WEATHER PREDICTION , BY NUMERICAL PROCESS CAMBRIDGE UNIVERSITY PRESS C. F. CLAY, MANAGER LONDON : FETTER LANE, E.G. 4 LONDON : H. K. LEWIS AND CO., LTD., 136, Gower Street, W.C. i LONDON : WHELDON & WESLEY, LTD.. 28, Essex Street, Strand, W.C. 2 NEW YORK : THE MACMILLAN CO. BOMBAY -I CALCUTTA I MACMILLAN AND CO., LTD. MADRAS J TORONTO : THE MACMILLAN CO. OF CANADA, LTD. TOKYO : MARUZEN-KABUSHIKI-KAISHA ALL RIGHTS RESERVED 048' 4S"E 3 37 30" 6 26' /5 Longitude from Greenwich ' An -arrangement of meteorological stations designed to fit with the chief mechanical properties of the atiup?pfier4. ."Other considerations have been here disregarded. Pressure to be observed at the centre of 'each 'sh'aded" chequer, velocity at the centre of each white chequer. The numerical coordinates refer to these centres as also do the names, although as to the latter there may be errors of 5 or 10 km. The word "with" in "St Leonards with Dieppe" etc. is intended to suggest an interpolation between observations made at the two places. See page 9, and Chapters 3 and 7. Contrast the existing arrange- ment shown on p. 184. WEATHER PREDICTION BY NUMERICAL PROCESS BY LEWIS F. RICHARDSON, B.A., F.R.MET.Soc., F.INST.P. FORMERLY SUPERINTENDENT OF ESKDALEMUIR OBSERVATORY LECTURER ON PHYSICS AT WESTMINSTER TRAINING COLLEGE CAMBRIDGE AT THE UNIVERSITY PRESS 1922 PREFACE THE process of forecasting, which has been carried on in London for many years, may be typified by one of its latest developments, namely Col. E. Gold's Index of Weather Maps*. It would be difficult to imagine anything more immediately practical. The observing stations telegraph the elements of present weather. At the head office these particulars are set in their places upon a large-scale map. The index then enables the forecaster to find a number of previous maps which resemble the present one. The forecast is based on the supposition that what the atmosphere did then, it will do again now. There is no troublesome calculation, with its possibilities of theoretical or arithmetical error. The past history of the atmosphere is used, so to speak, as a full-scale working model of its present self. But one may reflect the Nautical Almanac, that marvel of accurate forecasting, is not based on the principle that astronomical history repeats itself in the aggregate. It would be safe to say that a particular disposition of stars, planets and satellites never occurs twice. Why then should we expect a present weather map to be exactly represented in a catalogue of past weather ? Obviously the approximate repetition does not hold good for many days at a time, for at present three days ahead is about the limit for forecasts in the British Isles. This alone is sufficient reason for presenting, in this book, a scheme of weather prediction, which resembles the process by which the Nautical Almanac is produced, in so far as it is founded upon the differential equations, and not upon the partial recurrence of phenomena in their ensemble. The scheme is complicated because the atmosphere is complicated. But it has been reduced to a set of computing forms. These are ready t to assist anyone who wishes to make partial experimental forecasts from such incomplete observational data as are now available. In such a way it is thought that our knowledge of meteorology might be tested and widened, and concurrently the set of forms might be revised and simplified. Perhaps some day in the dim future it will be possible to advance the computations faster than the weather advances and at a cost less than the saving to mankind due to the information gained. But that is a dream. The present distribution of meteorological stations on the map has been governed by various considerations : the stations have been outgrowths of existing astronomical or magnetic observatories ; they have adjoined the residence of some independent enthusiast, or of the only skilled observer available in the district ; they have been set out upon the confines of the British Isles so as to include between them as much weather as possible ; or they have been connected with aerodromes in order to * Meteor. Office Geophysical Memoir, No. 16, deals mainly with types of pressure distribution but fore- shadows a more general indexing. t Printed blank forms may be obtained from the Cambridge University Press, Fetter Lane, E.G. 4. 498245 viii PREFACE exchange information with airmen. On the map the dots representing the positions of the stations look as if they had fallen from a pepperpot. The nature of the atmosphere, as summarized in its chief differential equations, appears to have been without influence upon the distribution. We shall examine in Ch. VII what would happen if these differential equations were the sole consideration. The result is represented in the frontispiece. The extensive researches of V. Bjerknes and his School are pervaded by the idea of using the differential equations for all that they are worth. I read his volumes on Statics and Kinematics* soon after beginning the present study, and they have exercised a considerable influence throughout it ; especially, for example, in the adoption of conventional strata, in the preference for momentum-per-volume rather than of velocity, in the statical treatment of the vertical column, and in the forced vertical motion at the ground. But whereas Prof. Bjerknes mostly employs graphs, I have thought it better to proceed by way of numerical tables. The reason for this is that a previous comparison f of the two methods, in dealing with differential equations, had convinced me that the arithmetical procedure is the more exact and the more powerful in coping with otherwise awkward equations. Graphical methods are sometimes elegant when the problem involves irregularly curved boundaries. But the atmospheric boundary, at the earth, nearly coincides with one of the coordinate surfaces, so that graphs would have no advantage over arithmetic in that respect. It has been customary to regard line-squalls and other marked discontinuities as curious exceptions to the otherwise smoothly gradated distribution of the atmosphere. But in the last two years Prof. V. Bjerknes and his collaborators J. Bjerknes, H. Solberg and T. Bergeron at Bergen have enunciated the view, based on detailed observation, that discontinuities are the vital organs supplying the energy to cyclones J. The question then arises : how are we to deal with discontinuities by finite differences 1 For such purposes graphs have a special facility which numerical tables lack. But it is not to be expected that a knowledge of the position and motion of surfaces of discontinuity will prove to be sufficient for forecasting, any more than " vital " organs alone would suffice to keep an animal alive. So probably the most thorough treatment will be reached by tabulating quantities numerically, where they vary continuously, and by drawing a line on the table where there is a discontinuity. The line will be a notification to the computer that one may interpolate up to it from either side, but not across it. This investigation grew out of a study of finite differences and first took shape in 1911 as the fantasy which is now relegated to Ch. 11/2. Serious attention to the problem was begun in 1913 at Eskdalemuir Observatory with the permission and encouragement of Sir Napier Shaw, then Director of the Meteorological Office, to whom I am greatly indebted for facilities, information and ideas. I wish to thank * Carnegie Institution, Washington, 1910, 1911. t L. F. Richardson, Phil. Mag. Feb. 1908; Proc. Roy. Soc. Dublin, May 1908; Phil. Trans. A, Vol. 210, p. 307 (1910); Proc. Phys. Soc. London, Feb. 1911. | Q. J. R. Met. Soc. 1920 April, and Nature, 1920, June 24. PREFACE ix Mr W. H. Dines, F.R.S., for his interest in some early arithmetical experiments, and Dr A. Crichton Mitchell, F.R.S.E., for some criticisms of the first draft. The arith- metical reduction of the balloon, and other observations, was done with much help from my wife. In May 1916 the manuscript was communicated by Sir Napier Shaw to the Royal Society, which generously voted 100 towards the cost of its publica- tion. The manuscript was revised and the detailed example of Ch. IX was worked out in France in the intervals of transporting wounded in 1916 1918. During the battle of Champagne in April 1917 the working copy was sent to the rear, where it became lost, to be re-discovered some months later under a heap of coal. In 1919, as printing was delayed by the legacy of the war, various excrescences were removed for separate publication, and an introductory example was added. This was done at Benson, where I had again the good fortune to be able to discuss the hypotheses with Mr W. H. Dines. The whole work has been thoroughly revised in 1920, 1921. As the cost of printing had by this time much increased, an application was made to Dr G. C. Simpson, F.R.S., for a further grant in aid, and the sum of fifty pounds was provided by the Meteorological Office. For the construction of the index we are indebted to Mr M. A. Giblett, M.Sc. The discernment and accuracy with which the Cambridge Press have set the type have been constant sources of satisfaction. L. F. R. LONDOK .1921 Oct. 10 GUIDING SIGNS For finding one's way about this book it is most helpful to realise that : (i) A complete list of symbols and notation is given in Ch. XII at the end of the book. (ii) An impression such as (Ch. 9/12/3) is intended to refer the reader to the 3rd subdivision of the 12th division of Chapter IX. (iii) The mark # is used to refer to equations, expressions, or statements which have had numbers assigned to them in the right-hand margin. Thus (Ch. 9/12/3 # 16) means the equation, expression or statement numbered 16 in the aforesaid subdivision. The mark # is often omitted where the meaning is plain without it. (iv) The physical units are those of the centimetre-gram-second system, unless a different unit is expressly mentioned. Temperatures are in degrees centigrade absolute. Energy, whether by itself or as involved in entropy or specific heats, is expressed in ergs not in calories. CONTENTS PAGE PREFACE vii GUIDING SIGNS x LIST OF CONTENTS 7~ ~ xi CHAP. I. SUMMARY 1 II. INTRODUCTORY EXAMPLE 4 III. THE CHOICE OF COORDINATE DIFFERENCES .... 16 3/1. Existing practice 16 3/2. The division into horizontal layers . 16 3/3. Effect of varying the size of the finite differences .... 18 3/4. The pattern on the map 18 3/5. Devices for maintaining a nearly square chequer . . . . 19 3/6. Summary on coordinate differences 19 3/7. The origin of longitude 20 IV. THE FUNDAMENTAL EQUATIONS 21 4/0. General 21 4/1. Characteristic equations of dry and moist air 23 4/2. The indestructibility of mass 23 4/3. Conveyance of water 25 4/4. Dynamical equations 30 4/5. Adiabatic transformation of energy. Entropy (See also Ch. 8/2/6) . 35 4/6. Uniform clouds and precipitation 43 4/7. Radiation 43 4/8. The effects of eddy motion (See also Ch. 11/4) 65 4/9. Heterogeneity 94 4/10. Beneath the earth's surface . 104 V. FINDING THE VERTICAL VELOCITY 115 5/0. Preliminary 115 5/i. Deduction of a general equation 116 5/2. Simplification by approximation 118 5/3. Method of solving the equation 119 5/4. Illustrative special cases 119 5/5. Further varieties of the simplified general equation .... 123 5/6. The influence of eddies 124 VI. SPECIAL TREATMENT FOR THE STRATOSPHERE . . .125 6/0. Introduction 125 6/1. Integrals of pressure and density 126 6/2. The continuity of mass in the stratosphere . . . 127 6/3. Extrapolating observations of wind 127 6/4. The horizontal dynamical equations in the stratosphere . . . 132 6/5. Radiation in the stratosphere 134 6/6. Vertical velocity in the stratosphere 135 6/7. Dynamical changes of temperature in the stratosphere . . . 140 6/8. Summary 147 CONTENTS CHAP. p AOE VII. THE ARRANGEMENT OF POINTS AND INSTANTS . . .149 7/0. General 149 7/1. The simplest arrangement of points 149 7/2. The arrangement of instants 150 7/3. Statistical boundaries to uninhabited regions 153 7/4. Joints in the lattice at the borders of sparsely inhabited regions . 153 7/5. The polar caps 155 VIII. REVIEW OF OPERATIONS IN SEQUENCE 156 8/0. General 156 8/1. Initial data 157 8/2. Operations centered iu columns marked "1"' on the map . . . 157 8/3. Operations centered in columns marked "M" on the chessboard map 179 8/4. Concluding remarks .180 IX. AN EXAMPLE WORKED ON COMPUTING FORMS . . .181 9/0. Introduction 181 9/1. Initial distribution observed at 1910 May 20 D 7 H G.M.T. . . 181 9/2. Deductions, made from the observed initial distribution, and set out on the computing forms 186 9/3. The convergence of wind in the preceding example .... 212 X. SMOOTHING THE INITIAL DATA 214 XI. SOME REMAINING PROBLEMS 217 n/o. Introduction 217 n/i. The problem of obtaining initial observations 217 1 1/2. Speed and organization of computing 219 11/3. Analytical transformation of the equations 220 11/4. Horizontal diffusion by large eddies 220 11/5. A survey of reflectivity ' . 222 XII. UNITS AND NOTATION 223 12/1. Units 223 12/2. List of symbols 223 12/3. Relationships between certain symbols ...... 228 12/4. Subscripts for height 228 12/5. Vector notation 229 INDEX OF PERSONS .' 230 INDEX OF SUBSIDIARY SUBJECTS . 231 ERRATA p. 5, 1. 14. For longitude read latitude p. 27, equation (8). In the coefficient of tan , for M H read p. 44, 1. 14. For C. M. K. Douglas read C. K. M. Douglas p. 57, 1. 25. For M. A. Boutaric read A. Boutaric p. 64, 1. 30. For M. A. Boutaric read A. Boutaric p. 65, 1. 8 from bottom. For conventional read convectional p. 87, 1. 14. For Rogers read Rogers' p. 105, 1. 10. For 1-92 read 1-72 p. 107, 1. 7. For Rothamstead read Rothamsted p. 110, 1. 9 and 1. 4 from foot. For Calendar read Callendar p. 126, footnote. For M. G. Gouy read G. Gouy CHAPTER I SUMMARY FINITE arithmetical differences have proved remarkably successful in dealing with differential equations ; for instance, approximate particular solutions of the equation for the diffusion of heat crO/dx" = dd/dt can be obtained quite simply and without any need to bring in Fourier analysis. An example is worked out in a paper published in Phil. Trans. A, Vol. 210*. In this book it is shown that similar methods can be extended to the very complicated system of differential equations, which expresses the changes in the weather. The fundamental idea is that atmospheric pressures, velocities, etc. should be expressed as numbers, and should be tabulated at certain latitudes, longitudes and heights, so as to give a general account of the state of the atmosphere at any instant, over an extended region, up to a height of say 20 kilometres. The numbers in this table are supposed to be given, at a certain initial instant, by means of observations. It is shown that there is an arithmetical method of operating upon these tabulated numbers, so as to obtain a new table representing approximately the subsequent state _oLhe atmosDhere after a brief interval of time, 8t_ say^_ The process can be repeated ADDITIONAL EERATA p. 9, equation 8. For cot read - . p. 42, equation 13. The value of a M is wrong. Please refer to pp. 158 to 162, especially to p. 161. p. 77, equation 22. For - ~ read + "^~ . I" t~ p. 77, footnote. After "a wrong sign" insert the words "before the second term of (5-2) and (5'3) and." p. 89. Please refer to Ch. 8/2/15 beginning on p. 171. p. 136, equation 9. The last form should read 8 / 1 \ _ 1 cot ad \sin ' p. 130, equation 8. Add to the second member the term tan c g d log 6 { bQ\ "A - ; - . I ft il.t I , ~ "A - ; - . i.t , a 2u> sin de \ g ) p. 137, equation 10. Change the sign of the last form of the second member. The two foregoing corrections in Ch. 6/6^8 and 10 imply that the expression ' ' wherever Oi it occurs in Ch. 6 should be replaced by . This change is necessary in Ch. 6/6 #11, 17, 18, 22, 23 ; in Ch. 6/7/2 $ ISA, 18 and again on the computing form Pxiv on page 201. p. 138, equation 22. Change the sign in front of . --^ from plus to mnus. L. F. Kichardson. Weather Prediction by Numerical Process. xii CONTENTS CHAP. PAGE VII. THE ARRANGEMENT OF POINTS AND INSTANTS . . .149 7/0. General 149 7/1. The simplest arrangement of points 149 7/2. The arrangement of instants 150 7/3. Statistical boundaries to uninhabited regions 153 7/4. Joints in the lattice at the borders of sparsely inhabited regions . 153 7/5. The polar caps 155 VIII. REVIEW OF OPERATIONS IN SEQUENCE 156 8/0. General 156 8/1. Initial data 157 8/2. Operations centered in columns marked "/"' on the map . . . 157 8/3. Operations centered in columns marked "J/" on the chessboard map 179 8/4. Concluding remarks 180 IX. AN EXAMPLE WORKED ON COMPUTING FORMS . . .181 9/0. Introduction 181 9/1. Initial distribution observed at 1910 May 20 D 7 H G.M.T. . . 181 9/2. Deductions, made from the observed initial distribution, and set out on the computing forms 186 9/3. The convergence of wind in the preceding example . . . .212 X. SMOOTHING THE INITIAL DATA 214 XI. SOME REMAINING PROBLEMS .217 n/o. Introduction 217 n/i. The problem of obtaining initial observations 217 1 1/2. Speed and organization of computing 219 11/3. Analytical transformation of the equations 220 CHAPTER I SUMMARY FINITE arithmetical differences have proved remarkably successful in dealing with differential equations ; for instance, approximate particular solutions of the equation for the diffusion of heat d 2 0/dx~ = d6/8t can be obtained quite simply and without any need to bring in Fourier analysis. An example is worked out in a paper published in Phil. Trans. A, Vol. 210*. In this book it is shown that similar methods can be extended to the very complicated system of differential equations, which expresses the changes in the weather. The fundamental idea is that atmospheric pressures, velocities, etc. should be expressed as numbers, and should be tabulated at certain latitudes, longitudes and heights, so as to give a general account of the state of the atmosphere at any instant, over an extended region, up to a height of say 20 kilometres. The numbers in this table are supposed to be given, at a certain initial instant, by means of observations. It is shown that there is an arithmetical method of operating upon these tabulated numbers, so as to obtain a new table representing approximately the subsequent state of the atmosphere after a brief interval of time, St say. The process can be repeated so as to yield the state of the atmosphere after 28t, 3$t and so on. There is a limit however to the possible number of repetitions, because each table is found to be smaller than its predecessor, in longitude and latitude, having lost a strip round its edge. Only if the table included the whole globe could the repetitions be endless. Also the errors increase with the number of steps. In Ch. 2 the working of the method is shown by its application to a specially simplified case. In Ch. 3 the coordinate differences are considered in relation to the average size of European cyclones, and the following differences are provisionally selected : in time 6 hours, in longitude the distances between 128 equally spaced meridians, in latitude 200 kilometres of the earth's circumference, and in height the intervals between fixed heights nearly corresponding to the normal pressures of 8, 6, 4, 2 decibars. Thus small-scale phenomena, such as local thunderstorms, have to be smoothed out. In Ch. 4 the fundamental equations are collected from various sources, set in order and completed where necessary. Those for the atmosphere are then integrated with respect to height so as to make them apply to the mean values of the pressure, density, velocity, etc., in the several conventional strata. Incidentally certain con- stants relating to friction and to radiation are collected from observational data. It is found to be necessary to eliminate the vertical velocity from all the equations, and in Ch. 5 it is shown how this can be done. Special difficulties arise in connection with the uppermost stratum on account of its great thickness and the enormous ratio of density * pp. 312, 313. K. 1 -' 2 : : . . I /"-. - : SUMMARY OH. i between its upper and lower surfaces. These difficulties are removed in Ch. 6, as far as high latitudes are concerned. In particular it is shown how the total mass transport above any level may be deduced from a pilot balloon observation which extends well into the stratosphere. In Ch. 7 the arrangement of the tabular numbers in space and time is discussed with a view to securing the best representation of differential coefficients by difference ratios. In Ch. 8 the whole system of arithmetical operations is reviewed in order. With regard to the horizontal differential coefficients the general method may be briefly described in the following four sentences : Take the differential equations and replace everywhere the infinitesimal operator 9 by the finite difference operator ft. Use arith- metic instead of symbols. Attend carefully to the centering of the differences. Leave the errors due to the finiteness of the differences over for consideration at the end of the process. With regard to the vertical differential coefficients, on the contrary, it is often possible to effect an exact transformation to differences, by means of a vertical integration. In arranging the computing, it has constantly to be borne in mind that the rate of change with time of every one of the discrete values of the dependent variables must be calculable from their instantaneous distribution in time and space, excepting only those values near the edge of the horizontal area represented in the table. We may refer to this necessary property by saying, for brevity, that the system must be " lattice-reproducing." In Ch. 9 will be found an arithmetical table showing the state of the atmosphere observed over middle Europe at 1910 May 20 d. 7 h. G.M.T. This region and instant were chosen because the observations form the most complete set known to me at the time of writing, and also because V. Bjerknes has published large scale charts of the isobaric surfaces, together with collated data for wind, cloud and precipitation. Starting from the table of the initially observed state of the atmosphere at this instant, the method described in the preceding paragraphs is applied, and so the rates of change of the pressures, winds, temperatures, etc. are obtained. Unfortunately this " forecast " is spoilt by errors in the initial data for winds. These errors appear to arise mainly from the irregular distribution of pilot balloon stations, and from their too small number. In Ch. 10 the smoothing of initial observations is discussed. In Ch. 1 1 is a collection of problems still waiting to be solved, with some suggestions for their treatment. Ch. 12 is a list of Notation. Pressures fictitiously " reduced to sea-level " are not used in the present method. Instead, the varying height of the land is dealt with by the variation of the lower limit of an integral with respect to height. See Ch. 4/2, Ch. 4/4. The problem of weather prediction is of the "inarching" variety. To explain this statement it should be pointed out that the ease or difficulty with which physical problems involving differential equations can be solved, depends on very different things according to whether symbolic methods or arithmetical differences are to be employed. In the former case the main facts in the situation are the "order" CH. i PROPERTIES OF FINITE DIFFERENCES 3 and " degree " of the equations and whether they are ordinary or partial. In the latter case what usually matters most is the relation of the " body equation " to the boundary conditions. By " body equation " is here meant the differential equation which holds throughout the region of space and the interval of time with which we have to deal. The " boundary " must be understood to be the limits of either this time or this space. According to the relation between the body equation and the boundary conditions, problems are divided into : (i) "Jury" problems in which the integral must be determined with reference to the boundary as a whole : for instance the problem of a stone thrown from a given point to hit another given point ; or that of the stresses inside a loaded dam. Cases like these frequently require troublesome successive approximations, before a statement is obtained with which the "jurymen," seated round the boundary, will all agree. (ii) "Marching" problems in which the integral can be stepped out from a part of the boundary : for instance the problem of a stone thrown with given initial vector-velocity, or that of the cooling of a body with given initial and superficial temperatures. Other things being equal, these problems are much more easily solved than those in division (i) above. Weather prediction falls into the " marching " category. Whilst dealing with the general subject of finite differences it may be well to mention two important properties brought to notice by Mr W. F. Sheppard. (a) The great gain in accuracy, in the representation of a differential coefficient, when the differences are centered instead of progressive; a gain secured by a slight increase of work. (b) That the errors due to centered differences, when small enough, are proportional to the square of the coordinate difference. This fact provides a universal means of checking and correcting the errors. For further information about centered differences the reader is referred to "Central- Difference Formulae," by W. F. Sheppard, Proc. Lond. Math. Soc. Vol. xxxi. (1899) and to "The Approximate Arithmetical Solution by Finite Differences of Physical Problems," by L. F. Richardson, Phil. Trans. A, Vol. 210, p. 307 (1910). 12 CHAPTER II INTRODUCTORY EXAMPLE BEFORE attending to the complexities of the actual atmosphere and their treatment by this numerical method, it may be well to exhibit the working of a much simplified case. Lest the reader, catching sight of numbers of 7 digits, should suppose that these are necessary, let me at once point out that they have been introduced in order to measure the errors due to finite differences, which in this example are very small. An intelligible picture of the sequence of phenomena would remain after the last 4 places of digits had been cut off everywhere. Suppose now that there is no precipitation, clouds or water vapour, neither solar nor terrestrial radiation, no eddies, and no mountains or land, but an atmosphere in which we can ignore or summarize variations with height moving upon a globe covered by sea. Further to simplify the problem, let us neglect all the quadratic terms in the dynamical equations. Then, in order to summarize the vertical velocity and the density, let us perform an integration with respect to height upon the horizontal dynamical equations and upon the equation of continuity of mass. If the limits of integration are sea-level, and a height so great that the density there is negligible, we thus arrive at a set of equations similar to those used by Laplace in his discussion of Tides on a Rotating Globe (vide Lamb, Hydrodynamics, 4th ed. 214): (') a l } (3) dn V "'" JV ~"" r/ ( (^ a ) Here M E , M N are the components of the whole momentum of the column of atmo- sphere standing upon a horizontal square centimetre at sea-level, p a is the pressure at sea-level, g is gravity, t is time, is latitude, a is the radius of the earth, cu its angular velocity, de and dn are distances to east and to north, and H' is an empirical height, used f h=x dp . dp G to convert f- dh into H' ~ . The data given by Mr W. H. Dines for the differ- J A=O e v & ence of pressure between cyclones and anticyclones up to 14 km, and at 20 km, when combined with the extrapolation into the stratosphere according to the method to be described in Ch. 6 below, indicate that H' = . To avoid an infinite geostrophic wind at the equator we can multiply by (sin <]>)-. So the selected form of the initial pressure distribution has been Ap G = sin X cos (j> (sin )' 2 x 10 5 dynes cm'", (4) where Ap G signifies the deviation from the general mean. The isobars are shown in Fig. 1 on the map of the globe. The equator and the meridian of Greenwich are isobars. There is high pressure over Asia, and over the Fig. 1. Isopleths of sin I cos (sin ) 2 corresponding to the isobars of the initial distribution studied in Chapter II. south Indian Ocean. Low pressure is at the antipodes of each high. The maxima and minima of pressure occur in latitude 547, that is 6'08 x 10 8 cm from the equator, and amount to 38*5 millibars above or below the mean. A small portion of this distribution of pressure near England is entered numerically on the table. To find the geostrophic mornenta-per-area, we next insert the above value of Ap G C H . 2 ARRANGEMENT OF COMPUTING 7 in equations (l) and (2), at the same time putting dMj;/dt = 0; dM N /dt = 0. Then it follows that initially TT> M E = - -sinX{i + f cos2<}xl0 5 grmsec" I cm" 1 , ............... (5) H' M N = + j. -cos Xsin (j> x 10" grm sec' 1 cm' 1 ......................... (6) These initial momenta-per-area, expressed in numbers, ai-e entered at the appropriate points of the numerical table. Next, inserting the numbers from the table into equations (1), (2), (3), and treating difference-ratios as if they were differential coefficients, we get the rates of increase of M E , My, p G . These rates have been multiplied by a St equal to 2700 seconds, that is by f hour, in order to get the increases in that time. An example of the calculation of the increase of M' E will now be given at the point having longitude 38X, latitude 6'4 x 10 s cm from equator, which will be found in the top left-hand corner of the table of initial data : M y = 827578-6 grm cm" 1 sec' 1 , 2.St = 0-3324746, a pure number ; multiplying these together we get 2w sin . M N = -I- 275148-9 grm cm" 1 sec" 1 , Nearest Ap tf to east = 3744'11 grm cm" 1 sec" 2 Nearest Ap G to west= 745217 ,, difference +3708 '06 but -^-= +74-17322 seconds; oe multiplying, we get i ~U f -~l-^ = -275038-8 grm cm' 1 sec" 1 ; adding, we get from equation (1), g-:-8= +110-1 grm cm' 1 sec" 1 . This change of HO'l is entered on the initial table in parenthesis under the value of M E . Its value is, in this case, a measure of the error due to the finite difference, for we took Jifysuch as to make dM s /dt vanish, when the calculation was performed exactly by analysis. It is seen that the error is in this example quite small, being only 1/7000 of the resultant of M E and M N . That is why such large numbers of digits have been taken. The changes in M E at other points are all worked in the same way, but of course the coefficients vary with latitude. The computation of 8t . dM y /dt from equation (2) is so similar that it need not be illustrated. Next, as to the pressure changes : it happens conveniently that cos (f> . Sn is a fixed TO TO in o I 00 00 c o S TO * m CM I TO nq 01 00 OC oj 01 CM in TO 9 10 O rt <3i in CD . ~ SP ? t- 9 S 01 O OS 2 C O TO 00 m OS [^ 00 - 00 TO - TO i ^< 9P oo 10 i CM 9 cb ? OooScM TO 00 CO CM CM OS 69 850-52 3- Ol i < Ol i^ 1 t ^ to 4- t- '-' O J a .a A.+ c t 8528 II ^ sr ?> Oo O TO CO with c CO 1 to - 00 00 c 7-1 6S " * S CM CD t! * CM""? ot? Ai TO "2 CM TO 00 2 I-H CD TO""?" 0? 00 Q v _ f t s^ os'P O OP O "P >n CM x-v g CD r ^ CO C CM CO 00 CM X 'O I Q - 1 OC? 1 ." K" s o 10 I ft 1 s-r-j, 9 to CD 00 00 00 c TO oc 00 1~ 00 o CD 00 OS M 01 CO OS n OS TO m CM op in CM 00 TO CO in S S~-P cT g in 1 3 O I-H x n I O & TO CD in cs -f TO CD CO CM OC o TO O TO OS TO CD TO in OS 01 X CD CM 00 1 00 CM 3: CD CO r~ p do TO 00 O TO m CD s -t op CD TO CM CD 8 CH. 2 INADEQUACY OF GEOSTROPHIC WIND multiple of Se, so that by combining the multiplier with M N cos , the arithmetic is considerably shortened.' Equation (3 a) may then be written (7) where the suffixes in S t> S A ., S v indicate the variable which alone is varied. As an example of the computation of these pressure-changes we may take the following, which refers to the point at longitude 38\, latitude 6'2 x 10 8 cm north. Cm from equator 10 8 x6'4 10 8 x6'2 10 8 x6'0 792967-7 grm cm" 1 sec" 1 728273-0 M N Se Bn 827578-6 692873-1 ' i * *-J^ vylMifx- Se Sn Sum - 35399-9 grm cm" 1 sec" + 5019-8 -30380-1 Sum x g . St/Se = + 2285-89 = S ( p expressed in dynes cm" 2 per 2700 sees. This time-change is written in parenthesis under the initial pressure. The pressure- changes at all the points are worked in the same way, except that the coefficients vary with latitude. The advantage of the chessboard pattern is now seen to be that the time-rates are given at the points at which the variables are initially tabulated. By adding the changes in f hour we obtain a new table which has the same pattern as the initial one, so that it can in turn be taken as a starting-point. In this way the process can be continued with no limit; except that set by the loss of a strip, round the edge of the map, at every step. If we had begun with p G , M E , M N all three tabulated in every square, the distri- bution might have been regarded as two interpenetrating chessboard patterns. In the subsequent steps these interpenetrating systems would have been propagated quite independently of each other. Let us now compare with observation the result so far obtained. In this deduction the distribution distorts and moves west. Actual cyclones move eastward. It is natural to expect a slip of a sign in the process, but that expectation may be very simply disposed of. For, substitute the geostrophic momenta from (1) and (2) into (3), then there results Spa_ 9H' dt .(8) - . _ * wv u *p *. 2o> sin

. \JV3 \if _ _ . . -> I , a T \dtd(f) cos0 9X / where O = 2 sin ^> and V 2 stands for 1 / 9 2 .3 1 i No wonder that isobaric maps look complicated, if this be their differential equation, derived from most stringently restricted hypotheses. I am indebted to Mr W. H. Dines, F.R.S., for having read and criticised the manuscript of this chapter. CHAPTER III THE CHOICE OF COORDINATE DIFFERENCES THE choice would have to be guided by four considerations : (1) the scale of variation of atmospheric disturbances, (2) the errors due to replacing infinitesimals by finite differences, (3) the accuracy which is necessary in order to satisfy public require- ments, (4) the cost, which increases with the number of points in space and time that have to be dealt with. CH. 3/1. EXISTING PRACTICE In general the distance apart of telegraphic stations in the existing distribution is the safest guide to the required scale of proceedings. In a " Map showing the position of the Meteorological Stations, the observations from which are used in the preparation of the daily weather report Jan. June 1918," the mean distance between a station and its immediate neighbours appears to be about 130 kilometres, if we confine attention to the British Isles, to which the map principally relates. Or to put it another way: the number of stations marked on the British Isles is 32, and the area of the polygon formed on the map by stretching a string round the outermost stations is about 56 x (100 km) 2 . So that if the stations were, in imagination, re- arranged in rectangular order, there would be enough of them to put one at the centre of each square of 132 kilometres in the side. From the open sea there are indeed valuable wireless reports of observations on ships (see International Section of Daily Weather Report of Meteor. Office}. But, in comparison with reports coming from land stations, they are scarce and irregular, and refer only to surface conditions. With regard to time intervals, the existing practice of making observations for telegraphic purposes every 12 hours, or sometimes every 6 hours, is again our safest guide. CH. 3/2. THE DIVISION INTO HORIZONTAL LAYERS In making a conventional division of the atmosphere into horizontal layers the following considerations have to be borne in mind. It is desirable to have one con- ventional dividing surface at or near the natural boundary between the stratosphere and troposphere, at an average height of 10'5 km* over Europe. Secondly, that to represent the convergence of currents at the bottom of a cyclone and the divergence at the topf, the troposphere must be divided into at least two layers. Thirdly, that the lowest kilometre is distinguished from all the others by the disturbance due to the * E. Gold, Geophys. Mem. v. ; W. H. Dines, Geophys. Mem. n. t W. H. Dines, Q. J. R. Met. Soc. 38, pp. 4150. CH. 3/2 CONVENTIONAL STRATA 17 ground. Thus it appears desirable to divide the atmosphere into not less than 4 layers. If the layers are of equal or approximately equal mass the treatment of many parts of the subject is greatly simplified, e.g. radiation, atmospheric mixing, etc. To facilitate comparison with V. Bjerknes' charts and tables, I have chosen 5 strata divided at approximately 2, 4, 6, and 8 decibars. This being granted, there are various ways in which the divisions may be taken : (1) Divisions at the instantaneous pressures of 2, 4, 6, 8 decibars. This is Bjerknes' system, except that he takes 10 sheets, not 5. The heights of the isobaric surfaces become the dependent variables in place of the pressures. This system readily yields elegant approximations. But it entails the inconveniences of deformable coordinates, for it is equivalent to taking p as an independent variable in place of h. The corresponding alterations in the equations can be carried out by means of the following set of substitutions. Let A be any variable and let the suffixes denote the quantities which remain constant during the differentiations. Then when and t are constant, fdA There is a similar equation in 3< when X and t are constant. Also when X, < and t are constant, dA dA Also when X and (f> are constant, fdA\ __ idA\ /SA\ Idh The result of these substitutions is to produce a large number of terms. The additional terms are small, but they are not always negligible in comparison with the errors of observations. As observations improve they are likely to become more significant. On this account I have preferred to use instead the following system : (2) The divisions between the five conventional strata are taken at fixed heights above mean sea-level, so chosen as to correspond to the mean heights of the 2, 4, 6 and 8 decibar surfaces. These mean positions* are at about 2'0, 4'2, 7'2, ITS kilo- metres over Europe. Gravity has a small north-south component parallel to a surface at a fixed height above sea-level, which is taken into account. The "horizontal" components of wind are taken parallel to the sea surface in the theory, which in this particular is much too good for the observations. (3) Another alternative, and a rather tempting one, would have been to have taken the divisions between the strata at fixed values of the gravity potential corre- sponding to mean pressures of 2, 4, 6, 8 decibars. This system would be particularly * W. H. Dines, Geophye, Mem. 2. Also Bjerknes' synoptic charts. 3 18 THE CHOICE OF COORDINATE DIFFERENCES On. 3/2, 3, 4 elegant in the stratosphere. The horizontal components of wind would be defined as at right angles to the force of gravity. The variation of the thickness of a stratum with latitude could be taken into account in the equation of continuity of mass. Analytically *\> would be an independent variable in place of h. CH. 3/3. EFFECT OF VARYING THE SIZE OF THE FINITE DIFFERENCES The reduction of the differences of latitude and longitude in the ratio n -would multiply by n~ the number of stations falling on any territory. The cost of maintain- ing observing stations is but little affected by a change in the number of points in a vertical line at which observations are required, and would only be affected by a re- duction of the interval of time, if this necessitated an extra shift of observers, as for instance a night-shift. Together we may reckon the effect of reducing both Sh and 8t in the ratio n, as increasing the cost perhaps in the ratio n instead of n 2 . We have then Cost of stations =An 3 , Cost of computing = Bn\ where A and B are constants. Administrative expenses would contain an element C independent of n, as well as other parts which may be debited to the stations and to the computing, by changing the constants A and B to A' and B', The whole cost would then be of the form A'n'+ffn'+C. Next, as to the accuracy obtained : it has been shown* that in the representation of a smooth continuous function by finite differences, which are in any case sufficiently small, the finite size of the differences produces errors proportional to l/n". Thus as n varied, the errors would be inversely as the square root of the cost of computing alone. The relation to the whole cost A'tf+tfrt+C cannot be simply expressed until we know the values of A', B', C. CH. 3/4. THE PATTERN ON THE MAP As has already been illustrated in the introductory example, it is very desirable to arrange pressure and momentum in a special pattern like that of a chessboard. The consideration of this question is deferred to Ch. 7 as it cannot be properly dis- cussed until we have the full differential equations before us. Suffice it to point out here, that the pattern is such that the least difference in latitude or longitude between similar quantities, such as pressure and pressure, or wind and wind, is twice the least difference between dissimilar quantities, such as pressure and wind. To what then are we to compare the distance of 130 km, which we have seen is the mean distance between existing British telegraphic stations? If we take 130 km as more than one alternative and as less than the other, we arrive at approximately J 00 km between dissimilar and 200 km between similar stations, for a densely populated area. * L. F. Richardson, Phil. Trans. A, Vol. 210, 1910, p. 310. CH. 3/5, 6 ECONOMY VERSUS PRECISION 19 CH. 3/5. DEVICES FOR MAINTAINING A NEARLY SQUARE CHEQUER There is some advantage in a chequer which is neai'ly a square ; for the degree of detail with which it is desirable to study the weather, is roughly the same in all directions at any fixed point. A square-shaped chequer might be maintained in all latitudes by making 8n decrease towards the poles in proportion to cos . But then the treatment would be the more detailed in the high uninhabited latitudes where detail matters less. A more economical plan would be to maintain 8n at the same value in all inhabited latitudes and to make it increase in the uninhabited polar zones ; and, when the chequers became too elongated, on -approaching the poles, to omit alternate meridians. To make this possible right up to the poles, it would be necessary that the number of meridians used on the equator should be divisible by 2 many times over. For instance, instead of 120 meridians 3 apart it would be better to have 2 7 = 128 meridians 2 48' 45"'0 = O'()490874 radian apart. Unfortunately this was not thought of, until after the 3 difference of longitude had been used in the example of Ch. 9. CH. 3/6. SUMMARY ON COORDINATE DIFFERENCES The adopted size of chequer might be a compromise between that indicated by the existing practices on land and on sea ; for abundance of observations on the former does not compensate for scarcity of observations on the latter. A satisfactory arrangement would appear to be to divide the surface into chequers by parallels of latitude separated by 200 km and by meridians spaced uniformly at the rate of 128 to the whole equator. The chequer is then nearly a square of 200x200 km in latitude 50. At the equator even, it is not too elongated, as it measures there 313'09 kmeastx200km north. In latitude 63 the chequer is about 142 km from east to west. As ^ = Vl/2 this would be the latitude for the first omission of alternate meridians. The discussion of the polar caps is deferred to Ch. 7. In the example of Ch. 9 the width of the chequers is taken as 3 of longitude ; but that must be considered as an inferior choice. The centres of the chequers are supposed to be the points for observing and recording the meteorological elements. If we imagine the chequers to be coloured alternately red and white, so as to produce a pattern resembling a chessboard, then it will be shown in Ch. 7 below, that the red chequers should bear the pressure, temperature and humidity, and the white chequers should bear the two components of momentum ; or vice versa. Horizontally the atmosphere may conveniently be divided into five conventional strata at the fixed heights of 2'0, 4 - 2, 7'2, 11 '8 km above mean sea-level, heights which correspond to mean pressures of 8, 6, 4, and 2 decibars. Ten layers would give four times the accuracy obtained with these five layers. A technically more perfect division would be to take the separating surfaces at constant gravity potential instead of, as in this book, at constant height. The interval of time t has usually been taken in what follows, as 6 hours. 32 20 THE CHOICE OF COORDINATE DIFFERENCES Cu. 3/7 CH. 3/7. THE ORIGIN OF LONGITUDE This might be determined so as to get the greatest number of stations on land, or on steamship routes. If the whole world were considered, the origin would be a matter of indifference, as the coast lines are so irregular. For a small group of islands like the British Isles, if they could be treated as independent (as they certainly cannot), choice of origin would be important. Thus taking Greenwich as origin and meridians spaced equally at the rate of 128 to the equator, we find two stations about 50 kilometres west of the Irish coast, and three others at a like distance west of the Hebrides, of the Isle of Man and of Cornwall. By shifting the origin to longitude 2 W. of Greenwich these five stations come on land, or within about 10 kilometres of it ; while only one other, which moves from Flamborough Head into the North Sea, is lost. The arrange- ment with the origin at 2 W. of Greenwich is shown in the frontispiece. The stations are supposed to be at the centres of the chequers. Postscript: A study of turbulence, in Ch. 4/8, has disclosed several reasons for taking thinner strata near the ground. The divisions might for example be at heights of 50, 200 and 800 metres. The dividing surfaces would need to follow, more or less, the slope of the land. Again difficulties connected with the stratosphere in the tropics may make it necessary to divide the atmosphere above 11*8 km into two conventional layers. CHAPTER IV THE FUNDAMENTAL EQUATIONS CH. 4/0. GENERAL THERE are four independent variables : t time. h height above mean sea-level. X longitude, reckoned eastward. latitude, reckoned negative in the southern hemisphere. Seven dependent variables have been taken, namely : v s velocity horizontally towards the east. V N north. v a ,, vertically upwards. p density. p. joint mass of solid, liquid and gaseous water per mass of atmosphere. temperature absolute centigrade. p pressure, expressed in dynes cm~ 2 . If an eighth dependent variable had been taken, it might perhaps have specified the amount of dust in the air. The rates of change of the seven dependent variables are given by the following seven main equations. The "tributaries" in the table supply the values of certain terms occurring in the main equations. Sp/dt S/J./SI Wfit Main equations eastward dynamical equation northward ,, upward indestructibility of mass ... conveyance of water conveyance of heat characteristic gas equation, SO/dt and dp/dt Tributaries eddy-viscosity > eddy-conduction of heat precipitation precipitation, stirring precipitation, stirring, radiation and clouds One of the first questions which had to be decided was whether to eliminate any of the dependent variables before proceeding to the numerical process. Now if a 22 THE FUNDAMENTAL EQUATIONS CH. 4/0 variable be eliminated between two differential equations the resulting equation is usually more complicated, so that the saving in arithmetical toil due to the absence of a variable, is partly or entirely compensated by the increase in toil due to the com- plication. There is also a clear advantage in keeping to the familiar variables which are observed, to the avoidance of stream-functions and other quantities which cannot be observed. Therefore I decided to do without analytical preparation of this kind except in two cases : (i) To eliminate temperature between the characteristic equation p = bpd and any other equations in which 9 occurs. This introduces no complications, because the characteristic equation is not differential. (ii) To solve for the vertical velocity. This is necessary, because sufficient obser- vations of the vertical velocity are not available, nor likely to become so. The solution can be obtained because the vertical equilibrium can be treated as a static one, that is to say DvjjjDt can be neglected. It is done in Ch. 5. After this change V H is given in terms of mixing, precipitation, and radiation, and of the instantaneous values of the five remaining variables V E , v x , p, fj,, p. The time rates of the remaining variables are then given by the following five equations : Main equations Tributaries 3 (pVv/jot ) two horizontal dynamical equations, V H , eddy-viscosity \pVtf) lot ) modified by equation continuity V H , eddy- viscosity dp/St indestructibility of mass' ... V H , precipitation dp/% conveyance of water v,,, precipitation, mixing Spjot vertical static 11,1 and dp/tit The equations are further modified so as to make them apply to conventional strata. The way in which they then fit and lock together will be explained in detail in Ch. 8. As the arithmetical method allows us to take account of the terms which are usually neglected, many of these terms have been included. But as all terms cannot be treated with equally small time- and space-differences (see Ch. 7), they cannot all be treated with equal accuracy, and so it is necessary to know beforehand which terms are likely to be the more important. Some figures, representing the extreme values ordinarily attained by the various terms, are set out underneath certain of the equations. These figures have been obtained by a casual inspection of observational data and they may be uncertain except as to the power of ten. They are expressed in c.G.s. units. They relate only to the large-scale phenomena which can be represented by the chosen coordinate differences of 200 kilometres horizontally, one-fifth of the pressure vertically and by the time step of six hours*. * Very much fuller data concerning the size of terms has been collected by Hesselberg and Friedmann Geoph. Insl., Leipsig, Spezialarb. Ser. 2, Heft 6. CH.4/i,2 THE EQUATION OF CONTINUITY OF MASS 23 CH. 4/1. CHARACTERISTIC EQUATIONS OF DRY AND MOIST AIR The units in the following are C.G.S. centigrade throughout. The equations are taken from Hertz's* paper. Dry air p = 2-870 xlO 6 ^, .................................... (1) moist unsaturated air p = (2-870 + 1746/t)xlOV0, .'.. ........................ (2) saturated with liquid water or with ice (p-2> a ) = 2'87QxlW(l-n) P 0, .......................... (3,) here p w is the saturation pressure of vapour in equilibrium with water or with ice. A table of p w , in 10 3 dynes/cm 2 as unit, is given in Sir Napier Shaw's Principia Atmo- spherica. For our purposes it is convenient to employ w,, the density of the saturated vapour, instead of p w . On taking account of the relative densities of air and water- vapour equation (3) yields the following equation in w s , p = 6> {/3(1 -/*)x2-87xl0 6 + w s x4'616 x 10"} ................... (4) We shall frequently require to find 6 from the characteristic equations, being given p, p, p.. Now in order to find 6 we must know whether to use equation (4) or equation (2), that is to say we must know whether the air is saturated or not. For this purpose we require the relation between p, p and p. when the air is just saturated. This relation is to be found by treating (2) and (4) as simultaneously true, and then eliminating 6 between them. Thus solving (2) for 9, and substituting in (4), there results w* = W, .................................... ...(5) in which it is important to notice that w g corresponds to the temperature given by (2). If the numerical values of p, p, p. are such as to make (5) an identity, then the air is just saturated. If w s is too large to fit in (5), then the air is unsaturated and the temperature already found from (2) is correct. If on the other hand w, is too small to fit in (5), then the air is saturated, and both w, and 6 must be recalculated by solving (4). CH. 4/2. THE INDESTRUCTIBILITY OF MASS This principle leads to the well-known "equation of continuity (of mass)." Let v be the vector velocity, p be the density, so that vp is the momentum-per : volume, which for brevity we denote by m. Then the equation runs as follows, in vector notation ^ - ~ = div (pv) = div m = vV/j + p div v, ........................ ( 1 ) or when expanded in spherical coordinates on a globe of radius a : _ dp _ dmy Smy _ m N tan (ft Sm H 2m a St ' Se " dn a dh a ................... \ > 10-' J 10- ? 10~ 7 10- 8 tan0 < 10~ 7 10-' * Miscellaneous Papers, No. xix. (Macmillari it Co.). 24 THE FUNDAMENTAL EQUATIONS CH. 4/2 The term - - is small, but the other term involving the vertical velocity is by no Qb means negligible. The term ~- is usually small in comparison with either -- or -~^ ; that is to say, the atmosphere, although compressible, flows almost as if the density were constant at a fixed point. This is a much better approximation as V. Bjerknes* and later Hesselberg and Friedmannf have pointed out, than to suppose that the atmo- sphere moves like a liquid. As cyclones and anticyclones sweep past an observatory, W. H. Dines has shown that the increases of pressure are compensated by increases of temperature in the troposphere, so that the density there has a standard deviation, at a fixed point, as small as 1'5/ of its mean value. In the stratosphere, up to 13 km of height, the variation of density at a fixed point is about the same in absolute mag- nitude as in the troposphere, but therefore much greater relative to its meanj. The fact that the momentum-per-volume m is nearly a non-divergent vector, has been the reason for using it in preference to the velocity. However, although dp/dt is small, it will not be neglected. To adapt the equation of continuity of mass to our conventional strata let us integrate it across a stratum with respect to height, and apply the rule for differ- entiating a definite integral, remembering that all the limits of integration are constant 3 ~> during ^- and r- except only the limit at the ground ; and that all, without exception, are independent of time. The limits are denoted by G for the ground and 8, 6, 4, 2 for the conventionally fixed heights which correspond approximately to mean pressures of 8, 6, 4, 2 decibars. Taking the lowest stratum as example, equation (2) becomes 3 P 3 9 P -r- 01 J G G dh tan tj> f s ,, , 2 P m N dh+ - m n dh ......................... (3) af1 I J G u J G Now [fj| 5- + m JVa -- m ff] i g the scalar product of the vector m and of a vector \ of. on Jo at right anglesto the surface of the ground, and therefore it vanishes since the ground is impervious to the wind ................................................................... (4) Now let capital letters denote the integrals, with respect to h, of the corresponding small letters across the thickness of the stratum. Here R is taken as corresponding to p. Then the following equation is precisely true (for a spherical sea-surface). In it ~\ r5 the differentiations r- , now follow the stratum as a whole, de dn SR * Dynamic Meteorology and Hydrography, Art. 117. t Geoph. Inst., Leipsig, Ser. 2, Heft 6, p. 164. J W. H. Dines, Characteristics of the Free Atmosphere, Meteor. Office, London, 1919, p. 76. OH. 4/2, 3 THE CONVEYANCE OF WATER 25 For any layer, except the lowest, there is a term m a at the lower limit to be subtracted from the right side of this equation. Because they fit in this equation, the quantities R, M E , MS have been chosen as dependent variables throughout this book. R is the mass per unit horizontal area of the stratum. M E , M N , M H are the components of the total momentum of the stratum per unit horizontal area. Since dh = * we have 9p R =~^, " ( 6 ) M E = - l l B v E dp. -.r. (7) 9 J a This formula for M E has been used in deriving M E from observations in which a registering balloon had been followed by a theodolite. The velocity V E was plotted against p and M E measured as an area on the diagram. It is rather difficult to say what is the best way to take account of precipitation in the continuity of mass. It depends on how one defines m H the vertical momentum per unit volume, when rain is falling. If m a is defined to be the total momentum including that of the rain, as well as that of the air, then the equation of continuity of mass is correct as it stands. If on the other hand m H refers solely to the moist air, then a correcting term is necessary. This is discussed in Ch. 4/6. CH. 4/3. THE CONVEYANCE OF WATER, WHEN MOLECULAR DIFFUSION IS NEGLECTED, AND EDDIES ARE NOT AVERAGED-OUT It is a familiar observation that winds coming from over a warm sea bring water with them. This large-scale conveyance of water will now be put into equations, which will be equally applicable to small-scale motions provided that the equations are applied to the actual detailed distribution of the velocities, humidities and the like, and provided also that the detail is not so small as to bring molecular diffusion into the question. The modifications introduced, when smoothed velocities replace the actual velocities, will be considered in the section on eddy-motion. Precipitation will also be reserved for separate treatment. Apart from these complications, the air carries the water along with it, so that the velocity of the water is V E , V N , v lt , the same as that of the air. The density of the water substance is w grarns/cc. We have therefore an equation of continuity of mass of water substance similar in form to the equation of continuity of total mass (Ch. 4/2), except that tv replaces p and that precipitation and convection come in. dw j. , , (increase of water per volume and per timel , \ - f +div(iw) = \ , * I (1) at \ due to convection and precipitation J This expands in spherical coordinates just as does the corresponding equation in p. R. 4 26 THE FUNDAMENTAL EQUATIONS CH. 4/3 Now w = up, where p. is the mass of water substance per unit mass of moist atmo- sphere. So div (MTV) = div (/j,pv) = div (/woi) = (2) = mV^t, + p, div m by the well-known vector formula. But divm=-| > (3) at So + divM = / + ^ = /0 + vV /A (4) t^L Now -~+vVp 1 is the increase in /u,, following the motion of the fluid, and is cus- oc tomarily denoted by -= . Thus, on dividing through by p, we get a second form of the Uv equation of continuity of water substance, -a/r = - + vVu =^- + V E ~+V N ^ + V H = rate of increase of water per unit mass and Dt dt dt de dn dh per unit time by convection and precipitation ......................................... (5) It is in fact immediately obvious that the relative proportions of water and air by mass do not change during the motion, and therefore that j- = except for convection J-fb and precipitation. Thus the equation is confirmed. By working backwards from (5) to (l) it is found that the second member of (1) is p -~- , instead of j)~ which might have been erroneously expected. Equation (1), expanded in spherical coordinates, reads Sw ?I(WVE) 9 (wvy) _ wv N ta,n d (WV H ] 2 (wv ff ) _ D^ , , dt^ Se~ dn 'a Sh a ~ p Dt ' Adaptation to Conventional Strata when W is given In seeking to adapt equation (6) to conventional strata characterized by M E , M N , R, we are met by the difficulty that neither w nor the velocity is independent of height ; so that, for example, I wv E dh, taken between the upper and lower limits of a stratum, is not simply and accurately expressible in terms of M E and I wdh between the same limits. However w varies* very much more rapidly with height than do V E , V N or V H . Therefore V E , V N are treated as constant across a stratum, when the integral is being evaluated. Taking the lowest stratum for example and integrating (6) we get le \ V ' \\ wdh ] + In {?* H wdk } + IX1 + WG ( V * f e + V In ~ Hann, Meteorologie, 3 Auf. pp. 227-234. OH. 4/3 CONVEYANCE OP WATER, WHEN W IS GIVEN 27 / ^b 7 ^1 Zi \ The quantity ( V E + V N r -- VH- ) vanishes, because the ground is impervious to on f 8 wind. Write I wdh W so that W is the total mass of water-substance, whether J G liquid, solid or gaseous, per unit horizontal area of the stratum. Then (7) becomes dW S IM E W\ S IM N W\ tend>lM u W\ 2 W + Se HH + 3n(^r)^ HH + a ^ W+ ^ V ^ = rate of increase of water per unit horizontal area of the stratum, by convection and precipitation R ~^ ...................................... ~.~ .............................. (8) In the terms - v ff W aud R -yr- we must take v a and p, as mean values for the Ct X/6 stratum. For any stratum, except the lowest, we must subtract from the left side of (8) the value of WV H at the lower limit. Equation (8) is convenient because it is expressed in terms of W, the mass of water per horizontal area of stratum. On the other hand, the vertical integrations by which (8) is derived take no account of the three main statistical facts in this connection, namely : That for the mean of many occasions log ju is nearly a linear* function of height. ." ...... (9) That between heights of 0'5 km and 8 km m E and m N are nearly independent! of height. This is sometimes known as Egnell's law ..................................... (10) That log p is also nearly a linear function of height ............................ (11) Now from (9) and (11) it follows that on the average p.= Cp B , ............... (12) where C and B are nearly independent of height. We shall probably improve the accuracy of all our operations with conventional strata if we assume the general type of relationship (12) to hold in such a way that (7 and B are independent of height in each stratum but vary from one stratum to another. The constants C and B are then to be determined, for each occasion, from the instantaneous values of p and p. Let numerical suffixes denote heights. Taking the stratum between A 8 and A 6 as a type, it follows from (12) that , L 4 , logj? 8 - \ogp 6 It is now possible to transform various integrals, in a more accurate way than before, by making use of C and B. Thus 9 * Hann, Meteorologie, 3 Auf. pp. 227-234. t Gold, Geophysical Memoirs, No. 5, p. 138, Meteor. Off. London. 42 28 THE FUNDAMENTAL EQUATIONS CH. 4/3 By this equation W, the whole mass of water per horizontal area of the stratum, can be found when p., the mass-of-water-per-mass-of-atmosphere, is given at the boundaries of the stratum. Conversely, given the value of W and of p. at one boundary, we can find p. at the other boundary, and so proceed to the next stratum and do likewise. This rather troublesome process has to be used to find the term w$v m in equation (8) when W is given for each stratum in place of /A at the boundaries. For a simpler method might hardly be accurate enough. Alternative scheme : ^ given at the levels where strata meet We have so far supposed that observation gives the masses of water in the form of vapour or cloud per horizontal area of the strata. Now let us suppose instead that the given qiiantities are the "specific humidities" /A G , p. L , p. s , /u, G , p. t , ^ at the levels where two strata meet. It will then be necessary to express equations (5) or (6), or their derivatives, in terms of p, G ...p,. 2 . This can be done by means of integrations with respect to height, during which much use will be made of (9) and its derivatives (12), (13), (14), (15). Some equations integrate more neatly than others, for instance if we attempted to integrate (5) with respect to h, the term I V H dh would prove a stumbling-block. On the other hand if (5) be multiplied throughout by p/p it integrates nicely with respect to h. However it seems best instead to transform (6), because the physical meaning of the result is simple, and in simple relation to equation (8), and to the equation expressing the diffusion of moisture by eddies. Put then fjL = B'e- Alt , .................................... (16) where, within any stratum, B' and A are independent of h. Then, for example, in the stratum h to h s A _ h G -h s where /Z is an appropriate abbreviation for i% r~ G / 1 n\ T -T - (19) log/i G -log)U 8 since JL is the mean value of p with respect to height in the stratum. Now integrate (6) term by term with respect to A. The term ,, ; \ _i , ,. , dh\ (ho - h,) P} + HG \m>s - = (Mire* -^)+P-G f W, ... .(20) Cn.4/3 CONVEYANCE OF WATER, WHEN p. IS GIVEN 29 \ The term in transforms similarly. Then de f* J < Next tan ,,_ , ~ ^ - ~ (21 f 8 3 (wVfr} I -^- L = ^m a ,-p. g m HG ............................ (22) The term in a l does not integrate nicely, since we do not know how m ff varies with height. But it is a very small term. So treating m a as independent of height we get .(23) a ^ a The term p^-dh= I -^- dp= - I pdp, provided that p is regarded Jo -Ml gj G jJt g ut ) o D [ s as constant during the differentiation yr-. Here I p.dp is given by substituting in J G (15) above, G for 8 and 8 for 6, so as to make it apply to the lowest stratum. But I p -~ dh may as well be left in that form until precipitation and eddy motion have been discussed ; as it is from these causes that the term arises. Lastly the term f 8 And here p.dp is given by substituting in (15) above, G for 8 and 8 for 6, so as to J G make the equation apply to the lowest stratum. Note that in (24) both p. snidp are affected by the differentiation d/dt. Collecting terms (6) transforms into dW ai = _l^(logp 8 -log2 dt gdt log (/*,#.) -] ,, -^G 8 - j", (25) where jil is defined by (19) above. Note that the term ^ G m HO has disappeared because the ground is impervious to wind, and that in the corresponding equations for the upper strata there would always be a pair of terms of the form p.m H . It will be shown in Ch. 4/8 that the effect of eddies can be inserted in (25). The fundamental equations on this alternative scheme are (25) and those obtain- able from (25) by changing the suffixes which indicate the height. As D^/Dt, M E , are supposed to be known they can be inserted in (25) which then yields 30 THE FUNDAMENTAL EQUATIONS Cn.4/3,4 In this alternative scheme W is not one of the principal variables, and finding 3 W/dt is only a stage towards finding 9/i/3i. For, to make any numerical scheme of prediction "self-contained," it is necessary and sufficient to find d/dt of each one of the variables which have been tabulated initially. Unfortunately 3 W Ge /dt does not im- mediately give d/J. G /dt and 3/x 8 /3, for it also depends on the pressure changes. It will be better then to wait until the new distribution of pressure has been found after the time step 8t, and then from the new set of W given by (25) to find the new values of /u, at the boundaries by means of equation (15), starting from a known value JJL G given by the conditions at the ground. The increase of accuracy, which is the aim of all these special proceedings, could of course be attained by the general, and possibly easier, method of using sufficiently thin strata. The stratosphere does not require special treatment because there is no appreciable amount of water in it. CH. 4/4. THE DYNAMICAL EQUATIONS, WHEN EDDIES ARE NOT AVERAGED-OUT AND MOLECULAR VISCOSITY IS NEGLECTED In vector* notation -Vf--V^ = ^ + (vV)v+2[w.v] + [to.[w.a]], ............... (l) C" where the operator d/dt refers to time changes at a point rotating with the globe. The angular velocity . a]] represents the acceleration of a point at rest relative to the rotating globe ; and this acceleration is customarily included together with Vi/>', the true acceleration of gravity ; since it is possible to derive both together from a potential \fi, so that V^ = V^ + [o.[a.a]] ............................... (la) Further, it is shown in Vector Algebra that ] ............................... (16) So that the dynamical equation (1) may be written 1 3v -V,/,- Vp = + (vV) v + 2[w.v] O Ob dv v 2 = XT + V - - [v . curl v] + 2 [o> . v] ............. (id) ot The method of deriving the corresponding equations in spherical or spheroidal coordinates is set out in Lamb's Hydrodynamics in the section dealing with Laplace's theory of the tides on a rotating globe. Lamb first modifies Lagrange's dynamical equations in generalized coordinates, to suit the case when the coordinate system is rotating. He then applies the result to spheroidal coordinates. An alternative method, * See, for example, Silberstein's Vectorial Mechanics, ch. vi. CH. 4/4 DYNAMICS 31 probably simpler, is first to work out Lagrange's equations infixed spherical coordinates X', (f), r having the same axis as the rotating coordinates X, . at. Thus, in one way or the other, we get the dynamical equations for a particle moving in any manner relative to the earth. To form the corresponding equations for a fluid we must express the quantities -j- 3 , -Jf , -4-^- belonging to a particle moving with the fluid, as the sum of (i) the changes in the velocity of the fluid at points moving with the earth, and of (ii) changes in the velocity of the particle due to the fluid- velocity varying from point to point over the earth. The second part corresponds to the vector (vV) v in equation (1). The final result is not given completely by Lamb, because, in the theory of the tides, various terms involving squares and products of velocities can be neglected, as the tidal velocities are so small. In meteorology we must retain some at least of these terms. Here in Ch. 4/4 all are retained, except that a is put for a + h. The complete equations are given by Bigelow in his Report on the International Cloud Observations 1896 to 1897. They are of the type But for the purposes of this book it will be necessary to have the equation giving - instead of -^ , and similarly for the other components. To obtain the required Ot GL forms : the dynamical equation beginning -v^is multiplied throughout by p; the equa- tion of continuity of mass in the form ^ = etc. is multiplied by V E ; then p is added vt ot to V E - to give -~ . There results : _<^E = S P, d_ ^ . dt de de i/ io~ 4 io- 3 io~ 4 io- 3 dm N dp d f. -, - j. jr .j ( on de fov ' 1 E V H/ ~ ^ w sin

depend also on the curvature of the parallels of latitude, in such a way as to become formidable near the poles. The terms in a" 1 are derived partly from the centripetal accelerations and partly from the effect of the crowding together of the coordinate lines in the equation of continuity of mass. A body rotating with the earth as if it were rigidly attached to it, experiences cen- trifugal forces per unit volume p (ca cos <) 2 (a + h) upwards and pco 2 cos $ sin < . (a + h) northwards. These forces are found from Lagrange's equations along with those de- pending on the winds, but the former are not allowed to appear explicitly in equations (5) and (4) because they are customarily included respectively in g and in the component of the acceleration of gravity towards the north, which is here denoted by g N . The other part of g N is a true gravity effect depending on the ellipticity of the earth. Values of g N , for heights and latitudes to fit our conventional division of the atmo- sphere, are given in the annexed table. 9s Component of the acceleration of gravity along a line running north and south at a constant height above mean sea-level. Directed towards the equator in both hemispheres. From Helmert's constants (1901) quoted in Landolt Bomstein Meyerhoffer'ft Tabellen Kilometres above mean sea-level 1-0 3-0 5-5 9-1 16-3 Distance from equator Approximate mean pressures in decibars 9 7 5 | 3 1 Degrees Megametres 10,000 times accelerations in cms/sec 2 9 1 3 8 14 23 41 18 2 5 14 26 43 78 27 3 7 20 36 60 107 36 4 8 23 43 70 126 45 5 8 24 45 74 133 54 6 8 23 43 71 126 63 7 7 20 36 60 108 72 8 5 14 26 44 78 81 9 3 8 14 23 41 90 10 Ce.4/4 DYNAMICS OF STRATA 33 The numbers under the terms in (5) show that the vertical equation is nearly equivalent to 0gp + J^, .................................................................. (6) as is well known. If the largest of the small terms be included, the equation runs = gp + -^ - 2 . m E ............................... (7) The term 2a) cos (f> . m E may produce a pressure of 0'5 millibar at sea- level under extreme conditions. Mr W. H. Dines points out that this term 2 . m E may be taken to mean that air moving eastward is lighter than the same air moving westward, and suggests that this may explain the fact that westerly winds commonly increase aloft much more than do easterly winds*. If the term 2 . m E really has such an important influence it ought not to be neglected when one is dealing with parts of eddies. He also adduces an observation of Nansen's as illustrating a similar effect in the horizontal the Siberian rivers, which flow northwards, erode the land more on their eastern sides, as if the water there moved faster than on the west. Now let us try to express the dynamics of a stratum, considered as a whole, in terms of the quantities R, M E , M N> which fitted so neatly into the equation of con- tinuity of mass (Ch. 4/2). Integrate the longitude equation (3) with respect to height across a stratum. Taking the lowest stratum with its limits denoted, for brevity, by G and 8 as example and considering terms separately ( 8 dp , d ( 8 I dh\ dh = ;- pdh+lpj-) ............................ (8) ) de de) G ^ VSeJ G ft Let us denote I pdh by P Gfl . Now it will probably be more convenient to use p a J G and p G as variables rather than P Gs . In that case we shall require to express P^ in terms of p s and p G , and this can be done by the following formula, which is strictly true if the air is dry and if temperature and gravity are independent of height, and is in any case a good approximation t p _ * - log e _p 8 3wi The important terms , 2w sin (j> . m N , + 2u> cos (f> . m H yield respectively, on dM integration across the stratum, ^ , 2o) sin $.M N , + 2a> cos (f>.M H . The remaining terms can only be expressed as functions of R, M E , M N , M H in special cases, because they contain squares and products of velocities ; but fortunately these terms are usually small. Consider the three terms 5- ( r m E v E ) + (m E v N } + ^r (m E v H }. G& Giv Git' * Manual of Meteorology, by Sir Napier Shaw, Part iv. p. 58. t No doubt, but a better one could easily be obtained. R. 5 34 THE FUNDAMENTAL EQUATIONS OH. 4/4 On integrating with respect to h they become 3 f 8 3 f 8 3eJ G 3nJ G * + * VH )\ .. (10) 06 on / j at the ground, and the part in {} vanishes, because the ground is impervious to wind, just as did the similar expression in the equation of continuity of mass. Now in the special case in which the velocities are independent of height V E = ^^ , V N = , V M = ff , and in -it J\r -L\i this case the two integrals in the last expression transform into " I T~I cn\ H I have assumed that this transformation is a fair representation in general, even if the velocities are not independent of height. The range of velocity in a stratum diminishes, as a rule, as the stratum becomes thinner. So if the above assumption should prove to be insufficiently exact, one course will be to take a larger number of thinner conventional strata. The other terms containing products have been dealt with similarly. Collecting terms, the dynamical equations are transformed into the following (11), (12), (13), which are suitable for use in computing, and which take account of the height of the ground. The differentiations and now follow the de dn stratum as a whole. The terms pdh/de, pdh/dn of course only occur when the equations relate to the lowest stratum. In the integrations g and g N have been regarded as independent of height. Mean values of g and g s for the stratum should therefore be used. K , aR aR ~aR M H M N tan aR aR aR~ aR * For any stratum except the lowest the corresponding quantity at the lower limit of the stratum is to be subtracted from this term. CH. 4/4, 4/5 FORMS OF ENERGY OF AIR 35 In the example of Ch. 9 the terms - and p are opposite in sign and numerically much larger than any other terms in the northward dynamical equation. dP It is therefore well to use exactly the same process in approximating to . 8 as in approximating to \p =- , lest a slight difference in their fractional errors should make L n -\G a serious error in their sum. Accordingly integrate both these terms, with respect to latitude through an interval of 200 kilometres. Then dh Now if we assume that the pressure at the ground is nearly an exponential function of the height of the ground, then the last integral above is of similar form to P given by equation (9). So that, if quantities with one dash and two dashes refer to the beginning and end of the interval Sn = 200 km, we have + represented by ~- 1 Pr:*" Pa* + ^-r , g [fdnje J 8n{ log, p " - log dn And a similar transformation applies to the corresponding terms in the eastward dynamical equation. CH. 4/5. ADIABATIC TRANSFORMATION OF ENERGY CH. 4/5/0. GENERAL THEORY This may be very simply expressed by saying that the entropy of a given mass of air is only changed by radiation, by precipitation or by eddy-diffusion. But the theory of entropy is usually developed with reference to substances in the cylinders of idealized engines, where the weight of the working substance and its kinetic energy are both negligible. And such rigorous experimental versifications of it, as have related to gases, have mostly either been carried out in closed vessels or else have referred to wave-motion. When we have to apply such indoor results to the free atmosphere, where observation is scarcely as yet able to provide a rigorous check, there is apt to be some confusion of mind as to the precise way in which gravity and kinetic energy fit into the scheme. The following pages have been written to make this clear. The result, which is stated above, is shown to be correct. Continuity of Energy. In counting up the total energy of a mass of gas we must beware lest we add on the same part twice over, under different names. Imagining the gas as a swarm of molecules we see four kinds of energy (i) gravitational due to earth, (ii) energy within the molecules, (iii) energy of forces between molecules, (iv) kinetic. 52 36 THE FUNDAMENTAL EQUATIONS OH. 4/5/0 Pressure does not appear. Now when we change to a scale on which molecules become invisibly minute, there remains simply under the same name only gravitational energy. It is denoted by pfy per unit volume, where i/> is the gravitational potential and p is the density. The kinetic energy is now split into two portions : (a) mean-motional, still called "kinetic energy" and denoted by ^pv~, (b) deviations from the mean. These deviations are grouped together with the energy within the molecules, and the energy of forces, other than gravity, between molecules. The group so formed is called the " intrinsic energy " and in this book is denoted by v (upsilon) per mass or pv per volume. Here dv y v dd where y v is the specific heat at constant volume. Schematically the classification of energy-per-mass runs thus MOLECULAR MOLAR Gravitational \f/ (Mean ^ 2 Deviations .... Within molecules.... Between molecules The point to be emphasized is that as I pressure . d (volume) did not appear in the molecular classification, it cannot appear in the molar classification unless we com- pensate for its appearance by omitting an equal amount of energy under some other name. For instance part of pv that comes from the translational kinetic energy of the molecules is equal to %p in an ideal gas. See Jeans' Dynamical Theory of Gases, 2nd edn. 161. Margules* no doubt makes some such compensation when he calcu- lates the energy associated with local irregularities of pressure. In the present scheme this energy appears under other names, and to include pressural energy in addition would be to count some of the energy twice over. It seems to be most natural to regard pressure only as concerned in the transfer across surfaces. Once across, energy appears in some other form : gravitational, intrinsic or molar kinetic. The rate of increase of energy in unit volume is therefore in the present scheme (i) Let us next examine the rate at which energy crosses a very small plane. Again imagining the gas on an enlarged scale, we see only a swarm of molecules, each carrying its energy with it. To simplify the problem, let the plane move with the mean velocity of the swarm in its neighbourhood. That is to say let the fluxes of mass across the plane from its two sides be equal and opposite. * "Mechanical Equivalent of Pressure" and "Energy of Storms" in Abbe's Mechanics of the Earth's Atmosphere, 3rd Collection. CH. 4/5/0 THE FLOW OF ENERGY 37 Then as the gravitational energy which a molecule possesses when it is crossing the plane is independent of its speed and of the direction of its motion, so on the average no gravitational energy crosses the plane. The internal energy which a molecule possesses when crossing the plane is likely to increase with its speed relative to the plane, but is independent of the side from which it comes, if the general state of the gas is the same on the two sides, so for that reason, no internal energy crosses the plane. (Note that we have here made an assumption equivalent to neglecting part of the conduction of heat. Let us neglect the rest of it also, as it can be allowed for afterwards.) Next, will the energy of inter-molecular forces be carried across the plane? Apparently not, for it also will be independent of the direction of motion relative to the plane, if conduction of heat can be neglected. But the same is not true of the mole- cular kinetic energy, for this has been reckoned relative to the earth and the motion of the plane produces a lack of symmetry in the flux of energy, so that the opposing parts do not balance, as will be shown. For simplicity let the plane be set at right angles to the mean motion of the mole- cules and let its velocity relative to any Newtonian framework be ~x, the mean velocity of the particles. If we distinguish mean values by bars and deviations from the mean by dashes, then any molecule has kinetic energy, relative to the same standard, equal to l m {( + xJ + ^ + yJ + (z + zJ}, (2) where m is the mass of a molecule. The molecule traverses the plane with a velocity x' '. If there are n molecules having this velocity in unit volume, a number x'n of them cross unit area of the plane in unit time and carry with them a kinetic energy ^mnx'{( + xJ + ^ + yJ+(^ + zJ} (3) The resultant flux of energy is the sum of this expression for all the molecules in unit volume and may be written as follows, n being omitted because 2 sums for the molecules individually, 1 m 2x' {(xY + (#) 2 + (I) 2 + 2M + 2yy' + 2lz' + X H + y'* + z' 2 } (4) Now the sum of the product of any mean into any deviation-from-a-mean vanishes. Also, in the molecular chaos, the correlations between x', y' and z' all vanish if the effects of molecular viscosity are negligible. Again 2x /3 is negligible if we can neglect the molecular conduction of heat. Consequently the flux of energy simplifies down to mx'Zx' 1 . Now in works on the Kinetic Theory of Gases* it is shown that mSx" is equal to the pressure, at least in an ideal gas. Thus if a very small plane be moving at right angles to itself with a velocity v such that on the average no mass is traversing it, then the fiux of energy across it is pv per area (5) * Jeans' Dynamical Theory of Gases, 2nd ed., 161 and 216 et seq. 38 THE FUNDAMENTAL EQUATIONS CH. 4/5/0 Thus is explained the appearance of pressure in the flux of energy and its absence in the energy contained in unit volume. The above treatment is sketchy and the reader should be referred to works in which the kinetic theory is treated thoroughly. The above theorem is a very special case of one by Maxwell set out in its generality in Jeans' book, 2nd edn. 340. Now if we add to the small plane, which we may imagine as square, five other planes, also moving with the fluid to form, at one instant, a unit cube, then we see that the rate of decrease of energy in the moving, distorting, swelling cube will instanta- neously be Or if we had taken a cube containing unit mass instead of unit volume, the rate of decrease of energy in it would have been (7) But the energy in unit mass of fluid is \ v 2 + 1/ + v, so if DJDt denote a differentiation following the motion of the fluid (8) There is another form, sometimes useful, into which (8) may be turned by means of the general theorem that if A is any scalar and p the density of a fluid moving with velocity v then DA d(pA) If we put A = (iy 2 + ^ + v) then equation (9) causes (8) to transform into ............. (10) Equations (9) and (10) are two forms of the equation of continuity of energy*. But we can change to a more convenient one. For, purely by geometry, in the ordinary vector notation, div (pv) =p div v + vVp ............................... (11) And the term vVp can be found and removed along with (^v 2 + t/<), by means of the dynamical equation, as follows. When the axes of reference are fixed to the earth the dynamical equation is Dv ' (12) where the square bracket denotes a vector product. Now form the equation of activity- per-volume by taking the scalar product of v into each term of the above equation. Since v and [wv] are at right angles to one another their scalar product vanishes. * Cp. Webster, Dynamics, 188 ; Lamb, Hydrodynamics, 10, eqn. (5). CH. 4/5/0 ADIABATIC LAW INDEPENDENT OF GRAVITY AND MOTION 39 And since the gravity potential does not change at a point fixed to the earth d\jj/dt = Q; so that vV\^ D^i/Dt (13) Thus the dynamical activity per volume is expressed by either side of (14) Now on substituting (14) in (11) we obtain as a form of the dynamical equation _ (15) And this when combined with (8) yields -/5-^=pdivv .................................. (16) We might advantageously bring in the equation of continuity of mass in the form which when substituted in (16) gives D That is to say, following the motion, the increase of v, the intrinsic-energy -per- mass, is equal to the pressure multiplied by the decrease of volume per mass. This is the ordinary adiabatic relation. We have neglected radiation, molecular conduction of heat, and molecular viscosity. The reason for deriving a familiar result by such a roundabout process, is to make it clear that neither the kinetic energy tjpv 2 nor the gravitational energy appears in (18). They are not neglected; they do not belong in this adiabatic equation. Equation (18) from its mode of deduction should apply equally to clear air or to clouds moving in the most unrestricted way occurring in nature, provided that radiation and conduction and precipitation are negligible, or allowed for separately. There is a query as to the flux being vp when intermolecular forces act, but in meteorological applications the error, if any, must be small. Entropy If unit mass, during its motion, is gaining energy by radiation at a rate De/Dt we have from (18) on bringing in the additional term and then transposing and dividing by \De_\Dv pDl\\ 6Dt~ dDt + dDt\ P )- Each side of this equation is the rate of increase of entropy -per-mass, a-, following the motion ; that is Da/Dt. 40 THE FUNDAMENTAL EQUATIONS CH. 4/5/0 To see that this formula agrees with those in common use (Hertz, Neuhoff, etc.) take, as a test case, that of dry air. Then There results = ~' as usual Thus the entropy is a- = y v log 6 b log p for dry air, and its constancy during the motion is entirely unaffected by changes in the molar kinetic energy, or by any gravitational effect. The Potential Temperature* r in a mass of air is denned as the absolute tempera- ture which the air assumes when brought adiabatically to a standard pressure p t . If in the equation which follows from (20) cr = y v log 6 b log p + const. we replace p by p by means of the equation p = bp6, applicable to dry air, we obtain cr = "/ p log 9 b logp + const. Thus if the air at 6, p be brought adiabatically to a standard pressure p it its final temperature 0, now called its potential temperature T, will be T = 6 ( } 0\-\ \PI \PI so that (r = y p log T + const, again independently of any kinetic energy generated en route. In dealing with eddy-motion the potential temperature is more convenient than the entropy in this respect : If two masses of dry air at different potential temperatures mix intimately and adiabatically by diffusion under the constant pressure of their surroundings, their mean potential temperature remains unaltered, although their total entropy increases f. This follows immediately from the formulae. It will be suf- ficient to consider the mixture of two equal masses, as any other case may be reduced to that. Let the initial temperatures be & ', 6". Then the potential temperatures are initially \0-289 / \0-289 \P/ \P1 On mixing portions of dry air under constant pressure, without loss or gain of heat, there is no change in the total intrinsic energy, so the final temperature is (0' + 0") and, as the pressure is fixed, the final potential-temperature is \ (/ + T") ; but the entropy, which is proportional to y p log T + const., necessarily changes. The Potential Density might prove to be a more useful conception than potential temperature, for it is density that is directly connected with stability. * Vide v. Helmholtz (p. 83), v. Bezold (p. 243) in Abbe's Mechanics of the Earth's Atmosphere, 1891 ; L. A. Bauer (p. 495) in Abbe's Mechanics of the Earth's Atmosphere, 3rd Collection, Smithsonian Institution, t Pointed out to me by Mr W. H. Dines. CH. 4/5/1 ENTROPY AND POTENTIAL TEMPERATURE 41 CH. 4/5/1. NUMEKICAL VALUES FOB THE ENTBOPY-PEE-MASS, P \ ) = + 8-41 x 10 6 log, p-\ 0-16 xlO 6 log, p /p.p const- - 10-16 xl0 6 log e {10 6 (r74G/A + 2-870)} -10M10-16/. + 9-92) ................ (13) We also require the quantity ft, denned as equal to -- - in Ch. 5. From (11) a p and (12) p 10-16^ + 9-92 = p y, p 8-41^ + 7-05 p'y,' By means of the characteristic equation (9), /8 can be expressed in terms of 6 and p.. The result is So that for dry air /8 is simply 4 '04 X 10" 9. Entropy in the Rain, Hail and Snow stages To fit with the rest of this work we require the entropy expressed as a function of pressure, density and moisture ; the temperature must not appear explicitly. How- ever, there is a difficulty in expressing the relation in formulae, because the vapour pressure is given experimentally as a somewhat complicated function of the tempera- ture. On this account it is simpler to proceed by way of graphs or numerical tables, of which those by Neuhoff* appear to be the best, although they need to be converted to millibar units. If p' is the partial pressure of the dry air, Neuhoff gives the adiabatic relation in the form log/ - ^~ ~ ^ (/*) log + const. = 0, where F l is a function only of the amount of water present, and varies slowly, while F, depends only on the temperature, with which it rapidly increases. Following Bezold, Neuhoff treats also the " pseudo-adiabatic " case in which the water is pre- cipitated as soon as it is condensed. * Abbe's Translations, 3rd Collection, pp. 430 494. OH. 4/5/2, 4/6 TEMPERATURE OF AIR SUPPLY CH. 4/5/2. THE CONVEYANCE OF HEAT ON A LARGE SCALE Hot winds from the desert and cool breezes from the sea are well known. F. M. Exner* has published a prognostic method based on the source of air supply. V. Bjerknes and his assistants, J. Bjerknes, Bergeron and Solberg, attach great impor- tance to the conveyance of heat. They go so far as to find a "polar front" where cold air from the polar regions meets warm air from the Tropics in European latitudes, and thereby causes cyclones. The present writer regards the "polar front" as a sketch, in black and white, of a reality, which these authors deserve much credit for having discovered, but which requires, for its proper representation, many delicate gradations of half tones, as well as the occasional sharp outlines. In other words we may not take the entropy-per-mass a- as unchanging following the motion, because a- must be altered by radiation, precipitation, and mixing. If these effects are known, then we know Da/Dt, the rate of change following the motion. And then the rate of change da-/dt at a point fixed relatively to the earth is given, in the usual way, by 3cr _ da- da- da- Da~ This equation is used in finding the vertical velocity. Having served that purpose it is found in the lower strata to be no longer necessary ; so that we need not integrate it to make it apply to a stratum as a whole. The uppermost stratum receives a special treatment in Ch. 6. On. 4/6. UNIFORM CLOUDS AND PRECIPITATION When the dependent variables are tabulated numerical quantities, it is as easy as account-keeping to deduct a numerical quantity of water from the amount present in any stratum, and to transfer it to the ground. Knowing the total amount of water-substance present, and also the temperature, we can find the amount of liquid or solid by means of a table, for it has been found that supersaturation does not occur f. As a general indication of what may be expected, the following table shows the limiting amount of water, which can just exist as gas, in the several conventional strata, for an average English temperature distribution }. Pressure in deciba,rs at limits of strata 1086420 Gaseous water in grams per square centi-1 metre of horizontal area of stratum } 1-34 0-67 0-25 0-04 1 water vapour vapour in equilibrium with ice * Exner, Dynaminche Meteorologie (Teubner, 1917), 70. t Sir Napier Shaw, Principia Atmospherica, p. 83. J Mean temperature British Isles, 1908 to 1911. Geophysical Journal, 1913, p. 91. 62 44 THE FUNDAMENTAL EQUATIONS CH. 4/6 Of the condensed water some will be precipitated and some will float as cloud. Several observers* have measured the amount of liquid in clouds on mountains and their results indicate a maximum mass of liquid of about 4 x 10~ 3 per mass of atmosphere. For the high cloud alto-stratus the maximum value of the volume of the particles per horizontal area has been estimated t by photometric measurements to be of the order of three times the diameter of the particle, regarded as spherical. The mean of value for the diameter found by Pernter, from measurements of coronae, was about 1 x 10~ 3 cm (Hann, Meteorologie, 3 Auf. p. 257). The mean thickness of such a cloud comes to about 500 metres (Hann, I.e. p. 284), and its mean height above ground to say 4 km (Hann, I.e. p. 280). From these data it follows that the ratio of mass of water to mass of air is of the order of 7xlO~ 5 . This is remarkably less than the ratio 4x 10~ 3 found at lower levels. However, only a few alto-stratus clouds show coronae. C. M. K Douglas, who has studied these clouds while flying through them, writes: "The snow particles of alto-stratus or false cirrus are usually of the order of 1 m.m. in length and of elongated form. Occasionally much smaller particles are met with, perhaps of the order of O'l m.m. These are usually thinly scattered and cause halos readily. They are occasionally met with quite low down. A layer of dense grey alto-stratus usually consists of larger particles. Occasionally alto-stratus consists of water drops, espe- cially in summer, or with warm damp upper currents in the autumn or early winter. The water drops may of course be super-cooled, but the water drop clouds more often have the appearance of alto-cumulus. The water drops are usually too small to be felt on the face." According to Douglas' observations we might take the volume of the particle to be that of a sphere of about 0'03 cm in diameter, and it would then follow, from the photometric observations, that the mass of water or of ice per mass of air is of the order of 2xlO~ 3 , which is about half of the maximum value observed on mountains. Thus the limiting amount of condensed water which can float in any one conventional stratum will be about 0'8 gram per square centimetre of horizontal area. What has been said above applies to the formation of extended sheets of cloud. Detached clouds are discussed in Ch. 5 below. In order to save labour I have supposed there to be a sharp distinction between rain which falls and clouds which float. Actually there is a gradual transition. All sizes of particles occur. Even the smallest ones observed by Pernter would fall, according to Stokes' formula, at a rate of 70 metres per day. Generally speaking a small sphere of ice or water falls relatively to the air at a rate of about J <# ff = O'SxlO 6 (diameter in cms) 2 cm sec" 1 (1) * Hann, Meteoroloyie, 3 Auf. p. 306. t Recent measurements similar to L. F. Richardson's Hoy. Soc. Proc. A, 96 (1919), p. 22. J Wrong sign of power of ten in Hoy. Soc. Proc. A, 96, p. 15. Cn.4/6 UNIFORM CLOUD AND PRECIPITATION 45 In discussing the conveyance of water in Ch. 4/3 above, we supposed that the water had the same velocity as the air. Let us now correct that assumption. The total mass of solid, liquid, and gaseous water, per mass of atmosphere has been called /A. Let the mass of the solid and liquid parts jointly per mass of atmo- sphere be v. Then (a v) has the velocity V H of the air) , . /, , ., Y relative to the earth. And v has the velocity V R + ^J B The horizontal velocities may be considered the same in both cases. So that the indestructibility of water-substance leads to If there are particles of different sizes and velocity the last term would have to be summed for all sizes so that where D, as usual, denotes a change following the motion of the air. Similarly in the equation of continuity of mass we have taken m H as equal to pv H . But if V H is the velocity of the air, and if p is the total density, then the mass per volume of all the gases jointly is p (1 v}, so that, strictly (4) Or, if the sizes are various, m H = pv B + ptv . OUo Ci , v/Uo Li> &(!+- -. dp]- *- . - V gr / gr TT r- So that the differential equation for the radiation passing in either direction through the parcel is dE A sec f ,-, O0* dA, . dA . cos L . cos I E* ! 2 = r dp g \ TT r' Now to apply these general theorems to Angstrom's observations, let dA t be one square centimetre of the horizontal receiving surface of his instrument. Then is the zenith distance of the narrow beam received from the direction dA a _. ~Let'dA 3 be an infinitesimal zone dt, of the sphere-at-infinity concentric with the instrument. Then dA,, cos 8 /r 2 is the solid angle of the zone as seen from the instrument, and is equal to lir sin . o?. So (6) becomes dE A sec f . . 7i ,-i ,_, - {E - 00 4 2 sin . cos . d} (7) i ts Now let / be the "brightness" of the sky for this invisible radiation, that is to say let / be the radiation per parcel of unit area. Then = /cos. 2irsinC.c# (8) So that (7) becomes |/ = Asec_n 7 _oJ dp g \ TT J But 3#Vir is the full radiation for a parcel of unit area. Expressed in words (9) means that : in traversing an infinitely thin horizontal lamina, of mass dR per unit area, the radiation gains A sec . dR of its deficit from the full radiation which would pass through a parcel of the same area in an enclosure at the temperature of the air. It would not be convenient to integrate (9) analytically even for clear air, be- cause we should have to take account of the variations of A and 6 with height. And when it is remembered that clouds have a great influence on /, and that' clouds can hardly be treated analytically, the necessity for a treatment by strata and finite differ- ences becomes evident. However, within the narrow range of single stratum there is much advantage in integrating (9) analytically. The integrals for the successive strata can then be fitted together at the boundaries. Within a stratum it is assumed that A may be considered independent of height, and that 0* may be considered to increase linearly with p at a known rate. Thus d6*jdp is about 65 (degrees) 4 cm 2 dyne" 1 in the troposphere and is zero in the stratosphere. 72 52 THE FUNDAMENTAL EQUATIONS CH. 4/7/1 Under these restrictions, (9) may be solved by the general analytical method for linear equations. When this has been done, and when the arbitrary constant of integration has been eliminated between the values of / and 6 at the boundaries of the stratum, which, for illustration, has been taken as the one lying between h s and /i 6 , the result is _ _ _ __ I ' fl lit I ^ 8 ^ V 7 TT\ A sec dpi] ir\ A sec 4 dp This equation is suitable for reducing observations made simultaneously in moun- tainous country at different levels. Angstrom's excellent California!! observations do not refer, unfortunately, to an isolated zenith distance, and so cannot be used thus. In (10), as in (5), sec is positive for ascending radiation, negative for descending. ...... (11) JQi The variation of 6 with height occurs in (10) in two ways: by the term in -j- and by the separate appearance of S and e . In the following calculation of A from Angstrom's observations, the variation of 9 with height has been treated by using two different sizes of coordinate differences. There was consequently little to be gained by retaining the temperature variation in (10). It was neglected so that (10) became simply < 12 > where ij = 1 - e -*> Beo t- s f air are found from the balloon observation and from them the full radiation Ofr/ir is deduced for each layer. Then the descending radiation in each cone or conical shell can be traced downwards by making successive applications of (12), frorn the top, where it is assumed to be zero, to the instrument. The computation is set out in the accompanying table. It gives 0'2267 cal cm" 2 min~ 1 at the instrument, for the stated trial value of the absorptivity- per-density A. Computation of Radiation from the Atmosphere Assuming A= 1-601 x 10~ 3 c. G. s. units and temperatures from balloon at Avalon, California on 28 July 1913 Pressure millibars Mean temperature of stratum. A Full Radiation per parcel of unit area. calcm~ 2 min~ 1 t= mean 1-, n approx. area of) parcel > percm 2 atinstr.) .0 3 15" 718 282 813 6 45 636 364 1-626 9 9 75 291 709 813 Sum for all zones VERTICAL CONTRIBUTION OF ZONES cal cm~ 2 min" 1 214 0510 f=0 30 60 90 {transmitted i x586 emitted 0117 -0302 0294 231 0701 total (transmitted 0117 0084 0302 0192 0294 0086 0712 |x586 emitted 0161 0415 0405 260 1118 total {transmitted 0245 0176 0607 0386 0491 0143 1343 586 emitted 0256 0661 0645 total 0432 1047 0788 2267 Next to estimate the errors due to finite differences, the calculation was re- peated in exactly the same manner, but taking 6 layers of equal mass instead of 3, and dividing the hemisphere into 6 parts by the cones =15, 30, 45, 60, 75. 54 THE FUNDAMENTAL EQUATIONS CH. 4/7/1 This gave 0'2210 cal cm" 2 min" 1 at the instrument. Now the errors are usually proportional to the square of the coordinate differences, when these are small enough. If this rule holds in the present case, the radiation at the instrument would be 0'2192 cal cm" 2 min" 1 , if calculated by infinitesimal steps for the same absorptivity- per-density A=r601xl0~ 3 c.G.s units. The observed value of the radiation was 0'210 cal cm" 2 min" 1 , which is slightly less than the calculated. To correct A to the observed value of the radiation, the variation of the radiation was recalculated with a slightly different value of A, namely A = 1'501 xlO~ 3 c.G.s. units. In this calculation the larger of the two afore-mentioned sizes of differences was used, to save trouble, and it was assumed that the small correction would be practi- cally the same, whether determined by comparing two coarse difference tables or two fine difference tables. For A= 1 - 501 X 10~ 3 C.G.S. units coarse differences gave a radia- tion at the instrument of 0'2195 cal cm" 2 min" 1 . The final result is that, with infini- tesimal differences, the observed radiation would be given when A = 1*48 xlO~ 3 C.G.S. units, for dry clear upper air, for its own radiation. We are now in a position to find the radiation emitted by a horizontal lamina of clear air dp thick. This may be shown to be ff-ir/2 A SBC f dp A 00 4 2cossin(l-e" ~^~) d^ = 2O& 1 - dp = 20& t Apdh, (16) J-fmt 9 in which both dp and dh are to be taken as positive. Putting in the value of A and taking #=980 cm/sec 2 it follows that: a horizontal lamina of air Sp millibars thick emits from each side radiant energy at a rate 0'00302Sp times that from an ideally black plane to one side, for the same area. For comparison with the experimental constants quoted by Gold on page 53 of Proc. Roy. Soc. Vol. 82, we require r^ of the radiation emitted by a layer of air 1 cm thick. Gold does not state the temperature or the pressure of the layer, so I have assumed them to be normal, that is 273 A and 1013 mb. It then follows from (1.6) that the increase in the radiation from the layer per degree is 0*143x10"* cal cm" 2 per hour, when A is given the value l.'48x 10~ 3 C.G.S. units. Thus the figures quoted by Gold correspond to much greater absorptivities than the one deduced above. In another paper* Gold has criticized the deduction of the absorptivity from observations of the cooling of surface air. In what follows I have preferred to rely on Angstrom's observations as reduced above. It would be interesting to repeat the computation of the absorptivity on the assumption that dry air is perfectly transparent except between wave lengths of 13 and 16 microns, which is the position of the band due to carbon dioxide. * Q. J. R. Met. Soc. Oct. 1913. CH. 4/7/1 WATER VAPOUR AND LONG WAVES 55 Moist Clear Air. No satisfactory treatment will be possible until we are able to group wave lengths into ranges according to the absorptivity. In the meantime we may note that the effect of the moisture is usually less than that of the dry air. This has been shown by A. Angstrom. He observed the total radiation falling from a clear sky upon a horizontal plane. Then having corrected all the observations empirically to a standard air temperature of 293 A at the instrument, he plotted them against vapour pressure at the instrument. A range of vapour pressure of to 10 mm of Hg caused the radiation to vary from 0"27 to 0'41 g cal cm" 3 min" 1 . That was when observations at all levels were combined in a single diagram. In view of the result which has been obtained above, that a stratum of dry-air 200 millibars thick allows as much as 0'7 of vertical radiation to pass through, it does not appear to be permissible to combine the results from levels differing by several kilometres. And for the same reason the humidity and temperature at the instrument are only of interest in so far as they are a guide to the whole distribution above. If we treat the levels separately it is found that the effect of humidity is as follows : Station Height (metres) Increase of radiation per 1 mm of vapour pressure at the instrument. Radiation being in g cal cm~ 2 min" 1 . Temperature constant at 293 C. Mt Whitney 4420) 016 Mt San Gorgonio ... 3500) Mt San Antonio ... Lone Pine Canyon 3000] 2500] 017 Bassour 1160 007 Lone Pine ... 1140 009 Indio 005 Thus the down-coming radiation is most increased by water vapour in the higher levels. That is as we should expect, knowing that the radiation of water vapour is concentrated in bands. For as soon as the full temperature radiation is attained in the band, increases will only follow the increase of temperature. It would be desirable to know the absorptivity A as a function of ^, the mass of water- vapour per mass of atmosphere. The relationship could be found, if of the form A = (value for dry air) {1 -I- (constant) fi}, by pursuing step-by-step computations like those made above for dry air. It would be necessary to have both p, and the temperature given as functions of height. A very rough calculation of this kind, based on the mean of the above data, indicates that the constant is of the order of 40, so that A=l-48xlO- 3 (l+40//,)cm 2 grm- 1 (17) 56 THE FUNDAMENTAL EQUATIONS CH. 4/7/1 The definition of A is contained in equation (l) above. The absorptances of the con- ventional strata might be taken to be increased in the same ratio (1 + 40/U,), where p. is put equal to W/R, the total mass of water divided by the total mass of atmosphere in the stratum. But no treatment will be satisfactory until the wave lengths can be grouped. Approximate simplified process. It is proposed to diminish the amount of computing by treating the hemisphere as a whole, instead of in separate zones. For the resultant flux of radiation is almost invariably vertical. Near the edge of a clouded area, the resultant could have a horizontal component, but this exceptional case has been neglected. The whole radiation falling on one side of a horizontal unit area has already been denoted by %E which is denned by equation (IS). We require to find, for 'S.E, an equation corresponding to (12), for the brightness 1. Now if we multiply (12) throughout by cos . 2ir sin . dt, and integrate over the hemisphere, the left-hand side transforms to 2-Ev The term in 77 / 6 does not however transform into one in ~S,E t . But if 77 were independent of , and equal to rj, then equation (12) would transform into 2 8 = (1-77) 2^ + 773^, (18) which is accordingly taken as an empirical equation used to define a mean absorptance 77 for descending radiation distributed in all directions in the actual manner. In the last term of (18) 3# 4 is the radiation emitted by unit area of a perfectly black plane, to one side. Now S-E 1 is given in the last column of the table on p. 53, which is based on Angstrom's observations. From this column, by use of (18), the values of rj have been deduced for the three strata shown in the table. They may be stated by giving the mean zenith distance which must be inserted in equation (13) in order to make 77 equal to rj. The advantage of giving the angle is that it is probably not affected by the small error in A in the table. The result is, for strata approxi- mately the same as our three upper conventional ones of 2 decibars thickness, Po P-2 P* Pe = 57-l 54-5 54-7 (1 radian nearly) It is seen that the descending radiation is more horizontal above, more vertical below. The mean angle I may also be different for ascending radiation. More obser- vational values are needed. This simplified process will be illustrated in connection with the data for 20 May 1910, which will be discussed in Ch. 9. There is taken as 55 throughout, A is taken as 1'48 x 10~ 3 (1 + 40ju) cm 8 grm~ l , A is calculated by putting in place of in (13), and tE is traced from stratum to stratum by means of (18). Mr W. H. Dines* has used this simplified process extensively to compute the radiation at all heights according to various trial values of the absorptance. * Q. J. R. Met. Soc. April 1920. CH. 4/7/1, 2 SOLAR RADIATION 57 Clouds. A. Angstrom in the introductory summary to his paper makes a state- ment which implies that low and dense clouds behave almost as black bodies to the long-wave nocturnal radiation. The recent observations of W. H. Dines confirm this. Surface of Sea. The radiation from water may be calculated by Kirchhoff's law from its reflectivity, which in turn may be deduced, by Fresnel's formulae, from its refractive index. The reflectivity varies from unity for grazing incidence to 0'02 for normal incidence. But for rays distributed as they would be in all directions A. Angstrom has computed, in this way, that a flat water surface gives out 94 / of the radiation from a black surface. Some Papers on Long- Wave Radiation E. GOLD. "The Isothermal Layer... and Atmospheric Radiation." Roy. Soc. Proc. A, Vol. 82, pp. 43 to 70. E. GOLD. Q. J. H. Met. Soc. Oct. 1913. o A. ANGSTROM. "A Study of the Radiation of the Atmosphere." Smithsonian Miscellaneous Collections, Vol. 65, No. 3(1915). o A. ANGSTROM. "Uber die Gegenstrahlung der Atmosphare." Meteor. Zeit. Heft 12 (1916) and Heft 1 (1917). F. E. FOWLE. " Water-vapour Transparency to Low-temperature Radiation." Smithsonian Miscel- laneous Collections, Vol. 68, No. 8 (1917). o A. ANGSTROM. " On the Radiation and Temperature of Snow and the convection of the air at its surface." Arkiv for Mat. Astr. och Fysik. Stockholm, 1918. A. ANGSTROM. "Determination of the constants of Pyrgeometers." Arkiv for Matematik. Stock- holm, 1918. C. G. ABBOT. " Terrestrial Temperature and Atmospheric Absorption." Proc. National Academy of Sciences, U.S.A., Vol. 4, pp. 104-106 (1918). M. A. BOUTARIC. " Contribution a 1'etude du pouvoir absorbant de 1'atmosphere terrestre." Theses a lafaculte des sciences de Paris. Gauthier-Villars et Cie, 1918. W. H. DINES. "Atmospheric and Terrestrial Radiation." Q. J. R. Met. Soc. April 1920. CH. 4/7/2. SOLAR RADIATION General What we require to find is the energy absorbed by each stratum of the atmo- sphere and by the soil, vegetation or sea. Scattering must be taken into account. The fraction of the direct solar beam transmitted by a horizontal layer, Sp units of pressure in thickness, is Vxp{-8p.sect(f(l) + Bl-% (1) where B is a quantity depending only on water and dust and I is the wave length. K. 8 58 THE FUNDAMENTAL EQUATIONS OH. 4/7/ The part in f(l) represents the loss by absorption. The loss varies irregularly with wave length, from a large value in certain bands due to oxygen or water vapour, to a much smaller value in the continuous spectrum between the bands. The other part in Bl~* represents the loss due to scattering. Scattering does not warm the air. The scattered radiation spreads in all directions, but not equally so. Its intensity in a direction making an angle A with the incident beam has been found* by Rayleigh, by Kelvin and by Schuster, to be proportional to 1 + (cos A ) 2 (2 ) The important point about this formula, for present purposes, is its symmetry: the radiations scattered in any two opposite directions are equal. In its course through the atmosphere, the scattered radiation is partly absorbed and partly repeatedly scattered. Some of it eventually goes off to space and the rest comes to earth as skyshine. It will be convenient to consider firstly the radiation reaching the upper atmo- sphere from the sun and secondly its distribution in space. Sunshine above the Atmosphere Abbot, Fowle and Aldrichf by extrapolating, for each wave length separately, from observations made at large and small zenith distances of the sun, found the distribution of energy in the solar spectrum outside the atmosphere; and then by integrating with respect to wave length they found the total energy. Their mean value for this total is T93 calories cm~ 2 min~' which is equivalent to l'16x 10 11 ergs cm~ 2 per mean solar day ...................... (3) By simultaneous observations at different stations Abbot | has found evidence of small variations in the total radiation, variations having a period of the order of a few days, weeks or months. Until some way of predicting these variations can be found, we must just neglect them. The small effect of the seasonal change of the sun's distance from the earth is easily taken into account by reference to astronomical tables. We require the radiation falling on a unit horizontal area. At any instant this is proportional to cos where is the sun's zenith distance. The mean value of cos during a time-step can be found by expressing in terms of the sun's hour angle and declination and then integrating. The result is that the mean value of cos , between two times t'_ and /, reckoned in mean solar days from local apparent noon, is / , v j i \ 1 / i si n 27rfc/ sin 2irtj' sm (north decl.)x sin <+ cos (north decl.) x cos - , -- ....... (4) ZTT I 2 *i From this formula the following numerical values have been calculated. They refer to the example of Ch. 9 which is based on observations made at about 8 h local apparent time ( = 7 h G.M.T.) on 20 May 1910. * Vide L. V. King, Phil. Trans. A, 1913, p. 376. t Smithsonian Astrophysical Annals, Vol. in. \ Nature, July 29, 1920. CH. 4/7/2 SIMPLIFYING APPROXIMATIONS FOR SOLAR RADIATION 59 Mean value of cosine of sun's zenith distance 1910 May 20 d. Kilometres north of equator 5 h to 11" ll h to 17 h local local apparent time apparent time 6200 0-522 0-693 6000 0-523 0-706 5800 0-528 0^718 5600 0-531 0-729 1 5400 0-535 0-740 5200 0-537 0-750 5000 0-539 0-759 4800 0-541 0-766 Transmission of Solar Radiation through the Atmosphere To be exact we should have to follow the Smithsonian observers in treating each wave length separately, and L. V. King in considering scattering as occurring repeatedly. All this might be practicable if the atmosphere could be treated always as a single stratum, but such a treatment would break down as soon as clouds inter- vened. Cloudiness or transparency are of course the most important atmospheric properties in connection with solar radiation, and everything else must be subordi- nated to the distinction between them. Thus it becomes necessary to keep account, separately, of the radiation passing through the several conventional strata; and then, to economize arithmetic, approximations must be sought elsewhere. The fol- lowing approximations, while greatly reducing the work, would appear to permit an accuracy of a few per cent in the final result. (i) Dust is left to be estimated statistically. That is to say it is not included, as water is, among the dependent variables, whose history we attempt to trace throughout. (ii) We neglect the radiation passing from any horizontal coordinate chequer, of 200 kilometres of latitude times 2 49' of longitude, to the neighbouring chequers. This point may be illustrated by the fact that the curvature of the earth prevents a cloud, above the centre of one of these chequers, from casting a shadow on the centre of the next chequer, unless the cloud is more than 3 kilometres above sea-level. Even for the highest clouds it would only be possible if the sun were so low as to have a negligible heating effect. (iii) As the zenith distance of the sun changes largely during our time-step of six hours, a mean value of sec must be inserted in all transmission formulae such as (l). A mean value of cos , for the same interval, has in any case to be calculated from (4), 82 60 THE FUNDAMENTAL EQUATIONS CH. 4/7/2 and its reciprocal may be used for mean sec . A much better mean value of the factor e --Bsec0 .2~~8 S X P I 1 9 . lif*" i 6 "Sfe CO 05 CO c 11 n^.S S 00 6 B f"" S T5 O -a ^ co O5 .3 Q) qj 3 HH 1 1 OJ g OS O a o ^ 03 O i i S o i i a o as H t 1 T 1 12 be Column number I -tf 1 a difference between Mt. Whitney and Mt. Wilson CD i O T3 s PS CH. 4/7/2 NUMERICAL CONSTANTS FOR SOLAR RADIATION The results are set out in the table below, and refer to clear air 63 Bands Remainder Water substance per nms-^ - /< See note below Absorptivity per density = A Scatterivity per density = jt Absorptivity per density = A Scatterivity per density = J* Air above Mt. Whitney 2-28 x 10- 4 0-76 x 10- 4 0-24 x 10- 4 1-60 xlO- 4 0-135 x 10- 3 Air between levels of Mt. Whitney and Mt. Wilson 12-8 x 10- 4 0-6 x 10- 4 0-43 x 10- 4 ^ 1-98 xlO- 4 2-83 x 10- 3 Note. The measurements of the absorption in the bands were made on isolated occasions when the water content may have differed from the mean value given in the last column. Clouds. Abbot states that thick clouds reflect about 65 / of the solar radiation falling upon them. Photometric measurements* indicate that the fraction of red light transmitted by uniform stratus is about 25 / o for an ordinary cloud and 8 / for a very dark one. Thus the effect of clouds appears to be mainly a scattering. How much they absorb is apparently not yet known exactly. Routine Process. The foregoing principles and data have been applied to the observations in Ch. 9. The procedure was as follows in any group of ranges of wave lengths. Take the "bands" group for illustration. First A and / were obtained for each stratum, by using the values given in the table above, and assuming that A and / were linear functions of the water-per-mass, p.. It was concluded that the sky was cloudless, because the water-content was every- where low, and the period considered, 5 h to ll h by local time, too early in the day for much cumulus cloud. Next the fraction of energy not transmitted was formed for each stratum, and thence the fraction absorbed A 860 and the fraction scattered These are all fractions of the energy incident on the particular stratum. They were entered on the computing form which is printed in Ch. 9. Next 15 / of the incident radiation was entered opposite A in the column headed "Direct beam, flux." The direct beam was then traced down stratum by stratum, deducting the amounts ab- sorbed or scattered, until the vegetation was reached. The downward flux of diffuse * Roy. Soc. Proc. A, Vol. 96, pp. 23 and 31 and similar measurements now in progress. 64 THE FUND AIM ENTAL EQUATIONS CH. 4/7/2 radiation was next traced, starting from zero at the top h a . In each stratum one half of the energy scattered from the direct beam was added to the downward flux of diffuse radiation. From this sum was deducted the amount of ditfuse radiation absorbed by the stratum, the amount absorbed being taken as a known fraction, corresponding to = 55, of the diffuse radiation entering the stratum from above. Thus diffuse radiation was brought down to the vegetation film. The portions of the descending radiation, direct or diffuse, which were considered as reflected by the vegetation, were added and transferred to the foot of the column of ascending diffuse radiation, at the level h L . The ascending radiation was then traced upwards, subtracting in each stratum the fraction absorbed, and adding half the energy scattered from the direct beam in that stratum, until h was reached, where the radiation left the earth for interplanetary spa-ce. In the last column of the computing form, the energy absorbed by the several strata, and by the earth, was added up. Finally the total energy received from the sun was checked against the total of all absorbed plus the loss to space. Publications on Solar Radiation ABBOT, FOWLE and ALDRICH. Smithsonian Astr&physical Annals, Vol. in. and Smithsonian Miscell. Collections, Vol. 65, No. 4. F. LINDHOI.M. " Extinction des Radiations solaires dans 1'atmosphere terrestre." Nova Acta Upsaliemis, Ser. IV. Vol. 3, No. 6 (1913). L. V. KING. Phil. Trans. A, Vol. 212 and Nature, July 30, 1914. R. S. WHIPPLE. "Instruments for the Measurement of Solar Radiation." Trans. Optical Society, London, 1915. W. H. DINES. " Heat Balance of the Atmosphere." Q. J. R. Met. Soc. Apr. 1917. SIR NAPIER SHAW. "Memorandum on Atmospheric Visibility," Feb. 1918. Hydrographic Dept., Admiralty, London. F. E. FOWLE. "The Atmospheric Scattering of Light." Smithsonian Misc. Collections, Vol. 69, No. 3 (May 1918). C. G. ABBOT. "Terrestrial Temperature and Atmospheric Absorption." Proc. Nat. Acad. Sci., U.S.A., April 1918. M. A. BOUTARIC. " Contribution a 1'etude du pouvoir absorbant de 1'atmosphere terrestre." Theses a la facult^ des sciences de Paris. Gauthier-Villars et Cie, 1918. F. LINDHOLM. "Sur 1'insolation dans la Suede Septentrionale." Kungl. Svenska Vet. Hand. Bd. 60, No. 2 (Wm Wesley & Son, 28 Essex St., Strand). A. ANGSTROM. " Uber die Schatzung der Bewolkung." Met. Zeit. Heft 9/10, 1919. C. G. ABBOT. Nature, July 29, 1920. CH. 4/8 TURBULENCE 65 CH. 4/8. THE EFFECTS OF EDDY MOTION Note: Numerical references attached to persons' names in Ch. 4/8 refer to the bibliography on p. 92. CH. 4/8/0. GENERAL The kinetic theory of gases has revealed to us that the properties of viscosity, diffusion, and conduction of heat, which are attributable to a gas enclosed in a small vessel, are in reality due to molecular motions which we cannot follow in detail. In the same way in the atmosphere, many varieties of motipii which we cannot or do not wish to record in detail, can be ignored; provided that their general statistical effect is taken into account by adding to the equations, describing the general motion, appro- priate additional terms. This was very clearly brought out by Osborne Reynolds (3) in connection with viscosity. He took the mean value of the dynamical equations and found that they were of exactly the same form in the mean velocities as in the actual velocities; except that the quadratic terms introduced, on taking the mean, the body force derived from a system of "eddy stresses." This was true however large was the interval of time or the volume over which the mean was taken, with the limitation that the eddies within this time and space must be sufficiently numerous*. Our theory and constants must be appropriate to the size of the element of the fluid which we treat as a "differential" in that we ignore the details of "any motions taking place wholly within it f . The upper limit to the size of an eddy is, like the length of a piece of string, a matter of human convenience. When an airman says that the wind is "bumpy" he is thinking in terms of a differential element probably comparable in size with one wing of a flying machine. For the present purpose any motion which disappears on taking the average over our coordinate intervals (St = 6 hours, SX=3, S< = 200 km, S/i= mass upwards) has of necessity to be ignored. If it occurs in large numbers as cumulus eddies do for example then its general effect can be satisfactorily represented by additional terms; although unfortunately this does not help us for example to say whether it will hail or not on Mr X's field. It has been customary to separate circulatory motions in the atmosphere into two main groups according as they derive their energy from the local heating of the lower layers of air ("convection") or from the kinetic energy of the wind ("dynamical instability"). It is easy to classify a thunder-cloud as belonging to the conventional type, or a gust in a gale as being an example of the other extreme. But recent researches have shown the existence of numerous intermediate forms, which derive their energy from both sources jointly. Thus Akerblom (8) found that the eddy viscosity at the Eiffel Tower was greater in summer, when the supply of heat for con- vection was greater. Hesselberg (13) obtained a similar result for the wind 500 metres above Lindenberg. Again Taylor (21) found a very large seasonal variation, in the same sense, for the diffusion of heat as deduced from the Eiffel Tower temperatures. * See also Ch. 4/9/5 below. t Bryan, Thermodynamics (Teubner), Art. 46; Jeans, Dynamical Theory of Gases, Art. 11. K. 9 66 THE FUNDAMENTAL EQUATIONS CH. 4/8/0 Exceptionally low diffusivities have been measured at night by L. F. Richardson (32) in the cold air near the earth. Airmen are very familiar with the increased bumpiness of the wind caused by sun shining on the ground below them. All these facts show that the production of eddies in the wind is greatly facilitated when the thermal equilibrium becomes less stable, although we may not suppose that actual thermal instability is reached in the majority of cases, because such an event is unusual among the collected observations made either by registering balloons or from aeroplanes. A quantitative theory of the criterion of turbulence has been given by L. F. Richardson (32). On the other hand we find that convectional motions are hindered by the formation of small eddies resembling those due to dynamical instability. Thus C. K. M. Douglas writing of observations from aeroplanes remarks : "The upward currents of large cumuli give rise to much turbulence within, below, and around the clouds, and the structure of the clouds is often very complex." One gets a similar impression when making a drawing of a rising cumulus from a fixed point; the details change before the sketch can be completed. We realize thus that: big whirls have little whirls that feed on their velocity, and little whirls have lesser whirls and so on to viscosity in the molecular sense. Thus, because it is not possible to separate eddies into clearly defined classes according to the source of their energy ; and as there is no object, for present purposes, in making a distinction based on size between cumulus eddies and eddies a few metres in diameter (since both are small compared with our coordinate chequer), therefore a single coefficient is used to represent the effect produced by eddies of all sizes and descriptions. We have then to study the variations of this coefficient. But first we must consider the differential equation. In doing so the aim has been to lay down theoretically only so much as can be determined with strictness, leaving all un- certainties to be decided by observation. In hydrodynamics or aerodynamics it is customary to speak of the motions of " definite portions" of the fluid, portions which may be marked by a dot of milk in water or of smoke in air. The capital D in D/Dt is commonly used to denote a time differentiation following such a definite element. It is customary to ignore the fact that molecules are constantly passing in and out of the element called "definite." When we have to deal with eddies, the interchanges are more conspicuous, for boundaries marked by smoke would rapidly fade and disperse. Yet some way must be found of specifying an element which follows the mean motion. The fundamental idea seems to be the following. When there are no eddies we are accustomed to compute the flow of entropy or water across a plane from the flow of mass across the plane. As the effect of eddies is to be treated as additional, it should not include any flow due to the mean motion of mass across a plane. Accordingly we should adopt some such definition as the following: Draw a sphere in the fluid. Let the radius be as large as is necessary to include a considerable number of eddies, but no larger. Let the sphere move so that the whole momentum of the fluid inside it is equal to the mass of the same fluid multiplied OH. 4/8/0 THE EQUATION FOR EDDY- DIFFUSION 67 by the vector velocity of the centre of the sphere. The centre may then be said to be "a point moving with the mean-motion." ............................................. (l) As bars denote mean values the capital letter with a bar over it in D/Dt may suitably denote a differential following the mean motion. Here we may usefully bear in mind the analogy with the conduction of heat in a solid. The total water in a portion of air, or the mean potential temperature of it, is not altered by gently mixing it at constant pressure, provided precipitation does not occur. If, then, Z stands for either of these quantities or for any other quantity, which Has its total, for a secluded portion of air, unchanged by the internal rearrangement of that portion ....................................................... (2) Or by delay, ................................................................................. (3) then the rate of increase of the total Z in the portion must be due to .Z flowing in over the sides. As the variations of moisture, entropy and velocity in the atmosphere are usually much more rapid vertically than horizontally, the ensuing treatment is confined to the vertical diffusion. Consider a large horizontal surface moving so that there is no net flow of mass across it. Let us define the "upward eddy flux" of the quantity Z across any such surface as the ratio of the amount of Z rising across the surface in unit time to the area of the surface, which is supposed to intersect many eddies ..................................... (4) Then rate of increase of Z per area in the layer dh bounded by two surfaces which rise and fall with the mean motion must be dh ^-(upward flux) ............................... (5) If now x be defined as the amount of Z per unit mass of atmosphere, then The amount of Z per area in dh is \pdh, ..................... (6) so that _ = _ dh j_ ( U p war d flux) ......................... (7) Now we can define a coefficient c such that jj upward flux of Z= c of > ........................... (8) where c might be called the "eddy-conductivity"* when the quantity transferred is moisture or energy; or it might be called the "eddy- viscosity" when we have to do with exchanges of momentum. The coefficient c corresponds closely but not exactly to the "Austausch" which W. Schmidt of Vienna denotes by the symbol A. When c is defined by (8) it will be for observation to show whether c is a constant, as would * Since I adopted this name (Bibliog. 32, p. 2) I see that G. I. Taylor (12, p. 4) had previously used it to denote another quantity. To avoid muddles it is best to refer always to the differential equation for diffusion. 92 68 THE FUNDAMENTAL EQUATIONS CH. 4/8/0 be nice, or whether c depends on various factors, and in particular on S-^/dh = 0. But, if c is to be useful, it must not be infinite when d-^/Sh = 0. Therefore we had better confine ourselves to conditions in which The upward flux vanishes when 3-^/Sh ................... (9) Now this implies that: X must be unchanged by the simple transportation of air to a different level. . . .(10) Otherwise expressed D^/Dt where D denotes a differentiation following the eddying motion ................................................................ (1 Oa) Now x is the total Z per unit mass of a definite portion of air. So, as the mass is unchanged by transport, (10) implies that: The total Z in a definite portion of air must be unchanged by its removal to a different level ................................................................... (11) Fortunately, (11) is satisfied by the same two quantities which previously satisfied (2), so that (10) is satisfied when ^ is either mass-of-water-per- mass-of-atmosphere or else potential-temperature ............................ (II a) If, in place of (8), we had made the flux proportional to c'd (p\)fih where p% is the total amount of Z per volume instead of per mass of atmosphere then, to avoid an infinite value of c, the flux would have to vanish when 3 (p^}/dh vanished. That would not be possible for the two given meanings of x> because rising air changes its volume. So it appears that we had better keep to the definition (8). Inserting (8) in (7) we have D ^ X ^ = dh ~ (M\ ............................ (116) for the layer is bounded by surfaces moving with the mean motion. Then (ll&) reduces, without approximation, to Dt It is sometimes more convenient to use pressure as a measure of height by means of the transformation dp= gpdh. At the same time, in order to get a single constant, let us put = ....................................... (14) then (13) becomes (15} where ^ may have either of the meanings ( 1 1 a). The dimensions off are (mass) 2 (length)" 2 (time)~ 5 . Pressure is peculiarly convenient if we can neglect any lateral convergence of CH. 4/8/0 DIFFUSION OF MOMENTUM BY EDDIES 69 wind, for then an isobaric surface moves up and down with the mean motion, so that when t and p are the independent variables, D/Dt is the same as d/dt and (15) becomes (16) The flux is given by (5), (8), and (14), that is to say, if x is the amount of Z per unit mass of atmosphere, then the amount of Z rising across a large horizontal surface per area per time, when no mass traverses the surface on the average, is -f* or equivalently,-CLJ| ......................... < 17 ) The quantity ^ may have either of the meanings (11 a). The conductivity c or the turbulivity f are left to be determined by observation, but we know beforehand that they are finite when d%/dh = 0. No attempt is here made to show that they are the same for various meanings of x- It should be pointed out* that entropy per mass cr cannot stand for x in the above equations. Because, when the variegated structure produced by eddies is smoothed out by molecular diffusion at constant pressure, then the entropy increases and so does not satisfy (2) or (3); although the mean potential temperature of a secluded portion of air remains unchanged. The equation for diffusion of entropy per mass a- has to be derived from that for potential temperature T. For dry air dcr = y p d log T, .................................... (18) so that (13) leads to Dt pdhV ah i (1 D', the deviation from the mean. The question is whether, if dv/dh vanish at any level, an upward flux of momentum can cross that level? * In view of a mistake of mine in Roy. Soc. Proc. A, Vol. 96 (1919), p. 9 from which paper the above theory is taken, with improvements. 70 THE FUNDAMENTAL EQUATIONS On. 4/8/0 In other words, can there be an eddy-shearing-stress where there is no rate-of-mean- shear? The converse, of course, occurs where there are no eddies. Or to put the question over again in another way : imagine two adjacent horizontal layers each say 100 metres thick and having the same mean velocity; would it be possible for the faster moving portions to sort themselves out and to flock together into one layer, leaving the slower moving portions in the other ? If molecules did that sort of thing, the occurrence would be one of the exceedingly rare exceptions to the second law of thermodynamics. We may reasonably expect that there is a corresponding law, on an enlarged scale so to speak, applicable to the statistical mechanics of eddy motion. The only other possibility seems to be that portions of air should wander in from quite distant regions, giving a part of the vertical flux of momentum proportional to 2 3 v/dh 3 and a term in (15) proportional to 3 4 t>/9A 4 . When V E plays the part of ^ then c plays the part of viscosity, as may be seen by imagining (13) as a dynamical equation from which pressure gradient and geostrophic wind have been omitted because they just balanced each other. To sum up: we may expect equations (13) and (15) to be an exact expression of the effect of mixing when ^ is either: potential temperature (but not entropy per mass), mass of water in all its phases jointly per mass of atmosphere, horizontal velocity in a fixed azimuth, or any other quantity satisfying (2), (3) and (9). But there is no evidence from this theory as to whether the coefficients c and are the same for these different meanings of x- The Eddy-Flux of heat behaves in a manner remarkably different from the flux of heat in a solid. If x in the above general theory be given its permissible special meaning of the potential temperature r, then Z is (potential temperature) x (mass) and the amount of Z rising per second across a unit horizontal area which moves with the mean motion is c . dr/Sh. The flux of heat may be said to be the flux of (mass) x (ordinary temperature) x (specific heat at constant pressure), and this will be y p (p/p')' w times the flux of Z. That is to say the flux of heat will be, in dry air, V'- 1 ' ST .................................. < 21 > Let us consider the simplest case. Suppose that the temperature does not change at points moving with the mean motion, and that the pressure does not change either. Then Dr/Dt Q and from (13) it follows that, as far as eddies are concerned - ( c -^ } = 0. so that c - 7 - is independent of height. dh \ dh] dh CH. 4/8/0, i THE EDDY-FLUX OF HEAT 71 Therefore from (21) we see that in this steady state the flux of heat depends on the height. The explanation of this distribution so different from that in a solid is that heat is being transformed into the kinetic energy of eddies or vice versa as it goes. The matter has been treated by the author (33, p. 354). If T increases upwards, then heat flows down ; but then also the atmosphere is stable, and as it is stirred kinetic energy is drawn from the mean wind and becomes, partly immediately heat, and partly first eddying kinetic energy. The latter part ultimately, by molecular dissi- pation, is converted to heat. If, on the contrary, T increases downwards, then the flow of heat is upwards; but also the equilibrium is unstable even in the absence of wind, and heat is being converted into eddying kinetic energy. Ultimately this is also dissipated as heat. But in the meantime, provided Dr/Dt 0, then whichever sign may have, the flux of heat, in conformity with equation (21), is greater below. CH. 4/8/1. COLLECTION OF OBSERVATIONS OF VISCOSITY AND CONDUCTIVITIES Although these are put down as values of c in equation (13) yet very many of the observers have gone on the unreliable hypothesis that c was independent of height so that ., [c -y ) was taken by them as - pSh\ oh) pdh- The most striking fact is the large value of the viscosity which eddies give to the atmosphere. Thus several observers have found viscosities of the order of 100 c. G.s. units at a height of 100 to 1000 metres above land. Glycerine in the same units has a viscosity of 46 at C., Lyle's Golden Syrup 1400 at 12 C. However, when the forces resisted by viscosity are produced by the inertia of the fluid, we should compare fluids by their ratio of viscosity to density. For the atmosphere this ratio attains values of the order of 10 5 C.G. s. units, a figure which lies between 10 s for golden syrup at 12 C. and 5xl0 6 for shoemakers' wax at 8 C. (The laboratory data are taken from Kaye and Laby's tables.) In the adjoining columns are set out various circumstances which are likely to have influenced the state of eddying. They will be examined later. I have endeavoured to pick them out to fit the same ranges of height, place and time as the observations of turbulence. The coefficients c v , C M , C T mean the values of c in equation (13) which apply when the diffusing quantity is respectively v, /u, or T. Thus c v is the eddy-viscosity and is not to be mistaken for the specific heat at constant volume, for which, following G. H. Bryan, I use the symbol y v . 72 THE FUNDAMENTAL EQUATIONS CH. 4/8/1 Observed values of the Eddy-viscosity c v for velocity The x axis is directed with the wind at the mean level e, prm (W ( 8p "Y g So- y p ;ih sec" 2 10-4 x Height above ground Metres Wind near surface Character of surface Remarks \Sh) seer* 10-4 x \Sh) sec" 2 10-' x Velocity cm sec" 1 At height cms secern F. AKER BLOM (8). From variation of s nnd with heigV it, allowin g for poss ble variation of baron letric gradient. 83 6-8 0-4? 2-4 ) 21 to 239 2100 Paris, winter | Eiffel Tower 113 4-1 0-8? 1-6 J 305 205 2100 Paris, summer j winds G. I. TAYLOR (11) and (12). From variation of wind with height, not allowing for variation of barometric gradient. 70 ) 950 | 56 to 1 1700 590 I 3000 Salisbury Plain G. M. Dobson's Balloons 32 J 330 J 0-9 > to 8-Of mean 2-0 mean 1-8 mean 6-0 to 200 Sea, Newfoundland "Scotia" kites 0-9 to 4-8 1-8 3-4 0-9 to 200 440 1000? Sea IDl^Aug^n 1 ' HKSSELBKRG and SVEBDHUP (14). From variation of wind and barometric gradient with height. 0-9 to 9 Stevenson and 40 2-6 1-7 !2-6 9 to 209 1 Hellmann obsns. 50 0-5 0-8 for to 109 to 309 50 0-1 02 500 at 8a.m. 209 to 4Q9 , 564 Anem. 900 Lindenberg 60 00 o-o 309 to 509 f not allowing for variation of barometric gradient 50 9 to 3009 ! W. SCHMIDT (18). Assuming that very near the ground the vertical flux of momentum is independent of height and therefore c^'jdh constant. [30 assumed as basis at height of 200] 2-6 9 Hellmann, 1-3 2 Anemometers 1-1 i H. U. SVEKDRUP (19). Variation of velocity with height in the North Atlantic Trade Wind, allowing for variation of pressure gradient. 140 0-08 0-00 1-7 to 1200 Trade wind (zero near sea) 260 0-01 0-32 2-6 1200 to 1800 . 640 1000? Sea Intermediate layer 90 0-07 0-02 2-6 1800 to 2000 Anti-trade wind CH. 4/8/1 EDDY-VISCOSITY 73 Observed values of the Eddy-viscosity c v for velocity (continued) The x axis is directed with the wind at the mean level gnu (*$? (rV \Sk) sec~-' 10-1 x /; dff Height above ground Metres Wind near surface Character of surface Remarks UJ seo~ 2 10-" x y,,Sh sec" 2 10 4 x Velocity cm see" 1 At height cms sec cm L. F. R ICHARDSOJ r (32). By integrating the loss of momentum with respect to height from a level chosen so that dv/Sh and consequently the stress there both vanish. Neglecting variation of pressure gradient. 15 110 parallel to wind horizontally across wind I 400 ) Lindenberg General mean L. F. RICHARDSON (32). From the speed of ascent of cumuli, the height through which they rise and the fraction of the sky covered. 1030 Zero Zero Nearl y to 2000 Zero zero Land Midday Observed values of the Eddy-conductivity c for dust, smoke or other floating solids c grm (3>'*\ ft*Y a S^ z y ^\ o) r^j ( 3i *" a 9 -^ V I\ JE N \ \ X ^ \ _@ 4 X s X ^ ^ *' i i 1 ^ i 1 -* i i 10 20' Notation for diffusing substance 30 80 f (/dVx\ 2 /^/X 2 ) / 7 3 where p 7 p 5 may be taken as ^ (p^p^ or as 5 (-^ 8 + ^)- Now / is approximately the same, and equal to 2 decibars, for all conventional strata except the lowest, so that the shearing stress at the h t . level is nearly And the differences between the shearing stresses at the different levels are the additional terms required in the dynamical equations quoted above. Water, when IF is given. Putting W, the mass of water per horizontal area, in place of M in (2) we have for the water crossing unit area of a horizontal plane at height h e IW W \ I "' 86 _ '' 64 \ & \ ^86 -&B4 / /,\ fl^(/k-/g ' This is the effect of eddies alone. The change due to any general motion of air up or down is dealt with under the "conveyance of water," Ch. 4/3. Water, alternative scheme, /A given at boundaries of strata. (Compare part of Ch. 4/3.) In the equation it is now most simple to put ~ approximately equal to (6) for the stratum h s to h K , which may serve as a type. Similarly for the stratum above. So that (5) transforms into f j. p s P-e _ f fa t CH. 4/8/2 DIFFUSION TREATED BY UPPER CONVENTIONAL STRATA 81 giving the time change of /x correctly centered at the boundary where we need it. Note that f is now required at the levels, with odd suffixes, corresponding to the mean pressure of the strata. As p 7 and p 5 are not tabulated variables we must put (8) Or indeed, in view of the uncertainty of f, it may be near enough to make both the 8p simply equal to their average value of 200 millibars. It hardly seems worth while making use of the fact that, on the average of many occasions. /j. = Cp B where C and B vary slowly with Jiejght, as was done in Ch. 4/3, until we know more about the variations off with height, as they also are involved. In the equation for the conveyance of water (Ch. 4/3 #25) there is a term in , which is shown in that section to transform thus 1 Dt g Dt Dt In so far as this term arises, not from precipitation, but from the eddy-flux, the values of D^/Dt for insertion in it are given by (7) above. Potential Temperature. The mean potential temperature of a stratum follows from the pressures at its boundaries and from its water content per horizontal area. Having found this mean it is to be inserted in place of W/R in the numerator of (4). The result is the rate at which temperature x mass is crossing unit horizontal area. Again the effect of any general vertical motion has been separately formulated in Ch. 4/5/2. As to the method of finding the mean potential temperature of the stratum when dry or at least clear. In the troposphere, on the average, T only varies by 5 or 10 / o of itself in the thickness of a stratum, so that it does not much matter whether we take an arithmetic, geometric, or harmonic mean. For dry air exactly, and very closely for clear air, T = (p'/p) ' 289 , ...................................................... (10) where p' is the standard pressure used in defining T, so that the vertical static equation may be written* *p_ _yp. 9Plp'\ ni) dh~ - Jf> - W~ ~br\p) ' ........ whence r= -0'289 (pj. ~^- ......................... (12) * Vide Exner, Dynamische Meteorologie, 70. 11 82 THE FUNDAMENTAL EQUATIONS CH. 4/8/2, 3 It follows that a mean potential temperature is given by the following fairly simple formula : ?= -0-289? 'pV*t_ p VK .(13) Tables of '2 8 9th powers would be a useful aid. For cloudy air we come in for more complications, as in Ch. 4/1 and Ch. 4/5/1. CH. 4/8/3. EDDY SHEARING STRESS ON THE EARTH'S SURFACE When x ls velocity we require the eddy shearing stress on the ground to be given by statistics as a function of such variables as are available, namely of : the position on the map, the date, the mean temperature 6 G8 of the lowest stratum, the tempera- ture of the sea, of the bare earth or of the air in the vegetation film, and the mean momentum per volume of the lowest stratum. For brevity the last-named quantity will be denoted by m. It has components m E = M EGB /(h s -h c ,); m N = M NG8 /(h s -h G ) (l) o Akerblom appears to have been the first to get a fairly reliable measure of the surface stress. He deduced it from Angot's statistics of wind observations at the top of the Eiffel Tower (305 metres) and at the Bureau Central (21 m) by assuming the eddy- viscosity to be independent of height. We now know that this assumption is hardly permissible. Akerblom, F. (8). Shearing stress on surface dyne cm~ 2 Velocity cm sec" 1 at Bureau Central 21 m Velocity at 305 m Eiffel Tower cm sec" 1 0-40 234 982 Winter 0-29 209 777 Summer Hesselberg and Sverdrup (14), treating the same data by a different method, confirm the above result. G. I. Taylor (16) illuminated the subject by comparing, by aid of the theory of dimensions, the friction on the earth with that obtained on small metal plates at much higher velocities in laboratories. By analysing G. M. Dobson's Pilot balloon observations made over Salisbury Plain, Taylor showed that the surface stress was more or less proportional to the square of the velocity "near the surface." Like Akerblom he assumed the eddy-viscosity to be independent of height. On. 4/8/3 G. I. Taylor (16). SURFACE FRICTION 83 Stress dynes cm- 2 Gradient wind cm sec" 1 m grm sec" 1 cm" 2 0-31 460 1-37 910 2-44 1560 1-70 As velocity varies so rapidly with height near the surface it may well be that the upper wind represented by m provides a standard more suitable than the wind near the surface. In the present scheme at any rate we have to use m, for it is the only measure of wind available. W. V. Ekman(6) in 1905 pointed out, with reference to the sea when a steady state has been attained, that the total momentum, produced by a tangential stress on the surface, is directed at right angles to the stress, and is equal in magnitude to the stress divided by 2w sin . Thus the momentum produced by the stress is quite independent of the value of the viscosity or of its variation with height. This is true provided that the quadratic terms in the dynamical equations produce only a negligible disturbance, and that will be so if we take, as our standard of what the momentum would be in the absence of stress, not the wind deduced from the isobaric map, but the actual upper wind. Even that approximation will fail near the equator. A high level should be chosen at which the stress vanishes because dv/dh vanishes there, and deviations of momentum from the aforesaid standard should be computed from the ground to the level of no-stress. One must either select observations in which the upper momentum becomes independent of height as I have done or else, preferably, compute the variation with height of the standard momentum from obser- vations of horizontal temperature-gradients. Here read the table on p. 84. Another way of measuring the eddy-shearing-stress is provided by 0. Reynolds' theorem to the effect that the stress dragging the lower air in the direction of x increasing is , ....................................... (2) where v x ' and V H ' are simultaneous deviations, at the same point, of velocity from the mean; and where the bar denotes a mean, over a long interval of time, of the product of these deviations. By observing the motion of thistledown and using Reynolds' formula, there was obtained on one occasion (L. F. Richardson (32), pp. 10 to 15 and p. 21) Height Mean velocity Stress dragging lower air to right of wind Probable error 200 cm 145 cm sec" 1 + 0-48 dyne cm~ 2 0-20 112 84 THE FUNDAMENTAL EQUATIONS OH. 4/8/3 T . b IW3-* g H _H b I-H X q> b X r-H b X b X co 1 1 X I-H rH _OJ "be 'S -M v ll ometer r 1 r-H b b * S 1 6 2 6 c6 T3 of r O^ B O &D C C) M & f. cS co r-H O2 l-H *e E 3 | rf |S 2 1 oc OJ U3 t> w ^ > b DO "o K a i tions in omitted |3 s Q? -^ qi r* t-~ ^ . TAYLOR' 1 03 O 50 5s 1 aj a, a. 2" 2^ OD C M ft. o ; & 1 hH C ^S "S H ^ H ^o Reference || O^ fcjo "g . W >' d 1 } If | o Q oi * -a ^H a cs> "P^ g ^ a a -a g to September mber to Februa d a S IN 1 5 iame observati( 49 OH O o *rt r" ^d ^ ^3 ^^ O . i CO S b jg o So | S g gto O e -i ^ OQ | >> t-s ID 4! 's B J3 H 8 r q j3 "3 co 10 32 l-H to *" .S ~ c d E IO CO -H, lO -w g CO CS O "p HH O3 00 o (f-1 10 i * 9 oo P O o o (M rH b aj 1 4-* O CO CT5 I-H in SS " + + + + + 1 P + QJ O ^* 5 r* ^ 1 -2 lal **. B I a o E So* 811 on S C co 1 O (7-1 T3 -Sj Si^ I-H 1 b CO b co l-H + 1 g O o bo o CO 10 1 r 1 oo T3 .0 9 i oo 2 OQ p 1 N 'C 8 I 0) 4) T3 o 3 1% r-3 o "B fi T* "ii S -3 a s II S O I-H 2 co I-H r-H I-H CO b O 3 s CO oo 6 b o 2 a I 1 CO I-H i-H *O s 08 X 9 B S s o n i i CO tt ,1, o '> i 2 g g 1 1 1 1 1 1 1 15 CH. 4/8/3 SURFACE FRICTION 85 On clear nights the air over land is often calm near the ground while the wind blows uninterruptedly at a height of some hundred metres. How does this cairn air remain unmoved under the joint action of friction from above and of the all-pervading barometric gradient? The answer must apparently be that the upper surface of the calm is not level, but is tilted so as to balance these forces. Seeing that the calm air always has a distinctly greater "potential density" than the air above it, a tilt would produce a horizontal pressure gradient. Observations of the height of the calm at neighbouring stations would be interesting, and would give a measure of the surface stress, which may be very small*. After sunrise the wind descends. Its junction with the calm is sometimes very abrupt, as is shown by shooting spheres up to various heights. The arrival of the wind at the level of the anemometer is often quite sudden. One gets the impression that the upper wind planes away the night calm a shaving at a time. Suppose that the lower boundary of the wind descends at rate of B, that the velocity of the wind is c and that the air-density is p. Then a mass of air equal to Bp grams per horizontal cm 2 acquires a velocity v in one second, so that the surface stress is Bpv (3) The mean stress will have this value whatever be the order in which the parts of the calm air are accelerated, provided the interval of time and their final velocity are given. Observation at Benson. Early data Anemometer data Mean Stress Date V say cm sec" 1 B dyne cm 3 Calm reaches to Velocity cm see" 1 Wind first reached Upper velocity reached anem. anem. 120m 1919 Oct. 22 70m. at 7 h 12 m 500 at 120m 10 h 55 m 12 h 30 m 500 0-39 5" 18 = 0'63 cm see" 1 250m 1920 170 m 300 13 h 50 m 15" 10'" 300 8 h 30"' 0-31 Feb. 23 at 6 h 50 m at 250 m = O'SScoi sec" 1 To these stresses must be added any due to the (as yet unmeasured*) slope of the upper surface of the calm. A cornfield provides a uniform elastic surface by observing which we can measure the shearing stress. For example at Benson, 1920 Aug. 16th to 22nd, observations were made on a field of ripe wheat. The wind acts mainly on the ear, but partly also on the stalk, as may be shown by lowering a glass jar over the ear. The centre of the forces due to the wind appears to be about 5 cm below the ear. The springiness of a number of growing stalks was measured on a cairn evening and it was found that 1 cm deflection of the centre of force corresponded on the average to a horizontal * Since this was set in type I learn that W. Georgii has made extensive observations (Ann. Hydr. Berlin, 1920, pp. 207222 and 241262). 86 THE FUNDAMENTAL EQUATIONS CH. 4/8/3, 4 force of about 130 dynes applied at that level. The number of ears per square metre was about 175. Therefore a deflection of the centre of force of about 1'5 cm on the average corresponded to a stress of 1-5 x 130xl75xlO- 4 = 3'4 dynes cm- 2 . Date Aug. 1920 Stress dyne cm* Wind 30 cm above ears cm sec" 1 Anem at 26m cm see" 1 At 600 m by balloon cm sec" 1 m grm cm 2 sec Other conditions Stress (m)2 cm 3 /grm 17 d 10 h 3-4 130 350 550 0-6 Lapse rate nearly normal. Overcast 9 22 d 16 h 16 400 550 1-21 2/10 cumulus 11? Thus a cornfield appears to be rougher than the average of the country-side. Thus estimates of the stress, by five or six different methods, give values clustering round 1 dyne cm" 2 . Suppose we have found the surface shearing stress on the air, for the particular locality, with due reference to the strength of the wind and to the vertical gradient of entropy-per-unit-mass. The stress will be specified by its components directed with M, and perpendicularly to the left of M. Denote these by A and B respectively. The components of stress on the air to the east and the north are then (4) .(5) where, as usual, M= (M/ + CH. 4/8/4. THE KDDY-FLUX OF HEAT AT THE SURFACE We have already noted on pp. 70, 71 that, when temperature remains steady, the flux of heat increases downwards. The curve showing the relation of temperature to height reminds one of a fishing rod, held aloft with its thin end pointing downwards, and shaken. The upper part is fairly steady both in slope and in position. The lower part switches to and fro, and its slope alternates between large positive and negative extremes. Measurements of 30/dh made at Benson by carrying an Assmann aspirated thermometer up a steel mast to a height of 16 metres show that on a sunny summer morning dO/dh may be at the rate of 45 C per km ; while just before sunrise it may be at the rate of + 130 C per km. Such up-grades as these are unheard of, if the differences are taken over a whole kilometre. We see thus that the relations of potential temperature to height are as detailed as the relations of wind to height would be, if the earth's surface had the habit of slipping to and fro parallel to itself, with a daily period, and with a velocity attaining 30 metres per second. The stratum from the ground to 2 km was rather thick for dealing with the varia- CH. 4/8/4 FLUX OF HEAT AT THE SURFACE 87 tions of wind, but the case of potential temperature is worse. Yet as thinner strata would mean more arithmetical toil, let us for the sake of economy try first to make the thick stratum do. The origin of most of our difficulties lies in the fact that the up-grade of potential temperature in the first 50 metres is frequently of opposite sign to the mean up-grade over the first two kilometres, so that the latter alone is of no use in estimating the flow of heat at the surface. Although the variation between day and night of heat produced at the surface from the balance of radiation has a large effect, yet we cannot ignore the consequences of the warmth or coolness of the upper air relative to the daily mean at the ground. We might attempt to separate the mean eddy-flux from the daily oscillation. There are some good observations of a damped periodic wave of temperature propa- gated with an amplitude which diminishes, and a time of maximum which lags, as it goes upwards. For instance there are Angot's statistics for the Eiffel Tower, and J. Regers for Lindenberg*. The amplitude of the daily wave decreases to l/e of its surface value at a height of about 400 metres above Lindenberg. Such statistics have been compared with theory by G. I. Taylor ('2,1) and by W. Schmidt (25), (34), (35) with a view to finding a measure of turbulence. Schmidt separates the effect of radiation absorbed by the air from the effect of eddy-conduction. Schmidt also treats of the ratio in which heat, generated from radiation at the interface, divides itself between the two media. These theories no doubt have some relation to fact and give us some new insight into it. But they begin by assuming f to be inde- pendent of time and of height a treacherous assumption, which we must try to avoid, seeing that f is observed to vary in a range of 100 to 1. However may vary, it is so far possible to separate out an oscillatory part, as in consequence of Ch. 4/8/0 # 16, which reads st being linear in ^, it follows that the sum of any two integrals of this equation is itself an integral. But there are other difficulties which will now be explained. W. Schmidt (18, p. 26) concludes that the flux is on the average downwards because, on the average, potential temperature increases upwards. His argument is convincing when it refers to a height to which the daily variation hardly penetrates. Referring to layers near the ground, W. H. Dines f, on the contrary, concludes that the eddy-flux of heat is on the average upwards, because heated air rises freely on sunny days, while on cold nights the air next the ground, by becoming stagnant, prevents downflow. Mr Dines' description suggests that the deviation of potential temperature from its daily mean will not in general satisfy the diffusion equation satisfied by the actual potential temperature. For, take the time-mean of equation (1). At any level let a time-mean be denoted by a bar, a deviation from a mean by a dash, so that *W+'; x = x + x' .................................. (2) * Lindenberg annual volume for 1912. t "Heat Balance of the Atmosphere," Q. J. R. Met. Soc., April 1917, p. 155. 88 THE FUNDAMENTAL EQUATIONS CH. 4/8/4 Be it noted that and x ar e both intrinsically mean values. We are now performing, over a longer interval, a second averaging. Then (2) implies that to^ + to ..................................... (3) So that the diffusion equation (1), when averaged with respect to time, becomes and as the product of a mean into a deviation becomes zero when its mean is taken %=*If%+i*\ (5) st dp t ? a ' dp)' 2 7 Now there is no expectation that ' -~- will vanish, at any rate if ^ is potential temperature, for the wind tends to become more turbulent as the lapse-rate approaches or exceeds the adiabatic. In fact on a sunny midday, when dr/dp is positive, the tur- bulivity at a height of about a kilometre was estimated by the present writer * to be 1-lxlO'grm 2 cm" 2 sec~ s for velocity and it would be greater still for potential temperature which is about ten times greater than the average turbulivity found by various authors at this height. Again smoke observations made at a height of 2 metres above ground show that is much greater on clear days than on clear nights. Also, as tends to increase with increase of velocity aloft, so ' PJ will not vanish if ^ is the horizontal velocity. The case when x = p. is not so clear. Thus we conclude : the daily mean of potential temperature does not satisfy the diffusion equation satisfied by the instantaneous value. Subtract (5) from (l), then it follows that + + e _ st dpdp^*dp^'dp dp) ' Thus the deviation from the daily mean follows the course described by this com- plicated equation. The effects of radiation, conveyance and precipitation are to be regarded as additional. On these grounds it appears useless to attempt to separate the daily oscillation from the mean distribution. Another reason against such a course is to be found in the non-periodic irregularities of the thermograph record which are produced by cloud. Turning away from harmonic analysis we come back to an extensive empiricism. From data uncomplicated by the effects of eddies, we know the rate at which heat is being produced or destroyed at the surface. This heat diffuses both upwards and downwards. Let the ratio of the upward flux to the downward flux be denoted by * Phil. Trans. A, Vol. 221, p. 26. CH. 4/8/4 PARTITION OF HEAT FORMED FROM RADIATION AT SURFACE 89 zz and let us make an empirical study of the value and variations of ex . The parti- tion coefficient, as we may call ^ , is not enough to fix the flux of heat, for sometimes there may be no net radiant energy to be divided in parts, and yet, when for instance colder air overlies a warmer sea, there will be an upward flux of heat. We need a second empirical coefficient, say JUL, defined so that the upward flux of heat when there is no net radiation is (7) JUL /_ 9 where T (; is the potential temperature of air in thermal equilibrium with the surface of the sea or land. It will be interesting to compare the numerical value of JUL, when it has been observed, with tlmt of the corresponding quantity for the diffusion of water, which is discussed in the next section under the symbol m , and of which some estimates are n given in Table on p. 84. See Angstrom's observation quoted in Ch. 4/8/1. The relation of zx to JUL and their combination or alteration to give the flux of heat, is a question which is deferred to Ch. 8/2/15. W. Schmidt (35), from whom I get the idea of the partition-coefficient, has written of it mostly from the point of view of integrals of S-^/St = Bd^x/dh?. Going instead directly to Homen's observations *, of the heat penetrating the ground, and of radia- tion, it is seen that in latitude 60 on 14th and 15th August z? had the following values : Surface Granite rock Sandy heath Grass moor By day 6 h to 18" ... 0-84 2-3J 2-80 By night 18 h to6 h ... -0-35 0-60 1-86 The two coefficients - and JUL might be classified as functions of such variables as are available in the scheme of numerical prediction, namely of v Ga , of T O T GS and of the rate of production of heat at the interface. It is probable that ^r increases as each of these variables increases. Hornen's observations show an increase with either T G T GS , or with the rate of production of heat, or with both of these variables. The classification should also take account of position on the map, for zz is much less over water than over land. W. Schmidt (35) estimates that zx =0'0037, on the average, over the sea. But that estimate is deduced from a theory of a simple harmonic oscillation. Perhaps the surest way to obtain zz and JUL would be to analyse air-temperatures observed from kites, aeroplanes and balloons. The gradual "depolarization" of the first two kilometres of "polar air" as it flows southwards would, when compared with the radiation, give us the information we need. Hann, Meteorologie, 3rd edn. p. 49. 12 90 THE FUNDAMENTAL EQUATIONS CH. 4/8/5 CH. 4/8/5. EDDY-FLUX OF WATER FBOM THE SURFACE The eddy-flux of water upward from the surface must, on the yearly average of the whole globe, be equal to the rainfall. But here we need to take a detailed view of the matter. The records of evaporation from tanks are not supposed to be applicable to lakes or seas, and do not help us much. The botanists and agriculturalists have measured the evaporation from plants and that will be referred to in Ch. 4/10. The lai'ge daily oscillation of relative humidity at head level above ground is nearly compensated by the variation of temperature, so that the diffusing quantity p. varies only some 1 5 / o from its mean. What concerns us more is that dp/dh, like dv/dh but unlike dr/Sh, does not ordinarily change sign near the ground during the course of a summer day. Thus Angot's statistics for the difference between the vapour pressure at the summit of the Eiffel Tower, and at Pare St Maur some 250 metres below it, show that dp./8h is normally negative, and throughout an average July day varies only 23 / o on either side of its mean. Again such variations of moisture as occur in the atmosphere have much less effect on the density than have the variations of temperature. Thus the distribution of /A, unlike that both of T and of v, is not so much a contributory cause of turbulence. Thus the exchange of moisture is likely to depend on the exchange of mass as if the instantaneous deviations of V H and of dp/dh were not correlated'*. In other words ob- servations on the dispersal of smoke should yield a value of the turbulivity more applicable to the diffusion of water than to that of heat. Possessing observed values of f near the ground we might combine them with 3/t/3p at the same level, to get the flux g/g . dp/dp ; but unfortunately our coordinate difference is too big to serve as dp in dp/dp near the ground. So the best we can do is to express dp/dp as an empirical function of the excess of the mean value /Z 68 for the first stratum over the value near the earth's surface say /%. Or when there is vege- tation p. L may advantageously replace /%. Let us now examine the compensation which must be made for the greatness of the coordinate difference. We have, from Ch. 4/8/0 #15, when D/Dt is expanded T = ~- ( I ~- ) + terms due to precipitation and transport, ( I ) dv x d I j.dv x \ 1 dp -ar = a- lT^ +2a sin < . ; F - (2) fft cp \ op/ p 8x Now the rapid vertical changes in dp/dp or dv x /dp occur mostly in the first 50 metres or less above the surface where they are due to the decrease of turbulence as the surface is approached. la this thin layer the mixing is probably more important than either precipitation or transport. And if we choose the x axis to be parallel to the isobar, then dp/Sx is zero and v r is small. Also it is probable that the flow by * L. F. Richardson, Phil. Trans. A, 221 (1920), pp. 9, 10. CH. 4/8/5 FLUX OF WATER TREATED BY THICK LOWER STRATUM 91 way of eddies through this thin layer is more important than the accumulation in it. In other words we may neglect dp./dt. If the same be done with So x /St then the above equations (l) and (2) both reduce to (3) which implies a flux of ^-times-mass which is independent of height. See Ch. 4/8/0 #17. By integrating (3) and putting in the limit G (4) Integrating again and putting in the same limit t ' "\"l Integrating once more from p a to p G and dividing by p G p 8 so as to obtain the mean denoted by x=~ - \X d P> ( 6 ) PG~P& j s g CP rln^ dp* - Po-Ps Solving for the flux, which we have assumed to be independent of height d x _ X-XG i (po-psY .(8) which shows that when we replace dxfip by its mean value (X~XO)/^(PS~PG) then, to get the flux, we must also replace f by the peculiar mean value G (Vflrf (9) 8 J G ? The idea that f was the same for all meanings of ^ was first suggested by Taylor (12), and has been widely used by W. Schmidt (18), (25). Although it is almost certainly not exact, yet we may try it here in order to obtain some preliminary esti- mate of the upward flow of water, for comparison with direct measurements. Now if f is the same for the diffusion of velocity as of water per mass, and if the assumptions on which (8) is based are good, then for these two diffusing quantities f tf8 will also be the same. W. Schmidt (18) using values of the Austausch, /g*p, deduced from wind at a height of 1/2 km, in combination with up-grades of water there, has estimated the mean flux of water at this level. Similarly we see that from the surface stress and wind distribution we can estimate Ge , and we can then use Ga in combination with the changes of p given by our large coordinate differences, to find the upward flux of 122 92 THE FUNDAMENTAL EQUATIONS CH. 4/8/5, 6, 7 water at the ground. For, replacing x by v x m ( 8 ) an ^ (9), the flux becomes the surface stress and 7 _ (surface stress parallel to the isobar) (pppa) 9 /IA\ (-^)x2 Some values of GS deduced in this way are set out in the last column of the Table on p. 84, ready for use in computing the flux of water. Here V xo has been taken as zero. For reasons already given, the analogy cannot be extended to the flux of heat. It would be desirable to obtain f ft8 directly from observations of humidity. CH. 4/8/6. CONCLUSION ON EDDY MOTION NEAR THE EAETH'S SURFACE For the sake of economy in arithmetic, an effort has been made to use a thick lower stratum extending to a height of 2 km above sea-level. Whether this plan will be successful or not depends on whether four empirical quantities, namely the stress xh G for velocity, xz and JUL for heat, and G8 for water, can be expressed with sufficient accuracy as functions of the data available in the numerical process. There is some hope of it, but in order to settle the question many more observations, and reductions of existing observations, are required. If this plan should fail, then an alternative is to be found in taking thinner strata near the surface. If for example we had divisions at heights of about 50 m, 200 m, 800 m, all our processes would be much more exact. Although xh G , cr , AA and G8 would still be required for the new lowest layer, since the first metre differs remark- ably from the first 50 metres, yet we should then have the velocity and the up-grade of temperature near the surface as aids to the estimation of that very variable quantity, the turbulence. These thinner layers would need to rise and fall with the height of the land above sea, and consequently the dynamical equations in them would need to be furnished with extra terms depending on the slope of the ground. As the land sometimes rises above 2000 metres, these surface layers would project above the h s horizontal. In avoiding that difficulty we might end by making all the conventional strata rise and fall with the height of the land. CH. 4/8/7. LIST OF PUBLICATIONS ON ATMOSPHERIC EDDIES 1. KELVIN and TAIT on work obtainable from an unequally heated solid. Quoted in Preston's Heat, 2nd edn., 341. 2. GULDBERG and MOHN. "Studies on the Movements of the Atmosphere." 1876, revised 1883. Abbe's Translations, 3rd Collection. 3. REYNOLDS, 0. "On the Dynamical Theory of Incompressible Viscous Fluids and the Determin- ation of the Criterion." Phil. Trails. A, 186 (1894), Papers, Vol. II. quoted in Lamb's Hydrodynamics, ivth edn., 369. 4. Numerous other references to theoretical papers by KELVIN, RAYLEIGH, and others in Lamb's Hydrodynamics. CH. 4/8/7 THICK STRATA VERSUS THIN 93 5. MAEGULES, M. "Energy of Storms," 1901. Abbe's Translations, 3rd Collection. 6. EKMAN, W. V. "On the Influence of the Earth's Rotation on Ocean Currents." Arkivfdr Mat. Astr. och Fysik, Stockholm, 1905. 7. BERNARD, HENRI. "Les Tourbillons Cellulaires." Ann. Chim. Phys. Paris, Ser. 7. 23, J. Phys. Paris, Ser. 3. 10. 8. AKERBLOM, F. "Sur les Courants les plus bas de 1'atmosphere." Nova Acta Reg. Soc. Upsa- liensis, 1908. 9. SANDSTROM, Kungl. Svensk Vet. Akad. Handl. Bd. 45, No. 10 (1910), or Bull. Mt. Weatli. Obs., Vol. 3, part 5 (1911). 10. DINES, W. H. "The Vertical Distribution of Temperature in the Atmosphere and the work required to alter it." Q. J. R. Met. Soc., July, 1913. 11. TAYLOR, G. I. In "Report of Work carried out on S.S. Scotia, 1913." Published by H.M. Stationery Office. 12. TAYLOR, G. I. "Eddy Motion in the Atmosphere." Phil. Trans. A, 215, pp. 126 (1914). 13. HESSELBERG, TH. "Die Reibung in der Atmosphare." Met. Zeit., May, 1914. 14. HESSELBERG, TH. and SVERDRUP, H. U. "Die Reibung in der Atmosphare." Oeo. Inst. Leipzig, Ser. 2, Heft 10 (1915). 15. Numerous other references in Exner's Dynamische Meteorologie, 38 to 41. 16. TAYLOR, G. I. "Skin Friction of the Wind on the Earth's Surface." Roy. Soc. Lond. Proc. A, Jan. 1916. 17. DOUGLAS, C. K. M. " Weather Observations from an Aeroplane." /. Scott. Met. Soc. 1916. 18. SCHMIDT, W. "Der Massenaustausch bei der ungeordneten Stromung in freier Luft und seine Folgen." Akad. Wiss. Wien, 1917. 19. SVERDRUP, H. U. "Der nordatlantische Passat." Qeo. Inst. Leipzig, Ser. 2, Bd. 2, Heft 1 (1917). 20. SVERDRUP, H. U. and HOLTSMARK, J. "Uber die Reibung an der Erdoberflache " Geo. Inst. Leipzig, Ser. 2, Bd. 2, Heft 2 (1917). 21. TAYLOR, G. I. "Phenomena connected with Turbulence in the Lower Atmosphere." Roy. Soc. Land. Proc. A, Vol. 94 (1917). 22. TAYLOR, G. I. "Fog Conditions." Aeronautical Journal, Jan. March, 1917, Vol. xxi, p. 75. 23. DOUGLAS, C. K. M. "The Lapse Line in its Relation to Cloud Formation." /. Scott. Met. Soc. 1917. 24. DOUGLAS, C. K. M. "The Upper Air, Some Impressions Gained by Flying." J. Scott. Met. Soc. 1918. 25. SCHMIDT, W. "Wirkungen des Luftaustausches auf das Klima und den taglichen Gang der Luft-temperatur in der Hohe." Akad. Wiss. Wien, 1918. 26. SVERDRUP, H. U. "Uber den Energieverbrauch der Atmosphare." Oeo. Inst. Leipzig, Series 2, Bd. 2, Heft 4 (1918). 27. ANGSTROM, A. "On the Radiation and Temperature of Snow and the Convection of Air at its Surface." Arkivfdr Mat. Astr. och Fysik, Bd. 13, No. 21 (1918). 28. RICHARDSON, L. F. "Atmospheric Stirring Measured by Precipitation." Roy. Soc. Lond, Proc. A. Vol. 96, 1919. 94 THE FUNDAMENTAL EQUATIONS CH. 4/8/7, 4/9, o 29. JEFFREYS, H. "Relation between Wind and Distribution of Pressure." Roy. Soc. Proc. 1919. 30. WHIPPLE, F. J. W. "Laws of Approach to the Geostrophic Wind." Q. J. R. Met. Soc., Jan. 1920. 31. BRUNT, D. "Internal Friction in the Atmosphere." Q. J. R. Met. Soc., April, 1920. 32. RICHARDSON, L. F. "Some Measurements of Atmospheric Turbulence." Phil. Trans. A, Vol. 221, pp. 128 (1920). 33. RICHARDSON, L. F. "The Supply of Energy from and to Atmospheric Eddies." Roy. Soc. Proc. A, 97, July, 1920. 34. SCHMIDT, W. "liber den taglichen Temperaturgang in den unteren Luftschichten." Met. Zeit. Heft 3/4, 1920. 35. SCHMIDT, W. "Worauf beruht der Unterschied zwischen See- und Landklima?" CH. 4/9. HETEROGENEITY* CH. 4/9/0. GENERAL OBSERVATIONS Not only is the velocity usually turbulent but the temperature is often patchy. An ordinary pen thermograph in a Stevenson screen shows variations of about 1C on a sunny afternoon. When motoring on a calm evening one can sometimes dis- tinctly feel alterations of warm and cool air not corresponding to height. E. Barkow (Met. Zeit. 1915 Miirz), using a thermometer of quick period, found variations of some tenths of a degree occurring at time intervals which could be interpreted as meaning that patches of linear dimension of 14 to 127 metres were moving past the instru- ment with the speed of the wind. Barkow's observations were made at Potsdam 41 metres above the ground on a wooded hill. The daily weather reports show that in a square on the ground of 200 km in the side the temperature at the screen on an anticyclonic morning may vary as much as 10 C. For the upper air an estimate of diversity has been extracted from the aeroplane records published in the London Daily Weather Report for 1920, January 1 to October 30. Only those pairs are included which are simultaneous to an hour or less. The aspect of the data to which attention is invited is that the square root of the mean square of the difference of temperature between two stations is not at all pro- portional to their distance apart, as it would be if the stations lay in a straight line and the horizontal temperature gradients were uniform. The differences may be in part due to observational error. If we ignore this unknown error we get the impres- sion that there are irregular variations in temperature, represented by a standard deviation of about 2 C, in distances comparable with the side of our co-ordinate chequer of 200 X 200 km. But, as the stations do not lie in a straight line, the state- ment cannot be tested rigorously. * In what follows for brevity I have written "diverse" for "heterogeneous" and "diversity" for "heterogeneity." OH. 4/9/0, i IRREGULARITIES OF TEMPERATURE 95 Height. Kilometres above M.S.L. Number of pairs Mean difference of temperature Arithmetic mean. Centigrade Square root of mean square. Centigrade Baldonnel minus either South Farnborough, Upavon or Andover. Distance about 400 kilometres. 3-55 3-05 12 16 + 0-8 0-2 3-7 3-1 2-44 15 -0-3 2-6 1-83 17 -0-3 2-5 1-22 17 0-0 2-9 0-61 17 02 3-1 0'30 14 -0-1 2-0 o-o 16 1-1 3-9 South Farnborough minus Andover. Distance 53 kilometres. 1-83 4 0-0 1-0 1-22 5 -0-7 2-3 0-61 5 -0-1 1-4 o-o 4 -0-7 1-5 In telegraphic meteorology the smaller diversities are smoothed-out. The tempe- rature is commonly measured by a thermometer with a bulb large enough to damp out rapid variations, and if the barometer is "pumping," owing to a gusty gale, the observer does his best to estimate the mean reading. In the scheme of this book it is proposed to replace the instantaneous and local values by means over times as long as 3 hours and spaces as great as 100 km horizontally or several km vertically. That this neglect of detail has important consequences we have already seen in the case of velocity. It now remains to enquire whether the diversity of density, temperature, pressure and moisture produces any statistical effects of like importance to eddy- viscosity. CH.- 4/9/1. NOTATION AND ASSUMPTIONS The theory of Osborne Reynolds concerning turbulence suggests a mathematical method. The bar-and-dash notation of H. Lamb (Hydrodynamics, ivth edn. 369) allows the theory to be put into a compact form. We begin by supposing that the actual distribution is replaced by a smoothed one, which is indicated by putting a bar over the symbol. Thus for any quantity whatever A, we have A = A + A' where A' is the deviation ( 1 ) 96 THE FUNDAMENTAL EQUATIONS CH. 4/9/1 The subsequent algebra is based on the following suppositions, no one of which is strictly accurate, but all of which tend to become decent approximations if the diver- sities are sufficiently numerous and random. These suppositions would become exact if it were possible to choose a smoothing-interval which could be regarded as an infi- nitesimal for the smoothed distribution, and yet as infinitely large as compared with the diversities. (i) A smoothed-value is unaltered by a second smoothing. Z = A (2) It follows from (1) and (2) that A' = (3) (ii) The smoothed-value of the product of a deviation into a smoothed-value vanishes AB'=0 (4) (iii) The product of a number of smoothed-values is unaltered by smoothing. A.B.3=A.B. C. (5) (iv) The smoothed-value of a differential coefficient is equal to the differential coefficient of the smoothed-value SB SB The mean of the product of any two diversified quantities A and B is accordingly *# (7) And the deviation of a product takes the following alternative forms, however large the deviations of the factors may be (AB}' = AB-AB (8) I'ff-^y (9) = AB'- A'B - A'B' containing A without bar or dash (10) = A'B + AB'- A'B' containing B without bar or dash (11) In other parts of this book a good deal of attention is given to integrations with respect to height across strata, and the normal variation with height is then taken into account. But any "normal" variation is essentially a smooth one, so that in those integrations we are concerned with the facts, which we here ignore, that A is not exactly equal to A nor A . B . C to A . B . C. Let us now apply these smoothing operations to each of the chief equations. CH. 4/9/2, 3, 4 PRESSURE OF HETEROGENEITY 97 CH. 4/9/2. SMOOTHING THE CHARACTERISTIC GAS EQUATION Under all circumstances for dry air p = bpO so that p = bp6 + bp'd' ( 1 ) We might give to bp'd' the name "pressure of heterogeneity." Suppose that the standard deviation of 6 were 2 '8 C that is to say 1 / of its mean value, and suppose that the pressure had no local variations. Then the "pressure of heterogeneity" would come to 10~ 4 of the mean, that is to O'l millibar. If the moisture is also diversified then b' will not vanish and p=b.p$+b.p'W+p. 0^+0.^7+ (2) The term 6'b'p' has been omitted as being of a smaller order. CH. 4/9/3. SMOOTHING THE EQUATION OF CONTINUITY OF MASS In the form given in Ch. 4/2 # 2 the equation is linear in the dependent variables, so that it transforms simply into dp dm E dm^ M N . tan dm H ~di = ~W ~~dn~ ~^~ 'IF" which is of exactly the same form in the smoothed variables as the original equation was in the nnsmoothed. The neglect here of the variation tan < is as justifiable as the statement that A A' = 0; for tan < is already smooth. But if the equation of continuity had been written in terms of velocities in place of momenta-per-volume, then smoothing would have produced a crop of new terms. That is another reason for preferring momenta-per-volume to velocities. CH. 4/9/4. SMOOTHING THE EQUATION FOR CONVEYANCE OF WATER If we begin with the form (Ch. 4/3 # 5) OLL OU, CLL 3/Z UlL then there results 3 . _ 3 . _ 3 . _ 3\ fJ)l* \ , up , i/ym, , up , , de dn dh The first member is a rate of change following in some sense the smoothed motion but is not quite the same as -=- . p. defined in Ch. 4/8/0 # 1. There are three new terms on the right. B. 13 98 THE FUNDAMENTAL EQUATIONS CH. 4/9/4, 5 On the other hand if we begin with the equation for div/dt which taken from Ch. 4/3 # 6, and written in Cartesians, runs | 8 0* m r) . a (^H) _ n ^ /Q\ ~~ ~ ~ P ' we obtain on smoothing 8w d(p.m x ) S(p.m r ) Here there are again three new terms, which in this case have a close formal resemblance to the eddy-stress terms in the smoothed dynamical equation, /u replacing each of the components of v. The new terms are the divergence of a flux which has components fj/m' x , p.'m' Y , pfm' H ; and that is the flux relative to a point which moves so that, relative to the point, m x = 0, m Y =0, m a =0. This point corresponds to the "definite" portion of a turbulent fluid as it was defined in Ch. 4/8/0 # 1. But according to the quite different view of Ch. 4/8 we have been accustomed to express the vertical flux as cdp,/dh (5) In this statement, as in so many others not in the present Ch. 4/9, there is an implied bar over the /A. We must also admit that there may possibly be different conductivities in different directions, say c xx> C YY> C HH- It follows, from (5), as thus modified, that "/* I i _ "/* >i. r" / / /c\ x dx * dy B dh There is a close analogy between these equations and (viscosity) x (space-rate of mean-shear) = (eddy -shearing-stress). CH. 4/9/5. SMOOTHING THE DYNAMICAL EQUATIONS For the present purpose Cartesian co-ordinates are preferable to polar ones. We get the equations in Cartesians from Ch. 4/4 # 3, 4 by making the polar-coordinate- radius infinite and by replacing E by X and N by Y. Only the terms containing products of dependent variables, namely ^- (fn x v x ) and the like, produce on being smoothed any additional terms. As Reynolds showed, these additions express the body-force produced by a system of eddy-stress. But, if the density is diverse, the components of stress are not quite of the form p . v' x v' x etc., which Reynolds first gave for an incompressible fluid, and which the present writer has elsewhere applied CH. 4/9/5, 6 EXACT FORM OF EDDY-STRESSES 99 to the atmosphere. On referring to Ch. 4/4 # 3, 4, 5 we see that the exact form of the additional terms is g ( m 'x v 'x) + g- (m'xv'r) + gT ('xv' H ) in the -XT equation, ^ __ p\ _ p\ (m'yv'x) + (m' r v' r ) +^r (wi' F v' H ) in the Y equation, <*< ^ + (m' 7 Xr) + gT (m' H v' H } in the // equation. Thus in place of the stress />. t>V'''A we have strictly m' x v' x . Furthermore the stress in the x direction on the plane normal to the y-axis is m' x v' Y which may differ from m'yv'x, the stress in the y direction on the plane normal to the #-axis. There are thus nine components of eddy-stress in contrast to the six found when the density is not diverse. Silberstein* gives a proof to show that any stress arising from causes which conform to d'Alembert's principle will have only six different components. The question then arises : does smoothing the equation, which expresses d'Alembert's prin- ciple, cause it to throw off extra terms which invalidate the aforesaid proof? We have here the suggestion of an analogy to the stresses in magnetised bodies. But the subject will not be pursued further now except to note that the eddy-stresses could be reduced in number from nine to six if we rearranged the dynamical equations, 3 d before smoothing them, so as to have in the first (in E v N ) and in the second ^- (jn E Vj^, Gil/ O& and so as to make corresponding changes elsewhere. But then the smoothed equations would have an ugly lack of symmetry. CH. 4/9/6. DETACHED CLOUDS AND LOCAL SHOWERS The scheme proposed in Ch. 4/6 for forecasting cloud was that there would be general condensation, if the mean value of the water-content of a stratum exceeded its saturation value at the mean temperature. If air is cooled by expansion the result of diversity is that the wetter parts of the volume become cloudy before the volume is on the average saturated, and after it has become on the average saturated the drier parts may still remain clear. Diversity thus converts a sudden transition into a gradual one. The more diverse the atmosphere the more gradual the change. If the air is cooled by radiation, the parts containing more water than the average absorb and emit more radiation, so that the question is more complicated than that of a simple expansion. But certainly the most familiar example of diversity is that connected with cumuli, and here the adiabatic cooling as well as the moisture is diversified. It is known that for cumuli to be formed there must be (i) in the upper air a lapse rate riot too stable, (ii) a supply of heat and moisture below. The problem of the statistical mechanics of cumuli deserves further attention. * Vectorial Mechanics (Macmillan &, Co.). 132 100 THE FUNDAMENTAL EQUATIONS CH. 4/9/7 CH. 4/9/7. THE PRODUCTION OF DIVEBSITY The measure of diversity which fits best with our mathematical habits is half the smoothed square of the deviation. For if the deviating quantity were the velocity the aforesaid measure would be the eddying-kinetic-energy-per-mass. We have already a theory of its production and dissipation in connection with the Criterion of tur- bulence 4 ". Our present task is to find what happens when v is replaced by p., T or other deviating quantities. Let us consider the diversity of a, the water per mass. That is to say, we seek to formulate the changes in {(ju/) 2 } the smoothed square of its deviation. The equation for the conveyance of water (Ch. 4/3 #5) may be written, after multiplication by p, (0 where DajDt is zero except in so far as precipitation and molecular diffusion come in. Now take the deviation of this equation term by term using the formula for the deviation of a product (Ch. 4/9/1 #10) in part of which the actual p and m appeal- without either bar or dash. da' ,9/5, ,da' .da , da . da - m' r ~ - m r - m H -- dx dy d/i Now multiply by a' and bring the terms in the complete m x , m Y , m H to the first member. Then * / / dp. t , dp. I Da\' /u/x(a smoothed quantity ) + a' (p-fr-j (3) On smoothing this equation various terms disappear, but it is necessary to split m and p into their means and deviations. Thus there results ' 2 (4) * O. Reynolds, Phil Trans. A, 186, p. 123 (1894); L. F. Richardson, Roy. Soc. Proc. A, 97 (1920), p. 354. CH. 4/9/7 PRODUCTION OP HETEROGENEITY BY EDDIES ;101 The first term resembles that which we seek, for the term is p multiplied by the time-rate of the smoothed square of the deviation of /A, following some sort of mean motion. To see what sort of mean motion, let the co-ordinate axes move so that This supposition is permissible because the equation from which we began was true for any motion of the axes. But if m = then no mass is crossing the co-ordinate planes. That is the same sort of mean motion as the one considered in Ch. 4/8/0 # 1 . Let us put a bar over a capital D to denote change following this kind of mean motion. We have already (Ch. 4/9/4 # 6) arrived at an interpretation of p!fn' H as the upward flux of water, equal to c mi rr Inserting this and similar terms in (4) there Gil results ' ~ r'tii j- = T at a r- r TM. IP -771 ; (5) 8as 9y 9/i J \ r l^l/ There is no obvious reason why m' H should be correlated with ~- - , nor M T hy p' should be correlated appreciably with // or with (p.') 1 . If these correlations and the ut corresponding ones in m' x , m' Y were to vanish, then This equation has a close formal resemblance to the one originally worked out by O. Reynolds to express the activity of the eddy stresses. We may leave the last term over for consideration in the next section. The equation signifies then, since c is always observed to be positive, that unless the mean distribution of /u, is originally uniform, any turbulence tends to increase the diversity. This one can believe, without the aid of mathematics, after watching the process of stirring together water and lime- juice. But we have obtained mathematically a numerical measure, which could con- ceivably be tested by observations such as Barkow's*. Obviously there would be a precisely similar theory in which potential temperature T replaced p. as far as (5). But the supposed vanishing of the correlations which cause the simplification to (6) would need more careful scrutiny. * E. Barkow, " Uber die thermiache Struktur des Windes." Met. Zeit. 1915, Marz. 502- THE FUNDAMENTAL EQUATIONS OH. 4/9/8 OH. 4/9/8. THE DISSIPATION OF DIVERSITY This is carried out presumably by the molecular agitation. The term p! Ip -} in \ Ln i the equation at the end of the last section has been put there as a reminder of the existence of dissipation. It now remains to examine the term in detail. Since the molecular velocities are enormously greater than the relative molar velocities of two portions separated by a distance equal to the mean-free-path, we may now ignore the molar velocity and treat the molecular dissipation as if it were proceeding in still air. For the effect of eddies is represented by the other terms in the aforesaid equation. The usual statement* is that the vapour density w tends to become uniform accord- ing to the equation dw /a 2 a 2 3 2 \ ^T = /C r- + r-i + ro W, ..... . ........................... ( 1 ) dt \5x- dlf dn?J where K is the molecular diffusivity and equal to 0'2cm 2 sec~' for the interdiffusion of water vapour and air. Smooth equation (1) and subtract the smoothed form from the original. So we get an equation in the deviation dw' / a 2 a 2 a 2 , "7T7 f ->~~i i~ ^ 9 ' ^~7~9 ) ^' St \da? dy* d Multiply by w' Then because, by the rule for differentiating a product, a / ,atiA ,aw , a~ ( w a~~ = w VT + UH 2 dxj on rearranging it follows that Inserting (5) together with similar expressions in the other two co-ordinates into (3) the latter becomes i a Now smooth this equation in order to obtain one in (w'"} the mean square of the diversity. It reads * Jeans' Dynamical Theory of Gases, 2nd edn., eqn (868). CH. 4/9/8 DISSIPATION OF HETEROGENEITY BY MOLECULAR DIFFUSION 103 The term V 2 (w'~) on the right represents a diffusion of mean-square-diversity from regions where it is lai'ge to those where it is small. The coefficient of this diffusion is K just the same as for the diffusion of w. But, as the rate of space-variation of the smoothed quantity (w' 2 ) is usually very much smaller than the rate of space-variation of the urismoothed w, the curious diffusion of (it/ 2 ) is likely to be of minor importance. We must look for the main effect in the second term on the right of (7) for in it the space-variation is taken before the smoothing. This term is composed of squares. Its sign is such as to correspond always to a decrease of the diversity. It may be contrasted with a rather similar term in Ch. 4/9/7 # 5, which has however the opposite sign and which contains the space rates of a smoothed quantity /Z in place of the deviation w'. The term in Ch. 4/9/7 # 5 expresses the production of diversity, when an atmosphere, non-uniform in its smoothed distribution, is stirred ; the present term expresses the dissipation of the diversity so produced. The two equations could be reduced to com- parable quantities, either p. or w, if density were independent of time and place. For then we should have, for the extra term in Ch. 4/9/7 # 5, This expression, although instructive, must be regarded as only a stage on the way to a working theory, for we have as yet no theoretical way of finding dw' Let us now examine the hypotheses. If the temperature distribution is variegated, will it be the vapour density which tends to become uniform ? The ordinary theory of distillation proceeds from the assumption that it is not w but the vapour pressure p w which tends to uniformity horizontally. In the vertical, according to Dalton's law, the tendency of diffusion alone is not towards uniformity of either w or p w but to a state such that dp. t/ ,/dh= gw. Then again the diffusion coefficient K which we have assumed to be a constant, really increases with temperature, and if temperature is diverse that increase should be taken into account. But to bring in all these cor- rections would make the equations very elaborate and would perhaps obscure their main features. The dissipation of temperature must follow very similar lines, replacing w, and the molecular thermal conductivity replacing K. 104 THE FUNDAMENTAL EQUATIONS OH. 4/10/0 CH. 4/10. BENEATH THE EARTH'S SURFACE CH. 4/10/0. GENERAL The atmosphere and the upper layers of the soil or sea form together a united system. This is evident since the first metre of ground has a thermal capacity com- parable with 1/10 that of the entire atmospheric column standing upon it, and since buried thermometers show that its changes of temperature are considerable. Similar considerations apply to the sea, and to the capacity of the soil for water. As it will not do to neglect the changes in the land or sea, two courses are open : (i) The variables expressing the temperature in the sea or land might, at least in imagination, be eliminated. The result of the elimination would be to yield a complicated boundary-condition for the atmosphere. (ii) A forecast for the land and sea might be attempted concurrently with that for the air. Let this be the ideal which we here set before us. One reason for this choice is that a forecast for the soil would itself be of value to agriculturalists. The quality of the forecast might range from a mere use of the normal variables for the time of day and year, to a thorough treatment by finite differences. Possibly a combination of the two may eventuate. We have already regarded the earth's surface in Ch. 4/8/3, 4, 5, 6, from above, now let us look at it from below. Ultimately these two points of view must be combined. In this connection there are three principal varieties of surface : the sea, bare earth, and earth covered by vegetation*. It will be convenient to consider these separately and afterwards to attempt to form average constants for our horizontal chequers of about 200 km square, by reference to the relative amounts of the three kinds of surface in each chequer ; due regard being paid to the season and to the customary times of ploughing, harrowing, sowing and the like. The changes in the soil may be described by two differential equations, one for the conduction of heat, the other for the transference of water. It is intended to treat these equations by finite arithmetical differences. The soil must accordingly be divided into conventional strata. Let z be the depth reckoned positive downwards from the surface of the soil or sea. It will be remembered that h the height is reckoned positive upwards, and always from mean sea-level. At what depths z shall we make the divi- sions between the conventional strata? Well on referring to Rambaut'st observations of temperature in the Oxford gravel, it is seen that at a depth of one or two metres the temperature depends simply on the time of year. The more rapid oscillations, associated with the passage of cyclones, scarcely penetrate to these depths. In other words it will suffice to consider a layer of soil having a thermal capacity comparable with that of the atmosphere. * And next in interest perhaps snow and rock, t A. A. Rambaut, Phil. Trans. A, Vol. 195, 1901. OH. 4/10/0, i TEMPERATURE OF SEA 105 Again, in the first few centimetres below the surface the down-grade of temperature is often steep so that a detailed treatment is required. Whereas, at a depth of a metre it would be wasteful to keep account of the temperatures at heights differing by so little as a centimetre. Accordingly it is proposed to change the independent variable from the depth z to some function of z, which is called j, and which, when divided equally, gives thin conventional strata near the surface, thick ones lower down. The choice of the particular function is arbitrary. A convenient form would be y = log.(z + l) (1) Giving toj the values 0, 1.2, 3, 4, 5 in succession- we get the following depths in centimetres as the boundaries of the conventional strata : 0; T92; 6'39; 19'1; 53'6; 147'5. To transform the differential equations we must substitute 1 d d I \tf d\ & and T~T\ u \ ~J^. ~ T- \ ' or Tl ............................... (*>) (2+1) W d J) dz Now the equations for the motion of water and heat are interlinked by various terms, so that it is convenient to use the same conventional strata for both. It is accordingly intended to make the substitutions (2) and (3) in all the equations for the soil. For simplicity however the independent z is written in what follows. On. 4/10/1. THE SEA A layer of sea 2'4 metres deep has a thermal capacity equalling that of the whole of a dry atmosphere standing upon it. The temperature of the sea surface is much steadier than that of the land. We might, for some purposes, assume the sea surface to have the mean temperature observed at the given place and date in previous years, by consulting, for example, the monthly Charts of the Atlantic Ocean*, published by the Meteorological Office. The error which we should thereby commit may be represented by a standard deviation. For areas on the Atlantic Ocean measuring 2 in longitude by 1 in latitude this standard deviation would be about 1C in the eastern North Atlantic, while on the western side, where the gulf-stream is narrow but not quite fixed in its course, the standard deviation might amount to 5 C. Hann (Meteorologie, in edn. p. 65) has computed the daily variation of temperature in the sea and in the air immediately over it. The average difference between the two, at any given time of day, rarely exceeds 1 C. So that the temperature of the air in contact with the eastern North Atlantic, is now predictable to 2 C. * Helland-Hansen nnd Nansen, Temperatur-Schwankungen Jes Nordatlantischen Ozeans und in der Atmosphare, Kristiania, Jacob Dybwad, pp. 50 84. English Edition published by Smithsonian Institu- tion, 1920. B 14 106 THE FUNDAMENTAL EQUATIONS CH. 4/10/1 If a more exact value of the surface temperature had to be predicted, the system of prediction would have to take into account : (i) The long-wave radiation which is given off or absorbed, on account of the great opacity of water, only by the uppermost centimetre*, (ii) The solar radiation which penetrates to much greater depths. O. Krlimmel t gives a table of absorption coefficients ranging from O'Ol per metre for a wave length of 450/zju, to 0'30 per metre for a wave length of 660ju,/x. (iii) The turbulence in the sea which depends on the down-grade of "potential density." The problem here is more complicated than that of the atmo- sphere on account of the variation of salinity with depth. Presumably the shearing of horizontal velocity must also be a prime cause of turbulence in the sea as it is in the atmosphere ; and the shearing will in turn depend on the wind. (iv) The turbulence and temperature of the air which convey heat to or from the surface. We have already discussed this in Ch. 4/8 and will not go into it further, except to note that the interaction of sea and air must be treated as one problem. (v) The ocean currents, their dynamics, and the heat which they convey. Presumably these five principles could be put into a scheme of prediction such as has been worked out for the atmosphere in this book. It may come to that, but let us hope that something simpler will suffice. Interaction of Sea and Air. If the sea is colder than the air at deck-level, the turbulence in the air becomes small and very little interchange of heat goes on. A low eddy-viscosity under these circumstances was observed by G. I. Taylor during the Scotia cruise J. Prof. Helland-Hansen to whom I am indebted for much of the information in this section tells me that when the sea is colder than the air at deck- level there may sometimes be a decided change of relative humidity in the first few metres above the water, indicating a protecting film so thin as that. The warmer air tends, if anything, to diminish turbulence in the colder sea and thus to cut off the flow of heat on that side also. On the other hand if the sea is warmer than the air at deck-level, as it tends to be in winter, there is much turbulence in the air, which tends to have no up-grade of potential-temperature nor of water-per-mass. This state of affairs is illustrated by observations in the North Atlantic trade wind and by the eddy-viscosity deduced therefrom by Sverdrup J. The sea, being cooled above, tends also to become turbulent, if the down-grade of salinity permits. Thus the sea surface acts as a leaky valve, which allows heat readily to flow upwards, but hinders its descent. More observations of turbulence over the sea and within it are much needed. The smoke method might be suitable. * Winkelmann, Handb. der Physik, 2 Aufl. Bd. in. p. 342. t Handbuch der Oceanographie, i. 389. J Vide p. 72 above. Phil. Trans. A, Vol. 221, pp. 526 (1920). CH. 4/10/2 EVAPORATION FROM FALLOW LAND 107 CH. 4/10/2. THE BABE SOIL In winter great areas of arable land are bare. As in the case of the sea, we require to forecast the temperature and humidity of the air in immediate contact with the surface of the soil. A wealth of observational material has been brought together by Warington in his Physical Properties of Soil (Oxford Press) from which I have drawn freely. The Motion of Water in Soil. The evaporation from bare land has been measured by the Rothamstead drain gauges'" with the remarkable result that in the six winter months, October to March, the evaporation is practically identical with that from a water surface, as measured by Greaves*, near London. Provided that we may assume the atmospheric conditions above Mr Greaves' gauge to have been the same as at Rothamstead, we may say : during that portion of the year in which any considerable fraction of the land is bare, the bare surface is so wet as to saturate the air in contact with it. This rough generalization might suffice, but it seems desirable also to develop the general equation applicable to summer as well as winter. In this equation it will be convenient to regard the soil as a continuous medium, that is to say the "infinitesimal" differences of the coordinates must be large compared with the soil particles and yet small enough to give a good representation of the variation of moisture with position. The theory of percolation in saturated soils has already been put in mathematical form by Boussinesqt. For unsaturated soil, such as is often found near the surface, I have not found the equation anywhere. It may be developed from the ideas of Briggs, according to whom the water in the unsaturated soil may be typified by a waist-shaped piece partly filling the crevice between two spherical soil particles. If the amount of water in the waist decreases, the curvature of its surface becomes more strongly concave and the negative pressure in the water is thereby increased. If the water in all the crevices is continuous with itself, the pressure will tend to become everywhere equal, in the absence of gravity. Denote the mass of water per volume of soil by w. From the point of view of our large infinitesimals, the pressure in the water will be a single valued continuous function of position, and will depend on w. Denote the pressure in the water by T (iv}. The form of the function Y can be determined, for any particular soil, by experiments similar to those of Loughridge. He put air- dried soil into vertical metal pipes closed at their lower ends by muslin. Water was supplied through the muslin and rose by capillarity. From the mode of its entry this water was probably all continuous. When the steady state was established, samples taken at different heights were weighed, dried and reweighed. This gave w. At the same time Y (w) is equal to g times the height of the sample above the free * Warington, loc. cit. p. 109. t Journal de Mathematiques, Paris, 1904, pp. 1 and 363. See also L. F. Richardson, Proc. Koy. Soc. Dublin, 1908, May. F. H. King, Irrigation and Drainage, Macmillan, 1882. J. M. K. Pennick, "Over de Bewegung van Grund water," De Ingenieur, 29 July, 1905. 142 108 THE FUNDAMENTAL EQUATIONS CH. 4/10/2 water-level outside the tube, as the density of water is unity. In the more general case, when there is not equilibrium, ^ * g will be the unbalanced pressure gradient 0% producing a flow upwards. Since the flow is in very narrow channels it is non-turbu- lent. It is therefore proportional to the unbalanced pressure gradient, and varies inversely as the viscosity. The flow also depends on the dimensions of the channels. These are bounded in part by the water-air surfaces, with the result that the con- ductance of the channels diminishes rapidly with diminishing w. Denote the con- ductance of the channels connecting the opposite faces of a centimetre cube of soil by cy, which is a Coptic letter pronounced "shai." Then the flow of water upwards is, in cm 3 sec" 1 per horizontal cm 2 , <') So the rate at which water accumulates to any point by creeping is dw 8 The conductivity cy could, I think, be determined experimentally as a function of iv by means of apparatus similar to that which gives the pressure T. For let a uniform slow current of water be established, either down the tube by dropping water on the top, or up the tube by promoting evaporation at the top. Let the current be measured and be maintained constant until a steady state has become established throughout the tube; and then let w be determined at a series of heights by drying- samples. From the distribution of w, that of the pressure Y (iv) could be found, since the form of Y is known from the previous experiment. Taking the gradient of the pressure Y (tv) and inserting it along with the constant flow in (1) we should have the conductivity cy for a series of values of w. Y and cy depend on the temperature in known ways; if, as usually happens, the soil is not uniform, they also depend on the depth. The possibility of the existence of isolated water must not be forgotten, although no simple way of treating it mathe- matically may be to hand. Nor must the loss of water by surface drainage, root-holes and fissures be overlooked. In the unsaturated soil, simultaneous with the creeping of liquid water, there goes on a distillation of vapour. In summer there is often a thin coat of dry soil on the surface; and the distillation through this may dispose of quantities of water which cannot be neglected. Where liquid water is adhering to the soil particles the vapour density will be saturated for the temperature and for water surfaces of the existing curvature. The curvature in turn depends on w, so that we may write for the vapour density F (w, 6}. Soil which feels dry to the hand may still contain considerable quanti- ties of occluded, adsorbed or "hygroscopic" water. From this cause iv may amount to as much as 0'05 grm cm~ 3 or more*. Now the adsorption of gases by solids has been the subject of many investigations!, which have shown that an equilibrium state is * Warington, loc. cit. p. 60. t Winkelmann, Handb. der Physik, 2 Aufl. Bd. II. p. 1524. OH. 4/10/2 CREEPING AND DISTILLATION OF WATER IN SOIL 109 reached in which the mass adsorbed depends on the temperature and on the density of the gas. So we may write here: vapour density = F (w, 6], thus prolonging the function F from the region of liquid water into that of adsorbed water. For such dif- ferent substances as charcoal *, glass* and peat dustf it has been found that F, in the region of adsorption, is more or less proportional to the mass adsorbed, and decreases rapidly with increasing temperature. df The rate of motion of vapour will be proportional to the gradient of density -=- and also to the porosity which we may denote by 2f, a Coptic letter pronounced "janja." The porosity will be diminished if the passages are partly choked with water, so that it should be written 2C (w), a function of iv. It may be defined so that the mass of water distilling upwards per unit of time and of horizontal area is And the rate of accumulation of water by distillation is therefore dw a < 4 > I have attempted to measure the porosity X of peat dust to water vapour in the following way. A little water was put into the bottom of a test-tube. Above the water-level a partition of brass wire gauze and linen was fixed in the tube. The dust was packed in the tube above the partition. The tube was stood upright in a desic- cator, and was weighed at intervals during several weeks, until the rate of loss became constant. The experiment was repeated with a column of peat dust of a different length. It became apparent that a length of 5 cms of peat dust allowed the water vapour to diffuse almost as freely as in a tube containing only still air. The diffusivity in still air is well known. To have obtained a good measurement special attention would have had to be paid to the constancy of the resistance offered by the open end of the tube. Adding the accumulation by creeping, given by (2), to that by distillation, given by (4), we obtain the total rate of accumulation of water in an unsaturated soil, as follows: If we were to add the two terms representing the horizontal components of motion, equation (5) would then apply to saturated soils as well as to unsaturated ones; for when saturation occurs x vanishes, cy is the porosity to water, and the pressure Y is no longer a function of w but depends on position instead. Natural soils are rarely uniform so that in (5) T, cy, /, x will depend on height. Temperature reduces the capillary tension, and therefore Y, by 0'002 of itself per 1C. The porosity cy varies rapidly with the temperature in a known manner, because it is proportional to the reciprocal of the viscosity. f Winkelmann, loc. cit. f Unpublished experiments by the author. 110 THE FUNDAMENTAL EQUATIONS CH. 4/10/2 The flow of water across the surface z = depends on precipitation and evapora- tion, and can be calculated at the initial instant, at which w is given, according to the methods of Ch. 4/6, Ch. 4/8/5, Ch. 4/9/6. Then the approximate distribution of iv after 8t can be found from (5). The Motion of Heat in Soil. Turning now to the temperature equation, numerous researches * have shown that it is, at least approximately, of the form w ve where the diffusivity s is of the order of O'OOS cnr sec~'. The treatment of this equation by arithmetical differences has been illustrated by the present writer}. Calendar* and Pott t have found that the diffusivity s is much greater for wet soil than for dry, so that we must regard s as a function of w. If the soil is stratified s will also depend upon z, or rather it will be necessary to regard the conductivity and thermal capacity separately as functions of z, so that, in place of (6) we have - where u is the thermal capacity of unit volume and k is the conductivity. We may improve upon equation (7) by taking account of the disappearance of heat wherever water is evaporating. The rate of condensation is given by (4). Multiplying the rate-of-condensation by the latent-heat-of-evaporation and dividing the result by the heat-capacity-of-unit-volume-of-the-soil, we get a measure of the rate of tempera- ture due to condensation, in the form 4-18 x 10' {598 -0-60 (0-273)} d f 9 - ;r- -(2 - u dz ( dz) where u is the number of ergs required to raise the temperature of one cubic centi- metre of soil by one degree. For sand, for example, if the grains are of pure quai'tz and occupy half the volume of the soil, then u will be ^ (density) x (specific heat of quartz) = 0"23 x 4'18 x 10 7 , when the sand is quite dry; and, when water is present u = (0'23 + iv) x 4' 18 x 10 7 , with an upper limit of t< = 0'73x4'18xl0 7 , when the sand is saturated. On the other hand, if the soil contains fine powders which attract water, as peat dust does, both the specific and the latent heat of the water may be expected to differ from their normal values. Calendar has found that heat is also carried through soil by percolating water. If, neglecting any rapid motion through root-holes and fissures, we assume that the water is moving slowly between the solid particles, then the temperature of the water at any point will be the same as that of the particles, so that the symbol 6 will serve * Encyc. Britt. xi edn. Vol. 6, pp. 893 to 894; A. A. Bambaut, Radliffe Observations, Vol LI. Oxford; Hann, Lehrbuch der Meteorologie, 3rd edn. pp. 48 to 55. t Warington, loc. cit. pp. 161 to 171. I Phil. Trans. Roy. Soc. London, A, Vol. 210, p. 313. Encyc. Britt. xi edn. Vol. 6, pp. 893 to 894. CH. 4/10/2, 3 THREE WAYS BY WHICH HEAT MOVES THROUGH SOIL 111 for both. The upward flow of water is given by (l) in grm sec' 1 cm" 2 . Multiplying this flow by and by the specific heat of water, which is 4'2 x 10 7 ergs gram" 1 degree" 1 we get the upward flow of heat in so far as it depends on the motion of water. Differentiating the flow of heat with respect to z, and dividing the result by the thermal capacity of unit volume of the soil, we get the rate of rise of temperature, due to the flow of water, in the form: (9) Collecting now the three parts of the temperature-rise due severally to conduction, condensation and convection, we have from (7), (8), and (9) d0 1 9 /, 90\ latent ht. of evapn. 9 [~ 9/"l = - - \k -- H X - m u dz \ m] u dz |_ 3zJ + ~1[>-*M] <"> This is the complete temperature equation. The latent heat of fusion of ice in the soil may be regarded as a very large increase in the thermal capacity in the imme- diate neighbourhood of the freezing point. The flow of heat across the earth-air surface, 2 = 0, depends on precipitation, eva- poration and radiation, and can be calculated at the initial instant at which is given, according to the methods of Ch. 4/6, 4/7, 4/8/4, 4/9/6. The approximate distribution of temperature after St can then be found from (10). Now as to the treatment of equations (5) and (10) by finite differences. The combinations of 6 and w which occur in these equations, mostly require to be tabulated at the same depths as w. The last term in (10) is an exception to this statement, but as we cannot have it both ways, this exception has been ignored. Again, in order to calculate the radiation and the evaporation from bare soil, it is desirable that and w should be tabulated actually at the surface, rather than as mean values for the upper layer which is l - 7 centimetres thick. These principles have been embodied in the corresponding forms in Ch. 9. CH. 4/10/3. EARTH COVERED BY VEGETATION Leaves, when present, exert a paramount influence on the interchanges of moisture and heat. They absorb the sunshine and screen the soil beneath. Being very freely exposed to the air they very rapidly communicate the absorbed energy to the air, either by raising its temperature or by evaporating water into it. The amounts evaporated by crops are considerable; barley for instance has been estimated to give off the equivalent of one millimetre of rain per day during its growth. A portion of rain, and the greater part of dew, is caught on foliage and evaporated there without ever reaching the soil. Leaves and stems exert a retarding friction on the air, so 112 THE FUNDAMENTAL EQUATIONS CH. 4/10/3 that within a forest it is very doubtful whether we can still estimate the "eddy diffusion" of entropy and moisture, from observations of the change of wind velocity with height, as G. I. Taylor and W. Schmidt have done in the free air (p. 76 above). For a numerical treatment I have referred to a series of papers by Brown and Escombe*, and Brown and Wilson*. Their observations are built around the fol- lowing framework of theory. The air in the intercellular spaces of the leaf is supposed to be saturated in equilibrium with pure water at the temperature of the leaf. The vapour diffuses out through the stomata at a rate proportional to difference of vapour density between inside and outside. The transpiration also depends on the size, shape and number of the stomata. By analogy with electric conduction, the rate of trans- piration may be said to be inversely as the "resistance" of the stomata. The resist- ance consists of two parts in series. The larger part is due to the air in the intercellular spaces and in the constricted passages of the stomata; this part is unaffected by wind. The remainder of the resistance is due to the air immediately outside the stomata and is reduced by wind. Thus the wind can only alter the transpiration in a limited range. The openings of the stomata are sensitive to sunlight. The careful experi- ments set out on pages 106 to 109 of Brown and Escombe's paper show that a reduction of sunshine to one-half of its intensity may increase the 7'esistance by perhaps 10 / c i n the case of Helianthus annus. The initial sunshine averaged about 0'4 cal cm" 2 min" 1 . The temperature of the leaf is above or below that of the surrounding air according as the radiation received by the leaf is greater or less than the energy absorbed by the water in evaporating. The difference between these two sets of energy, divided by the emissivity of the leaf for heat, gives the temperature difference between the leaf and the air. The emissivity here is the total rate of loss of heat to the surround- ing air by conduction, convection and radiation jointly, for 1 C temperature difference. Brown and Wilson measured the emissivity and found that it was a linear function of the wind velocity ; thus for Helianthus multiflorus emissivity in cal cm~ 2 min' 1 (degree C)"' = 0'015 -f O'OIO x (velocity in metres sec' 1 ). The emissivity is so great that although, in the absence of any loss of enei'gy, siin- shine would raise the temperature of a particular leaf at the rate of 35 C per minute, yet the temperatures of the leaves observed by Brown and Escombe f never differed by more than 2 C from that of the air, and were much nearer to the dry-bulb temperature than to that of the wet-bulb. To put this in a formula: Let the rate of loss of water from a leaf be denoted by <$, then )-w*} .................................. (1) Here K is the conductance of the stomatal openings and F(6) is the saturated vapour density at 6. * Hoy. Soc. Land. Proc. B, Jan. 1905. t Loc. cit. Tables vm. and x. CH. 4/10/3 WATER TRANSPIRED BY FOLIAGE 113 In (1) we must substitute (radiation) x (absorptivity) tf x (latent heat of evapn.) , . C'leaf ~ "air + ~* vT~ ~) (2) emissivity and then $ appears on both sides of the equation. However, to a first approximation, we may take the dry bulb temperature as equal to the leaf temperature so that 4 = K{F(e*)-w*} (3) The equations (l), (2), (3) apply to a single leaf. We require the corresponding expressions for a mass of foliage. The chief difference will be the large increase in the conductance K. We can estimate K for a crop from the following considerations. The total water transpired by a crop* is from 250 to 700 times its dry weight. For example for barley it is at the average of about 10~ 6 gram per cm 2 of land surface per second. The average increment of vapour density, which would just saturate the air at the temperature of the dry bulb, could be determined by keeping a recording psychrometer in a barley field. Taking this increment provisionally as of the order of 5x10"" gram cm" 3 , it follows that K is of the order of IQ- 6 -^- 5x 10~ 6 = cm 3 sec' 1 per cm 2 of land surface. In Brown and Escombe's experiments the lea,ves were supplied with abundant water. If the amount of water in the soil at the level of the roots falls below a certain small amount, the leaves lose their turgescence and transpiration diminishes. This critical amount of water has been determined by Heinrich f. It is, for example, about w = 0'02 for a coarse sandy soil, iv = Q'l for a sandy loam. Now w is already one of our de- pendent variables, so its effect on transpiration can be taken into account. A Conventional Film of Vegetation. In order to bring (1), (2), (3) into con- nection with the other equations of the atmosphere, we must take the temperature and vapour density of the air surrounding the leaves as dependent variables. Unfor- tunately it is not possible to estimate these quantities, even approximately, from the known instantaneous mean values for the stratum which extends up to h = 2 kilometres; because the estimate would be spoilt by the steep and changing temperature gradients near the surface. For example, at Eskdalemuir, E. H. Chapman found gradients of + 2 C per 10 metres near the earth. It is therefore necessary to introduce another conventional layer extending from the earth to the top of the foliage. It is called hereafter the "vegetation film" and its lower and upper limits will be denoted re- spectively by the subscripts G and L. This layer or film might be treated on the same general plan as are the conventional strata, with the distinction that the air must be considered as having its opacity and its thermal capacity increased by the leaves mixed with it, and that the friction does not vanish when -r vanishes. * Various observers quoted by Russell in Soil Conditions and Plant Growth (Longmans, Green . Let a^, a p , a p be defined to be such that dcr = a^ . dp + a p . dp + dp . dp ............................ (5) CH. 5/1 MANY CONTRIBUTORY CAUSES 117 For clear air the values of a M , a p , a p have been given in Ch. 4/5/1. In general they are functions of p., p, p and therefore may vary with position and time. Now multiply (2) by a,,, and (3) by a p , and subtract both products from (l). The result is fdm H 2m H \ a Da- Dp. Next dp/dt in (6) must be replaced by known quantities. This can be done by bringing in the hydrostatic equation P=\ J h (7) where h is a height so great that the pressure at it may be neglected. Take the time- rate of (7). The lower limit of integration is independent of time because we require dp/dt at a height which is fixed. Thus Substitute in (8) the value ofdp/dt from (3) then "~ ' * (9) Note that equations (8) and (9) are exceptions to the rule that the second members are completely known. Next substitute in (6) the value ofdp/dt given by (9), and in so doing separate the integral into unknown and known parts, and arrange them respectively in the first and second members. Thus we arrive at dp dp\ f^n H 2m a ^""dh ~ af \W + ~a C no = +a v g . div jV m.dh + a,. div MN m Jh /da- 8p.\ [da- dp. ~ V E (~^~, ~ a ii 57 I ~ v x \ z^ ~ a M a^ The equation (10) involves only the vertical velocity and quantities expressing the instantaneous distribution of other meteorological elements, a distribution which is supposed to be known. But (10) can be simplified. For in the first member And again for dp/dh we can put gp. 118 FINDING THE VERTICAL VELOCITY CH. 5/1, 2 Then (10) reduces to 2m a a riu = a. p \ g.div ) h dcr 3u\ /3cr So far the equation is very free from assumptions, almost the only one being that the pressures due to vertical acceleration are negligible in comparison with the pressure due to gravity. In particular in deducing (12) we have not assumed g to be inde- pendent of height. CH. 5/2. SIMPLIFICATION BY APPROXIMATION As (12) is inconveniently complicated, let us simplify it by making the following two approximations which will probably not affect the accuracy by one per cent. g is independent of height, (13) the terms in the reciprocal of the radius of the earth are negligible (14) There is then a cancelling, in the first member of (12), of C >>0 (j.ftl CL, I g - dh with a v .g.m a , Jh oh so that the first member of (12) reduces to Let us next divide through by a p . p so as to leave dv u /dh by itself. In so doing let us express a f and a p in terms of the more familiar quantities y v and y p , the thermal capacities, in erg units, of unit mass at constant volume and pressure. By Ch. 4/5/1 # 11, 12, for clear* air y*v c . fp With these substitutions equation (12) reduces to ft, i g . divj^ m . dh div EN m y p .p h in which div^f m is given by (4). * A temporary restriction for illustrative purposes. In Ch. 8 the unrestricted a p , a p> a^ are used. CH. 5/2, 3,4 SIMPLE CASES 119 It may be said that (18) would look neater if the integral were removed by differ- entiation. But in so doing we should also remove the valuable statement, which is implied in the definiteness of the integral, namely that the pressure becomes negligible at a great height. As a matter of fact the differential form was fi rst derived directly from dp/dh= gp and was employed throughout the example of Ch. 9 ; but a constant of integration kept on appearing inconveniently in places where it could not be deter- mined. This " hysterical manifestation " was eventually traced to the suppression of the limits of integration which are now explicit in equation (18). CH. 5/3. METHOD OF SOLVING THE EQUATION The approximate solution may be obtained by arithmetical steps. The explicit integral in (18) is begun at the top, where it vanishes, and is carried downwards to each conventional level in turn. The equation is then integrated for V H beginning at the bottom and working upwards, because at the ground v a is known from the hori- zontal winds and the slope of the surface according to the equation dh dh\ ae dn/ G which Prof. V. Bjerknes has used to prepare maps of the vertical velocity at the surface (see his Dynamical Meteorology and Hydrography, Part II). As equations (18) and (19) are linear in V H and dv H /dh, the sum of any number of solutions is itself the solution for the sum of the corresponding distributions of m E , m N ,p, a, //,. CH. 5/4. ILLUSTRATIVE SPECIAL CASES Case (i). No horizontal velocity anywhere, no radiation, precipitation or stirring. Then Ch. 5/2 # 18 reduces to Sv H /dh = 0. But we may derive this result independently. For the pressure on any definite portion of moving air, being simply the weight of air in a column of unit cross-section above it, remains constant. Therefore, as the motion is adiabatic, the density is constant for a moving portion. Thus the atmosphere rises or falls like a rigid body, so that dv B /dh = Q. If we bring in also the boundary condition (19) we see that V H is zero everywhere. Case (ii). No horizontal velocity, no precipitation or stirring, but radiation in progress. Then Ch. 5/2 # 18 reduces to W = ^ ~Dt W The change in entropy Da occurs under the constant pressure due to the air above. Therefore Da- = ^y p D0 and (18) implies that Sv H _lD0 120 FINDING THE VERTICAL VELOCITY CH. 5/4 But we may deduce this independently. For consider a portion 8h of a vertical column. If it changes in temperature by D6 while the pressure is kept constant, then it will expand by a fraction D6/9 of its original length 8h. The displacement of the upper end of the short column relative to the lower is therefore 8h . D6/8. If the displacement takes place in a time Dt, the velocity of the upper end of Sh relative to its lower end is s/ ID6 * h -eDt' But this relative velocity is Sv H . Equating and dividing by Sh we get &V H _1-D0 8h '" 6 ~Di ' which agrees with (2). Case (iii). Horizontal winds exist, but are such as to cause no divergence of hori- zontal momentum per volume, that is to say div EN m = Q at all levels. The air is dry and there is no stirring or radiation. Then Ch. 5/2 # 18 reduces to dv B 1 / So- da- Suppose that the wind is coming from a region where the entropy-per-mass is greater towards one where it is less. Then the quantity in the bracket will be negative, as is easily seen by considering the special case V K = and remembering that, contrary to sailors' usage, V E is positive when the wind blows to the east. In the case imagined dv a /dh will be positive. The arrival of air of greater entropy-per-mass causes an increase of upward velocity with height. If the motion happens to be at constant pressure then we may say : the arrival of warmer air causes a swelling. An independent check is lacking. Case (iv). At a level h ( air is being abstracted from the column. Moisture is absent. The entropy-per-mass has no horizontal variation and does not change with time following the motion. The general equation Ch. 5/2 #18 then reduces to dv, a?" ^r 9- d\\ EN m.dh-~div EN m (1) n 7v P i A P 8A Case (iv A). Let us suppose, for mathematical simplicity, that div EN m is zero every- where except in a very short range lying between A f Sh, and A,- + SA, and let us represent the total rate of abstraction of mass from a column of unit cross-section by A, where A=\ d\v EN m.dh (2) Then below the level of abstraction the integral in (1) includes the range in (2) so that dvrr y r qA /ON f = fS . e (3) aA y p p CH. 5/4 HORIZONTAL DIVERGENCE CONCENTRATED AT ONE LEVEL 121 Above the level of abstraction the integral in (1) does not include that in (2) and so Let the ground be flat and at a height h G . Then v a =0 at h h G (5) Now integrating (3) upwards from the ground it follows that, below the level of abstraction h t h * Next, when this second integration is carried upwards across the level h it there is an abrupt change in v a arising from the last term in (1), thus n, i+Sh l A -div, y m= (7) Jht-thp pi So from (6) and (7) it follows that, just above the level of abstraction, --~ w P Pi And (4) shows us that the vertical velocity remains constant for all greater heights. The integral in (6) and (8) has been computed from the observed* mean pressures over Europe, with the following results : Height above M.8.L. kilometres 20-5 19-5 18-5 17 -5 16-5 15-5 14-5 13-5 12-5 11-5 10-5 i ' - cm 3 grm - 8540 7220 6180 5240 4430 3750 3170 2670 2240 1870 1560 Height above M. 8. L. kilometres 9-5 8-5 7-5 6-5 5-5 4-5 3-5 2-5 1-8 0-5 [0-0 V 'o P cm d grm~' 1293 1064 866 694 545 415 301 201 112 35 0] Note g has here been taken as 980 and y,,/y p as 1/1-405. The latter is of doubtful validity in a moisture-containing atmosphere. * W. H. Dines, "Characteristics of the Free Atmosphere," Meteor. Office, London, Geophys. Mem. No. 13, Table X. B. 16 122 FINDING THE VERTICAL VELOCITY CH. 5/4 The curves in the adjoining figure have been drawn by using this table in equations (6) and (8). 14 13 12 11 10 . 9 8 a? o JD I 1 f 6 ffl 5 7 1000 down up 1000 down up 1000 down up 1000 down up 1000 up x (vertical velocity), where A is defined by (2). A. ... Four distributions of vertical velocity produced by the horizontal removal of mass at equal speeds but at different heights, namely at 2, 5, 8 and 11 km. Since div^m and V H enter equation (1) linearly, we may add together any fixed multiples of the abscissae of any of these curves in order to obtain a new distribution satisfying (t). For example if we multiply the velocities of the third curve by 1 and add them to those of the fourth curve we obtain the distribution due to the insertion of mass at 8 km and its withdrawal at an equal speed at 11 km. Below 8 km there is no vertical motion ; between 8 km and 1 1 km there is an upward velocity increasing with height ; above 1 1 km there is a small downward velocity. The latter is at first sight surprising, but it must be remembered that we are not discussing a steady state, but a sudden disturbance of the actual mean distribution, and that the air in the neighbourhood of 1 km would be in the course of replacement by colder air, on account of the difference between the actual and the adiabatic lapse-rates. The integral in (6) may be put into a variety of equivalent forms which are some- times useful. Thus by using the hydrostatic and characteristic equations P G vertically dh. .(9) CH.5/4,5 SOURCE AND SINK 123 By way of these it may be shown that the vertical velocity above a level of abstraction is downwards, unless the atmosphere has a degree of instability far ex- ceeding anything that is observed. To illustrate this exception we may take the hypothetical case in which the density is the same at all heights. Then the second form in (9) integrates, and when inserted in (8) gives for the vertical velocity above the level of abstraction .................................. (10) 7p p which is positive if p is small enough. Thus in such an atmosphere the removal of mass at a sufficiently high level would cause the lower part to raise the part which is above the level of abstraction. CH. 5/5. FURTHER VARIETIES OF THE SIMPLIFIED GENERAL EQUATION In some applications it is desirable to replace div BV m by an expression in velocity. ,, ,. (dv E 2v N tan d> } dp dp , . NOW div^m-,^*^ - J %} +*+** ................ (i) If this be substituted in Ch. 5/2 #18 then because, when different samples of air are compared*, ^ .............................. (2) it follows that a term in Ch. 5/2 #18 derived from div A rW combines with the P two terms following it giving us __ + . _ 7 P P JP 7p P And Ch. 5/2 # 18 becomes But, by the rule for differentiating a product 9 and there is a similar expression in the northward coordinate and component. If in the integral in (4) we substitute for gv ^ the value given by (5) we obtain a term C*6 * 9 / 9\ ~ transforms simply, because at the upper limit p vanishes everywhere, so that * See Ch. 4/5/1. Clear air is here assumed for illustration, but in Ch. 8 we revert to the more general form in a fl , a p . 162 124 FINDING THE VERTICAL VELOCITY CH. 5/5, 6 The value of the same integral at its lower limit cancels another term in (4). Thus we arrive at 1 I Do- Dp\ . . ^\Dt~ *Dt r ( ' y p \isi si / i_ ,. dv F dvj? tan d> where div EN v = ^ + -^- VN - ^ (8) f\f> rtiYi n x ' s - on a The above has been found to be the most convenient form of the equation when we have to do with the stratosphere, as in Ch. 6/6. In places where we measure height by means of pressure it is convenient to take p as the independent in (7) in place of h. In these differentiations with respect to p, it is implied that latitude and longitude are constant, whereas in a p = da-/dp it is p and p. which are constant. Where there is any risk of confusion the symbols of the quantities which are constant during the differentiation are added as suffixes to the differential coefficient. Then (7) transforms into /dp x - - x <*> a J v f p f -, aT" {div 3h y p .p) Po { D CH. 5/6. THE INFLUENCE OF EDDIES Smoothing the equation for vertical velocity. This equation is linear in the three velocity components. So if the other variables p, p, a p , a p , a^, a, p. are not heterogeneous, the process of smoothing, carried out as in Ch. 4/10, merely puts a bar over all the velocity components and over the capital D, which now denotes a differen- tiation following the mean motion. If the other quantities are also diversified great complications will arise. But as became evident in Ch. 4/10, the percentage variations in velocity are usually so much greater than the percentage variations of the other quantities that for many purposes the latter are negligible. Vertical velocity connected with eddy-motion. Wherever light warm air is dis- placed and pushed up by heavy cold air, the joint centre of mass falls, for gravity supplies g/y p of the kinetic energy of the motion*. Again wherever a thermally stable atmosphere is stirred by eddies derived from the wind, there heat is carried down and the centre of mass rises. These motions of the centre of mass imply a small, but highly significant, mean vertical velocity. In view of what has been said about smoothing, it follows that the equations already given in case (ii) in Ch. 5/4 for the vertical velocity due to radiation, will also apply to that due to eddy motion, if we place a bar over V H and over D. That is true provided p, p, a p , a p , a^, a; p, are not diversified. * W. H. Dines' theorem, Q.J.R. Met. Soc. 1913, July, p. 188, see also Roy. Soc. Proc. A, Vol. 97, 1920. CHAPTER VI SPECIAL TREATMENT FOR THE STRATOSPHERE CH. 6/0. INTRODUCTION THE equations developed in Ch. 4, between the integrated quantities represented by the capital letters P, R, M E , M x , hold good, with the approximations indicated, for any one of the conventional strata. But in the stratum, which has its base at 11 - 8 km and extends upwards to at least 40 km, the ratio of pressure is so great that the aforesaid approximations are all open to criticism and must be reexamined. This examination is one of the principal aims in Ch. 6. Another aim is to find a way of extrapolating observations made by balloons, which seldom penetrate into the upper tenth of the mass of the atmosphere, so as to obtain P, R, M E , M N which are integrals up to the top. The final aim is to choose a set of quantities, either P, R, M E , M N or some other equivalent ones, and to find a corresponding set of equations so that, when the quantities are given at one instant, the equations will give their time rates. This problem is for the most part carried over to Ch. 8. But its general features have already been described in Ch. 4/0. The quantities P, R, M E etc. are the definite integrals of p, p, m E etc. with respect to height, taken so as to include the whole thickness of the stratum. In such integrals, values of quantities at the lower limit will be denoted by the subscript 2, because the mean pressure there is 2 decibars ; thus we have p 2 , A 2 , m E2 etc. Similarly the subscript will be used to denote the upper limit. But while A resembles infinity in that it denotes a height so great that its exact value does not concern us, it must be understood to lie somewhere between 50 and 100 kilometres above sea-level, at a pressure between 1 millibar and O'OOl millibar. By this convention we free ourselves from the necessity for entering into difficult questions concerning that outer atmo- sphere* which is ionized, which may be escaping, and in which the variation of gravity, and the term - in the equation of continuity of mass, would cause mathematical Cv difficulties. By this convention also we assume that whatever the rare gas above h a may do, it has no influence on the surface weather. The vertical variation of gravity has been neglected. It may however be mentioned that the theory could be carried through with integrals with respect to gravity-potential instead of with respect to height, . M m<) 2M H ~~ ~ We assume here, as elsewhere, that there is no escape of air from the top of the atmosphere, so that m m = ........................................ (2) Now from Ch. 6/1 # 8 SR W _ \_ 9p 2 dt ~g~tt' so that equation (1) gives us pressure-changes at the base of the stratum. M E> M N , for insertion in (l), are main tabulated variables, while m m and M H are given by the equation for the vertical velocity to be discussed in Ch. 6/6 below. CH. 6/3. EXTRAPOLATING OBSERVATIONS OF WIND For this purpose, in latitudes not too near the equator, we can use a theory developed from " The Upper Air Calculus " * of Sir Napier Shaw. The dynamical equations may be written dv N dv Now since is independent of h and since p bp6, it follows that 9h\pde}~dh\ vv r ^T~/" w ~aA^~ < 3 But the hydrostatic equation may be written 3 log p _ g dk ~W " On inserting (4) in (3) and in (5) dn might replace de. Therefore on differentiating again a/ 1 !"" a )~ anc ^ s i mu ' ar 'ly a/i (~a ) = (6) Thus both sides of equations (1) and (2) are linear functions of height. * Nature, 1913 Sept. 18, also /. Scott. Met. Soc. 1913. 128 SPECIAL TREATMENT FOR THE STRATOSPHERE CH. 6/3 Near the equator it will be difficult to interpret this general proposition in any simple way, because neither the linear nor the quadratic terms are negligible on the right-hand sides of (1) and (2). But in European latitudes it has been shown"" that all terms except those in sin <}> may, with tolerable approximation, be neglected. In the stratosphere, as there is no vertical convection, there can be no appreciable friction between horizontal layers, so that one of the causes which disturb the "geostrophic wind" near the ground is here absent. Assuming then this simple geostrophic wind, with neglect of curvature of path, there follows from (5) and (l) = ' _ __ dh ' 2cu sin Se j -i i dvv q 3 log 6 and similarly ^= -- ^ -- ^ -- -- dh 2 dn thus in the stratosphere in not too low latitudes the horizontal component velocities are linear functions of the height^. To test this theory I selected the two highest balloon flights included in V. Bjerknes' Synoptic Charts, hefts 1, 2, 3. The ascents were at Zurich on 1910 Feb. 3 and May 19. One must resolve the velocities on rectangular axes, for (.9) does not imply that the resultant Jv^/lrVy is a linear function of height. The figures on p. 129 show that, in the stratosphere, the component velocities can be fairly well fitted to straight lines. The fluctuations may be in part attributed to errors made in observing balloons. For at a height of 16 km and at zenith distance 60, a standard error of 0'l in the measurement of the zenith distance produces a standard error of 2 '6 metres/sec in the radial velocity of a balloon, as deduced from successive differences of position observed at intervals of 60 seconds of time when the heights are correct. On the other hand, if the fluctuations are really gusts, we must suppose them to be due either to stable waves or else to eddies turning about vertical axes ; for if the axes were not vertical then a fall of temperature with height would result. (' I* f /' From the slopes of the lines corresponding to -^ and W^ on May 19th we may calculate the horizontal temperature gradients by equations (7) and (8) ; it is thus found that ~\/3 3/3 = - 0-48 C per 100km; ~= + 0'll C per 100km. de dn Unfortunately the temperatures obtained by sounding balloons are not precise enough to test these numbers, for according to Gold (Geophys. Mem. No. 5, p. 66) they are to be suspected of errors of + 2 C. WengerJ seems to be of much the same opinion. W. H. Dines finds a probable error of < 1C for a single ascent at night, but more in sunlight. * E. Gold, "Barometric Gradient and Wind Force," Report to the Director of the Meteorological Office. t First published in Q. J. R. Met. Soc. 1920 January, p. 63. I R.Wenger/'TJebsrden EinflussderInstrumentalfehler..."(reo/>/M/s./ws<. Leipzig, Specialarb. n. 1(1913). CH. 6/3 COMPARISON WITH OBSERVATION 129 130 SPECIAL TREATMENT FOR THE STRATOSPHERE CH. 6/3 CH. 6/3 LINEAR RELATION OF EAST OR NORTH VELOCITIES TO HEIGHT 131 As a further test of equations (9) one figure on p. 130 shows the highest theodolite- balloon record published in the international observations January to June 1912. It was made at Uccle 1912 Apr. 13. The remaining figure on p. 130 represents the highest ascent recorded in G. M. B. Dobson's paper* on " Winds and Temperature Gradients in the Stratosphere." This ascent was made at Uccle on 1911 Sept. 13. The straight lines fit moderately well. The selection of observations has been made by choosing the highest in order to eliminate any personal bias in favour of the theory to be tested. For average conditions over England, Shaw t has calculated from a formula, with which (7) is really identical, that the wind velocity at 12 kilometres falls off by 9 per cent of itself in one kilometre. The formula however shows that the fall in each kilo- metre must be the same in absolute amount, not in percentage of the wind at that height, and so it follows that the velocity changes sign at about 23 kilometres and that at an infinite height the velocity would be infinite, if we might press the logic so far. Of course if the velocity became very great the square and product terms in the dynamical equations would cease to be negligible and the hypothesis of the simple geostrophic wind would be no longer tenable. That high velocities do sometimes occur at great heights has been shown by the Krakatoa glow stratum (33 metres/sec at 35 km) and by the bolide of 1909 Feb. 22} (70 metres/sec at 75 km). Even if the velocities V E , V N did become infinite linearly, the total mass-transports M E , M s would remain finite, because the density falls off exponentially. Height-integral of density times a linear function of height. The momenta per unit volume tn E , m N are of the form p (A + Bh), where A and B are independent of f'<<> height. M E , M N are therefore of the form p(A+Bh}dh. In what follows we J h i shall frequently require the corresponding indefinite integral. To evaluate it, put p - - TT , integrate the coefficient of B by parts and take \pdh from Ch. 6/1 # 6. J/ There results {p(A+Bh)dh=-^\A+B(h + b -\\+ const ................ (10) Now - is a length, and is equal to 6'45 kilometres when the stratosphere has the J temperature of 220 A. If the upper limit of the integral be h a , the pressure there is very small and we have r (11) where the subscript i refers to any height in the stratosphere. But p. rh a pdh = the whole mass above h t ...................... (12) y j ''i * Q. J. R. Met. Soc. Jan. 1920. t J. Scott. Met. Soc. 1913, p. 170. J J. E. Clark, Q. J. R. Met. Soc. 1913. 172 132 SPECIAL TREATMENT FOR THE STRATOSPHERE On. 6/3, 4 And by taking the special case A 0, B=l we see that h^ + bOg' 1 is the level of the centre of mass of the atmosphere above h f (13) Thus we may say : " With reference to a column in the stratosphere extending from an arbitrary level to a region ivhere pressure is negligible, the integral up the column, of the product of the density into any linear function of height, is equal to the pressure at the base of the column, multiplied by the value of the linear function at a height bd/g above the base and divided by the acceleration of gravity. " (14) When applied to velocity, statement (14) is equivalent to the following: The total eastward momentum above a small horizontal surface, at any level in the stratosphere in high latitudes, is equal to the total mass above that surface, multiplied by the eastward velocity at a height about 6 '4 kilometres greater than the height of the surface. An exactly similar statement holds for the northward momentum, but not, in general, for the resultant momentum. Thus the initial values of M E , M N can easily be computed from balloon observations which extend into the stratosphere far enough to allow the constant values of -^ , ,- to be measured. CH. 6/4. THE HORIZONTAL DYNAMICAL EQUATIONS IN THE STRATOSPHERE In extrapolating the initial observations we have neglected all the small terms, but in tracing the course of the subsequent development it is both possible and desirable to retain them. In Ch. 4/4, dealing with the dynamical equations, it is shown that a term such as lm E v x dh transforms approximately into M E M N /R. In the uppermost stratum the ranges of velocity and density are so great that one might expect the approximation to fail altogether. It will therefore be examined in detail. The argument proceeds thus : If the small terms vanish, the component velocities are linear functions of height. If the neglected terms are finite but small, a linear relation of component velocities to height may still be expected to hold approximately, as indeed the balloon-observations show that it does. This relation may therefore be used in calculating, to a first approximation, averages with respect to height. Integrating by parts and using Ch. 6/3 # 9 \m E v N dh v N \m E dh- ^ \ \ m s dh* + const ................... (1) Now putting (A + Bh) = V E in Ch. 6/3 # 1 1 we get h f J CH. 6/4 DYNAMICS OF HIGHEST CONVENTIONAL STRATUM 133 And therefore ofo 16 f"<> VffidvsC** J mf dh>= --J pivft- -^-J pdh Collecting terms p b0S - -f Taking this between the limits p and j? 2 f^ M E Miy D b 2 ff > Svydv E m E v^dh = Q h R f- pjT^ -~T- (5) i\ Now , are known initially, but not subsequently, as the stratosphere is repre- oh oh, sented by a single conventional stratum. But we may replace -rr-, -~r by means of Ch. 6/3 # 7, 8 and thus obtain _M E M N 10? W d0 (6) IllJfUVUilU fi J-T . 7\., ^ . i . H (2w sin 9)" oe on The terms on the right are known both initially and subsequently and can be taken into account when dealing with the dynamical equation in P, R, M E , M N . Inserting (6) in the dynamical equations of Ch. 4/4 and making other corresponding transformations, we obtain : The dynamical equations for the uppermost conventional stratum*. 3 (M K M X \ . 3 f 2 tan (f> /M E M f Rtf l?i /Pi/)\2 a /fifi ?/l\ 9 tan eh rl, + sin ^) / \/ a p\2 /I J^ 2 /) Note that . - is not equal to . oe on once sin 3 2 + 2 millibars thick radiates by 0'00302Sp times the radiation for a black surface. This is for radiation scattered in all directions in the actual manner to one side. Doubling the figure, since the air radiates to both sides, we get for the loss calorie . cm 2 day But the thermal capacity of the layer, at constant pressure, is calorie - i-j- cm 2 degree * E. Gold, Roy. Soc. Land. Proc. A, Vol. 82, also Met. Office, London, Geophys. Mem. 5, p. 126; Humphreys, Astrophysical Journal, 1909; R. Emden, Miinch. Akad. Ber. 1913, S. 55, said to be an important paper: I regret that I have not seen it; A. Friedmann, Met. Zeit. 1914 March; Hergesell (Lindenberg, xm Bd.) reviewed in Met. Zeit. 1920 Aug. t Compare E. Gold's analysis of surface observations, Q. J. R. Met. Soc. 1913 October. OH. 6/5/2, 6/6 RADIATION IN THE STRATOSPHERE 135 Whence we conclude that " if a thin horizontal layer in the stratosphere has its temperature slightly raised above that which would be in equilibrium with its surround- ings, the disturbance will be decreased to e' 1 of itself, by radiation, in 8'3 days." (1) Now seeing that cyclones often sweep past a given place in a time short in com- parison with 8 '3 days, it is difficult to see how the uniformity of temperature with height in the stratosphere can be maintained by radiation alone. One is led to the conclusion that the distribution of velocities must be such, that the adiabatic changes of temperature are the same at all levels. This conclusion might be avoided if the stratosphere contained a larger proportion than the lower layers of some constituent which absorbs long wave radiation ; for in calculating the absorptivity in Ch. 4/7/1 it was assumed that this quantity was the same at all heights provided the air were dry. The percentage of ozone has been observed by Pring*. He concludes that no very large increase in ozone content occurs between 5 kilometres and 20 kilometres. Also, according to A. Angstrom "f", spectro- scopic observations have shown that in summer there is not enough ozone to have an appreciable effect. Is it possible that the presence of free ions increases the conversion of long wave radiation into heat ? Again, the inconsistency between (1) and a theory depending simply on radiation may conceivably disappear when enough is known about the selective absorption of dry air for certain wave lengths to enable calculations like those of Ch. 4/7/1 to be carried out strictly. CH. 6/6. VERTICAL VELOCITY IN THE STRATOSPHERE This will now be investigated by means of the general equation Ch. 5/5#7. It is assumed that dO/dh = 0, but only at an initial instant, not necessarily subsequently. In other words 32/3 dhdt is arbitrary (l) As the air is dry the precipitation-term vanishes, the specific heats are strictly con- stant and y p ly v = 1 '405. The second member of the general equation for vertical velocity must be expressed in terms of 6, j). 2 , M m<> , M N . 20 , for these comprise all the information that is provided by the general process of computing. The discussion of what should be and what should not be provided involves a review of the entire subject and is treated in Ch. 8. I here anticipate one of the conclusions. Further, although it would be possible to do without M m<) , 1/ A . 20 by expressing V E , V N and p at all levels in terms of 6 and p,, by an extensive use of the hypothesis of the geostrophic wind ; yet, remembering that this hypothesis is only an approximation, it will be much better to use it only to * Roy. Soc. Proc. A, 90, p. 218. t Smithsonian Miscell. Vol. 65, p. 87. 136 SPECIAL TREATMENT FOR THE STRATOSPHERE CH. 6/6 give the variation of wind with height, and to introduce, for the motion of the centre of mass, M E and M N as given directly by observations, or by the general process of computing. For the centre of mass of the uppermost conventional stratum is at the level A, + &#/ dn Combining (2) with (3) we get V E at any level in the stratosphere, thus _M En _ g 9 log e f bff\ VE "R^ 2S^"an r '~ 91 ................... ( Similarly from Ch. 6/3 # 7 M N ^ g alogfl/, W\ = -JT^ + , -^-[h-h.-- ...................... (5) 2fe> sin tf> 2e \ g / We have also from Ch. 6/1 # 5 ................................. (G) Let us now prepare to insert these expressions for the horizontal velocity, pressure and density in the general equation for the vertical velocity Ch. 5/5 #7. In so doing it is best to watch for any groups of terms which are independent of, or proportional to, height, for these can be integrated as in Ch. 6/3# 14. In the first place by Ch. 5/5 #8 / 3 tan i 5 = --- de \ R w / \dn a g , I ^ I \ -- ^ ^ - "I -- ^ - ^ - f I '*2 ~~ ~~ ^ I \ -- ^ ^ - "I -- ^ - ^ - f I '*2 ~~ ~~ 2w sin

] ad .9\. d(j> d \cos _ 1 d log B d I 1 a dX d \cos Collecting terms in (8) we arrive at (10) ,. U1V d IM EK \ d /J/ V20 \ EX v ~ a"J ~D~ + 5^ ~Zi~ ~ de\ R,J ^ dn\ jR 2 J a \ R M J g , , ' 'i -- ; , = -- . '*' 'i -- 2w sin de \ a J \ g Thus divgvi; is a linear function of height. The part dependent on height does not involve the northward gradient of temperature. Again, in the general equation Ch. 5/5 # 7 for the vertical velocity, there occurs an expression which transforms thus by the aid of (4) and (5) &VE 9p Sv^ dp_ g (" _3jog_0 dp Slogd dp dh ' de dh ' dn 2o> sin { dn ' de de dn = T say, for short ..................................... (12) But by Ch. 6/3 # 5 i\ dp\ dioed ( - ^ \ g , which is independent of h ................ (13) ~r - ^ So that dp/de is p times a linear function of height. By a similar argument the same may be said of dp fin and therefore also of T. It follows that we can write T/p as the sum of its value at any basal level, say at h 2 , plus a quantity proportional to height above h. 2 . This second term is obtained from (13) and from the similar expression in dn. Thus d\og_e 1 9p 9 logj? p 2w sin ( dn ' p 2 de de !_ dp,} ' /> 2 dn J The part of (14) which depends on h is seen to vanish. That vanishing may be traced back to its cause in the assumed connection with the geostrophic wind. R - 18 138 SPECIAL TREATMENT FOR THE STRATOSPHERE CH. 6/6 The preceding expressions (11) and (14) have been prepared for insertion in the general equation for the vertical velocity Ch. 5/5 #7. Consider first the definite integral in the latter. It is which may be written + B^h-hfidh, .............................. (16) h where A, and B l are independent of height and have the following values given by (11) and (14) /tanjH-cot^N _ . jn ~ g fa\ R w I dn\ RM I ~ a \ R& 2 ' de \ a 1 f 9 log 1 fo. 3 log l^A / 17 x 2o)sin 8n ' > 2 de ~ de ',dn' ^ w sn < e \ a The integral (15) can therefore be written down by the rule Ch. 6/3 #14 in the form (1.) so that the general equation for the vertical velocity, Ch. 5/5 #7, now runs <20) The terms in (20) can again be arranged so that where T and <& are quantities independent of height, and have the following values in terms of the known quantities M 20 , M NW , p z , 6 : g r e, s n a w_ fy^/ajog^ aiogp, 2osin^Ly P \ 3 & 3e 9w "* ' The dimension of T/g is the reciprocal of a time. 9 a so that */(/ is of dimensions : (time)" 1 (length)" 1 . These expressions T and $ appear on the computing form. As the air is dry CH. 6/6 VERTICAL VELOCITY A QUADRATIC FUNCTION OF HEIGHT 139 We must next integrate (21) to find the vertical velocity. Before we can do that we need the relation of Da-/Dt to height. As precipitation and stirring are absent Da- arises wholly from radiation. The stratosphere is obviously very transparent to solar radiation and, if the calculations of Ch. 4/7/1 are reliable, the stratosphere only absorbs about f- of the diffuse long- wave radiation incident upon it. Under the circumstances the rise of temperature and the gain o/entropy-per-mass will be roughly the same at all heights. Thus we assume that jY is independent of height ......................... (25) 7p -t>t Then (21) integrates, yielding .................. (26) where the arbitrary " constant " of integration F is a function of latitude and longitude determinable by the value of v n at h. 2 . Thus eliminating F by the introduction of v m it follows that ................ (27) Equation (27) shows among other things that the vertical velocity in the stratosphere is a quadratic function of height. It would become a linear function if <3> were zero, and (26) shows that would be zero if the temperature had no east-west variation, or if the earth could be regarded as flat. See the conclusion of Ch. 6/7/2. We shall subsequently require M HW , the total upward momentum of the air above the level 4 2 . To find this multiply (27) by p and integrate by parts. It follows that 9 99 y,- g) \g g g 3 f 3 c We are now in a position to evaluate the terms - I m E v a dh, - m N v a dh, which aj 2 aj 2 are required for the dynamical equations. It will simplify the analysis to reckon all heights from \ + b0/g, the level of the centre of mass. Then, as V E and V H are respec- tively linear and quadratic functions of height, we may put (29) (30) \ tJ ' \ tJ ' where A, A', B, B', B" are independent of height and need not be further specified. So V E V H =AB + (A'B + AB') (h -h 2 - -) + (A'ff + AB") (h - h t - - Y \ *j ' \ *? ' 3 h-h,-- ................ (31) 91 182 HO SPECIAL TREATMENT FOR THE STRATOSPHERE CH. 6/6, 6/7/1 Next multiply (31) by p and integrate the product " by parts " four times, remem- bering that by Ch. 6/6 # 7 ~ P = - ......................... (32) / */ Then, on putting in the limits, the linear term disappears, leaving \2A'B" ....... (33) t J " But, by similar integrations by parts, ro ro J 2 f ( fhf)\^ pv H dh = R, 9 \B + B"(~ u )\=M im . ...(34), (35) J 2 t \ 9 I } And on substituting these in the right-hand side of equation (33), it follows that 3 f J 2 3 MM . M im , 3 D /b6\ 2 dvj,(Sv H ., . b - . + tmce --)K... (86) -2o a \g m \ g which is in suitable form for computing. As v n , M H are only found directly over the points on the map where pressure is tabulated, the mean of four surrounding values must be used here. CH. 6/7. DYNAMICAL CHANGES OF TEMPERATURE IN THE STRATOSPHERE CH. 6/7/1. MEAN HORIZONTAL TEMPERATURE GRADIENTS Radiation-equilibrium does not explain why, when tropical are compared with polar regions or anticyclones with cyclones, the stratosphere is found in each case to be cooler where the air beneath it is warmer. Prof. V. Bjerknes* has published a dynamical explanation. Taking the case of the pole and equator he begins by assuming, as is more or less borne out by observation, that the lower part of the atmosphere is spinning about the polar axis more rapidly than the earth, while the upper part is spinning less rapidly than the lower part. From the distribution of velocity he finds the form of the isobar ic surfaces, assuming that the motion is steady, and that it is uncomplicated by minor circulations. The vertical separation of a pair of adjacent isobaric surfaces varies from place to place proportionately to the volume- per-weight of the air between them. Bjerknes takes the volume-per-weight to be proportional to the absolute temperature. This last step is unconvincing as the weight depends not only on latitude but also on the eastward velocity relative to the earth. However a closer examination confirms Bjerknes' remarkable result, unless the varia- tion of speed with latitude is unnaturally large. In any case it relates only to a steady motion along the parallels of latitude. Bjerknes' theory of the temperature difference between cyclones and anticyclones is similar to the above, except that the rotation considered, instead of being around the polar axis, is around the vertical. * V. Bjerknes, Comptes Jtendus, Paris, t. 170, p. 604 (1920). CH. 6/7/1, 2 DYNAMICAL THEORIES OF TEMPERATURE DISTRIBUTION 141 In the same publications Prof. Bjerknes explains the smaller height of the tropo- pause in cyclones by the fact that the angular velocity of the troposphere about a vertical axis is greater in cyclones than in anticyclones, while the angular velocity of the stratosphere is in both cases small. This explanation agrees with one given in other terms by Mr W. H. Dines*. It may easily be illustrated by floating oil on water in a glass vessel and stirring the lower layer. But we must pass on ; for in this book we are not concerned to explain mean dis- tributions, but to forecast. CH. 6/7/2. DYNAMICAL INFLUENCE ON VERTICAL ISOTHERMY It is indicated in Ch. 6/5 that radiation alone is probably insufficient to maintain the uniformity of temperature in a column. Let us enquire whether the compressions and rarefactions are so distributed as to produce the same adiabatic temperature changes at all heights. Sir Napier Shawf has discussed this problem in a special case that of the initial motion of a stratosphere disturbed from rest by the passage of a cyclone beneath it -and the conclusion is that the temperature change is the same at all points of a vertical. Let us now rework the problem more generally. Indeed it must be solved as part of the general process of forecasting for the uppermost stratum. The case of the lower strata is different. In them we find dR/dt from the equation of continuity of mass. Next by summing g . dR/dt downwards from the top we get dp/dt at the boundaries of the strata. Then by a process equivalent to interpolation a change of pressure is found for a mean level in the stratum so as to correspond to dR/dt, or to dp/dt which follows from dR/dt. Having thus obtained dp and dp at the same level, the temperature change follows at once from p bpd. This process fails in the uppermost stratum for there 3R m gives dp. 2 , and we have 8p = 0, but the^ range of pressure is far too great to allow us to interpolate between dp., and dp a a value for comparison with some mean density to be derived from dR w . The failure is thus not due to isothermal conditions but to the application of finite differences to a quantity varying in an infinite ratio. But we can proceed as follows. The equation for the conveyance of heat Ch. 4/5/2 # 1 is da- da- da- da- Da- ,, where Da- is the change due to radiation which has been discussed in Ch. 6/5. As the instantaneous distribution of velocity components may be supposed, in view of the discussions in Ch. 6/3, Ch. 6/6, to be known at all heights, and as the same applies to the entropy-per-mass, equation (l) may be expected to give us temperature changes at all heights. In the following process it is assumed that dO/dh = Q, all over the map at the instant considered, ............ (2) * W. H. Dines, Phil. Trans. A, Vol. 211, p. 276. t "The Perturbations of the Stratosphere," Meteor. Office London Publications, No. 202, 1909. 142 SPECIAL TREATMENT FOR THE STRATOSPHERE CH. 6/7/2 but not necessarily at neighbouring times before or after so that 9 2 , dhdt s arbitrary ..................................... (3) Begin with the expression for the entropy-per-mass Ch. 4/5/1 # 8 which may be written dcr = b.dlogp + y p .d\og0 ............................ (4) Then by (2) dcr 7 9 log p But by the hydrostatic equation _ 9 be So from (5) and (6) dh^0" Thus in the stratosphere the entropy-per-mass is a linear function of height in- creasing by g/0 per unit of height. The statement applies to our hypothetical initial instant or to any other instant at which dd/Sh is shown by observation to vanish everywhere. In these circumstances Scr/Se, dcr/dn are also linear functions of height although they increase at a different rate thus 9V gd0 ' It appears from (7) that temperature can be found more directly from 9cr/9/i than from a: Accordingly let us differentiate (1) with respect to height. The result is 9V dv E dcr dv N 9 ' de a so that is independent of height. On eliminating d 2 v n /dh 2 from (14) by means of (15) the result is since as d0/dh it follows that (dlog0\_i dt .(17) The condition that the vertical distribution should not be departing from isothermy is that the second member of (16) shoidd vanish. We may distinguish the terms in it calling -/9 the adiabatic term and the rest the radiation term. In the latter Da/Dt has already been discussed in Ch. 6/5 and its smallness suggested that the adiabatic term must also be small, for otherwise d' 2 6/(dhdt) would be noticeable in the observations. 144 SPECIAL TREATMENT FOE THE STRATOSPHERE CH. 6/7/2 For comparison with balloon records the adiabatic term may be put into a form which avoids horizontal temperature gradients, which have large observational errors. For, neglecting radiation, (13), (14), (15), (15 A) yield = 0-288. . 6 dhdt 2h a Here - 288 is the value of 1 y v lj p for dry air. The presence of an infinite tan< at the pole is of purely geometrical origin. The 8 / term in cot$, infinite at the equator, comes from - -. . and so arises from the dn \sm 97 known failure at the equator of the geostrophic approximation. But let us consider middle latitudes. Both d0/dt and dd/de have their ups and downs, so that on the average of many balloon ascents both sides of equation (16), apart from the radiation term, are likely to vanish, without yielding any check on the theory. Individual sets of observations, sufficiently comprehensive to test (18) are scarce, but I have found a few. One relates to 1920 May 19, the day before that of the example of Ch. 9. Strassburg is ISOkmNNW of Zurich and for the present purposes, on account of shortage of observations, we must regard them as one place. The theodolite observations yield the following up-grades of the north component of wind May 19 d 8 h Zurich -2-0 x IQ-'sec- 1 May 20 d 8 h Strassburg - 1-9 x lO^sec-' Zurich -0-6 x lO-'sec- 1 . The first of these is read from the diagram on p. 129. The last is very doubtful, being obtained by rejecting the highest reading. There is however an indication of persistent decrease of V N with height during this 24 hours, and we may put the mean at . Sv N /Sh= - 1'5 x ^""sec" 1 . At this latitude a" 1 (tan < + cot <) = 3'15 x 10~ 9 cm' 1 . It follows from (18) that 1 3*0 a... u ohm = -l'4xlO~*cm~ 1 seo The sign implies that the temperature would decrease more rapidly with time in the higher levels. Let us find the difference in temperature-increases during one day at two points separated vertically by one kilometre. The required quantity will, according to (16), be obtained by multiplying l'4x 10~ 2 cm" 1 sec" 1 by the temperature, by 86,400 seconds, and by 10 6 cms. It works out to = 2'7 C per km per day. Now the balloon observations provide a check on this figure, for the record at Strassburg reaches to 15 '5 km at 8 o'clock on both days. The differences between the CH. 6/7/2 OBSERVATIONAL CHECK ON GEOSTROPHIC THEORY 145 two temperature curves are within the observational error. On both mornings the lowest temperature is at about 12 km and there is a large recovery above that. It may be said with some confidence that between 12 and 1 5 "5 km the observations show that jjTjTT is less in absolute value than 1 C per km per day. The table summarizes this comparison and adds others given thus briefly. They all are taken from the international balloon ascents* for the year 1910. Place Time G.M.T. Year 1910 98 0-288 6 dVfr ( tan * + cot ^ 9 2 dhdt 8h on a. C km c C km . day km . day Zurich 12 to 14k Strassburg 12 to 14k Feb. 3 d 9" 2" 8 h to 3 d 8 h o-o -2-3 -4-0 Zurich ) I 10 to 20k StrassburgJ Strassburg 12 to 15 - 5 k fl9 d 8 h Ma y{20 d 8 19 d 8 h to 20 d 8 h + 1-6 -2-7 0-0 Lindenberg 14 to 1 7 k I May 19 d 4 h 18 d 8 h to 19 d 4 h + 0-6 + 0-9 + 1-7 Vienna 14 to 18 k The temperature t May 19 d 8 h 19"4 h to 8" s were taken fror hose by a differer + 0-6 Q two therm it pattern be -0-9 ometers of the same patl ing rejected. -2-6 wherever it occurs, both in the equation for temperature change and in that for vertical velocity. As 50/dt is taken as independent of height, we can find it at any one height. The height of the centre of mass h z + b6/g commends itself as being the one that would be chosen on general principles, if there were no theory at all concerning variations with height. The locus of the centres of mass of all vertical-sided columns, having their bases at a common level h.,, is not quite a level-surface, since b0/g varies. However that does not appear to lead to any error in the following theory, in which the east and north differentiations are made strictly on the horizontal. Similarly time-changes must be taken at fixed heights. Beginning again at the equation for the conveyance of heat Ch. 4/5/2 # 1 and ex- panding dcr we have /a a a a\_ , /a a a a\. D6/y so that by Ch. 6/3 # 14 and Ch. 6/1 # 8 v E = M m& /R. 2() , v s = MXW/RM (2) Since we now neglect we thereby take the vertical velocity to be, like the east and north velocities, a linear function of height and therefore v a = M ato /R so (3) In simplifying the term in (l) which contains log^> we must remember that the differentials of logp in latitude, longitude and time are to be taken at the fixed height h. 2 + bd/g, so that 6 in this expression alone must be regarded as fixed at its instantaneous local value, which we may distinguish by a dash. In all other con- nections 6 is variable and undashed. Thus W 9 logp ff --f = - (4) g dfi Whence dt dt + 0*'di = ~~dt h dt ' ' And dlogp _ dlogp., 6' dd _ d log p 2 d log 6 , , de de ~ffide,~ de de And similarly for the northward differentiation. These terms in log 6 combine with those in (1) which explicitly contain log 6 in such a way as to change the thermal capacity at constant pressure to one at constant volume, thus 7 P -b = 7 (7) On inserting these relations in (l) and rearranging terms we get what is required, thus dt R w de R. 20 dn y v p- 2 \dt R. 2U de R 2(> dnj y v R. i0 y v Dt' This gives d0/dt in terms of known quantities for dp.,/dt is found from the accumulation of mass by Ch. 6/2 # 1, 3, and M Ht(l is found from Ch. 6/6 #28 in which <& is to be neglected. CH. 6/8. SUMMARY An effort has been made to treat all the air above 13 '8 km as a single conventional stratum in the sense that the momenta, pressures, and densities in a column should each be represented by a single number. To do this, all quantities have to be integrated with respect to height. The integrals of p, p, a- come out simply because the tempera- ture is independent of height. But integrals involving velocity can only be obtained from the relation at any level of velocity to pressure. The strict treatment of this relation by analysis is too difficult, and so the geostrophic approximation has been 192 148 SPECIAL TREATMENT FOR THE STRATOSPHERE CH. 6/8 introduced. This is probably good enough for transforming the dynamical equation, but when applied to finding the temperature change it yields results which are unlikely. If on further consideration the single stratum has to be abandoned, another plan is ready. Divide the stratosphere into several conventional strata. For all of these except the uppermost the general processes of Ch. 4, Ch. 5 will apply. The mass of this one having been made small, any errors committed in treating it will be of little consequence near the earth where we live. Or again a conventional division may be made at a height of 20 km in order to benefit by the observed steadiness of pressure at this level, a steadiness to which Mr W. H. Dines has called attention*. * "Characteristics of Free Atmosphere," p. 71, Meteor. Office, London, Geophys. Mem. No. 13. CHAPTER VII THE ARRANGEMENT OF POINTS AND INSTANTS CH. 7/0. GENERAL IT will be convenient to have a brief distinctive name for the arrangement here to be discussed. For this purpose we may borrow from crystallography the term "lattice." The approximate representation of a differential coefficient by a ratio of finite differences is notably more exact when the differences are centered*. If A, B and u be any three variables, the ideal arrangement would be such that: 8 t (i) Wherever B has to be equated to -^- , then B should be tabulated at points on the it-scale half-way between the points where A is tabulated. (ii) Wherever two variables have to combine in an expression, not involving their differential coefficients, they should be tabulated at the same points and instants. (iii) When A is a function of -^ , special difficulties arise. See below under the arrangement of instants, Ch. 7/2. Unfortunately it is not possible to satisfy conditions (i) and (ii), in their entirety, for the given differential equations. The best we can do is to satisfy (i) and (ii) for the largest terms, and to leave the rest to be centered by interpolation. CH. 7/1. THE SIMPLEST ARRANGEMENT OF POINTS (a) To fit with ^ = gp it is convenient to tabulate p at the heights where strata meet, so that, for example, p a p e = gR es , where R u is the mass per horizontal area of the stratum bounded by h a and h 6 . (b) To fit with the equation of continuity of mass, R should be tabulated at points intermediate between M E and M N , when seen in plan; thus: to | north M M E R M K * to east M N * Vide W. F. Sheppard, Proc. Land. Math. Soc. 1899 Dec. 150 THE ARRANGEMENT OF POINTS AND INSTANTS CH. 7/1, 2 (c) To fit with the two horizontal dynamical equations, when all terms are neglected except; =*-- v -- ~ = ^- + 2w sin AM E at an M E and M N should be tabulated at points intermediate between P when seen in plan, thus : P P P M K P P M N P (d) Fortunately condition (c) is consistent with both (6) and (a). The frontispiece shows the arrangement adopted. It satisfies (a), (b), (c). The points at which pressure and momenta are to be tabulated are indicated respectively by P and M. The coordinate differences are 200 kilometres of arc in a north-south direction, and in longitude the intervals between 128 equally spaced meridians. (e) To fit with the characteristic equation of moist air, and with the expression for the entropy, it is convenient to have W the mass of water per unit area of a con- ventional stratum, become infinite and are balanced by an infinity of opposite sign in the terms in These infinities are artificial in the sense that cos .. i/20 See Ch. 6/1 # 7. The result is entered on Computing Form P I. CH. 8/2/3. MEANS ACROSS A STRATUM Where, in what follows, we require to know the mean values across the thickness of any stratum of the quantities p, p, w, m E> m N , they are usually taken respectively as P/8h, Rjh, W/Sk, M E /8h, M x /8h, where 8h is the thickness of the stratum. Similarly, mean values for a stratum of p., V E , V N are usually taken respectively as \\ I R, M E /R, M N /R. If some other process has to be employed in any case, it will be mentioned specially. 158 REVIEW OF OPERATIONS IN SEQUENCE CH. 8/2, 4, 5, 6 CH. 8/2/4. SATURATION For each stratum, except the uppermost, we find whether the air is saturated or not by means of Ch. 4/1 # 5, employing for p, p, ^ the mean values which have just been indicated. From the appropriate characteristic equation (Ch. 4/1) are next found the mean temperature of the stratum and also if necessary w s , the density of aqueous vapour saturated at this temperature. We may note that when the air is dry, the particular kind of mean temperature yielded by this process is I 6pd]i-^- I pdh. J J The temperature is entered on Computing Form P i. CH. 8/2/5. UNIFORM CLOUD AND PRECIPITATION We follow the scheme described in Ch. 4/6. For each stratum, in which the air is saturated, we nmlt'iply the density of saturated vapour w s by 8k so as to obtain W s the mass of vapour per horizontal area of the stratum. Then W W s is the mass of condensed water per horizontal area. If W W s exceeds 0'4 grm cm" 2 , we deduct the excess from the stratum and transfer it to the ground as precipitation. See Ch. 4/6. The cloud is entered on Form P i, the precipitation on Forms P I and P xvm. Local cloud and precipitation due to heterogeneity are treated separately, see Ch. 8/2/8 below. CH. 8/2/6. SHALL WE USE ENTROPY-PER-MASS a, OR POTENTIAL TEMPERATURE T? In the classical thermodynamics entropy occupies a central position, whereas potential temperature is rarely mentioned. In meteorological theories these positions are almost reversed. Let us adopt the definition of entropy favoured by G. H. Bryan as being most suitable in connection with irreversible internal processes such as the smoothing out . of heterogeneity: "If from any cause whatever, the unavailable energy of a system with reference to an auxiliary medium of temperature # ; undergoes any (positive or negative) increase and if this increase be divided by the temperature Q t the quotient is called the increase of the entropy of the system*." It should be noted that the choice of 6 t makes no difference whatever to the numerical value of the entropy provided that t is lower than any other of the temperatures concerned. If 6 t is not lower, the interpretation is unphysical. Let us define T as " that temperature which the air would attain on being brought to equilibrium at a standard pressure p t without loss or gain of either heat or moisture." The numerical value of T then depends upon the conventional p i . The entropy when condensation occurs. Error in Ch. 4/5/1 #13 and in the Hertz diagram. Even if the sky be cloudless, evaporation or condensation is usually in progress at the foliage. From general thermodynamics it is evident that a quantity de of radiant * Bryan, Thermodynamics, 71. B. G. Teubner, Leipzig, 1907. On. 8/2/6 DIGRESSION ON ENTROPY 159 energy absorbed by a leaf at A and given out again to the air at almost the same temperature will increase the entropy of the atmosphere by de/0 A irrespective of whether the energy goes in warming or in separating water molecules during the process of evaporation. Therefore when entropy is stirred upwards by eddies it may be con- sidered as going as the sum of two fluxes, one depending on sensible heat, the other on the water-vapour, and the latter flux must involve the latsnt heat. But we have shown that the flux of entropy, except for a possible modification depending on irre- versibility, is Therefore da- must involve the latent heat T, even if the air is clear. Now Hertz's well-known formula d]og^ + v -jj-^P. ( 4 ) where t, is the latent heat of fusion. At 180 'A the water per mass p. would be practically zero, so that da- is simply 1> . dlogp as above. As the isopleths of pc0 nst . > should be reckoned correspondingly. So to find a M we should take a pair of points on the Hertz diagram corresponding to the same p and p but to slightly different p,. Owing to an approximation which Hertz makes, these points coincide. In other words, Hertz's diagram makes (acr/a^ P HMk (5) But on tracing the samples down the adiabatic, they become saturated at different pressures, and so arrive at the snow stage with different values of a-. In this way the following values of o^ have been computed for unsaturated air : temperature pressure p. a^ A mb 10- 3 x 10" ergs/degree 300 1000 6 96 300 1000 18 84 280 700 5 98 (6) 272 600 3 101 263 450 3 117 253 500 1-5 120 roughly For air saturated with water the effect of adding more water is, according to the Hertz diagram, merely to increase the hail stage, and on this account a M = 17x10" ergs/degree (7) For air saturated with ice the addition of more ice is apparently to leave the entropy unchanged. That is because Hertz neglects the thermal capacity of the ice. Neuhoff's computations appear to be more accurate in this respect. But if great accuracy is to be attempted, it will no longer be possible to consider, as Hertz does, that p and 9 suffice to define the density, irrespective of p.. In the actual troposphere the entropy-per-mass is so very nearly independent of height much more nearly so than would be supposed if we took the observed temperatures to apply to dry air that quite small changes in procedure are enough to change the sign of the computed 3 /y cr. R. 21 162 REVIEW OF OPERATIONS IN SEQUENCE CH. 8/2/6 In reckoning the entropy of water from 180 A, we include the latent heat of fusion i- 1 of ice, divided by 273 A, so that the eddy flux of entropy in the form - =- would also \j y include the same. Now a flux of entropy found in this way at the level A 8 will often have to be compared with the flux at the foliage in order to find the accumulation in the air between and 2 km. If we were to treat the latter flux as the radiant energy divided by the temperature of the foliage, the water substance leaving the foliage would be supposed to carry with it the latent heat of evaporation, but not that of fusion. The proper course is evidently to reckon the ground as supplying, vid the plant stems, water having a certain entropy-per-mass which includes the effect of the latent heat of fusion. In general, wherever water-substance occurs, we must credit it with an entropy-per-mass reckoned from the same temperature, in this case ideally from 180 A, but with an approximation introduced by Hertz. The behaviour of entropy-per-mass and of potential temperature in regard to turbulence. One of the chief questions is whether, when condensation occurs, either T or a- satisfies the three conditions Ch. 4/8/0 # 2, 3, 9 which any quantity x must satisfy if it is to diffuse according to the equation (8) The first of these conditions, Ch. 4/8/0 # 2, is equivalent to, and perhaps finds a clearer expression in, the " mixing-rule " of W. Schmidt*, which states that samples having masses m ]; m., and XL X* mus t gi ye a mixture having x 3 such that The answer to this question becomes fairly clear if we adopt v. Bezold'sJ view of the process of mixture. Let us consider only processes which occur without loss or" gain of heat or moisture. Bezold replaces in imagination the natural process by two artificial ones in sequence. In the first artificial process there is supposed to be neither evaporation nor condensation. So that as the specific heat y v is nearly a constant, the temperature 3 of this mixture is given approximately by the mixing rule (9) above. And a similar rule will be followed, without approximation, for p. the total water substance per mass of atmosphere and for v the mass of water-vapour per mass of atmosphere, so that /x v represents the liquid or solid. So that if subscripts 1 and 2 denote the component, and subscript 3 the mixture, the mixing rule (9) may be written n ft ** ~ Ml _ V 3 P! _ P 3 P! _ "a., /JQ\ fr-fr v*-v 3 2 -0., m,' * Wm Schmidt, " Der Massenaustausch...," Sitzb. Akad. Wiss. Wien, Mathem.-n. Klasse, Abt. n a, 126 Band, 6 Heft, 4. 1917. t By an exception to the convention holding in the rest of this book, subscripts in Ch. 8/2/G do not denote heights. } Sitzber. Akad. Wiss. Berlin, 1890, pp. 355 390. English translation in Abbe's Mechanics of the Earth's Atmosphere (Smithsonian Institution, 1891). CH. 8/2/6 IRREVERSIBLE MIXING 163 This first process is not reversible because the mixture cannot be separated by reversing the motions. Therefore although no heat enters from outside, the entropy must increase. The second artificial process is one of adiabatic condensation or evaporation of the intimate mixture of moist air and liquid or solid particles. During it p, is fixed, but v and 6 change in such a way that (11) where T is the latent heat of evaporation or of sublimation. This change proceeds until either the air is saturated or /A v is reduced to zero. Bezold gives a pretty diagrammatic representation of it in the p., 6 plane, which however does not concern us here. Owing to the intimacy of the mixture this second process must be almost perfectly reversible, and therefore during it both entropy-per-mass and potential temperature are unchanged. The question arises whether in the natural adiabatic process of mixing, the changes in 7 and cr would be the sum of their changes in Bezold's two consecutive adiabatic processes. If we provisionally assume that this is so, then we can deduce the following important proposition. Take at one height a sample of cloud having properties T I , cr, , p. t , v l and at another height a sample of clear air having properties r. 2 , if the diffusion equation (1) is to hold, is that x should not be changed by delay. See Ch. 4/8/0 # 3. This applies obviously to cr or to T provided that the processes are really adiabatic or that we account separately for their non-adiabatic parts, as we do here in the case of radiation. The third condition to be satisfied by the diffusing quantity x is that the upward flux of (x x mass) should vanish when d^/dh vanishes ...................... Ch. 4/8/0 # 9. This is satisfied by either cr, T or /j. since they are not changed when a portion of air is raised or lowered without loss of heat or of moisture; and when these losses occur they are taken into account separately. As all three conditions are satisfied for T, the flux of (T X mass) is seen from % Ch. 4/8/0 # 17 to be cdr/dh or equivalently - . But now we come to a peculiarity. Suppose that a portion of cloud and a portion of dry air are in contact at the same temperature and pressure. Then it is impossible to say which has the greater potential temperature until the standard pressure p i has been fixed. For if p t be larger than p, then in changing from p to pi some cloud will evaporate and the dry air will have the higher T. Whereas if p t be smaller than p, then in changing from p to p { more water will condense in the cloud, so that the dry air will now have the lower T. Thus the flux of (rXmass) may be reversed in direction by a mere change in the conventional pressure p t . But the fluxes of latent or of sensible heat are not thus fantastical, 212 164 REVIEW OF OPERATIONS IN SEQUENCE CH. 8/2/6 and the explanation of the behaviour of T must be sought in its relation to heat. It 3 is therefore evident that - 5- does not in general measure the flux of sensible as dis- gdp tinct from latent heat, as it does at least approximately in the special case of clear air. The difference of entropy-per-mass between the cloud and clear air is free from the kind of artificiality that complicates dr. Thus to sum up : if we use potential temperature we must attend to any local and temporal variations of p it and to the relation of the flux of heat to that of (T x mass), but irreversible mixing does not matter. If we use entropy-per-mass we must add on the gains due to irreversibility, but the flux of entropy at a surface is simply the heat entering per area per time divided by the temperature, and to compute it we need not know the evaporation which the heat produces. Increases of entropy by irreversible internal processes such as the smoothing out of eddies or of patchiness by molecular diffusion. It has been noted on p. 40 that irreversible mixing of cold and warm air increases the total entropy but not the mass- mean of potential-temperature if the air is dry. For this reason on p. 69 the diffusion equation appears in a simpler form in potential temperature than in entropy. The latter equation Ch. 4/8/0 #20 may, for clear air, be expanded to read Dcr_ 3 / dcr\ f (dcrY ( } IK *(**)+ K\9r This equation is, for dry air, of the same form in cr as it would be in the potential temperature T except for the presence of the last term. Now in a paper 011 the "Supply of energy to and from atmospheric eddies*" a term of just this form %/y p . (9 + const (12) and y v and b are functions of p only. Therefore if air at 6, p changes to T, p t while and, as the moisture in the clear air increases, b and y p , though both increasing, do so in an almost constant ratio : thus /* = -000 -010 -020 -2881 "2870 I ) ......................... ' A formula for T will now be derived, which is convenient in the present scheme because we have given p as a function of h. On combining the hydrostatic equation dp/dh = (jp with p = bpO there can be obtained 0= -.9.1 dh } ' which is strictly correct provided b be given its value corresponding to the existing p. Now on multiplying (15) by ( p ; //>) 6/7 " and assuming that b/y p is independent of 166 REVIEW OF OPERATIONS IN SEQUENCE CH. 8/2/6, 7, 8 height, while admitting arbitrary variations in b proportional to those in y p , there results _^L ^(P'' /7 ")J ver ertically or approximately, with values of the constants corresponding to an ordinary value of p. = -0-983 Here h is the height in centimetres. The unit for p is arbitrary provided that the same is used for p t . A table giving jr 288 as a function of p will be found in the Quarterly Journal of the Royal Meteorological Society for July 1921. Cii. 8/2/7. AEBANGEMENT OF LEVELS FOB THE EDDY-FLUXES The moisture is tabulated at the mean levels of the strata. Its rate of accumula- tion is also required at these levels, and therefore the fluxes must be tabulated at the intermediate levels where strata meet. At the ground the boundary is somewhat blurred b}' the presence of vegetation ; but if we imagine it as viewed from the neighbouring boundary, 2 kilometres above, it would appear to be sufficiently definite. CH. 8/2/8. ESTIMATE OF TURBULENCE AND HETEBOGENEITY The estimate is intended to be based on the instantaneous distribution of entropy, of velocity, and of water-substance. The time taken to establish or destroy a state of turbulence or of patchiness is thus neglected. Common observation of the diurnal varia- tion of gustiness, or of the rising and dissipation of cumuli, show that this time is of the order of an hour, and is therefore unimportant in comparison with the time of passage of a cyclone, with which forecasts are usually concerned. An alternative plan, more correct and physically more interesting, but not attempted here on account of its probable toilsomcncss, would be to take suitable measures of turbulence and of heterogeneity as main variables in addition to the seven chosen in Ch. 4/0, and to trace their time changes step by step. The equations expressing their changes might include Ch. 4/8/1 # 22, Ch. 4/9/7 #6, Ch'. 4/9/8 #8, but this system is not in itself complete, and I do not know the equations required to complete it. The coefficient Ga developed in Ch. 4/8/5 for finding the amount of water moving up through the thick stratum is estimated from the wind measurements set out in the last column of the table on page 84. The coefficient JUL, introduced in Ch. 4/8/4 for finding the upward flux of heat in the thick stratum, is similar to G8 and is, pro- visionally, put equal to it. Observation and theory both suggest although they do not yet prove that these two measures ? G8 and JUL of turbulence near the ground could be expressed fairly well as functions of three variables, namely of the locality, of M Gs the momentum per area of the lowest stratum, and of T Ga T A , where r C8 is the mean potential temperature of the stratum and T A is the temperature of the interface where radiation is converted to * Compare Exner, Dynumisclte Meteuroloyie (Teubner), Art. 70. CFI. 8/2/9, i> " THE SEQUENCE OF COMPUTING 167 heat. This functional relationship could probably be made more definite by lowering the height denoted in these symbols by the suffix 8. But however much it is lowered we should still require to base the estimate on the temperature actually at the surface of the soil or vegetation. This temperature is computed below in Ch. 8/2/15, 16 from an equation which is non-linear, and which involves Gs and JUL in certain terms. Thus the best way to disentangle these processes appears to be to make first a trial estimate of T A , on which to base corresponding values of 6 ., and of JUt. These are then used, together with the radiation, to find a corrected value of T A and thence corrected values of f Gs and JU.. On the other hand the radiation cannot be computed until the amount of detached cloud has been estimated, and detached cloud, being an effect of heterogeneity, is naturally grouped together with turbulence. Thus after estimating the amount of detached cloud and of local showers by the aid of statistics (Form P in) we next compute the radiation, and come back to the diffusion produced by turbulence at a later stage (Ch. 8/2/17, 19). CH. 8/2/9. RADIATION From Ch. 8/2/5 we know in each stratum the density of any continuous cloud which there may be. In Ch. 8/2/8 we estimated the amount of detached cloud. The radiation can therefore be traced downwards and upwards. For long-wave radiation I have used the "approximate simplified process" of Ch. 4/7/1, for solar radiation the process and constants of Ch. 4/7/2. The processes have been so fully described there, that it should suffice here to refer to the computing forms Piv, v, vi and to note that the upward long-wave radiation from the interface, although to appear on Form P yi, is not com- puted until the subsequent Form Px has been filled up according to Ch. 8/2/15. Cn. 8/2/10. THE EVAPORATION FROM THE SEA The rate of evaporation depends only on the up-grade of /u, and on the distribution of eddy-motion in the air. According to Ch. 4/8/5 the rate of evaporation is taken as JG_S OzM = g for 9 where p. G is the value of p. which would be in equilibrium with the water at the surface of the sea. It is assumed here that the air actually in contact with the water is practically saturated. Observations made at the height of a ship's deck are no proper test of this assumption. Cn. 8/2/n. EVAPORATION FROM FOLIAGE At the mean level of the stratum (G, 8) we know by Ch. 8/2/3 a mean value fi as or say /i 9 of the water per mass of the atmosphere. In the intercellular spaces of the leaves \L according to Brown, Escombe and Wilson has, except in case of wilting, its saturated value at the temperature of the leaf. The latter is not a main variable but has to be derived from the temperature of the highest stratum of soil and that of the lowest stratum of air, taking into account the radiation. It will be convenient 168 REVIEW OF OPERATIONS IN SEQUENCE CH. 8/2/11 to defer this problem to Ch. 8/2/16. There is not room for much water to accumu- late in the air close to the leaves, so that the rate at which water is coming through all the stomata, above say a hectare of soil, may be put equal to the rate at which it is being carried aloft by eddies across the upper boundary of the vegetation film. There is thus an analogy to the electrical instrument known as the potentiometer in that there is a current through two resistances in series ; and /A plays the part of the potential. The transpiration equation Ch. 4/10/3 # 3 may be written thus in terms of /A (Mass of water evaporating per time) -, ,, <, \ c c r , ., c , if = K P Weaves /A o /,}=> ......... (<-) (from foliage above unit area of landj where \L QL is the value of /A for the air surrounding the leaves and Cleaves is the value within the leaves. For a field of growing barley Kp is of the order of 0'25 x 10~ 3 grm sec' 1 per horizontal cnr ...................... (2) But by Ch. 4/8/5 the upward flux of water between the vegetation film and the middle of the stratum next above is WJi (P-OL / o \ y po-p* where f i8 may be taken as the same as G8 given in Ch. 8/2/8 and where /Z i8 is practically the same as p, 08 since the vegetation cannot hold much moisture. Equating these two values of the same flux, f 1 __ GS (V-QL ~ P-Gs) / 4 \ Kp{ ^ es -* OL} -g ^(P -P*Y By the analogy to the potentiometer or by solving these equations, we see that and ff \P_~P S 1 pl a y the part of resistances in series. Call them A and B respectively. 2^08 Then the common flux is Cleaves /*C8 _ a / R \ A+B which is the formula used to determine it on Computing Form P vu. The water per mass in the air surrounding the vegetation is not required if our only object be to make a "lattice-producing" system. But this quantity may be of interest for its own sake. It is determined by -D . /^leaves + A ~ ~ A portion of rainfall is caught on vegetation and evaporates there without ever reaching the ground. From what is known about eddy diffusion we should expect that the amount evaporated would be proportional to p. A ^ p., the difference between the saturated water-per-mass at the vegetation and the actual water-per-mass at />-, and also to be directly proportional to 2tj a J{ 9 / fi'\ n* 1 "//} a'\ j_ A v ^ft'* \ ^ G + C . , 7- (T V V)- , (0 n - V } + 4 8 C / (20) This equation determines ^. and so by (18) the surface temperature 6 . Of course there are simpler less accurate ways of arriving at 6 G . One is to be content with the guess 0', that is to say to neglect ^. altogether. Another simplification would be to neglect various portions of the long equation for ^., but it seems best to draw up Computing Forms P ix, x for the full equation, and to leave the relative importance of the various terms to be decided by further experiencef. Terms such as 3C" Gi ,/90' imply that we have a table, based on observation, giving the conductance C G9 , which depends on turbulence, as an empirical function of M 08 , the momentum per area of the stratum, of (T, 6 G ), and of the locality. Observation and theory both indicate that such a functional relationship exists, at least approxi- mately. Then taking the fixed actual M a8 and locality, we pick out corresponding variations of C G9 , and of 6 G around a central value & ', and we call the ratio of the variations dC'^/Sd'. The best sequence of this portion of the computing-operations is thus seen to be the following. Estimate the detached cloud (Form Pin) and compute the solar radiation and the descending radiation from the atmosphere (P iv, v, vi), make a guess 6' at the surface temperature, and put on Form P in the corresponding measures of turbulence near the ground which we have denoted by G8 and JUL. From these find the trial values of the fluxes of evaporation and of heat at the interface. Find also the long- wave radiation which would leave the earth if it had the temperature & '. Put all these quantities into equation (20) and so determine the temperature correction 2^.. This is done on Forms Pix, x. Then correct the separate quantities to the true temperature of the surface, and proceed to other parts of the computing. The partition coefficient, which was discussed in Ch. 4/8/4, does not appear in this revised treatment. * Tk is a Coptic letter pronounced "dalda." t In the example in Ch. 9 the terms C'u.^ + C' A , } in the denominator are together ten times greater than the sum of the rest. 176 REVIEW OF OPERATIONS IN SEQUENCE CH. 8/2/16, 17 CH. 8/2/16. THE THERMAL BOUNDARY CONDITION FOE VEGETATION This may be discussed as a variation of the process described in Ch. 8/2/15 for bare soil. We may neglect any accumulation of heat in the vegetation film itself. Thus, for example, in a field of ripe wheat the mass* of the vegetation is about 2 "4 tons per acre, that is 6 tons per hectare, which is 0'06 gram cm" 2 . The mass of the vegetation is therefore of the same order as the mass of the layer of air between the tops of the stalks and the soil; and the sum of the two masses is negligible in comparison with that of the conventional stratum of air which extends up to 2 km . It is at the height where the foliage is densest that most of the radiation is con- verted to heat, and most of the water is evaporated. A suffix is needed for this height; let it be A. In place of the conductance C UiW of Ch. 8/2/15 we have now two con- ductances in series, one C lltW as before, the other (7 10j ^, so that T. Bedford Franklin (Edinburgh Roy. Soc. Proc. Vol. 39, Part 2, No. 10) has called attention to the great resistance offered to heat-flow by moss, by a carpet of dead leaves or by grass, instancing a primrose flowering with its roots and flowers in temperatures differing by 10C. This thermal resistance may come into either l/G lltll> or l/C w Ihis, expressed in symbols, is -~ + -*-^ My -- r and is next computed for each stratum. This expression will be denoted by div^-M"; the dash serving to dis- tinguish it from the corresponding expression with differential coefficients in place of difference ratios. The arrangement of the computing has been shown on p. 9. The computing form is P xnr. CH. 8/2/21. PRESSURE CHANGE AT THE GROUND Then &\v' EX M in each stratum is multiplied by the corresponding value of g and grdivjiylf is summed for all strata. This sum is equal to the rate of decrease of surface pressure. The pressure changes at higher levels cannot be determined until the vertical momentum has been found, but it has been thought convenient to collect all pressure changes on the same form P xm. CH. 8/2/22. THE UP-GRADE OF THE VERTICAL VELOCITY IN THE STRATOSPHERE This is found by Ch. 6/6#21 on computing form P xiv. Closely connected with the vertical velocity in the stratosphere is the temperature-change, which has therefore been placed on the same form. It is computed from Ch. 6/7/3 # 8. K. 23 178 REVIEW OF OPERATIONS IN SEQUENCE CH. 8/2/23, 24 CH. 8/2/23. THE VERTICAL VELOCITY IN THE FOUB LOWEB STRATA This is computed from an equation which is identical in effect with Ch. 5/5 # 9 if the air is clear, but which is more general in so far as expressions in a p , a p , a^ which are defined with reference to air that is either cloudy or clear, replace those in y p , y v which are correct only for clear air. This equation is obtained directly from the general form Ch. 5/1 # 12 by using the permissible approximations Ch. 5/2 #13, 14. It is here employed in the differentiated form as follows d (a. ^v tl \ , 3 K 1 3 f 1 j- \ gp af f = dlVgjytf + r- *- p. diVjtffW \ + r- 7T,- - V 7) dp (a p yr dp } dp (a p r j dp (a f \Dt * Dt / j Alternatively Ch. 5/2 #18 might have been used, after reinsertion of ct p , a p , a^, but I doubt if the computations would have been any simpler, especially as the varying thickness of the lowest stratum implies that fits f d\v EN m . dh is not equal to div JV I J h a J h s m Whatever process is employed we require it to yield V H at the heights where strata meet, for those are the heights at which V H has to be inserted in the equation of continuity of mass. That being so, dv a /dp must be tabulated at the middle heights of the strata, and d*v H /dp*, or other second derivatives, at the same heights as v a . As explained in Ch. 5/3 the first running sum is made downwards, because dv a /dh is known at the top from Ch. 8/2/22 ; but the second running sum is made upwards,. because v a is known at the ground in terms of the slope and surface wind, thus dh dh The surface wind is not a main variable and has to be estimated specially, for inser- tion in (2), from statistics of its relation to M G8 . The vertical velocity equation might be called the keystone of the whole system, as so many other equations remain incomplete until the vertical velocity has been inserted. CH. 8/2/24. THE PBESSURE CHANGES AT ALL LEVELS These are found, by way of the equation of continuity of mass, which can now be completed by the introduction of the vertical momentum. It gives first dR/dt for each stratum. Then forming g . dR/dt and making a running sum from above downwards we get dp/dt at the boundaries of the strata. Computing Form P xiu. CH. 8/2/25, 26, 27, 8/3/1, 2 OPERATIONS AT MOMENTUM-POINTS 179 CH. 8/2/25. THE CHANGES IN THE WATEB-CONTENT For each of the lower strata these are next calculated by means of the equation Ch. 4/3/# 8 which brings together the changes due to turbulence and to "advection." This equation spreads itself over four coordinate differences in both latitude and 8 I W\ longitude. Terms such as -^\M E ~ j involve horizontal interpolations, since M s is not given above the same points on the map as is p. = W/R. Form P xvn. CH. 8/2/26. CHANGES IN THE SOIL The changes in the water-content of the soil can now be computed by means of Ch. 4/10/2 # 5 and the change in its temperature by means of Ch. 4/10/3 #10. Com- puting Forms P xviu and P xix have been drawn up for this purpose. CH. 8/2/27. CHANGES IN THE SEA The change of temperature of the surface of the sea during our time-step will next need to be estimated. Its surface temperature may in many places be sufficiently forecasted by mean values for the same date in previous years. If that does not suffice, a more elaborate procedure has been sketched in Ch. 4/10/1. CH. 8/3. OPERATIONS CENTERED IN COLUMNS MARKED " M" ON THE CHESSBOARD MAP* CH. 8/3/1. EDDY-SHEABING-STEESSES The surface-shearing-stress is estimated from statistics with reference to the " roughness " of the surface and to the strength of the wind, as represented by M E , M N . The angle by which the vector (M E , M N ) is veered from the surface-shearing- stress is also estimated from statistics. The table on p. 84 was prepared for this purpose. Then the east and north components of the stress are calculated by expres- sions Ch. 4/8/3 # 4, 5, or more easily by a chart ruled with both polar and Cartesian coordinates. The shearing stresses at the upper levels h s , h n , h 4 are computed by equations such as Ch. 4/8/2 # 2, equations which are derived from the diffusion equation Ch. 4/8/0 #15. But the constant is not necessarily the same for velocity as for water or potential temperature. See the observations on pp. 72 to 76. The computing form is marked M I. The difference between the shearing stresses at the bottom and at the top of each stratum is transferred to the appropriate dynamical equation. CH. 8/3/2. DYNAMICS OF THE STRATOSPHERE The stratosphere has some special terms in its dynamical equations, as set out in Ch. 6/4 #7, 8. These are computed on Form Mil. They spread over six times the smallest coordinate difference in use. We also require V E , V N at the height h.,. To find these, an extrapolation is made to h. 2 from above, using equations Ch. 6/3 #7, 8. A * See the Frontispiece. 232 180 REVIEW OF OPERATIONS IN SEQUENCE CH. 8/3/2, 3, 8/4 second extrapolation is made to h., from below, assuming that m E , w% do not vary with height in the stratum h t to h,. The mean of these two extrapolated values is used in the dynamical equations. CH. 8/3/3. DYNAMICS OF LOWER STRATA Finally 3M E /dt and dM N /8t are determined from the dynamical equations Ch. 4/4 #11, 12. The terms such as =-/ E p N \ spread over four times the least Oil \ L\j j coordinate difference in use. They are best expanded in forms such as 1 S ,,, , x 8 R ^ ( M ' M ^ + M * M 8n CH. 8/4. CONCLUDING REMARKS The cycle is now complete, for it has been shown that the time-changes of each one of the initially tabulated variables can be computed approximately by means of the stated equations, without bringing in any outside information, except a few statistical data. The system is thus " lattice reproducing." The above applies to points not too near the edge of the region on the map for which the initial values were given. New values after 8t can only be obtained for a smaller area. Ways of avoiding this loss have been proposed in Ch. 7. It is curious that of the two very similar equations, one for the conveyance (or " advection ") of water-per-mass, the other for the conveyance of entropy-per-mass, given respectively in Ch. 4/3 and Ch. 4/5/2, only the one for water appears explicitly in the calculations for the lower strata. The equation for the conveyance of entropy- per-mass is used in finding the equation for the vertical velocity, and does not arise again. CHAPTER IX AN EXAMPLE WORKED ON COMPUTING FORMS CH. g/o. INTRODUCTION LET us now illustrate and test the proposals of the foregoing chapters by applying them to a definite case supplied by Natui-e and measured in one of the most complete sets of observations on record. Ch. 9/1 deals with the initial observations, Ch- 9/2 with deductions made from them. The computing forms which are used for this purpose may be regarded as embodying the process and thereby summarizing the whole book. In Ch. 9/3 a large error is investigated. CH. 9/1. INITIAL DISTRIBUTION OBSERVED AT 1910 MAY 20 D 7n G.M.T. The initial pressures are tabulated at the ground and at exactly 2'0, 4*2, 7'2, 11'8 kilometres above M.S.L. They are read from V. Bjerknes' maps for the instant in question. These maps give the "dynamic height" of the isobaric surfaces, so that various conversions were necessary. V. Bjerknes has provided suitable conversion- tables. In the first place 2'0, 4*2, 7'2, 11*8 km are equivalent to 1*959, 4-113, 7'048, 11*543 "dynamic kilometres" when <7 = 980'00 cm sec" 2 at sea-level. The pressures corresponding to these "dynamic heights" were obtained from the maps with the aid of V. Bjerknes' table "10M. " Then a small correction has to be applied for the variation of gravity. This was obtained from tables 2M, 4M and 10M and worked out as follows : Kilometres from equator Corrections in millibars to be SUBTBACTED from pressures at fixed " dynamic heights," to bring them to pressures at fixed heights corresponding to the dynamic height for g = 980-00 cm sec" 2 at sea-level p = 200mb p = 400mb /> = 600mb p = 800mb ^ = 1000mb 6000 04 0-5 0-4 0-3 o-o 5800 0-4 0-5 0-4 0-3 o-o 5600 5400 0-3 0-3 0-4 0-4 0-3 0-3 0-2 0-2 o-o o-o 5200 0-2 0-3 0-2 0-1 o-o 5000 0-2 0-2 0-2 0-1 o-o To find the pressure at the ground, the height of the ground was first assigned by reference to V. Bjerknes' maps of idealized topography"". The pressure at sea-level was next read from V. Bjerknes' map for the particular instant, and was then corrected * Dynamic Meteorolinjy ami llydroyraphy, Plates XXVIII and XXIX. 182 AN EXAMPLE WORKED ON COMPUTING FORMS CH. g/i to the height of the ground by means of the usual tables (Observer's Handbook). In the case of very high land the surface pressure was obtained from the maps of the heights of the 800 mb and 900 mb surfaces, with the aid of table 10M. At some points there is a large uncertainty as to the appropriate value of h G ; for example in Switzerland the uncertainty amounts to several hundred metres. How- ever, as the assigned value of h G will be used consistently throughout, the resulting errors will be small. The initial values of W, the mass-of-water-per-unit-horizontal-area-of-a-con- ventional-stratum, were obtained as follows. First the density, iv , of the water vapour was calculated from the temperature, and relative humidity recorded by the registering- balloon at successive heights. Then w was plotted against h. From areas on this diagram W = I wdh was obtained for each conventional stratum. These values of W above the observing stations were plotted on maps, and values above the points marked P on the map shown in this chapter were read off by interpolation. At Vienna and at Trappes observations were lacking and those for a time 24 hours previous were taken as a guide. At Pavia the relative humidity data cease at 7 km and values at greater heights were filled in by reference to statistics. The initial momenta-per-unit-area, M E , M K , were obtained from the data published in Verqff'entlichungen der Internationalen Kommission...Beobachtungen mit bemann- ten, unbemannten Ballons, u.s.w., edited by Hergesell. The first process was to undo the computing already done* by the observing stations, by reconverting the winds to components and the heights to pressures. The component velocities V E , v y were then plotted against the pressures. The divisions between the conventional strata, at 2'0, 4'2, 7'2 and 11'8 km were then marked off on the pressure scale, and areas on the diagram were measured with an Amsler planimeter so as to obtain \v E dp, \v N dp between the limits of the strata. Then, for instance, rs i p = pv E dh = - - v E dp. Jo 9)o The mean velocities ^-, '- were compared with the mean resultant velocities and K ' K directions published by V. Bjerknes for his 10 standard sheets, on this occasion, and in general there was good agreement. In the data for Pavia and Strassburg, however, there were some discrepancies for which I could not account. To extrapolate the velocities upwards to p 0, V E and V N were plotted against height, and straight lines were fitted to the curves in the stratosphere. In accordance with the theory of Ch. 6/3 the mean velocities M EW /E W and M Nt(s /R w were taken as the velocities on these straight lines at a height of h 2 -\ l = 11 '8 + 6'3 = 18'1 kilometres. \s The lines could be drawn satisfactorily for Vienna, tolerably well for Hamburg and Strassburg, and with considerable uncertainty for Copenhagen and Zurich. At Zurich * Probably, but I have no definite information. CH. 9/1 INITIAL OBSERVATIONS 183 the highest observation was neglected. The lines were drawn at Lindenberg by fjni ^1 assuming that gr ~^r were the same at Lindenberg as at Hamburg. This appears probable from the horizontal distribution of temperature in the stratosphere, taken in conjunction with equations Ch. 6/3 # 7, 8. Having thus obtained M E , M N for each stratum, they (or, alternatively, M E /H, MjffK) were plotted on maps at the observing stations, and values at the conventional points were estimated by interpolation, or, in some cases, by extrapolation. This process was an uncertain one, especially in the stratosphere, where the data were sparse; and near the ground, where the winds were irregular. It makes one wish that pilot balloon stations could be arranged in rectangular order, alternating with stations for registering balloons, in some such pattern as that formed by the points marked M and P on the frontispiece. In the present example, in spite of all the care taken, the tabulated values of the momenta-per-unit-area remind one of the stories which are "founded on fact." In consequence they give absurd values to 8 f^= -gdiv^^M. t all We shall return to this point in Ch. 9/3. The initial temperature of the stratosphere. The observations show that the mean temperature over Central Europe was about 214 A and that the temperature over South Germany, Switzerland and Lombardy was some 13 lower than that over Russia and England. But there were considerable irregular variations with height amounting to + 5 C at some stations. Probably the most exact method* of mapping the temperature in the stratosphere is to use ^ , obtained from the variation of wind with height according to equations o /\ o/l Ch. 6/3 #7, 8. The following table gives the temperature gradients -- , computed Ob Git from the observations of wind. They have been used to smooth the distribution of temperature. Station *9 3h in metres pe 10 kilo dvg W r second per metres 8 . 2w bin <(> ~ 9~ degrees sec cm Se de degrees pi 99 dn r 200 km Copenhagen ... Hamburg + 22 + 10 -19 - 6 + 3 -19 -14 + 22 + 15 + . 2 2-6 x 10- 5 2-6 2-4 2-3 2-4 + 1-2 + 0-5 -0-9 -0-3 + 0-1 + 1-0 + 0-7 -1-2 -0-7 - 0-1 Strassburg Zurich . Vienna Sir Napier Shaw, "Upper-Air Calculus," J. Scott. Met. Soc. 1913. 184 AN EXAMPLE WORKED ON COMPUTING FORMS CH. 9/1 The temperature of the soil is required at a series of depths. The depths proposed in Ch. 4/10/0 appear to be unnecessarily thin when their thermal conductance is compared with that of the atmospheric strata, and so the alternate divisions are here omitted, leaving those at z, 2 = 6'39, z u = 53'6 cm etc. The temperature is required at intermediate depths which, by following the process of gradation proposed in Ch. 4/1 0/0, are taken to be z n = 1'72, z, 3 = 19'1 cm. In default of direct observations the temperature at z n may be estimated from the statistics of E. Ebermayer*, for these refer to Bavaria in which the P point to be treated is situated. The mean of six stations for May 1868 shows that the air at a [continued on p. 186] MAP OF POINTS FOR PRESSURE (P) AND MOMENTUM (Jf) USED IN THE EXAMPLE of Ch. 9. NOTE: These points are placed at the centres of the chequers, and to the centres also the latitude and longitude refer. Each chequer measures 3 from west to east and 200 km from south to north. * Die physikalischen Eimmrkungen des Waldes, Berlin, Verlag von Wiegandt, Hempel und Parcy, 1873. CH. 9/1 TABLE OF INITIAL DISTRIBUTION 185 Obtained by interpolating or extrapolating from observations taken at 1910 May 20 rf 7' 1 G.M.T. longitude 5 E longitude 8" E longitude 11 E longitude 14 E longitude 17 E 1000 X 1000 X 0,214 C A Mf-20 - 65 jl/.vao + 8 0,216 A NOTE: The following strato- 31 KU + 127 JI/. V42 - 104 sphere temperatures have also been used .l/ft-so-81 il/. vt .f,zero long 11 E, lat 6200N, 216 M -198 il/ -- + 84 2E, 5600 N, 217 ,, 20 E, 5600 N, 216 A,;=(l 1000 X 1011 X 1000 X 0,215 A .!/,,,, ,-70 t 214 A MM> - 160 0,216 A p., 2047 I/ ' 62 ir 4 , o-o JZ^.^ + 40 p 4 4090 M M - 114 We40'2 M KM - 60 Pe 6086 MUM - 91 "'seO-4 jU^sn - 60 p 8 7983 !/.. 160 H",.(tO-9 MEGS ~ 219 p 9883 A s 15000 A,,. 20000 A B 10000 1000 X 1000 X 100 X 1000 X 1000 X 100 X 1000 X J1/ ff 20 - 30 J/.V20 - 110 0, 212 A -l/ ra > -56 Mxm - 18 0,214 A J ^20 - 100 Jlf.v 2 o - 32 p 2 2047 po 2049 J/, ;4 o-245 Af A 42 + 300 H'4-,0-0 MKW - 146 J/.v4 2 - 62 W t <, 0-0 JI/42 zero Af.v42 - 2BO p t 4083 /> 4 4091 .I/,..,;, - 223 M.V64 + 158 r ( ; 4 0-2 Jf^84-95 J-/ A0 4+29 W^O-l MM - 55 AT,.,,! - 135 p e 6067 p e 6087 31 /,-ng - 91 iV A 8o + 87 W'se 0-5 J/JE88 - 52 J/^ + 58 jr 88 0-4 J/A-SO - 25 Jf JVM + 48 PS 7950 PS 7979 31 EOS - 18 J/.VBS + 15 JFggl-2 ^08 1 10 M V()8 + 55 H r e8 0'9 J/jtes- 190 MMS + 160 Pe 9834 p 9763 A s 20000 AC 20000 A G 40000 AC 30000 h & 30000 100 X 1000 X 100 x 1000 X 100 X 0, 214 A Jfag+SW 0, 212 A Mf2o zero 0, 214 A p 2 2030 p 2 2050 p 2 2044 wvo-o 3/ M'> ~ 328 IP 42 0-0 M Eti - 166 W 42 zero pi 4049 p 4 4090 p 4 4082 (F 64 0-2 M, x , - 136 Wm 0- 1 1/AH4 - 95 ^640-1 p u 6044 p 6 6079 Po 6068 W so 0'7 jl/ 3S H'se 0-4 l/i - 19 Wge 0-4 p 8 7928 p 8 7960 PS 7978 Pc9744 " 81 ' .!/,,, + 48 PC 9626 "' e8 ' 9 M^cs - 65 N'csO'9 A c 20000 AC 40000 A ( , 40000 A 40000 h e 20000 100 X 1(100 X 1000 X 100 X 0, 214 A A/2<> ~ 50 Mfffd + 80 0,214 A p., 2039 p 2 2043 If 42 0-0 Jiy ff4 2-280A/jv 4 2+41 ^42 -0 ft 4062 p 4 4075 '^640-2 .I/TO, - 175 Jl/^ M + 150 ^640-1 p 6 6050 Po 6065 JFsoO-6 A/i-ae - 105 71/ iVS6 + 80 Wge 0-4 PS 7949 p 8 7967 1*8746 JFe8 ' ! .!/,.,,- 155 J7 V( ,. S + 40 PC 8458 ^^ A,.. 120000 AC 180000 h,, 150000 100 X , 0,213 A p, 2034 All the quantities in any one square refer to the same latitude and longitude, O 49 S " ^2 0-0 p 4 4034 namely to those stated in the margin. The notation is denned in Ch. 12. A numerical -Stf |-g c ^640-2 Po 6031 p 8 7957 value, when multiplied by the integral power of ten, if any, standing at the head of its column, is then expressed in c.G.S. units. To save space the equality symbols o OS * ^* 1000 X a PC 9972 are omitted, thus M f;m 66 stands for A,,. 10000 M EW = - 65000 grin cm" 1 sec" 1 . O O oo 10 I c3 I s Jl J3 o O O O O M E are usually the largest, but the "curvature of the path" has a con- siderable effect and so also has that much neglected term, the rate of change along the path of the square of momentum. In the lowest strata the terms dP/de, dP/dn are very large owing to the slope of the ground, and are for the greater part balanced by p -~ , p -~ Special care has to be taken in computing these, as described in Ch. 4/4 #14. * Q. J. R. Met. Soc. April 1920. t Of course there may have been larger vertical currents locally; the statement in the text refers to the mean over a chequer on the map. I Vide G. M. B. Dobson, Q. J. R. Met. Soc. Jan. 1920. 24 2 188 AN EXAMPLE WORKED ON COMPUTING FORMS OH. 9/2 o O co P O -P fl O O i s 02 a of -P 03 C 0) CD E co CO 05 II

(D w 05" Cb I I o> 1 S I V s s ^^ o e 11 iS- o ..4 g - - = X 5 , "3 .2^ q (a c s > 2 hH >~( ** "S g 1 "5 ; ~ .s 5 CC _3 1 TJ "o g i o 3 EH K I-C! ft lcc> X o 00 b "O 3 GQ 'S s i PL, a a J o S 5 so a 'So h-1 X o X o 7 05 4 + & cp co to (T-l P3 CH. 9/2 HETEROGENEITY AND SOLAR RADIATION 191 is so"-* 21 ll 9 * to ^ 01 2 a 1 3 E Incident radiation d erval in whole spect USE UPWARDS H 11 . - ^ - - + BSORBED, SCJM TO SPACE UM = INCIDENT 1 1 11 9. to G^ ! to * a 1 a < so . ^-, S Q M D 5S! C) K IQ t^ i o p* 1 T ""H "*H JC ^ 'b g H I . S. ^ ^ S a P*H 02 S .S II 3 o (^ H ^ *v^. &ri \ a S ' r d <; *^5 9 o ^ H Sc b || g 2 ^ w( s > * oo t> o 9 9 5j j **J P *H 11 H ^ ^ ^ ^ ^ ) ^ Vf^ 49 CO 7 S O D d +* *h" rj w d ll 9 g Q 3 S " 80 ^ N *~ 10 'g 92 C O o 5 O y .2 S II . 'O J 3 o> $ -S .(-H g 'o . ja to (Q *o 5 S to Li "o Q a 1 *4H 1 ^ 1* S L o DO Z < S A c'S CS ~ ^ tp a 02 co O 11 -1 1 CV T3 || S, IS g *"l O ^ | 4a H m g a p to to ~^- U 5 5* * < S 'S aj S 2 CD 1V( _S o M 2 .2 '3 = CO * * I 5 M g i ! oo 43 SB "9. 1* < a *Q * c u g to to 5 00 s- Oo t c a S 9> ^ ^ . w Op l| o ^ "^ ** ^ . 3 CD 1 K^ -T 4 ^f < s **< s o b r-l Q H 6 Q <; 5 5 b 5 5 1 i to OS to 49 OQ 0) "-O II 3 Q D d fl 3 to -tt P M a Q OS 'Cl a gj us _ 2 ^1 S S &5 ^. ^ *? ^i 1 IH CO T3 2 i & o 8 b V a 1 H 41 OS to i, O 8 S 02 1 _o 1 S S to 8 0) s i 1 a I "M > s fid Q CO f ** S o S *4-H -.j ^ <>-* H a "^ M to .g CQ Is <^ -H >* ~* oo IQ, 49 Q CQ S S ^ T ^H i ^*l cy\ *-ij "-P *; C y^ o -- S .2 sT 3 T3 to 5^ 6 _3 ^3 v; d >> PH g 1 . 8 JS to ^ N. g ?^ S 43 5s ^ - 3 ^( ^H to OS Oo ."^ H 1^ 1 >^ ^ 1 Q $M m i. 1 a Ob ! -a "^ "S * * e+4 O rj IS H o s 1 1 to ^ to fc ^ 'a S PH H H HH ^ a Q "3 'S 2 g c "S r 3 S3 < _o ,1 g OS OS 871 * fe 'S 1 1 to s OS i ll E f. * to * H PH *"M ^ M S QJ ^5 ^ PS ~^ 5: -C "* "* "^ ! i to 01 00 Ol .S 3 -8 . , i 1 1 o ^ II a S 3 g 3 -a S 7 to - GO || 7 i"a I en 1 ^ ^ & ) > 1 O, L " J3 "1 ^ So o ^ X + + H O goj^ a? , ? O . - H 2 + * 1 ^ Co OQ Oi Oi Oi >-H ^o $ ^ ^o 1 1 05 to oo OJ to to oo G P CO ^ 00 nil IS 1 i - 1! 9 f^s < 0> ^~ S 3 OJ 3 0*3"^ ?N H 111 5 w "S i 3 IT ^ i k I a-^3 0-0 3 " 'S 6 '3- 1 7 il 01 1 01 oo 1 01 to 1 ^7 ^3 ^N H CS c I* o ?->. ^o Oi QO S5 Oi to Ol to ^b & o o A o ^ 8 oo 1 - x' Oi rS* PH I 5 3^3 III ^ a 1 1" 1 " H 1 1 to "8 !!% * ? ta'2 O ^ 01 Ol ~^ ^2 ij 0) IP" 01 to oc a ^ g ^ 11 -^ >5 a CD n 31 a " a 'i -S c o i ff D O3 S *" ,2 5 ^ |l 1 | Sb * 03 E 6 so .2 g - I'-i o o 0, f I 1 i 1 H 1 O *C3 g c^, o ^j o >^ '-i >-l ^H . *^ ^ l sO N H i* g g X X X X 0} X X g J: S 1 ^ ~* s* S J^ ^H ** 2" >? ca b 43 *H ^ ** ^ * 1 fe 4* 2 s a g CO o o ^. ^ ,_ w H 2 JS CD CO ? ^ ^ "B 1 1 o aj a 1 s ^ r r 1 'n i * to OT o u Ci3 a O a Q fl *3 03 1 | s g o II & So 6C g C a u a E 60 o S 1 "1 "a o 1 s. 4> _ | a 41 O fl ^ 1 be bO b ^Bj B -M ^ <^ j "o S w~ 1 J CD o* 1 PH 03 ^ 3 CO I s o PH II a. 1 i^ 1 ^ 1 a f^ 1 , 1 *N CO C^l <^^^ 5 -13 ^i. CD ^ s S a s E 1 'S II C s 8, * 1 c, J O J ^ t "o 1 o 1 O- B E CD & a H te; 1 tb Trial value of surface temperature, as on " Resistance" of half air stratum =g (p II 7 II 1 S | 1 1 CO CD "o ^ I II be a 2 0. g -+3 8 a 3 a? 1 "S 3 1 1 m .2 3. II Difference p -/x 9 . For /x^ see Form P i 7 1 S For evaporation from wet vegetation add This portion is provisionally taken to where n 1'2 x 10 5 for beech forest in i Total rate of evaporation = H' S CO p O 43 PH CO a 1 a o "T^Pn 1 ~* o 03 CD p* ID 1J3 03 S. 2 ^ 1 * o i! O 0) .2 ^ o ^ 1 a * 1 1 o a rate of evaporation = H f CD J "o L .2 X 1 I O 8 ions of coordinate chequer having the sev i evaporation over the whole chequer in ot J ,0 = r O V s sanjBA I'Bux 2 09 K ! 5 ^ o o XI H .2 cS -5 2 a CH. 9/2 FLUXES OF WATER AND OF HEAT AT THE INTERFACE COMPUTING FORM P vin. Fluxes of Heat at the interface Long. 11 E. Lat. 5400 km N. Time from 1910 May 20* Jf to 10" G.M. T. Note : Water and bare soil are treated by the same process as is the vegetation, but terms marked with a heavy vanish. 195 Character of surface forest other crops bare soil water Height (h A h g ) at which radiation converted cms 1000 30 Part of depth of top soil stratum = z n cms 1-7 1-7 1-7 Thermal conductivity of soil =k erg cm" 2 sec" 1 degree" 1 10 5 x 1-6 1-6 1-6 very large Trial value of 6 A the temperature of the surface = ff Abs 290 Trial Resistances above h A W.-ffc-AVpJUl'r,) cm.gm-'seelO-'x 0-780 0-780 0-780 l/C' Ai = 0-005 \ o-oio\ 000 Sum of previous pair = l/C' Ait 0-785 0785 0-780 Trial Resistances below h A ^/^ 11, 10 ~ ^ll/rC ,, j, ,, 0-106 0-106 0-106 l/C'lO, vl = )! !! !I 1-5* 0-05} Sum of previous pair = 1 /C",^ ^ 1-61 0-156 0-106 Here fill up Forms P ix, P x to find corrected A , which is 294-3 Corrected value C M of C' M \ 10* x t TiYir fl/3fl' Qpn TiYirm P Y 1-70 Corrected value C llA of C' UA I 10 4 x 7-27 Temperature at 2 n in soil = 6 U Abs 288 290 r a = - to P g 05 Z Oi op ks s > \^ "^ ^O ^ to ^, to to f^ *J ^ >^ '"H M J *^" $ -2 1 1 + % O 1 to -s Ql | Ol I X 6 T 1 X X 6 r t X I-H X X X o I-H X X o X b i t X o R i 87 1 1 o b ^- 1 *o CO I'l - bn 1 f CO S 3 .a o J t- r O 4^ J f a to S | '3 1 T 1 05 O 1 (2, ?B c/f w> r CH^ 1 4 3 II 5t spending to : a) ^ c b O i f J fll g's 1*8 3 * 1. i o g * ^ o o ^ '^ *~ H ^H I-H 3 M -g So>: & _o *-*3 \ ^ > is E - >2 a >" .3 x - "* " a J ^ a 50 j PH cfi 1 O S. 1 Cu 3 "8 8 1 II 1 II 1 1 b ^PH 2pH o M ^ g ce o B?| SJD g II 11 I-H tn 1O = a ^ rj II 'f CO 3 f 1 X! Ii "s r II a - II e "ill o "B X) * ! 'B {B C^ ^ ^ ^ ^ ^ (^j Oi fl S + * I * e* ft> CO "o a If to ^ c '-I g 2 fl a 2 2 .9 o H X o X O X X O X X o o o X 6 X o X o 3 | $ ^ 1 I N 5 II II II II II CO CD M P DO c j I 2 5 *4-l O 1 3 1 ^ fli s ^ 2 * ""i. S o3 V 1 ^ ^ ^^ V 1 o 5* -! I ^ 3 ! ^ * * - o _B b ^ i *^ x. 1 ( 3 5 i | " 'a ^ ^ J S i * * 3 i C B, B -w E ^"^ > ^ i N < i ^^ ^Q PH ** t j> i J -d o aj ^ "o 1 s Si 1 1 s i & 1. 1 1 ^) CO 1 T b" a ^ CO & > PH 01 O -o X) 1 x> 3 'i 1 ~i I j 4 E ^: iiT 1 j 9 1 3 1 3 i 5 2 3 j 2 3 4 S b.' ^ o f X t-| ^^4 X 1M OQ ^ 1 <6> II II II ^ > >^ CD *^ CD CO |e ] li tSr r 1 PE< 1 i i (M oo O 'o X J 1 1 1 Denominator = algebraic sum of all num ^ = " numerator " from Form P IX divid< Corrected surface temperature = ff -f -a. Second approximation AN EXAMPLE WORKED ON COMPUTING FORMS CH. 9/2 1 1 '1 8 = ^ 1 I.I "-t-H M) el | bio 53 2 .5 -g a Jz2 ^ >> so '3 ? 1 .2 '~ l M .3 1 US ' 60 g> -e - 'O Co *>> 1 03 v*l 1^^^ a 2 g - s ^.1 . be *"^^ -*^ OO e * 1 ^s O ^ -fl G ?"~1 lp ^ 55 2 ^ ^ "i S 3 o g O be co lijl 5 a, M . * 02 bfi >-< ^| | ,3 P fe S tt! pu IS ' rt t PH ^ .-S ^ x fi ^ S .g Illf * *- '-*^3 i i 00 6 o6~ 6 ^ 0> T3 "So I ' 1 I ! "i i 1 ! i l - u * u ^_ D 1 \ H O 5 > H :O 3 H i 3 1 ! Q a 1 ji *-j ^ ^ & iq "3 *o CO rH 00" ^ O o> X cc rH uo'tjvBifsaam fiuipund paniwQ 3 3 > & (0 5o to 1 b = % 4* X s rH - 6 i to XJ 1 1 b 1 S b X s t-l 2 I ** 1 4f- | ^ *0 t^ ^L i > X 1 o ^ *** ! N -*j < f | - ! 1 * Oj ! 1 S 3. ft 1? to X T o rH 1 op O a. X 1 o -H S S S M PH o 4 |n X a rH 1 1 1 1 1 1 M 3 X s fH I 1 05 >-l oo GO Oi oo 1 1 1 4* [8 B X 00 rH t^- * 6 f * * - - CH. 9/2 EDDY-DIFFUSION, AND GAINS OF ENTROPY 199 CO a> 'I o S 05 03 I PH I 1 PH O C cc a S S X CM S O fe o J5 H O O & C Oi >-H I C4H O S S - o s. bo OJ . s 3 US S-' (X, a S. a 'a O = - jo ui ptIB JO Ul pnu uoijuipug iq SBBUI 43d jC3aaug jo ure8 \ jo UIBQ a 53c p. o o 00 G I PUB augqdsoiuiv spn^g n; OH CH H PH X X a oo X I o a 1 a H op oo a 00 to j 4 oo l-H 3 00 >~H Ob oo to to oo V) 10 to 1C 1 200 AN EXAMPLE WORKED ON COMPUTING FORMS CH. g/z 3 >^ *; yj 3 a ^ .5 e X ^"^ Qo f^ ^i> lO\ M *"H '> H S ~* tx -< w^g J 00 vious umn *o X ^ K. to g *0 ce S g a 1 1 4 03. - v 03 10 1:0 x ^ I* ZQ. H^J &Q $D en 1 3 O ^ 03 ,d t, t-i ' a o nq CO H u k 1 ^ 06 1 ? * "S i fit ? A M a o a) M g" 8 CO I'S X I *1 to S S3 1 1 "o si ISJ 3 1 oo oo ci ^ I O cv ^ (M e ^ ^ cS ~ x X US PM to i * g o QO 1 ^ 03 e ^ N. pu M *H ^ g CU ? i c*i ^ JJI ^ *^j a | < | 1 g s s - Kr, c g o to IN ^ : bC ""H eg Oi -2 H S 03 H S -e- I 1 M ^" .1 00 i ; 0N X I o S 8 S . E a c > "*. ti cS sJ 5 o *> iB e h- 1 Pi 1 1=1 0> o _ ,-H lAXJ IUJOJ g |^ suiunjoo qnanbgsqns 8i[^ QABB^ *"^ c8 "i* C*-i E ^3 2 O 3 Q> ^ ; I!- 3 03 ^ fO -W e* ^ 00 M ^^ O TO S 1 1 -* S ^ g 5 V X to to c^ oo r* 01 S o ? 03 s ^ _o S o E >( i S to 9 ! ^- p ^-^liS jj~ aj s to s "3 | hH 1C 2 5 I s "o u t 9 i ^ * - >, M O fi . r-.* 3 S C . a e X to 85 to =0 O 1* gas -S ^ OO J"H -5 5 T Q i c . O t^J 5 rH 3 **H ^ ^ u ^** C5 T 1 ^ ^ ^ =8 | l-H D **H ^ X 1 '" s?l * E-i J -^H. S^ ^ Ico D f PH H 1 " j 4 1 i ! oo 5 .^ + 1 I" 1 CO to g x |u ^ ^ to i 1 i 5 -f -T -? -T -F -f a CH. 9/2 CHANGES OF PRESSURE. STRATOSPHERE 201 o to terva Qj *-l 05 1 s 1=5 0) be o x oo -0- 1 3 I 09 d r^i '& 6 !,.3 I M- + O ? 'I % fl X 3 ^> 60 O > | S C) X =tfc oj 5> Y Is . o I 2 to E. 26 202 AN EXAMPLE WORKED ON COMPUTING FORMS CH. 9/2 !? 2 11 r o> Ll CM co o - 03 s 0) I d ce o EH o X PH o o Q Cb I Ss 05 I 5 a be o "a 2 'I'l I ^ !* q B o /M & Jo" ? 1 2 I ^1, a ^ s * I s tS to + 05 + eo + , 05 1 Its 1 5 I oo ~* 1-1 to oo 9* oo -H I oo I 8 1-1 I 2 00 to + to 1 . 95 + N. I a $ >Q a B fe" '> a H | a > 3 previous column ation ove ious mn H S X OH CM 3 g .2 9 > a S. M On X I O i s r/ a to i 05 I * s a 05 00 00 X I o X 1 o * <3* S QO to 1 y. ft* i O X t* I O 4 05 <3^ 1 1 00 ep l-S 1 op I X 1 o 9S 00 to 3 i 05 sauepunoq -a a x:f Aip d ' o & x i o X t* I o 9 00 1 oo 05 i 05 I 1 Ja e* 10 9 I 05 1 to *-l I OH s oo to to 204 AN EXAMPLE WORKED ON COMPUTING FORMS CH. 9/2 1 to i 5 * 1 i 1 0) II 1 1 S * X CD X 1 "3 I I ( 1 1 00 1 g >r) Q ^ r* e-I + * > -IJ a *i + ^ 00 1 | S -s X 00 I O 1 X & a- f-> T3 . *: O> cp Sj 4-3 ;~~l ^ 00 ^ 1 ? 1 + II + oo ^J Q^ S5 Oi to 1 + 'o H x J? -0- a ^< *c If ^ J S _ o 2 ^ 1 i * ~ ^""^ e 00 s C ro <5 1 e S a s X OO | 1 i 1 i o X 1 j -2 '^ | O t-H w 8 1 S ^ s + H i? + | J* M M .I 4. rs g ! q * I? B 13 w PH ^ ii S | 1 g * 3 -e- " s 1 ro + 41- ^ *2 | 1 i CO ^ CO !? o % ^ j 5 s s 03 ^ $ n CO 6fe fC S 1 1 i 1 ^^ 1 1 CM i e 1 x o ^ H H CD g + 1 1 r -1 B & M M X '5b ll PH PH PH PH Q M g s g g 1 1 1 1 CH. 9/2 MOTION OF WATER WITH AIR AND THROUGH SOIL 205 10 G O O aj C h -4^ te o H D AH S O O I t 1 J* t < | i j> ^ I i ^ * 1 "5 fc. X 1 it > > 5 1 I (C i 11 1 ) i < 4 -t- 'a C I C PH 3| * 4 iT Instant 0'^ -J * S 2' &g >-T3 u o 81 te * 1> << - . >2 Vegetatiol I Deficit of pressure II " ca ,c : i a 1 "8 1 i Mass of Is f! i * g .1 3 1 - 3 J O O t- 4 a i i r i> >!j r- : 1 EH H P3 o F 1 3 -j 1 o S CO PH S i I I 206 AN EXAMPLE WORKED ON COMPUTING FORMS CH. 9/2 O) .< ro 0) ro ro ^ ^^ 1 1 os S 3 <0 o SIS 1 3i~ gst II " o Ji 53 1 Sro "Sro !' 5 a , 1 ** iS 73 |- B h- 1 4-l _g IS 18 a "-& 0) I's a If 1 s a a 3 ^ - -^ ^_, 1! e a 3 S '43 g J ^^ ^,2 K W |^ g s? d 3 a X * 9 j^ 3 a 2 O A, S S u 1-1 eo g 1 ,| >~i * i to H n3 *~i o li|i" 1 I 5 e r -t- o 1 1 5 I 3 1 I s 01 <5< 6 j. u S II t rH l-H OJ r 1 If g " gg SH aj o * cb cp CO r Cu m id 1 '&, 5. M o > PM M O .S b I S H CH. 9/2 TEMPERATURE IN SOIL. EDDY STRESSES 207 .t^ OJ g 3 ^ ^ ^ CO ^ o fl C^> c^> (JU *^ . CO 1 o ^ Qj C^j (^ Qj c^ ' s 1 + C i . e i fw a -|s . S 9> ?* X ej, "^ p 3 . % | ^ o Q ^. "3 *JU* Ci o Tl + + *H IN l"* + g 6s co + IS ^ ^ ^S X oo 1-1 1^ i ^ a 5 < a, f ^ S3 00 *>( g? ii ii 'C o >! ^ 5. 1 + + T f -1 -C _o ^j 1 * . Srj ^ i j ~i p-*""" 'to * ~-v p^2 1 s H H O s ^ ^C -ijji ~C J Oi "% 4 s X s 1 Oi s i 1 '* ^ &D o ? ^, ^ | 43 d CO ^ "& a ^ " S * -2 SH JL r> 3 5 =5 co * r - S _L- fcJD to X 23 sj 5 1-H pt^ M CO x i j? s . f' C ' ,3 "g 5 S a 3 ^ tr _ 1 1 ' *** "^ ' [i t ) m ~ B -R 00 CO ^ T3 S <, CD PH .fl *3 o - r-r . CO QJ J 1 1 1 oo I 1 1 1 ^" 1 i i. 3 ? .. _2 t- ^' >^ ^ CO *" Q _ IS , O i ^ H -a IS w H 1 Z ,2 a P, 01 CD ^ pJ O< OO OH P- -> P S S T5 fll!l w - t -^r 1 J^J ^r 1 H y -^ 2 6 1 6 O CD te x II * Eg ii s " ?' ^ "3 j ^ a 2; * - 1 " "la ns CD aa "eS x -* 3 ^J. oo CT ^" 2 II 'so 3 4a - 1 s Ci 9 1 06 *> I.JP.B ? H 3 -2 d 11 J jj ia **l * ^ o CD ~i 'g a ll o -S .2 "^ H CO M ^ X & ( 1 1 s* I 22 oo to "-i g> 1 g" t 1 J-8' 3 8 "Si 9 ** 2 s^ ^ fl - 4 ^ tf 'B "3 I 1 8 -r -f ^ -I 1 if rf -? a d S -i 1 B a H O -I O O H fe a? O 208 AN EXAMPLE WORKED ON COMPUTING FORMS CH. 9/2 l 43 2 - I -, 0) -P eg ~p o 0> 03 CU M 00 03 0) O O 13 -e- ~ S3 3 60 H i 2 + -2 II ^ 8 co co X Op O S . I S CO \" CO 1! ; CO . > 1 ~ 5 s G ? O lc| I ^j PC CO .2 O I 3 a !S s o ribution a I? 17 to g OS II <* * <* .^ co X 00 ^c> I V co I co -<" -e- d 1 1 + =2 i> o s ,-*. s i 5 ce > a t- + + C eo I - o & < -fl 6C S> 1 S5 9> + life & Ob 00 3 OH. 9/2 INCREASE OF MOMENTUM 209 M 3 o o II K ^ II 6- de 5 Lat E e -a 49 'So J ?> S3 i 0} 9J 01 a ft; <~ i r* '-O s 19 * 09 O O. 3 PH B I PH 00 4 g I J ej *i x X o 01 P + X PM M PH pecial phe stra 13 i R. 27 210 AN EXAMPLE WORKED ON COMPUTING FORMS CH. 9/2 00 1 s jg ~~ 6^ I 1 i op S IT s II ^- 0> H to 11 ^ o^ ^s p" *i CO s JL Q ^^ IS ^S 5? 1 I to I 1 to to o -e- 1 s 5 to ^ 1 | Hi * 8g T! h 5 ^^ 1 C3 to ri 0) _j_ *^h ^ "^h to to ^ 1^ ^3 *"i Ci t^ to to 0^0 _a >. fc O g 1 ~ ^ e 2 ^> 1 >^ O co s >^I fs. to ^2 d a i ^^1 M i i ^| ^o ^J ^o *3^ Q .2 N S J -+J I- 43 . C: j^- 3 r- 00 1 11 ^> f^ -JJ. to^ q-j H D to .C a- 1 s > * H jf . g II e- S 1 N 55 S 9 X 'S J^j oo + =f 3 1 |5 ' < 1- Q co "^ " ^ -1 "^ t-4 <$ QJ - a to X rf + *9 st to I 1 g + to -* - 05 gs ^ op at . o ^ h II - to X fe ^ * i^T" + Is . ^IcS S > + *00 ^ \ 1 A CO O 1 E Latitude 5600 ki " t- T e- 3 } g J a e- 3 43 viscosity terms stratosphere, special 1 1 a CO 1 J CO si CT 1 o X 5 - . f i. - g ^ 1 5f j g p p 1 1 |i i u.' 6 == X PM PH PH g a eg W C a^ 'be ~ e 5<~ = g 3 S s q C - C H I o P 5 H 1 1 1 i E-i CH. 9/2 THE RESULTING "PREDICTION" 211 Summary of changes deduced from the Initial Distribution A here means the increase of the quantity concerned, during the period of six hours, which was centered at 1910 May 20 d 7 h G.M.T. At longitude 11 East, latitude 5600 kilometres North From computing forms M in and M iv 103 x 1Q3 x o - 730 AM"jv 20 - 337 - 196 AMy^ + 238 AM" 64 - 89 AM^+138 " - 153 AM^ - 43 63 At longitude l\ East, latitude 5400 kilometres North From computing forms P xm, P xiv, P xvn 100 x A#! 19'6 Ap, 483 APF" 42 0-007 Ap, 770 AJF M 0-024 Ap 6 1032 ATF^ 0-149 A 8 1265 It is claimed that the above form a fairly correct deduction from a somewhat unnatural initial distribution. 272 212 AN EXAMPLE WORKED ON COMPUTING FORMS CH. 9/3 CH. 9/3. THE CONVERGENCE OF WIND IN THE PRECEDING EXAMPLE The striking errors in the "forecast," which has been obtained by means of the computing forms, may be traced back to the large apparent convergence of wind. It may be asked whether this spurious convergence arises from the errors of observations with balloons, or from the finite horizontal differences being too large, or thirdly from the process by which the winds at points, arranged in a rectangular pattern, are inter- polated between the observing stations. We may examine this question by eliminating the third source of error by computing the convergence in a triangle formed by three observing stations. The formula for a triangle was given by Bennett in an appendix to the Life History of Surface Air Currents, by Shaw and Lempfert. Bennett's formula can be expressed for our purposes as follows : Let the triangle formed by the three balloon stations be drawn on a map. The momentum per horizontal area of a conventional stratum at each station is next to be resolved along the perpendicular to the opposite side of the triangle. Each resolved part is to be divided by the length of the corresponding perpendicular and the quotients are to be added. Their sum would be the rate of increase of mass per unit area of the stratum if there were no vertical motion. In selecting three stations to form a triangle I have chosen those at which the extrapolation of momentum in the stratosphere was the best. These were Hamburg, Strassburg and Vienna. They form a triangle which is nearly equilateral but rather large, the sides being roughly 700 kilometres long*. The computations are set out in the adjoining tablet, the interesting column of which is the last one headed "sums for all three stations," for the figures in it would be the rates of increase of mass per horizontal area of stratum, if there were no vertical motion. Unfortunately we have no direct observations of vertical motion for this occasion, but thanks to the extra- polation to the upper limit h a of the atmosphere, we can find the total rate of increase of mass for all strata by adding up the figures in the last column. Their sum is + <305xlO~ 5 grmcm~ 2 sec~ 1 . As the whole mass of the atmosphere is about one kilo- gram per horizontal square centimetre, this rate of increase implies a barometric rise of 0'003 millibar per second, that is about 60 millibars in six hours. Actually at BayreuthJ, which lies near the middle of the triangle, the barometric readings were as follows : May 1<) May 20 hour 7 14 21 7 14 21 700 mm of Hg + 25-8 25-0 25-3 25'6 24-7 25-6 Other neighbouring stations observed a similarly constant pressure, and the hourly values for Hohenpeissberg;}; and for Potsdam, which lie outside the triangle on opposite side of it, tell the same tale. Thus there is a marked disagreement between observa- tion and calculation. * See the map on p. 184. t The second table on p. 213. J Deutsches Meteoroloyisches Jahrbuchfur 1909, Bayern. . 9/3 ACCUMULATION OP MASS AND DETAIL OF WIND Convergence of Winds in a Triangle 1917 May 20 d 7 h 213 HAMBURG VIENNA STRASSBUBG Length of perpendicular Azimuth of perpendicular 5-7 x 10 7 cm S (exactly) 6-1 x 10 7 cm W19N 4-8 x 10 7 cm N50E Inward component of M along perpendicular divided by length of perpendicular Unit = 10~ 5 grm cm~ 2 sec" 1 Sums for all three stations Hamburg Vienna Strassburg A, to A + 33 - 20 19 6 A 4 to A 2 + 74 + 136 -215 5 A to A 4 14 + 102 . 29 + 59 A 8 to & + 7 + 36 + 48 + 91 h g to A 8 -142 + 243 + 65 + 166 Sums for alll five strata / - 42 + 497 -150 + 305 It is possible to suppose that the marked convergence in the lower strata was actually balanced by a large divergence in the upper part of the stratosphere, a divergence which does not appear in the table and which would have to be explained by casting the blame on the newcomer: the extrapolation in the stratosphere. That explanation would imply an upward current in the middle layers of some 700 metres in 6 hours at a height of 4 '2 kilometres. So large an upward speed would probably have produced cloud, whereas the sky remained almost clear'". So we turn to the alternative explanation, which is that stations as far apart as 700 kilometres did not give an adequate representation of the wind in the lower layers. That appears almost certain when one thinks of the irregularities of the surface wind exhibited on the daily weather reports. The wind stations in the proposed rectangular pattern are nearer to one another, being 400 kilometres apart, and that distance might well have to be halved in practice. This being admitted, let us refer again to the con- vergence of the horizontal winds set out in the last column of the table. It is seen that for the layers above 7 '2 kilometres the divergence is quite small and therefore credible. The suggestion is that at these great heights the flow was so smooth and lacking in detail that observations even as far apart as 700 kilometres gave a fair representation of it; and further that the extrapolation for the upper part of the stratosphere was, at least, not strikingly in error. * An uplift of 1-4 km would have produced general cloud, whereas the mean cloud amount at 16 Bavarian stations was below 2-5/10 until after May 21 d 7 1 '. CHAPTER X SMOOTHING THE INITIAL DATA WE are not concerned to know all about the weather, nor even to trace the entangled detail of the path of every air-particle. A judicious selection is necessary for our peace of mind. For some such reason it is customary, at stations which report wind by telegraph, to replace the instantaneous velocity by a mean value over about ten minutes. An extension of this process must be contemplated, for there is a good deal of evidence to show that the wind is full of small "secondary cyclones" or other whirls having the most various diameters. The arithmetical process can only take account individually of such whirls as have diameters greater than the distance between the centres of the red chequers in our co-ordinate chessboard, and this length has been taken provisionally as 400 km. If we smooth out these whirls we shall have to make amends by introducing suitable eddy-diffusivities. So far meteorologists do not appear to have attended to eddy-diffusivities of this kind. We shall refer to them again in Ch. 1 1/4. The evidence for the existence of such eddies includes the following : (i) The impossible rate of accumulation of air deduced in Ch. 9/3 from observa- tions at three stations. (ii) The irregular wind-arrows shown on the unusually detailed wind-maps prepared by the Norwegian weather service. In Norway many of the irregularities are obviously due to mountains. They none the less come under consideration here. (iii) The irregularities noticeable on the British Daily Weather Report between the pilot balloon observations at neighbouring stations""". (iv) The errors of forecasts which have been attributed by G. M. B. Dobsonf mainly to small irregular variations in the pressure distribution. Let us now consider various ways in which the smoothing could be effected. A. Space Means. If instead of one station observing wind at the centre of each red chequer on our chessboard we had a number of stations distributed, preferably regularly, over the area of the chequer, then some sort of mean of their observations would be the proper quantity to choose as an initial datum for the computing. This plan would be technically preferable to the processes described below, but as observa- tions are scarce and stations costly we should explore other ways. B. Time Means. Suppose that there is only one station on each red square of the chessboard, but that the observations of wind at it are made every hour during say n successive hours, and that their mean is taken and used as the initial datum of the computing. If the large eddies are distributed at random, the result will be much the * See also especially "The variation of wind with place," by Capt. J. Durward, M.A., London Met. Office Professional Notes, No. 24. t "Causes of Errors in Forecasting Pressure Gradients and Wind," Q. J. R. Met. Soc. 1921 Oct. CH. 10 VARIOUS WAYS OF SMOOTHING 215 same as if we took a space-mean along the line travelled by a point moving with the mean wind during n hours. We want this line to be equal in length to the distance between the centres of our red co-ordinate chequers. So for an ordinary wind velocity of 10 m/s and a chequer 200 km in the side the observations would need to be con- 400 xlO 3 tmuedfor 10^3600= lhoUrS ' The advantage of this scheme over A is that it would require fewer observers. It appears also to be more practical than C, D, E which follow. C. Potential Function. The irregularity in the observations which has forced itself on our attention is the large value of div^m. It may be possible by slight adjustment of m to remove large values of div^^m especially if the latter are scattered at random and if they vary, as is to be anticipated, symmetrically around a mean near to zero. Thus we might introduce a potential function f lt and replace the observed >K, m y by Then div^y of these new momenta per volume would be where V* EN is what V~ becomes when dfjdh is ignored. If then this expression be put equal to zero or to some function of the observed "barometric tendency" we should have a problem to be solved to determine f^ from ^ EN fi = a given function of position. Either there is a boundary of the sort contemplated in Ch. 7 in which case we have a "jury" problem*. Or else the region covers the whole globe, as if the boundary had contracted to a point. There must be no discontinuity of yi at this point and for that reason we have to do, not with a "marching" problem *, but with one more akin to the "jury" variety. The solution of such problems by analysis is the subject of an extensive literature f and their solution by arithmetic has been illustrated by examples else where +. In any case it is rather troublesome. D. Stream Function. The method C would leave the distribution of curl m un- altered by the smoothing. But curl m is almost certainly irregular and in need of preliminary smoothing if we are to avoid awkward consequences in the application of the dynamical equations. We might imagine a stream function f., introduced and m E , m N replaced by 3/, a / 2 m +~-, m^-f- 2 . dn oe Proceeding as with f i we could by the solution of a jury problem find a distribution f a _ which would remove or diminish the irregular curl, without affecting the divergence. * See p. 3 above. t For example Byerly, Fourier Series and Spherical Harmonics. Wiley & Co., New York. I L. F. Richardson, Phil. Trans. A, Vol. 210, p. 307 (1910). 216 SMOOTHING THE INITIAL DATA CH. 10 E. Smoothing during the forecast. While beginning the forecast with the un- smoothed velocities we might temporarily introduce into the dynamical equations terms representing a considerable fictitious viscosity. These would have the effect of smoothing out irregular motions whether waves of compression or whirls. As is well known, the shorter the wave length or the smaller the diameter of the whirl, the more rapidly would the corresponding motion be damped out. Arithmetical processes, somewhat analogous to that here suggested, have been used* to smooth an arbitrary function of position and to make it approach gradually to/ w where ^f n = 0. In the atmospheric case we should aim to remove the fictitious viscosity after it had smoothed the irregularities and if possible before its effect on the larger motions had become noticeable. * L. F. Richardson, Phil. Trans. A, Vol. 210, p. 307 (1910). CHAPTER XI SOME REMAINING PROBLEMS CH. i i/o. INTRODUCTION THE two great outstanding difficulties are those connected with the completeness necessary in the initial observations and with the elaborateness of the subsequent process of computing. These are discussed in Ch. 11/1 and Oh. 11/2. The scheme of numerical forecasting has developed so far that it is reasonable to expect that when the smoothing of Ch. 10 has been arranged, it may give forecasts agreeing with the actual smoothed weather. When that stage has been attained, the other difficulties will tend to group themselves with questions of the desirability of weather forecasts and of their cost. We need here an estimate of the economic value of a forecast reliable for n days ahead, given as a function of n. As with improved methods n is likely to increase, so forecasts will become of more value to agriculturalists. Now the annual value of the world's food crops is at least 1000,000,000, so that a very tiny fractional saving would correspond to a large sum. CH. u/i. THE PROBLEM OP OBTAINING INITIAL OBSERVATIONS Pattern When the observations are taken at stations scattered irregularly, an interpolation has to be made to find the initial data at the centres of the chequers of our chessboard. It has been mentioned in Ch. 9/1 that this interpolation was found to be both trouble- some and inaccurate, as may be evident from the map on p. 184. An existing meteorological station in the British Isles has been either an outgrowth from an astronomical or magnetic observatory, or it has adjoined the house of an enthusiast who lived there for reasons unconnected with meteorology, or it has been pushed out to the confines of the islands to grasp as much weather as possible, or it has been placed in charge of coastguards because they are on duty at night, or it has been set on a mountain to test the upper air. Excellent practical reasons all these, but it is remarkable that the properties of the atmosphere, which are expressed by its dynamical equations and its equation of continuity, appear to have had no influence on the selection*. There would be a great advantage, from the point of view of meteorological science, if observing stations for pressure and for velocity could be arranged alternately in rectangular order in the pattern shown in the frontispiece, modified where necessary by devices such as those proposed in Ch. 7/3, 7/4, 7/5. Wind Velocities given, as is customary, to O'l metre/sec, and at stations 400 km apart, have been found to be nearly but not quite sufficiently accurate (supposing the decimal * See resolution xxi in the Report of the International Commission for the Investigation of the Upper Air, Bergen 1921. R. 28 218 SOME REMAINING PROBLEMS CH. n/i figure correct) to give reasonable values of dp G /dt when they are inserted in the equation of continuity of mass. Greater accuracy could, of course, be obtained by using larger theodolites. Further, a pilot balloon observation naturally gives \v E dh, \v N dh and if these integrals were published for certain limits of h, the calculation of M E =\v E p.dh, M N =\v N p. dh could be improved. There remains the problem of gusts and local eddies of larger size which has been discussed in Chapter 10. The circumstances, in which wind can be observed, are extending. Thus during the war it became customary to observe pilot balloons at night by attaching to the balloon a candle in a small paper lantern. The wind data available at present relate mainly to clear air. But for observing the wind above fog, or low cloud, kite-balloons can be used, or the elaborate method of location by sound*. For the same purpose projectiles have been used at Benson up to heights of 600 m. The projectiles have been spheres of about the size of a cherry and they have been projected nearly vertically but in a direction slightly inclined towards the wind so that the returning sphere struck earth near to the hut which protected the observer. Temperature The example of Ch. 9 sets out from the records of registering balloons. But these would not serve as a basis for actual forecasts because the balloon is often not found until a week or more after its ascent. During and since the war temperature observations by aeroplanes have been taken in great number up to about 5 kilometres. Also kite-balloons have been utilized. Experiments have recently been conducted at Benson Observatory with a view to finding methods of observing temperature, or its equivalent, which should give im- mediate records and which should be cheaper than aeroplane ascents. The results obtained are described in publications entitled " Lizard Balloons for signalling the ratio of pressure to temperature t," "Cracker balloons for signalling temperature f " and "Sun-flash balloons for continuous signalling J." It is hoped also to describe some experiments in which the time of flight of a projectile shot upwards served as an indication of the temperature aloft. Water in Clouds Another gap in the existing observational data relates to the amount of water in clouds. Attempts have been made to measure this photometrically. And it has been shown to be possible when the cloud particles are all of one known size. But in actual cases we are often without definite information concerning the size of the particles. * Schereschewsky, Report Intern. Commiss. Upper Air, Bergen 1921, p. 22. t Meteorological Office, London, Professional Notes, Nos. 18 and 19. f Q. J. R. Met. Soc. 1920 July, p. 293. L. F. Richardson, "Water in Clouds," Boy. Soc. Land. Proc. A, Vol. 96 (1919), p. 19. CH. ii/2 A FORECAST-FACTORY 219 ' CH. 1 1/2. THE SPEED AND ORGANIZATION OF COMPUTING It took me the best part of six weeks to draw up the computing forms and to work out the new distribution in two vertical columns for the first time. My office was a heap of hay in a cold rest billet. With practice the work of an average computer might go perhaps ten times faster. If the time-step were 3 hours, then 32 individuals could just compute two points so as to keep pace with the weather, if we allow nothing for the very great gain in speed which is invariably noticed when a complicated operation is divided up into simpler parts, upon which individuals specialize. If the co-ordinate chequer -were 200 km square in plan, T,hDre would be 3200 columns on the complete map of the globe. In the tropics the weather is often foreknown, so that we may say 2000 active columns. So that 32 x 2000 = 64,000 computers would be needed to race the weather for the whole globe. That is a staggering figure. Per- haps in some years' time it may be possible to report a simplification of the process. But in any case, the organization indicated is a central forecast-factory for the whole globe, or for portions extending to boundaries where the weather is steady, with indi- vidual computers specializing on the separate equations. Let us hope for their sakes that they are moved on from time to time to new operations. After so much hard reasoning, may one play with a fantasy "? Imagine a large hall like a theatre, except that the circles and galleries go right round through the space usually occupied by the stage. The walls of this chamber are painted to form a map of the globe. The ceiling represents the north polar regions, England is in the gallery, the tropics in the upper circle, Australia on the dress circle and the antarctic in the pit. A myriad computers are at work upon the weather of the part of the map where each sits, but each computer attends only to one equation or part of an equation. The work of each region is coordinated by an official of higher rank. Numerous little " night signs " display the instantaneous values so that neighbouring computers can read them. Each number is thus displayed in three adjacent zones so as to maintain communica- tion to the North and South on the map. From the floor of the pit a tall pillar rises to half the height of the hall. It carries a large pulpit on its top. In this sits the man in charge of the whole theatre ; he is surrounded by several assistants and mes- sengers. One of his duties is to maintain a uniform speed of progress in all parts of the globe. In this respect he is like the conductor of an orchestra in which the instru- ments are slide-rules and calculating machines. But instead of waving a baton he turns a beam of rosy light upon any region that is running ahead of the rest, and a beam of blue light upon those who are behindhand. Four senior clerks in the central pulpit are collecting the future weather as fast as it is being computed, and despatching it by pneumatic carrier to a quiet room. There it will be coded and telephoned to the radio transmitting station. Messengers carry piles of used computing forms down to a storehouse in the cellar. In a neighbouring building there is a research department, where they invent improvements. But there is much experimenting on a small scale before any change is made in the complex routine of the computing theatre. In a basement an enthusiast 282 220 SOME REMAINING PROBLEMS CH. 11/2, 3,4 is observing eddies in the liquid lining of a huge spinning bowl, but so far the arith- metic proves the better way. In another building are all the usual financial, correspondence and administrative offices. Outside are playing fields, houses, mountains and lakes, for it was thought that those who compute the weather should breathe of it freely. CH. 11/3. ANALYTICAL TRANSFORMATION OF THE EQUATIONS It is conceivable that by a change of variables the equations could be much shortened. But as we are always required in the end to arrive at quantities of direct interest to the public, namely wind, rain, temperature and radiation, so it may be that analytical simplicity does not simplify the arithmetic. There is a tale of a philosopher who suc- ceeded in reducing the whole of physics to a single equation II 0, but the explanation of the meaning of H occupied twelve fat volumes. The sort of transformations that suggest themselves are those to log p, log 6, instead of p, p or else to stream functions and velocity potentials*. No use has been made in this book of Mr W. H. Dines' correlations between temperature and pressure, and it is felt that while it would be very rash to assume the correlation to be exactly unity, yet its proved approach towards unity might suggest an economical choice of variables. Then again experience must decide whether the various transformations, from equations true at any level, to equations true for strata as a whole, are worth the extra trouble they involve. Would it be easier and more exact, for example, to com- pute with 20 strata using un-transformed equations, rather than with 5 strata and the equations given in the computing forms \ CH. 11/4. HORIZONTAL DIFFUSION BY LARGE EDDIES We are led to consider this by the proposal in Oh. 10 that small secondary cyclones should be smoothed out. For any smoothing process requires compensation by the introduction of eddy-diffusion of an appropriate kind. There is the hypothesis t that we may measure the diffusion of heat, water, momentum, dust, etc., by first measuring the diffusion of mass of air and afterwards considering how much heat, water, momentum or dust is carried by unit mass. This process is easy and the numerical estimates for vertical diffusion show that it gives results at least of the right order. Let us apply it to horizontal diffusion. The smoke trails from cities have been observed by aviators to be hundreds of miles long. If aviators would also take note of the horizontal breadth of the trail at various distances from the source, and of the speed of the mean wind, it might be possible to extract a measure of the horizontal diffusivity. The formula has been given by the author j thus : * Compare A. E. H. Love, "Notes on the dynamical theory of the tides." Proc. Lond. Math. Soc. Ser. 2, Vol. 12, Part 4. t Due to G. I. Taylor and W. Schmidt. See p. 76 above. { Phil. Trans. A, Vol. 221, p. 6. Compare A. Einstein on Brownian movement, Ann.Phys. xvn. 1905. CH. 1 1/4 HORIZONTAL DIFFUSION AFTER MANY HOURS 221 Square of standard deviation of smoke from its middle line Eddy-difiusivity = - ^ r~i IT ~~i = i ~ ~> iwice time taken by smoke since leaving source (1) provided always that the time is long enough. There might be difficulties owing to the precipitation of the smoke, or to its becoming too faint to be seen. In these respects manned balloons are better. The Gordon Bennett races furnish suitable data. The balloons simply drifted, the control of the aeronauts being limited to letting out gas or ballast. Even that amount of control rather confuses the data for the present purpose. But as there is no other information this is not to be despised. On 1906 Sept. 30, between 16 h and 17 h 20 in the ^afternoon, sixteen manned balloons started from Paris. The landing places of all of them* and the log of one t are printed in the Aeronautical Journal. From these records I estimate, by means of the above formula, that the horizontal eddy-diffusivity J was of the order of 2 x 1 8 cm 2 sec" 1 . This figure is also the ratio of viscosity to density, and it lies between the observed values for shoemakers' wax and for pitch. The eddy-diffusivity when multiplied by the density of the air, which we may put at 10" 3 grm cm" 3 , gives the eddy viscosity 2 x lO'cm^grmsec' 1 . The mean velocity of one balloon on this occasion was about 400 cm sec" 1 . To explain the geometry let us take rectangular axes Ox, Oy, Oh. Let Ox be -drawn horizontally in the direction of the mean wind, so that x increases for a par- ticle moving with this smooth motion. Let the y axis be drawn horizontally to the left, and, in accordance with the right-hand screw rule, let the h axis point upwards. Let v x , V Y , V H be the corresponding components of the smoothed wind velocity. Then in a viscous liquid free from eddies there would be three shearing stresses, yh, hx, xy, connected with three rates of shear by the following equations in which C x , CY, C a would be equal to one another and would be called the viscosity : > (2) V h ~ -A ) | a.. | ' sh hx Ct xy = C H \-~ + ~[ (dx dy ) In the smoothed motion of the atmosphere the two terms in square brackets are usually negligible. The eddy-viscosity which has usually been measured is either C x or CY or some combination of C x and C r for it has been customary to assume that these viscosities are equal. However in one case the author has found some evidence that CY was seven times greater than C x . (See p. 73 above.) It has now been shown that C H must be taken to be 1000 or more times greater than either C x or C Y if we * Aer. J. 1906 Oct. f Ibid. 1907 Jan. J The corresponding race of 1921 Sept. gives diffusivity = 3 - 6 x 10 8 cm sec" 1 . Lamb, Hydrodynamics, 3rd edn. Arts. 30, 311 to 314. 222 SOME REMAINING PROBLEMS CH. 11/4,5 are going to smooth out not only gusts but also "secondary cyclones" as has been proposed in Ch. 10. All eddy-viscosities imply some conventional coordinate element. The element chosen in this book is shaped like a railway ticket as its edge extends about 200 km on the map, while it is only a few kilometres thick. Apparently the flat shape of the element is the cause of the great excess of C H over C x or C Y . Equations (2) are valid if the fluid is isotropic. But, now that we have proved that C x , C Y , Cf[ to be unequal, the form of the relationship between the six eddy- stresses and the six rates of mean strain becomes again an open question. In Ch. 4/9/5 it has even been suggested that there may be nine independent components of stress. But apart from that possibility, the general theory is to be found ready-made in connection with the elasticity of crystals. May we perhaps liken our coordinate element to a crystal having three unequal axes at right angles to each other ? If so equations (2) would still be correct* and the complications would be confined to the direct stresses. In order to form some idea of the order of magnitude of the eddy-shearing stress xy which would be produced by this enormous viscosity C a , an attempt has been made to estimate the rate of shear dv Y /dx + dv x /dy. For this purpose daily weather maps have been taken, local irregularities have been smoothed out of the isobars, and then the wind has been assumed to be geostrophic. In the well-marked cyclone of 1919 March 27 at 7 h the rate of shear on the slope of the depression across the British Isles appears to have been of the order of 4 x 10~ B sec~ 1 . Taking the viscosity C H as 2 x 10 5 cm' 1 grm sec" 1 we find by multiplication an eddy shearing stress xy of 8 dyne cm" 2 . This is only some 1 to 10 times greater than the shearing stress on the ground, so that the large viscosity is associated with the small rate-of-mean-shear, and vice versa, CH. 11/5. A SURVEY OF REFLECTIVITY We have been led to attribute considerable importance to the fraction of solar radiation which is absorbed at the surface. The mean value of the reflectivity over a large area could perhaps best be observed from aeroplanes. A very light and simple photometer f would serve to compare the brightness of a uniform stratus cloud above the aeroplane with that of the ground beneath. * Winkelmann, Handb. der Physik. t See, for example, Roy. Soc. Lond. Proc. A, Vol. 96 (1919), p. 25. CHAPTER XII UNITS AND NOTATION CH. 12/1. UNITS EXCEPT where otherwise stated, centimetre-gram-second units are employed. Tempera- tures are in degrees absolute centigrade. Energy, whether by itself or as involved in entropy or in specific heat, is expressed not in calories but in ergs. A power of ten standing at the beginning of a row or column of figures, and followed by a multiplica- tion sign, is intended to multiply each number in the row or column. CH. 12/2. LIST OF SYMBOLS The following list shows the meanings which have been used throughout the book. Where the symbol requires an extended definition reference is made to the place where the definition will be found. Mathematical notation is international, so that a foreigner, who is unable to read the letterpress, may yet grasp the purport of the book if he knows the meanings of the symbols only. So here I should like to explain the symbols in " the second language for all mankind," if there were such a one. Unfortunately there are several rivals, each apparently easier to learn than any national language. Thus there are Esperanto*, Idol, EsperantidoJ , all much alike, and differing considerably from Interlingua. A comparative study of these languages is being made by a committee of the Inter- national Research Council (at Washington, U.S.A.)H and the choice of one language should rest with some supremely authoritative body. Here without expressing any opinion as to which language is best, one namely Ido is selected for illustration. In making the translations I have been guided by my brother Gilbert H. Richardson and by the large " Dictionnaire Fran$ais = Ido par Beaufort et Couturat||." Words marked with an asterisk * are not in the dictionary and so are merely suggestions. * British Esperanto Association, 17 Hart Street, London, W.C. 1. t International Language (Ido) Society of Great Britain, Hon. Sec. J. W. Baxter, 57 Limes Grove, Lewisham, London, S.E. 13. J "Esperantido," 10 Hotelgasse, Bern, Switzerland. Headquarters in Turin, Italy. || 1915 Paris, Irnprimerie Chaix, 11 Boul. St Michel. H 1701 Massachusetts Avenue, Washington, B.C., U.S.A. 224 UNITS AND NOTATION CH. 12/2 a b c d, 8 e 3e f .dX = distance eastwards f Various functions Acceleration of gravity Height above mean sea-level Subscript for arbitrary height Special coordinate in soil Thermal conductivity Length. Distance Momentum per volume Mass of molecule Number = a8 = distance northwards t Pressure Radius. Correlation coefficient Diffusivity of soil for temperature Time Thermal capacity per volume Velocity Mass of water-substance per volume .-Horizontal rectangular coordinates Depth in ground 73 2 t Caution : ^^- is not equal to dedn' IDO Radio di la tero Gasala konstanto Viskozeso efektigata da vortici Infmitezima kreskuri di Bazo di logaritmi naturala Disto vers esto t Diversa funcioni Acelero efektigata da gravito Alteso super la meza surface dil maro Subskribajo indikanta alteso segun-vola Specala mezuro di profundeso en la sulo Konduktiveso kalorala Longeso. Disto Rapideso multiplikata per denseso Maso di molekulo Nombro o numero Disto vers nordo t Preso Radio. Korelatala koeficiento Difuziveso di la sulo ye temperature Tempo Kalorala kapaceso po volumino Rapideso Maso di aquo-substanco po volumino Horizontala koordinati ortangula Profundeso en la sulo t Atencez: - ne egalas r- r- . (in tin Sedn FUBTHEB DEFINITION OB BEFEBENCE Ch. 4/5/1 Ch. 4/8/0 2-71828 Ch. 4/10/0 Ch. 4/10/2 p. 37 Ch. 4/1 0/2 #6 Ch. 4/1 0/2 #7 Ch. 4/10/0 CH. 12/2 LIST OF SYMBOLS ENGLISH B C D G H I J K M N Q R Various meanings Infinitesimal increase, accompanying the motion, of... Subscript for eastwards Radiant activity in a "parcel" Various functions Subscript for ground level Subscript for upwards Brightness Subscript for upper surface of vege- tation Momentum per area of stratum Subscript for northwards = Ipdh across stratum U "V w X Y Liquid water per area of stratum Mass per area of stratum Entropy per area of stratum Time Velocity of cloud particles relative to air Water-substance per area of stratum Subscripts indicating horizontal rectangular components In theory of stirring IDO Diversa signifiki Tufiniteziraa kreskuro, akompananta la movo, di... Subskribajo indikanta ulo direktata vers esto Radiada energio trairanta "pako" po tempo Diversa funcioni Subskribajo signitikanta ulo ye la surfaco di la sulo Subskribajo indikanta vers supre Grade di brilo Subskribajo indikanta ulo ye la supera surfaco di la vegetantaro Denses-opla rapideso po areo stratala Subskribajo signitikanta vers nordo = Ipdh tra strato Liquida aquo po areo stratala Maso po areo stratala Entropio* po areo stratala Tempo Rapideso, relativa al aero, di nubala partikuli Juritata maso di vaporo, aquo e glacio po areo di strato Subskribaji indikanta horizon tala kompo- zanti ortangula Ye teorio pri vortici 225 FURTHER DEFINITION OR REFERENCE pp. 50, 51 pp. 50, 51 Ch. 4/2 # 7 Ch. 4/4 #9 Ch. 4/2 # 6 Ch. 4/6 p. 27 E. Ch. 4/8/0 29 226 UNITS AND NOTATION CH. 12/2 ENGLISH Coefficients relating to entropy ^hermal capacities per mass inite difference operator inergy per mass Zenith distance Absorptance of stratum Temperature, absolute Molecular diffusivity Longitude, always eastwards Joint mass of vapour, water and ice per mass of atmosphere Mass of liquid water per mass of atmosphere Turbulivity * 3-14159... Density Entropy per mass of atmosphere Potential temperature Internal energy per mass of atmo sphere Latitude (reckoned negative in the southern hemisphere) In theory of stirring Gravity potential (increasing upwards Angular velocity of earth IDO ioeficienti pri entropio* valorala kapacesi po maso ''inite-mikra kreskuro di... nergio po maso Angulo inter zenito ed ula direciono ?raciono di radiada energio absorbata da strato Temperature de - 273- 1 C. Molekulala difuziveso Longitude (sempre vers esto) Juntata maso di vaporo, aquo e glacio po maso di atmosfero Maso di liquida aquo po maso di atmosfero Specala mezuro di vorticado Denseso t Entropio* po maso di atmosfero Temperaturo ye ula preso normala se nek kaloro nek aquo esas perdita Interna energio po maso di atmosfero Latitude (negativa en la suda mi-sfero) Ye teorio pri vortici Gravitala potencialo (kreskanta ad-supre) Angulala rapideso di la tero FURTHER DEFINITION OR REFERENCE (Ch. 8/2/6 1(Ch. 4/5/1) Ch. 4/5/0 Ch. 4/7/1 #13 Ch. 4/9/8 Ch. 4/8/0 #15 (Ch. 4/5/0 (Ch. 8/2/6 /Ch. 4/5/0 \Ch. 8/2/6 Ch. 4/5/0 Ch. 4/8/0 #6 0-729211 xlO-'sec- 1 CH. 12/2 LIST OF SYMBOLS II 2 f (dalda)f JUL (mi) T (he) ~J (shai) X (janja) O A ENGLISH Radiant energy absorbed at interface per area and per time Increase of Eddy-heat per mass Mass of water evaporating from inter- face per horizontal area and per time Summing operator Pressure in water in soil } Relate to vertical velocity in the stratosphere = 2u sin Vapour density in soil Rate of evaporation from leaf Correction to the estimate of surface temperature See flux of heat at the interface Latent heat of evaporation per mass Conductivity of soil to soil-water Porosity of soil to vapour Partition coefficient of W. Schmidt Stefan's radiation constant Absorptivity per density Scatterivity* per density Emissivity of interface for long waves IDO Radiada energio absorbata an interfaco po areo e po tempo Kreskuro di Energio di vortici po maso Maso di aquo vaporeskanta de interfaco po horizontala areo e po tempo Sumigilo Preso en aquo en sulo i Relatas vertikala rapideso en la supra J strato = 2o> sin < Denseso di vaporo en sulo Vaporeskala rapideso de folio Korektilo al konjekturo di temperaturo interfacala Pri fluado di kaloro ad o de 1'interfaco Energio vaporigiva mezur-unajo di maso Konduktiveso di sulo por aquo sulala, Porozeso di sulo por vaporo Koeficiento di W. Schmidt pri divide Konstanto di Stefan pri radiado Absorbiveso po denseso Dissemiveso po denseso Emisiveso di interfaco por ondi longa 227 FURTHER DEFINITION OR REFERENCE Ch. S/2/15 p. 77 Ch. 8/2/11, 12 Ch. 4/10/2 Ch. 6/6 # 23 Ch. 6/6 # 22 p. 15 Ch. 4/10/2 Ch. 4/10/3#l Ch. 8/2/15 Ch. 4/8/4 #7 Ch. 4/10/2 #1 Ch. 4/1 0/2 #3 p. 89 Ch. 4/7/1 # Ch. 4/7/2 #5 Ch. 4/7/2 #5 Ch. 8/2/15 t I am indebted to Prof. Flinders Petrie for the names of these Coptic letters. 292 228 UNITS AND NOTATION On. 12/3, 4 CH. 12/3. RELATIONSHIPS BETWEEN CERTAIN SYMBOLS The symbol in the third column below is equal to p times the corresponding symbol in the second column. The symbol in the fourth column is the integral, with respect to height across a conventional stratum, of the symbol in the third column, or if that is absent, of p times the symbol in the second column. I II III IV Per mass of atmosphere Per volume of atmosphere Per horizontal area of conventional stratum Mass 1 P R Momentum V m M P P Mass of water in all forms! jointly } V- w W Mass of condensed water V Q Entropy ... dWjofV = ^ + ^--V N - r . de dn a In describing spatial variation it is very desirable, as Sir Napier Shaw has pointed out, to have a word which makes a clear distinction between vertical and horizontal directions, and for this reason he uses " gradient " for the horizontal, " lapse-rate " for the vertical. This entire change of term gives no reminder that the horizontal and vertical changes are components of the same vector. Again " lapse-rate " is to be reckoned positive when the quantity which varies is greater below than above, and this convention of signs is not always convenient. Lastly, some writers use lapse-rate without specifying the quantity which lapses ; one assumes it to be temperature ; but having done that, one hesitates to speak of the lapse-rate of anything else. As a way out of these difficulties the following notation is suggested. Let p be any scalar = up-gradient of p ; ~- = east-gradient of p ; - = north-gradient of p. -31 down-gradient of p ; - j- = west-gradient of p ; , = south -gradient of p. In the foregoing pages "grade" is sometimes used for "gradient," but that was- perhaps a mistake. INDEX OF PERSONS The numbers refer to the pages Abbe, C., 36, 40, 41, 42, 92, 93, 162 Abbot, C. G., 46, 57, 58, 60, 63, 64 Akerblorn, R, 65, 72, 82, 93 Aldrich, L. B., 46, 58, 60, 64 d'Alembert, J. le R., 99 Angot, A., 82, 87, 90 Eraden, R., 134 Escombe, R, 112, 113, 167 Exner, F. M., 43, 81, 93, 166 Forsyth, A. R., 6 Fourier, J. B. J., 1 Angstrom, A., 46, 48, 49, 51, 52, 54, 55, 56, Fowle, F. E., 46, 47/48, 57, 58, 60, 62, 64 57, 64, 76, 89, 93, 114, 134, 135 Aschkinass, E., 46, 47 von Bahr, E., 47 Barkow, E., 94, 101 Bauer, L. A., 40 Beaufort, .,223 van Bemmelen, W., 84 Bennett, G. T., 212 Berek, M., 115 Bergeron, T., vi, 43 Bernard, H., 93 von Bezold, W., 40, 41, 42, 162, 163 Bigelow, F. H., 31, 41 Bjerknes, J., vi, 43 Franklin, T. B., 171, 174, 176 Fream, W., 176 Fresnel, A. J., 57 Friedmann, A., 22, 24, 134 Georgii, W., 85 Giblett, M. A., vii Gold, E., v, 5, 16, 27, 46, 54, 57, 84, 128, 134 Gouy, G., 126 Greaves, C., 107 Guldberg, C. M., 92 von Hann, J., 26, 27, 41, 44, 48, 89, 105, 110, 170, 186 Heinrich, ., 113 Bjerknes, V., vi, 2, 17, 24, 43, 119, 128, 140, Helland-Hansen, B., 105, 106 141, 181, 182 Boussinesq, J., 107 Boutaric, A., 57, 64 Briggs, ., 107 Brown, H. T., 112, 113, 167 Brunt, D., 94 Bryan, G. H., 65, 71, 158 Byerly, W. E., 215 Callendar, H. L., 110 Cave, C. J. P., 73 Chapman, E. H., 10, 113 Chapman, S., 125, 126 Clark, J. E., 131 Couturat, L., 223 Dal ton, J., 103 Davis, W. M., 41 Dines, J. S., 115 Dines, L. H. G., 84 Dines, W. H., vi, vii, 4, 15, 16, 17, 24, 33, 40, 46, 56, 57, 64, 87, 93, 115, 121, 124, Krummel, O., 106 Hellmann, G., 72 Helmert, F. R., 32, 188 von Helmholtz, H. L. F., 40 Hergesell, H., 134, 145, 182 Hertz, H., 23, 40, 41, 159, 160, 161, 162, 165, 198 Hesselberg, T., 22, 24, 65, 72, 82, 93 Holtsmark, J., 93 Homen, T., 89 Humphreys, W. J., 134 Jansson, M., 114 Jeans, J. H., 36, 37, 38, 65, 102, 125, 160 Jeffreys, H., 10, 94 Kaye, G. W. C., 71 Kelvin, Lord, 58, 92 King, F. H., 107 King, L. V., 46, 58, 59, 60, 64 Kirchhoff, G., 46, 50, 57 Koref, .,160 128, 141, 146, 148, 187, 220 Dobson, G. M. B., 72, 74, 82, 84, 131, 187, 214 Douglas, C. K. M., 44, 66, 93 Durward, J., 214 Ebermayer, E., 184 Egnell, .,27 Einstein, A., 220 Ekman, W. V., 83, 93 Kurlbaum, F., 50 Laby, T. H., 71 Ladenburg, ., 47 Lagrange, J. L., 30, 31, 32 Lamb, H., 4, 5, 30, 31, 38, 92, 95, 221 Laplace, P. S., 4, 30 Lehman, ., 47 Lempfert, R. G. K., 212 Lindemann, F. A., 160 INDEX OF PERSONS 231 Lindholm, F., 64 Loughridge, R. H., 107 Love, A. E. H., 220 Maclaurin, C., 6, 12, 151, 152, 153 Margules, M., 36, 93 Maxwell, J. C., 38 Meyerhoffer, L. B., 32 Milne, A. E., 125, 126 Mitchell, A. C., vi Mohn, H., 92 Nansen, R, 33, 105 Nernst, W., 160 Neuhoff, O., 40, 41, 42, 160, 161 Newton, Sir I., 175 Paschen, F., 46 Pennick, J. M. K., 107 Pernter, J. M., 44 Petrie, F., 227 Planck, M., 46, 49, 50 Pott, E., 110 Pring, J. N., 135 Rambaut, A. A., 104, 110 Rayleigh, Lord, 58, 92 Regers, J., 87 Reynolds, O., 65, 83, 92, 95, 98, 100, 101 Richardson, G. H., 223 Rubens, H., 46, 47 Russell, E. J., 113 Sandstrom, J. W., 93 Schereschewsky, Ph., 218 Schmidt, W., 67, 72, 74, 75, 76, 87, 89, 91, 93, 94, 112, 162, 220, 227 Schuster, Sir A., 58 Scott, J., 127, 131 Shaw, Sir N., vi, 9, 10, 23, 33, 43, 64, 127, 131, 141, 146, 183, 212, 229 Sheppard, W. F., 3, 149 Silberstein, L., 30, 99 Simpson, G. C., vii, 114 Solberg, ., vi, 43 Stefan, J., 172, 174, 227 Stevenson, .,72 Stoermer, C., 125 Stokes, Sir G. G., 44 Sverdrup, H. U., 72, 78, 82, 93, 106 Tait, P. G., 92 Taylor, Brook, 175 Taylor, G. I., 65, 67, 72, 75, 76, 77, 82, 83, 84, 87, 91, 93, 106, 112, 170, 220 Thomson, Sir J. J., 45 Warington, R., 107, 108, 110, 113 Webster, A. G., 38 Wenger, R., 128 Whipple, F. J. W., 94 Whipple, R. S., 64 Wigand, A., 74 Wilson, W. E., 112, 167 Winkelmann, A. A., 47, 106, 108, 109, 222 INDEX OF SUBSIDIARY SUBJECTS (For the main topics the reader is referred to the table of contents on p. xi.) Absorptivity-per-detisity of air, for long-wave radiation, 50 et seq. in stratosphere, 134, 135 for solar radiation, 60 et seq. Accuracy, gain of, by centred differences, 3, 149 Adiabatic expansion, 35, 158 et seq. and clouds, 99 and mixing, 162, 163 in stratosphere, 135, 141 et seq. law, independent of gravity and motion, 39 Adsorption of water by soil, 108, 109 Advection (nee Conveyance) Aeroplanes, observations from, 44, 65, 66, 94, 218, 222 " Austausch " of W. Schmidt, 67, 91 Balloons, free, and horizontal diffusion, 221 registering, precision of observations, 128 Barley, evaporation from, 111, 113, 168 232 INDEX OF SUBSIDIARY SUBJECTS Carbon-di-oxide, absorption of long-wave radiation by, 47, 49 Characteristic equation, 22, 23, 41 smoothing of, 97 in stratosphere, 126 Cloud, continuous, 44, 188 detached, 99, 190 particles, size, 44, 45, 218 rate of fall, 44, 45, 225 and radiation, 57, 63, 167 (see also Cumulus) Computing, cost of, 18, 219 Condensation, and entropy, 158 -nuclei, 45 Conductance, of stomata of leaves for water, 112 of crops for water, 113, 168 thermal, of surface strata, 173 et seq., 196, 197 Conductivity, of soil to soil water, 108, 109, 227 thermal, of snow, 114 Continuity, of energy, equation of, 38 of mass, equation of, 23, 178, 200 allowance for precipitation, 45 smoothing of, 97 in stratosphere, 127 of mass of water substance, equation of, 25, 179, 204 allowance for eddy-flux, 80, 81 allowance for precipitation, 45 smoothing of, 97 Convection, 65 Conventional strata, 1, 17, 19, 30, 34, 220, 228 in soil, 104, 105, 172, 184 in stratosphere, 20, 125, 147, 148 thick versus thin near ground, 20, 92, 156, 171, 174 vegetation film, 113 Conveyance of entropy-per-mass, 43, 180 of water (see Continuity) Cornfield, roughness of, 85 Correlation, between temperature and pressure in troposphere, 24, 220 time-rate and east-gradient of pressure, 10 Cost, of computing, 18, 217, 219 of observing stations, 18 Criterion of turbulence, 77, 100, 190 Crops, evaporation from, 111, 113, 168 Crystallography, term borrowed, 149, 156 and eddy-stresses, analogy, 222 Cumulus eddies, 65, 66, 73, 76, 99, 166 " Definite " portion of turbulent fluid, 66, 98 Density of air, time-rate at fixed point, 24 Diffusion of atmospheric eddies, 77, 78, 79 Diffusivity, thermal, of soil, 110, 224 Distillation of water in soil, 109 Diversity (see Heterogeneity) Dust, 21, 45, 59, 220 Dynamical equations, 30, 180, 209, 210 allowance for eddy-flux of momentum, 79, 80 importance of small terms, 10, 132, 187 smoothing of, 98 in stratosphere, 132, 208 Eddies, atmospheric, diffusion of, 77, 78, 79 energy of, 65, 71, 77, 100, 164 Eddy-motion (see Turbulence) INDEX OF SUBSIDIARY SUBJECTS 233 Eddy-viscosity, lack of isotropy in, 222 Electrical analogy, to evaporation, 112, 168, 170 to thermal conditions at surface, 172, 173, 176 Energy, atmospheric, 35 of eddies, 65, 71, 77, 100, 164 -theory of stability, 77 Entropy derivatives, 42, 117, 159 et stq., 189 ; see especially 159 to 166 -per-mass, 35, 39, 40, 41, 42, 177, 198, 199 behaviour in regard to turbulence, 40, 69, 162 conveyance of, 43, 180 when condensation occurs, 158 linear function of height in stratosphere, 142 versus potential temperature, 40, 69, 158, 187 Erosion by Siberian rivers, 33 Errors, due to finite differences, 1, 3, 4, 7, 12, 13, 14, 18, 53, 141, 151, 153 in initial data, forecast spoilt by, 2, 212 Evaporation, 90, 194 from sea, 167 soil, 107, 169 vegetation, 111, 112, 158, 159, 162, 167, 176, 227 - crops, 111, 113, 168 and surface temperature, 174 Finite differences, analytical preparation, 22, 220 centering of, 3, 149 et seq., 174 errors due to, 1, 3, 4, 7, 12, 13, 14, 18, 53, 141, 151, 153 " step-over " method, 150 Forecast, spoilt by errors in initial data, 2, 212 -factory, 219 Forecasts, economic value of, 217 for sea, 104, 106, 153 for soil, 104 Free balloons and horizontal diffusion, 221 Friction, surface, 82, 179, 207 in stratosphere, 128 in vegetation film, 113, 114 Gas constant in stratosphere, 126 Geostrophic hypothesis, test of, 5, 145 inadequate, 9, 14, 146 -approximation in stratosphere, 128, 131, 135, 143, 144, 146, 147 Gradient, use of term, 229 Gravitational energy of gas, 35 et seq. Gravity-potential, 17, 19, 30, 36, 125, 226 Ground, account taken of height of, 2, 24, 34, 92, 181, 182, 187 Heat, conveyance of, by air, 43, 180 eddy-flux of, 70, 76, 177, 198 transference of, by precipitation, 45 motion of, in soil, 110, 179, 206 in snow, 114 flux of, at interface, 86, 170 et seq., 195 emissivity of leaves for, 112 Height of ground, account taken of, 2, 24, 34, 92, 181, 182, 187 of tropopause, 16, 141 Heterogeneity, 76, 94, 166, 190 " pressure of," 97 " Horizontal," defined, 17 Horizontal diffusion, by large eddies, 214, 220 Hydrostatic equation, 81, 115, 117, 164 in stratosphere, 126 "Hygroscopic " water, in soils, 108 R - 30 234 INDEX OF SUBSIDIARY SUBJECTS International languages, 223 Intrinsic energy of gas, 36, 226 lonisation of upper atmosphere, 125, 135 Ions and condensation, 45 Irreversible mixing, 162 fit seq. Isobaric and level surfaces, 17 map, general equation of, 15 " Jury " problems, 3, 215 Kinetic energy of gas, 35 et seq. of eddies, 65, 71, 77, 100, 164 Lapse-rate, use of term, 229 Latent heat, 170 and entropy, 158 et seq. and temperature in soil, 110 and surface temperature, 174 Lattice, term borrowed, 149, 156 -reproducing system, 2, 156, 168, 180 Leaves, emissivity of, for heat, 112 turgescence of, 113 (see also Vegetation) Level and isobaric surfaces, 17 Local showers, 99, 167 Magnetic stresses, 99 "Marching" problems, 3, 215 " Mixing-rule " of W. Schmidt, 162 Momentum, behaviour in regard to turbulence, 69, 70 eddy-flux of, 79, 179, 207 -per-volume versus velocity, 24, 97 Nitrogen, absorption of long-wave radiation by, 47 Observations, upper air, method of obtaining, 218 by registering balloons, precision of, 128 from aeroplanes, 44, 65, 66, 94, 218, 222 of wind, extrapolation in stratosphere, 127, 179, 182, 208 Observing stations, cost of, 18 distribution of, 16, 183, 217 Occlusion of water by soil, 108, 109 Oxygen, absorption of long-wave radiation by, 47 Ozone, absorption of long-wave radiation by, 48, 135 in stratosphere, 135 " Parcel of Radiation," 50, 225 Partition of heat at surface, 89 coefficient, 89, 175 Percolation of water, through soil, 107 transference of heat by, 110 Photometric measurements, of light transmitted by clouds, 63 of reflectivity of surface, from aeroplanes, 222 of water in clouds, 44, 218 " Polar-front," 43 Porosity of soil to water vapour, 109, 227 Potential-density, 40, 85 in sea, 106 -function, 220 smoothing by, 215 -temperature, defined, 40, 158 benaviour in regard to turbulence, 40, 67, 68, 70, 87, 88, 162 INDEX OF SUBSIDIARY SUBJECTS 235 Potential-temperature, versus entropy-per-maas, 40, 69, 158, 187 computation of, 81, 165, 188 diversity of, 101 peculiarity .arising from choice of standard pressure, 163 Precipitation, 44, 158, 188 and surface temperature, 174 transference of heat by, 45 Pressural energy of gas, 36 et seq. " Pressure of heterogeneity," 97 Pressure, not reduced to sea-level, 2, 181 and temperature in troposphere, correlation, 24, 220 time- rate and east-gradient, correlation, 10 in water in soil, 107, 108, 109, 227 Radiation- equilibrium theory of stratosphere, 134, 140 " parcel " of, 50, 225 and surface temperature, 174 vertical velocity, due to, 119 long-wave, 46, 167, 193 and sea, 57, 106 and snow, 76, 114 in stratosphere, 134, 135, 139, 146 solar, 46, 57, 167, 191, 192 and sea, 106 and snow, 114 transparency of stratosphere, 139 Rain, evaporation of, from vegetation, 111, 169 (see also Precipitation) Reflectivity of surface, 222 of sea, 57 of snow, 114 Registering-balloons, precision of observations, 128 " Reststrahlen," 46 Rivers, Siberian, erosion by, 33 % Salinity and turbulence in sea, 106 Saturation, test for, 23, 158 " Scatterivity "-per-density, of air for solar radiation, 61, 227 Sea, 104, 153, 179 evaporation from, 167, 194 and radiation, 57, 106 , thermal boundary condition, 105, 106, 170 " Secondary-cyclones," and horizontal diffusion, 214, 222 Smoke-trails from cities, and horizontal diffusion, 220 Smoothing, operations, 95, 96 of fundamental equations, 97 of vertical velocity equation, 124 of initial data, 214 Snow, 76, 114 Soil, 104, 107, 179, 184, 205, 206 evaporation from, 107, 169, 194 thermal boundary condition, 171 Solar constant, 58, 191, 192 Space-means, smoothing by, 214 Specific heat, at very low temperatures, 160 Stability, 65, 66, 71, 165, 190 and potential density, 40 " energy" theory of, 77 Statistical facts, employment of, in numerical scheme, 27, 59, 82, 92, 104, 105 153 167 171, 178, 179, 180, 182, 184, 190 " Step-over" method of integration, 150 Stratosphere, 125, 179, 187, 201, 208, 209, 210 236 INDEX OF SUBSIDIARY SUBJECTS Stratosphere, horizontal temperature gradients, 128, 140, 183, 187 Stream-function, 22, 220 smoothing by, 215 Surface-friction, 82, 179, 207 reflectivity of, 57, 114, 222 -temperature, denned, 172 of land, 171 of sea, 105, 170, 179 determination of, 174, 175, 196, 197 Temperature, diversity of, 94, 101, 103 and pressure in troposphere, correlation, 24, 220 up-gradient in air near surface, 86, 87, 113, 172 Thermal capacity, of soil, 104, 110 of sea, 105 conductivity, of snow, 114 conductance, of surface strata, 173 et seq., 196, 197 diffusivity, of soil, 110, 224 resistance, of moss, etc., 176 Tidal theory, 4, 5, 30 Time-means, smoothing by, 214 Tropopause, height of, 16, 141 Turbulence, 65, 106, 162, 166, 190 in sea, 106 criterion of, 77, 100, 190 Turbulent fluid, " definite " portion of, 66, 98 differentiation following mean motion of, 67 Turbulivity, denned, 68, 69 Turgescence of leaves, 113 Vegetation, conventional film, 113 evaporation from, 111, 112, 158, 159, 162, 167, 176, 194, 227 thermal boundary condition, 176, 195 Velocity versus momentum-per-volume, 24, 97 Vertical velocity, 1, 22, 41, 43, 115, 178, 180, 202, 203 smoothing equation for, 124, 165 in stratosphere, 124, 135, 177, 201 Water content of clouds, 44, 218 -substance, behaviour in regard to turbulence, 67, 68, 70 eddy-flux of, 80, 176, 198 (see also Evaporation) motion of, in soil, 107, 179, 205 diversity of, in atmosphere, 100, 102 -vapour, absorption of long-wave radiation by, 47 -content of atmosphere, 43, 48 porosity of soil to, 109, 227 Wheat-field, roughness of, 85 Wind, extrapolations of observations in stratosphere, 127, 179, 182, 208 high velocities at great heights, 131 increase of west-component with height, 33 methods of observing, 218 PRINTED IN ENGLAND BY J. 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