or low areas of pressure. The closed isobars form a rough vortex, which is supplied by the two-branched stream-lines and gradually dies out in the higher levels. meters per second, is called the eastward drift in middle latitudes. This is the normal velocity component which must be eliminated from the observed component to produce the local disturbing component of velocity due to the cyclones and anticyclones proper. The eastward and the westward drifts, in the middle latitudes and the tropics, respectively, are shown on Figs. 33, 34, 35. Typical abnormatlsobars, (8,500 foot) FIG. 316. Typical-abnormal Isobars, (10,000 foot) FIG. 3 ic. 226 CONSTRUCTION OF VORTICES IN EARTH'S ATMOSPHERE TABLE 56 SUMMARY OF THE VELOCITIES OF THE MOTIONS OF CLOUDS IN THE DIFFERENT LEVELS FOR THE MIDDLE ATLANTIC STATES. THE UPPER CLOUDS INCLUDE Ci., Ci.S., Ci.Cu., A.S., A.Cu., WITH THE MEAN HEIGHT 8.4 KILOMETERS, AND THE LOWER CLOUDS INCLUDE S.Cu., Cu., S., AT THE MEAN HEIGHT 2.4 KILOMETERS. Symbols: Ci. = Cirrus. S. = Stratus. Cu. = Cumulus. A. = Alto. L TOTAL VELOCITY IN HIGHS AND Lows WITHOUT REGARD TO DIRECTIONS Clouds Ci. Ci.S. Ci.Cu. A.S. A.Cu. S.Cu. Cu. S. Wind Height in kilo- meters 9.8 9.8 8.1 5.9 4.5 2.5 1.5 0.9 High Areas 34 9 39 1 33 5 30 2 23 5 23 3 11 2 11 4 4 8 Per cent Northern Southern Low Areas Total motion 38.3 30.4 40.8 44 6 42.6 34.8 39.8 42 5 33.9 30.5 39.3 43 8 31.1 24.1 36.0 39 4 26.6 19.7 29.2 32 6 22.7 18.5 28.6 32 9 10.9 10.4 14.6 17 4 12.2 9.5 11.1 13 2 4.9 4.8 5.4 5.3 "19" 15 Southern 28.3 36.3 34.8 30.5 24.4 21.1 11.8 8.6 5.9 28 II. SOUTHWARD AND EASTWARD COMPONENTS OF VELOCITIES IN HIGHS AND Lows Clouds Ci. Ci.S. Ci.Cu. A.S. A.Cu. S.Cu. Cu. S. Wind High Areas + 1 Q7 + 1 ft** 60 37 07 32 13 1 22 69 +0 IN 1 T? W 4-33 7 +32 +32 6 +27 2 +22 1 +16 + 51 + 58 + 11 Low Areas 10 VT c OR 9 24 3 00 4 60 2 38 4 00 11 1 32 40 1 c> VXT _i_qq A iqc q +37 2 +31 3 +24 3 4-24 3 +11 4 + 78 + 15 III. MEAN NORMAL COMPONENTS OF VELOCITY FOR THE UNITED STATES 1 6 3 8 1 8 2 5 1 2 2 2 1 1 3 05 or war +36 6 +34 4-34 9 +29 2 +23 2 +20 2 + 83 + 68 + 13 IV. COMPONENT VELOCITIES IN SELECTED AREAS BETWEEN HIGH AND Low CENTERS Selected Areas + 66 2 11 + 4 95 4-2 79 + 6 24 +10 22 + 6 52 + 5 25 + 2 23 +40 1 +36 9 +38 7 +26 5 +23 7 +22 1 + 96 + 75 + 32 Selected Areas 3 75 3 89 7 34 7 47 7 78 11 13 8 13 7 97 3 25 +32 7 +36 9 +32 1 +31 +21 9 +17 1 + 65 + 51 + 02 V. SEASONABLE VELOCITIES OF THE UPPER AND LOWER CLOUDS Clouds Upper clouds 8.4 Lower clouds 2.4 Seasons June Sept. Dec. March Ann. June Sept. Dec. March Ann. 20.0 17.2 27.7 19.6 High Areas Northern Southern Low Areas Northern Southern 30.8 24.2 39.7 25.9 33.7 35.6 45.2 30.9 37.5 36.0 47.4 33.1 41.7 27.7 37.1 34.8 35.5 29.1 42.6 32.4 12.8 16.3 19.2 12.9 23.2 17.8 32.8 21.9 24.9 20.0 31.5 18.7 21.8 18.2 27.5 16.1 All velocities in meters per second (1 m.p.s. = 2.2 miles per hour) NORMAL AND LOCAL VELOCITIES IN STORMS 227 The scale on Fig. 35 is twice as great as on Fig. 33 or Fig. 34. It is seen that at San Juan, and generally in the West Indies, the westward drift in the lower levels reverses into an eastward drift in the upper levels, the transition occurring in the A.Cu. and A. St. levels. Hence, the westward trade winds are shallow, Height Velocity Scale Metere Feet Miles m.p.h. 22.2 44.7 67.1 90 10000 9000 8000 7000 6000 5000 4000 3000 32808 m.p.s. 10 20 30 40 Ci.CiS Ci.Cu "A.St. A.Cu S.Cu Cu St. Wind 6.21_ 29527- - \\ 6.59- 26247 - 4.97 _ 22966- $ 1 4.35 _ 19685- 16404- 1 I - 3.11- 13123- } J 2.49 _ 98431 1.86 _< 6562- x^ / / - 1000 1.24- 3281- V 0.62 _ / FIG. 32. The total eastward velocity in high and low areas. Cloud heights. the greatest velocity westward being in the S.Cu. level (2,000- 3,000 meters), and that they give way to the eastward drift which prevails over the sub tropics in the higher levels. At Key West, in the midst of the North American high-pressure belt, the eastward and the westward drift is small in velocity, a similar reversal taking place in the middle levels. In the middle latitudes of the United States the eastward drift prevails in all 228 CONSTRUCTION OF VORTICES IN EARTH'S ATMOSPHERE levels, the velocities increasing from the surface upward. On these diagrams / stands for January and D for December, and all the intermediate months of the year are given in the line of FIG. 33. The eastward drift above Washington, D. C., for each month in the year. SCALE OF VELOCITY ^ 20 40 60 METERS PER SECOND ^ ^D Ci,Ci.S,Ci.Cu,. J ^^^, D A.S, A.Cu,. ^S 1 *. D S.Cu, Cu,S,. ** <-*x Wind ^ \ FIG. 34. The eastward and the westward drift above Key West, Florida. vectors. In summer the velocities in the tropics for the upper levels are small and disturbed in direction, showing that the circulation is diminished when the sun is north of the equator. In all cases circulation depends upon contrasts in temperature, so that a vigorous circulation occurs in winter rather than in ANALYTIC CONSTRUCTION OF RESULTANTS 229 summer, when the temperatures of the air in the northern hemis- phere are much more nearly equal than they are in winter. SCALE Of VELOCITY V *" METERS ER SECOND FIG. 35. The westward drift in the lower levels at San Juan, W. I., revers- ing into the eastward drift in the A.Cu., A.St. levels and the eastward drift in the upper levels. The Normal and the Local Isobars in Cyclones and Anticyclones The Analytic Construction of the Resultants The general theory of the separation of components can be illustrated by the following figures. Draw circles about the pole (see Fig. 36) representing baro- metric pressures at some level above the surface, ranging from 25.4 inches near the pole to 27.2 in latitude 20. At two points 230 CONSTRUCTION OF VORTICES IN EARTH'S ATMOSPHERE superpose a series of local circles, 1, 2-8, representing a defect of pressure at (1), and an excess of pressure at (4). Add the respec- tive values together at every point of intersection, and connect up the pressures having the same isobaric value. The resultant lines for (1) are seen at (2), and for (4) they are seen at (5). Latitude 20 c FIG. 36. The formation of cyclones in the general circulation about the poles of the earth. Putting (2) and (5) in continuous figures the resultant disturbed values are found at (3) and (6), the ordinary form of the isobars observed in the atmosphere somewhat above the surface, as on the 3,500-foot and the 10,000-foot planes. The same facts can be determined analytically as illustrated by Fig. 37. Take the co-ordinate systems as shown on the diagram. Let R = the radius of the circle, (a. b) the co-ordinates ANALYTIC CONSTRUCTION OF RESULTANTS 231 of the center, (x. y) the co-ordinates of any point on the circle. The general equation of the circle is (644) (x - a) 2 + (y - b) z = R*. Take 5 = and transpose the terms, so that, (645) y 2 = - x z + 2ax + R 2 - a*. The equation of condition for the isobar which is the resultant of successive circular isobars added to successive straight-line -1 FIG. 37. The composition of right lines and circles where the gradients are twice as great on the lines as on the circles. normal isobars, is that the sum of certain pair numbers shall be constant on the same line. Thus, A + B = constant, where A = n x, some multiple of the ordinate x, and B = the gradient number on the circles. For example, take the gradient on the normal straight lines one-half that on the normal circles, so that n HJ which is about the average in highly developed storms. Take successive circles, R = 6, 5, 4, 3, 2, whose gradient numbers are respectively = 0, -1, -2, -3, -4. Take a = 6, A = Y^ x, A + B = for the 0-line, and n = %. Similarly, by taking the proper groups of R, B, x for the 1, +1 . . . 2, +2, . . . lines in low and high areas, we 232 CONSTRUCTION OF VORTICES IN EARTH'S ATMOSPHERE obtain the resultants shown on Fig. 37. Curves can be com- pounded analytically through trieir equations, when the equations of the lines are known, but as this is not usually the fact for the lines representing meteorological gradients, and lines of equal meteorological values, resort must be had to some graphic proc- ess of construction. Graphic Construction of Resultants The first step is to determine the normal isobars, isotherms, and wind vectors on certain selecte'd planes. For the purpose we have chosen, as appropriate to forecasting requirements in the United States, the 3,500-foot and the 10,000-foot planes, besides the usual sea-level plane; and the Barometry Report contains the normals of pressure, temperature, and vapor pressure on these three planes, while the Cloud Report contains the normal vectors for the velocities of the air motions. We have, therefore, to reduce the observed data to these three planes, and subtract graphically the normals from them to obtain the local disturbing terms of the given element. Thus, in Fig. 38, if we obtained the oval lines from observation and subtracted the right lines from them, we should recover the circles or local components. Practically, lay down the normal values of the isobars or isotherms on a given plane, using transparent paper, superpose this upon the chart of observed values on that plane, and draw the diagonals of the quadrilateral figures that cover the diagram. In this way the charts of barometric pressure for the year 1903 on three planes (of which an example is given in Fig. 38) have been decomposed into the elements of the general and the local circulations from which we can study the general circulations on the one hand, and the local cir- culations on the other hand, without confusion. The observed pressures at the stations of the United States can be reduced to these planes by means of suitable tables, so that they are easily embodied in a telegraphic report without delay to the forecast message. This was done for a few weeks in a preliminary study. It is to be noted that the closed isobars of the lower planes soon expand into loops in the upper level. 115~ 110 105 s 100 3 95" 90 5 85 80 125 120 115 110 105 100 95" 9Q- 1 85 80 7 115 J 110 J 105 J 100 J 90 J 85 80 75 FIG. 38. The systems of isobars on three planes for the storm of February 27, 1903. 115 110 -.1 105 100 90 90 85 ,".-- J6^ 70 65 FIG. 39. The normal isobars (continuous lines) and the disturbing local isobars (dotted lines) in the storm of February 27, 1903. The normal lines were laid upon the observed lines of Fig. 38, and the dotted diagonals of Fig. 39 drawn. There are five closed isobars on the sea- level plane and only two closed isobars on the io,ooo-foot plane. The system of high and low areas on the sea-level charts soon opens up into sinuous lines in the upper levels. A general view of this fact is given on Fig. 40. ISOTHERMS IN CYCLONES 235 The Normal and the Local Isotherms in Cyclones and Anticyclones The isotherms as observed at a given time in the United States are separated by the same process into the two compo- nents, the normal and the local disturbance isotherms. Lay a chart containing the normal isotherms over the observed map and draw diagonal lines connecting up points having the same Observed Isobars Local Component Isobars Pressure Pressure Low High FlG. 40. Scheme of the distribution of the pressures in high and in low areas, in the observed and in the component isobars, on the levels up to 10,000 meters. These isobars are somewhat ideal, but they conform to conditions existing up to the top of the local disturbances in the atmosphere, that is to the cirrus region. The winter storms are cut off at 6,000 meters and even lower, while the summer storms can be traced much higher, on account of the relative retreat of the low temperatures to the higher levels. The U-shaped loops of the high areas open southward, and those of the low area open northward, so that in the upper levels there are sinuous, not closed isobars. Progress in forecasting con- sists in studying these upper plane auxiliary charts, in connection with the corresponding sea-level charts. 236 CONSTRUCTION OF VORTICES IN EARTH'S ATMOSPHERE 30.30 30.20 ,29.80 7() '29. 90 30.00 30 ' 10 Isobars and Isotherms on the Weather Map for February 27, 1903 FIG. 41. The weather map of February 27, 1903, showing the observed isobars and isotherms. -10 -10' Temperature Components for February 27, 1903 FIG. 42. The normal isotherms (full) and the local disturbance isotherms (dotted) which added together produce the observed isotherms of Fig. 41, Fahrenheit degrees of temperature. Height in Meters 6000 High Low High 5000 4000 3000 2000 1000 000 FIG. 43. Distribution of the high and low temperatures in cyclones and anticyclones up to the height of 6,000 meters, showing the tendency to divide into two branches with the maximum departure near the border of the high- and low-pressure areas. Centigrade degrees of temperatures. 238 CONSTRUCTION OF VORTICES IN EARTH'S ATMOSPHERE difference of temperature, which are the local disturbing iso- therms. Such normal charts of temperature on three planes are found in the Barometry Report. On Fig. 48 the composite of such a disturbance temperature system is given for nine cyclones, similar to that of February 27, 1903 (Fig. 4l). Fig. 42 shows how the disturbance isotherms cover adjacent high- and low-pressure areas. These cold and warm areas, as distinguished from the normal temperatures of the season and place, are what accompany all anticyclonic and cyclonic disturbances. The wind currents are simply the effect of the force of gravity trans- porting these masses of air of different temperatures, so that the cold mass underruns the warm masses, and the warm mass over- runs the cold masses to the right and the left hand, on all the levels simultaneously from the surface to the top of the disturb- ance. The campaign of extending the temperature observations into the higher levels is going on in different parts of the world, but definitive results have not been reached. There is no very general and fixed system of temperature values to be expected, because the incessant circulation, due to the annual change in declination of the sun, prevents the atmosphere from settling down into a simple thermal equilibrium. Fig. 43 gives an example of the distribution of the warm and cold areas in cyclones and anticyclones up to 6,000 meters. There is a tendency for the warm area to divide into two branches to the northward, and for the cold area to divide into two branches to the southward. The maximum departure of the temperature is somewhere between the centers of the low- and high-pressure areas, and it is not distributed symmetrically about the center as was assumed by Ferrel, and by the early German meteorol- ogists, in the construction of their theories of vortex motion. This asymmetric theory of vortices was first discussed in 1894 and in the Cloud Report of 1898, and the defects of the other theories were pointed out. The pressure and temperature data of observation were then entirely lacking in the upper strata, so that several years were allowed to elapse before the subject could be properly resumed, as was done in 1906 in the series of papers on the "Thermodynamics of the Atmosphere," Monthly ISOTHERMS IN CYCLONES 239 Weather Review, and continued in 1907, 1908, in the series of papers on the vortices in the atmosphere. The labor of securing Observed isotherms Local component isotherms Temperature Temperature High Low High High Low High. 000 FIG. 44. The observed isotherms and the local disturbing isotherms in high and low areas of pressure from the surface to 10,000 meters. There is a ten- dency for the warm mass to ascend and rotate through about one quadrant, changing the direction of the horizontal axis from N.E. to N.W., and for the cold mass to descend and rotate through one quadrant changing the direction of the axis from the S.E. to the S.W. The sinuous lines in the upper levels deepen in the lower levels, chiefly because the rapid eastward drift in the upper levels, which smooths out all kinds of disturbances, relaxes in the lower levels, and permits the disturbance components to dominate more fully. Compare these diagrams with the Figs. 41, 42, 4.8, and note the position of the line of 0- departure. Further observations will improve the accuracy of these diagrams. a sufficiently large amount of data in the upper levels, by balloon and kite ascensions, in order to eliminate temporary local con- ditions and secure average values, is so great that many years 240 CONSTRUCTION OF VORTICES IN EARTH'S ATMOSPHERE must elapse before meteorology will possess data for computa- tions of precision in this field of research. Meanwhile it is very important for students to have in mind a picture of the general phenomena so that useless discussions may be avoided. It is particularly necessary that the temperature values due to the location of the observatory, as a hill or mountain, should be thoroughly eliminated, because the temperatures on an elevation of land are not the same as that in the free air at the same height. Otherwise, errors at the base station would go into the entire series of gradients through a mistaken process of computation. The Normal and the Local Velocity Vectors in Cyclones and Anti- cyclones The results of the observations made on the velocities of the cloud motions at Washington, D. C., are embodied in Fig. 45 for the anticyclones, in Fig. 46 for the cyclones, and in Fig. 47 the entire system of stream lines is laid down as a whole. The complete vector as observed in different areas surrounding the center is first plotted, and then the component vector after the mean eastward drift has been eliminated. The resultant vectors are long in the upper levels and short in the lower levels; the currents are slightly sinuous in the upper levels, and as they escape from the eastward drift they become more nearly cyclonic. In the component vectors there is a maximum velocity in the 3,000-meter level, and a decrease in velocity upward and down- ward. The cyclonic disturbances often penetrate to the 10,000- meter level, but in many cases they cease before the cirrus level is reached, and as a practical matter they do not retain an im- portant value above 4,000-5,000 meters, because the eastward drift there predominates. It should be noted that in the cyclonic components the U-shaped opening at A is in the northwest quadrant of the upper levels, but in the northeast quadrant of the lower levels; in the anticyclonic components it is in the south- west quadrant in all levels. As can be readily inferred, it becomes a very difficult task for the meteorologist to construct analytical formulas to cover these complex curves, and to include the press- ures, temperatures, and vectors under the general equations High Area Vectors Anticyclonic Components 76 M P 4.66 miles 0.62 mUea FIG. 45. Mean vectors of velocity and direction in high areas. I cm. = 500 kilometers for the distances, i mm. = 2 meters per second = 4.48 miles per hour for the velocity vectors. Low Area Vectors Cyclonic Components 4.66 milet 0.62 miles Surface FIG. 46. Mean vectors of velocity and direction in low areas, i cm. = 500 kilometers i mm. = 2 meters per second = 4-48 miles for the distances. per hour for the velocity vectors. THE LAND CYCLONE 243 of motion requires unusual skill. Some years may, therefore, elapse before a satisfactory general theory can be perfected. In what follows there is only possible a series of fragmentary prop- ositions regarding circulations in the atmosphere. Observed total wind vectors Local component vector Velocity Velocity High Low High Low High FIG. 47. Observed total and observed local component wind vectors connect- ing high and low areas. The Land Cyclone It has been shown that the ocean cyclones are imperfect vortices of the dumb-bell-shaped type, which depart from the nearly perfect forms found in hurricanes and tornadoes, through the effect of certain resistances that are represented by com- ponent reversing vortices. These departures may become very 244 CONSTRUCTION OF VORTICES IN EARTH'S ATMOSPHERE irregular, leaving only the remnants of pure vortex motion from point to point in the cyclone, in proportion as the new system of thermodynamic forces, due to gravity acting on masses of air of different temperatures, are not symmetrically distributed about an axis. In the pure vortex motion of the tornado and the hurricane the* e was no need to consider specifically the action of gravity on the vortex motion, because of the symmetrical disposition of the air masses in superposed horizontal layers. In the cyclones, on the other hand, the differential action of gravity on adjacent air masses of different densities becomes the primary consideration, as demonstrated in Chapter II, so that the vortex action, though still of influence, becomes of secondary dynamic value. The study of temperature distributions in cyclones and anticyclones, together with the corresponding velocity vectors and pressures, must be first determined by observations before the dynamic theories can be suitably applied. The ocean cyclone has been used as a transition be- tween the hurricane and the land cyclone, in order to bring out the method of the composition of vortices. In the land cyclone the departures from the perfect dumb-bell vortex are very great, especially in the upper strata, where the head of the vortex is depleted by its intrusion into the rapid eastward drift whose average velocities increase with the height above the surface of the ground. This subject is so very voluminous that only the leading features can be brought out in this place. The cloud observations, made by the United States Weather Bureau, 1896-1897, showed that when the true cyclonic com- ponents of velocity are eliminated from that of the eastward drift, there remain a cold current on the western side of a cyclone and a warm current on the eastern side, and that this arrange- ment persists in a general way from the ground up to the cirrus levels, 10,000 meters. The tangential velocities v are at a maxi- mum in the strato-cumulus level, 3,000 meters, and they decrease downward and upward, the lower part being the truncated portion of the vortex, while the longer upper part is gradually destroyed by degradation in the eastward drift. The radial velocities seem to be inward from top to bottom, taking the THE LAND CYCLONE 245 cyclone as a whole; or rather the inward flow on the west and the outward flow on the east side do not appear to balance in the different levels, so that the mean velocity shall become inward below and outward above, as required by the perfect dumb-bell vortex. In this respect the funnel-shaped vortex, with the tube pointing upward, was suggested as the proper mode of analysis, but the analogy does not hold in its details. The determination of these radial velocities, upon which so much depends, in the upper strata is really very difficult, and some suitably located observatory might properly devote several years of observations to the elucidation of this point with precision. It has been proper to make a resume of the observations, so far as was required to bring out the theory of the subject. We began with the pressures and then took up the temperatures and the velocities in the levels up to 10,000 meters. The land cyclone differs from the ocean cyclone especially in the fact that it is not so highly developed as a dumb-bell- shaped vortex. The barometric pressure in the ocean cyclone sometimes falls to 28.00 inches (711 mm.), while in the land cyclone it seldom falls below 29.00 inches (737 mm.). This deficiency of the central areas in the vortex tubes is due to a variety of causes, but the principal fact is that the air masses of different temperatures are placed side by side on the same horizontal plane instead of being superposed; and the second point is that the penetration of the head of the vortex into the eastward drift of the general circulation is followed by its de- pletion, which is caused by stripping away from the vortex of fragments of the masses of ascending air. The meteorological data that serve to illustrate these facts can be briefly presented. A study has been made of the location of the isobars, the variations in the temperature, and the wind velocity and direction in a large typical land cyclone, by constructing the mean values for a composite of nine selected cyclones, March 16, 1876, March 27, 1880, April 18, 1880, January 12, 1890, December 3, 1891, November 17, 1892, April 20, 1893, January 25, 1895, November 22, 1898. They were chosen such that the cyclonic center occupied nearly the same place in the United States, 246 CONSTRUCTION OF VORTICES IN EARTH'S ATMOSPHERE namely, the lower Ohio Valley, and they were about equally developed at the axis. On the weather map the scale is 1 mm. = 10,000 meters. The linear distances of the radii to each isobar TABLE 57 The Land Cyclone I. THE RADIAL DIMENSIONS (i) (2) (3) (4) (5) (6) B 760 755 750 745 740 735 25 1250000 975000 733000 508000 300000 110000 log ztr 6.09691 5.98900 5.86510 5.70586 5.47712 5.04139 logp 0.10791 0.12390 0.15924 0.22874 0.43573 II. A/ VARIATION OF THE TEMPERATURE FROM THE MEAN DISTRIBUTIONS s -1.3 -1.5 -1.1 -0.5 +0.2 +0.9 S30E + 1.3 +1.7 +1.9 +1.4 +2.3 +2.7 S60E +2.9 +2.8 +2.9 +2.9 +3.0 +3.2 E +2.1 +2.1 +2.8 +2.8 +2.4 +2.1 E30N +1.7 + 1.8 +1.7 + 1.7 + 1-2 +1.0 E60N +2.2 +2.3 +2.1 + 1.8 +1.6 + 1.1 N 0.0 +0.1 +0.1 +0.7 +0.8 +1.3 N30W -1.1 -1.2 -1.0 -0.4 +0.4 +1.3 N60W -3.2 -3.0 -2.8 -2.4 -1.0 +0.8 W -6.4 -5.5 -4.3 -2.7 -0.7 +1.7 W30S -7.1 -6.1 -4.7 -3.3 -1.1 +1.0 W60S -5.6 -4.6 -3.3 -2.1 -0.4 +1.1 III. WIND VELOCITY AND DIRECTION WITH THE ISOBAR S S30E S60E 6.4 47 6.0 45 6.7 37 7.3 44 7.6 46 7.8 41 10.0 43 9.1 46 8.7 44 11.9 43 10.3 44 9.4 45 12.6 44 10.8 41 9.9 39 12.6 46 11.0 38 10.1 35 E E30N E60N 6.9 40- 7.9 42 7.8 29 7.6 42 8.1 45 7.6 35 7.2 43 8.5 46 7.8 41 6.8 41 8.2 46 8.0 44 7.3 39 7.9 45 7.8 43 8.5 37 8.2 45 7.6 40 'N N30W N60W 6.5 54 6.6 55 8.1 49 7.3 54 8.7 51 10.2 46 9.0 54 10.8 48 11.0 43 11.1 53 11.6 45 11.0 41 10.9 50 10.4 42 10.4 39 9.3 47 9.6 39 10.2 38 W W30S W60S 8.9 44 6.9 39 6.7 51 10.8 40 8.4 36 9.4 51 11.4 38 9.8 35 10.7 52 11.5 35 10.8 34 11.2 50 11.3 33 11.1 34 11.1 45 11.5 31 11.2 34 10.7 39 THE LAND CYCLONE 247 were scaled in the N.W. - S.E. and S.W. - N.E. directions, and the means were taken for the equivalent circular isobars. All the data of Table 57 are given in the metric measures. The section I contains the barometric pressure B, the circular radii N30 W E60N N60VV E30N -7 W30S S60E -3 FIG. 48. Land cyclone with circular isobars equivalent to the elongated cyclones of the United States, with the temperature disturbances and the wind vectors, the center being located in the central valleys. ZD-, and the log p = log ^ n . In a pure vortex log p = constant, but in the land cyclone log p is not a constant. Hence it follows that the pure vortex laws no longer prevail, though their influence continues to be felt. On Fig. 48 the isobars are laid down, and radii are drawn for every 30 degrees, making points of intersection where the com- putations can be concentrated. The isobars are spaced too widely near the center of the land cyclone, that is, the barometric pressure does not fall near the axis sufficiently to conform to 248 CONSTRUCTION OF VORTICES IN EARTH'S ATMOSPHERE the outer isobars from which the vortex is to be constructed. The temperatures were scaled from the weather maps at the 72 points of intersection just indicated, and the means for each point were taken and plotted on diagrams, one for each radius. As a matter of fact, the compilation was made in the English system, and this involved 132 readings for each of the nine cyclones. Similarly, the wind directions relative to the isobars, and the velocities were measured at the same points. The re- sults transformed to metric measures appear in sections II, III, of Table 57. After the temperatures at the several points had been found, it was necessary to subtract from them the average undisturbed temperature of the region, that is, the normal tem- perature for the average of the dates of the years in question. It is desired to know what disturbance of temperature accompanies the cyclonic movements of the air, as distinguished from the normal temperatures which are due to the general circulation taken by it- self. The section II contains these differences, which are also plotted on Fig. 48, together with the vectorsof the wind circulation. It shows that the maximum departure for the cold area is on the S.W. edge of the cyclone, and that the maximum depart- ure for the warm area is in the S.E. quadrant generally, the line of 0-departure running nearly due north and south through the center of the cyclone. The mean angle of the vector is i = 43, though it ranges from 27 to 54 in an irregular fashion from one point to another. It has been shown that a similar asym- metric distribution of the temperature prevails within the lower levels, being at a maximum of departure in the strato-cumulus level, 3,000 meters, and disappearing above in the cirrus level, 10,000 meters. When there are masses of air of different tem- peratures on the same level, the densities are different, and the action of gravity is to set up currents which cause the cold cur- rents to underflow the warm currents, and the warm currents to overflow the cold currents. The effort of gravity is to restore the isobars to a normal value when they have been disturbed by abnormal temperature densities. The air masses are trans- ported from the north or from the south into some middle latitude, where this underflowing and overflowing process sets THE LAND CYCLONE 249 up the cyclonic and the anticyclonic circulations. This prin- ciple can be illustrated by a vertical section running from west to east through a series of cold and warm masses of air. In a cold mass the isobars are concentrated near the surface and opened in the upper levels; in a warm mass the upper isobars are concentrated and the lower are opened. When these cold and warm masses alternate with one another the cold underflows in two opposite directions, and the warm overflows in two opposite direcions. In effect in nature, the cold mass from the "Warm Vertical Section West to East Cold Warm Cold Warm W E. Horizontal Section South to North FIG. 49. Model of the action of gravity G in forming streams of air which underflow and overflow the warm and cold masses on either side. The isobars in warm masses are relatively open below and closed above ; in cold masses they are relatively closed below and open above. Gravity tends to restore them to the same barometric levels, and the cyclones and anticyclones are the effect of this process of circulation in the impure vortices of the dumb-bell-shaped type. A thermodynamic discussion of the cyclone and the anticyclone has been given in Chapter II, showing the interplay of the general forces: gravity, pressure, circulation, and radiation. We shall next give a summary of the data for the land cyclone, which will include the entire series of terms in the equa- tions of motion, transformed to the vortex type, as in Formulas (561), and their various modified forms. north flows southward and divides, underflowing two warm masses on either side, while the warm mass flows northward and over- flows two cold masses on either side. The result of this complex system of currents is to produce the cyclones and anticyclones, and the tendency is to approach a dumb-bell vortex, though the resistance is too great in general to permit this to be done. 250 CONSTRUCTION OF VORTICES IN EARTH'S ATMOSPHERE Recapitulation of the Formulas for the Dumb-B ell-Shaped Vortex, (526)-(550), Fig. 18 We resume the formulas for the dumb-bell-shaped vortex in connection with the cylindrical equations of motion, (526) to (550) and (406), illustrated by Figs. 14, 18, including all the terms: inertia, expansion, deflection, friction, radiation, circula- tion, pressure, and gravitation. If + i is the angle between the tangent and the horizontal velocity of motion, positive (+ ) on the outside, negative ( ) on the inside of the circle, we have az = 90 + i, the angle from the radius, which in the complete vortex passes from a z = and i = 90 on the lower reference plane for the air inflowing along the radius, to a z = 90 at the middle height and i tangential, to az 180 and i = + 90 on the upper reference plane for air outflowing along the radius. The intermediate inflowing and outflowing angles are all de- termined by the relation of the line integral of the air flowing radially to or from the axis, and the surface integral of the air that rises in the vortex from the surface upwards. For a z = 60, i = - 30 and for az = 120, i = + 30. Hence, we have for (646) Angles, cos az = -f- sin i, + sin a z = + cos i, u cot a z = + tan ^ = . (647) Velocities, u = A aw cos a z = -f A a w sin i, v = A a iff sin az = -j- -A a iff cos i, w = 2 A sinaz = -f 2 A cos i, u = cot a z = + tan ^, u aw aw = -- cot a z = tan ^, w Z Z 2u 2u w = --- tan a z = cot i. aw aw (648) Line Integral, 2x w u = 2 ir A a w 2 cos a z = 2 IT A aw 2 sin i. (649) Surface Integral, xwi w = + 2 n A w* sin a z = 2 x Aw 2 cos i. RECAPITULATION OF FORMULAS 251 (650) Ratio, = - - = a cot <z z W TffW tan u a. v 1 2 (651) Tangential Angle, tan i = . 2u u cot az = . v Meaning of the Tangential Angle i The line integral of velocity is the product of the closed line, circle, ellipse, or any other boundary line, multiplied by the mean velocity at right angles to it, or more properly the integral of the mass velocity per unit length around the boundary, ff uds where u is the velocity perpendicular to d s at every point. For a circular vortex symmetrically disposed to an axis at the radius uf the velocity u is radial and the same at every point of the boundary circle 2 TT tcr, so that the line integral is, S = 2 TT iff u It is assumed that the inflowing air at the bottom of a vortex, for example in a tornado, is not congested and compressed, and therefore the inflowing mass must escape by rising upward from one level plane to another in the surface integral I, = J J w dS. The inflowing mass 2 iriff u escapes vertically with the velocity w through the plane whose area is irttr 2 , so that the surface integral is simply Trttr 2 w, not now counting the impermeable bottom, or the ^ cylindrical surface. The same mass of air entering the vortex ra- dially and horizontally on one plane escapes vertically on the next adjacent horizontal plane, and it is the vortex-constant a and the tangential angle i that controls this flow, through the equation, w r 2 . w FIG. 50. The line integral and the surface integral in vortices. 252 CONSTRUCTION OF VORTICES IN EARTH'S ATMOSPHERE u aix v = tan i = - tan ^ w 2 w 1 Zirlffu 2u u Hence, tan * = _ . -- r = - - = -, and a tan i = the " 2 ' ratio of the line integral to the surface integral. Similarly from (523) to (525) for the funnel-shaped vortex, and from (527) to (529) for the dumb-bell-shaped vortex, we may summarize, Funnel Vortex Dumb-bell Vortex , . u 1 u (652) - = - . - = cot a z = tan ^. V Z V u iff 1 u -or -of = .- = a cot a z a tan ^. w 2 z w 2 2 v w v tcr w ~ ~ 2 ' w ~ 2 a ' 2 21 w = - . z .u. w = -. - . . u. zcr -ay a tan i The connecting link between these vortices becomes, (653) - -- = tan*', a z the difference of sign depending upon the fact that the vertical axis was assumed in opposite directions in these vortices. The tangential angle i varies from one plane to another, 90 on the lower reference plane, gradually changing to on the middle plane where there is no inflowing or outflowing air, then con- tinuing to + 90 on the upper reference plane. These are due entirely to the supply of air needed to balance the inflowing or outflowing line integrals with the increasing or decreasing vertical surface integrals over the same planes in succession. It has been customary for meteorologists to explain these tangential angles in cyclones as the effect of the deflecting or the friction forces. Thus, by equations (480) (481), AM n _ s* (654) tan 2*1 = - = j , for the inner part, tan i 2 = - = -T-, for the outer part, but in fact the theory is erroneous. The deflecting forces de- pendent upon A are small, and those depending on the friction MEANING OF THE TANGENTIAL ANGLE * 253 k are nearly negligible. The inflowing tangential angle in cyclones at the surface is due to the supply of air necessary to compensate for the rising air over the entire surface of the closed isobar, and therefore depends upon the integral of the entire thermodynamic system. In the hurricane the inflowing angle on the ocean shows the amount of air that is required to supply the mass of air that is flowing away in horizontal radial directions in the high levels, underneath the cold stratum that has flowed as a sheet over the tropic region. Since tan i = cot a z, the vertical distance of the azimuth plane from the ref- erence planes can at once be found and thence the structure of the entire cyclonic vortex can be deduced by the preceding methods. Since we are not dealing with pure vortices in the case of cyclones, these simple laws must be modified from point to point according to conditions, and we proceed to evaluate the land cyclone in the several complete terms of the equations of motion, applied to the dumb-bell system. Making the substitutions indicated, we find the following system of equations for a symmetrical circular vortex with (406) in cylindrical co-ordinates. 3P du du du v 2 (655) Rad 1 al ) -^ = r ,+ ^+ W g- --- 2 w 3 cos 6 . v + ku+dQ & . = + A 2 a 2 tff 2 o> 3 cos .A atxrcosi + dv 8 v 80 uv Tangential, =- + 2o> 3 cos . u + k v + = r-T + . . . + 2 us cos B.A a -& sin i + ot dw dw dw , Vertical, - = -^ + u - + w + g+ kw+dQ z . -- ^- = - - 4 A 2 a sin i cos i +g+k w+d Q z poz o t 254 CONSTRUCTION OF VORTICES IN EARTH'S ATMOSPHERE These equations contain the inertia, convergence, centrifugal, deflecting, friction, and radiation terms in succession, reduced to a form for computation at any point where the data are known. Example of the Evaluation of the Terms in the Equations of Motion for a Cyclone As an example in the evaluation of the several terms in the radial, tangential, and vertical equations of motion for a cyclone, I have taken the data of Table 57 for B. P. T. p. R. -or. q. - i } as in Table 58A, and have computed a, a ttr, A , u, v, w, A iff, A P, p m , in succession. In Table 58s is a summary of the several terms, as indicated in the equations of motion, for the pressure, convergence, deflection, inertia, and friction combined with radiation. In the vertical component the data and results are from Table 25, (Qi - Qo) in the first column for the levels 000 to 500 meters, these data being divided by 500 to give the heat losses per meter in a vertical direction. The other data of Tables 58B are reduced to the unit length in all cases for compari- sons. The pressure term radially is very much larger than the sum of the convergence, deflection, and inertia terms, so that the remainder friction plus radiation amounts on the average to .0014582 per meter in mechanical units (M. K. S.) Since press- ure acts inward these two terms act outward along the positive radius. The result shows how impossible it is to balance the terms of the equation of motion without friction and radiation. It is not now known how to separate friction energy from radia- tion energy directly, and thus evaluate them separately. These terms in the tangential component are much smaller, and act in the antirotational direction. The radial component averages 0.00146, and the tangential component is 0.00038, that is about one-fourth the amount. It is probably true that most of the tangential term is due to friction alone, and since i = about 45 we may suppose that the same amount of friction energy applies to the radial component, leaving 0.00108 for the transported heat energy towards the axis of the cyclone. The vertical component is without friction and amounts to 0.11648, so that more than 100 times as much heat is transported upward vertically as inward radially. EVALUATION OF TERMS IN EQUATIONS 255 TABLE 58A SUMMARY OF THE DATA OF OBSERVATIONS FOR THE VORTEX TERMS IN A TYPICAL CYCLONE Quantities (1) 1 (2) (3) (4) (5) (6) Formulas Barom. pressure, B 0.7600 0.7550 0.7500 0.7450 0.7400 0.7350 m mercury Force pressure, P 101323 100656 99988 99323 98655 97990 (M.K.S.) system Temperature, T 290.7 287.1 284.9 283.2 281.9 281.1 South 281.8 281.8 283.1 283.0 282.6 281.7 East 262.9 266.3 269.2 272.7 275.8 279.0 North 266.5 268.1 270.1 273.1 276.6 281.2 West Density, P 1.2143 1.2215 1 . 2227 1.2219 1.2193 1.2145 South 1.2526 1 . 2444 1 . 2305 1.2227 1.2163 1.2119 East 1.3427 1.3475 1.2941 1.2689 1.2462 1.2237 North 1.3246 1 . 3080 1.2897 1.2613 1.2427 1.2140 West Gas coefficient, R 287.033 287.033 287.033 287.033 287.033 287 . 033 Constant Radii, u> 1250000 975000 733000 508000 300000 110000 Meters OA1 K 180 180 Vortex constant, a . UlO = 12000 " 9000 + 3000 45 truncated Vortex product, a w 18750 14625 10995 7620 4500 1650 Tube constants, A . 000341 .000499 .000910 .001562 .002802 . 007633 g _ u cosec i a co 357 519 655 986 1621 5156 E 347 500 829 1457 2422 5635 ^ ^ v see i a to 475 738 1037 1510 2513 6966 W Velocities, Q 6.4 7.3 10.0 11.9 12.6 12.6 S Meters per sec. 6.9 7.6 7.2 6.8 7.3 8.5 E 6.5 7.3 9.0 11.1 10.9 9.3 N 8.9 10.8 11.4 11.6 11.3 11.5 W Tangential angle, i 47 44 43 43 44 46 S Inflowing angle 40 42 43 41 39 37 E 54 54 54 53 50 47 N 44 40 38 35 33 31 W Radial velocity, u -4.68 -5.07 -6.82 -8.12 -8.76 -9.06 S u = q sin i -4.30 -5.08 -4.91 -4.46 -4.59 -5.12 E -5.26 -5.91 -7.37 -8.87 -8.35 -6.80 N = A a 75" sin i -6.19 -6.94 -7.02 -6.60 -6.16 -5.92 W Tangential velocity, v 4.36 5.25 7.31 8.70 9.06 8.76 S v = qcoai 5.29 5.65 5.26 5.13 5.67 6.79 E 3.82 4.29 5.35 6.68 7.09 6.34 N = A a oT cos i 6.40 8.27 8.98 9.41 9.47 9.86 W Vertical velocity, w .000465 . 000718 .001458 .002283 .004027 .010618 s.-lt a <o 564 773 957 1346 2520 8230 E 408 587 973 1753 3116 7685 N = 2 A cos * 683 1131 1633 2470 4209 11952 W = cot * o5 Differences A w 275000 242000 225000 208000 190000 The mean values of two successive Differences A P 667 668 665 668 665 tubes placed under the second of the Means Pm 1.2179 1.2221 1.2223 1 . 2206 1.2169 pair from which they 1.2485 1.2375 1.2266 1.2195 1 2141 are computed. 1.3451 1.3208 1.2815 1 . 2576 1 2350 1.3163 1.2989 1.2755 1.2520 1.2284 The ratio of the line integral to the surface integral checks. 2u a u a tan * = = = . taw v 256 CONSTRUCTION OF VORTICES IN EARTH'S ATMOSPHERE ^S j| en O 8 1 d TTI + en ^?^ |^' jj 3 8" i 1 " W 3 S to ^ s e <S S s I S i ^ 8 ^ (M rH CO rH 1 1 ' + l> 00 CO rH 00 O5 00 00 O . ^ CO CO e i 8 8 ! i i 1 rt< CO CO CO CO (M CO o u ^"^ c^ ^^ rH rH ^ 8 8 8 8 a i 10 a rr. H W S so' g ?22go. (M CO w 8 ^ ^^ i ^- ^ rH rH 3 51- 8 8 8 8 H 1 1 ' & 00 ^O 00 CO S 2 S S ^ If g 1 8 D 3 ^ ^s lO I>- t^* ^^ **"t^ i-H > to W (C f 1 1 rH CO O b- O5 Tj< IO I iS e 8 I 8 CX 1 ]a rH "O I d o | H 2 &l rt m > cu'S d O d "u .- 3 >- . .0017365 00 CO rH O rH |>. TjH CO CO *O IO CO rH CO O 00 rH rH I IO CO TjH 00 00 b- Oi co rH ? l> 10 CO Ci 8 3 8. 8 CO CO CO *O O T-H CO O O^ *H IO 00 1 I 8 8 00 Oi CO ^O 1 1 i 8 i IP IO rH CO 00 IO CO 8 I 8 8 d .2 i 8 i 1.2 3 b .S tg > <U -M d .2 8 * | UQrS 3 'i- (/) b EVALUATION OF TERMS IN EQUATIONS 257 5l 8 8 rH 8 Tt< rH IO rH t^- CO (M 00 T^ CO OS CO CO t> c^ S CO C^ C^l CO 00 tO <O I rH CO ^^ ^H ^^ 1 t^ rH 1 i 8 8 I i 8 8 00 10 l^ to CO Oi rH rH tO rH T^ <M tO CO CO I O5 CO Cl to -^ to ^f 00 CO 00 s i 8 8 = o 8 8 CO O CO rH CO t>- O5 CO (M CO O5 Oi t*^ " o O t^- CO to CO rH IO rH 8 0) g rH rH CO rH 8 1 1S1 t > * ^H rH >O 00 CO IO 00 00 CQ CO to O5 rH 8 8 8 8 i O *H 8 8 <M CO rJH C^l CO "^ ^D CO O5 CO N. to CO CO rH O i ^ O 5 '^H OO rH CO CO 00 O CO ^ ^ o 8 8 O rH 8 8 rH O 8 1 1 8 | \ \ j , j a d o .2 *o "O 8 2 c s g K -S rt 1 ||l 1 all 1 1 S c <u bfl.2 3 V t5 .2 c ~ C^ flj QJ 0^ f ^) O II d o en s .a 3 u. C/3 (x< 258 CONSTRUCTION OF VORTICES IN EARTH'S ATMOSPHERE 8 colco ff ^ O CO Tt^ Oi (N CO ^.oS 1 1 tj ctf o a; '-M '^ w co o oo O5 (N CO IQ <N 10 ^H 1 1 O5 + \ + l (M CO CO t^* 1 1 rt o II + l 1 1 CO 1-1 1 1 + EVALUATION OF TERMS IN EQUATIONS 259 8 a|| II 58 H.S bfl "^ G s. 3 0) 'o ^ _ -G 0) O Jt & | l^lllll 3 TO <-> ^ V LH f3 . C Cti ^ ""^ ^^ 2 E 3 ,, 'S)>3 ^ 88-^ Ctf c/3 CO ^'^ S 260 CONSTRUCTION OF VORTICES IN EARTH'S ATMOSPHERE Similar studies of the relation of the heat and radiation terms to the mechanical dynamic terms may be extended to the higher levels of the cyclone, and to the anticyclone, but the final conclusion to be remembered is that there is no possibility of balancing in a dynamic system of equations the several terms of motion, without including the radiation of heat energy, and its convection from point to point. This branch of meteorology will require much further study along the lines that have been developed in this Treatise. CHAPTER V Radiation, lonization, and Magnetic Vectors in the Earth's Atmosphere THE incoming solar radiation separates into two parts, the first the irreversible heat that cannot be transformed back into the original energy, the second the reversible energy which appears as electrical and magnetic forces. The heat energy is observed as the air temperature at different points, and its effects are found in the general and the local circulations of the atmosphere. The short waves of the solar radiation, at very high temperatures, as 6700 to 7700, are capable of producing ions of positive and negative electricity by the disintegration of the atoms and molecules of the gases that compose the air, whereby a part of the radiation energy reappears by transforma- tion as free ions, or free electric charges, more abundantly in some strata than in others. These ions tend to move in electric streams, in certain general lines as controlled by a series of physical conditions, and in their movement they induce magnetic deflect- ing vectors, which disturb the earth's normal magnetic field, through whose lines of magnetic force the electric ions move. We have, therefore, to study the distribution of the solar radia- tion in the atmosphere, the production of free electric charges through ionization, and the dependent induced magnetic de- flecting vectors. In spite of prolonged researches in these sub- jects by many students, there is a wide discrepancy in the results, as whether the solar intensity of radiation is 2.00 calories or 4.00 calories, whether the absolute coefficient of electric conduction is 2X10~~ 5 or 6 X10~ 5 , whether the vectors that produce the diurnal variations of the magnetic field originate in the higher or the lower strata of the atmosphere, whether the sun is a star with a variable periodic output of radiation or practically con- stant, and whether the annual changes of the weather conditions are dependent upon solar variability or are merely accidental. 261 262 RADIATION, IONIZATION, AND MAGNETIC VECTORS The Determination of the Intensity of the Solar Radiation by Observations with the Pyrheliometer and the Bolometer Measurements of the intensity of the solar radiation are made by the pyrheliometer, which integrates in bulk the rays received by the instrument, and by the bolometer which measures the individual lines in the energy spectrum. The two instruments supplement each other, because while the pyrheliometer gives no account of the selective absorption of lines and bands in the spectrum, the bolometer defines these depletions, and permits the comparison of the observed spectrum energy with that of a full radiator at the given temperature. Neither instrument gives any account of that portion of the incoming solar radiation which is reflected back to space, as the albedo of the earth, but this can be found indirectly by thermodynamic computations on the temperatures of the atmosphere, as observed in balloon ascensions to great elevations. The Pyrheliometer. This instrument consists of a chamber for receiving a bundle of the solar rays, whose temperature can be accurately measured at any time. The temperatures are measured when the solar rays are shaded by a screen, and again when exposed to the radiation, the sum of the changes of tem- peratures in a given interval of time, as one minute, being the effect of the radiation in temperature degrees. A factor can be found by experiment which will convert these temperature changes per minute into calories per square centimeter per minute. There are many types of actinometers or pyrheliom- eters, in which the different materials used for receiving the radiation are involved with certain conversion coefficients. The earliest form of pyrheliometer, by Pouillet, 1838, consisted of a silver vessel rilled with a known volume of water, the surface being blackened to absorb all the radiation, whose increase in temperature in a given time could be measured by a thermometer whose bulb was embedded in it. Silver box with mercury, copper box with mercury, silver disk with no liquid, and many other combinations have been employed. The electrical resistance thermometer, the Angstrom double strip compensated pyrheliom- DETERMINATION OF INTENSITY OF SOLAR RADIATION 263 eter, in which the heat absorbed is measured by a compensating electrical resistance, are used with success. We shall confine our attention to Abbot's silver disk pyrheliometer, which has been standardized against an elaborate Primary Standard No. Ill, 1911, of the Astrophysical Observatory of the Smith- sonian Institution, and furnishes the comparison factors for copies of the secondary pyrheliometers. This instrument leaves little to be desired for durability and accuracy of its operation. More time is required for the observations with a thermometer system than with an electrical resistance apparatus, but the latter needs much more elaborate auxiliaries, battery, galvanometer, current, and resistance apparatus, so that it is less readily portable, and more liable to accidental inaccuracies of adjust- ment. Theory of the pyrheliometer. Let S = the entire surface of the body receiving radiation. 5 = the cross-section of the rays falling upon it. c = the coefficient of heat received referred to water, h = the coefficient of heat lost by radiation, dQ 2 = q = the intensity of the radiation received in dt, dQi = qsdt. Hence, the general equation of equilibrium is, (656) dQ = dQ 1 -dQ 2 = qsdt-hSTdt = cdT. The shaded or cooling term. If the body is in the shade, g = 0, and we have, 7 rr\ 1 Ql (657) -=-= ---dt. Integrate for T=T , when /=0. J. c h S* h S* (658) log T = - - - 1+ const. = - - - 1+ log T . c c T -* * -* t (659) TT = e '< or T = T Q e ' ^ This is the cooling correction for T= the excess of the tem- perature above the surrounding medium, when / is the interval of time elapsed, and T is the initial excess at which the cooling begins. 264 RADIATION, IONIZATION, AND MAGNETIC VECTORS The exposed or heating term. Divide the general equation by c, and we have, (670) dT= q dt---Tdt. c c From (656) the maximum temperature T m is obtained when d T = 0, (671) = qs-hST m , ~ = ~ T m . Substituting in (670), (672) d m . C C L m L C Integrate for T = T when dt = 0, since dT=-d (T m -T), (673) log(r, n -r)= - ^l+const.-- --t+log(T m -T,). C C T m T hS T m T - t (674) log ~^ ^r = - - t, and -~^ ^- = e c . Hence, 1 m 1 o c l m l o _h^ t _hS t (675) T=T e c +T m (l-e c ) for the total heating effect. This is the equation for the total effect of radiation when exposed to the solar rays. It consists of two parts: (1) the cooling term from the initial temperature T 0} and the heating term reckoned from the possible maximum temperature T m relative to that of the surrounding medium, where T is the effective difference of temperature at the time / above the sur- rounding medium. It depends upon the total surface S of the receiving body, the cooling coefficient h, and the heating co- efficient of the body c, referred to water. An Example of the Practical Observations with the Silver Disk Pyrheliometer, S. I. No. 7, 1911 Several auxiliary tables must be prepared. 1. The table for the equation of time for every day of the year. 2. The table of the declination of the sun for every day of the year. 3. The table of 2 log r, for the radius vector of the earth. 4. The secant of the zenith distance, secz, should be com- puted through the formulas leading to z. DETERMINATION OF INTENSITY OF SOLAR RADIATION 265 *o <u *o <u *o III 1 1 13 B g S cog || > m 4 rH rH rH N rH C 5 rH c i s ss s ss 3 r if f r + r r ^s K < ri N N 00 CO O N IO lOrHOJ t-t-CO OO^ NrHrH rHOO COCON OJOO ****&> i <i 1 1 1 If"-* 1 + III gco oo w COrHO OOOOOIO COOOt-00 00 UO <M lO 0000CTj< O 00 . W5NN NtMINN INNNIM CJNNN to OOOO OOOO OOOO OOOO OOOO coco coco coco coco coco c oo ^ *, I "rji 10CO t-00 00 rHM ".CO COCO COCO CO 1 ^ 1 '*^< *s OrH rH rH o. 00 v (N 1 8 8 o co o i I r-H -t r- t- <o o c rH r-i < D IO H CO 5 i '+ f r + r r I < F> ! ggrH OOOrH rH C, 3 ^ ^ ^ ^^ < 1 + + 1 |' ' *COCOTJ OOCOU3CD IflTjtNOO OOtOOON OOrj<t>]O ^,^^4 t.3 fH NrHOOO TTrHOOiO USCOrHO "^ odoO*Oo'od OOOOrH rHrHrHo' rHNNCO COCo'co'w c^ O 00 00 S us ^ COM CO COCO COCO CO lO l> i NCO T}<IO Ot- OOO OrH *+ = rH 3 | * a j * o v. 1 IN 266 RADIATION, IONIZATION, AND MAGNETIC VECTORS t x tan 5 tan'/ cos M tan(< M) (676) tan M = , tan A = ^ ( 1/r , , tan z cost cos A A carefully constructed diagram of curves is required for the interpolations to two decimals of sec z. Take four readings of the thermometer every 30 seconds for the beam shaded and exposed alternately during ten minutes. Compute the mean A7 for each group; take the means by pairs of the shaded terms, and subtract from mean exposed term lying between them; the sum of the two corrected A7 gives the rise in the thermometer degrees per minute; multiply by the pyrheliometer factor 0.610 to obtain I in calories; take log 7 for the ordinate on the diagram of the Bouguer formula. Adopt as epoch the middle minute of the watch, running on mean time, and add the equation of time +7 W 8 s to reduce to apparent time; subtract this from 12.00.00 for the hour angle t; take the sun's decimation of date 6=+0 27', and with /. 6 as arguments read off sec z on the scale diagram for La Quiaca. Plot the points (log 7, sec z) on the diagram, and draw the best mean line through the points of the day. Scale off the point (log 7 , sec 2 = 0), and reduce to the mean solar distance by the radius vector factor 2 log r to obtain log 7 . r and 7 . r . TABLE 60 LA QUIACA, ARGENTINA, SEPTEMBER 22, 1912 Similar pairs for September 22, 1912, at La Quiaca are as follows: Time Log/ Sec z Time Log/ Sec z A.M. 6.48.00 0.018 4.65 P.M. 2.0.00 0.205 1.24 Plotted as o for the A.M. 6.58 0.060 3.89 2.10.0 0.182 1.33 Plotted as X for the P.M. 7.8 076 3.37 2.20.0 0.190 1.38 8.21 0.157 .82 4.0 .0 0.141 2.30 8.31 0.180 .73 4.10.0 0.124 2.52 8.41 0.180 .64 4.20.0 0.114 2.73 10.28 0.202 .17 10.38 0.205 .15 10.48 0.202 .13 THE BOUGUER FORMULA OF DEPLETION 267 The most successful hours of observation are, weather per- mitting, 1. Group of five, ten minutes each, 7.00 to 7.50 A.M. 2. Group of three, ten minutes each, 8.30 to 9.00 A.M. logU 0.250 0.200 0.150 0.100 0.050 0.000 \ loglo lo.r 0.263 \ V lofiflo.r 0.267 1849 * \0 c IN logic Ii I 2 Is -Alogl \ log p 0.263 for .210 .157 .104 -.053 9.947 0.885 secz=0 =1 = 2 = 3 P \ o^v \ sec z 0.00 1.00 2.00 3.00 4.00 5.00 FIG. 51. Plotting the pyrheliometer observations. 3. Group of three, ten minutes each, 11.00 to 11.30 A.M. The corresponding computations can be completed in one hour, so that three hours suffice to obtain I .r on any good day. The Bouguer Formula of Depletion The incoming solar radiation is subjected to two types of depletion, (1) the reflection and scattering of the rays, on the molecules, ice crystals, dust, and other constituents of the at- mosphere, whereby a certain amount of energy is reflected back 268 RADIATION, IONIZATION, AND MAGNETIC VECTORS to space as albedo, and is not subject to measurement by the pyrheliometer. This type of depletion affects chiefly the very short waves in the energy spectrum, and its region of operation is especially in the high cirrus region; (2) the other type of depletion is due to selective absorption of certain lines and bands of the spectrum, which can be determined specifically by bolometer observations, with the sun at different zenith distances for the same station, or by observations on the radia- tion at the same zenith distance, from stations having different heights above the sea level. The law of depletion is generally expressed by Bouguer's formula, (677) / = / sec2 , where /o = the source of radiation before depletion, ^ = the fraction which is transmitted for the unit dis- tance, sec z = the secant of the angular distance from the zenith to the incoming ray. / = the radiation measured at the instrument. The formula is only correct for homogeneous rays. I. Passing to logarithms. (678) log / = log 7 + sec z. log p. Taking two successive observations at Zi, z 2 , (679) log /i - log 7 2 = (sec Zi - sec z 2 ) log p, hence, (680) log p = 1 ~ 2 f or p constant in the interval. sec zi - sec z 2 II. If it is assumed that sec z is constant and p variable, First line, log 7 = log 7 + log p. sec z. Second line, log /' = log /</ + log p f . sec z. (681) (log / - log /') = (log 7 - log 7o') + (log p - log p') sec z. (682) log j, = log -p + log j, sec z. By the first type of formula, it is seen that log I decreases as sec z increases. Taking (log /i log 1 2) for unit differences in (sec Zi sec z 2 ), as in the example, we easily find its mean value THE BOUGUER FORMULA OF DEPLETION 269 0.053 for the given line as drawn, by reading the successive values of log/ as the line crosses the sec z lines, 0.1.2.3 . . . whence p is found from log p. As the aqueous vapor rises from the surface upwards in the diurnal convection the absorption (1.00 p) increases, and p decreases towards the midday, so that the line joining the observed values of log / is likely to be a curve which is slightly convex toward the origin. In the dry climate at La Quiaca such a curvature was not observed, though it is common at such stations as Washington and Mt. Wilson. A refined treatment of the variable p can be made for the same zenith distance by the second form of the formula. The Bouguer formula indicates that the depletion of the incoming radiation is proportional geometrically to the length of the path m which it traverses in the atmosphere, and it is common to the large class of physical formulas which correspond with similar conditions. It has been generally assumed, in applying the formula to the earth's atmosphere, that the unit distance is the depth of the atmosphere in the zenith mo, and that what is measured by the pyrheliometer is the intensity of the solar radiation at the actual distance of the earth, before any depletion takes place. It is proposed to give several arguments which show that this is a clear assumption, which does not correspond with the facts of observation. 1. The Bouguer formula contains two unknown terms, since sec 2 is a ratio of two distances, m m' m" (684) sec s = = = 77 =...- mo mo mo in which the denominator m 0) m f , mo" is undetermined; and since only log / and log p can be deduced from the observations, log IQ is also to be considered an unknown term. For example, m may refer to one plane above the observer as the cumulus stratum, m ' may refer to another plane as the cirrus stratum, and m " may refer to any other plane as the outermost layer of the earth's atmosphere. The two unknown terms (log 7 , m ) refer to the source of the radiation 7 n on that plane whose unit 270 RADIATION, IONIZATION, AND MAGNETIC VECTORS distance above the surface m may be any plane in the atmosphere satisfying these conditions. If the incoming radiation is partly reflected back to space as albedo on the cirrus levels m Q ', then Outer I ( layer Cirrus layer Cumulus layer FIG. 52. Illustrating the use of the Bouguer formula. IQ is already diminished to that extent, and is not subject to observations by the instruments. /^o^\ W l W l W l (685) sec zi = = 7 = 77 = . . . . mo mo m m<t m z ' m 2 " sec z 2 = = 7 = 77 = . . . . sec Zz = = 7 mo mo m The Bouguer formula contains, therefore, a double ratio, (686) log -r = log p, that is -7- and & /o m IQ mo and it is quite indeterminate in itself, unless some means can be found to fix the unit distance m , whether 7 emanates from the cumulus, the cirrus, or the outermost layer of the atmos- phere. It has been customary to make m = 1, refer it to the outermost layer, and thus assume that the pyrheliometer meas- ures the so-called "solar-constant" /o", or intensity of the sun's radiation as it falls upon the earth's atmosphere, at W. THE BOUGUER FORMULA OF DEPLETION 271 s !" 00 00 t^; GO 00 OOOOO 1> t^ GO 00 00 ooooo OOOOO-^ rH rt< Oi O iO 00 Oi O > 1 <N (N (N CO COOiC^ i-i T-H (M CO OO ^O I s ** s ! C^ rH O5 CO IO OO 03 *H O-l l>- >O IO t>- Ttl TH O5 CO oooooooooo oq oq oq oq i> oq ooooo ooooo d i-<O <NCOC5t*-O OO Oi'* (N<MOO<MCO O lOrHI>C<lTH COCO^OIOCO CD OOOOOOOOOO OOOOOOOOOO OO ooooo ooooo d COOOCOt>-rH OO CO1>OOOO5 ^O OOOOO OOOOO 0* t^t^I>cD l>l>t^t^t^ t>- COOSOJCOt^ 00050^-100 rH rH rH C<l <N rH i-l II, - 4J 1-4CO1OCOO5 OOiOOOOOiO OOr}H '-ieO oqoooooo osoooiOioo oo ooooo o'oo'do o rf-rtiT^tOiO ^(M-^^tO Tt< OOOOOOOOOO OOOOOOOOOO 00 O f- 1 <N CO IO 00 Oi O rH C^ (MC^<M<M<M<M<MCO <N CO CO 00 i-H i I <M CO ^ Tj< *^H CO ^~* ^D '"^ CO C^ ^^ + + r + 1 i r + + + r + r r + r 11 +-H i I C^ ' i CO i I T-H CO i I (N r r i'++ +++ r C<IO ++++ CO 00 00 T IT- I ++-H i-H IO rH Tt< IO I> t- 8O T-H T-H Q O T- ( o o o o o o l-H CO 1-H CO T I i I o o o I ++ I + + + + I I -H II I I I + I I ++-H f ++ r r++++-H 8iO T-H CO <M ^ CO o o o o o o ++ r ++ r r T-H I I + I -H B8S88 88888 f + + + + + r + + + 272 RADIATION, IONIZATION, AND MAGNETIC VECTORS Si" QO CO O5 b~ oo iO' CO <N jo co t> . r* <N co : b- S- |> <MC5Tt<O<NCOOOT^^ COCOOOOOO(NCOCOCO ^1 O^ CO t^** CO JLQ ^H CO (N <M rH TH ^(N(MCO^T^OiT-(O5c35 CO<N(M(MCO(NCOCOCOCO +1 1 1 + 1 + 1 ++ C C oo : : 8 O5 b- I> CO ^^ O5 O5 O5 O5 O5 O5 O5 O5 3 TO aB nJ C fl TO C !> !X J k.fc TO -t-> " > THE BOUGUER FORMULA OF DEPLETION 273 1 s * 00 OO t^* 00 CO O^ O5 rH O5 rH oo "o3 > O5O5O5G5O5O5O5O5O5G5 s s ^OJ 1_1 <i> 00 CO t> -CO O5 rH rH rH ^ rH O5 O5 O5 O5 O5 CO rH t> oo rH O5 I rH rH rH rH rH ^ rH '8 jg rH rH . . . . O5 rH 05 rH 1 1 rH rH O5 rt GO rH rH rH O5 O5 CO - <N ^f O J^ C^l OS O5 O5 ^ O5 rt rH rH : :| : i CO 8 10 1 ~ IP ; rH rH rH rH rH 0) en 0> " ' " * o? " " " ^ t^ CO 0) ^x. c^ * 2 ^R ^^ ^ a OJ o l> ,_, .- rH rH ed 35 ' IN rH O5 O5 O5 1 s rH en o" rH rH en" 2 1 1 : : : :^ : : : 2 o -CU <-t-c r* o rH rH rH -" rH -(J c <U ::::::: :g i w Q ^C 00 00 r> ^J O 3 cd O <u ... co CO TjH OJ O5 l^ 7^ K oo GO H ?-H ^~ ^ > ^ ^^ C^ O^ "^ ^^ C^ CO 00 "^ "^ (NCOCOOOOOO(NCOCOCO '& X i OO01>rHtflrHlOOCOCO COCOCOOOOJO'-iiMIMfN 1 a 2 : : : : 1 *o3 o >* ^3 > 4J en 3 0) 8 s w etf - d C Sfii-S 1 |2 ^^3 v g^ s^^ 1 ^ > *s "5 rt > * > *S .S .2 !> ^, g g > >,> o ^ ^ . Of c - .3 . ^ tn en s^si^al! Means on the Means on the Ui i j < rHC^OOOOOJOJCO 1 ^ O5oooooooot>oooo dddddddd CS| rH 00 00 d d oo oo oo 05 ^ GO 00 00 t~ -00 ] d d d ] d d | :::::::: ; :::::::: :1 : : : : : : 8 '. : d : O5 O5 :::::::: t> 00 d d ; i ! ! oo ! ; ; d co oo 00 00 : : : : o : : : iO 00 rH O5 00 t> d d JO CO 2 rH d (NO5COOOOOOCNCO c^c^ '.'.'.'.'.'. '. * rt - Isi<| 1 Illli^l S/jffi^S.H.Sc'j II 274 RADIATION, IONIZATION, AND MAGNETIC VECTORS We shall now give strong evidence for thinking that the effective radiation Id is on the lower levels, its value being less than 2.00 calories, while about 2.10 calories is reflected back to space and escapes direct observation, so that the true "solar-constant" is really more than 4.00 calories per square centimeter per minute. 2. The depletion of the incoming radiation from a maximum value on the cirrus levels, as determined by observations at different heights. Several series of pyrheliometer observations have been made at stations having different heights above the sea-level, and after they have been reduced to homogeneous data, and referred to a given primary standard, they are comparable, and show an increase in the value of 7 . r with the height. The primary standard adopted is Abbot's Standard No. Ill, 1911, Smith- sonian Astrophysical Observatory. The data of Table 13, Vol. II, Annals, have been reduced by the factor 0.951 and the data of the United States Weather Bureau (Mt.Weather Bulletin) by the factor 1.039, those observations having been there re- duced previously to the Angstrom pyrheliometer No. 104. In order to exhibit the average conditions that prevail at different elevations, several 10-day series have been impartially selected from the large amount of material in hand. The individual values (/o- r > P) are given in Table 61, and in the second section their variations on the 10-day means. It is noted that the variations of the coefficient of transmission p, do not materially differ with the height, but there is a change in the mean values of p, approximately 0.825 for all stations below the 1,500-meter level, and 0.900 for all stations above that level. The variations of A /o. r decrease steadily with the elevation, and 7 .r increases with the height to a maximum value 1.918 which is to be identi- fied as belonging approximately to the cirrus level. This is explained more fully on Table 62, where the available annual values are collected for eight stations. Special attention is directed to the remarkable results at La Quiaca. THE BOUGUER FORMULA OF DEPLETION 275 It contains in section I, the station, observer, latitude <, altitude in meters z, the individual annual values for the 10 years 1903-1912, and the mean for each station increasing from 1.709 at Washington to 1.848 at La Quiaca and 1.861 at Mt. Whitney. These means are plotted on Fig. 53, as /o. r , alongside of p the vapor pressure of the aqueous vapor in grams per cubic meter /*, and the relative absorption of A/ . r by A/* = 1.00 gram per 500 meters of vertical path length as will be explained. In order to determine the maximum value of 7 . r for the pyr- heliometer observations, we need to know the bolometer factor Cirrus Mt. Whitney La Quiaca Maimara Mt.Wilson Jujuy Mt. Weather Cordoba Washington Vapor press Radiation in Relative ab 10000 9000 /* [lo.^a \ I 8000 u 7000 st I^Sfc, 6000 X5- ^ X. 5000 \ \ \ ( " 20) 4000- - \ X \ 5 ^ 3000 - \ X^ c (2384) 2000 - x \, ^^--*. x ^ C^ -^ (1780) (1302) 1000- - ^1, "-v^ N\ (526) ( ?I1! ooo: : "" "\\ are /J. 0123456789 10 Calories Io.r 1.900 1.850 1.800 1.750 1.700 sorption 0.1000 -0.0750 -0.0500 -0.0250 -0.0000 FIG. 53. The relation between the loss of radiation per i.o gram per cu. meter for 500 meters, and the ratio between them, a. F which will supply the amount depleted by line and band absorp- tion. Mr. C. G. Abbot has determined these for Washington, (1.123), and Mt. Wilson (1.094) from a long series of carefully executed bolometer observations. These factors refer the depletion to the adopted curve of a perfect radiator at 6,000 temperature. At present, similar factors are not available for the other stations in this list, so that the data of the table in reduced form are instructive rather than definitive, and it is very important to extend accurate bolometer observations to the other stations employed in making pyrheliometer observa- 276 RADIATION, IONIZATION, AND MAGNETIC VECTORS tions. In section II, the annual values are all reduced to the sea level by the bolometer factors FI, and in section III to the cirrus level or approximately to the 10,000-meter plane by the factors F 2 . The mean value of I . r on the sea-level is 1.711 calories; and on the 10,000-meter plane it is 1.918 calories. Section IV contains the mean annual values of p, approximately 0.900 above 1,500 meters, the top. of the diurnal convection, and approximately 0.825 below that level. This difference in the transmission factor p is due to the increase of density, p, in the lower atmosphere. An example is taken from the data derived from balloon ascensions for summer in the temperate zone, which gives a fair distribution for the year in the low latitude zones. TABLE 63 RELATIVE EFFICIENCY OF 1 GRAM OF AQUEOUS VAPOR PER CUBIC METER IN ABSORBING THE INCOMING RADIATION PER 500 METERS, MEASURED IN CALORIES Elevation in Meters Vapor Pressure Radiation Absorption z P A/z I .r l ^I .r a 10000 0.03 0.02 1.918 -0 .002 -0.1000 0.05 0.02 1.916 - .002 - .1000 9000 0.07 0.02 1.914 - .002 - .1000 0.09 0.03 1.912 - .003 - .1000 8000 0.12 0.03 1.909 - .003 - .1000 0.15 0.05 1.906 - .004 - .0800 7000 0.20 0.10 1.902 - .006 - .0600 0.30 0.13 1.896 - .007 - .0539 6000 0.43 0.17 1.889 - .007 - .0412 0.60 0.20 1.882 - .007 - .0350 5000 0.80 0.28 1.875 - .009 - .0321 1.08 0.36 1.866 - .011 - .0310 4000 1.44 0.46 1.855 - .015 - .0326 1.90 0.60 1.840 - .025 - .0417 3000 2.50 0.79 .815 - .030 - .0380 3.29 0.93 .785 - .025 - .0269 2000 4.22 1.54 .760 - .016 - .0104 5.76 1.69 .744 - .014 - .0083 1000 7.45 1.91 .730 - 014 - .0073 9.36 0.87 .716 - .006 - .0069 000 10.23* 1.710 /* = the vapor pressure in grams per cubic meter. A /* = the variation of n for 500 meters. THE BOLOMETER AND ITS ENERGY SPECTRUM 277 7 . r = the observed radiation in calories per square centi- meter per minute, with maximum 1.918 on the 10,000- meter plane. A 7 . r = the variation in the radiation or absorption per 500 meters. A/o . r (687) a = the absorption by 1 gram per 500 meters. The coefficient multiplied by the observed change in /* for 500 meters gives the amount of radiation which has been absorbed by it. The curve of a shows that there is a gradual diminution of A 7 . r per A /* = 1 gram from 10,000 meters to the sea level, with a maximum of principal absorption, chiefly of short wave lengths, in the stratum 10,000 to 8,000 meters, and a secondary maximum in the stratum 4,000 to 2,000 meters. The laws of selective absorption in the atmosphere for different wave lengths are very complex, and they will require much more research for their complete explanation. It is quite evident that the curve of 7 . r , as derived from the pyrheliometer and bolometer observations at different elevations up to 4,500 meters, at Mt. Whitney, cannot be extended much above 10,000 meters without exaggeration. Furthermore, since the aqueous vapor, upon which the absorption chiefly depends, does not extend above the 10,000-meter, or cirrus levels generally, we are compelled to place the effective source of the radiation observed by the instruments at the cirrus levels 10,000 to 20,000 m, rather than on the outermost layer of the atmosphere. m f Hence, in sec z = - f the unit length md is something like 13,000 m$ meters, rather than the full depth of the atmosphere, and the depletion, or albedo by reflection, is not observed. 3. The Bolometer and its Energy Spectrum of Radiation The bolometer is a complex apparatus, of which the reader can find excellent accounts in the Annals of the Astrophysical Observatory of the Smithsonian Institution. It consists of a siderostat or ccelostat for directing a beam of solar light in a 278 RADIATION, IONIZATION, AND MAGNETIC VECTORS fixed horizontal position, upon a slit which can be adjusted to alter the quantity passing through it; a converging and a collimating mirror to focus the slit upon a prism, which directs the resulting spectrum upon a reflecting flat; a bolometer which consists of a very fine filament of platinum, forming one branch of a delicate Wheats tone balance, including a minute galvanom- eter in one branch; the lines of the heat spectrum falling upon the bolometer thread modify the current of electricity in the circuits, and deflect the galvanometer which is registered photo- 'graphically in a manner to act in synchronism with the position of deviation of the spectrum. The movement of the entire spectrum across the bolometer produces a spectrum energy dia- gram according to the prism, which must be transformed into a normal spectrum of uniform dispersion, and upon this there are very numerous line and band deficiencies of the ordinates due to the selective absorptions that may have occurred. There are numerous coefficients of absorption and reflection in the mirrors and other parts of the apparatus, so that an unusual degree of skill and experience is required for its successful manipulation and the correct interpretation of the resulting ordinates. This more or less defective energy spectrum is to be compared with the full energy spectrum of a perfect radi- ator, and that of 6,000 C. has been used by the Smithsonian Observatory. The energy spectrum of a full radiator at a given temperature T may be computed by the Wien-Planck formula, in which / = the energy of radiation at the wave-length X (x in microns, V = 0.001 mm.), ci = 575,000, and c 2 = 14455, the adopted constants. (688) Wien-Planck formula,*/ = a /~ 5 (eTr- l)' 1 . * Transformations of the (M. K. S.) and (C. G. S.) systems for the mechanical and heat units. (C. G. S.) System. 1 large calorie = 1 Kilogram degree 1 C. water = 426.8 Kilogram meters = 426.8 X 1000 X 100 X 980.60 = 4.1851 X 10 10 ergs (Log. 10.62171). 1 small calorie = 1 gram degree 1 C. water = 426.8 gram meter = 426.8 X 100 X 980.60 = 4.1851 X 10 7 ergs (Log. 7.62171). THE BOLOMETER AND ITS ENERGY SPECTRUM 279 The total energy of radiation is proportional to the fourth power of the temperature and would be expressed by (689) Stefan formula, / = a T* = . 6 4 T* Watts /cm. 2 deg. 4 (C. G. S.). iv c% Multiply by 10 7 for ergs/ cm. 2 (C. G. S.) Mech. units, and divide by 4.1851 X 10 7 for gr. cal./cm. 2 sec. c\ in absolute (C. G. S.) units is about sixty times too large, and depends on the block (A A) included in the spectrum measures. The constant is therefore more appropriate to a system with the minute for the unit of time. The displacement of the maximum ordinate is inversely proportional to the temperature. (M. K. S.) System. 1 large calorie = I Kilogram degree 1 C. = 426.8 kilo- gram meter = 426.8 X 1 X 9.806 = 4.1851 X 10 3 (Log. 3.62171). 1 small calorie = 1 gram degree 1 C. = 426.8 gram meter = 0.4268 X 1 X 9.806 = 4.1851 (Log. 0.62171). Kilogram Gr. X 10 3 Gr. N Transformalvn Factor,. ^^, = ^^ = , X 10- M i. KS ; ) X ^X6- . !iCaL_ Factor =0.000014336 (5.15644-10). mech. units/ 41851000 cm. 2 mm. Dimensions and transformation of the gravity equation from (M. K. S.) to (C. G. S.}. Gravity equation, g fa - ) = - Pl ~ P - Y^ (tfi 2 - tfo 2 ) - Pio (Qi - Qo), (M. K. S.). All these terms as computed in the (M. K. S.) system are transformed into the (C. G. S.) system of mechanical units by the Factor 10 4 , as can be tested by substituting the terms dimensionally. Similarly the equations (330) to (337) have the same factor 10000. This equation is transformed from (M. K. S.) mechanical units by the factor 1.4336 X 10- 5 into _ cm 2 , mm, Dimensions and transformation of the radiation equation. Radiation equation. Qi ~ Qo - P 10 = Ul ~ Uo = K lo = c T*. v\ VQ Vi VQ All these terms as computed in the (M. K. S.) mechanical units, are transformed into (C. G. S.) mechanical units by the factor 10, and into -M*. by the factor ",*!* = 0.000014336. cm 2 , mm. 4.1851 X 10 7 The transformation of the coefficient in the Stefan Law, J = a T*. The co- <c 3 X 10 10 6 ci , ci = 8 TT c h , ., , , efficient a = 7 X p, where _ , ,, for the Planck constants h = 6.545 X 10- 27 , k = 1.3606 X lO" 16 (C. G. S.). 280 RADIATION, IONIZATION, AND MAGNETIC VECTORS 2891 (690) Wien displacement formula, ^ max = ~^~- The constants used in these formulas have not been found quite the same at all temperatures, especially c z seems to have increasing values for higher temperatures, but they serve suffi- ciently for illustrating the principles now discussed. If the sun is a body which emits radiation at a given tem- perature, it will have an efficient energy of radiation per square centimeter per minute at the surface of the emission. Take R = 694,800 the radius of the sun in kilometers, D = 149,340,900 the distance to the earth in kilometers and the effective radiation falling upon the earth's outermost layer of atmosphere, before any reflection or absorption takes place, is found by, (691) 7.90 X 10" 11 T 4 . (C. G. S.) ci = 8 TT c h = 8 X 3.14159 X 3 X 10 10 X 6.545 X 10~ 27 = 4.93456 X 10- 15 (-15.69326) C2 = ch/k = 3 X 10 10 X 6.545 X 10- 27 /1.3606 x 10- 16 = 1.4433 cm. (0.15928) (Mech. units) a = 3.10 10 X 6 X 4.93456 X 10- 15 /4 X 10 7 X (1.4433) 4 = 5.1210 X 10- 12 (-12.70935) (Heat units) a = 5.121 X 10- 12 X 10 7 /4.1851 X 10 7 1.2236X 10- 12 (-12.08764) (M.K.S.) ci = 8 * c h = 8 X 3.14159 X 3.10 8 X 6.545 X IO- 34 = 4.93456 X IO- 24 (-24.69326) C2 = c h/k = 3 X 10 8 X 6.545 X 10- 34 /1.3606 X 1Q- 23 = 0.014433 (-2.15928) (Mech. units) <r = 3 X 10 8 X 6 X 4.93456 X 10- 24 /4 X 10 7 X (0.014433) 4 = 5.1210 X 10- 13 (-13.70935) (Heat units) a = 5.121 X 10~ 13 X 10 7 /4.1851 X 10 3 1.2236 X 10- 11 (-11.08764) In the Wien-Planck Formula % in cm. becomes 10 4 ft. Since Ci is the coefficient of A 5 = 10 20 /", and c 2 that of 2, = 10 4 /*, we shall have, (C. G. S.) ci = 493456 (5.69326), c 2 = 14433 (4.15928). (M.K.S.) cj. = 4934560 (3.69326), c 2 = 14433 (4.15928). These values are computed for dry air. Kurlbaum's G for ether, from recent experiments may be taken provisionally: C. G. S. Log. M. K. S. Log. a in cal./sec. 1.3167 X 10 -12 12.11948 1 3167 X io- 11 11.11948 a in cal./min. 7.900 X io- n 11.89763 7 .900 X 10 -io 10.89763 a in ergs/sec. 5.510 X io- 5 - 5.74119 5 510 X 1(H - 8.74119 a in ergs/min. 3.3063 X 10- 3 3.51934 3 3063 X 10-* 6.51934 THE BOLOMETER AND ITS ENERGY SPECTRUM 281 The following Table 64 gives the solar energy of radiation for certain selected temperatures T. It is computed with the value of ff = 1.316 X 10 C. G. S., which is not far from the value given by Kurlbaum. In order to make the numbers comparable with the usual convention by which the solar con- stant is stated in equivalent gram calories for the time-unit of 1 minute, the C. G. S. value of ff is multiplied by 60 in (691). TABLE 64 I EVALUATION OF THE SOLAR RADIATION AT THE MEAN DISTANCE OF THE EARTH FOR SELECTED TEMPERATURES T /o T /o T /o 8000 7.004 7000 4.106 6000 2.216 7900 6.661 6900 3.876 5900 2.072 7800 6.329 6800 3.656 5800 1.935 7700 6.011 6700 3.446 5700 1.805 7600 5.705 6600 3.245 5600 1.682 7500 5.410 6500 3.052 5500 1.565 7400 5.128 6400 2.869 5400 1.454 7300 4.856 6300 2.694 5300 1.349 7200 4.595 6200 2.527 5200 1.250 7100 4.345 6100 2.368 5100 1.157 According to this formula the equivalent "solar-constant" 1.918 would correspond with a temperature 5,787. At the time the bolometer factors were determined it was supposed that the value IQ.J. = 2.200 so that 6,000 was assumed to be the proper temperature for the full radiator, with which to compare the actual observed ordinates of the bolometer energy spectrum. Successive improvements in the standardization of the pyrheliom- eter has, however, reduced that value to 1.918, so that the 5,800 curve would be the proper one to use in determining the bolometer factor at the several stations. Table 65 contains the evaluation of the Wien-Planck formula of radiation for temperatures from 7,700 to 5,800, together with the sum of the ordinates from /I = 0.020 p to A = 2.500 p, the maximum ordinate I max , and the maximum wave-length A m , in the displacement formula. Abbot has determined the or- dinates in certain arbitrary units for a number of wave-lengths, 282 RADIATION, IONIZATION, AND MAGNETIC VECTORS S a S^ H* ^ <TO005-*0>^OOCOU300(NOOCO<NlOO>t-r-IOO>lOt-TJ<OONlO 0(NOOCO<NlOO>t-r-IOO>lOt-TJ<OO <ot-'^<oaooooiot-T-i?OT-(ooiOD3i-io NCU5NC<IC<lTj<O>Tj<lOOeO < '*t-lOOC>eOr-lTl<*O OOiCiOS05CO(Nt>(NO>T--ICOi35NOOCOCOO>OOCSC<I iHO>COOOOO>OC<IOlOOO5>OrHOt-lOTl*eOCO THE BOLOMETER AND ITS ENERGY SPECTRUM 283 and we have finally decided to reduce them to calories corre- sponding approximately with 6,900, and for this purpose have divided them by 1,000. This will be fully explained. T 6700 6600 6500 6400 6300 6200 6100 6000 5900 5800 A. =0.20|* 0.779 0.662 0.559 0.470 0.393 0.326 0.270 0.222 0.181 0.146 .25 2.209 1.938 1.694 1.474 1.277 1.101 0.945 0.807 0.685 0.579 .30 3.743 3.357 2.999 2.671 2.370 2.095 1.844 1.616 1.411 1.226 .35 4.844 4.411 4.006 3.626 3.273 2.944 2.639 2.358 2.098 1.859 .40 5.382 4.958 4.555 4.177 3.817 3.478 3.160 2.862 2.584 2.322 .45 5.458 5.073 4.705 4.353 4.017 3.698 3.395 3.108 2.838 2.583 .50 5.232 4.896 4.574 4.264 3.965 3.681 3.408 3.147 2.898 2.661 .55 4.841 4.557 4.280 4.014 3.757 3.509 3.270 3.041 2.821 2.610 .60 4.378 4.140 3.909 3.683 3.465 3.253 3.049 2.851 2.660 2.477 .70 3.451 3.287 3.125 2.968 2.815 2.664 2.517 2.375 2.236 2.101 .80 2.662 2.548 2.436 2.326 2.218 2.112 2.009 1.907 1.807 1.710 .90 2.046 1.965 1.886 1.808 1.732 1.657 1.583 1.509 1.438 1.367 .00 1.577 1.521 1.464 1.409 1.353 1.298 1.245 1.191 1.139 1.089 .10 1.226 1.185 1.144 1.103 1.063 1.C23 0.983 0.944 0.906 0.869 .20 0.963 0.932 0.901 0.871 0.840 0.811 .782 .752 .724 .695 .30 .763 .740 .717 .694 .672 .649 .626 .604 .582 .560 .40 .611 .593 .576 .558 .541 .523 .506 .489 .472 .455 .50 .495 .481 .467 .453 .439 .426 .413 .399 .386 .372 1.60 .403 .392 .381 .371 .360 .349 .338 .328 .318 .307 1.70 .332 .323 .315 .306 .297 .289 .280 .272 .263 .255 1.80 .276 .268 .262 .255 .248 .241 .234 .227 .220 .214 1.90 .231 .225 .219 .214 .207 .202 .197 .191 .185 .180 2.00 .196 .189 .185 .180 .176 .171 .166 .162 .157 .152 2.10 .165 .161 .157 .153 .149 .145 .141 .138 .134 .130 2.20 .140 .137 .134 .131 .127 .124 .121 .118 .114 .111 2.30 .121 .118 .115 .113 .109 .107 .104 .101 .099 .096 2.40 .104 .102 .099 .097 .095 .092 .090 .088 .085 .083 2.50 .090 .088 .086 .084 .082 .080 .078 .076 .074 .072 Sums. 52.718 49.247 45.950 42.826 39.857 37.038 34.392 31 . 873 29.515 27.281 Max. 5.474 5.077 4.704 4.354 4.024 3.714 3.424 3.153 2.899 2.661 * max. 0.432 0.438 0.445 0.452 0.459 0.466 0.474 0.482 0.490 0.498 Fig. 54 contains the diagrams of the energy curves for 7,700, 6,900, 5,800, and Abbot's observed ordinates, which closely agree with the 6,900 curve from wave-length A = 0.50 /* to A = 1.00 p. It is seen that there is a heavy depletion from X = 0.00 to 0.30 {J. or 0.35 ft, a slight excess from A = 0.50 n to 0.70 fi, a slight deficiency from A = 0.70 j to 1.00 j, and a rapid increase in the ordinates to agree with those of the 7,700 curve for A = 1.50 j to 2.50 /*. Whether the matching of the observed with the computed ordinates is made for the curves 6,500, 6,600, 6,700, 6,800, 6,900, 7,000, the preceding description of results holds true, but the best coincidence seems to be for 6,900. 284 RADIATION, IONIZATION, AND MAGNETIC VECTORS 0.00/X 0.50 M 1.00/A 1.50 jU, 2.00 P 2.50 /* 3.00 /* FIG. 54. Observed bolometer ordinates in their relation to the perfect radiation spectrum at different temperatures. THE BOLOMETER AND ITS ENERGY SPECTRUM 285 Now this corresponds with 3.876 for the solar constant at the earth, instead of 1.918 as given for the maximum from pyrheli- Q Q7t ometer observations, and this requires - - = 2.02 times as many calories to fall upon the surface of the outer layer of the atmosphere as upon the pyrheliometers, so that at least 1.958 calories must have been reflected back into space as albedo, which would therefore be about 51 per cent of the radiation of the sun received at the earth. If the bolometer ordinates can be more accurately determined this ratio may be definitely found; if the ratio varies from year to year at the same, or at several stations, the proper distribution between the variability of the solar radiation and the variability of the terrestrial reflection can be further discussed. At present, one must be very con- servative in attributing to solar variation the entire apparent variations of the radiation measured by the pyrheliometer and the bolometer. The remarkable fact appears to be established by Abbot's bolometer ordinates, Table 65, that these ordinates do not cor- respond with any single temperature of emission. By comparing the Abbot ordinate at any given wave-length with the ordinates computed at different temperatures for the same wave-length, it is not difficult to interpolate for a temperature of emission that would produce the ordinate. Thus, they are 6,850 at A = 0.40 /*, 6,960 at A = 0.55 /*, 7,260 at A = 1.30 /*, 7,800 aU = 1.60 / to 2.00 p. On the face of it, the Abbot ordinates range through 1 ,000 temperature, 6,700 to 7,700, and this may have several inter- pretations. (1) The solar envelope may consist of layers of different temperatures, 7,700 at the photosphere, which would be the general source of emission for all wave-lengths, gradually diminishing to 6,700 at the top of the chromosphere, or possibly the inner corona, in which envelope there is gradual selective absorption of certain wave - lengths, so that the effective emission of the sun to space is very complex, and corresponds with the observed bolometer spectrum; (2) a similar selective depletion of a uniform spectrum of 7,700-energy may occur in the middle and lower region of the earth's atmosphere, so that 286 RADIATION, IONIZATION, AND MAGNETIC VECTORS i g W 8 3 S u I 8 J j g 1> CO CO CN I 00 OS T-I CO <N IO rH CO TjH i-H CO CO CO CO CO C^ I I I I I 00 t- CO T*< rH 1 1 1 1 1 i-t CO 1> 00 t- O> CO 00 rH O) i 1 IO i 1 CO CO "* Tj< O CO CO CO CO 1 1 I 1 t- CO 10 O C~ rjl 777 i i oo TF CQ o co OS OS t^ JO O3 CO CO CO <M C* I I I I I CO CD t- IO 77 i i i ^ 00 00 OS O CO CO CO CO C^ I I I I I U3 OS O OS O i i i i i CO t^. CO <M <N ^ oo i> co ^ CO CO CO CO CO 1 1 1 1 1 O O> vH O) IO IO 1 1 1 + 1 CO CN CO "tf O O (M (M CO 10 CO CO CO CO CO I I I I I 1-1 CO T* CO O CO ' I CO O t* 05 O ^ t- 10 I I I I I CO co os co (M CO CO CO CO CO <M co o eo os i I CO IO I I I I I I I I I I O CO 1 1 CO CO CO 1 1 1 l> O iH IO 00 1 1 CO C<1 iH 1 1 1 COJ>(M(NOO COCNCOiHO OOI>COU7)Tf< i iiOrfr-iiH <M<Mcococo CO<MTHTHTH Mill I I I I I 00 O5 (M (N 1 1 (M CO CO CO 1 1 00 CO CO O CO 10 os o o eo 1 1 1 1 1 THE BOLOMETER AND ITS ENERGY SPECTRUM 287 rH C<l O 00 CO 00 rH OS CO t-i (M <N <M <N O lllll OS CO <* C* CO 1 1 1 1 1 O CO CO *O (N i> t~ co 00 00 t* ^1 CO CO l> N 77 i i IO 00 Tfi 3^3 I I I M CO <N CO <N lllll O O O> t iH 00 O 00 CO 00 77 i i i i 7777 iO 1 1 1 1 C^ CO <N rH CO t>- tO tO O CO 00 rt^ O3 OS rH 777 ~ iH i I i I T^ O5 co co co co c5 lllll \a *& a c* 00 1-1 -* 5 lllll lllll to oo <M eo 777 i 1 1 1 00 Tfl rH O O to rt< to I-H tO t- 00 ro co co co (M lllll I I I I + lllll to OS OS IO |g! 777 i CO to 00 !> c<i oo os os n< C^ rH Tfrl rH O rH 777 w N Os OS OS CO OS IO CO rH i-H CO c<i co i> eo eo O5 Oi CO J-t 1 1 1 1 1 CO CO CO 10 * I I I I + IO tc i 7 CO (N IO oo >o TJ< III I I I I CO OS OS CO 00 l> CO COrH I> (M O rH O rH (N IM' <N' OSCOCOlMtO rHlOiOOSCO |> IO 00 i-H 00 %%%%% lllll <N t> (N T}< IO CO CO ^t^ CO CO I I I I + t- tO tO 00 O I-H (N (N tO CO O5 CO to to lllll d r77 1 00 rH 1> to 06 <M' CO COrH co to n< rH rH rH ' <N <N' rH OS tO (N IO Cjl (N<N <N CO CO OO O t- CO CO -^ CO O IT- CO C> I I I I + lllll I I I I O l> CO to oo co CO COrH to to co co oo tO t~- !> OS O C^ (N (N (M CO lllll co co oo t> t> rH N. CO 00 tO 1 1 1 1 1 00 CO CO rH IO r^^^% Sc^^$H lllll I I I I (M l> to 00 (N 00 CO (N rH I I I rH rH 1> 00 O CO rH 00 CO <N C<J (N <M (M lllll O^ O^ i~4 TH ^^ CO CO *O CO OO 11111 77777 (N rH 00 t^ T^l IO C^ t^- CO CO CO (N rH Tfl ^^ O^ 59 i i o 10 lo5^S I I I I CO rH rH 00 OS 00 CO <N rH I I I ^ ^ JO rH CO (N (N (N (M (N lllll t^ CO OS S S J2 SS Q CO CO CO CN 1 1 1 1 1 (N rF OS O O ^0^ rH rH rH rH Ol lllll O OS OS ^i !i i i i i T-H CO ^ = (2 s 288 RADIATION, IONIZATION, AND MAGNETIC VECTORS the observed spectrum is only the remainder that has escaped from the terrestrial depletion in the lower levels; (3) a mix- ture of these two cases may be more closely related to the facts, but if so, there will be great difficulty in disentangling the ele- ments. Corresponding with these temperatures we may compute the value of the radiation at the earth, on the first supposition, that a depleted radiation escapes from the solar envelope, and we find 3.765 calories for X = 0.45 /<, about 4.000 for A = 0.50 fJ. to 0.60 /*, a lower variable amount from A = 0.70 /* to 1.00 /*, 4.751 at * = 1.30 /*, 5.705 at /l = 1.50 /, 6.329 at A = 1.60 p, 6.169 at A = 2.00 p. The evidences indicate that the solar intensity of radiation at the earth is from 4.00 to 6.00 calories per square centimeter per minute. The isothermal region radiates 2.11 times as much heat as does the convectional region. The isothermal region is separated from the convectional region by a wedge-shaped layer, which FIG. 55. The courses of the relative radiation. partly reflects the incoming ray and partly absorbs it. The remainder proceeds through the convectional region, suffering depletion by minor absorptions to the cumulus layer, where it is more rapidly absorbed, and the balance reaches the sea level THE BOLOMETER AND ITS ENERGY SPECTRUM 289 or land areas. The outgoing ray experiences a similar complex series of absorptions and reflections. The absorption is accom- panied by transformation of energy into ionization currents in the cirrus layer and in the cumulus layer, the electric streams of the former flowing to the poles cause the auroras, and by induc- tion the aperiodic variations in the magnetic field; the ionization currents of the cumulus region are controlled by the diurnal convection circulation, and induce the diurnal variations of the magnetic field. TABLE 67 dQ APPROXIMATE MEAN VALUES OF -~f IN THE RADIATION EQUATION jy- = dz at k 2 - PER 1000 METERS a z 3 90 80 70 60 50 40 30 20 10 19000 -245 -250 -256 -271 -270 -286 -310 -325 -288 -248 18 -245 -266 -275 -275 -282 -304 -354 -367 -310 -241 17 -277 -272 -286 -283 -296 -302 -345 -386 -308 -259 16 -282 -289 -294 -284 -309 -321 -341 -357 -303 -244 15000 -294 -303 -309 -317 -336 -314 -320 -288 -254 -207 14 -306 -321 -322 -333 -327 -311 -284 -196 -176 -170 13 -322 -343 -348 -361 -379 -353 -233 -145 -148 -142 12 -335 -352 -384 -396 -427 -361 -113 - 87 -100 -103 11 -352 -329 -309 -200 -180 -110 - 49 - 40 - 37 - 52 10000 -262 -278 -236 -182 -108 - 25 - 23 - 79 - 59 - 52 9 -167 -163 -144 - 96 - 58 - 51 - 92 - 85 - 78 - 76 8 -168 -165 -136 - 78 - 86 -151 -145 -114 -113 -110 7 -172 -152 -149 -143 -121 -154 -160 -149 -145 -148 6 -192 -163 -166 -146 -151 -152 -156 -152 -162 -178 5000 -215 -180 -188 -164 -163 -146 -130 -157 -168 -173 4 -226 -202 -196 -134 -159 -162 -131 -159 -169 -164 3 -234 -211 -216 -140 -154 -170 -174 -182 -171 -164 2 -260 -239 -232 -210 -189 -140 -147 -126 -103 - 78 1 -140 -125 -126 -110 - 92 - 62 - 46 - 35 - 19 - 24 0000 The mean values of -r* are collected in Table 67 and plotted (l Z in Fig. 55. The radiation increases downwards to a maximum at the 12,000-meter level, then there is a large increase in the 290 RADIATION, IONIZATION, AND MAGNETIC VECTORS heat supply by absorption in the cirrus region, then an increase in radiation to the cumulus level, and a second supply of heat is at hand extending to the surface. d 2 The Values of -y in the Radiation Equation J dz 2 4. There is another important argument leading to the same conclusion, that the so-called " solar-constant" exceeds 4.00 calories, derived from the thermodynamic computations described in Chapter III, the results of the same being more fully published in Bulletin No. 3, Oficina Meteorologica Argentina, 1912. The value of (Qi Q ), the loss of heat by radiation for every 1,000 meters difference of level, and for every 10 degrees of latitude from the equator to the pole, was derived from the temperature data of the balloon ascensions. Table 66 contains /d 2 O -7-7, which are found in the radiation formula, r692) *Q__ k2 <?Q (692) dt -- k dz2 , and from which, having once obtained the coefficient k 2 , the true radiation -~, loss of heat in the unit time, may be computed. For our present purpose we need only admit that the rate of radiation is proportional to the second differences A 2 (Qi Q Q ) per thousand meters, to be able to make important inferences. The two cases of low I\ and high 7 2 temperatures in the isothermal region are considered in connection with average conditions in the convectional region C; between the isothermal and the convectional region there is a region of absorption and reflection of the radiation R, wedge-shaped, about 6,000 meters deep over the equator, and thin or vanishing in the arctic zone; below the convectional region there is a second stratum of absorption and reflection, less than 2,000 meters deep, and occupying the region of the specific diurnal convection R 2 . We are not now concerned with the absorption regions RI, R%, but VALUES IN THE RADIATION EQUATION 291 only with the radiation regions /i, 7 2 , C. Taking the mean values of A 2 (Qi Q 2 ) in these regions, respectively, and the ratios I\/C and /2/C, together with the means, it results that about 2.11 times as much radiation is passing through the iso- thermal region as passes through the convectional region. This can only signify that of the incoming radiation 52 per cent is reflected before reaching the surface of the earth, and that a nearly equal amount returns to space as albedo. If we again admit that the 1.918 calories determined as the max- imum passes below the cirrus levels to the lower stations, it follows that, 2.11 X 1.918 = 4.047 calories, is the approximate average value of the "solar-constant." This confirmation of the results from the bolometer ordinates also strengthens the conclusion that the solar temperature of emission is about 6,900. Since I . r = 1.918 calories, and this corresponds nearly with the temperature 5,800 in Table 65, the sum of the ordinates for a full radiator at 5,800, summing them from /I = 0.20 // to A = 2.50 //, is about 27.28. Now, this sum is proportional to the area of the energy spectrum between these wave lengths, beyond which limits the energy not included is very small in amount, and since we require to employ 2.11 times this energy, 2.11 X 27.28 = 57.56, it is seen that the sum of the ordinates equal to this amount lies between 6,800 and 6,900 solar temperature. If an allow- ance be made for selective absorptions in the several zones of the earth's atmosphere, as in the cirrus region, it is evident that these general conclusions may be extended to many important special researches. We have, therefore, besides the negative argument that the Bouguer formula is indeterminate, so far as fixing the level from which the observed radiation proceeds to the surface, the three positive arguments that the solar constant is 4.00 calories up- ward, and the temperature of the solar photosphere, 6,900 292 RADIATION, IONIZATION, AND MAGNETIC VECTORS upward, namely, (l) The maximum 1.918 calories on the 10,000-meter plane, from stations at different elevations above the sea level. (2) The form of the bolometer ordinate curves of the energy spectrum for 6,900 solar temperature. (3) The thermodynamic ratio 2.11 of the radiation in the isothermal and convectional regions. Compare in addition Chapter VII. The Measures of the lonization of the Atmosphere There are two types of instruments for measuring the ioniza- tion of the atmosphere, (1) the Elster and Geitel apparatus for the coefficient of dissipation of electric charges from a body connected with an electroscope, (2) the Ebert ion-counter with fixed capacity and variable charge, and the Gerdien ion-counter with fixed charge and variable capacity. The former measures the coefficient of dissipation A, and the two latter measure the component parts of /, namely, the number of ions per cubic centimeter of air n, and the velocity of motion of the ions which dissipate the charge on the body attached to the electroscope. For each kind of electricity, we have, A = (e n u + where e = 3.4 X 10~ 10 , the constant charge on the single atom. These subjects have been discussed in many papers, and the results are collected in: Die Atmospharische Electrizitat, H. Mache und E. V. Schweidler, Braunschweig, 1909; Die Luftelek- trizitat, Albert Gockel, Leipzig, 1908; Electricite Atmospherique, Observatoire de 1'Ebre, 1910, par le P. /. Garcia Molld, S.J. There is a fundamental discrepancy, amounting to about 300 per cent, between the resulting values of A observed and computed by the two methods indicated, and as it is our purpose to come to this problem, we shall as briefly as possible summarize the formulas and the conditions of the instruments leading to these conclusions. The elementary electrostatic formulas can be verified by reference to numerous treatises, of which one of the most convenient is, Elements of the Mathematical Theory of Electricity and Magnetism, by /. /. Thomson, Cambridge, 1895. IONIZATION OF THE ATMOSPHERE 293 It is not necessary further to explain the individual formulas which follow by simple transformations from the elementary conditions that have been stated. Notation and Elementary Relations The sign -f signifies repulsion along the line joining two charges of the same kind Q Q f . Q = charge on a point, Qi = charge on unit line, Q 2 = charge on unit area. 00' (693) Force = F = ^^-. F n = the normal, and F t = the tangential components. (694) Total intensity = I = S F n S = / F n d S = 4 IT Q on the closed surface S. The induction is zero when the charge is outside S. Special applications to given surfaces for unit lengths. Sphere Cylinder TotalforceF=F g .47rr 2 = IwQ F = F c 2irr = lirQi (695) F.= - F c - Infinite Plane F p = 47r<2 2 The difference of potential A V = the work done in moving the unit charge against the electric field of intensity F n through the distant A r. (696) Work. W = f*F n dr = F n (r 2 - r,) = (Vi - F 2 ) for the unit charge. for the charge Q. 294 RADIATION, IONIZATION, AND MAGNETIC VECTORS (697) Force. F n = - (F2 ~ Fl) = - ^. Mechanical f 2 f\ a^ force F = F n Q. (698) Work. W = /F n dr = -- f^dr = y = (Fi - F 2 ) for the sphere. (699) Inner energy. U = J JF = J Q (7i - F 2 ) = The following quantities, Q, Z>, 77, <r, w, are synonyms for the same physical process in a dielectric, and differ only by coefficients. E = the electric force, c = the specific conductive capacity, H = the magnetic force, /* = the magnetic permeability, D = cE = the electric displacement; B = /J-H = the magnetic induction. 77 = - F n = the polarization = <s the surface den- sity = Q the charge. (670) U = W = D E = \cE^=\^=\E(V, - F 2 ). L> In a medium other than air the specific inductive capacity is K. The number of Faraday lines = n. (671) K K (672) Ui = | . F n 2 = . n F n = the longitudinal tension. K K (673) U t % . j . F^ 2 = . w ^ = the transversal tension. A very complete collection of electrical and magnetic for- mulas may be found in Bulletin I, U. S. W. B., 1902, "Fclipse Meteorology and Allied Problems." ELECTROSTATIC RELATIONS PER UNIT LENGTH Electrostatic Relations per Unit Length Q = c V = <*S 295 Formula Sphere Cylinder Plane (674) Charge Q = C V =rF Ol= 2lol~r -if, (675) Potential V = Q/C (676) Capacity C = Q/V (677) Potential coefficient = V/Q dV (678) Force F = - a r (679) Work U = i F C =r 1 * = 7 *-I| F = 2 Oi log e r F =4* r. Qj ~21og e r = 2 log e r I/ = Oi 2 log e r I/ = 2 * r Q,* I/=i>Q'=*CF' Formula Two Concentric Spheres Two Coaxial Cylinders Two Parallel Planes (680) Potential Fi - Fj Vl _ Vt = Q !ii n Vi - Fz = F! - Fz = 2 Q, i g 11 4 TT Qz (ri rz) Q nr* 1 1 (681) Capacity C - Fl _ Fj C ' rt-n 21ojr 4 TT (ri - rz) (682) Charge Q = C (Fi - Fz) Q= ^ (VI _ VI} 1 1 Ol " ri 4 TT (n - rz) 2 log e (Fi - Fz) (Fi - Fz) (683) Potential Fi - Fz ri - r\ P = 4 TT (ri - rz) coefficient Q nrj p - 2 log e rj (684) Work U = $ Q (Vi -Fz) n rz U = log,, Q. 2 U - 2 rr ( n _ rj ) Qj* U = i P Q 2 = i C (Fi -^Fz) 2 Q Oi 02 (685) Surface density <r = 2 IT (ri - rz) / ~ 5 (686) Force F = 4 TT o- r Q F = (n^i 4^Qz (rt-r-P) "S Quantity neutralized in the unit time (687) j idS =4?rAQ =oQ fidS-i*(e*u)Q fidS = Mnu) Ql JW5 -4- (.)& = 4 * [() -t + (nu)_jQ = 4irA . C F - Derivation of Coulomb's Law of Dissipation (688) fidS = 4 TT A . Q = 4 w A . ff S = A. F. 5 = a (X 296 RADIATION, IONIZATION, AND MAGNETIC VECTORS (689) fidS = - -jj . The loss of charge with the time dt. (690) 4 * * . Q = - ^2 . Combining (689) and (690) in Coulomb's Law, (691) nx.dt= (692) Vi = V Q e~ a(tl ~^\ (693) a (/i - t ) = C. log. nat -~ = ^ . C log ^- Conduction lonization Currents Number of ions per cu. cm. n + n__ * Charge IP E. S. U. e + e, e = 3.4 X 10 ~ 10 . Velocity of ions in cm/sec at - , u + u _ (694) Conductivity = the positive current through 1 cm 2 in unit time, A = (e + n + u + ) + (e^n_ w_). Example of an Insulated Charged Sphere of Radius r (695) Charge. Q = CV = rV. = <?S. (696) Surface density, a = - - = -^. S = the surface of the sphere. (697) Potential. V = = 4 TT r. <r. (698) Capacity. C = r = . (699) Force. F n = -^ = 4 TT a, acting on the unit quantity at distance r. * Result of many experiments. Compare Table 95. THE ELSTER AND GEITEL APPARATUS 297 (700) Ion current i = A F = [(e n u) + + (e n w)_] from the unit area. (701) On whole sphere = 4 w r 2 . A F n = 4 TT [(e n u) + + This ionization current neutralizes the charge of the body at a certain rate by 4 w (e n u) + Q acting outward, and 4 x (e n u) _ Q acting inward, as to the surface. The dissipation coefficient A is the measure of the rate of the neutralizing the charge. (702) A + = 4 TT (neu)_ neutralized negative charge. /_ = 4 TT (n eu) + neutralized positive charge. (703) Quantity neutralized = XQ = Xr V = 47r Xr* <r. Large bodies neutralize faster than small bodies. The neutralization amount is proportional to a. The Coefficient of Electrical Dissipation of the Atmosphere X The Elster and Geitel apparatus consists of an electroscope, whose capacity by itself is C, and a small dissipating body, generally a copper cylinder 10 cms. long and 5 cms. in diameter, whose capacity by itself is K. The dissipating body charged with the quantity of electricity Q, in the time d t, loses the quan- tity d Q, by the formula for voltage, (704) --(C + *). Since it is required to know the rate of dissipation from the cylinder alone, where, (705) Q = KV, we construct the Coulomb law, by division, , 7r ^ 1 dQ C + K dV (706) 4,* = =--- -= -- -. 298 RADIATION, IONIZATION, AND MAGNETIC VECTORS Multiply by d i and integrate for common logarithms, (707) ,-<)= ^J^ "^ ' log f. This is the formula for dissipation from the body to the free air, provided there is no leakage into the interior of the electroscope. This is expressed, from parallel reasoning, for C and K separately, (708) <*(/!-/<>) =-^ lo g^7- Combining these two parts into one formula, (709) 4^-.-^.*..(iog;- C F ' If the instrument is charged with the electroscope alone for Fo' at the time / , then the V\ is to be read at the time fc, gener- ally an interval of 15 minutes for (/i / )- When the insulation is good this correction is small, and the compensation can be made by (Fi) = Vi + A Fi. It is customary to express the constant a in terms of percentage of loss of charge per minute, so that, .. , 100 K + C 1 Fo (710) a lg - T'M' log (FO- Finally, the general formula for A becomes in E. S. U., -2- 4 TT 4 TT X 60 X 100 4 TT X 60 X 100 K - log = ( e n+ u++ e n_ u,) 300. The Ebert ion counter consists of an electroscope to which a condenser is attached as an integral part, and through this a known volume of air A in cubic centimeters per second is drawn by a revolving turbine. The capacity C of this combination is known. In the axis of the long cylinder of radius r, a small THE EBERT ION COUNTER 299 cylinder of radius r z and length / is supported by an insulating rod connecting with the electroscope. If the apparatus runs for a given time (t\ t Q ) while the small cylinder is without charge, there will be the loss in voltage (712) C (Fo' - F/), (without charge). When the rod is charged the loss in the same interval is, (713) C (Fo - FO, (with charge). . This change in the potential is caused by the passage of N ions, each with charge e, in the volume of air A per second, so that in volts with the factor 300, we have the equation, con- taining correction for the leakage by (695) , for positive ions, (714) + e N + A 300 = C [(F - FO - (F ' - F/)] in cubic meters. (715) n+=NX l<r 6 = C [(F. - F:) - (V t f - V,')} jj~^ in cu. cms. The passage of n + positive ions per cu. cm. has neutralized a certain amount of the charge of the opposite sign, so that by charging F , F ' for one sign, the number of ions of the opposite sign is determined. Example of the Computation A 1 run = 0.185283 cubic meters. C = 9.69. A b (560 seconds) = 0.926417 cubic C X 2941 meters. 300,4 (+) (Fo - FO = 9.0. (Fo' - F/) = 0.6. For positive charge, w_ = 84 X 102.54 = 861. (-) (F - FO = 10.2 (Fo' - F/) = 0.5. For negative charge, n+ = 9.7 X 102.54 = 995. The Formulas for the Velocity of the Ions The velocity of motion of the positive and the negative ions respectively depends upon a special proposition, as follows: 300 RADIATION, IONIZATION, AND MAGNETIC VECTORS J E* (716) =-- U ' Gdr=-Fudx. (717) F=-==- ^ by (678) and (682). 2 log e H *~^fc * FIG. 56. The parabolic motion of the ions in a condenser of co-axial cylinders charged to V. u = the velocity of motion in cm/sec. F = the radial component along r. G = the axial component along x. (718) T~ = "TT ' ~ 1 > f r the unit charge. Integrate ax Lr r\ r log e after transposing, ^2 2 ^ 7 _ Vi (719) r 2 = -^r . ; x + C. C = r 2 on ion entering log e the condenser field at r - * (720) The maximum value of x is at a distance less than / from the origin, where the ion entering at r Q falls finally upon the charged surface of the inner electrode. 2 log e [Malls inside 11 Condition , v r - x - , x ^2 , , that all the (721) fr - (fi'-r^) rrr- 1 =falls on end / ionsfallupon electrode. L < falls outside / ri (722) ~ '= (r 2 - r 2 2 ) i ', equation of condition. velocity apparatus consists of the primary condenser and electroscope, used for counting the number of ions, and in EXAMPLE OF VELOCITY COMPUTATION 301 addition an auxiliary condenser in the same axial line, the inner electrode charged to a few volts A V, and the outer connected to earth. Let VQ Vi = the loss in voltage with no charge on the auxiliary. Fo' Vi = the loss when the auxiliary is charged to about A V = 30 volts. As some ions fall on the auxiliary electrode, it follows that (F - FI) > (F '- F/) The amount of electricity drawn down upon the auxiliary electrode by its charge of opposite sign is, (723) We have, also, e n = C (F - 7i) = F ~ V \ Hence, (Vo - Fi) - (Fo' - FiQ Example of the Velocity Computation A 0.85065 X 10 6 _ cubic centimeters /i 1 515 per second 1 1 1 47T/.AF" 47TX12X27 4071* ~ 2 log ^ = 2.3026 X 2 X (log 1.50 - log 0.25) = 3.584. 1652 X 3.584 Constant = - - = 1.45. Q-CFo'-F/), Fo - V l auxiliary T7 -, 7 Vo V\ not charged. Fo' - Vi' auxiliary charged. 302 RADIATION, IONIZATION, AND MAGNETIC VECTORS (190.7 - 177.1) - (184.9 - 177.6) ' (-) Charge. , + - 1.45 1.45 f = 0.67 cm/sec. lo.O (192.8 - 179.0) - (183.9 - 176.1) (+) Charge, u. = 1.45 * - 192 .s _ \ 79 . - '- = 1.45 ' = 0.64 cm/sec. lo.o The Gerdien apparatus for the number and velocity of ions. This consists of a double condenser, fitted with a turbine for a measured quantity of air to be drawn through the tubes in the unit time. There are two electrometers, / with a variable capacity C", // with a constant capacity C. The outer cylinder has the radius r\ = 4.9 cm., and the length / = 65 cm.; the auxiliary with variable capacity C' has the radius r% = 0.5 cm., and / = 20.1 cm.; the principal with constant capacity has the radius r 2 = 1.0 cm., and / = 35 cm. C = 16.7, C = 20.2, in some instruments. The number of ions. The capacities are both at a minimum, and the number of ions falling on both electrodes neutralizes a quantity of electricity of the opposite sign, expressed by the equation (725), where A = the number of cubic centimeters of air that passes in the time (/i - J ) = 10800000 cm. 3 for 80 revolutions. (725) e n A 300 = C (F ' - F/) + C (V Q - Fi). (726) n = [(C (Fo' - F') X C (F - Fi)] 0.908. The velocity of the ions per second. The capacity C' is increased to C/, from 16.7 to 124, by raising the capacity cylinder 25 divisions. From (722), v ' v ' 2 lo S* 7" (727) K^ZL = ( , 2 _ , 2) __- . The quantity of electricity on w-ions entering the field through the area TT (r 2 rf) falls on the electrode, so that, GERDIEN APPARATUS 303 F' (728) TT (>o 2 - rf) = 4 TT /. u. C V = 4 w I. u. 2 log e 7-2 Hence, by substitutions, and for the interval d t, V (729) en .TT (r 2 - rf) G d t = e n . 4 TT A w 2 **. C/ 7 T// i a V . (731) u + = en + u_ = (e n u) - Velocity in a (732) A + = (en + w_j_) 300. en- field 1 volt/cm. E. S. U. ?w_ _) 300. A = A , +A_. EXAMPLE OF THE GERDIEN COMPUTATION COMPARE MOLLA, TORTOSA OBSERVATORY ( ) Negative charge for the positive ions neutralized. I. Ci = 16.7 II. C = 20.2 H m s Index L B S I ri log yi L R S V to 9 40 18 ti 10 6 33 26 15 li-h 1575 log (ti-io) 3.19728 logcn_j_3.7254 log_j_ 0.0912 log 300 2.4771 60 140 80 A I 3 2 11.2 10.7 21.9 19 9.3 9.8 19.1 174 yi^yi 2 log. log VVVi 1 [Constant] >.8 2.30060 LO 2.24055 5.8 0.06005 8.77851 8.23544 12.0 15.6 27.6 9.0 10.5 19.5 Vo-Vi C i (V( C Vl \ 209.4 145.5 63.9 1290.8 430.9 1721.7 Factor 0.908 n_j. 156.3 Iogn + 3.19396 loge 0.53148 log (ti-to) 3.19728 log. Const, log VoVW 7 . 01395 log en i (<i-fo) 6 . 92272 log + 0.09123 tt + 1.234 = 1.967 X KM log\ + 6.29379 X + log. en _j_ (h-to) 6 . 92272 Read the scales for V ' (auxiliary), and F (principal), while C is at minimum value 16.7. Raise the C to C/ = 124 by 25 divisions. Start the turbine and read the hour and index; after an interval for 80 A read the hour and index. Lower the capa- 304 RADIATION, IONIZATION, AND MAGNETIC VECTORS city to minimum C' 16.7 and read V\ and V. The formulas become by evaluation : 1 u = e n ' ti A = e n u 300. 4 TT A = a. 1 V r 0.0172 log T /7 (auxiliary). Example Continued. ( +) Positive charge for the negative ions neutralized. 16.7 II. C = 20.2 H m s Index 1 R S V logy L R i ? V to 10 11 30 h 37 5 65 10 145 9 .0 10.7 20.7 .5 9.5 19.0 188.4 173.4 2.27508 2.23905 12.5 15.8 28 9.9 11.7 21 Vr-Vi C(VoVi) C l (Vo l Vi Factor w_ log u loge log (fa-fo) .3 213.9 .6 162.0 25 35 ti-to 1535 log '(h-to) 318611 log en 3.60299 logtt 0.00300 log 300 2.47712 logv_ 6.08311 80 Vo^-Vi A log. log. V^/Vi [Constant] log Const, log. log en_ (ti-to) log u_ u_ A_=A 1.211X10-4 A=A + i 15.0 0.03603 8.55666 8.23544 51.9 1048.4 250.5 WVi 1 6.79210 6.78910 0.00300 1.007 + A_ =3.178X10- 4 1298.9 0.908 1179 3.07151 0.53148 3.18611 log e n_ (h-to) 6.78910 The Cause of the Discrepancy in the Values of the Conductivity of the Atmosphere, as Determined by Two Methods It will be seen from the collection of results in Table 68 that we have arrived at two very different results for the coefficient of atmospheric conductivity X, as determined by the two methods that have been described. Section I is compiled in part from Mache and Schweidler's volume, and in part from Bigelow's obser- vations of 1905, made during the U. S. Eclipse Expedition to Spain and Algeria, and of 1912 made at Cordoba, Argentina, with the resulting value of /I = 5.575 X 10 ~ 5 E. S. U. Section II is compiled from the same sources, with the result A = 20.49 X 10 ~" 5 , so that the ion counters give a value for X which is 3.68 times greater than by the dissipation apparatus. The cause of this DISCREPANCY AS DETERMINED BY TWO METHODS 305 great discrepancy is not far to seek. The dissipation apparatus is generally worked under a cylindrical hood, which converts it into a condenser, through which the air current is not freely passing, with the effect that the free dissipation is transformed into a saturated or stagnant circulation of the ions. It is easy to show that this apparatus with the hood does not respond to the capacity formula upon which the computations are based. We have used cylinders of different dimensions, with the hood TABLE 68 COMPARISON OF THE COMPUTED VALUES OF A FROM THE DATA OBSERVED BY THE Two METHODS Section I. Elster and Geitel dissipation apparatus Stations a l + a l _ 2 a + 0_ A + A_ A Lugano ... 1.87 4.22 3.02 3.76 2.88 1.17 1.26 0.84 0.72 0.97 1.26 0.55 3.33 1.36 1.45 3.63 1.87 0.73 1.50 2.59 1.87 4.44 3.38 5.86 4.62 1.54 1.34 1.33 1.00 1.29 1.43 0.60 3.82 1.39 1.55 3.77 1.90 0.79 1.56 2.53 1.00 1.05 1.11 1.55 1.60 1.32 1.06 1.58 1.43 1.36 1.13 1.09 1.15 1.02 1.07 1.04 1.02 1.08 1.04 0.98 3.12 7.04 5.03 6.27 4.80 1.95 2.10 1.40 1.20 1.62 2.10 0.92 5.55 2.27 2.42 6.05 3.12 1.22 2.50 4.32 3.12X10" 4 7.40 5.63 9.77 7.70 2.57 2.23 2.22 1.67 2.15 2.38 1.00 6.37 2.32 2.58 6.28 3.17 1.32 2.60 4.22 2.48 5.59 3.99 4.98 3.81 1.55 1.67 1.11 0.95 1.29 1.67 0.73 4.40 1.80 1.92 4.80 2.48 0.97 1.99 3.43 2.48 5.87 4.47 7.75 6.11 2.04 1.77 1.76 1.33 1.71 1.89 0.79 5.06 1.84 2.05 4.98 2.52 1.05 2.06 3.35 4.96 XlO- fi 11.46 8.46 12.73 9.92 3.59 3.44 2.87 1.28 3.00 3.56 1.52 9.46 3.64 3.97 9.78 5.00 2.02 4.05 6.78 Capri Tromso Spitzbergen, land. . . Spitzbergen, sea .... Juist Wolfenbuttel Misdroy Ostsee Potsdam Kremsmiinster Triest .... Karasjok Daroca, 1905 Porta Cceli Guelma. . Bona Casar, bulkhead. . . . Ccesar, hatch Cordoba 1912. 5.575XKT 8 and without the hood, leading to the following results: cylinders are as indicated, The 306 RADIATION, IONIZATION, AND MAGNETIC VECTORS TABLE 68 (Continued) Section II. Ebert's and Gerdien's Ion counters. A = (en+u+ + en_u_) 300 E. S. U. Stations n + n _ u + u_ * + A_ A . dv * = *dh Helgoland, dune. . . . Helgoland, oberland. Swinemiinde .... 382 735 823 1088 1147 1323 1029 1264 1558 1793 559 1176 828 1117 1000 588 206 382 647 882 794 1117 853 382 1238 823 529 1205 676 970 735 588 Potsdam Atlantic Ocean Seewalchen Mattsee 1.02 1.25 13.76 14.24 28.00X1Q- 8 9.33X10- 7 Santis Miinchen Bayern Alpental Golf von Lyon IMallorca 0.83 0.90 9.96 11.06 20.02 6.67 Barcelona. . . . Karasjok Freiburg 1.00 1.11 10.20 8.32 18.52 6.17 Stiller Ocean Gottingen 1 32 1 40 Daroca 1905 999 2940 1155 621 1013 1275 1558 1733 819 2960 972 626 1166 1206 1542 1522 0.684 0.771 6.97 6.43 13.40 4.47 Guelma. Bona CcBsar, bulkhead. . . . CcKsar hatch Casar, outward Cordoba, Sep. 12 to Oct. 10, 1912 Oct. 11-26 0.701 0.715 0.613 0.766 10.77 11.83 10.30 10.08 21.07 21.91 7.02 7.30 20.49X1Q- 8 6.83X10" 7 The value of A by the Ebert or Gerdien methods is more than three times greater than by the Elster and Geitel method. (1). r = 0.25cm. (2). r = 2.50cm. (3). r = 2.50cm. / = 5.00 cm. / = 10.00 cm. / = 20.00 cm. K = 0.65. K = 5.46. K Q = 10.91. The capacity of the electroscope is 4.36 cm. The capacity coefficients without hood become, K+C K + C K + K 7.71. K 1.80. K 1.40. DISCREPANCY AS DETERMINED BY TWO METHODS 307 C C K + C 0-87. -0.44. = 0.29. The capacity coefficients with the hood become: K = 1.34. K = 7.50. K =15.00. K + C K C = 4.25. = 0.76. K + C K C K + C = 1.58. 0.37. K + C K C K + C = 1.29. = 0.23. Two sets of experiments have been carried out, one in 1905 and the other in 1912. In the Eclipse expedition the hood was used throughout the observations, but the dissipation bodies (1) and (2) were, used in frequent interchanges. The following table shows the average fall in scale divisions during fifteen- minute intervals, with (2) the cylinder, and (l) the small rod, respectively: TABLE 69 MEAN Loss IN SCALE DIVISIONS IN FIFTEEN-MINUTE INTERVALS Station Daroca Porta Coeli U. S. S. Ctesar Bulkhead U. S. S. Casw Hatch Cylinder 10x5 cm. Rod 3x0.5 cm. No. (+) (-) 657 -5.4 -5.4 139 -5.3 -5.5 No. + (-) 354 -5.8 -6.3 80 -5.1 -5.8 No. (+) (-) 147 -3.4 -3.6 32 -3.3 -3.4 No. (+) (-) 126 -6.5 -6.7 25 -6.6 -6.6 The observations at Daroca and Porta Cceli were continued throughout the twenty-four hours. It is seen that the size of the dissipating body under the hood does not effectively control the rate of dissipation, and that the computed capacity coefficient does not properly enter into the computation. A series of experiments was made at Cordoba, 1912, with the three dissipation bodies, in part with, and in part without, the hood. The mean values of the loss in scale divisions are proportional to the following data: 308 RADIATION, IONIZATION, AND MAGNETIC VECTORS TABLE 70 MEAN LOSSES IN SCALE DIVISIONS WITHOUT AND WITH HOOD Dissipation Body Without Hood With Hood (+) (-) Mean (+) (-) Mean (1) Rod 1 58 1.32 1.45 2.07 1.73 1.90 (2) Small cylinder . . . 4.82 3.98 4.40 1.63 2.07 1.85 (3) Large cylinder . . . 5.60 5.12 5.36 2.20 1.97 2.09 1.98 1.92 1.95 It is seen that with the hood the size of the dissipation body is an indifferent matter, but that without the hood the size of the body and the rate of dissipation progress together. A special observing shelter at Cordoba was constructed of canvas and netting, so that the dissipation observations could be made without using the hood, and the results for the series are recorded in Table 68. Referring to the capacity coefficients /(" j C* for the three bodies, it is seen that increases as K dimin- ishes, the upper limit for K = being <*> , and for K = <*D being 1. We may, therefore, increase the coefficient by decreasing the size of the dissipation body. Since most of the experiments of Table 68 were made with the hood and for K = 1.58 (on the average), while, without the hood in Cordoba, 1.80, it is apparent that the comparatively large value A K + C K 6.78 X 1CT is due to this fact, at least in part. We can make a cylinder for a proper dissipation body which will give 5.575 X 3.68 = 20.49. Its approximate size can be obtained by computation, but its actual and correct size is a matter for experiment. Such experiments were executed at La Quiaca, 1913, with the result that a dissipating cylinder 52 centimeters long by 1 centimeter in diameter will give nearly the same value of A as the ion-counters. Similar results have been obtained in Potsdam. Such a long cylinder is very difficult to charge to a sufficiently high potential V and it is impractical. The Elster and Geitel apparatus with- DISCREPANCY AS DETERMINED BY TWO METHODS 309 out hood is valuable for giving relative values of (a+ a_) (A + A_), and not their absolute values. These must be obtained by means of the Ebert or Gerdien ion-counters. In Terrestrial Magnetism, Vol. XVIII, No. 4, Vol. XIX, No. 1, No. 2, No. 3, 1914, Mr. W. F. G. Swann discusses the theories of the several ionization apparatus, and indicates that several cor- rections must be applied in order to remove certain errors. These depend upon the following conditions: 1. The variation 'of the atmospheric potential gradient with the height above the surface to the axis of the horizontal cylinder. The change may amount to 20%. 2. The change in the shape of the electric stream lines on en- tering the cylinder, due the electrostatic attractions and repul- sions, whereby the number of ions entering may vary as much as 20% to 30%. 3. The modification of the capacity of the inner condenser, which should contain the stub of the supporting rod outside the electroscope as well as the small inner charged cylinder. This may increase the "computed capacity" into the "measured capacity" by as much as 30% to 40%. 4. The value of the unit electric charge, as determined by several different lines of experiments, has been taken 3.4 X 10~ 10 , but the electrical constant of Table 95, e = 4.774 X 10~ 10 , seems to require this increase in the computation of the coefficient of conductivity ^. It is evident that the entire subject is in need of further discussion. The Atmospheric Electric Potential' The distribution of the electricity in the atmosphere consists normally of a heavy positive charge of several thousand volts at about 5,000 meters altitude, which diminishes toward the earth, with an increasing potential fall near the surface, the latter being charged with a negative potential. There are continual fluctuations among these potentials in diurnal, annual, and other periods, or in aperiodic variations as in thunder- storms and minor changes. 310 RADIATION, IONIZATION, AND MAGNETIC VECTORS Let V = the potential at any height. d V (733) Force F = -rr = 4 TT <r, where <r is the surface den- sity. (699) (734) -T^T = + -jjp = 4 TT p, where p is the volume density. _&V__ volts volts E. S. U. d h ~ ~ meter ~ " 100 cm. " 300 X 100 cm.' Surface density for the average potential fall of 100 volts /m. 100 4 TT X 300 X 100 1 00 (735) * = - 7 = - 2.65 X 1<T 4 E. S. U. d 2 V The potential gradient changes at about 1/1000 = -T-VT. d fl (736) ^=-f|-4 1 ;=-1^56(l^)35o = +27X 10~ 9 E. S. U. The total surface charge of the earth is computed from <r with the radius of the earth r = 6.37 X 10 8 cms., by (696), (737) Q E = 4 TT r 2 . * = 12.56 X (6.37 X 10 8 ) 2 . (- 2.65 X 10~ 4 ) = - 1.35 X 10 15 . Since 1 coulomb = 3 X 10 9 E. S. U., Q E = - 4.5 x 10 5 E. S. U. 7^2 1 c* 2 2 The electric pressure U t = U t = - - = ^ - 2 TT <r 2 , by O 7T O 7T (672). (738) 2 TT ** = 4.43 X 10~ 7 dynes. The vertical electric current for A = 20.5 X 10~ 5 E. S. U. - E. S. U. by (700). i = 2.05 X 10~ 15 amperes/cm. 2 , since 1 ampere = 3 X 10~ 9 E. S. U. (730) ^.^xHT.-e. 83X10 ATMOSPHERIC ELECTRIC POTENTIAL 311 The theory adopted in this work of the electric potential and its gradient observed in the lower atmosphere is that the in- coming radiation ionizes the aqueous vapor in the strata within Height Z 5000 4000 3000 2000 1000 000 Gradient per meter = Voltage at different heights 1602 x 100 1236 x 100 60 Volts per meter = -r-^- Volts FIG. 57. The electric potential fall and the voltage at different heights. a few thousand meters of the surface, so that the normal charge is 150,000 volts at 5,000 meters, 100,000 volts at 1,400 meters, and volts at the surface, with negative induction in the earth itself. A study of the radiation data of Table 67, those of Table 63 for the distribution of the aqueous vapor, together with those of Fig. 57 for the electric potential, may lead to the function connecting radiation, ionization, aqueous vapor contents, electric currents, and diurnal magnetic deflecting vectors. This subject will require prolonged research in observation and analysis. Table 57 gives the distribution in heights for two cases of the voltage (160200, 123600), and the voltage gradient per 100 meters (-8 -5) at 5000 meters, with (-130 -90) at the surface where V = o in both cases. CHAPTER VI Terrestrial and Solar Relations The Five Types of the Diurnal Convection in the Earth's Atmosphere THE further analysis of the problems of electric and magnetic variations depends upon the determination of the types of the diurnal convections in the earth's atmosphere. There are five of these types, distinct from one another: (1) In the Arctic zone; (2) in the North Temperate zone, Lat. + 66 to + 30; (3) in the Tropic zone, Lat. + 30 to - 30; (4) in the South Temperate zone, Lat. - 30 to - 66, and (5) in the Antarctic zone. What is needed is a complete determination of the diurnal deflecting wind vectors, for each hour of the day and night, and on several planes from the surface to 3,000 meters, as 000, 200, 400, 600, 800, 1,000, 1,500, 2,000, 2,500, 3,000 meters. These can be obtained by kites or captive balloons, but the labor will be not inconsiderable. Unfortunately the available material is very meager, and it is almost wholly lacking during the night. Without the night observations those made during the day, 6 A.M. to 6 P.M., are of quite subordinate value. At present we have data made during the day and night only at the Blue Hill Observatory, 1897-1902, analyzed in my papers, " Studies on the Diurnal Periods in the Lower Strata of the Atmosphere," Monthly Weather Review, February to August, 1905; at several Mountain Observatories, J. Hann, K. Ak. Wiss., Wien, Bd. CXI, Abth. Ha, December, 1902, and April, 1903; there are several stations on the surface which can be utilized for pro- visional discussions, Wien, Mauritius, Batavia, Cordoba, Cha- carita, Laurie Island in the South Orkneys. The circulation in the North Temperate zone can be quite accurately constructed, while that in each of the other zones can only be provisionally inferred. The available data have been thoroughly recomputed with the results collected in Table 71, where s = the velocity in meters per second, and & = the azimuth angle from 5 = 312 FIVE TYPES OF DIURNAL CONVECTION 313 | M E H H B I ex 6 3 o O 2 <N I O CO c/5 E L <NIOCOOOC5<MCOOCOOO<NIO OOOOOOOOOOOO w- vq w. ,.^ - co ^ ^ CO C^| CO C^ OcoT^coeocot^.Oicot-HO5(N iOCOCOiOOOGO^COCOCOCX)TtH o'o'o'oooooo'ooo o I^i-i^COtNC^fNCN OOOOOO'-ioOOOO oooooooooooo oooooooooooo OOOOOOr-i^OOOO rHi IT IOO*~<'-<OOOO 314 TERRESTRIAL AND SOLAR RELATIONS I H u PQ < I 6 a s H w ri. 8 I>OOOO i 1 0} CO 00 CO CO r-(i-( rHTtlOOOOlNC'O-tf rH CO ^^ ^O ^O CO C^ ^^ CO ^H ^H *"^ C^ CO CO OOOOOOOi-HrHOOO OOOOO^O iOcOCDcOrJH COlMOO^Oi OOOOOO'-H'-HOOOO ^ <M (N <M <N <M r-i oooooooooooo 88 S 8 S 848 g 2 3 fc 8 CO (M (N C^ (N -r-i oooooooooooo FIVE TYPES OF DIURNAL CONVECTION 315 I a i\ S i PQ o S C/5 i a 8 s M C S s (/) 2* H ^CO CO <N 000000000000 T~^ O^ CO CO ^H OO i (JO^fCOOOs COCOCMfNr-t 0000 ^D (M *O "^ CO CO ^"* co co 1-1 Ot-HT-iC<ICO<N'*iOiOC5 THi IdCOCOlOi IT (OOS THi-Hi li IT ICMCOCOCOC^ t > * CO O^ i^ *O C^l ^O CO C^ CO O5 i^ (N<N<MCOCOCOCOCO(N'-<COCO ^Hr-li-lT-lrH(NCOCOCOCO i-l CO CO CO CO CO 316 TERRESTRIAL AND SOLAR RELATIONS through E = 90, N = 180, W = 270. The vector direction is that toward which the stream is moving. Having the mean observed hourly vectors, D, and the mean 24-hourly resultant R, these deflecting vectors D are such that R + D = 0, con- structed as true vectors of velocity and direction. MN. 10 Noon 2 10 MN. 2 From Stations in the Northern Hemisphere \ \\ \ \ \ \ \ \ \ \ \ From Stations in the Southern Hemisphere FIGS. 58 and 59. The probable types of the mean diurnal circulation. The data of Table 71 were plotted in diagrams and the apparent circulation for the North Temperate zone was con- structed, as in Fig. 58. By analogy, from the surface data of Cordoba and Chacarita, the circulation is constructed for the DIURNAL VARIATIONS OF ELEMENTS 317 South Temperate zone, Fig. 59. From Mauritius and Batavia, we have that from the Tropics; and those for the Arctic and the Antarctic zones are made from other data to be mentioned later. In the two Temperate zones the circulation is oppositely directed, in general, with turning-points at 10 A.M. and 8 P.M., with divergence at 2 to 4 P.M., and convergence at 6 to 8 A.M., and 8 to 12 P.M. The air is rising in the afternoon and falling in two streams during the night, as 10 to 12 P.M. and 2 to 8 A.M. This is the circulation which is caused by the diurnal heating of the earth's surface and the lower strata of the atmosphere with rising air during the daytime, and with cooling and descending air during the night. This circulation is limited to 3,000 meters from the surface, and it is not vigorous above 2,000 meters. The local conditions of mountain stations introduce many minor modifications, and such stations are never fully equivalent to ideal free-air conditions. The diagrams of Figs. 58, 59, contain the horizontal component chiefly, but the vertical component can be approximately inferred from the general stream lines. Similarly, we have the horizontal and the vertical circulations in the several zones, as may be seen in Fig. 64 in connection with the magnetic vector systems which depend upon them. The vectors of Fig. 64 are the ends of the stream lines at the surface, as determined by the data of Table 71. These data were actually applied to globe models, and from them the adopted circulation was derived. The Diurnal Variations of the Meteorological, Electrical, and Magnetic Elements The effect of the diurnal circulation on the several meteor- ological elements is very complex, and especially so in view of the incessant interchanges between the diurnal and the semi- diurnal periods in the lower strata. Table 72 summarizes some examples of this interconversion, which can be profitably studied by transferring the data to suitable diagrams. 1. The temperature data for the lower strata, 000 to 2,500 meters, are taken from Table 24; that for B is from the same 318 TERRESTRIAL AND SOLAR RELATIONS table, by the conversion from P to B. The temperature has a simple diurnal wave at the surface, as heretofore explained, but a semidiurnal wave above 500 meters, diminishing to extinction at about 3,000 meters. These results conform to the Blue Hill direct observations of temperature in the free air. TABLE 72 EXAMPLES OF THE TRANSITION FROM THE SEMIDIURNAL PERIODS TO THE DIURNAL PERIOD OF DIFFERENT ELEMENTS. 1. Temperature T 2. Barometric Pressure B \ jf _4._ Cordoba TVyT^4- Cordoba Meters 000 500 1000 1500 2500 Meters 000 500' 1000 1500 2500 A.M. 292.8 295.0 291.0 287.7 280.9 A.M. 760.35 717.64 676.95 638.18 565.99 2 291.7,295.4291.9 288.6 281.0 2 760.13717.27676.72 638.10 565.98 4 291.1295.3,291.6288.0 281.0 4 760.40 717.42 676.85 638.18 565.98 6 291.0294.9290.6287.5 280.8 6 760.80 717.84 677.16 638.29 565.99 8 293. 2|294. 0290. 0287.1 280.8 8 760.93 718.22 677.30 638.33 566.00 10 297.71293.5289.5287.0280.7 10 760.95 718.28677.34 638.34 566.01 P.M. 300.2295.0290.7287.7,280.8 OP.M. 760.30 717.80677.06 638.28 565.99 2 301.3298.0292.61288.8 280.9 2 759.08 717.05676.71 638.13 565.97 4 300.7 298.6293.8289.7 281.2 4 758.56 716.72 676.43 637.93 565.95 6 299.0297.7 293.5 290.0 281.5 6 758.62 716.49 676.25 637.79 565.94 8 296.4296.1 292.2 288.9 281.2 8 759.70717.15676.60 638.02 565.97 10 294.3 294.7 290.7 287.1 280.8 10 760.58 717.91 677.21 638.27 566.00 3. Vapor Pressure e d 4. Vapor Pressure ej Tower Sal ton Sea, Tower No. 1 Tower Salton Sea, Tower No. 4 Pans (1) (2) (3) (4) (5) Pans (1) (2) (3) (4) (5) Feet 00 10 20 30 40 Feet 2 10 20 30 40 A.M. 11.7 11.7 11.6 11.8 12.9 A.M. 16.0 14.3 14.8 14.2 14.3 2 10.7 11.0 11.0 11.0 12.7 2 14.8 13.8 14.1 13.3 13.5 4 10.0 10.2 10.3 10.6 12.5 4 13.7 12.8 13.2 12.0 12.7 6 9.8 10.2 10.1 10.4 12.3 6 12.6 12.7 12.9 11.6 12.5 8 12.4 13.2 12.2 12.3 13.3 8 14.6 14.6 14.0 13.7 14.1 10 14.3 14.7 13.6 13.7 13.9 10 16.9 16.5 15.8 15.0 15.4 P.M. 12.1 12.1 10.6 11.2 12.0 P.M. 18.0 17.7 16.1 13.8 14.0 2 9.7 9.6 8.2 8.5 9.5 2 18.6 17.3 15.5 12.0 12.0 4 9.3 9.3 8.4 8.2 8.9 4 18.8 16.4 14.6 11.8 11.3 6 9.9 9.7 9.8 8.7 8.6 6 18.3 15.1 14.2 11.8 11.2 8 11.0 11.3 11.0 10.3 10.8 8 17.8 15.0 14.8 13.3 13.3 10 12.1 12.3 12.0 12.1 13.3 10 17.4 15.0 15.8 15.4 15.3 DIURNAL VARIATIONS OF ELEMENTS TABLE 72 (Continued) 319 5. Vapor Pressure e 6. Electric Potential Fall. d Blue Hill (Summer) Kew Kremsmiinster Greenwich 000 200 400 1000 Sum. Wint. Sum. Wint. Sum. Wint. A.M. 11.07 12.15 8.78 7.53 A.M. + 3 - 8 -12 -33 + 6 - 1 2 10.91 11.40 8.41 6.80 2 -14 -33 -19 -51 + 1 - 9 4 10.75 11.16 8.08 6.44 4 -20 -58 -24 -54 - 3 -13 6 10.88 10.74 8.15 6.44 6 00 -36 + 2 -23 + 1 -13 8 11.25 10.75 8.45 6.67 8 +23 + 8 +16 + 4 + 1 - 6 10 11.14 11.11 8.92 7.47 10 +14 +34 +13 +22 + 5 + 5 P.M. 10.67 11.62 9.48 8.19 P.M. -10 +10 + 7 +19 - 3 + 4 2 10.50 11.53 10.08 8.84 2 -24 - 6 + 4 +29 - 9 + 3 4 10.46 11.12 10.27 9.02 4 -21 + 8 + 5 +23 - 8 + 6 6 10.74 10.85 9.72 8.81 6 - 2 +25 + 4 +35 - 3 + 8 8 11.15 10.79 9.15 8.31 8 +24 +29 +11 +32 + 1 + 7 10 11.17 11.38 8.87 7.81 10 +27 +21 - 7 + 3 + 8 + 7 2. The pressure B has semidiurnal waves from the surface upward, diminishing to extinction on the 3,000-meter level. The morning crest of maximum is smaller than that of the afternoon. There is not the least evidence that the semi- diurnal pressure waves embrace an oscillation of the entire atmosphere, as Kelvin's theory of the forced oscillations demands, and therefore several discussions and other inferences depending on that theory are really without proper foundations. 3. The vapor pressure is subject to this interchange of periods. At Salton Sea, Southern California, Tower No. 1 was located in the desert, 1,500 feet from the water, and the semidiurnal period is clearly denned at every stage from the surface to 40 feet. Tower No. 4 was located in the sea, at one mile from the shore, and it was observed, as in section 4, that the diurnal vapor pressure e d near the water converts itself into a semidiurnal wave within 40 feet of the water. At Tower No. 1 the diurnal convection could not obtain vapor from the surface sufficient to fill up the diurnal wave, while at the water this deficiency did not exist near the surface, in consequence of the rapid evaporation. Section 5, for the vapor pressure in the free air at considerable heights above the surface of Blue Hill, shows a recombination 320 TERRESTRIAL AND SOLAR RELATIONS of the semidiurnal waves at the surface into diurnal waves at less than 1,000 meters above the surface. Here the diurnal convection carries the vapor upward to cooler strata and there concentrates it into a single maximum at about 4 P.M. The details of the physical conditions of these periodic interchanges must be left for more minute researches into the prevailing forces that are at work. Section 6 gives some examples of the well-known change from the semidiurnal waves of the electric potential fall, prevail- ing generally in the summer where there is vigorous convection in the lower strata, into the diurnal or approximately diurnal wave which is characteristic of the winter months. Here the positive ions in the atmosphere appear to move up and down, relatively to the surface, upward in the convection of the after- noon, thereby diminishing the potential gradient, and downward at 8 A.M., and 8 P.M., thereby increasing the potential gradients. Table 73 contains a collection of the coefficient of dissipation a f , in percentage per minute, from observations made throughout the 24 hours at Daroca, Porta Cceli, Guelma, Stations of the U. S. Eclipse Expedition, 1905, also at Bona, and on the U. S. S. Ccesar during the voyage across the Atlantic Ocean. Daroca is on the Aragon plateau; Porta Coeli is near Val- encia; Guelma is in the interior of Algeria and Bona is the port. The observations on the U. S. S. Ccesar were made during the voyage from Gibraltar to the United States: (i) on the forward hatch, and (2) under the shelter of a large iron bulkhead. The values of a' + and a' _ at Guelma have been multiplied by the factor ^2, and those at U. S. S. Ccesar , bulkhead, by the factor 2, in order to reduce them to the scale of the other series in taking the general means. These data are plotted in Fig. 60 in percentage per minute a' . Fig. 60 shows that the diurnal variation of the dissipation coefficient in percentage per minute has a maximum at about 3 P.M., corresponding with the vertical convection, and secondary maxima at 10 P.M., and 2 to 4 A.M., corresponding with the two descending branches. The minor crests occur at 1 A.M., 4 A.M., 8 A.M., P.M., 3 P.M., 5 P.M., and 9 P.M. Each of them corresponds DIURNAL VARIATIONS OF ELEMENTS 321 U UK 1 UK O OOOi-HOOOSOOO OOOOSO O>OMC500-#001010-<J<CO-<JI 10 10 <0 t-OOOOt-t-00<Ot>t-OOC-<0 T}< 05 <N T* 10 l> T}< > D tO ^ O O * 00 O O O rj< O C^OOkO^t~D t-t- > -<t-kOOO cgoooor-iaseo oooia>Deo CJCO^O^^OCO LClO^UdUD otO(N<ocoeo o eo ^ eo o TJI 322 TERRESTRIAL AND SOLAR RELATIONS with some feature of the diurnal circulation, as it transports the ions from one level to another in this complex system of local Oa.m. 2 10 Op.m. 2 10 a.m. 1.60 1.50 1.40 -CtA 1.60 1.50 1.40 3 1 FIG. 60. Mean diurnal variation of the coefficient of electrical dissipa- tion in percentage per minute. *1 c AT +1.0 Temperature o.O .m.2 4 6 8 10 Op.m. 2 4 6 8 10 \ Cordoba / __5 / \ ^ ^~ \ 2 \ 600 meter level -1.0 m.m. AB -0.50 Pressure o.OO \s 4 X^_ _s/ 400-1000 meters Cordoba " ' / ^ N ~7 \ / + 0.50 A m.m. A -1.00 Vapor pressure 0.00 ^ "X / \ Surface Salton Sea ^^^ . 7 ~ ^ / \ / \ ^ -\ / +1.00 V AV -25 Electric Potential 0.0 _^--' ^^ ^ \ _-^i California Grenwich Paris s^^ X . / \ / \ . x + 25 a' + 0.10 Electric "X. / V / Potsdam Potsdam , X. ^~ 1 ^^ / > / ^ ^^ Dissipation -0.10 ^ ^ ' Daroca FIG. 61. Summary of the various semidiurnal waves. circulations. These data should be much more extensively studied. It is generally found that the coefficient of dissipation varies DIURNAL VARIATIONS OF MAGNETIC FIELD 323 as follows: (1) Greater in clear air and less in cloudy, dusty air; (2) Greater with increase of the wind velocity; (3) Greater with increase of the temperature; (4) Greater with the higher vapor pressure. We may finally compare the diurnal curves of the several elements with the temperature waves in the strata 400-1,500 meters above the surface. The distribution of the evaporation in the soil from the sur- face to 100 c.m. has been carefully worked out at Cordoba, with the result that the evaporation from the water table in the soil takes place in a diurnal curve-, exactly agreeing with that of the vapor pressure. It is, therefore, thought that A e, A F, a' of Fig. 61 should be inverted and referred to subsurface evap- oration of ground water. The Diurnal Variations of the Terrestrial Magnetic Field There is another effect of the diurnal circulation in the earth's atmosphere, first, in the generation of electric currents, and (-) North Pole 'Day Night / \ I Day Night (+) South Pole FIG. 62. Scheme of the diurnal circulation in zones. secondarily, in the induction of the diurnal magnetic deflecting vectors that cause the variations of the normal magnetic field. 324 TERRESTRIAL AND SOLAR RELATIONS In Bulletin No. 21, U. S. W. B., 1898, were published the results of a computation on the observed elements H. D. V. at 30 stations. TABLE 74 THE MEAN MAGNETIC DEFLECTING VECTORS IN FOUR ZONES Arctic Zone North Temperate Tropic Zone South Temperate Zone Zone Hours of the Observations Mag. Lat. Mag. Lat. Mag. Lat. Mag. Lat. 78 to 62 61 to 28 + 10 to - 15 30 to 55 Stations 7 Stations 13 Stations 5 Stations 5 s a p s a p 5 a f sa0 Midnight 60 -36 345 15 -30 111 20 -33 5 19 +27 259 1 A.M. 63 -44 355 14 -35 109 19 -32 16 19 +31 250 2 69 -43 5 14 -32 102 20 -36 7 17 +35 251 3 74 -44 16 14 -33 108 20 -42 6 18 +36 243 4 75 -42 25 15 -35 112 18 -34 10 20 +36 226 5 77 -42 30 17 -33 110 17 -37 6 21 +33 223 6 78 -40 32 20 -31 112 19 -36 4 24 +31 222 7 76 -40 36 22-6 107 21 -37 339 26 +24 235 8 65 -37 45 25+3 99 24 -30 297 28 +28 248 9 54 -18 68 26 +24 66 26 +23 228 28 +33 256 10 39 +31 117 27 +37 49 35 +25 210 26 -27 296 11 47 +44 195 25 +38 312 43 +22 204 25 -37 327 Noon 56 +43 200 33 +35 287 43 +30 193 28 -41 47 1 P.M. 64 +42 204 32 +26 277 40 +31 163 32 -36 53 2 73 +41 206 29 +23 268 34 +27 156 30 -35 72 3 83 +39 206 25 -24 263 17 +27 121 30 +30 82 4 89 +39 206 22 -39 259 16 -31 40 28 +30 85 5 87 +36 209 19 -45 260 16 -25 18 24 +39 84 6 78 +37 209 18 -54 234 19 -22 12 20 +40 84 7 62 +34 212 17 -44 183 21 -30 3 17 +45 78 8 54 +32 219 16 -40 183 23 -30 2 17 +46 108 9 51 +11 256 14 -39 103 23 -30 358 17 +47 283 10 50 - 6 279 15 -36 96 23 -28 2 18 +41 269 11 51 -37 336 13 -33 105 23 -28 6 18 +36 264 s = the vector in units of the 5th decimal (C. G. S.), o.ooooi dyne. a = the vertical angle, positive above the horizon P = the azimuth angle, from the South through E. N. W. (740) s = (dx*+d y *+ </z 2 )*, o- = (dx z + dy 2 ^, tana = ~ tan/? = ~ Table 74 contains a condensed summary of these vectors, s, a, /3, in the four principal zones (Fig. 62), that for the Antarctic being omitted for lack of observations. Fig. 63 contains a diagram DIURNAL VARIATIONS OF MAGNETIC FIELD 325 of the vectors, which illustrates the system to some extent, though they can be properly studied and appreciated only by Midnight 6a.m. 6 a.m 6p.m. Midnight Arrows with for angles above the horizon, + a. Arrows with > for angles below the horizon, a. FIG. 63. Scheme of the directions of the deflecting forces causing the diurnal variations of the magnetic field in five principal zones. (Figure on page 90, Bulletin No. 21, U. S. W. B., 1898.) reference to the original 30-inch globe model. The reader is directed to Bulletin No. 21 for further discussion of these data. 326 TERRESTRIAL AND SOLAR RELATIONS This magnetic system has constituted a difficult problem for solution, as it is necessary to have a simple, world-wide cause capable of producing these diverse effects. The most prominent fact is the inversion of vectors as be- tween the two hemispheres, and it is easy to show that the diurnal convection is oppositely directed in reference to the normal magnetic field, positive in the southern hemisphere and negative in the northern hemisphere. In the Tropic zone the air rises nearly vertically by day and falls by night; in the Temperate zones it flows toward the poles by day, and toward the equator by night, being oppositely directed in each hemisphere relative to the positive direction of the magnetic field; in the Arctic and Antarctic zones the movement is upward by day toward the sun and downward at night. These five zones of circulation are marked off from each other by the high-pressure belts in latitudes + 30 and 30 and by the low-pressure belts in latitudes + 66 and 66. The zones of circulation agree with the zones of magnetic vectors as defined in 1892. Fig. 64 contains a scheme of the circulation vectors (black), and the magnetic vectors (dotted), as derived from the two sources indicated. There is remarkable agreement so far as the observational data extend, and the corresponding portions of the circulation adopted by natural inference agree with the parts that are known. It is generally true, (l) that the circula- tion vectors and the magnetic vectors are at right angles to each other, and (2) that the turning points in both systems coincide in all parts of the five zones. The conclusion is almost imperative that the circulating vectors, through the generated ions in streams, induce the observed magnetic deflecting vectors. While there is much to be done by observations fully to verify this theory, it is clear that the main features of both the systems are in remarkable conformity to the known facts of the observa- tions. The horizontal and vertical components of the two sets of vectors in Fig. 64 should be united in one set of spacial vectors, in order that this system may be properly comprehended. The evidence is very strong that the magnetic variations depend DIURNAL VARIATIONS OF MAGNETIC FIELD 327 Mn 2 4 6 8 10 Noon 2 4 6 8 10 12 *'** S' *'"" f --- X r "' Arcti !\ ^> x< cZon * \ i e i \ \ k \ V- \ \ *''' /x Vertical \ \ \ \ \ / V 1 / / X X ^* Planes \ \ A -A N >*\ arth2 --^ remp /\ eratu reZo ^---. ne /*. / ^ /\ ,'\ ^ Horizontal / J / / \ V /N /" N ^^ 2 / Planes / Vertical \ \ X "* / >.' N 4. s * Planes ^ ^ .^ *^ > \ Iropi \ cZon e f / s / y s Horizontal * / \ / \ I 1 / V / ^ ^ r^ ' \ \ \ </ \ / \ J ^ Planes / Vertical \i \ v x ^ v y / * '"-*. X A % A NN A Planes > x \ A, / \ / \ s /\ outh / N Temp erati / ireZc / me i / \ * p Horizontal \ \ \ \ x-' 'r*. J N A ft x ^ ^*. SZ N \ Planes Vertical *.' tf *-' if *.' \ ^ i "7 ** / -A *' >-' Planes / / / / A / a tare f ticZo ne '^i 4 /\ X / i ,-V / ^Vertical ^^ Planes ^-.^ "*** *"*x **ik *-x A "^X FIG. 64. The probable types of the diurnal wind vectors as generators of the diurnal magnetic deflecting vectors. Full-lined vectors = the elec- trical currents in the streams of the diurnal convection secondary vectors. Dotted vectors = the induced magnetic deflecting vectors as computed from the observations and given in the U. S. Weather Bureau Bulletin No. 21, page 87, 1897. 328 TERRESTRIAL AND SOLAR RELATIONS upon ionization in the lower strata, and not upon any system of ionization currents in the upper strata, as has been claimed. Besides the vector directions of the magnetic forces we may approximately obtain the forces by the formula, (741) A# = 4wen + u + , where e = 3.4 X 10~ 10 , n+ = the number of positive ions per cubic centimeter, and u+ the velocity of the circulation in centi- meters per second. The available data for the level 400 meters are probably approximately as follows, in the South Temperate zone: TABLE 75 THE DEFLECTING MAGNETIC VECTORS AS COMPUTED AND OBSERVED A H = 4 TT e n u in C. G. S. units. A.M Formula 2 4 6 8 10 47T 12 56 e 3.4X10~ 10 n+ u + AH Observed 1045 35 .00016 .00019 1226 38 .00020 .00017 1374 40 .00023 .00020 1284 36 .00028 .00024 1245 34 .00018 .00028 1223 46 .00024 .00026 P.M . Formula 2 4 6 8 10 47T e n + u + AH Observed 1195 75 .00038 .00028 1232 70 .00037 .00030 1287 55 .00030 .00028 1225 50 .00026 .00020 1242 45 .00024 .00017 1274 38 .00021 .00018 The number of ions per cubic centimeter was obtained at the surface at Daroca and Guelma, 1905, and from the other available published data; the velocity of the moving medium in cms/sec, was adopted from the study of the Argentine data. The results are so far in harmony with the observed A H = s of Table 74, South Temperate zone, equivalent approximately to THE APERIODIC MAGNETIC VECTORS 329 5 of the North Temperate zone, that we must admit that there is a close causal connection. The magnitude and direction of the deflecting magnetic vectors are so far in harmony with the convectional vectors, in all parts of the earth, that the subject will deserve to be further studied, especially in the determina- tion of the wind vectors in the lower strata of the atmosphere. The Aperiodic Magnetic Vectors Along the Meridians According to Tables 66, 67, and Fig. 55, there are two principal regions of the absorption of the incoming solar radiation, the cirrus region and the cumulus region, in both of which there is transformation of energy into heat or into electric ions. The consequences of such ionization have been studied in the cumulus region, in the induced periodic diurnal or low-level variations of the magnetic field. It remains to give some account of the effects of the ionization in the cirrus region upon the earth's normal magnetic field. In order to analyze this subject the hourly variations are eliminated by taking the mean daily values of H the horizontal force, D the declination, and V the vertical force, as commonly published. As an example of the world- wide correlation of these daily movements of the magnetic field the horizontal force is transcribed in scale divisions, or units of force, for Greenwich, Toronto, Singapore, St. Helena, Cape of Good Hope, Hobarton, very widely separated in latitude and longitude. It is seen that substantially the same sort of varia- tions, + A H, A H, occur nearly simultaneously all over the earth. Similarly, there are + A D, - A D, + A F, - A V variations occurring from day to day. These rectangular variations must first be all transformed to C. G. S. units d x, dy, dz, and these are to be combined in polar co-ordinates s, a, 8, which give the magnetic deflecting vectors disturbing the normal field of the earth. In computing A H, A D, A F from day to day, since there is an incessant secular or long-period variation of H, D, V, it is necessary to secure a proper base line, with appropriate slope, to which A H, AD, A F may be referred. This is best done by constructing the 10-day con- 330 TERRESTRIAL AND SOLAR RELATIONS secutive means, H 0) D , V for every day of the year, so that the consecutive mean plus the variation is the observed value, H + A H = H , D + A D = D, F + A V = V. In passing the dates of excessive magnetic storms, it is proper to substitute a minimum A H min = 0.00025 C. G. S. It has been proposed to obtain the normal field by taking out the " quiet' 7 days for .0370 Greenwich .0360 Abstract, p.7 .0350 560 Toronto 550 p. 395 540 .1050 Singapore .1000 p. 15 .0950 60 St.Helena 8 p. 81 56 66 Cape Good Hope 64 p. 161 62 120 ."Hobarton 118 p. 371 116 7 +20 +10 Direct Type -10 -20 1 8 a i 5 7 8 9 10 11 12 i; 14 15 1C 17 18 19 20 21 2~ 23 24 25 2fa 27 -^^, ^ s~ \ / s S S \ /" -N / r N / "^ ^ ^ -^ / x ^ ^\ / \ / N - \ ^ ^ ^ s ^ N s~ \ S -^. / ^ -- ^ V. / *-* *"' V ^s ' ^^ , / V / V ^ ^^ / "* \ / t J \ \ ^ / ^. -~N 1 / ^s / J ^^ ~^ V ^s 1 / \ V "^ ^s \ -\ f / \ / - A V / ^_s "^ V s ^ ' \ \ -, '-. t , -.. .... -. s '. -.. \. . ! /' FIG. 65. Example of the variations of the ^-component of the magnetic field in all latitudes, and showing the derived normal type curve in the 26.68-day period (direct), beginning August 3.58, 1845. y = 0.00001 C. G. S. unit. each month, and computing the means from these selected days. Unfortunately the fact that a day is "quiet" does not guarantee that the day is near the normal, because "quiet" days are as likely to run on one side of the normal as are the rough or moderately disturbed days. The best daily and monthly means are derived by taking all the observations as they occur, THE APERIODIC MAGNETIC VECTORS 331 except that all variations greater than 0.00025 C. G. S. shall be counted at that value for the sake of taking out the consecutive means. There has been great confusion in the instrumental data, in the manner of discussing the variations, and in the interpretation of the results. The computed magnetic vectors have been found to possess .0380 Greenwich .0370 Abstract.p.7 .0360 620 Toronto 610 p. 401 600 .1050 Singapore .1000 p. 18 .0950 52 St.Helena 50 p. 87 48 M Cape Good Hope 52 p. 1C7 50 114 Hobarton 112 p. 377 110 y + 20 +10 Inverse Type -10 -20 1 \ 10 11 \-z la H ir> 10 17 18 lt> M i\ i-i i\\ ,U -> M 27 s- S f -~^ N ^ , . / \ \ S N /* ^ / \ J * s -^ ' / - \ / s~ ~\ ^ .^ *\ /~ v_ / V / f ^ / * . . , ** \ ^^ "> / " ~x s *-" 3 " ' ^ ^ ^ji V, ^s f~ ^_j /^ ^ ^ V ^~ -v, Vx s~ y ^^ ' ^ ^ /~\ _y \ / \ jS- ^ f > ^ v^, v_ *s J \ V / v_x ^ -^ ~s \ / s- ^_ y / \ / ^ */ s^^ ~J~ / > . / ^-^ / ^ _^ \ \ \ ' f "^ s~. / " ' \ ) v> I / r */ . \ C \ I t ,* N / : \ : X. .-' .' ; \ / : / FIG. 66. Example of the variations of the /^-component in the 26.68-day period (inverse), beginning November 18.30, 1845. A careful study of these curves, 1843-1905, shows that this type (reversed) occurs semiannually. Direct type, Feb. 1-April 20 and July 15-Oct. 15; inverse, April 20-July 15 and Oct. 15-Feb. 1. interesting and important characteristics. Bulletin No. 21, U. S. W. B., 1898, contains a full explanation of these data, to which the reader is referred. (1) These vectors of deflection are generally closely confined to the magnetic meridians, and depend chiefly upon A H, A V. According to the latitude of 332 TERRESTRIAL AND SOLAR RELATIONS the station the vector, s, a, /3 has well-defined lengths and vertical angles, similar to those seen in Figs, 68, 69. Corresponding with the waves in Figs. 65 66 these vectors point first south- FIG. 67. Generation of the ions in the cirrus and cumulus regions. They flow alternately toward the north and south poles. FIG. 68. Closed circuit for southward magnetic vectors. The + ions flow toward the north pole. ward, Fig. 68, and then northward, Fig. 69, alternating about every three days. (2) A very extensive study of these vectors for the years 1843-1910 shows that they have a well-defined period of recurrence, on the average 26.68 days in length, and witji two THE APERIODIC MAGNETIC VECTORS 333 types, the direct as in Fig. 65, and the inverse, Fig. 66, the relative intensity from day to day being shown in the lower section of each figure. The recurrences are complicated with many irregularities, but the periodic action is unmistakable and corresponds with the synodic period of the rotation of the sun on its axis, as observed in the equatorial zone. The inference follows that these magnetic meridian deflecting vectors depend upon certain variations in the solar radiation, distributed in FIG. 69. Closed circuit for northward magnetic vectors. The + ions flow toward the south pole. solar longitudes in such a manner that certain areas of the solar surface emit stronger radiations than do others in different longitudes. The equatorial period, 26.68 days, is exactly the same as the period determined from numerous direct observations on the sun spots, the faculae, and certain spectrum lines. From a least square solution of the magnetic data, an ephemeris was constructed on the period 26.679 days, and epoch, June 13.72, 1887. (3) The periodic reversal of the type curve occurs in semi- annual periods, as determined by the records, 1841-1894. Take the successive periods by years and match the type curve with the observations as in Figs. 65 and 66. 334 TERRESTRIAL AND SOLAR RELATIONS TABLE 76 SOLAR MAGNETIC EPHEMERIS, PERIOD 26.679 DAYS, EPOCH JUNE 13.72, 1887 1840 Jan. 16.87 1870 Jan. 24.05 1900 Jan. 5.55 41 Jan. 24.38 71 Jan. 5.88 01 Jan. 14.06 42 Jan. 6.21 72 Jan. 14.39 02 Jan. 22.57 43 Jan. 14.72 73 Jan. 21.90 03 Jan. 4.40 44 Jan. 23.23 74 Jan. 3.73 04 Jan. 12.91 45 Jan. 4.06 75 Jan. 12.24 05 Jan. 20.42 46 Jan. 12.57 76 Jan. 20.75 06 Jan. 2.25 47 Jan. 21.08 77 Jan. 1.59 07 Jan. 10.76 48 Jan. 2.91 78 Jan. 10.09 08 Jan. 19.27 49 Jan. 10.42 79 Jan. 18.60 09 Jan. 26.78 1850 Jan. 18.93 1880 Jan. 27.11 1910 Jan. 8.61 51 Jan. 27.44 81 Jan. 7.94 11 Jan. 17.12 52 Jan. 9.27 82 Jan. 16.45 12 Jan. 25.63 53 Jan. 16.78 83 Jan. 24.96 13 Jan. 6.46 54 Jan. 25.29 84 Jan. 6.79 14 Jan. 14.97 55 Jan. 7.12 85 Jan. 14.30 15 Jan. 23.48 56 Jan. 15.63 86 Jan. 22.81 16 Jan. 5.31 57 Jan. 23.14 Epoch 87 Jan. 4.64 17 Jan. 12.82 58 Jan. 4.97 88 Jan. 13 . 15 18 Jan. 21.33 59 Jan. 13.48 89 Jan. 20.66 19 Jan. 3.17 1860 Jan. 21.99 1890 Jan. 2.49 1920 Jan. 11.67 61 Jan. 2.82 91 Jan. 11.00 21 Jan. 19.18 62 Jan. 11.33 92 Jan. 19.51 63 Jan. 19.84 93 Jan. 27.02 64 Jan. 1.67 94 Jan. 8.85 65 Jan. 9.18 95 Jan. 17.36 66 Jan. 17.69 96 Jan. 25.87 67 Jan. 26.20 97 Jan. 6.70 68 Jan. 8.03 98 Jan. 15.21 69 Jan. 15.54 99 Jan. 23.72 THE SEMIANNUAL REVERSAL OF THE DIRECT AND INVERSE TYPES Period 1 2 3 4 5 6 7 8 9 10 11 12 13 14 D 17 32 42 41 18 7 13 33 43 41 30 16 10 12 I 37 22 12 13 36 47. 41 21 11 13 24 38 44 42 Type / D W. D / /max. / D >max. D D / /max. / The direct type prevails annually, February 1 to April 20. The direct type prevails annually, July 15 to October 15. The inverse type prevails annually, April 20 to July 15. SYNCHRONOUS ANNUAL VARIATIONS 335 The inverse type prevails annually, October 15 to February 1. These facts of periodic action from the sun in the equatorial period of 26.68 days, together with the semiannual inversion of the type, indicate that the problem of the solar radiation at the sun, and in its effects throughout the earth's atmosphere is an exceedingly complex phenomenon, which will require extensive researches of various kinds. By way of suggestion it may be seen on Fig. 67 that if the incoming radiation transforms a part of its energy in the cirrus region into positive (+) and negative ( ) ions, it may be supposed that they seek the poles of the earth in opposite direc- tions, as (+) to the north pole and the ( ) to the south pole, completing their circuit through the outer shell of the earth. This generates the magnetic vector system pointing southward, and the corresponding earth electric currents; at another time the (+) ions seek the south pole and the ( ) ions the north pole, thus producing the northward vectors, and the corresponding earth electric currents. This reversal of direction from time to time depends upon the physical condition of the atmosphere as a conducting medium for the ions, its congestion of ions, its accumulation of ice and vapors, producing the magnetic vectors, auroras, magnetic storms, electric currents, in the well-known conditions as observed. The energy expended at the earth is that transformed from the solar radiation; it is inexhaustible in amount, and depends for the observed aperiodic irregularities upon the prevailing states of the solar and terrestrial atmospheres. The Synchronous Annual Variations of the Solar and the Terres- trial Elements The possibility of a scientific forecast of the type of weather likely to prevail in a large country as the United States or Argentina, whether the coming year is to be rainy and cool, or dry and warm, depends upon the establishment of the following two propositions: (1) The radiation output of the sun is a variable quantity, as 4 or 5 per cent, each side of the mean; (2) The meteorological elements, temperature, barometric and 336 TERRESTRIAL AND SOLAR RELATIONS vapor pressures, and the precipitation synchronize with the solar changes in their annual variations. The evidence at present enables us to affirm that both are true, and that the synchronism exists, though in a very complex form, because the prevailing local conditions depend primarily upon the general circulation, and therefore only indirectly upon the solar varia- tions. It is not possible in this place to do more than summarize the general principles that have been established in a research extending over twenty years, and embracing the available solar and terrestrial data. The first task is to procure homogeneous material of the several observed quantities, extending over a long series of years, sun-spot frequencies, solar -prominence frequencies, amplitudes of the terrestrial magnetic field, baro- metric pressures in all parts of the world, temperatures, and vapor pressures in all countries, precipitation in many districts, direct observations of the solar radiation in calories per square centimeter per minute. Unfortunately the difficulties of secur- ing such homogeneous data of any of these elements is greatly complicated by the irregular and inconsistent methods that have been employed by meteorologists. In consequence of the necessity of substituting a few selected hours of observing for the twenty-four hours of each day, it is necessary to reduce the means from selected hours to the mean of twenty-four hours, which involves a long, special research for each country. The selected hours are different in different countries; the series are broken by changes in the selected hours in consequence of some admin- istrative requirement; the corrections change from place to place when the same hour is made the basis of the work, as where the 75th meridian of the United States is made the hour of observing, which involves a range of three hours locally between the Atlantic and the Pacific States; or where the Greenwich noon is the basis of simultaneous world observations, involving variations up to twelve hours in local conditions; the altitudes and locations of the instruments in great cities have been not infrequently changed, and the instrumental equipment and the methods of computing have never been uniform for the long series. It is necessary to overcome these obstacles by SYNCHRONOUS ANNUAL VARIATIONS 337 setting aside a reasonable number of permanent stations for long series of fundamental work in meteorology, just as astrono- mers dedicate certain observatories to fundamental star places upon which the National Ephemerides are based. Cordoba, in 1875 1885 1900 1905 "" "^ - ~~-^ / ._ _J ^ / S f_ ^ 11.1 year \ 2 2 *\ Z 's ^ . / s-^^s^ ^ ^ ^ "- 4 f- -i - - - 7 /^ ~~ / ^ ' \/ y / _/ r- ' ~f~ "^ C 7 \^ 52 ^ +400 + 200 z V <c 1 !^/> y^ _^ 1 ^ i 'i E 5 " r 7 L Z. ^^ a 1 -0.50 -0.25 / 5 I Z ^v^ \- T "~ ~ T ~/ "^" Argentine 0.00 L- -X ~ \- 4 7 Barometric +0.26 / Pressure - ~? /. s _ + 050 -4 " +o'.25 7 n/N /~\ ^_v ^ J Ji X Centigrade "O* 25 ^ Z ^ ' T +100 /I f r 'Sv L " x \2. \ 'A o.i / H A Z J Z _ PL ^2 w S / 1 / \ y v / / / 4.Yft u y y .~ + 05 ./ L "u 6 3 /x ^^ ^ ^ -/^ o.o T t * , / V 'United States _ 5 T \ 7 , } 7 ^ J Fahrenheit S Z * + 15.0 + 12.0 T + 9.0 " " " + 6.0 C- V -^ + 3.0 .-^ ^/ V T r United States o.O / "I \_\ /. / ... Excess Precipitation _ 3 ' r r j^ ^. r" - 6.0 r _, j 7 7 -9.0 _ ^ v/ -12.0 \/ -15.0 \i_ \T FIG. 70. Synchronism between the annual variations of the solar and terrestrial phenomena. Argentina, is such a first-class meteorological station, because, since 1870, the instruments have had the same natural exposure, and practically the same apparatus has been used continuously, subjected to numerous tests for normality. There is no station 338 TERRESTRIAL AND SOLAR RELATIONS TABLE 77 THE SYNCHRONISM BETWEEN THE SOLAR AND THE TERRESTRIAL ANNUAL VARIATIONS OF THE METEOROLOGICAL ELEMENTS Year Sun-spots Promi- nences Horizont Mag. Am. Argentina United States CM R CM R CM R AB AT A.e Free. &B AT Free. mm. C mm. mm. inch F. inch 1872 720 +500 1800 +845 2033 +679 -0.94 +0.21 +0.29 - 76 - 9.60 73 645 +151 1700 +353 2054 -251 .58 + .15 + .72 + 32 + 9.35 74 578 - 43 1569 -154 1894 +168 + .05 - .58 - .48 + 78 - 7.46 75 364 -159 1188 -332 1630 - 61 + .47 + .09 - .31 - 50 - 0.89 76 213 - 77 865 + 11 1681 -358 + .04 + .04 + .26 + 94 - 0.99 77 120 + 27 695 + 47 1715 -320 - .52 + .72 + .70 -104 + 2.65 78 177 -136 791 -353 1948 +110 + .25 - .62 + .20 +252 -0.021 +0.6 +22.11 79 260 -188 1015 -454 2256 - 23 + .33 - .14 - .51 -327 + .005 +0.4 -15.18 1880 373 + 14 1332 + 6 2729 + 1 - .26 - .22 - .05 - 69 + .023 -1.4 - 6.15 81 518 +133 1064 +394 2946 - 80 - .41 + .37 + .46 +107 - .010 +0.4 + 4.47 82 656 + 60 2063 +264 8144 +613 + .22 - .18 - .49 -108 + .003 -0.2 + 0.60 83 704 + 61 2252 -456 3193 - 47 + .20 - .01 + -17 + 73 + .021 -0.3 - 5.39 84 634 +127 2170 +686 3204 + 19 - .18 + .37 + .36 - 22 - .024 -0.9 +18.36 85 523 +103 2080 +204 2920 + 53 + .14 - .48 - .19 +174 - .011 +1.6 - 0.81 86 386 - 81 2098 -513 2779 +142 + .33 - .25 - .36 - 87 + .009 -0.3 -14.84 87 249 - 92 1671 +206 2569 -233 - .17 + .35 - .11 -204 + .010 -0.6 - 5.76 88 140 - 59 1352 +535 2298 +144 - .43 + .54 + .48 + 83 - .014 +0.6 + 7.59 89 165 - 90 1402 -678 2220 - 48 + .11 - .45 + .15 +217 - .009 +0.6 - 0.31 1890 309 -224 1521 -836 2473 -854 + .50 - .26 - .77 -186 + .016 -0.6 0.44 91 496 - 69 1592 +244 2613 - 82 - .57 + .29 + .49 +108 - .002 +0.3 - 0.51 92 669 +207 1740 +732 2795 +804 + .04 - .10 + .08 - 31 - .003 +0.4 + 3.69 93 805 +214 1931 +310 2993 +151 + .65 - .42 -1.03 -185 + .003 -1.0 + 0.33 94 820 +116 1806 -339 3073 + 7 + .05 - .43 - .29 + 60 + .006 -0.5 - 5.31 95 708 + 60 1534 +106 2806 -195 - .47 + .48 + .15 + 35 .000 0.0 - 7.48 96 568 - 67 1230 - 192650 +280 - .08 + .91 + .83 + 21 - .007 +0.7 + 2.15 97 410 - 95 1037 + 76 2441 -177 + .52 - .14 + .09 - 17 + .005 +0.2 + 8.38 98 279 + 42 817 - 96 2236 +130 + .09 -1.10 - .91 -112 + .001 -0.4 - 5.39 99 185 - 40 604 -106 1925 +107 - .45 + .21 + .28 +129 - .003 -0.5 - 2.80 1930 135 - 21 398 +145 1765 -176 - .25 + .59 + .61 +169 - .002 +1.0 + 6.63 01 129 - 96 295 -150 1717 -343 + .11 - .04 - .67 -211 + .008 +0.3 - 7.98 02 201 -140 305 -222 1853 -389 - .44 + .16 + .11 -134 - .017 -0.7 + 2.16 03 330 - 38 300 - 96 1896 +230 + .22 - .32 + .17 + 76 + .014 -0.5 - 0.85 04 453 + 51 400 +150 2000 -213 + .01 - .03J+ .06 +229 + .008 +0.7 -11.53 05 550 +212 600 - 81 2180 + 14 - .08 - 1 .19 + .12 + 41 - .009 0.0 + 7.81 06 650 4 .17 -l_ 74' 01 197 +19.11 07 _ .08 .02; + .15 - fi + 0.26 08 + .41 .26 .17 + 26 09 + .20 - .01 - .47! -115 1910 .16 + .45 .20 66 11 - .11|- .46 - .07 +178 CM = consecutive mean; R = residual; C M + R = Observed. A B = variation of barometric pressure; A T = variation of temperature. A e = variation of the vapor pressure; Prec. = excess of precipitation. in the United States that compares with it, because of changes of one kind or another in the instrumental conditions or the hours of observing. The writer spent many years in adjusting the imperfect observations in the United States, and finally SYNCHRONOUS ANNUAL VARIATIONS 339 produced a set of series of Pressure, Temperature, Vapor Pres- sure, and Precipitation, that are fairly homogeneous, and form the fundamental basis from which the annual variations may be computed. Similar reductions to homogeneous data are being made in Argentina, and in other countries, and in time it is hoped that world-wide comparable series of reduced observa- tions may be made accessible to the scientific public. Table 77 and Fig. 70 contain a series of examples of the results of such a comparison of the solar and terrestrial annual data, enough to give the reader a fair idea of the possibilities of this important subject. The sun-spots are from Wolfer's data, and the consecutive means C. M. added to the residuals R produce the observed annual means O, C. M. + R = O; the solar prominence frequencies are from the data of the Italian observers; the amplitudes of the horizontal magnetic force were compiled from several European observatories: the Argentine Meteorologica can be found in Bulletin No. 1, Oficina Meteorologica Argentina, 1911; the United States data may be found in the Barometry Report, U. S. W. B., 1902; Temperatures and Vapor Pressures in Bulletin S, 1909; Temperature and Precipitation Normals, Bulletin R, 1908; Temperature Departures, Bulletin U, 1911; Climatological Summary in 106 sections, Bulletin W, 1912, all prepared under the writer's supervision. The consecutive means represent a long periodic cycle averaging 11.1 years in duration, but very irregular in length, as from 8 years to 14 years between certain maxima; the residuals represent a short periodic cycle averaging 3.75 years, but ranging between 3 years and 5 years. The first curves of Fig. 70 represent the 11.1 years and the 3.75 years cycles between 1875 and 1905. Following them are several curves for Argentina and the United States in the short period cycle. The synchronism in the short period is pronounced, in spite of certain irregularities, demonstrating the general fact that the sun has a variable output of radiation which persistently modifies the earth's circulation and climatic conditions. The barometric pressures and temperatures were studied in all parts of the world and the result summarized in Monthly Weather Review, October and November, 1903. 340 TERRESTRIAL AND SOLAR RELATIONS Barometric Pressure. The net work of barometric pressures for the world, taken for the annual variations, shows that the stations must be divided into two classes: (1) Those where the synchronism is direct between the pressure and the prominences, as surrounding the Indian Ocean, and those where the synchron- ism is inverse as in North and South America. Under the external impulse from the sun an increase of the annual radiation accelerates the general circulation in such a way that the pressure is simultaneously higher in certain large regions and lower in others. This is due to the fact that the total pressure of the earth's atmosphere is an invariable constant, so that if the pressure in one region is relatively high, that in another region is relatively low at the same time. The wandering cyclones and anticyclones, added to the more permanent centers of high and low pressures, should sum up to the same constant for the world. The oscillation of regional pressures is, therefore, a fundamental fact leading to an extensive study of the pressure conditions in various localities. The Temperatures. Similar studies of the annual tempera- tures divide the stations into two groups, (l) Those in the Tropics with direct synchronism, (2) those in the Temperate zones, on the poleward side of the high-pressure belt, with inverse synchronism. There are many places of mixture or disintegrated effects which it is still difficult to classify. An increase of solar radiation increases the vertical convection of the Tropics, with increase of the surface temperature; this is followed by an in- crease of downflow in the Temperate zones, with an extension of the high areas and cooler temperatures. The temperature integral of the entire earth's atmosphere must be nearly a con- stant, or else the earth's rotation period of twenty-four hours would indicate variations of an astronomical value, which have never been detected in the observations. Precipitation. The changes in the general and the local cir- culations, depending upon the solar variations, carry with them the rain-bearing currents, as from the oceans to the continents, and thence the annual amounts of the precipitation in the regions concerned. There are great irregularities in these precipitations SYNCHRONOUS ANNUAL VARIATIONS 341 from one region to another, from one year to another, and for the same station. The results from Argentina and the United States indicate clearly that the precipitation synchronizes with the solar variations, and that the variations are of large amounts, ranging through 400-500 millimeters in Argentina at the same station, and several inches in the United States. Partial Formations. Fig. 70 shows that the annual crests in certain elements occasionally fail to form completely in the 3.75-year period, and for this cause irregularities appear in the series of curves. It is easy to see how this may occur in many cases by a sort of self-contradiction in natural causes and effects. Thus, if in a certain region the excess of solar radiation of the Tropics has produced higher temperatures, this has resulted in spreading a rain and cloud sheet over another region at a distance from it, both due to the same cause. This very cloud sheet, however, acts as a screen upon the surface temperatures, so that lower local temperatures are registered at the surface, while they are really higher above the cloud sheet. The rain currents may precipitate so much aqueous vapor on one side of a mountain range that the overflow on the other side is dryer than usual, so as to give opposite effects for the same efficient increase of circulation, excess in one region, and defect in another region. The observations of the solar prominences depend upon the number of clear days per month. Hence, an increase in solar radiation, following an increase in the frequency of the promi- nences, may locally produce a 'cloud sheet, and hence a lower annual count in the number of the prominences. It is quite irrelevant to attempt to discredit the facts of synchronism, by presenting irregularities or inconsistencies in certain localities, unless the trouble is taken to understand the full series of causes and effects between solar action and final local conditions. Since opposite results, inversions of effects, are inevitable in terrestrial meteorology, from the same solar cause, it will be necessary to study carefully the history of each region, before attempting to arrive at any conclusions. The magnetic field presents similar synchronous variations, as may be seen by plotting the amplitude curves. This element 342 TERRESTRIAL AND SOLAR RELATIONS is very sensitive to many radiation and ionization influences, and it is our purpose to pursue the research into the function connecting these several elements. The radiation in calories per square centimeter per second does not yet present annual variations which seem to be reliable. The cause of this result is seen in the section on radiation, and may be verified by studying the divergent annual values on Table 62. The possibility of annual forecasts of the weather conditions is being tested in Argentina by projecting forward the normal 3.75-year curve from 1911 to 1915. The results for 1911, 1912, and 1913 are entirely successful, the precipitation being quite the same as indicated in Bulletin No. 1, 0. M. A. It is certainly possible to make similar forecasts for the United States as to precipitation in different districts, wherever the sequence of the rainfall in each district is studied in relation to the funda- mental solar 3.75-year period. Compare Abstract No. 3, U. S. W. B., 1909. with the data of Fig. 70. The Aqueous Vapor in the Atmosphere It is evident from the discussions on radiation, on cloud formation and precipitation, and on evaporation of aqueous vapor from areas of water, as in lakes and oceans, that the presence of aqueous vapor in the atmosphere is of primary significance. We can compute the number of grams of aqueous vapor per cubic meter of air, or per kilogram of air, according to convenience. Grams of Aqueous Vapor in 1 Cubic Meter of Saturated Air (742) ^ = The full form most used at all elevations above the sea level. By substituting the observed vapor pressure e, the tempera- ture 7", the barometric pressure B, at any other point, the corresponding /* can be computed. Extensive tables have been AQUEOUS VAPOR IN THE ATMOSPHERE 343 prepared for // where T ranges from 50 C. to +50 C., and B from 800 mm. to 20 mm. Grams of Aqueous Vapor in One Kilogram of Saturated Air. (743) ft = 0.622 ^ + 0.235 ^ e Q is the saturated vapor pressure for temperatures ranging from - 50 C. to + 50 C. When the air is not saturated the following formula serves: e Q = the saturated vapor pressure, / = the dry-bulb temperature, /i = the wet-bulb temperature. Vapor Pressure in Millimeters when the Air is not Saturated (744) e = e Q - 0.00066 B (t - tj (l + g^) . Tables applicable to practical work may be found in Bulletin. No. 2, Oficina Meteorologica Argentina, 1912. In the free air the aqueous vapor is distributed approximately by Harm's formula, h (745) e = e, 10 6517 , where h is the height in meters. The Laws of the Evaporation of Water from Lakes, Pans, and Soils with Plants The subject of the evaporation of water has been very ex- tensively studied, and there is a large literature on the results. These, however, are unsatisfactory as concerns the terms and the coefficients of the proposed formulas. Another research was undertaken by the writer in 1907 for the U. S. Weather Bureau, at Reno, Nevada, where the proper type of formula was determined; it was continued in 1908 at Indio and Mecca, So. California, and at the Salton Sea, 1909, 1910, in co-operation with numerous stations in various parts of the United States, during which the coefficients were approximately computed; the 344 TERRESTRIAL AND SOLAR RELATIONS work was continued in 1911, 1912, at Cordoba, Argentina, and extended to include evaporation from soils, and soils with plants of different kinds, and the final coefficients with the necessary working tables for the computations were constructed. The results of this work are summarized in Bulletin No. 2, Argentine Meteorological Office, 1912. Several special pieces of apparatus have been invented: Bigelow's micrometer hook gage for measur- ing the water height, Bigelow's dial gage for measuring the water height in soil tanks, Wilcken's self -registering apparatus for continuous records of every position of the water surface. The principal difficulty in arriving at conclusions has been due to the necessity of using pans for evaporation, in which case the wind in blowing over the pan greatly complicates the action of the evaporation. Pans of different sizes in the same wind evaporate different amounts during the same interval of time, because the wind carries away the evaporated vapor at different rates, according to the size of the pan, and thus produces a varying mixture of dry air and vapor. A large body of water in a wind, and a small pan in a calm, produce the same effect as an evaporat- ing medium, because the vapor is actually the same in density near the water on a lake in a wind, which merely transports it from place to place without really removing it, as in a calm air over a small pan. The result is that lakes evaporate only at about two-thirds the rate from pans near by in moderate winds. In certain places it was found that a small pan evaporates three times as much water as does a lake in the neighborhood. For example, there were three towers built in the Salton Sea, No. 2 near the shore, No. 3 about half-mile from No. 2, and No. 4 TABLE 78 EXAMPLES OF THE ANNUAL EVAPORATION AT THE SALTON SEA Tower No. 1. Pan (5), 40 feet above the desert, 195 inches. Tower No. 1. Pan (1), on the ground of the desert, 165 Tower No. 2. Pan (5), 40 feet above the water, 138 Tower No. 2. Pan (1), 2 feet above the water, 109 Tower No. 4. Pan (5), 40 feet above the water, 140 Tower No. 4. Pan (1), 2 feet above the water, 106 The evaporation from the Salton Sea itself, 72 FORMULA OF EVAPORATION 345 about one mile from the shore, while No. 1 was 1,500 feet inland from the sea in the desert. These towers carried pans near the surface of the water and at every 10 feet up to 40 feet above the water. The evaporation for a year was as follows at several pans, as summarized in Table 78. The evaporation was registered at other stations from pans of different sizes on the ground, and on a stand 10 feet high, of which annual examples follow. EVAPORATION AS RECORDED IN SEVERAL PLACES Station Indio Mecca Brawley Mammoth Height Size of pan Evaporation Ground 10 feet 6 feet 2 feet 119 200 Ground 10 feet 6 feet 2 feet 108 170 Ground 10 feet 6 feet 2 feet 104 164 Ground 10 feet 6 feet 2 feet 126 179 Station N. Yakima Cincinnati Birmingham Lake Tahoe Lake Kechess Height Size of pan Evaporation Ground 10 feet 4 feet 3 feet 68 86 Water 10 feet 4 feet 3 feet 46 62 Water 10 feet 4 feet 2 feet 51 64 2 feet 4 feet 42 10 feet 3 feet 33 The formulas that have been found to be adequate to follow the course of evaporation in all climates, that is, in all conditions of temperature, vapor pressure, and wind velocity, are as follows : Hours of observation for 4-hour intervals (2, 6, 10) A.M. (2, 6, 10) P.M. t = the temperature of the dry bulb on the whirling psy- chrometer as usually employed. ti = the temperature of the wet-bulb thermometer. e d = the computed vapor pressure at the dew point d. S = the temperature of the water surface. e s = the computed vapor pressure at saturation S. de -rz = the rate of change of the vapor pressure with the tempera- d o ture change of the water. w = the velocity of the wind in kilometers per hour, derived from the successive anemometer readings. 346 TERRESTRIAL AND SOLAR RELATIONS Formula of Evaporation from Large Water Surfaces (746) -T = - 0230 ~ ^ + ' 084 ) ; ( Ar S entine anemometer) . Formula of Evaporation from Pans of Different Areas = 0-0230 F - l 1 + 0.084 w ). F (w) = a factor depending on the area of the pan, which varies with the wind velocity up to about 10 kilometers per hour. FI (w) applies to the Dines' system of wind velocities, used by the Argentine Meteorological Office, and a pan of 1.0 meter 2 area; F 2 (w) to the same wind system and a 0.5 m 2 area pan; F (w) to the wind system used in the United States, where- in the same wind velocity is recorded higher in the ratio 1.21 to 1.00, and a 1.0 m 2 pan; F 4 (w) to the U. S. wind system and a pan 1.17 m 2 area or 4 feet in diameter; F 5 (w) to the U. S. wind system and a pan 0.29 m 2 area, or 2 feet in diameter. There are two wind systems in use: (1) that based upon the Dines' pressure-velocity, and (2) that based upon the whirling machine velocities. Thus the anemometers by Casella, Negretti, and Zambra, U. S. Weather Bureau Freiz, Richard, are approxi- mately in agreement together, but they are about 20 per cent higher than the Dines, Hess of the Oficina Meteorologica Argentina, Munro, and Tschau system of anemometers. Mar- vin's table of corrections to the Robinson anemometer gives about 20 per cent correction to reduce from the indicated to the true wind velocity, Monthly Weather Review, October, 1906, Table 64, so that the first group becomes equivalent to the second group after making this reduction. Unfortunately, it is customary to omit these reductions, so that the published wind velocities of the United States, and other countries using the above-mentioned anemometers, are about 20 per cent too great. It is indispensable in evaporation reductions that the coefficients should be adjusted to correct wind velocities. For this purpose the following factors F (w) are introduced into the working Tables: FORMULA OF EVAPORATION 347 TABLE 79 THE FACTORS F (w) FOR ADJUSTING THE EFFECTS OF THE WIND VELOCITIES FOR PANS OF DIFFERENT AREAS System Argentina United States Wind 1.0m 2 0.5 m 2 1.0 m2 1.17 m 2 0.29 m 2 Velocity Mw) F(w) F,(w) F 4 (w) F t w = 1.000 1.000 .000 1.000 1.000 1 1.150 1.148 .120 1.150 1.160 2 1.265 1.274 .240 1.212 1.290 3 .376 1.392 .320 1.289 1.410 4 .463 1.493 .400 1.367 1.520 5 .542 1.592 .480 1.433 1.630 6 .600 1.667 .540 1.480 1.710 7 .617 1.712 .590 1.523 1.760 8 .627 1.746 .615 1.542 1.810 9 .629 1.762 .623 1.552 1.830 10 1.629 .777 1.629 1.561 1.840 15 1.629 .782 1.629 1.561 1.850 2Q 1.629 .782 1.629 1.561 1.850 25 1.629 .782 1.629 1.561 1.850 30 1.629 .782 1.629 1.561 1.850 The complete tables for evaporation computations may be found in Bulletin No. 2, Argentine Meteorological Office, 1912. It has been found that about 90 per cent of the computed results are less than 0.30 mm. from those as observed. This difference includes the errors of measurement as well as of computation. The computed difference from the observed amounts for entire months in Cordoba is about 4 millimeters, and the total difference for the year on one pan was 3 milli- meters, and on another pan 10 millimeters, the total in the first case being 1,091 millimeters, and in the second case 1,945 millimeters. Our experience leads us to conclude that pans need not be employed in work on evaporation, but that compu- tations are quite as accurate provided observations of the water temperature S, the vapor pressure of the air e dj and the velocity of the wind w are made. As it is impossible to float pans on large bodies of water, lakes and reservoirs, except under restricted 348 TERRESTRIAL AND SOLAR RELATIONS conditions that injure the observations, it becomes necessary to dispense with pans entirely and depend upon simple computations. There are several methods of abbreviation for computing the mean monthly amounts of the evaporation from lakes and reservoirs, which make the computations an insignificant labor. Studies on evaporation from soil, sand, soil planted with alfalfa, wheat, barley, beans, have been carried on, which analyze successfully the amount of water lost under all conditions throughout the year, from soils by themselves, and from the plants by themselves. Thus, the transpiration of plants is subject to accurate measurement and analysis, and the results, when sufficiently verified, will be of great value to meteorological agriculturalists and botanists. The Polarization of Sunlight in the Atmosphere Common sunlight vibrates indifferently in every plane perpendicular to its wave front, but when it falls upon any object, large relatively to its wave length, a portion of the light is refracted and a portion reflected in the plane of incidence containing the incident and reflected rays. The vibrations in the reflected ray become at least partially constrained to vibrate parallel to the surface of reflection, and it is plane polarized. The plane of the polarization is at right angles to the plane of vibration, and therefore contains the incident and the reflected rays. If an observer looks at any point in the sky he will receive certain reflected rays that have proceeded from the sun to the reflecting particle and to the eye, this plane being the plane of polarization, and the vibrations are at right angles to it. If the polarization is partial, and the motion circular, elliptical, or of any other figure, components of plane polarized light may be constructed for this plane and another at right angles to it, so that partially polarized plane vibrations in two directions at right angles may more or less neutralize each other between the limits per cent and 100 per cent. A turbid medium, such as air mixed with small solid particles of dust, ice, or even mole- cules, whose diameters are small relative to the wave lengths of light, scatters and polarizes light by Rayleigh's Laws, in which POLARIZATION OF SUNLIGHT 349 |8 = the angle of departure from the line of incidence for the reflected ray. (748) Intensity of scattering = 1 -f- cos 2 /3. (749) Fraction of light polarized = : _ ^. Hence the maximum scattering at = 90 from the sun is twice as much as in the direction of the sun, /3 = 0; the amount of polarization is 100 per cent at (3 = 90 from the sun, and it is per cent at /3 = in the direction of the sun. In the atmosphere with the sun on the horizon, as at the equinox in the east, the maximum polarization is in the zenith, and in the vertical plane passing through the zenith and the north and south points. If the solar point is east, the antisolar point is west; as the sun rises the antisolar point sinks below the horizon; as the sun moves to any other usual point the plane of polarization is that which includes the sun, the point of reflection in the sky, and the eye of the observer, the vibrations being generally at right angles to this plane. Besides the primary scattering and polarization on the small particles in a turbid atmosphere, it is found that the light is only partially polarized, so that a secondary polarization exists at right angles to the primary, primary and secondary vibrations, and polarizations at right angles to each other, thus tending more or less to complete neutralization of plane polarized light as the primary and secondary components approach equality. There are several such points of neutralization: Babinet's neutral point about 15 to 25 above the solar point, Brews ter's neutral point about the same distance below as the sun rises above the horizon, and Arago's neutral point about 15 to 25 above the antisolar point when the sun is on the horizon. The positions of these points vary with the position of the sun in the heavens, and the relative turbidity of the at- mosphere. Since the dust particles accumulate chiefly in the lower atmosphere, in a stratum less than two miles thick, there is an apparent ring of special turbidity close to the horizon, which causes the light to be horizontally polarized within a 350 TERRESTRIAL AND SOLAR RELATIONS few degrees of the horizon. Generally, polarization is a maxi- mum in the zenith, and diminishes to the north and south horizon points, and from these to the east point, for a sun in the east and on the horizon. There are numerous variations of these principal results, due to change in the intensity of solar light from radiation, and change in the contents of turbidity in the atmosphere. The subject of polarization is discussed fully in "Tatsachen und Theorien der atmospharischen Polarisation," Friedr. Busch and Chr. Jensen, 1911. The literature of the observations and discussions is very extensive during one hundred years. The observations are made by a polarimeter, consisting of a grating of parallel bars and spaces, from which the light falls upon a Rochon prism which separates it into the ordinary and extraordinary rays. These fall upon a Nichol, and by its rotation, there is extinction, or flattening of the appearance of the field, at four angles of observation. Thus for four angles of observed extinction, the computation is of the following form. Cordoba. (1) (2) (3) (4) (2-1) (4-3) Mean P% Feb. 8, 1912, 11.3 94.0 190.0 278.0 82.7 88.0 85.4 73.4% A convenient table is prepared for obtaining the mean percent- age of polarization P% from the mean angles (2-1) and (4-3), where (1), (2), (3), (4) are the successive readings. Various relations have been traced out, such as: (1) Movement of the altitude of the neutral points with the frequency of the number of the sun-spots. (2) Minimum polarization at the time of maximum tempera- ture and maximum convection. (3) Maximum polarization in winter rather than summer. (4) Water drops have little effect on the polarization at 90 from the sun; ice crystals and large particles in that direction decrease the polarization and increase the natural scattered light; light that is reflected from the earth's surface, or from snow areas, diminishes the polarized and increases the reflected common light. The relations between these several terms have numerous interesting optical considerations, and they serve to measure POLARIZATION OF SUNLIGHT 351 to some extent the state of turbidity in the atmosphere, and hence have value in connection with the absorption and radiation of solar energy in the atmosphere. The polarization at Daroca, Spain, August 19, 26, 1905, was relatively high following rains, but it often fell to 40 per cent or Date 10 11 Noon 1 August 19 August 26 August 30 lipse Aim Sept.21-Oct.10 U.S.S. Caesar FIG. 71. Percentage of polarization of the sky light at Daroca, Spain, during August, and on the U. S. S. Casar, Sept. 21-Oct. 10, 1905. 50 per cent on account of fine dust in the air; that on August 30 shows the effect of the passage of the shadow of the solar eclipse, the sky having been thoroughly cleaned of dust by a rain on August 29; the observations on the U. S. S. Casar during the voyage from Gibraltar to Norfolk, frequently in the clear spaces between cumulus clouds, showed a normal high percentage of polarized light at 90 from the sun. Impurities from solid particles produce natural light by reflection, fine particles and gas molecules produce polarized light. 352 TERRESTRIAL AND SOLAR RELATIONS Solar Physics It will be possible merely to summarize a few important points in the subject of solar physics, in this connection, because it is very extensive in amount, and in consequence of the fact that much of the theory is in a conjectural stage of development and is still indecisive. (1) It is evident that the thermodynamic equations employed in the discussion of the earth's atmosphere are applicable to the sun's atmosphere, by changing the data in a proper manner. Thus, gravity becomes G = g Q X 28.028; the pressure on the photosphere is about five atmospheres, so that P = 5 X G p m E n ] the temperature at the photosphere is apparently 7500 C.; at the top of the chromosphere or lower layers of the inner corona, 10" arc = 7260000 meters, 6900: and top of the inner corona, 35" arc = 25410000 meters, 6500. From these data for a hydrogen atmosphere, or a calcium atmosphere, the various thermodynamic terms can be computed as far as the united terms of the velocities' and radiations' energies. (2) The probable velocities in the sun-spots, assuming that they are the stream lines of the funnel-shaped or the dumbbell- shaped vortices on the upper plane of reference, on a level with the layer of the photosphere, can be computed from the general dimensions of the penumbra and umbra, and checked to some extent by the spectroscopic observations on velocities. The vertical and the horizontal velocities in different layers of the sun's atmosphere are being studied with the prospect of ultimate success in a few years. It may be hoped that the radiation output from the sun, computed from such data, may be found to conform to the radiation energy at the earth as derived from the pyrheliometer and the bolometer records, but much research will be required to accomplish this result. (3) The rotational velocity of the sun's atmosphere in different latitudes on the level of the photosphere, and in other higher layers as already determined, indicate a very complex kind of circulation, of an entirely different type from that in the earth's SOLAR PHYSICS 353 atmosphere. The latter consists of a thin shell heated on the tropics, and acquiring an approximately steady type of equili- brium, as heretofore explained, while the sun has maximum velocity at the equator diminishing to the poles on the level of the photosphere, and increasing upward in all latitudes. Since the integral in every small column along a radius extended must conform to the gravity integral, which is the sum of the pressure, circulation across it, and radiation through it, there is an op- portunity to determine these terms through an approximation by trials. (4) Table 80 contains a convenient series of transformations between sidereal and synodic periods. Table 81 contains a collection of the observed synodic periods of rotation in different latitudes. Bigelow's data from the prominences refer to the higher levels of the sun's atmosphere, because they are seen projected above the chromosphere. The acceleration in the polar region over the velocities devised from spectrum dis- placement lines is probably correct, because the spectrum lines are all located at lower levels. Some of the Mt. Wilson data are in conformity with this result. (5) The magnetic data at the earth, as already indicated, produce a synodic period of 26.68 days at the sun's equator, conforming closely to the general mean value 872' or 26.58 days from the eight researches quoted. The Zeeman effect has been detected by Professor Hale in the sun-spots, due to the rotation of electric ions in the tube of the vortex. This proves that electric ions in circulation produce magnetic field at solar tem- peratures. Hence, the interior of the sun, if polarized into rotation filaments by its circulation, by rotation on its axis, and processes of radiation, is probably magnetized throughout its mass, in much the same way that the earth carries an internal and external magnetic field, though its interior is at a high tem- perature. There is evidence of such spherical magnetism in the shapes of the polar rays seen in the minimum activity of the coronal formation, where the observed rays from the sun, seen in projection, conform to the lines of force surrounding a sphere, supposing that they are generated chiefly in a polar ring about 354 TERRESTRIAL AND SOLAR RELATIONS TABLE 80 THE SIDEREAL AND SYNODIC PERIODS OF THE ROTATION OF THE SUN FOR CERTAIN ASSUMED DAILY ANGULAR VELOCITIES. -= = -$= k n. 1 L O Daily Angular Velocity X Daily Angular Velocity in Degrees f Sidereal Period in Days r Angular Velocity of the Sun in Days h* Angular Velocity of the Earth in Days 1 ~=n Synodic Velocity of the Sun in Days j = k-n Synodic Period in Days S 1 700 o 11.67 30.86 .03241 .00274 .02967 33.70 705 11.75 30.64 .03264 " .02990 33.44 710 11.83 30.42 .03287 " .03013 33.19 715 11.92 30.21 .03310 " .03036 32.94 720 12.00 30.00 .03333 " .03059 32.69 725 12.08 29.79 .03357 " .03083 32.44 730 12.17 29.59 .03380 " .03106 32.20 735 12.25 29.39 .03403 " .03129 31.96 740 12.33 29.19 .03426 " .03152 31.73 745 12.42 28.99 .03449 n .03175 31.50 750 12.50 28.80 .03472 " .03198 31.28 755 12.58 28.61 .03495 " .03221 31.05 760 12.67 28.42 .03518 11 .03244 30.83 765 12.75 28.24 .03542 " .03268 30.61 770 12.83 28.05 .03565 " .03291 30.39 775 12.92 27.87 .03588 " .03314 30.18 780 13.00 27.69 .03611 " .03337 29.97 785 13.08 27.52 .03634 " .03360 29.76 790 13.17 27.34 .03657 n .03383 29.56 795 13.25 27.17 .03681 11 .03307 29.35 800 13.33 27.00 .03704 11 .03430 29.15 805 13.42 26.83 .03727 " .03453 28.96 810 13.50 26.67 .03750 " .03476 28.77 815 13.58 26.50 .03773 11 .03499 28.58 820 13.67 26.34 .03796 11 .03522 28.39 825 13.75 26.18 .03819 " .03545 28.20 830 13.83 26.02 .03843 " .03569 28.01 835 13.92 25.87 .03867 " .03592 27.83 840 14.00 25.71 .03889 "- .03615 27.66 845 14.08 25.56 .03912 " .03638 27.49 850 14.17 25.41 .03935 11 .03661 27.32* 855 14.25 25.26 .03958 11 .03684 27.14 860 14.33 25.12 .03982 " .03708 26.97 865 14.42 24.97 .04005 " .03731 26.80 870 14.50 24.83 .04028 " .03754 26.64f 875 14.58 24.69 .04051 " .03777 26.48 880 14.67 24.55 .04074 it .03800 26.32 885 14.7.5 24 .41 .04097 " .03823 26.16 890 14.83 24.27 .04120 " .03846 26.00 895 14.92 24.13 .04144 it .03870 25.84J 900 15.00 24.00 .04167 11 .03893 25.69 930 15.50 23.22 .04306 n .04032 24.80 *Lat. .13, spots. f Equator of sun. $ Terrestrial. Hydrogen. SOLAR PHYSICS 355 t^t^t^t^t^t^t^r^t^r^t^oooooo t^ 00 si a tS l>lSt^l>J>l>t^l>l>GOOOOOOOOOOOOOOOOO i s CO ^O ^.(N CO^fO^OOOCOOCOOOOOO^ ^H s ! CO "^ *O !> O^ ^H ^H CO Oi i^ CO Tt^ CO !> 00 OO COCOCOCOCOcOCOl>t>I>l>OOGOOOOOOOOOOO < t>-t>-l>l>- '-HC s QCOCO^t l ^t ( ^t l ^t | iO OOOOOOOOOOOOCXDOOOO GO O I-H (N l> 00 00 00 CO ^t^ O^ ^O CO C^l CO ^^ t** THi-HT-iC'qcO^iOCOt^ 000000000000000000 II O^ CO CO 00 ^^ Oi !> 00 O i i C^ ^ ^ ^O I s * 00 00 00 00 GO 00 10 O 356 TERRESTRIAL AND SOLAR RELATIONS 23 degrees distant from the coronal pole. This system has been traced from one epoch to another, through several eclipses from 1878 to 1905 at least, as if the synodic period 26.68 days was also fundamental in producing the aspect they present upon the series of eclipse photographs. A model was constructed of such a magnetic field, and turned by its astronomical co- ordinates into the required positions on the days of the several eclipses. The coincidence in position between the pole of the sun, pole of the earth, pole of the corona, and its stream lines as parts of a spherical magnetic field, are too striking to be over- looked. Superposed upon this deep-seated magnetic field, em- bracing the entire interior of the sun, is a strong electrostatic surface field with its rays in normal directions, and in many dis- torted positions. These two fields interplay among the forces of circulation and radiation to produce the numerous fantastic forms seen on the edge of the sun. Astron. Soc. Pac. No. 27, 1891. The Spherical Astronomy of the Sun It is necessary to give a brief account of the variable relations due to the rotation of the sun on its axis, and the revolution of the earth in. its orbit about the sun. If the spherical conditions of Fig. 72 be transferred to a small rubber ball, it will greatly facilitate the study of this complicated branch of solar physics. The photographs of the sun give pictures which must be inter- preted in terms of spherical co-ordinates, and this is a great labor of computation, where any large number of points are to be considered. Some mechanical devices have been used for securing heliocentric co-ordinates approximately, but for defini- tive work the micrometer measurements must be employed with accuracy. As the earth passes around the sun the aspect of the disk undergoes an annual periodic change which must be followed, and as the sun rotates on its axis the positions of the spots, faculaa, and prominences change from day to day. The following definitions and formulae can be very readily verified from the diagram, and by Chauvenet's treatise on Spherical Trigonometry. The angle H is the apparent projection of SC K on a plane SPHERICAL ASTRONOMY OF THE SUN 357 perpendicular to the plane of sight, the angle G is the apparent projection of K C E on the same plane, the angle P is the position angle of the spot 2 from the north. E, counted positive eastward, so that X = H + G -f P, the position angle from the sun's pole S. At the same time p is the angular distance of S from C, as measured at the center of the sun. The prime meridian is the central meridian at one adopted epoch, mean midnight, Decem- ber 31, 1853 (Carrington) ; Greenwich mean noon, June, 13.72 (1887), Bigelow. The rotation periods of the sun change in latitude, from a maximum, 26.68 days synodic at the equator, to a minimum, about 30.00 days at the poles. Carrington's adopted period of rotation is applicable to latitudes = 12, and O=Center of Sun C = Center of Disk 2= Position of Spot '' (l.d.) Sun's Equator Ecliptic (cC.5.) Earth's Equator L from N to C >- Center I from N to "2 -* Spot /' from N to M-> Prime Meridian FIG. 72. Spherical positions on the sun's equator. represents the average period of rotation for the sun-spots in that latitude alone. The radiation effects measured at the earth are for the equatorial period. Poles. 5 = sun's equator; E = earth's equator; K = Ecliptic. Inclinations. 7 = sun's equator to the ecliptic, 7. 15' = KO'S. co = earth's equator to the ecliptic, 23.27' = EO' K. 358 TERRESTRIAL AND SOLAR RELATIONS H + G = earth's equator to the sun's equator, 26 20' = EO'S. H = the projection of S 0' K at the center of the disk C. G = the projection of K O' E at the center of the disk C. P = the position angle of 2 = EC 2. X = H + G + P. The positive direction of the angles H, G, P } X is through the east. p = the heliocentric angle of 2 from C = C 0' 2 . Co-ordinates of the center of the disk C. at L = longitude from the node N on the sun's equator, NO ' L - D = latitude from the sun's equator = C. L. f E = right ascension from the node ^ on the earth's equator, TO' E. [ F = declination from the earth's equator = C E. f O = sun's celestial longitude from the node T on the Ecliptic] ecliptic, T O f C. (^ ... = the latitude is zero = C C. Co-ordinates of the spot or point 2 on the disk. c , / = longitude from the node N on the sun's equator, NO'L equator . , . , ^ a == latitude from the sun's equator = / S. . , ( a = right ascension from the node T on the earth's Earth's ^ ^ -j equator T a. [ 8 = declination from the earth's equator, a S. A = heliocentric longitude from the node T on the Ecliptic -j ecliptic, TO r A. /? = heliocentric latitude from the ecliptic = A 2. /'= longitude of the prime meridian from N on sun's equator. SPHERICAL ASTRONOMY OF THE SUN 359 L-l = heliocentric longitude from the central meridian, or difference of heliographic longitudes of center of disk and spot. L-V heliographic longitude of center of disk from the prime meridian. /-/' = heliographic longitude of 2 from the prime meridian. Reduction of the photographic plate. R = solar radius on the photograph. R' = solar radius corrected for distortion on the plate. R" = solar radius as given in the ephemeris. r = the perpendicular distance of S from the line COO', = 02. r f = the measured distance of the spot from the cen- ter of the sun-picture, corrected for distortion. p' = the angular distance 2 as seen from the earth. (750) p' = jr R". ~r = sin 0' 2 2' = sin (p + p'). (751) p + P '= sin" 1 -. p = sin' 1 ~- - p'. From the right spherical triangle TEC (Chauvenet, p. 172, 88), (752) tan G = tan co cos O ; from cos O = cot co cot (90-). From the right spherical triangle N L C (Chauvenet, p. 172, 88), (753) tan H = cos (O - AQ tan 7; from cos (O - N) = cot I cot (90-#). From the right spherical triangle N L C (Chauvenet, p. 171, 86), (754) sin D = sin (O - N) sin 7. From the right spherical triangle N L C (Chauvenet, p. 171, 87), (755) tan L = tan (O - N) cos 7. From the spherical triangle S C 2, two sides and the included angle known, 90-Z), X, p (Chauvenet, p. 179, M), 360 TERRESTRIAL AND SOLAR RELATIONS (756) sin d = cos p sin D + sin p cos DcosX. From the spherical triangle S C 2, from the two sides and angle opposite one of them (Chauvenet, p. 193, 148), (757) Sin (L - /) = sin X sin p sec d. From the right spherical triangle T E C (Chauvenet, p. 171, 86, 87), (758) sin F = sin O sin co. (759) tanE = tan O cos co. T = the fraction of a revolution executed by the prime meridian at a given date. / = time from the epoch, June 13.72, 1887. K = the mean angular velocity of the sun on its axis. n = the mean angular velocity of the earth around the sun. K n = the synodic angular velocity of the sun. m = the complete number of sidereal rotations of the prime meridian M since the epoch. (760) T = - m (Bigelow): (761) l' = TX = T X 14.4783 (Bigelow). To transform (A . /?) to (L . D), compute the auxiliary angle a, (762) tan a = sin A cot 0. Then, / \ si* 1 (I + 0) (763) tan L = - ^ -- tan A. sin a ( 764 ) si nZ? = cos a The Magnetic Fields of the Earth and the Sun The sun presents many aspects of a magnetized sphere, with the positive pole on the south side of the sun's equator; the earth is a magnetized sphere with its positive pole on the south side of the equator. An isolated (supposed) positive (+) magnetic pole tends from the south side to the north side of the sun's SPHERICAL ASTRONOMY OF THE SUN 361 equator, and from the south side to the north side of the earth's equator. (x) Magnetic pole near the south geographical pole. FIG. 73. Magnetic coordinates and component forces. The following formulas are convenient for reference: (765) Mass = M = - x R*. o (766) Mass Potential P = y = -g TT R s - dP 4 r R* (767) Vector Potential V e = - I -^ = -g * I -^ * = TT / cos 6 (external). (768) (internal), Exterior Forces e , (770) X., --.+ */ a* 4 I? 3 TT -r / . 3 sin e cos O IT* 362 (771) TERRESTRIAL AND SOLAR RELATIONS dV. 4 R* Zzy + s" 71 " 7 ^ '~^~ = 0.since;y = 0. dy 3 r 3 r* (772) Z = F n = Z cos + X sin = - TT 7 o T" (- cos + 3 cos 3 5 + 3 sin 2 cos 0) r = x / 2 cos 0. (773) X = F t = - Z sin + X cos = -| TT 7 ^ (sin 0-3 cos 2 sin + 3 sin cos 2 0) (774) Z 2 = r = TT / r . sm 0. o f- 5 = TT 7 (l + 3 cos* 0). Surface R = r (775) Z e = - TT 7 . 2 cos o (776) X e = - T 7 . sin ' Equator 8 r o- * /. Pole - 'e = 0. 4 = 0. Interior Forces C778) ^ - - 7? T * (780) Line of Force = constant. SPHERICAL ASTRONOMY OF THE SUN 363 (781) Equipotential Surface V = K R 3 <r~ = constant. O T A magnetized sphere may be induced by two layers of positive and negative masses, circulating surface electric currents, two positive and negative poles placed near the center, and by other physical devices, as rotating Ampere electric currents in the interior around lines parallel to the axis, in which case the sphere is polarized. This is probably the case for the earth and for the sun. For surface exterior currents, as in the earth's atmosphere, it is evident that variation in the east- west direction changes the strength of the existing magnetic field along the lines of force. The vectors of Fig. 68, 69, can only be produced by north and south ionization currents, as there indicated, distinct from east and west currents on the normal field. The asymetric position of the sun's magnetic poles deduced by Bigelow in 1891 has been confirmed by direct observation at the Mt. Wilson observatory, 1913. Conclusion The general functional relations between the incoming solar radiation, the portion of it transformed into ionization currents, the magnetic disturbing vectors depending upon them, all re- main to be discovered. It has been possible to indicate in this Treatise some of the important elements in these problems, and it is hoped that the formulas and methods of discussion here adopted will greatly facilitate the pursuit of such researches by many students. The practical side of the matter consists in the development of the branch of Meteorology and Solar Physics which will culminate in the ability to predict the seasonal climatic conditions likely to prevail during the coming year in the several large agricultural regions of the earth. The extent and scope of these subjects are so great that the co-operation of many institu- tions and national offices will be essential for the success of so important an enterprise. CHAPTER VII Extension of the Thermodynamic Computations to the Top of the Atmosphere Remarks on the Bouguer Formula THE computations on the thermodynamic data of the earth's atmosphere have heretofore in the examples been limited to about 20,000 meters, but it is very desirable to extend them to the vanishing plane of the gaseous envelope. The following summary illustrates a method for accomplishing this purpose, and the results afford approximate material for further important researches. These concern the problems of the solar radiation and its absorption, together with the correlative terrestrial radiation and its absorption. It is especially proposed to test the result that the "solar constant," as derived from pyrheliometer observations is about 1.92 -^-~- r-, or whether cm. 2 mm.' it is about twice as much, in conformity with the bolometer data. The Bouguer formula of reduction, in the preceding notation, has been shown to contain two unknown terms /o, mo, which together determine the source of the effective radiation, so soon as their details are understood. It has been customary to identify / with the solar radiation falling on the outer limit of the atmosphere for which it is assumed that m = i and disappears from the equation, enabling it to be solved for 7 . This is, in fact, a special interpretation by assumption, and doubt is thrown upon it by the bolometer ordinates, since these require an energy spectrum curve of about T = 6,900 (3.88 calories), while /o = 1.92 calories is satisfied with 5,800 for the effective solar temperature. It has, furthermore, been shown by the pyrheliom- 364 FIRST DISTRIBUTION OF TEMPERATURE 365 eter observations that /o is not a constant quantity, that is, a source of radiation independent of atmospheric conditions as to aqueous vapor, dust and ice contents, not now referring to the small variations of the radiation energy emitted at the sun itself. The evidence is that as the coefficient of transmission p dimin- ishes in value the term 7 is progressively depleted. The extra- polated points /o on the axis of ordinates do not concentrate at /o = 1.92 calories. It follows that /o is a terrestrial variable, and it is the purpose of this chapter to investigate further the properties of the terms, 7 , m Q . I. FIRST DISTRIBUTION OF TEMPERATURE The balloon ascension, Huron, So. Dak., September 1. 1910 The balloon ascension, Huron, So. Dak., September 1, 1910, reached a height of 30,000 meters, recording temperatures 7", but without velocities q. These have been provisionally sup- plied by taking the mean values for ten other ascensions in order to balance approximately the terms of the gravity equation, (196) g (* - *) - - ^-p^ - J fo 2 - <?o 2 ) - (Qi - Co). ^10 Table 82 contains the adopted T, q, from which all the other terms were computed up to 30,000 meters, and the results were plotted on large diagrams of which Figs. 74, 75, are reduced copies. The method of extending the T data to the higher planes is as follows: On the 30,000-meter plane T = 232.3, P = 1352 (B = 10.14 mm. as computed, while the automatic register recorded B = 10.70 mm.), p = 0.05512, R = 105.61, the check equation P = p R T being perfectly satisfied. At first the P and p curves were extended graphically, and from these values up to 40,000 meters approximate values of T were com- puted, carrying the T curve around the corner of the isothermal region into a very rapid gradient. These temperatures are necessary to continue the P, p, curvatures at that elevation. Then all the other terms were computed directly from T and the diagrams were extended. At 40,000 meters T = 170.5, P = 366 EXTENSION OF THERMODYNAMIC COMPUTATIONS fl c/3 a gs M S3 t* g *!? II u I s OK 5 11 O ^ I M to Tempe ture rococococococococococococoro.cocococococococo 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 CO o T-H i i O i-H*O i ICOCOO5(NCOO5(M>O COO5(M>OOOi IIO I li (C^tNfNCOCO *OI i(MOO i IIOO5OO COOOiOOi<MOi IT iiOrH .T OOOOO<NCOOOO iOi lOSGOOOOiOi i<N^f>O CO CO C^l C^ C^ Ol CO CO CO CO CO ^ vsx^f'OiO' !COOiCOTtiCO y\/N c>q-^iu7if>.cOcOOiOt^ O51>iOCOCO(NOO COO'-HCOcO' i i I rH <N X X ^ GO 00 i 1 CO l> O5 CO rH TH t^ 10 TJH CO CO <M FIRST DISTRIBUTION OF TEMPERATURE 367 OO 00 ^2 * 4 C^ CO cococococococococococococococococooocococococococococococccococo I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I CO <N OiCOOO5C<l ^ CO co <N C5 as t Tt< <M i- o o o o <o co o t * o o C^COCOC^C^C^C<lC^C<IC^C<JrHrHrHrHC^C^CO^TjHUOO^ 368 EXTENSION OF THERMODYNAMIC COMPUTATIONS 279 (B = 0.002095 mm.), P = 0.01796, R = 91.20, and P = p R T is again exactly satisfied. These data indicate that the vanishing plane of the atmosphere is approaching, and a trial was made of z = 50,000 meters for T = the approximate temper- ature of space. Further experience with the data indicates that 50000 40000 FIG. 74. The dynamic data to the top of the atmosphere at 50,000 meters, from the balloon ascension, Huron, S. D., Sept. I, 1910 P = Pressure in kilograms per square meter. K = Radiation energy. p = Density in kilograms per cubic meter. T= Temperature in absolute degrees. r j = Temperature before the formation of the isothermal region. Cp = Specific heat ratio at constant pressure. The balloon ascension, Huron, S. D., September I, 1910, reached the elevation 30,000 meters. The lines for P, K, />, Cp, and T (up to the iso- thermal level) converge at a point on the vanishing plane 50,000 meters. By ,a series of trials T was determined such as to make the P, K, p curves run out smoothly to a common vanishing point. FIRST DISTRIBUTION OF TEMPERATURE 369 the vanishing plane is nearer 90,000 meters, but the example is reproduced because several interesting points are illustrated. The temperature T at any level, whether observed or adopted, can always be fully checked by the combination of these two equations. If there is an important residual for g (zi z ) it must be due to the fact that an erroneous T was ascribed to the level. In this case the residuals are large from 41,000 to 50,000 meters, but this can be remedied by further trial computations in 50000 40000 30000 20000 10000 000 (QrQul/v*' (Oi-O (u,- Uo) (W : -W ) (Ui-Uo) 2000. 4000. 6000. 8000. 10000. 12000. 3.50 3.&0 3.70 3.80 FIG. 75. The thermodynamic and radiation data to the top of the atmos- phere at 50,000 meters, from the balloon ascension, Huron, S. D., Sept. I, 1910. (Qi Qo) =The free heat transmitted by the solar radiation. (WiWo) =The external work of expansion. (Ui Uo) =The inner energy of molecular motion. a=The exponent in the radiation equation. (Ei Eo) =The total incoming free heat. (A i Ao) =The absorbed portion of it. (Qi- <2o) = (W, - W,} + ( Ui - C/o) 370 EXTENSION OF THERMODYNAMIC COMPUTATIONS connection with other high-level balloon ascensions. It should be noted in this example that the value of R = 268.46 at 46,000 meters has resumed nearly its surface value, 287.03. This means that the thermodynamic system has been subjected to great heating influences in the levels above 40,000 meters, and this is easily seen by studying the data of Table 82, and Figs. 74, 75. The P, K, p, curves run out smoothly to the vanishing plane, but the Cp = R ~^ (Qi - ft), (Wi ~ W ), (U,- Z7 ), log c, a, curves all are much distorted in the strata above 40,000 meters. This is the thermodynamic result of heating the rarefied gases of the outer levels from T = on the vanishing plane to T = 208.2 at 38,000 meters. The so-called isothermal layer from 11,000 to 38,000 meters is merely that part of the atmosphere where the surviving solar radiation nearly balances the terrestrial radiation. Above 38,000 meters is a region of very powerful absorption of the solar radiation, in which the energy of the short waves of the solar spectrum, * = 0.00 /* to 0.40 M, are absorbed, the energy being used in heating this stratum through about 208 degrees. This is the true albedo of the earth's atmosphere, and it amounts to nearly one-half of the total solar radiation falling on the earth's outermost stratum of atmosphere. If a line be drawn, T 1 , from the lower part to the upper part of the T-curve, the enclosed area between T and T 1 represents the region of absorption, and the isothermal curve to the right of it is one of its three boundaries. This area can be studied for several thermodynamic conditions. The divergence between the pressure curve P and the radiation energy curve K is due to the free heat per volume ?i ~ Co) - (Wi - TFo) + (tfi - Z7o), where (ft - ft) repre- sents the excess of the inner energy (Ui U ) over the work of expansion (Wi WQ) . At the points where the P and K curves coincide, as at the surface, there is no free heat at liberty to move. In the case of an adiabatic atmosphere, (ft ft) = 0, and there is no radiation. It should be carefully noted that meteorological discussions and the tables in common use have been heretofore uniformly based upon strictly adiabatic formulas. Since the FIRST DISTRIBUTION OF TEMPERATURE 371 adiabatic and non-adiabatic systems are identical at the surface, this distinction is not important there, but because these systems diverge to a large amount in the upper atmosphere the inference follows that all meteorological tables based upon R = constant produce erroneous values of the density above the surface. For example, with R = constant, g (zi Z Q ) = Cp a (T a T ) = , and from this comes the adiabatic gradient 77 = PaO Cp a - ( a ~ T ^ = - 9.870 per 1,000 meters. If, on the other (21 - z ) hand, the terms for circulation \ (qi 2 q<?) and for radia- tion (Qi QQ) are to be introduced, this can only be done by changing p ao to pio, since the pressure term (Pi P Q ) is fixed by direct measures at heights which do not correspond with any adiabatic temperature gradient. It follows that we have another value of the specific heat ratio Cpio which differs from p -p Cp^ so that Cpio (T a TO) = - and the non- |10 adiabatic pio differs from the adiabatic p a . By subtraction we find that, (198) - (Cp a - Cp lQ ) (T a - T ) = -\ (qi 2 - qo 2 ) - (Qi - Qo), as heretofore indicated. If for the moment we assume that ~" i (<?i 2 ~~ #o 2 ) = 0, as is the case at the top and at the bottom of the atmosphere, there are two extreme values of (Qi Q ) : 1 . At the vanishing plane, for Cpio 0, - (Qi ~ <2o) = - (Cpa - Cpio) (T a - To) = 993.58 X 9.87 = 9806, per 1,000 m. 2. At the surface, for Cp w = Cp a , ~ (Ql - <2o) = - (Cpa ~ Cpio) (T a - T ) = 0. The adiabatic formula requires that Cpio = Cp a from the surface to the vanishing plane, which is in disagreement with the observed temperature gradients. 372 EXTENSION OF THERMODYNAMIC COMPUTATIONS Illustrations of the Use of Erroneous Densities In order to illustrate the amount of divergence between p a and p, we have for this balloon ascension TABLE 83 THE DIFFERENCE BETWEEN /> (NON-ADIABATIC) AND p a (ADIABATIC) DENSITY z ft Pa 50000 0.0000 0.0000 45000 0.0035 0.0033 40000 0.0180 0.0057 35000 0.0324 0.0099 30000 0.0551 0.0203 25000 0.0929 0.0426 20000 0.1586 0.0935 15000- 0.2754 0.2073 10000 0.4760 0.4190 5000 0.7730 0.7151 surface 1 . 1485 1 . 1485 Since p is greater than p a it follows that, (782) Pi-Po. Pi - is smaller than pio Pao and this is compensated in the gravity equation by the terms for the circulation and radiation. It has been tested by numerous p r> computations that g (zi z ) = Cp a (T a T ) = Pao without important residuals from the surface to the vanish- -p __ r> ing plane, and that - Cpu (T a - T ) = - holds true pio throughout the atmosphere. No system of meteorology k founded on R a \ _ ^ = Cp a = constant is applicable in prac- tical studies of atmospheric conditions. There is another important matter in which erroneous den- sities have been introduced into studies of radiation phenomena. ILLUSTRATIONS OF THE USE OF ERRONEOUS DENSITIES 373 It has been thought proper to modify the Bouguer formula in such a way that the observed variations in I,p, shall be dependent upon the nature of the gaseous path m, so that the integrated amount of aqueous vapor in the form of liquid water may be made a factor attaching to the exponent m. Thus it has been computed that the water mass S/* = 2e Q , where S/* is the integral of the water contents in grammes and CQ the vapor pressure in millimeters of mercury at the surface. The hair hygrometer measures the relative humidity of the air at the different levels, and the corresponding e in mm. is computed. The full formula is easily shown to be, (783) M = p f -f - 4 (0.622 -I + 0.235 f). It has been customary to assume R = 287 and to omit the e 2 term in -57, so that there remains, &* 273 6 (784) M = po-^-X 0.622. This formula diverges from the true one in the same way that p differs from p a at the different levels, so that the sum of /*, S/t, is erroneous in the formula 2/z = 2e, being too small. Furthermore, the solar radiation suffers two sorts of deple- tion, the first as true absorption upon which p depends, the sec- ond as scattering upon which /o partly depends. We have shown how /o should be modified for vapor contents (7 0.0214 e) before introducing it into the Bouguer Formula. The ice crystals of the upper strata, 8,000 to 25,000 meters, are effective reflec- tors of the radiation energy in these strata. The hair hygrometer takes no account of the ice contents in the (3-stage), nor the water contents (f and ft stages) which may be present, but only of the vapor contents (<*-stage). It follows that since water and ice exist throughout the lower atmosphere up to 25,000 meters, but not always in cloud forms, though these have been observed up to 16,000 meters, that the a-stage Formula is inadequate to the purpose which has been imposed upon it. 374 EXTENSION OF THERMODYNAMIC COMPUTATIONS The remedy for these difficulties is being considered in a special research. The Thermodynamic Terms Table 82 and Fig. 75 give the relations of (Qi - Q ) = (Wi - Wo) + (Ui - Uo). (ft - Qo) begins at the surface without value because the external work and the inner energy are in equilibrium; it increases by a ci*rve determining the amount of the lower absorption to a point between the eleva- tions 26,000 and 27,000 meters, this being the level where 2 (Qi - Qo) = (Ui - Uo) - (Wi - Wo), which defines the true isothermal level where there is no absorption of solar energy, in the same sense that there is no absorption at the top and at the bottom of the atmosphere. This gives three points on the line of total solar energy, 9,806 on the vanishing plane, about 5,780 at 26,800 meters, and at the surface. The (Ei - E ) line of total solar radiation energy on the several levels was drawn by connecting these three points. The area between the axis of ordinates and the (Qi Q Q ) curve represents the free heat of transmission, the area between the (Qi Q ) and the (Ei EQ) curves that of absorption, while (E\ E ) = (Qi Qo) + (Ai AQ), the total area. The data on the several levels are for 1,000-meter areas, except below 3,000 meters, where the vertical height is 500 meters. The sum of these areas from the surface to 50,000 meters apparently represents the true " solar constant" of radiation energy, with the constit- uents of transmission and absorption. The (Qi Qo) curve continues to increase in value upward, as (Wi Wo) and (Ui U ) separate, till a maximum value for the free heat of transmission is reached at 6963 on the 38,000-meter level. It will be shown that the pyrheliometer receives an amount of heat at the surface such as results from a summation up to about this elevation, as if this were the general efficient source of radiation at about 1.92 calories. The (Qi - Qo) and (Ei - E Q ) curves begin to diverge at 27,000 meters, and very rapidly after passing 38,000 meters. (Qi - Qo) falls to a minimum at 48,000, and rises immediately to its THE THERMODYNAMIC TERMS 375 primary maximum, 9,806 at 50,000 meters, in this computation. (Wi Wo) and (Ui Z7 ) pass through entirely similar changes above 38,000 meters, and they all correspond to the very rapid increase in the temperature of the gases of the upper absorption level. It is evident that the pyrheliometer measures the trans- mission and the minor absorption up to about 38,000 meters, but is not cognizant of the great absorption area above it where the absorption is nearly complete. The bolometer measures the relative ordinates of the true solar spectrum, unless some of them are absorbed, but for its total energy depends upon the computations and integrations of the spectrum curve at given temperatures. This is shown by the spectrum to have lost nearly half the incoming energy in the short waves having long ordinates, these corresponding with the upper absorption region, besides some minor absorptions attributed to aqueous vapor selective absorption bands belonging to the lower region. The general slope of the (Ei E ) line indicates that the equal terrestrial* radiation is required to balance the solar radi- ation, generally large below but small above, in accordance with the prevailing temperatures and the equation of equilib- rium, as will be explained. The corresponding values of K\Q, log c, a, in KIQ = c T a , have been computed. The exponent a = 3.819 at the surface gradually falls to a minimum 3.564 at 38,000 meters; it rises to a maximum 3.905 at 45,000 meters, then falls to about 3,600 at the vanishing plane. Further computations will be under- taken in respect to the region above 38,000 meters. (Fig. 79.) The Constituents of the Solar and the Terrestrial Radiations in the Earth's Atmosphere It will be convenient to transform the thermal data which have been computed in the (Meter, Kilogram-second) (M. K. S.) system into the corresponding values in the (Centimeter-gram- second) (C. G. S.) system, before proceeding to a further discussion of the transmitted and absorbed constituents of the solar and the terrestrial radiations. The term (Qi Q ) in 376 EXTENSION OF THERMODYNAMIC COMPUTATIONS o (Zi - 80) = " Cpa (Ta - To) = - ?l ~J* - J fe 2 - g 2 ) - (Ci ~ Co) must satisfy the dimensions of all these terms, as well as the equations P = P R T, and (ft- g ) = (PFi - Wo) + (/!- tf ). (61 - &) was derived from - - (Cp a - - Cp 10 ) (T a - T ) + i (^i 2 ~~ ?o 2 ) = (Qi Qo) so that it has the same dimensions k 1 P Pv SLS R = T Q>, and since R = = -~r they are the same as, those for work, (Wi W ) = PW (vi V Q ), and for the inner energy (Ui U ) = Cv (T a To). In transformations from the (M. K. S.) to the (C. G. S.)system the dimension factors are, [M] = 1000 = 10 3 , [L] = 100 = 10 2 ; in transformations from the (C. G. S.) to the (M. K. S.) system, [M] = 1Q- 3 , [L] = 10~ 2 . All are in mechanical units. Pressure, kilog./met., [P 1 -Po]=ML~i _ _ Density, kilog./met., [ft ] =ML~M P 10 ~ ' L Velocity square, [gi 2 g 2 ] = i 2 = 10 4 = .................... = 10000 Specific heat, [Cp.Cv.R.K.] = U = 10 4 = .................... = 10000 Heat, [<2i - <2o = (C - Cp 10 ) (T a - Tc)] = L 2 = 10 4 = ........ = 10000 Work, [(TFj - TFo) = P 10 (i - o5] = ML~\ M~ ! L 3 = U = 10 4 = . = 10000 Inner Energy, [(Z7 t - Z7 ) = Cy (r - T )] = L 2 = 10 4 = ........ = 10000 The common factor for reducing all the terms of the gravity and heat equations from (M. K. S.) to (C. G. S.) in mechanical units is 10 4 = 10,000. From the general equations (Wi Wo) = PW (VIVQ) and (Qi ~ Qo) = (W 1 - W ) + (/! - J7 ), we have, U, - U ft-Qo a AIO = " - = - X = C L V which is the radiation equation employed in the computations. The dimensions of all these terms are the same in reducing from 10000 the (M. K. S.) to the (C. G. S.), the factor being - - = 10, as 1UUU SOLAR AND TERRESTRIAL RADIATION 377 is commonly used in the pressure P 1Q . The reduction from mechanical units to heat units (C. G. S.) is by the factor 4 1851 X 10 7 ' and the factor for chan S e from second to minute is 60. Hence, to reduce (Qi Q Q ) in M. K. S., mechanical units gram calorie , into - , the factor is, cm. 2 minute 1000^1851% - Sil 5 =0-^14336; Log.-5.15644. Compare the constants, coefficients, and dimensions of Table 95, where the formulas afford many other combinations in conform- ity with the kinetic theory of gases. The Total, Transmitted and Absorbed Amounts of the Solar and the Terrestrial Radiations Huron, September i, 1910 The method of separating the total radiation (Ei E ) received at the earth from the sun into the transmitted (Qi Q ) and the absorbed (Ai A ) constituents on each level has been explained. The coefficient of transmission is p = ~ ^=, L\ ~ L>Q and of absorption is k = ~ ^r. The terrestrial radiation h,\ Lo is obtained as follows : Let JQ c T 4 , the black body radiation of the atmosphere at the point whose temperature is T; let J a = c T a the actual radiation for c and a as computed from the thermodynamic data, so that the actual absorption between two planes is / a -i Ja-o = c\ T\ l CQ T Q a . The general equation of radiation equilibrium is 2 J = E + D, where D is the total terrestrial radiation. Hence, D = 2 J Q E, and B = D J a , where B is the trans- mitted terrestrial radiation. The coefficient of terrestrial 7? - - 7? transmission is p 1 = 77 ^, and that of absorption is k 1 = U\ L/o ~^ pp- . The results of these computations appear on Table 378 EXTENSION OF THERMODYNAMIC COMPUTATIONS w o rt rH u 2 c/) s ^ IcS <X 8 I <3 O H Jo g ^ I I " I B I 5 a * 2 '31 1 'J3 S CO CN CO CO CO CO CO CO CO CO(MOOOSCOCOC s Q OOS'*t t t > - OSO5OSO01>-COiOCO CO CO i t rHCOOiOCOcOCOcOcOCOCOcOcOCDcOcO-OiO I I I I I I I I I I I I I I I I I I I I I I I ^J^ CO O : ll> rHOSOO rH CO rH OS O CO I I I I I CO O Tt< CO O CN *H O 00 - CO 00 OS Tt< l> l> l> l> 00 10 00 1> I I I I I I I I I I I I I I I I I I I I COCOOOt^OOOSOOCOOOO lO(MOSTtlrHI>.COrHlCOb- rHrHrHrHrHOOOOpp r r rHCOb-OSOTh'OOC^OOrHlOCOOOCOOOrHlOGOCMCO I I I I I I I I I I I I I I I I I I I I I>-OTH|>.O'~"i OSl lO5OOl I O^ l CNC^ir-i' I rH CO CN rH CO *O CO *O CO SCOOO5Tt<rHTt<COrHOOSb-TtlGO OS*OGOrHCOTt<lOCOb-b-OOOSOS !> b* OO 00 Os Os Os Os Os Os OS Os OS os rH OCJ O b- O CO rH rH rH O O O O O 00 00 I I I I I I I I I I I I I I I I I I I I I I + CO <M OS OS CO O CO CO 00 00 IO CO rH OS b- OS OS OS 00 00 o o o o o OOOlOCNOI^^fNOSb-TtHrHOSCOT^rHGOCOCOrHOOJOCO tfCOCOCOCO<N<N<NrHrHrHrHOOOOO5O5OSOSOOOOOO ^^1 IrHrHrHrHrHrHrHrHrHrHrHrHrHOOOOOOO r r r \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ SOLAR AND TERRESTRIAL RADIATION 379 CMrHCSt^-tf't^^t^OOiOt^cOr- icDCOfNiO^COCOTttcOC^^iOOO-^ COTfi i OS l> -^ r- (OOiO<NOOiO<NOOcOTficN'-HOO5001>t>'COCN>-i uooO^T^T*Tt<COCOCO<NcN<NrHrHrHrHrHrH I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I ' b- OO CO rH ^H <M OOOOCO rfi rH OS OS i i <NOOO5t-ia500l> <N CO OS i i > i CO GO 00 i-H CO l>* "^ i-H <N . CO (M O i-4 i-H CO GO O OS -00 - ^^ rH T-H ^^ i-H T-H T-H C^J rH rH C^ rH CO CO CO "^ CO ^O t^ 00 t^* ^^ OS CO ^ f^ IO ^O CO CO CO CO CO CO CO CO CO CO CO CO CO CO CO CO CO CO CO CO CO i"H ^^ TH CO CO CO CO I I I I I I I I I I I I I I I I I I I I I I I I I I I I ' iO : b- :s +++ i ++++ i i i i i i i i i i i i i + i i i i T (Ol>- T tiOO5rHrHOCO OOOOi i i iCOCO^^f . CO I . 00 ; oo 01 rH CO CO O t^ rH l^ rH 00 00 l> l~~ 8888 + + I I i 1 I I I I I I I I I I I I I I I I I I I I I I CO CO CO rt< b CO 00 l^ 00 rH CO rH CO rH CO GO CO Tt< CO CM rH O o o o o o o o 8 cooooco ^f "^ ^t^ CO CO r r i i r i i i i i i i i i i i i i i i i i i i i i i i i t^-f>-i x *t > r>-iv d C^l Ol i-^ ^H i-^ CM <N <N <N <N <N CN o o 380 EXTENSION OF THERMODYNAMIC COMPUTATIONS 84, inj^'a^ units. The values of (Q l - Q ) in M. K. S. mechanical units are found in the last column. The summa- tions are as follows for the entire depth of the atmosphere. Total sum of solar radiation, 2 (Ei - ) = - 3.6942 gr \ Ca .' cm. 2 mtn. Transmitted solar radiation, 2 (Qi - Q ) = - 2.7177 " Absorbed solar radiation, S (Ai A ) = -0.9101 " Atmospheric (black) radiation, S (/O.I-/O.Q) = - 3.8181 " Total terrestrial radiation, S (A -A) = - 3.9419 " Transmitted terrestrial radi- ation, S (Bi -B ) = - 2.5744 " Absorbed terrestrial radiation 2 (J a >i Ja-o) = 1.3675 The minus ( ) sign is due to the fact that while the positive (+) direction is along the axis z outward, the temperature and thermal gradients decrease in this direction; a positive (+) sign would indicate an inversion of temperatures and thermal quantities outward. In the case of p, k, the coefficients are all positive; in the case of p l , k 1 they are generally positive in the lower levels, but sometimes (+) and sometimes ( ) occur in the upper levels. Since the original observations of the velocity are lacking, those used being supplied by analogy, it is probable that the data of Huron, September 1, 1910, are inade- quate to produce accurate values of p l , k 1 . II. SECOND DISTRIBUTION OF TEMPERATURE The Balloon Ascension, Uccle, November 9, ign As already stated the gravity residuals in the check equation (196) g (z, - zo) = - Pl ~ P - (c Pa - C#io) (T a - T ) Pio + A g ( Zl - z ) were quite large above 40,000 meters in the Huron ascension. It was supposed that this was due to the fact that the velocities q were assumed, that g was taken constant in the adiabatic SECOND DISTRIBUTION OF TEMPERATURE 381 gradient, and that the formulas might begin to fail when the P, p terms became very small. Accordingly, a new series of computations has been undertaken for Uccle, June 9, Septem- ber 13, November 9, 1911, the necessary data including veloc- ities having been courteously supplied by Dr. Vincent, Directeur de 1'Institut Royal Meteorologique de Belgique. In these, besides using the observed velocities, the gravity ranges with the height. It is known that a temperature T introduced into the series of computations, building upward from the surface from level to level, whether observed or assumed, can always be checked by the control gravity equation. If the series of T is correct, A g (^ z ) vanishes. Hence, it is practical, by a set of trial computations, to arrive at such temperatures through- out the atmosphere as will satisfy the entire group of ther- modynamic equations-. The balloon ascensions to 20,000 or 30,000 meters give the necessary foundations upon which the entire structure can be built up to 90,000 meters. Up to 40,000 meters the four ascensions give about the same residuals, with occasional wide pairs depending on an erroneous T. The trials to reduce the check residuals were therefore limited to the region above 40,000 meters. It appears that positive residuals are to be diminished by increasing the temperatures on the several levels. The general outcome was to introduce a second tem- perature region beginning at about T = 170 on the 40,000- meter plane and terminating near the 90,000-meter plane. The remarkable result is that the formulas are rigorous, however small the values of P, p, R, may become for the successive T, and that small residuals above 50,000 meters imply rather large changes in the temperatures T. Extending the temperatures from 40,000 meters to levels higher than 50,000 meters, the residuals have the following mean values: Huron, Sept. 1, 1910, 40,000 to 55,000, three trials, A g (zi-z ) = +163.0 Uccle, June 9, 1911, 40,000 to 69,000, four " " = + 17.8 Uccle, Sept. 13, 1911, 40,000 to 80,000, four " " = + 4.8 Uccle, Nov. 9, 1911, 40,000 to 90,000, two = + 2.5 Table 85 contains a summary of the Uccle, Nov. 9, 1911, ascension, where the gravity differences may be examined in the 382 EXTENSION OF THERMODYNAMIC COMPUTATIONS I f 1 !| R 1 g 8 l< S 1 g w I I ^ H S W B 2 fe O a> 8? s ^^ ^ ^ W o Efl S .2 * 11 Ml* II <NOasooooooooot-t-i>t-t-t-t-t>t>t-t-t-<o<o<o 77 I I I I I I I I I I I I I I I I I 1 | | | ^sss'sskssss^S xxxxxxxxxxxxxxxxg^ rHt-mcooio-'*a>oix>-'*?ooQ'<f-rt<ooc<ioo 1 1 1 1 1 1 i 1 1 1 1 1 1 1 1 1 1 1 1 i imioicioiomiijioioioioicus I I I I I I I I I I I I I I I I I I I I I I I O^f^^fCDCiOQiCOi TH(N(N(N(N03eOCO f?f?TTTTTTT7TTrrrrrttrT ooooooooooooooooooooooo XXXXXXXXXXXXXXXXXXXXXXX TTTTTTTTTTTTTTTTTTrr oooooooooooooooooooo XXXXXXXXXXXXXXXXXXXX! THE THERMODYNAMIC TERMS 383 eoeoeoeocoeocoeoeocoweococoeocoeocoeoOTcoeocoeoeoeococo xxxxxxxxxxxxxxxxxxxxxxxxxxxx kO t> ai r-i rH N <N ^' 10 t> 00 I-! T-! i-( 384 EXTENSION OF THERMODYNAMIC COMPUTATIONS sixth column. They are smaller from 90,000 to 65,000 than from 65,000 to 40,000 meters. The inference is that the entire thermodynamic system up to vanishing quantities is reliable for numerous researches depending upon the data. In order to illustrate how small these quantities really are, we have at 65,000 meters, B = 0.001225 millimeter of mercury, p = 0.00000009629 C.G.S., while hydrogen at normal surface con- ditions is p H = 0.00008924 grJcm 3 . The remarkable precision of the computations is proved by the system of checks from the surface to 90,000 meters. These data above 40,000 meters, therefore, modify the data of Table 82, and may be substituted for them. The results of further studies on these computations will be published in Bulletin No. 4 of the Argentine Meteorological Office. It may be here remarked that the thermal efficiency of the atmosphere begins at about 65,000, there being little absorption above it. The shape of the absorbing area above 40,000 meters has been changed by this extension, but the amount is not very different, as will be illustrated. Compare Figs. 74, 75, 79. Summary of the Computations for Twenty-one Balloon Ascensions In order to improve the data as far as practical, similar computations were extended to twenty other balloon ascensions. (10) United States, Omaha, February 21,22,23, 1911; Sep- tember 28, 1909. Huron, September 1, 4, 7, 16, 1910. In- dianapolis, October 6, 30, 1909. (6) Europe, Lindenburg, July 27, 1908; April 27, May 5, 6, September 2, 1909. Milan, September 7, 1907. (5) Atlantic Tropics, Otaria, June 19, 1906; July 29, August 29, September 9, 25, 1907. The mean values for the several quantities appear on Table 86, where the data can be conveniently examined. (Ei Eo). The same values were adopted throughout, but since balloon ascensions now reach 28,000 meters the critical value at that elevation can be further examined. (61 - 60), (Ai - 4 ). These areas are seen on Fig. 75, and SUMMARY FOR TWENTY-ONE BALLOON ASCENSIONS 385 they have been described. It should be noted that the course of (Qi - Qo) in the lower absorption region does not in the least follow the distribution of the aqueous vapor contents, which is at a minimum where (Qi Q ) is at its maximum. This problem is very difficult to solve satisfactorily. p increases rapidly up to 2,000 meters, then more slowly up to p = 1.000 in the true isothermal level at 27,000 meters; above that level p falls to a small value p = 0.053 at 46,000 meters, and then rises to about unity on the vanishing plane. k passes through inverse relations in respect of p. J Q has a maximum value at 4,000 meters, and proceeds irregularly to zero at the top. Its value can always be recovered from J = $ (D + E) in the tables. D has a large negative maximum at the surface, passes through zero with change of sign near 18,000 meters, and gradu- ally increases to its maximum + 0.1406 on the vanishing plane. B falls from a negative maximum at the surface to zero at the 13,000-meter level, at the bottom of the so-called isothermal layer, and then increases irregularly to the top of the atmosphere. J a begins with a maximum at the surface and gradually falls to a small quantity, finally vanishing at the top. p 1 has a nearly constant value of 0.800 in the convectional region and probably about 0.500 above it. k l is correlative to p 1 . Uccle, November 9, 1911 The data are fully computed for the Uccle ascensions, as extended to 90,000 meters, and the results above 40,000 meters will be briefly summarized. The total atmospheric radiation energy, I Jo = 2 ( Cl TV - c TV) from the surface to 90,000 meters, is equivalent to the "solar constant" at the distance of the earth, because it represents the amount of heat required to maintain the existing temperature distribution in equilibrium with the incoming and outgoing radiations that are in operation day and night. Summarizing the Jo data, we have, 386 EXTENSION OF THERMODYNAMIC COMPUTATIONS e C 3 % ri o CQ <; 73 O Tj-OOO t^Ti-(N Os t^oo O> O Oirft^O* O> O O\ rf t- oo M 1-1 M\O MrO'<frOr-i 88 f I O\t^OOOOOOOOOOO f*J fO O MOO N OOO ^t O\ M Tf IH t^w O O O\ O> O\ <fr o 00 O vO <N row O\OO s^ 'OO Ov 100 i^ O O O\O\OvO\OO\OiOOoO t^t^OO >O 10 O\tO*frfOOOMOO>OO < <tNt- Ol^OOt^i/5^t>-cO\OO>O/jTt MWMWHMMOOOOO ooooooooo Ssssssssf Illi I I I I I I I I I f f I I S G a c ION H t^rorooo OvrO" O\O VO\O OOOOOo"o'oMwwi-ii-ii-iMM)-i OOOOOOOOOOOOOOOOO I I I I I I I I I I I I I I I I I I I I I I I I I I I o o r r f r f r f r f f f f 888 i f f :.:.!": :.'LJ:::1 888 i i i I I I O M r^t^t^-vOOO ioio^t"T}--, oooooooooooooooooooooo I I I I I I I I I I I I I f I I I I f f I I 1 I I I I I TjMoo ION O>\O roo r^Tj-M TT^J-rOrOPONNNNWMM oooooooooooo ooo' v o v o v o o'o'o'o' I I I I I I I I I I I I I I I I I I I I III I I I I I I 80800^ ^OO^OOLOs SUMMARY FOR TWENTY-ONE BALLOON ASCENSIONS 387 ^ I O\ O 00 O\ O 00 00 00 00 00 Oi O O\ 00 t^ O O 00 1/5 < I ::::::::: : ? *!". ^ 1 ".^ ".". ".". i i i i \O PO >/5 00 N OlOO /5 1/5 Ov I ! ! I I '. . '. \ot^t^oot^oooooooooooo oooot^i^oo '. '. ' lSt^MOOi5o-"i-i . . . .t^^Ot^Ot-OOOOOO O O O>M t-OO HI roroN>O t^oooo o> Ovoooo lO'tfO'O oi ooooooooooooooooooo r i* r r r r r r i" f r r \ \ \ \ r r i fOt^-t^t^ooOvOvt^-O TtOOON OOOOOOOOOOOOi-iOOOOO i' i' \ \ \ \ \ \ \ \ \ \ \ \ i i i \ MMNMMMMMNMT-'v^OO OOOOOOOOOOOOOOOOOOMMMNWMOMMO r r r r r r r r r r r r \ \ \ \ r \ \ \ \ \ \ \ \ \ \ \ \ r^^*OoQWC*OM^' N OO^"OvOOQf^iot > -oo : : : : : : : : : :?yo$l^|*5-t'S5 : r r r r r r r r r r r r r .r r r i* r i ^^^J^^-r^M M N o 2 Tj-oo r^M POOv^-w Ttoo fOOO N O roo rt-N OOOOOOOOOOOOOOwwMNro^tWfOfO^'-J 1 ^ 1 ^^ 1 ^ ' r r f f r r r r i' \ \ \ \ \ \ \ \ r r M \O *-" O\ t^-oOOO 388 EXTENSION OF THERMODYNAMIC COMPUTATIONS ' 66,000 ' 76,000 " 56,000 ' 66,000 " 46,000 1 56,000 " 36,000 ' 46,000 " 26,000 ' 36,000 " 16,000 ' 26,000 ' " 6,000 ' 16,000 ' " Surface ' 6,000 From the 76,000 to 90,000 meters, 2 J = -0.0001 -0.0032 -0.0372 -0.1981 -0.2952 -0.2914 -0.3275 -1.0151 - 1 . 5658 Total atmospheric radiation energy 3.7336 From the Huron ascension we obtained 3.8181. It is apparently a question of distribution rather than the amount that is concerned in these two temperature systems. The total " solar constant" derived from the black atmospheric radiation is about twice the amount measured by the pyrheliometer. More data on this important subject will be accumulated. First Method of Computing the Pyrheliometric Data. The Effective Energy of Radiation and the Solar Constant. When the preceding Chapter V, on radiation, was written it was supposed that the extension of these computations would be limited to the lower levels in the atmosphere where direct temperatures were obtained in balloon ascensions. But 'it is now evident that temperatures may be measured, or assumed by trial, and they will be correct provided they produce such values of P. p. R. T. in successive stages as will satisfy the gravity equation with small residuals, Pi -Po (196) g (zi - ZQ) = - - J (qi 2 - 2 ) - (Qi ~ Co). Pio Hence, by a series of trial computations, it has become possible to extend the data up to 90,000 meters, and four balloon ascensions have been computed up to 70,000 or 90,000 meters. We thus obtain by (335), (333) (Qi ~ Co) = (Wi - W ) + (Ui - U ) = P 10 (vi -%) + (tfi- Z/o), = K -Co = Ja for selective radiation, and THE CONSTANTS AND COEFFICIENTS FOR DRY AIR 389 (344) J = ci 2V - c TV for black-body radiation, in each 1,000-meter stratum from the surface upward, and thence the total thermodynamic and radiation energies throughout the atmosphere by taking the summations for all of the strata. The results are summarized in Tables 82, 85, and Figs. 75, 79. Similarly, the computations include the data of the Kinetic Theory of Gases, (H 9 U) external and internal energies; (q. f) velocities; (n. N) number of mole- cules, l m ax free path length, y number of collisions, W H mass of the hydrogen atom, e^ the negative ion charge, in all strata to 90,000 meters. Tables 96, 97. See BuUetin No. 4, O. M. A. TABLE 87 SUMMARY OF THE RESULTS OP THE COMPUTATIONS ON THREE BALLOON ASCENSIONS FOR THE VALUES OF THE THERMODYNAMIC AND RADIATION ENERGIES Station and Date of the Observations Height in Meters of Temperature Uccle June 9, 1911 70,000 Uccle Sept. 13, 1911 80,000 Uccle Nov. 9, 1911 90,000 2%(zi 20)] gravity acceleration .... 2[ (<?i 2 <7o 2 )] kinetic energy of circulation ... 9.5792 0.0000 5.4250 4.1542 4.1146 3.9690 11.0895 -0.0006 6.6676 4.4219 4.1608 3.9261 12.4565 0.0002 8.6590 3.7975 3.9386 3.5373 2[ (Qi Qo)] free heat (non-adia- batic) S[0(2i-2o) +M?i 2 -?o 2 ) -f- (Qi-Qo)] summary L PlQ J " S[/o = CiTV - CoTV] black body ra- diation 2 [K i KO] radiation energy . . 1.4528 0.0000 1.4602 1.4603 1.4672 -0.0087 1.4404 1.4600 1.4802 0.0352 1.4085 1.4395 t 2 rQi Qo~| free heat per volume l_ v\. #o J change 2 [Pi Po] pressure differences 2[/ = CiTi al - c 7> ] selective ra- diation General mean of the thermodynamic data General mean of th&black body radiation data / General mean of the radiation and pressure data . General mean of the selective radiation data J a . 4.0979 3.9476 1.4516 1.4536 390 EXTENSION OF THERMODYNAMIC COMPUTATIONS Table 87 shows that the thermodynamic state of the earth's atmosphere is such that about 4.00 gr. cal./cm. 2 min. is required to raise it from the frozen and solid state of air at A into the expanded state it now has with its P. p. R. T. in all strata. This is, also, shown to be equal to the black body radiation J Q as derived from the thermodynamic data. The selective absorp- tion J a in a state for continuous emission is about 1.46 calories, and this is equal to the temporary amount of energy in the (K! - KQ) and (Pi - Po) terms. Table 88 contains a summary of the terms in the equation of equilibrium, (785) S = /i + B + J a and h = R. TABLE 88 THE TERMS IN THE "EFFECTIVE" RADIATION FORMULA, WHERE h = R R J a z 5 Ja h =R P Pw Wo 90000 4.00 0.000 2.000 1.000 14000 76000 4.00 0.000 2.000 1.000 10000 66000 4.00 0.000 2.000 1.000 10000 56000 4.00 0.000 1.999 0.999 10000 46000 4.00 0.007 .993 0.998 10000 36000 4.00 0.034 .983 0.996 6000 36000 4.00 0.070 .967 0.994 . . . 3000 27000 4.00 0.100 .950 0.992 . 3000 24000 4.00 0.130 .935 0.991 3000 21000 4.00 0.170 .915 0.988 3000 18000 4.00 0.220 .890 0.984 3000 15000 4.00 0.295 .853 0.975 3000 12000 4.00 0.410 .795 0.961 3000 9000 4.00 0.580 .710 0.950 3000 6000 4.00 0.776 .617 0.929 0.920 3000 3000 4.00 1.050 .475 0.887 0.890 3000 000 4.00 1.461 .270 0.840 0.840 /i = the "effective" radiation = 2.00 at 90,000 meters. S = the solar constant = 4.00 calories. R = the " reflected" radiation = 2.00 at 90,000 meters, and neutralizes one-half the solar constant, /i = R on all levels, except during changes in T. J a = the absorbed radiation in the lower levels, small at 40,000 meters, and increasing by variable values of p THE EFFECTIVE ENERGY OF RADIATION 391 in the Bouguer Formula, for mo about 3,000 meters, / = /.#-. p = the values of the coefficient of transmission formed by p = using the values of 7 for different 3,000-meters. lo p w the general observed values of p at the sea level, at 3,000 meters as La Quiaca, with its probable value at 6,000 meters. m = the adopted depth of the stratum in the zenith. It is evident that p is not constant throughout the atmos- phere, and that mo is not to be taken as unity for the 90,000 meters. On the other hand, 7T ^ m > ^ m2 A. m * x 1* n = /O pl p2 p3 p n , for variable (p. m.). /i= the effective radiation as observed by the pyrheliometer. /o = 2.00 cal. at 90,000 meters, /o penetrates to 30,000 meters undepleted, and then diminishes by a variable p, constant through m = 3,000 meters, in the Bouguer Formula, to 1.46 calories at the surface. R = /i at every level. This return current neutralizes one-half of the solar constant S = 4.00 calories. J a = the absorbed energy in the lower levels, and together with R neutralizes J S. The Bolometer measures the transmitted parts of S. The Pyrheliometer measures the transmitted parts of 7 , that is, one-half the "solar constant." Figure 76 shows the relations of these terms throughout the atmosphere. The effective radiation /o proceeds downward till it is deflected by J a acting outward, arriving at the surface as 1 1. Simultaneously, the returning radiation R = I\ in every level, during temperature equilibrium, reaches the top of the atmosphere, with the addition of J a in the lower levels, having the value R = 2.00 cal. which neutralizes one-half of the incoming S = 4.00. It is easily seen that the pyrheli- 392 EXTENSION OF THERMODYNAMIC COMPUTATIONS E = 1.950 S = 3.950 I 0.000 0.500 1.000 1.500 Scale for the density p 1.200 2.000 0.960 2.500 0.720 3.000 0.480 3.500 0.210 4.000 0.000 FIG. 76. Two methods of discussing the pyrheliometer observations. p A B Area Area Area Ja J = the solar constant 3.95 gr. cal./cm. 2 min. = the radiation energy absorbed in producing temperature T. = the density curve. = the probable pyrheliometer curve. = the probable bolometer curve. S B = the temperature radiation. B A = the scattered radiation. A = the free-heat radiation. absorbed energy in 1000 meters. black body energy in 1000 meters. intensity of zenith sun. = I \ / pa- THE EFFECTIVE FORCE OF RADIATION 393 ometer measures only the effective radiation 7 = J S, while the bolometer measures such ordinates as are necessary for the construction of a 4.00 calories curve at solar temperature be- tween 6,900 and 7,000. If we supply by simple interpolation the missing ordinates on the energy spectrum curve for 6,900, so that the wave-length interval is A A = 0.05 /*, and similarly fill out Abbot's ordinates for Washington, D. C. (34 m), Mt. Wilson (1,780 m), Mt. Whitney (4,420 m), (Bui. No. 3, O. M. A. Tables XXVI, XXVII), we have the following results for the sum of the ob- served ordinates, that is, the relative areas of transmission. TABLE 89 COMPARISON OF THE ABSORBED ENERGY BY THE BOLOMETER AND THERMO- DYNAMIC DATA Energy Spectrum 2J D t Ratio Ja fci Ratio 6900 80.295 4.00 Washington 51.19 29.105 0.363 1.45 0.363 (34) Mt. Wilson 59.55 20.745 0.258 1.15 0.287 (1780) Mt. Whitney (4420) Final curve . . . 61.38 67.26 18.915 13.035 0.236 0.162 0.90 0.65 0.225 0.163 (8000) S / is the sum of the ordinates of transmission at each station. D is the absorbed part =27 (6,900) - S / (station). k is the ratio of absorbed part at each station to J (6,900). J a is the computed thermodynamic absorption at same levels. k 1 is the ratio of the absorbed part at each station to 4.00. It is seen that these two ratios (k, k 1 ) are substantially the same, except for minor variations. It must be concluded that the amount of absorption in the lower atmosphere J a is about the same as the amount of deple- tion in the bolometric energy spectrum, referred to a 6,900 curve, or 4.00 calories for the solar constant. This analysis points to a very different method of discussing pyrheliometer 394 THE EXTENSION OF THERMODYNAMIC COMPUTATIONS observations from that commonly practised, and it promises to harmonize the several branches of this hitherto conflicting subject. SUMMARY 1. The bolometer curves as observed are best satisfied by an energy spectrum of 6,900 or a solar constant of 4.00 gr. cal./ cm. 2 min. 2. The thermodynamics of the atmosphere requires an ex- penditure of 4.00 calories to produce and maintain the existing pressures, densities, temperatures, and thermodynamic values up to 90,000 m. 3. The pyrheliometer measures only one-half the solar constant, that is, the efficient energy 2.00 calories, because one- half of the incoming radiation is neutralized by the returning energy stream of 2.00 calories. 4. The amounts of absorbed radiations, as measured by the bolometer ordinates or by the thermodynamic conditions, agree in giving about the same ratios (k, k 1 ). 5. The conclusions derived from the extensive computations summarized in this section verify, generally, the text in Chapter V. Change of Theory The foregoing analysis is based upon the view that the pyrheliometer measures the efficient incoming solar radiation at 2.00 calories, and upon the formula that half the incoming ray advances to the surface while half of it is scattered back to space. Further experience brings both these ideas into doubt, and we proceed to give evidence that the I curve of Fig. 76 should be made the I a curve, thus enclosing the scattering of radia- tion between the curves A and B. The Second Method of Discussing Pyrheliometric Data The discussion of pyrheliometer observations begun in Chapter V has been continued by the development of a new A NEW METHOD OF DISCUSSING PYRHELIOMETRIC DATA 395 method of computation, which will be briefly described. Com- pute from the observed pyrheliometer readings the values of the intensity /i, for the sun in the zenith, and with p w) the unconnected coefficient of transmission, extrapolate I the intensity reduced to the mean solar distance. Abbot multiplies I by a bolometer factor, 1.123 for Washington, and 1.094 for Mt. Wilson on the average, to which a small correction is then added for the effect of the aqueous vapor pressure e. We pro- ceed to develop the data in another way. Collect the individual observations in convenient groups, according to the observed values of p w , as (0.900 - 0.880), (0.880 - 0.860), ... for I OJ /i, 6j and take the mean values, such as appear in Table 90 for Washington, D. C., at 34 meters, Bassour at 1,160 meters, La Quiaca at 3,465 meters. Plot on diagrams with p w for abscissas, and values of 7 , /i, in gr. cal./cm. 2 min. for ordinates. It is seen that in the case of Washington, Fig. 77, that I is a line sloping downward to meet h sloping upward in the contact point on the ordinate I a = 1.528; in the case of Bassour I is a hori- zontal line meeting the upward sloping h in the contact point 1.680; in the case of La Quiaca I is an upward sloping line to meet /i in the contact point 2.010. In each case the contact point is on the ordinate axis to which corresponds the coefficient of transmission p w = 1.000, which is that for perfect transmission. Whatever may be the physical cause of the sloping of the I lines, below or above the horizontal, the contact point has eliminated that cause from the system. In Case III, Wash- ington, it is necessary to depress the sloping line I into a hori- zontal position I a ') in Case II, Bassour, this is already done; in Case I, La Quiaca, the sloping line must be raised to the horizontal position 7 a . In Case III, if Ii/p w = I , then Ii/p a = /a, where p a is larger than p w ; in Case I, if h/p w = I , then I\/pa = fa, where p a is smaller than p w . It is necessary to determine an equation for each station depending upon the aqueous vapor pressure e which will convert p w into the required p a . This is done in Section 2 of Table 90. Assume p l w at con- venient intervals 1.000, 0.980, 0.960, . . . ; take the contact point I 1 from the diagram, and 7 1 ! from the mean line on the 396 EXTENSION OF THERMODYNAMIC COMPUTATIONS <N <N (N >O 00 <N O i-i rH CO <M' >O cd "3 I IO CO O CO rH TJH ^ l> 1> t> CO CO *O CO O5 C^l 1^ CO rH O5 t^* CO C^Q CO Oi 00 00 00 00 00 *> <N iO (N rH O5 O5 >O O Oi O5 00 CO 00 00 00 OO O5 1 rH (N (N rH CO1>I>OOOOCO001>1>1>CO tt^-C^i 1 l^- O5 l>- O CO CO CO t>* CO CO CO CO CO CO CO 1 1 I I I I Ot>COOOTjHOCD<N O rH CO TjJ cd 00 CJ i-i !> ^ cq O5 CD co o SCO O CO CO O !> O5 O5 00 00 OO t*- CO ^^ O O5 00 00 1> c oooooooo oooooooo rfi <N O 00 CO O^ O^ O^ 00 00 CDcOcOcOcococO CO^C^OOO O^ O^ O5 O5 00 00 -O5 -T}HCO -Oi(M !> -O -COO -COO -O5 -COOi 'ii -OcO C^COOCOGOC5CO O5 GO '"^ 1 O i~H TJH T-H I^-CO^COCfli (OlOOt^- O5 O5 O5 O> O5 O5 00 OO 00 s -^ o I oooooooooooooooooooooo (N(M(M(M(N(M(M<N(N(N(N COMPUTATION OF P a AND I a 397 diagram; then compute p a = y. Plot down e on an auxiliary lo ordinate scale for the same axis of abscissas, and draw the mean e. Now it is evident that if the aqueous vapor effect is to be elimi- nated the line e should pass through the origin where p w = 1.000. In order to do this, draw a line through the origin parallel to e, that is (e + 2.4) mm. for Washington, (e 7.5) mm. for Bassour, and (e + 5.0) mm. for La Quiaca. Take p a p w = F (e + B\ where F = the coefficient, in fact the ratio of the ordinate of e to the ordinate of h counted from the horizontal line 7 a . Hence, we find the equation, where B is the vertical ordinate between the two vapor-pressure lines. In type II, p a = p w ; in type III, p a = p w + F (e + B) ; in type I, p a = p w F (e + B). At each station there is to be computed such a value of the coefficient of transmission p a as will make the values of the extrapolated /i fall upon the hori- zontal line / a , instead of upon the sloping line I . If they do fall upon a sloping line, it follows that the resulting mean values of a series, winter series, for example, will differ radically from the summer series, because the means will pertain to different groups of p w . Take the mean values of each half of the groups in Section 1: Station Pw /o Pw /o Difference Washington Bassour 0.854 0.869 1.633 1.676 0.772 0.763 1.687 1.674 + 0.054 Type III -0.002 Type II La Quiaca 914 1 893 876 1 819 -0 074 Type I This is evidently one cause for the incessant variations that characterize pyrheliometer mean values. It is plain that many such fluctuations as have been attributed to solar action are, in fact, due to the imperfect elimination of the terrestrial effects of aqueous vapor and dust as well as density from the intensity of radiation at the station. 398 EXTENSION OF THERMODYNAMIC COMPUTATIONS Summary of the Correction Equations Calories Washington D.O., Height 34m. Ba 88 ourAlgeria,HeightllCOm. ^ Quiaca Height 2.000 Type III Type II Type I 1.900 1.800 1.700 1.600 1.500 1.400 1.300 1.200 1.100 1.000 O O 00 e'=(e- 7.5) = O pw. 1.000 0.900 0.800 1.000 0.900 0.800 1.000 0.900 FIG. 77. Three types of pyrheliometric data Type I. La Quiaca, height 3,465 meters, p a = p w - 0.0080 (e + 5.0). Mt. Wilson, " 1,780 " p a = p w - 0.0020 (e + 2.0). La Confianza (4,483), Mt. Whitney (4,420), Humahuaca (2,939), Maimara (2,384), have similar equations, but the series of observations is too short to determine the constants Type II. . Bassour, height 1,160 meters, p a = #,. Mt. Weather, " 526 " p a = p w . There is some uncertainty about the equation for Mt. Weather on account of the indecisive data of I . Type III. Jujuy, height 1,302 meters, p a = p w + 0.0060 (e - 6.0). Cordoba, " 438 " p a = p w + 0.0087 (e - 2.6). Pilar, " 340 " p a = p w + 0.0087 (e - 2.6). Potsdam, " 89 " p a = p w + 0.0090 (e + 0.0). Washington" 34 " p a = p w + 0.0100 (e + 2.4). The coefficient F diminishes from + 0.0100 at Washington REDUCTIONS TO SEA LEVEL 399 (34) through 0.000 at Bassour (1,160), to - 0.0080 at La Quiaca (3,465); the amount of the B is very variable with the local conditions of the vapor-pressure e at the station. Apply these equations to the values of /i and compute 7 fl . TABLE 91 WASHINGTON, D. C., STATION EQUATION, p a p w + 0.0100 (eX2A) No. Pw e e +2.4 A Pa * la Isea A/ 7 .884 3.78 6.2 .062 .944 1.410 1.497 5 - 16 .874 3.00 5.4 .054 .928 1.402 1.511 *t Js 20 .865 2.09 4.5 .045 .910 1.409 1.548 rH ^ 16 .854 2.60 5.0 .050 .904 1.392 1.540 *0> <fl 22 .846 3.34 5.7 .057 .903 1.386 1.535 > B 24 .834 4.06 6.5 .065 .899 1.410 1.568 "J fi 17 .824 3.80 6.2 .062 .886 1.353 1.527 '~ rt 17 .815 3.69 6.1 .061 .876 1.418 1.619 ti 10 .804 4.30 6.7 .067 .871 1.334 1.532 ^ JS mean 1.542 14 .795 5.02 7.4 .074 .869 1.349 1.553 *-, ** 18 .784 5.10 7.5 .075 .859 .316 1.532 0) 1 8 .773 5.10 7.5 .075 .848. .303 1.537 9 .764 5.19 7.6 .076 .840 .335 1.590 > d o 11 .756 6.10 8.5 .085 .841 .272 1.512 bit) ' 7 .743 7.03 9.4 .094 .837 .260 1.506 g E 5 .736 7.69 10.1 .101 .837 .262 .508 8 5 .714 8.85 11.3 .113 .827 .252 .514 <u o 3 .703 7.40 9.8 .098 .801 .170 .461 H H 4 .696 6.06 8.5 .085 .781 .212 .552 mean 1.527 .534 1.525 -0.009 The computed values of I a from p a , in place of I from /?,, plot near the dotted line in the diagram. The mean of the first nine values is 1.542, and of the last ten values 1.527, showing that the variations in p w , and the effect of the vapor-pressure e, are quite well eliminated. A longer series would probably prove that the elimination is complete. Table 92 contains the summary of the computed I a for nine stations and their mean values. If these mean values of I a are plotted, for the height z on the axis of ordinates and calories on the axis of abscissas, the points fall nearly on a straight line, so that I a diminishes in proportion to the height. This line cuts the sea level at 1.525 calories, and the differences which corre- 400 EXTENSION OF THERMODYNAMIC COMPUTATIONS II 8 8? H c3 o > M < i f* T ( SSS 3 en 5J > fli (N i I (N t^ 1 1 CO f s&M? CO CO 1 I CO f (6862) BOB l|i '* - s .2 O i I 1 (W8Z) ial (N t^ 1 oo Tt^ BOBinQ B1 -H | 1 Cti //\r i-N s ^u ^E uos lJAV}p^ . . l> ! i> T l> ! l> !* > T <u . 11 ,c I ^^^s^^^^s^ - ^ S jnmf r^t-t^ S t^t^^tS o t^ M rH 11 3 *U 1 C (0911) <MOOQOT-HI>-l>COC^OOThOO 03 s anossBg . i-H T3 1 *i a , . tj 52 S^ 2 ^^ 2^ ' ' ,^i! ^5 < *]S 2 ro rr^ rr^ 10 10 CO CO .0 CD CO CO CO T3 < T-H 1 . . i^ ^n ^^ nS ^^ ^4 *o r^ r^ tjqopSo lO^OcoOcOcOiOxO * T 1 1 co^ioco^^^t- 1> (0?8) * O TH 1 Sw ^io ^^ rS (68) JL| . . . ^H 1 f3 I>i !OOO>OCX)l>-O(NCOC^l>OC4cDOO < *'-'C<I i ^ 8 i-H 1 ANNUAL MEAN VARIATIONS 401 spond to the station height are readily computed. It is equally easy to determine the normal value of I a at any station, and several examples are added for stations having only short series. The Annual Mean Variations The station equations can be applied to the individual p w groups, or to individual observations, during different years, and their mean sea-level values are collected in Table 93; also, the annual means are taken for all stations which were observed during the same year. We have such means extending from 1903 to 1914, and they are plotted on the upper curve of Fig. 78, giving minima in 1903, 1907, 1912, 1914. The 1907 and 1912 minima are already well known. In the second curve of Fig. 78 is reproduced the adopted mean meteorological curve from Bulletin No. 1, Oficina Meteorologica Argentina, written in 1910, which contained a summary of data till 1910, and a forecast from 1910 to 1915. It should be noted that the forecast for 1911, 1912, 1913, was very well verified, as it has actually been in all other of the Argentine meteorological data. In 1914 the mini- mum, which the forecast placed in 1915, seems to have come in 1914, but this is very unexpected, because there is an interval of only two years following 1912, whereas the ordinary periodic interval is 3.75 years. Another forecast is added with maxi- mum in 1917 and minimum in 1919. It is necessary to maintain suitable solar observations, in order to study the causes of such irregular fluctuations in the output of the solar radiation, and several solar physics observatories, adapted to meteorolog- ical purposes, are indispensable in the interests of long-range forecasts. The observatories at Pilar and La Quiaca, Argentina, are well adapted to supplement the work of Washington and Mt. Wilson in the United States. The General Summary It has been shown that the mean values of I a plot along a straight line with a given slope. On Fig. 76 is plotted S = 3.95 the solar constant; S J a the curve of the absorbed radiation from 402 EXTENSION OF THERMODYNAMIC COMPUTATIONS s i : ill : rH rH i H rH rH CO : i : : : 1 a s ^4 C^ tO t^ t^* " CO rH rH rH O rH rH o to to o o to rH . 1 , 1 r ( rH iH s iill : : : iH rH rH rH rH O irn : : : : :ooco s tij tij 41^ iH rH rH rH 00 -OS -* 1 "* ^ tn . -HH Tj< CO "* .10 .10 to to ^ /-*"" eS O *? H rH i 1 rH rH V E o g to -to to to 2 -'' M ^n en iH i 1 rH rH rH s v ^> X ^ s :':::: iJl 3 ( k C bX) --- 3 \ ^ H CO 5 r ^c > C 03 OS y !( rH rH g s >> s jJ S5 PQ < < w 1 i 1 O rH !>. CO CM <N <N rH rH rH rH 11 c 7 ^ 7 r? en 1 P"* ^ oo GO 5=; ^ g H R 1 iO -^ to '. ' ' ' ' : ; "i -] ^ 1 1 1 i i i i 1 1 ! 'H .2 w ooo -a ' '. ' '. i o 2 "3 o oo 2 + + 1 1 2 M w i I rH - s s Si ~* CTJ a> J i i r i r r r III "3^3 oSi C C sis r>! OO t^ H to o o co GO o ^ CO GO CO OQ CO ^ CO CO rH rH ! z kH S Jo 3 l "5 ^ " I comes i early, th 12-14 bei O ta CO 0^ O o 3 o S o *H ** s -3 2 B j:|. I 1 ilsl-8il i O< w "O i- cS 2 cd ^ S 5 S .-5 jg J jSCQ^UOn^ < GENERAL SUMMARY 403 the outer limit to the station level, or it may be taken as (S 2 J a ) = B y plotting S J a from the ordinate S; the density curve p is plotted with the abscissa as indicated; the pyrheliom- eter curve I a = A is plotted up to 4,500 meters, and its course follows the p curve so closely as to suggest that these two func- tions run out together, that is, at the solar constant 3,950 calories. If Abbot's ordinates for the bolometer are filled up by proper interpolations for the several intermediate wave lengths, and the sums taken, we shall have good relative values of the several bolometer energy areas. TABLE 94 SUMMARY OF THE BOLOMETER DATA Bolometer Data Abbot's Scale Sum. Calories Factor of Reduction Calories B Height z Black body 6900 A Abbot's extrapolated 80.30 67.26 3.876 20.7 3.876 3.25 23000 6500 Mt. Whitney 61 38 2 97 4420 Mt. Wilson 59 55 2 88 1780 Washington, D. C 51.19 2.47 34 At the solar temperature 6900 A, for black body radiation, the value at the outer limit of the earth's atmosphere is 3.876, and corresponding to this the sum of Abbot's ordinates on his arbitrary scale is 80.30. The factor of reduction is taken 20.7, and the calories B for the several stations at the height z follow. Plot these on Fig. 76, and they seem to belong to the S J a curve. We conclude that the pyrheliometer data approximate the p density curve, and the bolometer data represent the selective radiation curve S/ a . The total solar radiation now divides itself into three parts: (1) Radiation intensity S J a used in producing temperature. . 1.46 (2) Radiation energy scattered back to space 1.00 (3) Free radiation energy penetrating to the sea level , . 1.52 Total or solar constant energy in calories 3.98 The corresponding temperature would be 6930 A. The 7o curve of the first method has been changed into the I a - curve by the second method of computation. 404 EXTENSION OF THERMODYNAMIC COMPUTATIONS GENERAL REMARKS On Fig. 76 are drawn the typical curves from the data of Chapter VII: J a the selective radiation absorbed in each 1,000- meter stratum, and 2 J a the total amount absorbed down to the given level; J the black body radiation in each 1,000-meter stratum/ anpl S/ the total amount down to the given level; /i the computed radiation for the zenith sun on each level, I the computed Ii/p w , and I a = Ii/p a ; the density curve p; the solar constant S = 3.950, and Abbot's solar constant = 1.950. (1) Abbot multiplies I by a small bolometer factor, and adds a small function of e to produce E = 1.950 on each level. Bigelow's curve I a , derived from the contact points for p w = 1.000 at perfect transmission, crosses I at an angle, because Abbot's mean values of I are too large at low-level stations, and too small at high-level stations, both systems agreeing for p w = pa at about 1,200 meters, the level of the cumulus cloud base. The pyrheliometer curve A, following closely upon the density curve p, will run out at about 3.950 colories, in conformity with the S/ = 3.950 calories. (2) The bolometer ordinates indicate an amount of energy well represented by the curve S/ or B, so that Abbot's small bolometer factor should be greatly increased, up to 2.00 in the higher levels. The theory which led to his small bolometer factor, namely, that the pyrheliometer registers the same amount of heat as that indicated by the bolometer, when the small corrections have been applied, is erroneous. The curve I does not represent the true pyrheliometer intensities, which are 7 fl , nor does it take any account of the energy scattered in the area between A and B, nor of the heat absorbed in making the temperatures T in the area between B and S. (3) t The pyrheliometer does not directly measure the solar constant at low levels, and the Bouguer formula is incapable of making the complete extrapolations; the bolometer does not measure the solar constant except by an approximate inference; the thermodynamics of the atmosphere affords data for com- GENERAL REMARKS 405 puting S/a and S/ in all levels, and the final sum for S/ at the sea level is the solar constant, about 3.950 calories. (4) The thermodynamic computations are in accord with this interpretation of the pyrheliometric data and the bolometric data, hi requiring about 3.950 calories to do the work actually measured in the atmosphere, and it would be impossible to maintain the existing temperature, pressure, and density distri- butions by expending only 1.950 calories. The pyrheliometer is useful in determining the relative annual variations of the solar radiation at any station; the bolometer is necessary in studying the relative absorptions of different wave lengths in the atmos- pheres of the sun and of the earth; but the thermodynamic data are required for the mutual interpretation of these two types of data. This subject is treated at greater length in Bulletin No. 4, Oficina Meteorologica Argentina, 1914. 406 EXTENSION OF THERMODYNAMIC COMPUTATIONS The Constants and Coefficients of Dry Air in the Kinetic Theory of Gases throughout the Atmosphere It facilitates the discussion of the problems of the upper layers of the earth's atmosphere to have the data of the Kinetic Theory of Gases for Dry Air in a form for reference. Such a table of constants and coefficients for the (M. K. S.) and the (C. G. S.) systems is given in Table 95, and the results of their ap- plication from the bottom to the top of the earth's atmosphere as computed in Tables 96, 97. The fundamental data for N = 6.062 X 10 23 ,e = 4.774 X 10 ~ 10 , conform to the summary of con- stants by R. A. Millikan, Physical Review, August, 1913, the values for n, EQ, e, k, h, cz, differing a little in consequence of slight variations in the adopted meteorological data for P, p, R. It is interesting to note that these fundamental physical quan- tities, and many others, can be computed, not only at the surface in standard conditions, but throughout the atmosphere. Sim- ilarly, the sun's data can be computed provided P, T, p, R } can be secured at the required points in the solar atmosphere. The formulas of Table 95 indicate the method of the computations, and numerous subordinate formulas can be derived from them. The transformation factors between the (M. K. S.) and the (C. G. S.) systems have been checked throughout this group of Formulas. In making transformations from (M. K. S.) to (C. G. S.) M = 10 3 , L = 10 2 ; and from (C. G. S.) to (M. K. S.) M = 10~ 3 , L = 10- 2 . In Table 96, the kinetic energy per unit volume H y and U the inner energy per unit volume, decrease in a curve similar to those of P and K to vanishing values; q, the mean square velocity, and T, the arithmetical mean velocity, diminish till they disappear at the outer limit; N, the number of molecules per kilogram-molecule, and n, the number of molecules per cubic meter, diminish slowly, but do not vanish. If the term N ought to remain constant, this diminution may go back to some additional physical forces not included in these simple formulas. Compare Fig. 70 for typical curves of the Dynamic, Thermo- THE CONSTANTS AND COEFFICIENTS FOR DRY AIR 407 dynamic, and the Kinetic Theory data, throughout the atmos- phere. A second more refined computation, with improved tempera- tures, has been undertaken. The data of Table 97 should be substituted for those of Table 96 above 40,000 meters, and the following remarks are based upon it. l max , Maxwell's free path, increases from the surface upward, rapidly above 45,000 meters ; v, the number of collisions per second, diminishes with the height. The supposed height of the atmosphere, as derived from meteors at 100 kilometers, finds complete verification in these computations. m Hj the mass of the hydrogen atom, as computed, shows a small increase in value, probably due to some defects in our adopted simple formulas. Similarly, e_, the elementary negative electric charge, assumed in this formula to hold a constant ratio to m H , 3.4554 X 10~ 13 (M. K. S.), follows the fortunes of m B . 408 EXTENSION OF THERMODYNAMIC COMPUTATIONS aS a -*-> H (/3 s*. c/5 O ^ p x->> c/5 M* 1 (OOcOCOI>-cO''^ t i cx> -^ co o > i<N 00 CO O5 CN rH oo 10 r- 1 I CO CO 05 05 COO5CO OdfNCOCO CO i I -<N c^SSoo CO (N i I O O 00 lO l> o o o o b i-H rH 1-1 T-H rH /\ /\ s\ I><NaO(N (NO 1>CM X X ~ i-<OO 05 ^H s 1> CO O 00 O (M rH 00 t>- Tfl CO O5 (N i-J Sb 8 ^ S 8 rH rH rH rH rH rH XX X X X X U *. II 8 (^ II II II II II II OH Q. R ^l 5 ICQ r S S fc o Q fa PC i? S ^ sl| c -^ liiili THE CONSTANTS AND COEFFICIENTS FOR DRY AIR 409 t>- O5 % % 05 00 CO rH 10 d -H I (N rt< O5 (M rH <N T^ O rH (M l> rH -^ t>. CO CO CO <N OS O1 l> <* - d TJH 06 co i> * I rH <N <N i-H rH 'ill X X X 82 rH O co XXX rH <N CO O5 rH O 00 X^XX TjH "* CO O5 O5 co oo oo co oo co o 05 CO rH IO rH 00 i-H (M -^ O5 (M rH C^l Tf O CO t^ C^^ O 00 CO O >O iO O5 O5 O5 33 ^ ^ttcoS COC<ICO (N OSCNt^^ O5O51O I s - H CO lO CM I rH (M CO rH rH I I ~ 1 o o X X O V r, o *O O5 CO rH 2 T 5 - 2o x C O5 O rH CO rH rH CM CO O5 O5 rH 00 CO l> oo S 2 XX O5 O5 co oo 10 ci co ICM^T^CM ico II II W 3 33 irkungs q g 3 >* i3 % ** cj cu bo~ O | < >_ <*H & W g J ^ .a w ^ & I b| ^ B-g t . a B M i I ! | ^& I a c 2 2 d ha 2i C ^ w S W ^> e g C o C o b 03 > -5 J h 3 >.*i "J o " c .t! g PL, -2 g ^ c/) (U *7? ^ O O> UQW S ^ ass hydrog ass electric a 410 EXTENSION OF THERMODYNAMIC COMPUTATIONS & . ail S m H ass of ydrogen Atom Freep Leng Maxw S O 3 !_ K S X X X THrl<r}<T-lt-Tj<eOCO(MC^NNNN ns s f, ii i rH iH r-l r-t l-i XxXX X r i- Tj C< T- i- r-( O> 00 00 t- t- t- <O > tf> kO US 000 X X XxX t*OOOCOOO 00 11 1 r-H iH T-l T-H tH XX XX X CO 00 t- N * T- 10 CO d rH T- r T- tH OS 00 t- <0 >0 ' XXXX T-<0t-C<|T}lOt-C5HT-lT-lT-lr-lT-lT-lT-lT-ll-lT-ll-e o o o T^ T I rH r- 1 r^ T-^ xxxxx x <NI r-iio to o T- OS U5 T- O3 00 * O IO s X 11 000 r-l T-H T-l XXX nt-^Tfcococowco THE KINETIC THEORY OF DRY AIR SYSTEM OF UNITS 411 wi to T* -<t <* ri * eo eo T "I X X X -^' CO 09 fri C3 M' fri -i r-l r-l i-l i-l 00 I * I ^X X CO Tj< Tli >O tO t^ 00 rH r-i i-i i-i i-i CJ frj IN i CO Tji -^ IQ (Q t> 00 ^i ,-i r-i rH rH i-i gj QJ M < N gj 9-* n O O> ^ 0> ^' CD .H 00 O t> O U5 fcS8S88Sfe8?SSSS ^< O N O . o O I- O T* iH g2 412 EXTENSION OF THERMODYNAMIC COMPUTATIONS I 0> w <3 cT 8 w S S 8 Q S | i S S o II* RS s Is o, s s 2 kzkz xxxx T)<t-rJii-^ t-ic>T}<c<i imi XXXXX lllllll I s g S XXXXXXX X X X X X OOJi-li-lcoOt-t-OC<100Omi-IC<It-ieOrHa>i-^'<ft~-l t- O U3 Wt- OCOCMU5t>^OlO^lCOiU5Ot-O5OST-(C<lOr-l t-' t-' O5 t-' t> tO W i 00 N D rH CO' l> rH (N 1C i-J r-i <N i *' 00 i-4 <N CO iim S s - XXXXX XX X t-t-COOiHTjttOOOt-CSI^Oii 3ioeocooooco^t-^0' XXXXX 00010000 O O XXXXXXX X X OOOOOOO O XXXXXXX X I I i XXXXXX XXX ot- THE KINETIC THEORY OF GASES 413 CDlOWOT^^^C^-OOOCOOrHi-HO^U^ 1 O iH CO 00 <* |8SS s s s s xxx x O) ' I I T-l fH 1-1 XX X I X 2C<lt-lOOa>Oa5Oa}t-000C<lO'*J<'*i-IC<IO^OU3U5Tj<OOC<l i-H5DcOi-IOT-lCOOOCOi-<THCOOOU3r)<COrHOOOOOWCOeOU3C<IOi T-ICO-^SOOOON^tDOSOlUJOOiHlOOiCOOONt-COOO-^OtOOO I I s XX X CJrJttOOJr-li-HCSJeOlOt^OiHrHMCO^USt-Oi-li-INNeO^lO^ g'ico>cot-i-Jioa>cooddi>c>]t-(Mt-(Noocooi 05t-t-0000000>0500r-ltH(M(MeOCO^<T)< -i(M<M'-IOOT}<a5r}<<MTJ<00OOOlMINeOT-IOOCOO>eOCOeD)OOCO cocoTi<^oo<N^c<iait><)t>oicoooT)<<Ni-io^HC<iioooi-i-<j<5t- rH X Cl-^USOOf-ti-INCO-^CCOONCDNOiOlNOlOooeOt-i-ICJCMOkO i-(THi-IC<lCOTl<W5O X OJOJiHCDO^i-IOOW5COlCOOimOCCDlOT)<OOOeQOia>OI^l < ooit-eo<NOicon<ocoiMNU5cococDio ~ 414 EXTENSION OF THERMODYNAMIC COMPUTATIONS Summary of the Dimensions with Special Reference to the Equivalents In transformations from the (M. K. 5.) to the (C. G. S.) systems, the following dimensions should be observed: Mass = M=10 3 . Length = L = 10 2 . Time=r=l second. Velocity = L T~ l . Momentum = M.L T~ l . Acceleration = LT.~ l T.~\ Force = M.L T ~ l T~ l = M L T~ 2 . Impulse = M.L T~ l . Force per unit mass or force of acceleration . = L T~ 2 Force per unit volume = L T~ 2 . M L~* . . = M L~ 2 T~ 2 I Specific Volume = Volume per unit mass . . = M~ l L 3 I Specific Density = Mass per unit volume . . = M L~ 3 Work = force X length = M L T~ 2 . L . . = M L 2 T~ 2 Kinetic Energy = Mass times velocity squared = M L 2 T~ 2 Increase of kinetic or potential energy = work done ..... ........ = ML 2 T~ 2 I Kinetic Energy per unit mass ...... = L 2 T~ 2 | Gravity potential = work per unit mass . . = L 2 T~ 2 Kinetic Energy per unit volume = L 2 T~ 2 . M L~* = M L~ l T~ 2 Pressure = force per unit area =M L T~ 2 . L~~ 2 = M L~ l T~ 2 Pressure = work per unit volume =ML 2 7^ 2 . IT 3 =M L~ l T~ 2 In the equations of Condition, we have for T = 1, p _ p , g ( Zl -z )= - -% ( qi * - g 2) - (Q l - Qo), s Pio L.L = M L~\ M~ 1 L 3 = L 2 = L 2 = 10 4 = 10,000. i =L 2 .M L-*= Pressure has two definitions: (1) force per unit area, as Kilo- grams per square meter, or grams per square centimeter; (2) work per unit volume, as free heat per unit volume, hydrostatic pres- SUMMARY OF DIMENSIONS 415 sure, inner energy per unit volume, radiation energy, and these all have the same dimension, M L~ l . Since the Erg and the Joule are units of work they must refer to the unit volume and not to the unit area. Hence = ML i T~\ L~* = M ZT 1 T~* = 10 3 X 1(T 2 = 10, While -~ = ML* T~\ L~* = M T~* = 10 3 = 1,000 is distinctly erroneous. Summary of Dimensions In the conversion from (M. K. S.) mechanical units into gr. cal./cm. 2 minute, the correct factor becomes: (M.K.S.) X 10 X 60/4.1851 X 10 7 = 0.000014336 On the other hand, should the area L? be placed in the denomi- nator under Joule, instead of the volume Z, 3 , the factor (M. K. S.) X 1000 X 60/4.1851 X 10 7 = 0.0014336 would be incorrect, in- asmuch as the - - is the work energy per unit volume and not per unit area. The equations of condition assume two typical forms: (a) - (Pi - Po) - = g (zi - z ) fto-j, M L~ l = L.L. M L~* = M L~ l = 10 , M (Pi - Po) 60 60 (b) -^r"^ = g(Si " 2o) ^ M L~ l . M~ l D = L . L = D = 10 4 For the equation (a), from (M. K. S.) Mech. through (C. G. S.) Mech., (M. K. S.) X 10 X 60/4.1851 X 10 7 = 0.000014336 For the equation (a), from (M. K. S.) Mech. through Kil. Cal./ m. 2 min., (M. K. S.) X 60/4.1851 X 10 3 X 10 3 = 0.000014336 For the equation (b), from (M. K. S.) Mech. through (C. G. S.) Mech., (M. K. S.) X 10 4 X 60/4.1851 X 10 10 = 0.000014336 For the equation (b), from (M . K. S.) Mech. through Kil. Cal ./M 2 min., (M. K. S.) X 60/4.1851 X 10 3 X 10 3 = 0.000014336 416 EXTENSION OF THERMODYNAMIC COMPUTATIONS TABLE 98 SUMMARY OF DIMENSIONS Formulas Dimensions M. K. S. C. G. S. Work ,_kilog.m 2 gr. cm. 2 P pR T ML 1 M kilog. Joule sec. 2 gr. erg p (m) = mol. wt. U. T 2 L 2 L T 2 ~L? L 3 m. sec. 2 m 3 kilog. m 3 m* cm. sec. 2 cm 3 . gr. cm. 3 cm. 3 (M) = M (m) R Cp Cu M L* kilog. gr. cm. 2 K = R(m) T^deg. L 2 (m) sec. Meg. m 2 (m) sec. 2 deg. cm. 2 (wi) A = mech. eq. H T 2 deg. LML sec. 2 deg. 4 13*51 V 1fV kilo &- m * sec. 2 deg. 4 IS'JI V 1010 /oor 2 Watt T^cal (M) sec. 2 kilog. Joule 1 Watt. 107 erg L 2 T N 1 r 3 deg. 1 sec. 3 deg. 4 m 2 sec. deg. 4 1 cm. 2 deg. 4 cm. 2 sec. deg. 4 1 n M I mass hydrogen atom 1 mass hydrogen atom 1 L ML-i (m)U (m)L 2 m 3 (mol. wt.) (mol. wt.) m 2 cm. 3 (mol. wt.) Pv RT T 2 M T 2 M L~i L 3 L 2 sec. 2 m 2 sec. 2 cm 2 . 2 3P0 N T 2 M~ T 2 sec. 2 , , m* M-mol. ; sec. 2 m 2 sec. 2 M-mol. ' sec. 2 cm. 2 2 V- 77 3 ^ T 2 sec. 2 sec. 2 2 n (M)L 2 m sec. 2 sec. 2 (M) cm. 2 t ** T 2 deg. sec. 2 deg. kilog. m 2 Joule sec. 2 deg. gr. cm. 2 erg. AT L T 2 deg. ML 2 ML* sec. 2 deg. deg. kilog. m 2 sec. 2 deg. deg. gr. cm. 2 Ci = Sirch f y2 ML? _, J ouie sec. sec. kilog. m 3 sec. gr. cm. 3 C2=ch/k Ldeg. ."z. r~2 , s/ / i y 2 sec. 2 Joule meter meter deg. kilog. Joule sec. 2 cm. deg. gr. erg H 2 P TJ C n T pi M L. 1 - L 2 M , M met. sec. 2 m 3 kilog. Joule cm. sec. 2 cm. 3 gr. erg C g a = T 2 deg.' L 3 deg> ~ L T 2 M L\ 1 M m sec. 2 m 3 w/sec. w/sec. 2 kilog. Joule cm. sec. 2 cm. s cm./sec. cm./sec. 2 gr. erg a c T 2 L 4 deg. 4 LT 2 deg. 4 M L M m. sec. 2 deg. 4 m 3 deg. 4 kilog. Joule cm. sec. 2 deg. 4 cm. 3 deg. 4 gr. erg 4 cr L r 2 deg. 4 ' T r 3 deg. 4 M T 2 cal. M cal. sec. 3 deg. 4 m. 2 sec. deg. 4 kilog. cal. sec. 3 deg. 4 cm. 2 sec. deg. 4 gr. cal. ^ A A <r r'deg. 4 L 2 " L 2 r d eg 4 m 2 sec. deg. 4 kilog. Joule cm. 2 sec. deg. 4 gr. ere; ^ T 3 deg. 4 sec. 3 deg. 4 m 2 sec. deg. 4 sec. 3 deg. 4 cm. 2 sec. deg. 4 GENERAL PROBLEMS 417 It should be noted that a X j, = cr, so that a is the work energy per unit volume. When this gains the velocity L Ti, it becomes cr or radiation through the unit area. The equation (b), page 414, KIQ = cT a gives c of the same dimension as a, and it must be multiplied by a velocity to become comparable with o- in the Stefan Law. The computations for the radiation coefficients conform to these dimensions. Similar computations have been successfully made for hydrogen and calcium vapor in the atmosphere of the sun, and the results are in conformity with the indications of the well- known observations. This knowledge of P.p.R.T. in the superficial layers of the sun's envelope is of great value in in- terpreting the lines of the visible and radiation spectra. Indeed, the thermodynamic data are so complex that it will be im- practical to determine them by direct observations. The change of R = constant to R = variable, that is, the transition from adiabatic to non-adiabatic conditions, converts the so-called constants k, h, c, c 2} a, a into a series of coefficients which change from level to level. It follows that the laws of radiation for each chemical element are much more complicated than is implied in the use of the Stefan or Wein-Planck Laws with simple constants. This subject is of such far-reaching im- portance that it is reserved for further study and research. General Problems in Atmospheric and Solar Physics The purpose of the development of non-adiabatic meteorology has been to discover a method of computations applicable to all atmospheres, as those of the earth and the sun. The process consists in fixing an initial P , /o , Ro t T on some level z 0j and then, by assuming 7\ on the level 2i, determine the corresponding PI, Pi, TI on that level, all to be checked by the gravity equation. From these values of T the entire system can be obtained. In the case of the sun the initial data can be fixed approximately as follows: If the Boyle- Gay Lussac Law in the atmosphere of the earth 418 GENERAL PROBLEMS is, P U = R T, this can be converted to the sun's atmosphere by multiplying each term with 28.028 = G/g = T, the ratio of the surface gravity on the sun to that on the earth, so that, PT X UT = RT X Tr. The corresponding factor for all other terms follows very readily, and the entire thermodynamic system can be developed for each element by the method of trials. The check is found in the gravity equation. g p _ p - ZQ) = - l ~ (C p a - C #10) (T a ~ T ). The analogy between the physical condition in the atmosphere of the sun just above the photosphere, and in that of the earth between the levels 37,000 and 44,000 meters, is very interesting. Compare Fig. 79. In both there is apparently a very rapid change of temperature with the height, and a transition from ex- cessively rarefied media to the tru egaseous condition. In the earth's atmosphere there is passage through such conditions as : z in meters 37,000 44,000 T (absolute) P B (mercury in M M) 211 437 3.3 124 98 0.7 M. K. S. Sys- tem of units P 0.025 0.0087 R 82.77 91.49 The physical transformations in this transition at 44,000 meters correspond with those sensitive states in the Geissler Tubes at which the electric and electro-magnetic phenomena are especially active. From this it is inferred that the origin of the auroral atmospheric electricity consists in the transformation of a portion of the incoming solar radiation of short wave lengths into ions, or free electric charges, and that these pursue their paths to the polar regions, as indicated in Fig. 69, generating the corresponding magnetic deflection vectors. In Table 95 the mass of the hydrogen atom is m H = -r = 1.6496 X 10~ 27 , and GENERAL PROBLEMS 419 if this is a constant, N = 6.062 X 10 26 should, also, be a con- stant throughout the atmosphere. But by Tables 96, 97, N diminishes with the height, especially above 40 km. z = 90,000 N = 1.665 X 10 20 80,000 1.064 X 10 24 70,000 7.023 X 10 24 60,000 2.216 X 10 26 50,000 5.364 X 10 25 40,000 1.092 X 10 26 30,000 1.730 X 10 26 20,000 2.671 X 10 26 10,000 4.161 X 10 26 000 No = 6.147 X 10 26 The differences between No and N may be taken as the number of transformed ions on the several levels, to be used in generating the auroral electric currents. Furthermore, the region above 44,000 meters in the earth's atmosphere, with its excessively rarefied media, in electric sensitiveness under the impact of the in- coming solar radiation of short wave lengths, is distinctly coronal in its physical conditions, and it corresponds with the deep corona that exists above the sun's chromosphere. The rapid changes of temperature near the sun's photosphere produce a layer which is the source of great radiation energy outward, while the rapid temperature increase in the levels 44 to 37 km. of the earth's atmosphere is the evidence of an important absorption of radia- tion energy. Hence, the sharpness of the sun's visible disk is a phenomenon depending upon the thermodynamic conditions existing between the rapid decrease of temperature and the in- crease of radiation outward to space. The coronal and the pho- tospheric regions in the atmospheres of the sun and the earth are, therefore, the physical counterparts of each other. The laws of the absorption of the incoming radiation in the earth's atmosphere need further study. In place of interpreting the bolometer data in terms of the pyrheliometer data, and ascribing the great depletion of 4.00 gr. cal./cm. 2 min. down to only 2.00 calories, as the result of a very imperfect action of the sun as a black body radiator, the result of the examination of the pyrheliometer and thermodynamic data seems to indicate 420 GENERAL PROBLEMS. that the pyrheliometer data should be interpreted in terms of the data of the bolometer, so that the "Solar constant" is about 4.00 calories, with the sun's effective radiation tempera- ture 6950 A. In the Bouguer formula, the coefficients of trans- mission p w as directly observed, and p a as computed, have an interesting relation to the wave lengths in the spectrum. Abbot's values of p w at different wave lengths were grouped together, at Mt. Wilson and at Washington, and diagrams of the mean values of p w , as 0.936, 0.925, 0.915 . . . were plotted on the abscissas, 0.20 M, 0.30 ju ... 1.60 M. If the values of p w and p a are plotted on these curves, they intersect at about wave length 0.57 M. If 0.57 M on p w = 1.00 is taken as the maximum ordinate on the energy curve, this corresponds with about 5000 and 1.07 gr. cal./cm. 2 min. at the distance of the earth. Now, this is the average direct reading of the pyrheliometer at sea level, so that we infer that not only the aqueous vapor and dust, but also the density of the atmosphere, must be removed in discussing the coefficients of transmission. This conforms to the result shown in Fig. 76, where the second method of computa- tion identifies the depletion of I a = I\/p a with the density P of the atmosphere. This research is not yet ready for final statement. It is very desirable that the excellent condi- tions prevailing at La Quiaca, at the elevation 3465 meters (11,037 feet) on the Bolivian Plateau, should be utilized for a solar physics observatory, equipped with a bolometer, spectro- heliograph, and direct-vision spectroscope for prominences, to be operated in cooperation with Mt. Wilson and Washington, D. C. The conditions of living in La Quiaca are comparatively easy for observers, and there are complete railroad facilities for transportation from Buenos Aires. La Confianza, at 4483 me- ters (14,070 feet), can, also, be occupied the year around, as the village of San Vincente and the neighboring mining camp pro- vide the necessary accommodations. It is important that the bolometric data should be obtained at these high levels, and there is an excellent opportunity for an expedition to make the necessary observations at those stations. An inspection of the data of Chapter VII suggests that there GENERAL PROBLEMS 421 are a very large number of problems in general physics that can be studied to advantage with the data which have been already developed. The laboratory is quite incapable of furnishing the fundamental relations that are shown to exist above the 45,000- meter level, where the pressure is a small fraction of one milli- meter of mercury. Fortunately, the balloon ascensions to 30,000 meters cover a very interesting region, the so-called " isothermal layer," and they provide quite accurate observations of the temperature. The most important outstanding problem is to find a thermodynamic function for the kinetic energy of circula- tion, ]4 (<?i 2 <?o 2 ), in order to separate it from the free heat - (ft - 90- It is hardly possible, in opening up so much new research material in the physics of the atmosphere, to have escaped imperfections and even errors, but it is thought desirable to indi- cate to meteorologists and astro-physicists some of the possible channels of investigation that appear to be accessible to such studies as are here illustrated. INDEX Adiabatic atmosphere, V. Bjerknes' system, 51 Margules' system, iii Adiabatic and non-adiabatic systems, differences, 51 Angular velocities, cylindrical, 140, 142 earth's, 160 polar, 140, 142 rectangular, 140, 142 Aqueous vapor in the atmosphere, grams per cubic meter, 342 grams per kilogram, 343 Atlantic Ocean, meteorological observations, 56 Atmosphere, homogeneous, 2 Avogadro's Law, number of atoms, 28 Azimuth, coordinates, 7 B Balloon ascensions, results, 115 Huron, So. Dak., Sept. 1, 1910, 365, 377, 381, 410 Twenty-one, 384 Uccle, June 9, Sept. 13, Nov. 9, 1911, 380, 385, 412 Barometer corrections, for local effects, 45 for mean column temperature, 43 for plateau temperature, 45 for removal of station, 42 for standard gravity, 42 for standard temperature, 41 for vapor pressure, 45 reduction tables, 46 to normal instrument, 42 Barometric formulas, general, 39 formulas, 23 pressure, annual variations, 338 transformations, 38 V. Bjerknes' paper on Dynamic Meteorology, 133 Bolometer, absorbed energy of, 393 energy spectrum, 278, 283 instruments, 277 observed ordinates, 284 Bouguer formula, application in atmosphere, 270 indeterminate development, 267 remarks, 364 423 424 INDEX Boyle-Gay Lussac law, P =P R T, 1, 28 Boyles' law, Mariotte, 28 Calories, large and small units, 15 Carnot's function, law, 71 cyclic process, 72, 73 examples, 77 Circulation, direction, and velocity on the earth, 190 general in the atmosphere, 189 transformation of energy in Case I, change of position of two layers, 180 Case II, adiabatic and non-adiabatic mixture, 184 Case III, overturn of strata, 185 Case IV, masses of different temperatures on the same level, 187 Case V, discontinuous temperatures, 188 work of, 153 Clausius' law of kinetic energy, 28 Coefficient of Conductivity, discrepancy of results, 304 examples, 305 formulas, 295 (c), in formulas of the planetary circulation, 126 Components, analytic construction, 229 general and local, 220 graphic construction, 232 Concentration, in mixtures, 30 Consecutive means, 338 Constants, of dynamic meteorology, 9 of static meteorology, 9 of thermodynamic meteorology, 10 three systems, 13 (M. K. S.) and (C. G. S.), 406, 408 Continuity, equation of in rectangular, cylindrical, and polar coordinates, 145 Convection, diurnal in lower strata, 98 semi-diurnal waves, 99 Co-ordinate accelerations, 136 cylindrical on the rotating earth, 148 forces, 137 polar on the rotating earth, 149 velocities, 136 Co-ordinates, rectangular, cylindrical, polar, 135 systems of, 8 Coronas, solar and terrestrial, 419 Current function, in angular velocity, 171 in the total pressure, 171 Cyclone, example of the evaluation of the several terms, 254 INDEX 425 Dal ton's law, of partial pressures, 28 Data, Thermodynamic, 389, 393 Deflecting force of the earth's rotation, 178 Density, in mixtures, 20 in the isothermal region, 96 in the planetary circulation, 118 of the atmosphere, 13 use of erroneous, 372 variations of, 17 Differential, total of -j- f 146 Dimensions, Conversion Factors, 376 Summary of equivalent, 414 Diurnal convection, five types in the atmosphere, 312 examples, 313 magnetic vectors, compared with convection vectors, 328 variations, electric dissipation, 321 magnetic field, 323 meteorological elements, 318 Dumb-bell-shaped vortex, equations of, 177 formulas collected, 250 tangential angle i, 251 Dyne, unit of force, 14 Eastward drift, 227 Ebert ion counter, formulas, 299 example, 299, 301 velocity formulas, 300 Electric potential of the atmosphere, formulas, 309 examples, 310 Elster and Geitel dissipation apparatus, formulas, 297 Entropy, (S\ So), in mixtures, 30 in the isothermal region, 93 in the planetary circulation, 124 Ephemeris, solar magnetic, 334 Equations, general, 6 of motion, cylindrical, 147 in cylindrical vortices, 168 in Ferrel's local cyclone, 164 in the German local cyclone, 166 in thermal energy, 151 First form, 147 Second form, 149, 150 Third form, 152 Fourth form, 161 Erg, unit of work, 14 426 INDEX Evaluation, of -jg, 159 of f in terms of temperature, 154 for standard density, 157 for the non-adiabatic atmosphere, 156 through the Boyle-Gay Lussac Law, 156 183 Evaporation of water, laws, 343 examples, 344, 345 factors for pan effect, 347 formulas, 346 from soil and plants, 348 Expansion heat, coefficient of, 69 Exponent (a), in the planetary circulation, 126 F Forces, of expansion and contraction, 137 of inertia, 137 of rotation, 139 Formula (172), differentiation of, 58 Free energy, of thermodynamic potential at constant volume and pressure, 67 heat, (Qi Qo), calories as unit of, 15 applications of formulas, 80 evaluation in balloon ascensions, 61 Friction, force of, 179 Funnel-shaped vortex, equations of, 175 G Gas coefficient R, as a constant, 1 as a variable, 16 in a mixture, 29 in the planetary circulation, 118 Gerdien apparatus, for number and velocity of ions; formulas, 302 examples, 303 Gold's paper, on radiation and absorption, 134 Gradient ratio n, 5 variable values of, 57 Gradients, adiabatic and non-adiabatic, 4 Gravitation, force of, in altitude, 19 in latitude, 18 in metric and English measures, 20 Gravity potential, 67 H Heat, evaluation through the entropy, 69 in anticyclones, 109 in cyclones, 108 INDEX 427 Heat energy, application of formulas, 77 formulas, 66 in the isothermal region, 92-96 inner energy of, 67, 69 losses in the convection region, 103 mechanical equivalent of, 15 Historical review of the theories of cyclones and anticyclones: Bigelow's asymmetric cyclonic system, 218 Ferrel's cold-center anticyclone, 216 Ferrel's warm-center cyclone, 216 Hann's dynamic cyclone and anticyclone, 217 Horizontal magnetic force, semiannual variations, 333-335 I Inner energy, applications of formulas, 77 bound energy, 67 formulas, 66 in mixtures, 29 in the isothermal region, 93-95 in the planetary circulation, 124 kinetic, 26 potential, 26 total heat and work, 26 Integral mean, of temperature, 36 '1L37 T of pressures, 22 lonization, in atmosphere, 292 conduction currents, 296 electrostatic relations, 295 notation and formulas, 293 Isobars, observed and local components, 224 Isotherms, normal and local, 235 observed and local, 239 Isothermal region, in Europe, 91, 94 in the Tropics, 95 K Kinetic energy, of motion, 67 in planetary circulation, 120 theory of gases, for the atmosphere, 30 constants and formulas, 31 Huron, Sept. 1, 1910, 410 Uccle, Nov. 9, 1911, 412 L Land cyclone, observed temperatures, 255 observed velocities, 255 428 INDEX Latent heats, definitions, 69 examples, 78-80 second form of the equations, 75 Leakage currents, Bigelow's, 219 Lindenburg observations in balloon ascensions, 54-64 Local circulations, general equations applied, 163 M Magnetic aperiodic vectors along the meridians, 329 semiannual inversions, 334 field, coordinates and forces, 360 horizontal force, observed variations, 330-331 sphere, theory, 363 Margules' paper, on the theory of storms, 133 Mass, in mixtures, 29 Meteorological elements, annual variations in Argentina, 338 in the United States, 338 Meteorology, constants of the static, dynamic and thermodynamic, 9-10 status of, 1 Mixture of gases, formulas, 23-29 N Non-adiabatic atmosphere, application of formulas, 77 general formulas, 50 working equations, 53 O Operator V 2 , in rectangular, cylindrical, polar co-ordinates, 146 P Partial formations, in the annual curves, 341 Physics, fundamental laws of, 28 Polarization, of sunlight, description, 348 examples, 350 percentage observed, 351 Potential gradient, definition, 144 Precipitation, annual variations, 337 Pressure, compared with temperature observations, 103 dynamic and barometric units, 112 gradients, rectangular, cylindrical, polar, 143 evaluations, 157 in anticyclones and cyclones, 107, 108 in isothermal region, 92-95 in mixtures of gases, 28 semi-diurnal waves, 100 three types of units, P, p, B, 2-3 variations of, 17 INDEX 429 Problems, general, in physics, 417 Processes, adiabatic, 68 Carnot's cyclic, 72 cycle for vapors, 73 irreversible and reversible, 70 isenergetic, 68 isodynamic, 66, 68 isoelastic, 66, 69 isopiestic, 66, 69 Prominences on the sun, annual variations, 337 Pyrheliometer, instrument, 262 annual mean variations, 402 example, 264 first method of discussion, 364 reduction to sea level, 399 second method of discussion, 395 theory, 263 Radiation, calories of, 288 co-efficients and exponents, 82-90 constituents, 375 depletion from the cirrus maximum, 274 determination of the intensity, 265 effective, 388 energy, in the planetary circulation, 126 formula, -*, 290 mean observed values, 291 function, K = c T, 1 formulas, 82 intensity at different elevations, 271 intensity at the earth, 280 in the isothermal region, 86 in the planetary circulation, 124, 129 maximum value, 274 mean annual values, 272 relative absorption by aqueous vapor, 275-276 selective and black, 388 solar and terrestrial, 375 total, transmitted, absorbed, 377 Relative humidity, observed in the atmosphere, 33 Remarks, general, 404 Residuals, 339 Rocky Mountain Plateau, temperature gradients, 34 Rotating earth, equations of motion, cylindrical, 148 equations of motion, polar, 149 430 INDEX Solar constant, derived from the bolometer, 291 derived from the pyrheliometer, 276 derived from thermodynamics, 292 physics, magnetic data at the earth, 353 siderial and synodic periods, 354 thermodynamic equations, 352 velocities of motions in sun-spots, 352 velocity of rotation in latitude, 352 Specific heat, at constant pressure, 1 examples, 77-79 in isothermal region, 93-95 in terms of latent heat, 77 variations, 38 variations in monatomic gases, 27 Spherical astronomy, application to the sun, 356 formulas, 357 positions on the sun's equator, 357 Stefan's radiation formula, 279 Stoke's current function, Eq. (494), (495), for funnel-shaped vortex, 172 for dumb-bell-shaped vortex, 172 Strata, interflow of warm and cold, 248 Summary, 394 general, 403 of dimensions, 408 Sun-spots, annual variations, 337 Synchronism, solar and terrestrial phenomena, 338 Synchronous variations of the elements, 335 Tangential angle i, in circulations, 251 Temperature, annual variations, 338 first distribution, 365 gradients, observed in the atmosphere, 32 computed for the plateau, 34 in cyclones and anticyclones, 105, 237 in semi-diurnal waves, 100 in the isothermal region, 92-95, 115 in the planetary circulation, 118 of the sun, probable values, 281-285, 403 second distribution, 380 standard, 3 variations of, 17, 37 virtual, 37 Theory, change of, 394 Therm, unit of heat, 16 Thermodynamics, first law, 65 INDEX 431 Thermodynamics, second law, 70 terms, 374 Transmission of energy, in the isothermal and convectional region, 287-292 U Units, equivalent values, 1 of heat and work, 14-1G systems of: (M.K.S.), (C.G.S.), (F.P.S.), 2, 408-409 transformations of, 12-14 Variations of P, p, R, T, 17 Vector, definitions of velocity and direction of motion, 7 Velocities, in cyclones and anticyclones, 106-109 in different latitudes (velocity function), 123 in the planetary circulation, 120 linear absolute and relative, 161 normal and local in cyclones and anticyclones, 240 normal and local in storms, 221 observed in different levels, 226 Viriol, of inner energy, 26 Vortex formulas, dumb-bell-shaped tube, 172 funnel-shaped tube, 172 general formulas, 174 tubes, successive, 173 Vortices, composition of, 212 connection between the funnel and dumb-bell, 252 cylindrical equations, 168, 175 evaluation of equations, 256-259 examples of the structure in the atmosphere, 196 examples of (1) Cottage City waterspout, 199 examples of (2) St. Louis tornado, 202 examples of (3) DeWitte typhoon, 205 examples of (4) Ocean cyclone, 209 general equations, 213 reversed dumb-bell, 213 W Westward drift, 224-229 Wien displacement formula, 280 Wien-Planck formula, of the spectrum energy, 278 evaluations, 281 Work, energy, formulas, 66 applications, 81 erg as unit of, 15 heat equivalent of, 16 in the isothermal region, 93-95 in the planetary circulation, 124 total work done, 27 THIS BOOK IS DUE ON THE LAST DATE STAMPED BELOW AN INITIAL FINE OF 25 CENTS WILL BE ASSESSED FOR FAILURE TO RETURN THIS BOOK ON THE DATE DUE. 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