V UC-NRLF C 3 550 Efli4 W. B. No. S7'2. U. S. DEPARTMENT OF AGEICULTURE, WEATHER BUREAU. STUDIES ON THE THERMODYNAMICS OF THE ATMOSPHERE Reprints from the Monthly Weather Review, January, February, March, June, July, August, October, November, and December, 1906. FRANK HAGAR BIGELOW, M. A., L. H. D., Professor of Meteorology. Prepared under the direction of WILLIS L. MOORE, Chief U. S. Weather Bureau. WASHINGTON: WEATHER BUREAU. 1907. W. B. No. 372. U. S. DEPARTMENT OF AGRICULTURE, WEATHER BUREAU. STUDIES ON THE THERMODYNAM ATMOSPHERE. Eeprints from the Monthly Weather Review, January, February, March, June, July, August, October, November, and December, 1906. BY FBANK HAGAK BIGELOW, M. A., L. H. D., //; Professor of Meteorology. Prepared under the direction of WILLIS L. MOOKE, Chief U. S. Weather Bureau. WASHINGTON: WEATHER BOKEAU. 1907. OOITTE1ITTS. I. Asymmetric cyclones and anticyclones in Europe and America Introductory remarks The supposed difference in the temperature distribution be- tween American and European cyclones and anticyclones The temperature-falls at Blue Hill and Hald-Berlin Explanation of Tables 1, 2, 3, and 4 Temperature-falls from the surface Temperature differences at the same elevation Vertical temperature gradients per 100 meters Vertical temperature gradients in the warm and cold areas Results II. Coordination of the velocity, temperature, and pressure iu the cyclones and anticyclones of Europe and North America The adopted mean temperatures and gradients on the 1000-meter levels for American and European cyclones and anticyclones The variations of temperature in the several sectors of cyclones and anticyclones on each 1000-meter level The mean temperature, T, in cyclones and anticyclones. . . Resume of the typical distributions of the velocity, tem- perature, and pressure in cyclones and anticyclones. . . . III. Application of the thermodynamic formula to the nonadia- batic atmosphere The nonadiabatic atmosphere Development of the thermodynamic formula I. The first form of the barometric formula II. The second form of the barometric formula in an adiabatic atmosphere III. The barometric formula in a nonadiabatic atmosphere IV. Construction of the primary differential equation. . . . V. Application to the general equations of motion VI. Four systems of constants for the atmosphere VII. The thermodynamic constants for the sun IV. Numerical computations in the vertical ordinate Three general theories regarding the formation of cyclones and anticyclones Comparison of the numerical results of computations by formula (38) and the general barometric formula of the cloud report (59) Computation of mean values of P , />, R a , from T on the 1000-meter levels Collection of the data showing the distribution of the dis- turbances on the 1000-meter levels Values of the temperatures T, T , and T r o Values of the ratios n, n a and (n n ) Values of the terms C p n (T T ) and (2- z c ) Distributions of the velocities q, q a \ (if qj) The distribution of the heat Q Distribution of the pressure BB V. The horizontal convection in cyclones and anticyclones . . Some of the difficulties in this problem The horizontal circulation The horizontal pressure gradients The horizontal interchange of heat energy Some cases of restricted conditions General thermodynamic equations Case I. Change of position in the layers in a column of air Case II. The temperature is a continuous function of the height, T ! =T l ah Case III. For local changes between two adjacent strata of different temperatures, where on the boundary the pressure P=P', = .P' 2 , and the temperature is discon- tinuous Case IV. The overturn of deep strata in the column Case V. Transformation of two masses of different temper- atures on the same level into a state of equilibrium .... Case VI. Continuous horizontal temperature distribution with adiabatic vertical gradient Case VII. Position of layers of equal entrophy when the pressure at a given level is constant and the tempera- ture at this level is a function of the horizontal distance and a linear function of the height 9 9 9 10 11 12 14 14 15 16 74 74 75 77 77 110 110 111 111 111 112 113 114 115 116 265 265 265 265 267 267 267 270 271 271 271 562 562 563 564 563 566 567 568 568 569 570 570 571 571 Page. Case VIII. Final condition of two air masses under con- stant pressure with given initial linear vertical tem- perature fall 571 VI. The waterspout seen off Cottage City, Mass., in Vineyard Sound, on August 19, 1896 307 The sources of the data used in the discussion 307 Letters and reports of observers 307 The photographs 313 Position of the waterspout in the Sound 314 Dimensions as measured on photograph 2d A, fig. 27 314 VII. The meteorological conditions associated with the Cottage City waterspout 360 Meteorological conditions for August, 19, 1896 360 Probable conditions near the waterspout 360 Computation of the pressure B, temperature t, vapor ten- sion e, and height H, for the n, /3, y, 6 stages 362 (A) The o-stage, or unsaturated process 363 (B) The /3-stage, or saturated process 365 (C) The > -stage, or freezing process 366 (D) The rf-stage, or frozen process 366 The cause of the formation of the waterspout cloud, and the vertical convectional velocity 369 VIII. The meteorological conditions associated with the Cottage City waterspout continued 470 Relations between wind velocities and atmospheric pres- sures 470 Marvin's correction to observed wind velocities 476 IX. The meteorological conditions associated with the Cottage City waterspout continued 511 The maximum falling velocity for rain in the lower atmos- phere 511 The probable vertical velocity in the cloud 512 Approximate position of the isotherms and isobars in the Cottage City waterspout cloud 513 The building of hail : 514 Theories of the formation of hailstones 515 TABLES. Table 1. Temperature-falls at Blue Hill 10 High areas in winter 10 Low areas in winter 10 2. Temperature-falls at Blue Hill 11 High areas in summer 11 Low areas in summer 11 3. Temperature-falls at Hald and Berlin 12 High areas in winter 12 Low areas in winter 12 4. Temperature-falls at Hald and Berlin 13 High areas in summer 13 Low areas in summer 13 5. Surface temperatures in centigrade degrees for day and night at Hald and Berlin 14 High areas 14 Low areas 14 6. Distribution of the sectors by warm and cold areas. ... 14 7. Temperatures and gradients in the American and Euro- pean cyclones and anticyclones arranged by the warm and cold areas 15 8. Temperature differences, cold minus warm areas 15 9. Temperatures and variation of temperature on each 1000-meter level in American cyclones and anti- cyclones 74 American winter high areas 74 American winter low areas 74 American summer high areas 74 American summer low areas 74 10. Temperatures and variation of temperature on each 1000-meter level in European cyclones and anti- cyclones ' 75 European winter high areas 75 European winter low areas 75 European summer high areas ; . 75 European summer low areas 75 iii IV Page. Table 11. Adopted mean temperatures and gradients on the 1000- meter levels for American and European cyclones and anticyclones 75 12. Adopted mean temperature variations, AT, in American and European cyclones and anticyclones. . 76 Winter 76 Summer 76 13. Mean temperature, T, in cyclones and anticyclones. . 77 Winter 77 Summer 77 14. Mechanical systems of consonants for the atmosphere in gravitational units 115 15. Comparison of the formulae 266 16. Computed values of the ratio, n, between successive 1000-meter levels 266 17. Computed mean values of pressure, density, and gas factor from the temperature at several elevations, European winter 207 18. European summer 267 19. American winter 267 20. American summer 207 21. Values of T, T a , T- '!' derived from Tables 12, 13, winter high areas 268 22. Winter low areas 268 23. Summer high areas '. 268 24. Summer low areas 268 25. Distribution of the values of n, n c , n - n , winter high areas 268 26. Winter low areas 268 27. Summer high areas. . . . ; 268 28. Summer low areas 268 29. Distribution of the heights z z . (z z a ) g= - t n a , (T T ), winter high areas 269 30. Winter low areas 269 31. Summer high areas 269 32. Summer low areas 269 33. Distribution of the velocities q, q a , 4 ( H o o 3 3 ,0 S ci O f ci Hi ! a* I o I a I o o -r-t a o -e 9 o a i 01 43 1 a o 1-1 15 <0 y>- I FEBRUARY, 1906. ' MONTHLY WEATHER REVIEW. 78 levels. The anticyclone has a system of outward components from top to bottom, and the cyclone a system of inward com- ponents from bottom to top, but in neither case can there be any true inversion in the type of the system. The tempera- tures show that the wave motion is intensified on approaching the surface, as the strong eastward drift is gradually dimin- ished in the lower levels. The pressure, on descending from one level to the other, in the same way gradually takes on the well-known features of the high and low pressure areas, the high areas standing with the " saddle" toward the south, and the low areas with the " saddle " toward the north. The closed isobars decrease in density from the surface upward, and dis- appear at two or three miles above the ground, being depleted at the top by penetration into the eastward drift. Whenever closed isobars occur there is a vertical component of the circulation, downward in anticyclones, upward in cyclones. There is evidently very little vertical movement in the upper levels of the atmosphere, where the isobars are mere wavy lines, unless some unobserved closed isobars occur, as is prob- ably the case in the development of hurricanes in the Tropics. In fig. 10 the disturbing components are given for the velocity, temperature, and pressure. In the velocity of the auticvclone there is a gradual transition of the known outflow- ing structure at the surface into a simple loop in the upper levels, the orientation being changed only a little; in the cyclone the inflowing components are better preserved from the surface to the higher levels, but there is a distinct rotation of the structure through about one quadrant. The tempera- tures show the maximum disturbances on the boundary of the high and low areas, with a distinct rotation of both the cold and warm areas through one quadrant. The pressure dis- turbances consist of closed isobars gradually diminishing into loops in the higher levels and rotating through one quadrant, especially in the cyclone. In one aspect the analytical solu- tion of this dynamic structure is simpler than that demanded in Ferrel's or in Guldberg and Mohn's adopted types of vor- tices, but it is certainly different from either of them. It is evi- dently necessary to distinguish carefully between the cyclonic system proper and the resultant system formed by its combina- tion with the general eastward drift, so that the mathematical analysis shall not deal with the components and resultants indiscriminantly. It is not proper to appeal to observed resultant motions in the atmosphere in verification of a theory applying solely to the components, namely, the cyclonic and anticyclonic gyrations as examples of a special form of vortex. Having thus found at least an approximate system of correlated velocities, temperatures, and pressures in the atmosphere, it will be possible to approach the mathematical analysis of the structure with some prospect of a satisfactory solution. MARCH, 190G. MONTHLY WEATHER REVIEW. 110 STUDIES ON THE THERMODYNAMICS OF THE ATMOS- PHERE. By Prof. FRANK H. BIUKI.OW. III. APPLICATION OF THE THERMODYNAMIC FORMULAE TO THE NONADIABATIC ATMOSPHERE. THE NONADIABATIC ATMOSPHERE. In the preceding papers of this series it has been shown that in the latitudes of the temperate zones the atmosphere is not arranged in such a way that the thermal gradients conform to the adiabatic rate of change along the vertical, dT - -3- = 9. 867 C. per 1000 meters, but that they depart from that rate, being generally much less. In the tropical zones the few available observations indicate that in the lower strata the temperature gradient exceeds that amount, or is equal to it. Thus O. L. Fassig 1 found the mean of four ascents at Nassau, in June-July, 1904, to be 28.3 C. at the surface and 18.3 C. at 1000 meters, evidently the adiabatic rate. H. Hergesell* found for 16 ascents on the Atlantic, in the region between the African coast, the Canaries, and the Azores, the following temperatures: , Height. T AT Meiers. C. c. 5000 (-10.0) 8.5 4000 1.5 -10.5 3000 9.0 Adiabatic gradient. 9.0 2000 18.0 - 8.4 1000 26.4 + 3.4 23.0 This is an average adiabatic rate from the lower cloud level to 5000 meters, but differs widely from that rate from the sur- face to 1000 meters. He also reports* an adiabatic rate, for the ascensions of 1905, from the surface to 1350 meters, then a zero or even a positive temperature gradient to 3550 meters, above that a rather rapid fall to 13,000 meters, and higher still in the atmosphere a slower rate, indicating an intrusion of warm air. As the result of my kite work from the U. S. S. Caesar, over the North Atlantic Ocean between Hampton Roads and Gib- raltar, during the Spanish Eclipse Expedition, I found the temperatures as follows, for the dates June 24, 26, 28, 29, 30, July 5, and September 22, 1905: Height. Mean of 5 ascents. Julyo. Sept. 22. Mrlfrx. 1000 a 16.9 C. 7.9 C. 15.6 800 17.1 9.3 17.9 600 17.6 11.1 18.5 400 18.5 13.2 14 6 200 19.6 15.6 17.8 22.1 18.0 20.9 These evidently approximate the adiabatic rate on July 5, but depart from it on the other dates, notably on September 22, when the kite ran through a warm stratification, probably blown from the peninsula of Spain over the ocean. These examples show plainly that meteorologists must be prepared to discuss the problems of the circulation of the atmosphere 1 Kite flying in the Tropics. 0. L. Fassig. M. W. R., December, 1903. 1 Sur les ascensions de cerfs- volant executees sur la M6diterranee et sur 1'ocean Atlantique 1904. H. Hergesell. Note in Oomptes Ren- dus, Jan. 30, 1905. 3 Die Erforschung der freien Atmosphare fiber dem Atlantischen Ocean 1905. H. Hergesell. Met. Zeit. November, 1905. whether the thermal vertical gradients are adiabatic or not, and since our common formulae are confined to the adiabatic case, it is an important study to learn how they can be practi- cally modified and rendered flexible enough to meet the actually existing conditions. FIG. 11. I have made an attempt to indicate the probable arrange- ment of the isothermal surfaces in the earth's atmosphere by means of fig. 11. In the tropical zones the adiabatic rate pre- vails up to a certain height, as the dotted line, and above that a slower rate. In the temperate zones there is an intrusion of the adiabatic rate into the lower levels and a mixing area, but generally the temperature-fall is less than the adiabatic rate, resulting in a small gradient near the surface and up to 3000 meters, a more rapid fall to 10,000 meters, and again a slower fall due to a second intrusion of warm air from the Tropics. In the polar zones the temperature gradients are probably small, the air being generally cold, and having only small changes from the surface upward. These suggested iso- thermal lines should be compared with the circulation de- scribed in my paper, MONTHLY WEATHER REVIEW, January, 1904, fig. 19, where the results of this intrusion of the types I and II between the temperate and the tropical zones are ex- plained. The arrows are reproduced on tig. 11, where it is seen that three circuits are proposed for each hemisphere; (1) Ill MONTHLY WEATHER REVIEW. MARCH, 1906 the tropic, circulating anticlockwise; (2) the temperate-tropic, circulating clockwise; and (3) the temperature-polar, circula- ting feebly anticlockwise for the Northern Hemisphere. In the temperate zones the local cyclonic and anticyclonic sys- tems represent the products of the vertical as well as the horizontal mixing of the currents of air derived by transpor- tation from different latitudes. The excess of heat of the Tropics, producing an adiabatic distribution of temperature in their lower strata, works out poleward at the top and at the bottom by irregular streams, which produce a varying system of temperature gradients in the atmosphere of the temperate zones, standing about midway in value, namely, 5.0 C. per 1000 meters, between that prevailing in the Tropics, 9.87 C. per 1000 meters, and that probably prevailing in the polar zones, as 2.0 to 3.0 C. per 1000 meters. The inter- change, of heat between the Tropics and the polar zones is by means of these three more or less irregular circuits, which produce primarily the well-known masses of permanent high or low pressure areas standing over the oceans and continents, and secondarily the rapidly migrating cyclonic gyrations of the temperate zones. We shall make an effort to approach our study of this complex circulation by a transformation of the thermodynamic formulae into forms which will be suitable for computations in the actual atmosphere, as distinguished from an adiabatic but fictitious atmosphere, which has com- monly been discussed by meteorologists. DEVELOPMENT OF THE THEBMODYNAMIC FORMULA. In the formulae derived for discussing the circulation of the atmosphere, it is important that the velocity should be ex- pressed as a function of the temperature in a nonadiabatic atmosphere. It has been generally the custom to treat the velocity as a function of the pressure P, the density p, and the gravity g, but it will be equally valid and more valuable to make it a function of the temperature T, the specific heat at a constant pressure G f , and the gravity g. We must in doing this assume the applicability of two physical laws in the atmos- phere. There has been a difficulty in connecting the results obtained by these two methods, which will be pointed out in this paper and their reconciliation will be explained. I. THE FIBST FOBM OF THE BABOMETBIC FORMULA. The special feature of this formula is that the density p is eliminated by the following process: Assume the Boyle-Gay- Lussac law, P=pBT, and the pressure law, dP=f>gdz, gral mean temperature is, dP (1) Then, -p- RT= gdz Since '*r; (2) dP By definition P^B^,, and * =^f, for the standard conditions, we have, so that, (3) dB gdz, for common logs. * /'o- J o For the hypsometric formula the gravity g is computed from the standard gravity g a by the factors, (1 + ?)= (1 + 0.0026 cos 2^) for latitude, and ('l + 1.25 ^ = (1 + 0.000000196) h for altitude, \ snce <7= In integrating for an atmosphere composed of dry and moist air between the heights z^ and z, the temperature term T, which is variable, is taken as the mean temperature of the air column the moist air is accounted for by the factor, T, = T m , and = ( 1 + 0.3670 ) = (1 + a o). zc We must pass from P * = (l + 1.25 h ~ h '\ to logarithms, P K \ R l log ~ =(1 + .00157) log E " =(l + r y ) log B by adding the fac- tor Finally, -.K= 18400, for 7? =0.760 meter Pm = 13595.8 /> =1.29305 3f=0.43429 Hence, by integration, (4) - fdB *J jj B Pm in the meter-kilogram system. ' 378 = I f* B] = I gdz, because, 0.378 If = K! is computed as a new barometric constant, and T m (1 + 0.378 * B \ = T r , the virtual temperature, then, (6) K t T r log -jj = g m (z z c ) in mechanical units. I have computed the logarithmic tables 91, 92, 93 of the International Cloud Report, 1898, in such a form that the dry air temperature term m, the humidity term ftm, and the gravity term -fin, are kept separate from each other in (7) log B = log H + m mil mr, for the sake of accurate and flexible applications in all possible meteorological computations. Auxiliary tables can be con- structed from these primary tables for any desired applications, by way of shortening the work in special cases, such as in numerous reductions to any selected plane, or in computing the pressures from point to point in the atmosphere, using as arguments the temperatures and humidities observed in bal- loon or kite ascensions. Especially, they can be used to com- pute the dynamical units of force, or work required to pass from point to point, by simply extracting (z z ) from the tables, wifh the temperature, humidity, and pressure as the arguments and multiplying (z z ) by g m , so that (8) /dP . T- *<>' when there is no circulation or velocity term, \ (gdz, with which we began this discussion. II. THE SECOND FOBM OF THE BAROMETRIC FORMULA IN AN ADIABATIC ATMOSPHERE. In formula (108) of my collection in the International Cloud Report the abnormal form for dry air was written: p (T8 m h\ (9) p = ( '^ ) , which is B_ B. (f* \ ft ^ 1 + 0.378 R \, where e is the vapor tension. Hence, the inte- where ^r- is the actual vertical gradient of temperature and MARCH, 1906. MONTHLY WEATHEK EEVIEW. 112 the exponent is undetermined. We acquire from the observa- of the same type which will admit other temperature gradients, in a quasi-adiabatic atmosphere. It has been assumed , that there was no addition or subtraction of heat in the varia- the adiabatic gradient, -rf , and seek to determine the tion of the pressures and temperatures, but as this is only a special case it will be proper to take the general case, where proper value of the exponent m. the quantity of heat dQ is added or subtracted, besides that From my formula (73), in the adiabatic state for dQ = 0, acquired or lost during the expansion and contraction pro- dP cesses. Since in the stratifications of the atmosphere by cur- rents possessing different thermodynamic properties, there is departure from the adiabatic state by the term dQ, we shall resume the full equation for discussion. Fig. 12 will make our treatment clear. we have = C p dT RT -5 , in mechanical units. dP (W) Hence, -p- = dP C 1 dT = ~ > and integrating, (11) =log T (12) Again, -^ = - a (13) (14) dz TS_ dz 7 n = 9 J 71 a R dB T., an E .-, and o Substituting in (10) Tfr. Hence, Zo (15) C j BqPmda 1 C Jgdz = - rgpr-jj- B '=]fdT, T * 0. as before. Hence, we see that - supplies the constants for FIG. 12. The relation of the observed to the adiabatic gradient. Let T = the initial temperature at the height T a = adiabatic temperature at the height z T = observed temperature at the height z the barometric constant K in the adiabatic case only. These Then the adiabatic gradient is a =- -?- = -" - -", substitutions, (12), (13), can be verified by referring to the dz zz a formulae of Table 14. It is well known that the use of the dT T. T and the observed gradient is, a = -j- = CL Z Z ~ ~ Z. is not applicable in the actual atmos- T , , . , ., dT a T T a Let the ratio of these gradients, n = i T = ^5 /p- o Having regard to the adiabatic thermodynamic equation, (18) 0=C p dT a - d ^ B formula D = k_ k-l 'T Z -\T a phere, except to give what is called by von Bezold the poten- tial temperature T , corresponding with (R . T) when reduced to the standard pressure 5 . Making the following substitutions, we observe that the thermal mass passes from to (T a z) B q = p J^ = B, and l - C d T = T m , we have bv tbe obli q ue P at h marked, T to T v in conformity with the // A T n z J formulae iust discussed; it can then be carried from the point ( 16 ) g (z - z ) = R T m log V in Naperian logarithms, and formulae just discussed; it can then be carried from the point T a to the point T at the same level z by changing the tempera- ture through (T T a ), and the addition of the heat Q=C P ( T-T a ). This result can be obtained again by another process. P Assume, - dP = + g p dz, and p = Q=C p (T-T a ), Now we have, from (18), and adding, we obtain, Then, _ dP = , 9 dz Tj D fJ~J 'dz , and by integrating, >-9_ C dz - ~ RJ T~ . R III. THE BAROMETRIC FORMULA IN A NONADIABATIC ATMOSPHERE. In the preceding case it has been assumed that the temper- ature varies witn the height by the adiabatic law, which is, " = '^ = - = 0.0098695 C., so that the temperatures dz U p r k ''TXi^i (19) (20) (21) /oo\ rff) __ (^ fi'p ^^ p ' Since dT a =ndT, we have dQ=G p (dT-dT a )~C p dT-C p ndT. Subtracting this value of dQ from equation (22) we find, Q = C p (T T ) C dP , or in the differential form J p (23) = C p ndT- dP in a quasi-adiabatic form, p [T\k-i of the formula} of section II, of which p = ( yf ) ls " le representative, must have this relation. Now it is known that this formula in the atmosphere does not apply, except in occasional instances, and we, therefore, shall seek a formula which is true in a stratum where n is constant, that is, where dT the gradient - is not changing. dz 1 RT Substituting, = -5- > we have, (24) and 113 MONTHLY WEATHER REVIEW. Groater. Adiabatic. MARCH, 19 / 7' \ '* i * \ *' ^1 / * \ * 9.877 ^o~ \ 2' +4.94 7 3. 46 6. 9 r \ / r \ T+ 9.87y " \jr+ 4.947 > /A 00 *- 1 == \Tj !"=! FIG. 13. The variations of the ratio n 2i. (25) (26) - R T T -p, t so that, g T\Jia The last forms are found as follows: By definition, dT a : a<> = the ratio, dT a '' dT a a g dz Since k C,, Icl ~ = R ' (27) k-l g dz RdT' Our formula, therefore, differs from the adiabatic formula k by the factor n in the exponent with^.^ This ration between the adiabatic and observed gradients, depends upon the amount of heat added or subtracted from an adiabatic atmos- phere to produce the given observed atmosphere within the stratum z where the gradient remains a constant. We can evidently pass from one stratum to an adjoining stratum either continuously by changing n gradually, or discontinuously by changing n abruptly. The ratio n is a new variable to be introduced into the thermodynamic equations in their ap- plication to the atmosphere, so that all the standard thermo- dynamic equations and discussions become available with this simple modification. Such an exposition as was given by M. Margules in his admirable paper, " Uber die Energie der Stiirme," which is limited to the adiabatic case, may be modi- fied in this way and be made very useful in practical meteor- ology. It is rarely the case that computations of T to T, from one level to another, z to z, can be made by general dynamic formulae, but they must usually be observed with balloons and kites. can range between dT. adiabatic gradient Ihe ratio, n = -T = -^ - 3 - 3^ 7 , dT observed gradient the limits n = and n = o> ; for n = 1 the gradient is adia- batic; for n < 1 the cooling is more rapid than in the adiabatic gradient, as in summer afternoons when the ground is super- heated and cumulus clouds are forming; for n > 1 the cooling is less rapid than in the adiabatic gradient, as generally in the temperate and polar zones; the Tropics probably conform to the adiabatic gradient in the lower strata of the atmosphere. IV. CONSTRUCTION OF THE PRIMARY DIFFERENTIAL EQUATION. Under the assumption that n is variable we now differentiate the equation with the variables P, T, n, (28) P P,~\T. Passing to logarithms, P k IT (29)log p -=n /l ,_ 1 loj ' T\ n k-i or for one limit, (30) log P = n-j^^ log T. Differentiate, dP k dT k 11 ( 31 ) p- = n _ --j- -y- -(- ^- ^ log Trfn. Substitute j, = 7J dP f,RT k dT n - Substitute/? - = ( 33 ) :> - = n O p d T + G p T log Tdn. In common logs and to dz, for the vertical direction. j rrt Again, since nC p -j = g , by this substitution we have, < 86 > '-j-, and hence, (36) dP= pgdz + (>C p TlogTdn. We see then that the effect of the change from an adiabatic atmosphere to any other gradient is accomplished by adding the term />C p Tlog Tdn. If it should happen that besides the strictly mechanical velocities thus indicated there is a further expenditure of heat by radiation, it would be necessary to add the special term, Q Q, making, from (33), (37) - =(Q - nC p (T- T t ) It is better to say that the full term C p T log T (n n ) has a radiation part, (Q Q ), and a velocity part, C p T log T(nn Y. The factor n, due to an addition or subtraction of heat other than by adiabatic expansion and contraction, fully accounts for the presence of a nonadiabatic gradient, through the strati- fication of the layers of air due to transportation horizontally from one latitude to another, or generally from one place to another; or else through the addition or subtraction of latent heat in the condensation of aqueous vapor to water, or by the MARCH, 1906. MONTHLY WEATHER REVIEW. 114 vaporization of water to aqueous vapor. In effect, by the practical use of the factor n, we can dispense with the difficult computations which occur in making an allowance for the action of the vapor contents of the atmosphere; or, on the other hand, we can substitute for ?i its equivalent in terms of such other computations as may be found convenient for par- ticular purposes. The corresponding formulae involving P, T, R, t>, and n, in terms of the temperature T, become, the static state there considered to the circulating state here computed. Since we have ( 45 ) g (2 z ) = G p n ( T T ), it follows that (46) i( 9 _ g ' o ) = _C p riog T(n-n ),torQ-Q = 0, so that the circulation can be computed directly in terms of T and (n /? ). This proves that the energy of circulation is derived from the difference of temperature gradients in neigh- boring masses of air, where n ?i is not equal to zero. More- over, since the integral of gdz around a closed curve is zero, (38) J- = (^Y *5 logP-logP.+ nz^jflog T- log T ). s C(gdz) ds = 0, and we have the remaining, * \ -^ o/ u (39) = T. V ^ log ,- log ,. + ^(1087- log r.). (47) -/^.*-/f *+/<*.)*-/*. ,/ o r o o (40) - T. ; log .R = log R + (n - 1) (log T - log T ). ; log p = log p tt + k (log P - log P ). It is evident that R is not constant except in the adiabatic system for n = 1 ; and that only that density determined through the use of n is generally valuable in the atmosphere. V. APPLICATION TO THE GENERAL EQUATIONS OF MOTION. We will now make the connection between this system of equations and the general equations of motion which have been employed in meteorology. From the equations (200) of the Cloud Report, we have, in connection with the differentia- tions of equation (37) along the axes x, y, z, for the acceleration, dQ dT dx f dx p - = ~ + sin (2 + w v) w + cos 6 (2 w + v ) u , 40 \ P y "* dQ dT dn = dy" '*dy~ Jf g dy 13P dw U* . Multiplying by dx, dy, dz, the equations for work are, QP uwdx = udu cos (2 u> + v) vdx -) = &QC p nd T G p T log Tdn = vdv + sin 8 (2 u> -\- v) wdy -f cos 6 (2< -f v) udy = fjQ_ C p ndT C p T log Tdn = wdw sin (2 to -f v) vdz dz + gdz = d Q _ C p nd T C v Tlog Tdn. Since, by substituting vdx = udy, wdx = udz and wdy = vdz, the terms in (2 u> + v), the angular velocity of the earth and the atmosphere relative to it, disappear in the summation, they represent a deflecting force at right-angles to the direc- tion of motion at the velocity q, which does not modify the circulation but only the path of motion. The integral, there- fore, becomes between two places, (44) /'o P : f This is the equation employed by Bjerknes in his discussion of the circulation of the atmosphere, and is applicable only in closed curves, along all points of which P, p, q, or qdq must be known by observations. The difficulty of securing such ob- served data simultaneously along the circuit at a given time is so great that this special case of the general equation will seldom be serviceable. In ordinary meteorology it is required to integrate between the two points, as in the same horizontal plane, or in a vertical direction. Since the term (9* q' ) is expressed in mechanical measures and represents work done, then it may be taken as equivalent to \ ()> so that (48) q'= g z', and q'= 2 g z'. The circulation is therefore always equivalent to a falling velocity through the height z', which may be computed. Furthermore, since Q Q is also given in mechanical units, it may be taken as equivalent to (49) Q- Q =g (z"-z".) t so that (50) Q=gz" and the stored up energy of radiation is equivalent to a verti- cal work. It follows from these considerations that we obtain It is noted that the term for the circulation \ (q' q' a ) must be added to the equations of sections I, II, III, IV, to pass from (51) ,dn dq _ d Q _ dx dx d 3= d Q-C. > n u * -C P T log 2"t" , in longitude. dy dy dy dy C p T\og T$!L in vertical. dT G^Tloe T , in latitude. dx ,dn dq = dQ 1 j j -, Since P= ft[> m g , we obtain in a stratum of mean p, (52) = < w- Q ' } - Gp n ( T ~ T } - CpT log T ~ n } I J It is readily perceived that the introduction of the factor n and the correlation of the pressure, velocity, gravity, radiation, specific heat, temperature, and gradient, in this double equa- tion leads to an innumerable number of special combinations, taken in connection with the equations of thermodynamics. These embrace the first and second laws of thermodynamics, cyclic processes, the entropy S, the inner energy U, the ther- modynamic potentials (F. o Bo 287. 0334 2. 45793 2870334. 6. 4579S 29. 2712 1. 46644 1716. 43 3. 23463 _ J? Cp To 273. 2. 43616 273. 2. 43616 273. 2. 43616 491. 4 2. 69144 00 f,dh dT Po A 1. 29305 0. 11162 101323. 5 5. 00571 0.00129305 7.11162 101323.5 6.00571 0.0012935 7.11162 101.3235 2.00571 0. 080529 8. 90595 67923. 5 4. 83202 *-*/* Po 10332. 8 4. 01422 1033.28 3.01422 10. 3328 1. 01422 2111.23 a 32454 C^ k lf> ff It 3. 461645 0. 53927 3. 461545 0. 53927 3. 461545 0. 53927 3. 461545 0. 5392 v k 1 TO tcl - * R t, 7991. 04 3. 90260 799104. 5. 90260 7991. 04 3. 90260 26217. 3 4. 4185 t-1 ffo 9. 8060 0. 99149 980. 60 2. 99149 9. 8060 0. 99149 32. 172 1. 50741 SoPo t 1 1 | 1 F 2?3 7.56384 273 7 ' 56384 273 490 7 ' 3085< = go dh <4 993. 5787 2. 99720 9935787. 6. 99720 993. 5787 2. 99720 5941. 57 3. 77391 *0=0. lo 7991. 04 a 90260 799104. 5. 90260 7991. 04 3. 90260 26217.8 4.4185 -B e 0o 9. 8060 0. 99149 980. 60 2. 99149 9. 8060 0. 99149 32. 172 1. 5074 ^" 90 1 1 1 1 1 - A 278 7 ' 56384 273 7.56384 273 7 ' 86384 490 7 "' i085 PO r. -RO 287. 0334 2. 45793 2870334. 6. 45793 287. 0334 2. 45793 1716.43 3.2346. C^-R, c. 706. 5453 2. 84914 7065453. 6. 84914 706. 5453 2. 84914 4225. 14 3. 6258 p k 1.4062486 0.14806 1. 4062486 0. 14806 1. 4062486 0. 14806 1. 4062486 0. 14801 c. t-1 0. 40C2486 9. 60879 0. 4062486 9. 60879 0. 4062486 9. 60879 0.4062486 9.6087 t t^l 3. 461545 0. 53927 a 461M5 0. 53927 a 461545 0. 53927 3.461545 0.5392 1 l-l 2. 461545 0. 39121 2.461545 0.39121 2. 461545 0. 39121 2. 461545 0. 3912 dT go dk C, dT ~ dh 0.0098695 7.99429 0. 000098695 5. 99429 0. 0098695 7. 99429 0. 0054147 7. 7335 1 ^ go 1 4185.57 a 621 75 41855700. 7. 62175 4185. 57 3. 62175 25027. 7 4. 3984 A* A 2, A - A A m 0.0002389 6.37829 2.389X10" 8 2.37829 0.0002389 6.37829 0. 00003995 5. 6015 0o e a 968 0. 59851 I'r. Th. U. F 1000. 3.00000 100 2.00000 1 0.00000 367. 8 2. 56561 A 0. 002343 7. 36978 2.343X10~ 6 5.36978 0. 002343 7. 36978 0.0012855 7.10901 1 A 426. 837 2. 63022 42683. 7 4. 63022 426. 837 2. 63022 777. 9 2. 8909 MARCH, 1906. MONTHLY WEATHER REVIEW. 116 P = pressure in units of force, g a . p a = the weight of a given mass of atmosphere, p m B n = /> Z . C p = the specific heat at constant pressure. C v = the specific heat at constant volume. G> j rp -jj- = the temperature fall per unit height in adiabatic state. Ufl r = the mechanical equivalent of heat, 426.8 and 777.9. A -^ = the factor to change mechanical units to heat units. A F = the factor connecting the thermal gradient and P a . tt = the number of British thermal units in 1 kilogram- degree. VII. THE THERMODYNAMIC CONSTANTS FOR THE SUN. There is much difficulty in passing from the thermodynamic conditions on the earth to the corresponding thermodynamic conditions on the sun. I have already approached this sub- ject from the side of radiation ill my " Eclipse Meteorology and Allied Problems," 1902, and from the method of Nipher's Formulce, in my studies on the "Circulation of the Atmos- pheres of the Sun and of the Earth," 1904. I shall briefly present the same subject as the immediate development of the fundamental formulae introduced in this paper. It is not so difficult to produce a self-consistent system of quantities as it is to find one which conforms to the actual physical state of the sun, and I conceive that it is proper to discuss this sub- ject in several ways. Specific heat. (53) From the preceding formula;, we have, dTgF F F dz _g a _ "(7. " Hence, (54) ti" Since f>,,J> n = /',/ is a given mass, and F is constant for a given system of units, it follows that C p is proportional to the square of the gravity. Taking the force of gravity on the sun, ( 55 ) (g) , un = g a x G = 9.806 x 28.028 = 274.843 it follows that the specific heat on the sun is (56) (Op) sun = Cp x G s = 993.5787 x (28.028)' = 780524. Adiabatic rate of temperature-fall. ( 57 ) For the earth - - -3- = ^ - = 9.8695 per 1000 meters. dz op / of the earth's atmosphere, then the proper modification of the preceding quantities can be readily com- puted from terrestrial data. Specific heat at constant volume. For the earth, G v = G p R, and hence, for the sun, (72) C V G' = CpG' RG> (73) (C v ) sun = C V G* = 706.5453 x (28.028)' = 555040. ( 74) Finally, k = = 1.4062486, as a check. This system throws the entire emphasis upon a change of gravity depending upon the mass of the central body, rather than upon the change of physical conditions implied in alter- ing the ratio of the specific heats k. Since the temperature of the photosphere may in this way be taken as about 7652, and the temperature gradient 0.32862 per 1000 meters, it follows that the effective temperature of radiation as determined by bolometer measures, 6100, will be reached at the height of 4418 kilometers, or 2745 miles above the surface of the photo- sphere. This change of 1552 may be sufficient to meet the requirements of the spectroscopic observations in regard to the absorption and reversal of the spectrum lines. The gra- dient, 0.32862 per 1000 meters, is 28.028 times greater than that obtained by my other methods, the difference arising from the different distribution of the gravity factor G, which seems to be fully accounted for in these formulae. JUNE, 1906. MONTHLY WEATHER REVIEW. 265 STUDIES ON THE THERMODYNAMICS OP THE ATMOS- PHERE. By Prof. FRANK H. BIGELOW. IV. NUMERICAL COMPUTATIONS IN THE VEKTICAL OKDINATE. THREE GENERAL THEORIES REGARDING THE FORMATION OP CYCLONES AND ANTICYCLONES. There seem to be only three important general theories regarding the formation of cyclones and anticyclones in the earth's atmosphere, which may be referred to those authors who have been conspicuously associated with their mathe- matical developments: (1) Ferrel's cold center and warm center cyclones and anticyclones; (2) Oberbeck's symmetrical central cyclones and anticyclones; and (3) Bigelow's asymmetric cy- clones and anticyclones. In my International Cloud Report, 1898, I reviewed the mathematical analyses of the first and second theories, and gave my reasons for thinking that they are inconsistent with the air currents as mapped out by the cloud observations, as well as with the distribution of temperature found in the lower strata. These theories start with the sys- tems of isobars which near the surface are distributed sym- metrically about a central axis, and they assume that the tem- peratures are similarly arranged, which is, however, not the case, as we know. The two central systems have their origin in the fact that the second equation of motion can be dis- cussed in two ways. Thus, in the case of no friction, k = 0, the equation dv = -r: + (2 n cos + v,) u + k v, can be integrated by introducing the idea of a boundary cyl- inder about the system at the radial distance zw o , whence is derived, /2w' \ u = I , Ijrttn COS 6, \ o / which is the tangential velocity at the distance w. This is Ferrel's method and several difficulties regarding it are mentioned on page 615 of the Cloud Report. The second equation of motion can be given another form retaining the friction term, where A = 2 n cos 6, so that, do uv from which are derived two solutions, c First X Second - H c k m These form the basis of the theory developed by Guldberg and Mohn, Sprung, Oberbeck, Pockels, and others. My specific objections are summarized on page 623 of the Cloud Report. The construction of a better theory was at that time very diffi- cult, for two reasons, the first, that it involved breaking away from the large mass of current literature in meteorology, and that it introduced many new ideas concerning the general and the local circulations of the atmosphere, the two being inti- mately bound up together; the second, due to the lack of definite pressure and temperature observations in the higher strata of the atmosphere. In the course of chapters 8 and 11 of the Cloud Report the leading ideas regarding the asymmetric cyclone and anticyclone were sketched out, and a fairly clear idea was given of the probable truth regarding the formation of these circulating structures. Since that time the Weather Bureau has secured daily pressure maps for the United States on the three planes, sea level, 3,500 feet, and 10,000 feet, throughout an entire year, 1903. This valuable material has been carefully studied, and a report presented on the subject, with a summary of the results in the MONTHLY WEATHER RE- VIEW, May, 1904. The recent publications of the temperature observations, made during balloon and kite ascensions in Europe and America, have in some degree supplied this de- ficiency, and we are, therefore, now trying to discuss more definitely the entire subject by means of these several data velocity, temperature, and pressure than has been possible heretofore. In the preceding papers of this series we have given the temperature data and the thermodynamic formulae, and in this paper we shall confine our attention to formula (44) in the vertical ordinate. COMPARISON OF THE NUMERICAL RESULTS OF COMPUTATIONS BY FORMULA (38) AND THE GENERAL BAROMETRIC FORMULA OF THE CLOUD RE- PORT (59). Since we have introduced a new system of formulae for the computation of the pressures, densities, and the gas factors, from the temperatures, through the use of the ratio n, the ratio of the adiabatic temperature gradient to the observed gradient, it will be desirable to compare the numerical results by some examples, showing the relation of the thermodynamic formulae to the well-known barometric formulae. Formula (59) can be written as follows : log |. In our new thermodynamic formula we have made no at- tempt to refine it by introducing corrections due to the vapor term, lt!_ - , the gravity term, ? , nor the land-mass term, B 9 ( 1 + - - ). It is evident that these will require a \ .' small change in the value of n, and the subject is worth an investigation, but for our preliminary studies of cyclones and anticyclones these refinements have been omitted. We retain, then, simply z z =18400 (1 + .00367 0) log 3> ......... (II) B where 8 = 273, the mean departure of the tern- perature of the air column from zero centigrade. This is to be compared with the formula, '- P fi '-log T ) = log = log 3, (I) Jr n and it will be sufficient to show that 18400+67.5 *-!*.), The examples are taken at random with sufficient range to test the formulae severely. The observed gradient is found r ITI , per 1000 meters. The computation is given from a = 2 z in full as an example of the numerical quantities involved. While the agreement in the logarithms is not perfect, the dif- ferences I II are small for so great ranges of temperature and height when translated into millimeters of mercury. If 5 o = 760.00 mm. in the fifth example, for the difference 0.00084 the value of B is 294.00 and 294.57, respectively. As my only purpose is to illustrate the numerical validity of the n formula, it will not be necessary to inquire further into the causes of the small differences between I and II. COMPUTATION OF MEAN VALUES OF P a , />, E^, FROM T O ON THE 1000-METER LEVELS. In making an application of the formula (44), 266 MONTHLY WEATHER REVIEW. JUNE, 1906 TABLE 15. Comparison of the formula. Formula l=n^ r - 1 (log 7- log TO) T To T-T 271.5 275.8 4.3 253.5 275.8 22.3 259.0 268.7 9.7 219.6 253.5 33.9 233.2 271.5 :i8.:i 290.0 273.0 +17.0 300.0 280.0 +20.0 275.0 300.0 25.0 280.0 310.0 -30.0 Adlabatlc. Observed. -9. 8695 4.30 4.46 1.85 -6.78 -5.47 +4.25 +4.00 5.00 5.00 log n k/k 1 -0. 36082 0.53927 -0.344% -0.30855 -0. 16306 0.25630 0.36590 0. 39223 0. 29532 -0. 29532 io g r log T log r log T log (log r log T ) 2. 43377 2. 44059 0. 00682 -7. 83378 2.40398 2.44059 0. 03661 -8. 56360 2.41330 2. 42927 0.01596 -8.20330 2. 34163 2. 40398 -0.06235 8. 79484 2. 36773 2. 43377 0.06604 8. 81981 2.46240 2.4I161G 0. 02624 8. 418% 2.47712 2.44716 0.029% 8. 47654 2.43933 2.47712 0. 08779 -8.57738 2. 44716 2. 49136 0. 04420 -8.64542 log I 8.73387 0.05418 9. 44783 0. 28043 9.05112 0. 11249 9.49717 0.31417 9. 61533 0. 41246 9.32413 0. 21092 9.40804 0.25588 9.41197 0. 25821 9.48001 0.30200 Formula II Z ~ Z 18400 + 67.5 r-L-To Z-Z 67.59 K JT+67.59 273.65 +0.65 1000 + 44 18400 18444 264.65 8.35 5000 564 17836 263.85 9.15 2000 618 17782 236.55 36.45 5000 - 2460 15940 252.35 -20.65 7000 1394 17006 281.5 + 8.5 4000 + 674 18974 290.0 +17.0 5000 + 1148 19548 287.6 +14.5 5000 + 979 19379 295.0 +22.0 6000 + 1488 19886 n 0.05422 0.28034 0.11247 0. 31368 0.41162 0. 21082 0.25578 0.25801 0.30173 i ii .00004 +.00009 +.00002 + .00049 +.00084 +.00010 +.00010 +.00020 +.00027 to the earth's atmosphere, it is evident that we must first com- pute the values of P , f> , -ff , corresponding to T as observed in the air while it is undisturbed by the local cyclonic and anticyclonic circulations. The observations of temperature were actually made in the midst of the prevailing local disturb- ances, but the average temperature of the air on the 1000- meter levels was found by taking the mean temperatures of the eight sectors, four in the high areas and four in the low areas, as in Tables 10 and 11. (See MONTHLY WEATHER REVIEW, February, 1906, Vol. XXXIV, p. 75.) Thus, Table 11 gives the adopted mean temperatures and gradients on the 1000-meter levels for American and European cyclones and anticyclones, and these data are used in the following computations. The values of n are found on dividing 9.8695 by the gradient per 1000 meters, assuming that the gradient is a constant be- tween the two levels. In a later section we shall compute, also, the values of n on the 1000-meter levels themselves, taking as the gradients those found on fig. 8 at the points indicated. We use the mean gradient between two levels in computing P , |0 , jR , and then the gradient at a given level to compute P, p, E, the abnormal values of pressure, density, and gas factor produced by the local cyclonic and anticyclonic dis- turbances. Having found n from level to level the following formula are applied in succession, including 41 as a check: (38) (39) (40) (41) log P = log P + log p = log /> + k kL (log T- log r.). k1 (log T- log log R = log*. + (n 1) (log T- log log p = log Pt + I (log P - log P.). A special point should be noted in connection with the symbols. P is the pressure on one level, as the 1000-meter level, and then P is the pressure on another level, as the 2000- meter level, corresponding with T and T, respectively. In this way a succession of values of P is found in the several strata of the undisturbed atmosphere, applying to the gen- eral circulation only. In studying the pressure variations in cyclones and anticyclones these P-pressures become P in formula (44), the P-values of that formula referring to dis- turbances within the local circulation on a given level. I have preferred to make this explanation rather than complicate the formulae with additional symbols for all contingencies. In computing the tables following, the pressure P = 101323, for B = 760 mm., was taken as the initial value. The initial value of the density on the sea-level plane was computed from, where R = 287.0334, and T is the value from Table 11, T = 275.8, 277.3, 290.2, 287.1. While it is true that the atmosphere is seldom in the state represented by these tables, yet it fluc- tuates about these mean values, just as it does about the mean pressure, temperature, and density at sea level, and it is con- venient to have reference values from which to conduct our discussions. TABLE 1G. Computed values of the. ratio n between successive 1000-meter levels. Height in meters. American. European. Winter. Summer. Winter. .Summer. 16000 3.037 2.820 3.037 3.589 14000 4.886 2.820 4.386 3.037 12000 2.078 1.716 2.078 1.645 10000 1.518 1.473 1.518 1.410 9000 1.390 1.410 1.410 1.316 8000 1.410 1.518 1.410 1.410 7000 1. 451 1.678 1.410 1.518 6000 1.518 1.518 1.410 1.618 5000 1.794 1.410 1.673 1.702 4000 1.974 1.316 1.794 1.936 3000 2.100 1.653 2.014 2.295 2000 3.525 1.828 2.243 1.673 1000 2.295 1.828 2.467 1.645 000 JUNE, 1906. MONTHLY WEATHER KEVTEW. 267 _ q Since n = T T , the variation of n in Table 16 is a func- J J o tion of AT. Where AT is large, n is small, and inversely. Hence, in the lower and the higher levels n is larger than in the middle levels, the change of temperature being slower below and above, for the reasons already given. In the middle levels w = 1.5 approximately, and it may become twice as great in higher or lower strata. Tables 17, 18, 19, and 20 contain Computed mean values of the pressure, density, and gas factor from the tem- perature at several elevations. TABLE 19. AMERICAN WINTER. the values of P , T f on the several levels, and their loga- rithms which are useful in computations. Since P a =g p m B m , we have B>1 ~ <7o P m ~ 9 - 806 X 13595.8 : The pressure is higher in summer than in winter on the same level; the density is higher in winter than in summer; the gas factor is higher in summer than in winter; and the tem- perature is higher in summer than in winter, the difference diminishing in the upper levels. The gas factor is not a con- stant in any system except the adiabatic, where n = 1. COLLECTION OF THE DATA SHOWING THE DISTRIBUTION OF THE DISTURB- ANCES ON THE 1000-METER LEVELS. We will collect the data in a form suitable for the discus- sion of the distribution of the energy in cyclones and anti- cyclones, leaving the reader to make his own inferences by an examination of the tables in their relation to one another. Computed mean values of the pressure, density, and gas factor from the tem- perature at several elevations. TABLE 17. -EUROPEAN WINTER. Height in meters. Po logPo PO logPo *o log R To log TO 16000 9547 3. 97987 0.23733 9.37535 203. 59 2.80874 197.6 2. 29579 14000 13414 4.12756 0.30226 9.48038 217.45 2.33736 2011 2.30984 12000 18679 4. 27135 0.38250 9.58263 234.12 2. 36943 208.6 2.31931 10000 25735 4.41052 0.48040 9.68160 245.63 2. 39029 218.1 2.33866 9000 30097 4.47753 0.53610 9. 72925 249. 40 2.39688 224.6 2.&5141 8000 34881 4.54259 0.59636 9. 77551 252.55 2.40235 231.6 2.36474 7000 40335 4.60569 0. 66127 9.82038 255.65 2.40765 238.6 2. 37767 6000 46450 4. 66699 0. 73108 9.86397 258.70 2. 41280 245.6 2.39023 5000 53276 4.72653 0.80595 9.90631 261.70 2.41780 252.6 2.40243 4000 60897 4. 78461 0.88636 9. 94761 265.80 2. 42455 258.5 2. 41246 3000 69403 4.84138 0. 99536 9. 98798 270. 28 2. 43181 264.0 2.42160 2000 78902 4. 89709 1. 06559 0. 02759 275.37 2. 43991 268.9 2. 42959 1000 89502 4. 95183 1.16551 0. 06652 280.98 2. 44867 27&3 2. 43664 000 101323 5. 00571 1.27300 0.10483 287. 03 2. 45793 277.3 2. 44295 TABLE 18. EUROPEAN SUMMER. 16000 10187 4.00804 0.24003 9.38027 210.20 2. 32262 201.9 2. 30514 14000 14224 4. 15302 0.30434 9.48336 225.34 2.35283 207.4 2. 31681 12000 19674 4. 29388 0. 38329 9.58353 239.% 2.38013 213.9 2.33021 10000 26816 4. 42888 0. 47811 9. 67953 248.55 2. 39542 225.9 2.35392 9000 31156 4. 49355 0.53152 9. 72552 251.68 2. 40085 232.9 2.36717 8000 35994 4. 55623 0.58896 9.77009 254.21 2.40520 240.4 2.38093 7000 41408 4. 61709 0.65069 9. 81337 267.22 2.41031 247.4 2. 39340 6000 47453 4. 67627 0. 71690 9. 85546 260.70 2. 41614 253.9 2. 40466 5000 54201 4.73401 0.78802 9. 89654 264.55 2. 42241 260.0 2. 41497 4000 61730 4. 79050 0.86440 9. 93671 268.68 2.42924 265.8 2.42455 8000 70118 4. 84583 0.94638 9. 97606 273.50 2.436% 270.9 3. 43281 2000 79464 4. 90017 1.03443 0. 01470 279.14 2. 44582 275.2 2. 43965 1000 89846 4.95350 1. 12882 0.05262 283.15 2. 45202 281.1 2.44886 000 101323 5.00571 1.22956 0.08975 287.03 2.45793 287.1 2.45803 Height in meters. Po l"g.Po Po logPo /?(, log *o TO log To 16000 9651 3.98456 0.24045 9. 38103 201.69 2.30446 199.1 2 29907 14000 13527 4.13120 0.30571 9.48531 215.22 2 ^289 205.6 2.31302 12000 18797 4.27408 0. 38630 9. 58691 231.60 2.36497 210. 1 2 32243 10000 . . . 25833 4.41217 0.48430 9.68511 242.91 2.38544 219.6 2 34163 9000 8000 30114 34915 4.47876 4.54338 0. 54010 0. 60037 9. 73247 9.77842 246.61 249.60 2.39200 2.39724 226.1 233.2 2.35430 2. 36773 7000 . . 40369 4.60605 0. 66524 9.82298 252.64 2.40250 240.2 2.38057 6000 . 46450 4. 66699 9. 73505 9. 86632 255.84 2. 40797 247.0 2. 39270 5000 53251 4.72633 0.81006 9.90852 259. 31 2.41382 253.5 2.40399 4000 60836 4. 78416 0. 89052 9. 94964 263.76 2.42121 259.0 2.41330 3000 69321 4.84087 0. 97718 9.98997 268.71 2. 42929 264.0 2, 42160 2000 78817 4.89662 1. 07059 0. 02962 273. 99 2.43773 268.7 2.42927 1000 89440 4. 95163 1. 17127 0.06866 281.25 2.44910 271.5 2. 43377 000 101323 5.0D571 1.27994 0.10719 287.03 2.45793 275.8 2.44059 TABLE 20.-AMERICAN SUMMER. 16000 10232 4.00995 0. 23822 9.37698 213.88 2. 33017 200.8 2. 30276 14000 14298 4. 15529 0.30223 9.48033 227.65 2.35727 207.8 2.31765 12000 19754 4.29565 0.38031 9. 58014 241.79 2.38344 214.8 2.33203 10000 26929 4.43022 0. 47407 9.67584 250.99 2.39966 226.3 2.35468 9000 31252 4. 49488 0.52701 9. 72182 254.48 2.40566 233.0 2.36736 8000 36107 4.55759 0.58401 9.76642 257.59 2.41093 240.0 2.38021 7500 41554 4.61861 0.64537 9.80981 261.18 2. 41694 246.5 2. 39182 6000 47652 4. 67808 0. 71138 9.85210 265.37 2.42385 252.4 2.40209 5000 54449 4.73599 0. 78213 9. 89328 268.88 2.42956 258.9 2. 41313 4000 62023 4. 79255 0.85802 9.93350 271.84 2.43431 265.9 2.42472 3000 70402 4. 84758 0. 93892 9.97263 274. 24 2. 43813 273.4 2.43680 2000 79712 4. 90152 1.02563 0.01099 278. 16 2.44429 279.4 2.44623 1000 69968 4.95409 1.11782 0. 04837 282.60 2.45117 284.8 2.45454 000 101323 5.00571 1. 21642 0.08508 287.03 2. 45793 290.2 2.46270 VALUES OF THE TEMPERATURES T, T O AND T T O - The temperature data given in Tables 12 and 13 (see MONTHLY WEATHER REVIEW, February, 1906, Vol. XXXTV, pp. 76, 77) have been reproduced in Tables 21, 22, 23, and 24, in a form more convenient for carrying on the computations de- pending upon them. T is the temperature in the several sec- tors; T is the mean temperature on the same level computed from the eight sectors of the correlative high and low areas; T T a is the departure of the disturbed area from the assumed mean undisturbed atmosphere without any cyclonic and anti- cyclonic action, and is shown on figs. 5 and 6. VALUES OF THE RATIOS n, n, AND (n nj. These are the ratios on the several 1000-meter levels, and they are used for computing the energy of the local disturb- ances on a given plane, rather than in reducing the elements from one plane to another, as was required for constructing Tables 17, 18, 19, and 20. The values of n on the several levels for each sector were scaled from fig. 8, for which purpose it was constructed, and from them the values of n in Tables 25, 26, 27, and 28 were computed. The values of n , the mean ratio, were found by taking the means of the eight values of 7i in the sectors on the given plane. Tables 25, 26, 27, and 28 contain the data n, n a , and the differences n n a , the dis tribution of n n being given in fig. 14 on the several -9.8695. levels. Since = ,_ , is a certain average gradient, if n 268 MONTHLY WEATHER REVIEW. JUNE, 1906 Values of T, T , T T derived from Tables 1, IS. TABLE 21. WINTEK HIGH AREAS. Distribution of the values of n, n c , n n . TABLE 25. WINTER HIGH AREAS. 10000 c a c. c. -55.2 56.2 68.7 -52.2 48.2 60.2 46.8 -48. 3 40.6 48.6 39.8 -37.6 82.9 37.1 32.1 29.6 26.8 80.7 25.6 -21.7 18.0 24.6 20.0 14.8 11.5 18.6 15.5 8.7 6.8 18.0 10.3 4.2 2.2 8.0 6.9 0.2 + 0.9 -8.7 - 8.1 + 2.7 + 4. 2 + 1. 5 + 0. 1 + 8. 6 C. -54.2 47.7 40.6 83.6 26.7 -20.0 14.3 - 9.0 - 4.2 0.6 + 8.6 C. C C C. 10000 1.607 1.778 1.574 1.835 1.339 1.537 1.867 1.319 1.282 1.495 1.348 1.246 1.283 1.523 1.404 1.234 1.323 1.567 1.613 1.272 1.420 1.623 2.100 1.430 1.747 1.725 2.317 1.974 2.274 1.894 2.804 2.443 2.920 2.285 2.937 3.056 3.184 2.179 2.903 3.439 2.991 1.769 2.467 3.290 1.600 1.444 1.396 1.403 1.459 1.602 1.819 2.105 2.554 2.661 2.403 -.093 +.178 .026 .065 -.105 +.093 .077 -.125 .114 +.099 .048 .180 -.120 +.120 +.001 .169 -,136 +.108 +.154 .187 .182 +.021 +.498 .172 -.072 .094 +.498 +.156 t.169 .211 +.699 +.338 + 866 269 + 383 + 502 1.0 -2.0 +0.5 +2.0 9000 . 9000 -0.5 2.5 +0.9 +2.4 8000.. 8000 0.0 3.0 +1.3 +3.0 7000 7000 +0.7 -8.5 +1.8 +4.0 0000 6000 +1.4 -4.0 +1.2 +5.0 5000 5000 +2.0 -4.6 0.0 +5.7 4000 4000 +2.8 4.3 -1.2 +5.6 3000 8000 . +2.7 4.0 1.3 +4.8 2000 2000 +2.0 8.8 2.7 +4.0 1000 +.528 .482 +.242 +.778 +.688 -.634 +.064 +.887 1000 + 1.5 3.1 2.5 +3.3 000 000 +0.6 -2.1 3.6 +2.0 TABLE 26.-WINTER LOW AREAS. TABLE 22.-WINTER LOW AREAS. 82.0 51.2 66.2 67.0 46.7 44.7 49.9 80.7 39.1 37.9 48.1 48.9 82.4 81.1 6.2 37.1 25.6 24.5 29.6 30.6 18.8 18.0 22.8 24.0 12.6 12.5 16.7 17.8 6.8 7.6 11.3 12.4 1.8 8.0 6.5 7.2 + 1. 1 + 0. 7 + 0. 5 2. 8 + 4.5 +4.7 +6. 5 +1.9 54.2 47.7 -40.6 33.6 -26.7 -20.0 -14.8 - 9.0 4.2 -0.6 + 3.6 1.687 1.547 1.613 1.659 1.528 1.458 1.600 1.500 1.471 1.447 1.437 1.443 1.430 1.471 1.430 1.447 1.426 1.518 1.458 1.498 1.489 1.629 1.564 1.562 1.629 1.766 1.707 1.690 1.848 1.974 1.725 1.876 2.903 2.518 1.667 2.150 8.123 2.611 1.637 2.213 2.632 2.367 1.643 2.065 1.600 1.444 1.3% 1.403 1.459 1.602 1.819 2.105 2.554 2.661 2.403 -.013 .053 + .013 +.059 +.084 +.014 + .056 +.056 +.075 +.051 + .041 +.047 + .027 +.068 + .027 +.044 .033 +.059 .001 +.039 -.113 +.027 - .038 .040 -.190 .053 .112 .129 .257 -.181 .380 .229 +.847 .0.% .887 .404 +.462 .050 1.024 .448 +.229 .036 . 758 . 338 9000 9000 +2.0 +3.0 2.2 3.0 7000 7000 +1.2 +2.5 -2.6 3.5 4000 4000 +1.8 +1.8 2.4 3.5 TABLE 23.-SUMMER HIGH AREAS. TABLE 27.-SUMMER HIGH AREAS. 10000 -48.1 48.4 46.7 46.9 40.7 42.1 39.6 38.6 32.8 85.3 -31.8 81.0 26.4 29.1 24.6 24.1 18.6 23.2 18.7 17.5 11.7 -16.8 18.7 10.8 4.9 10.4 7.7 r 5.0 + 1.0 3.8 - 1.8 + 1.1 + 5.7 +2.1 +8.9 +5.9 +10.5 +8.1 +8.8 +11.0 +15.9 +14.1 +13.7 +16.1 46.9 -40.1 -32.8 -26.1 -19.9 -18.6 7.2 0.9 + 4.3 +10.0 + 15.7 j 2 15 +0 2 +10 10000 1.371 1.623 1.430 1.886 1.233 1.482 1.307 1.328 1.265 1.491 1.309 1.360 1.392 1.662 1.537 1.469 1.443 1.667 1.873 1.678 1.447 1.589 1.905 1.725 1.542 1.505 1.577 1.631 l.!)47 1.569 1.645 1.818 2.113 1.681 1.909 1.974 1.905 1.637 1.974 1.909 1.788 1.645 2.014 1.902 1.603 1.382 1.396 1.539 1.628 1.607 1.563 1.728 1.869 1.753 1.688 .132 +.120 .073 .117 .149 +.100 .076 .054 8000 00 25 +10 +18 8000 .131 +.095 .087 -.036 .147 +.123 .002 .070 6000 +14 83 +12 +2 4 6000 .185 +.039 +.245 +.045 -.160 .018 +.298 +.118 -.021 -.058 +.014 +.068 +.219 .159 .083 +.090 +.244 .188 +.040 +.105 +.152 .116 +.221 +.166 +.097 .043 +.326 +.214 4000 . . +2.3 -3.2 0.5 +2.2 4 OW TABLE 24. SUMMER LOW AREAS. TABLE 28. SUMMER LOW AREAS. 10000 42.1 45.7 48.9 49.4 -36.5 -88.9 -42.4 -42.9 28.4 31.8 -36.5 35.8 21.9 24.9 -29.1 29.3 15.9 18.5 22.9 23.4 10.8 12.2 16.3 16.6 4.4 5.9 - 9.2 9.8 + 1.6 0.6 1.9 3.5 + 6.2 + 4.5 +4.3 + 1.9 + 11.2 +10.4 +11.0 + 7.8 + 16.5 +16.5 +18.5 +14.2 -46.9 40.1 32 8 +4.8 +1.2 2.0 2 5 10000 1.607 1.500 1.572 1.535 1.443 1.394 1.445 1.422 1.439 1.398 1.478 1.426 1.559 1.516 1.597 1.582 1.725 1.574 1.564 1.507 1.744 1.569 1.430 1.447 1.684 1.719 1.352 1.491 1.769 1.943 1.386 1.744 2.109 1.855 1.557 1.750 1 974 1.673 1.384 1.572 1.781 1.597 1.279 1.502 1.508 1.382 1.396 1.539 1.628 1.607 1.563 1.728 1. 80!) 1.753 1.688 +.104 .008 +.069 +.032 +.061 +.012 +.063 +.040 +.043 +.002 +.082 +.030 +.020 .023 +.058 +.043 +.097 -.054 .064 .121 + 137 038 177 160 9000 + 4.6 +1.2 2.3 28 9000 8000 +4.4 +10 27 30 8000 7000 26.1 19.9 13.6 7.2 - 0.9 + 4.3 +10.0 + 15.7 +4.2 +1.2 30 32 7000. 6000 +4 0+14 30 35 6000 {000 +8 3 +14 27 3 5000 4000 +2 8 +13 20 26 4000 +.121 +.156 .211 -.072 +.041 +.215 .342 +.016 + .440 .014 .312 -.119 + .221 .080 .369 .181 + .093 .091 .409 .186 8000 +2 4 +0 3 10 26 3000 2000 + 19 +0 2 00 24 2000 1000. .... 000 +0.8 +0 8 +2.8 1 5 000 JUNE, 1906. Distribution of the heights z V (z s ts )g = C f n v (TT <> .) TABLE 29. WINTER HIGH AREAS. MONTHLY WEATHER REVIEW. Distribution of the velocities q, q , J (g* g *). TABLE 33. -WINTER HIGH AREAS. 269 Height in meters. N. Cpn.,1 E. T TO) S. W. a N. z E. *o S. W. 10000 1590 3180 795 3180 9.786 +162 +325 81 325 9000 717 3587 1291 3443 9.789 + 73 +366 132 352 8000 4161 1803 -4161 9.791 +425 184 423 7000 976 4879 2091 -5576 9.793 100 +498 213 569 6000 2030 5799 -1739 7248 9.795 207 +592 178 740 5000 3183 7322 -9075 9.796 325 +748 926 4000 5060 7772 2169 -10121 9.798 516 + 798 +221 1033 3000 5647 8.366 2719 -10039 9.800 -576 +884 +278 1024 2000 5075 9643 6851 -10150 9.802 -518 +984 +699 1036 1000 3966 8196 6610 8725 9.804 ^405 +836 + 674 890 000 1433 5014 8356 4775 9.806 146 +511 +853 487 TABLE 30. WINTER LOW ABEAS. 10000 3497 4769 3180 4451 9.786 357 487 +325 +454 9000 2869 4304 3156 4304 9.789 -280 440 + 322 +440 8000 2081 3745 8468 4577 9.791 213 382 +354 +467 7000 1673 3485 3624 4879 9.793 171 356 +870 +498 6000 1595 -3189 4059 5509 9.795 -163 -326 +414 +563 5000 1910 3183 4457 6367 9.7% 196 325 +455 +660 4000 -3253 3253 4338 6326 9.798 332 332 +448 +646 3000 1601 2928 4810 7111 9.800 591 299 +491 +726 2000 6090 3045 3299 7613 9.802 621 311 +337 + 777 1000 4495 3437 2908 5817 9.804 159 -351 297 +593 000 2149 2626 6924 4059 9.806 219 268 706 +414 TABLE 31. SUMMER HIGH AREAS. 0000 1794 2240 299 1492 9.786 +183 +229 31 163 9000 824 2746 687 2060 9.789 + 84 -t-280 70 211 8000 3468 1387 2497 9.791 o. +354 142 255 7000 1071 4587 2294 3058 9.793 109 +468 234 312 6000 2265 5338 1941 3882 9.795 231 +646 198 397 5000 3034 5109 160 4471 9.796 -312 +522 + 16 457 4000 3572 4970 777 3416 9.798 364 +607 + 79 849 3000 3262 4979 687 3434 9.800 333 +608 + 70 350 2000 2600 4085 743 2971 9.802 265 +417 + 76 803 1000 871 3309 2090 1742 9.804 88 +338 +213 178 000 669 2683 3354 671 9.806 68 +274 +342 68 TABLE 32.-SUMMER LOW AREAS. 0000 7168 1792 2987 3733 9.786 732 183 +805 +381 9000 6316 1648 3158 3845 9.789 645 168 +322 +393 8000 6103 1387 3745 4161 9.791 623 142 +382 +425 7000 6422 1835 4587 4893 9 793 656 169 +468 +500 6000 6470 2265 4864 5661 9.795 661 231 +496 +577 5000 . ... 5269 2236 4311 4790 9.796 538 228 +440 +489 4000 1348 2019 3106 4038 9 798 444 207 +317 +412 3000 4120 515 1717 4464 9.800 -420 53 +175 +455 2000 3528 371 4457 9.802 360 38 +454 1000 2090 697 1742 3832 9.804 213 71 178 +390 000 1342 1342 46% 2481 9.806 137 137 479 +253 Height in meters. N. E. 9 S. W. o Mean. N. E. S. W. 10000 44 39 31 34 37.6 +261 +54 227 127 9000 43 38 28 38 36 5 +259 +66 274 122 8000 41 37 26 32 85.0 +228 + 72 275 101 7000 39 35 23 30 32.9 +220 +72 277 91 6000 37 32 20 27 29.9 +238 +65 247 83 5000 34 30 17 24 26 9 +216 +88 218 74 4000 32 28 16 23 25.3 +192 + 72 192 56 3000 30 26 14 21 23.6 +172 +60 181 58 2000 25 20 12 18 19.6 +123 + 10 118 28 1000 19 15 10 14 14.8 + 71 + 3 60 12 9 9 6 7 8.0 + 8 + 8 14 8 TABLE 34. WINTER LOW AREAS. 10000 31 35 45 42 37.6 227 95 +306 +175 9000 30 33 45 42 36.5 216 122 +347 +216 8000 28 31 44 41 35.0 221 132 +356 +228 7000 26 29 42 39 82.9 203 121 +342 +220 6000 23 27 38 35 29.9 183 88 +275 + 166 5000 21 25' 34 30 26.9 142 50 +216 + 88 4000 19 24 32 28 25.3 140 32 +192 + 72 3000 18 23 30 27 23.6 117 14 +172 + 86 2000 15 18 26 22 19.5 78 28 +148 + 62 12 13 20 16 14 8 38 25 + 96 + 19 000 6 7 10 10 8.0 16 8 + 18 + 18 TABLE 36. SUMMER HIGH AREAS. 10000 37 33 26 29 83.6 +124 12 223 141 9000 35 32 24 28 32.5 +120 16 240 136 8000. 35 31 22 27 31.1 + 129 - 3 242 119 7000 33 29 19 25 29.0 +124 240 108 6000 31 27 17 23 26.5 +130 +14 207 87 5000 29 25 14 20 23.9 +185 +27 188 - 86 4000 27 24 13 19 22.4 +114 +37 166 71 3000 25 22 12 18 21.0 + 92 +22 124 59 2000 21 17 10 15 17.3 + 71 5 100 37 1000 16 12 8 12 13.1 + 42 14 54 14 000 8 7 5 6 7.3 + 6 - 2 14 9 TABLE S6.-SUMMEK LOW AREAS. 10000 . . 29 33 42 39 33.5 141 -17 +321 +200 9000 28 31 42 39 32.5 136 47 +354 +233 ROOD 26 29 41 38 31.1 -146 -63 +357 +239 7000 24 27 39 36 29.0 183 56 +340 +228 6000 21 25 35 S3 26.5 231 39 +262 +194 5000 20 23 32 28 23.9 86 21 +227 +107 4000 18 22 30 26 22.4 89 9 + 199 + 87 3000 .... 17 21 28 25 21.0 76 + 172 + 92 2000 14 17 24 20 17.3 - 52 5 + 139 + 51 1000 11 12 19 15 13.1 26 14 + 95 + 27 000 6 7 9 9 7.3 9 2 + 14 + 14 270 MONTHLY WEATHER REVIEW. JUNE, 1906 Distribution of the heat, Q , and the pressure, B B a . TABLE 37 WINTER HIGH AREAS. Height in meters. Q-% N. E. S. W. B-B a N. E. S. W. 10000 11.2+21.7 8.8 8.0 10.1 18.2 + 8.5 + 19.5 9000 18.2 + 11.7 9.8 15.8 5.7 22.8+ 9.6 +23.4 8000 14.9 + 13.0 6.4 19.7 1. 1 80. 1 + 14. 7 + 32. 1 7000 16.2 + 16.3 + 0.1 22.9 + 7.1 89.5 + 19.2 t- 49.2 6000 19.0 + 15.1 + 21.5 26.2 +17.5 51.7 + 17.8 + 71.7 6000 26. 3 + 3. 1 + 71. 9 24. 9 +34.673.9+ 1.3 +104.8 4000 10.7 18. 9 + 73.9 + 23.0 +60.1 75.4 23.6 +128.3 3000 +25.7 82.1 +106.0 + 51.4 +69.1 94.8 80.6 +129.8 2000 +66.8 41.7 + 59.0 + 77.9 +83. 2 140. 102. 1 +176. 9 1000 +82.6 76.0 + 88.1 +122.5 +58. 1 112. 7 91. 4 +137. 2 000 +94.4 101.6 + 10.8 +142.5 +15. 9 57. 1 92. 9 + 55. 6 TABLE 38. WINTER LOW AREAS. 10000 1.6 6.5 + 1.7 +7.2 + 21.8 +12.0 19.1 25.6 9000 +10.6 1.8 + 7.1 +7.1 + 19.9 +29.3 21.3 26.3 8000 + 9.8 + 6.6 + 5.6 +6.2 + 16.8 +29.0 26.4 33.4 7000 + 8.6 + 9. 2 + 8.8 +6.0 + 15. 2 +30. 6 31. 1 40. 2 6000 4. 7 + 8. 2 0. + 5. 5 + 16.0 +30.9 37.9 49.8 6000 16. 4 +89 5. 6 5. 8 + 21. 6 +36. 1 46 9 57 3 4000 28.2 7. 9 16 6 91 + 39 9 +39 2 50 6 70 7 3000 39 1 19 9 57. 7 34 8 + 57 9 +35 9 56 9 81 6 2000 +53 9 5. 6 137 6 62. 7 t!02 7 +53 9 51 9 113 3 1000 +72 8 79 161 2 70 6 + 67.2 +61 +41 9 81 4 000 -f 36. 8 6. 8 121. 5 54. 3 + 25.2 +30.4 +80.7 46.0 TABLE 39. SUMMER HIGH AREAS. 10000 16 7 +15 2 98 14 8 10 8 122+2.1 +94 9000 19.5 +13.1 9.9 7.1 6.0 17.9 +6.3 +14.5 8000 17 8 +12.9 11 8 50 076 26 7 +11 8 +20 7000 20 6 +17 S 03 99 -(-85 37 3 +20 8 +26 9 6000 26 7 +56 +35 4 +65 +18 9 43 8 +17 9 +34.1 6000 28.8 2.8 +44.8 +17.6 +26.6 43.6 03 +409 4000 31.9 9.0 + 2.1 +10.4 +33. 1 45. 6 6 +41 3 3000 +84.5 26.0 13.1 +14.2 +34 6 51.2 6 4 +37 5 2000 + 39 2 30 4 + 6 7 +16 9 +32 2 49 3 84 +37 8 1000 +25 1 19 2 +36 5 +25 7 +12. 7 48 2 30 3 +26 5 000 +16 4 72 +65 +36 2 1 4. 5 36 1 41 9 ]92 TABLE 40.-SUMMER LOW AREAS. 10000 +13 1 04+88+41 +46 1 +11 2 18 2 21 9 9000 -(-80 +16 +83 + - 5 3 8000 + 68 +03 +11 2 +42 7000 . +28 38 +82 +61 6000 . + 14 78 92 17 5 6000 +20 4 57 26 3 23 8 >f48 4 -)-20 38.1 40 4 4000 +18 5 +23 9 32 3 11 3000 + 65 +33 9 53 8 + 2 8 +45 1 +55 19 2 16 6 2000 +38 6 23 50 2 19 2 1000 |365 13 60 9 - 9 9 000 + 15,7 15 4 69 1 31 4 is larger than M O and n n c is positive, it follows that the temperature gradient is smaller for n than for n , and if n is smaller than n the temperature gradient is larger in the dis- turbed region than in the normal undisturbed stratum. An examination of the tables and the fig. 14 shows that while the distribution of n n a is similar to that of T T as to the sec- tors, yet there is a distinct reversal between the lower and the higher strata, occurring near the 4000-meter level. The positive (+) values of n n in the lower strata, showing a decrease of gradient of temperature or an inflow of heat on the western side of the high areas, become negative ( ) values in the upper strata, where they indicate a more rapid loss of the temperature. The reverse conditions hold on the -western half of the low areas from the surface to 10,000 meters. Hence an inflow of warm air in the lower strata, that is the southerly current, diminishes the normal gradient of tempera- ture and produces n which is a ( + ) quantity ; similarly, an inflow of cold air in the higher strata, that is of cold air from the north, diminishes the normal temperature gradient and produces a value of n ? which is also positive. These condi- tions are therefore in harmony with the observed temperature distribution in cyclones and anticyclones. The reversal be- tween the lower and higher strata of the same sectors indi- cates the thermodynamic effect of the air masses striving to return to an equilibrium by reversing the gradients above and below the level of 4000 meters. VALUES OF THE TERMS C f H (T T O ) AND (z Z Q ). Having determined the distribution of ( T T ) and (n ) in the several strata, since all the terms of the equation, except the velocity, depend upon them, we proceed to compute these terms for the sake of ultimately finding the heat variations Q Q , and the barometric pressure variations B Z? . The first condition to be found is g (z z ) = C f n a ( T T ) which represents the available potential energy of the air mass at a given level, to be expended in producing the kinetic energy found in cyclonic and anticyclonic circulations. Tables 29, 30, 31, 32, and fig. 15 contain the several terms. C p n ( T T ) is the potential energy, due to the fact that the mass of temperature T on the level where T is the normal temperature can be reduced to equilibrium by rising or fall- ing through a given height under the gravity acceleration g, where g (z z c ) is the work to be expended in falling or rising through the height (2 z ). The value of g is given on each 1000-meter level. The height (z z ) is computed from the formula, and by fig. 15 one can see that it has the same distri- bution as (T T n ), upon which it depends. Thus, the posi- tive ( + ) sign for (z z o ) indicates that the air mass is too cold for its level and that it can fall through a given height before reaching the normal temperature of its stratum; similarly, the negative ( ) sign for (z z ) shows that the air mass is too warm for its level and can rise through a given height to reach equilibrium. The cold sectors have the ( + ) sign, and the warm sectors the ( ) sign, and hence the entire cold column is able to fall and the entire warm column can rise through (z z ) meters under the influence of gravity. This is the primary source of the energy of motion in cyclones and anti- cyclones, this potential energy being converted into pressure differences and motions. An inspection of the tables and the fig. 15 shows that the maximum potential energy is on the east and west sectors, at the boundary of the high and low pressure areas. Hence, there is a gradient of potential within the cold areas from the north toward the southeast or south, and within the warm areas from the south toward the northeast and north at all the levels except that next the surface. The difference between the lowest level and those above it must represent a reaction from the ground, and an accumulation of the dynamic effects from the other forces yet to be considered, such as those from JUNE, 1906. MONTHLY WEATHER REVIEW. 271 the horizontal components, the deflecting force, the friction, and the other dynamic effects. Primarily, we must recognize that we deal here with a couple, one branch from the north on the west of the cyclone, and the other from the south on the east of the cyclone. The complex interactions which occur in consequence of these dispositions of warm and cold air masses form an important and difficult subject of study in hydrody- namics, which must be considered in a later paper. DISTRIBUTION OF THE VELOCITIES , by the formula log p = log/-,, + ^ (log T- log T ) the n being taken as the mean n of Table 16. These values are also omitted because of the magnitude of the tables that would be required to reproduce them in this place. We finally obtain the values P P , the variation of the pressure in mechanical units, which may be converted into millimeters by the formula, _ P 3 = 100 X 4' The resulting values, B B , are given in the second section of Tables 37, 38, 39, and 40 and they are plotted on fig. 18. The distribution is again such as has been made familiar in the preceding figures of these papers. These pressure differences are given in millimeters, and they represent a potential energy which can be converted into cyclonic and anticyclonic motions and pressures. The fact that the observed pressures are not so great as those here given shows that the efficiency of the kinetic structure is not so great as the potential energy would indicate. Not all the available energy goes into storms, a portion being carried along in the circulating structures without transformation and a part being frittered away in internal work agitations. We have, however, shown that there seems to be an abundant sup- ply of energy in warm and cold masses of different tempera- tures in the neighborhood of each other, sufficient to account for all the phenomena observed by meteorologists. It has been proven that the primary distribution is asymmetrical in re- spect to the centers of the low and high pressure areas. It is now one of the difficult problems to show, mathematically, how the action of these cold and warm masses, arranged as couples between the dynamic centers, is transformed from the thermo dynamic structures here indicated into the hydrodynamic structures actually existing in the atmosphere. It should be remembered that the closed isobars of the lower strata, prac- tically symmetrical about the high and low centers, are quickly modified above the surface into loops wherein the distribution of the pressure is entirely different from that at the surface as shown on the sea-level weather charts. The construction of daily and monthly isobars on the 3500-foot plane and the 10,000-foot plane for the United States during the year 1903 made this change of the structure of the isobars familiar to me. It is next in order to discuss the equations in the hori- zontal plane, namely, dq dQ dT dn 9 rf-;=^- c ^o^- c " r lo ^ r o^ where + v) ud y, which are equal to zero, so far as the circulation is concerned, since vdx= udy, though they have a decided effect upon the position of the resulting isobars; and, finally, the unknown terms representing the secondary or vortical motions induced by the dynamic motions in the sensitive hydrodynamic medium of the atmosphere. XXXIV 80, FIG. 14. Distribution of the values of n n in the high and low areas. _J-fa/?. Jfegbf. Jtfqh. Low. /n Steler^ . XXXIV 81. FIG. 15. Distribution of z a. = *_ -. ^ ^ ^x N 7 & y a -9.87 n, l.o /,5 2.0 25 3.0 0. ~>,87 -6.3$ -4.94 -3.95 -3.29 as the motion of the atmosphere is concerned. The function uniting (Q l - Q) - k(q?-q*) + (R}+ (J) being unde- termined, it is very difficult to make satisfactory progress in this direction, and the problem must wait for further develop- ments. Reviewing columns I + II in calories, whiah is the heat energy available from the temperature distribution, it is seen that it is positive and diminishes up to the 5000-meter level, above which it is small and negative. Comparing this column with Tables 37 and 38 it is observed that the vertical heat potentiality is about the same as the horizontal capacity for motion. If a kilogram of air is moved as noted by the condi- tions of the problem, this amount of heat must be interchanged. In the actual atmosphere this transfer is not so simple, and hence only a portion of the Q-energy is actually produced. How much less is really generated depends upon the efficiency of the thermodynamic engine in the practical physical opera- tions of the air. TABLE 44. Valties of the terms in the formula, I II C, n (T-T) + G p T log T (n, - n) = ( q 1 d u=i m q'. External potential energy Internal kinetic energy ) Internal potential energy j P to = Cp(- r * TJ- ~ m (molecules) + H, (atoms) ) J M (molecules) + J a (atoms) [ T , Quantity of heat Work of expansion A to-W= _ Cdt C p ' 'rfu= fa J J P d t J 567 MONTHLY WEATHER REVIEW. Afargutes. Bigelow. Potential energy + centrifugal force W to V t = gr+% w\ta\ Friction Velocity Volume Density Ratio of specific heats Adiabatic constant Height Surface Entropy temperature Potential temperature Drive temperature (R)= - c, V to q k to v IL to /> f to k= 1 to _*>_ =s C p = (/ k 1~~R = 7i~o ' c to h to S 9 to T T to T .? to T GENERAL THERMODYNAMIC EQUATIONS. Il q cos (R q) ,, d r. (1) Conservation of energy. Q=8U+3 W+ (R) = S(K) U + (R). Q = \ti(K) + SV] external + [dH+ SJ] internal = S W+ 5 U. External work. Internal heat. (2) Variation of heat. dQ= (3) External po- tential energy. >ST-* + A C p dT- in mechanical units. dQ= pdv vdp -\- pdv ~TS~ p" v= V =Cp dz+ (Zz) p h = RCTdm+ const. (4) Internal energy. U=C CTdm+ const. (U+ F)= (C +R) Crdm+ const. = O p CTdm + const. (5) Transforma- f -<1(U+ V)=(U+V) a (imtial)-(U+V) e (&nal)=C ]i C(T-T l )d tion of J ^ energy. 5 (.ff) + (^) = M ? J == C7 p J ( T- T l )dm= C p (T- T 1 ) M. _ r^(? T v "t I rn = C v log -fp -f- R log *r * -i a % (6) Entropy vari- - ations. (7) Potential tem- perature. 5S_1 dQ_O p dT Rdp dz~ T dz == ~T dz~ pfTz' ^. P. DECEMBER, 1906 (8) In linear verti- C 1 exchanges. J Tdm = (P. T ~P DECEMBER, 1906. MONTHLY WEATHEE KEVEEW. 568 (9) Auxiliary equations. nk T\'i^i Adiabatic. (7. kl R 1 RT -=- p RT. Observed. nk nC v 1=1 7? 1 rfP -,- = 0- _ Rd fl O ^_^ i7 n nC ldP CASE i. CHANGE OF POSITION OF THE LAYERS IN A COLUMN OF AIR. P f T V become PJ, T*, and the function must be integrated thru- In consequence of the general and local circulations of the out the mass M *' the temperature of the mass M h is not affected by the mutual transfer of m l J/ 2 , but rises or falls like a piston in the chamber, while its lower surface maintains the pressure P h . Hence, we have the conditions, 77Z/, JVto FIG. 24 A. Initial. Final. atmosphere, a certain gradient a = prevails at a given lo- cality in a column above the earth's surface. This requires an amount of heat (? and a temperature T at each level z c to maintain the stratum in equilibrium. If the heat energy changes to Q for any reason or the temperature is altered to T there must follow a change in elevation to z to restore the equilibrium. The equation of equilibrium, (10) i (9 ! -9o 2 ) = ( /TT k-l Final. P 1 T 1 2 2 * 2 Pressure. kl T > ' - kl Substituting in the equation, (15) Kinetic energy = C p [ f( T t - T, 1 ) d m, + J"( 2 1 , - T, 1 ) d Jf,j . (16) Jm.g'- since *5 f lA f > = C d ^ =C^ = h . P t J n " J n/)^ J n n The gravity terms in these equations disappear, because the mechanical work in each case, g h M l and g (Z 3 Z l t ) M t ( where Z^ is the height of the center of gravity of M t ) is of the same amount and oppositely directed. Every expansion or contraction of air masses begins on an adiabatic gradient, and hence the for- mulas must be founded on that basis. But minor interchanges of energy as heat Q and velocity J^ 2 almost immediately begin in the mixing process, so that the theoretical conditions soon suffer modifications which it is quite impracticable to follow out. CASE II. THE TEMPERATURE IS A CONTINUOUS FUNCTION OF THE HEIGHT, T^T-ah. It is important to eliminate the pressures from the formula and express the function in terms of g, h, T, and the gradients. Several forms of the function for the temperature distribution may be employed to represent the atmosphere, but it is only occasionally that these formulas can be used to replace the actual pressure and temperature observations at different lev- els. For the observed gradient we have (18) Observed gradient. 569 Hence, (19) Adiabatic gradient. (20) (21) MONTHLY WEATHER REVIEW. DECEMBER, 1906 ah\' TJ K * " = l ~ ah\9j Ca 'TJ CASE III. FOR LOCAL CHANGES BETWEEN TWO ADJACENT STRATA OF DIF- FERENT TEMPERATURES, WHERE ON THE BOUNDARY THE PRESSURE P = P, 1 = P, 1 , AND THE TEMPERATURE IS DISCONTINUOUS. Then, Finally, Take the following conditions: + i . a-fl O J iyer. Initial. Final. m, P 2 T 2 P l T ] a a m, p,r, P l T * 1 1 Pressure. P, 1 Temperature. 7"-T ~ ' k-1 nk The equation of equilibrium becomes, for P > l =P l ^= (23) The mass m, is driven from its position with a velocity- energy inversely proportional to the temperature, so that warm air has less driving power than cold air. The drive depends upon the departure-ratio n and vanishes when n=l, that is, for an adiabatic expansion in an adiabatic gradient. When a > a c the mass m l is in unstable equilibrium- is too cold for its position and tends to fall. Example, for w=0.5, o c = 9.87. When a' +1 ] *} Ha oo ^/ jfd e o |^i_|_p| jta IRa Ra (P t T -P T). For any gradient other than the adiabatic we have, (28) 1+ nk DECEMBER, 1906. MONTHLY WEATHER REVIEW. 570 CASE IV. THE OVERTURN OF DEEP STRATA IN THE COLDMN. Let the pressures, temperatures, and heights be arranged in the initial and final states as indicated in the diagrams (fig. 24 B). The greatest entropy in 1 is less than the least in 2, so that the cold mass 1 will fall beneath the warm mass 2. The heights of the masses will change as well as the pressures and temperatures. Assume P , T 02 , h } , T {l , h v as known in the initial state. Tr, h.f A Pressures. (29) (30) Temperatures. ri a =r oj - P / r *A { IT^-J \-*< i / nk t-1 - T ~ ~ iCooV Ft Tiz 2'Wa -pf rp' -ti yz.2 Fl Tl, 1 Cool> TO rjr' -o J- Ot /nttia/ rinef FIG. 24 B. Substitute in G p ( Cldm CT 1 dm l j successively. (31) Initial, ( V + U) a = C p (Td m = 2z - -1- (P. T m - P i T it + P t T tl - P h T hl ) + const. " " 1 + (32) Final, (V+l') e = nk T (P. T K ~ p i T it l + P^ ZV - P h T^) + const. 9 1 + / nk (33) Kinetic energy = (F+ E7) (F+ U) e = \ M q 2 = \ (34) Heights, A 1 '=^(T 01 '-T il '), V= ^ (T (35) Approximate solution of Case IV. \ (f = -r- A 3 s - ^-l- h T- CASE V. TRANSFORMATION OF TWO MASSES OF DIFFERENT TEMPERATURES ON THE SAME LEVEL INTO A STATE OF EQUILIBRIUM. TA Tfa T*, P*, Tz 21 Ti, h l Poi 7oi Pn To, Pi j, Ti, Fo To J3, /nifial FIG. 24 C. Given as data at the height h, T hl , T hi P h , the areas jB,, B v the entropy #, < S f Hence by the formulas, nk nk (36) P m =P h (j (o7) P m = P /, I ip I (38) Initial. ( V+ U) a = C p gh l - -^-^ B 9 i i _ _ " nk (39) P\ =P h - P h ). (P n T ot P h T hl + P u T 0) P h T h2 ) + const. (P n ~ (40) Final. (T^+ U).= C p l - T B(P\ T\-P\ T\ + P\ T\,-P h T h} ) + const. Q t . # J- 9 1 + nk (41) Kinetic energy. \ M T >. T' = T, T f M= BP h = B ,> h (approximate). (44) i MY = \M. ' ' ff fcr. (45) Assume (46) (47) (48) T, CASE VI. CONTINUOUS HORIZONTAL TEMPERATURE DISTRIBUTION WITH ADIABATIC VERTICAL GRADIENT. A, & & Co >t Fr * Fox Initial ELnat, FIG. 24 D. f(P al -P h )dx = P-\P-P h -} hp^-P^dx]. I J I J l-x l-x T- T>= T- T = z. *= C p J"(T~ T>) lP h (50) (51) CASE VII. POSITION OF LAYERS OF EQUAL ENTROPY WHEN THE PRESSURE AT A GIVEN LEVEL IS CONSTANT AND THE TEMPERATURE AT THIS LEVEL IS A FUNCTION OF THE HORIZONTAL DISTANCE AND A LINEAR FUNCTION OF THE HEIGHT. Let the gradient ratio which distinguishes one stratification of the air from another having a different temperature gra- dient be n. (52) P = f (53) The curves. F(xz)=n log T k (nl)log T= const. (54) Angle of d F Id F n TM. Th, 111 s r, T 2 1 2 i i TV to! -2'- curves. (56) Entropy^ Initiai EinaL FIG. 24 E. , = C p [n, log T hl - (n, - 1) log TJ + const ^C,, (n, log 7', 2 - (n, - 1) log 7 1 ,) + const. CASE VIII. FINAL CONDITION OF TWO AIR MASSES UNDER CONSTANT PRES- ( 57 ) \ O g -jjf __ J Q ?s_ SURE WITH GIVEN INITIAL LINEAR VERTICAL TEMPERATURE FALL. Z 7 , ~ ' T h , On removing the partition the layers 1 and 2 spread out, change their heights, and there is a mixed stratum between them. (58) Heights n n 1 _/.* \, ( 55 ) Temperatures 9 If the vertical temperature fall of the masses 1 and 2 is smaller than in adiabatic equilibrium, then the entropy in- creases with the height, and it can happen that in the colder DECEMBER, 1906. MONTHLY WEATHER REVIEW. 572 mass (1) the entropy at the height A, will be as great as in the warmer mass (2) at the ground. The higher layers in (1) form a series with an entropy equal to the layers in (2) up to the height h h r In the final state the under part of (1) will spread out on the ground, above it will be layers which are mixtures of (1) and (2), and farther up will lie the masses of (2) which initially were between (h h t ) and h. On the bounda- ries of the three layers the temperature transition is continu- ous. It will be convenient to approach the dynamic equations of motion in cyclonic vortices thru a study of the Cottage City waterspout of August 19, 1896. It should be recognized that in ordinary cyclones the vortices are not perfect and it is only rarely and in highly developed storms that anything like pure vortex motion is attained. The waterspout, therefore, offers a good example of vortex motion in the atmosphere with which to test the above equations. I may remark that the theory first advanced in my International Cloud Report, 1898, for the generation of cyclones and anticyclones in the general circu- lation seems to be practically confirmed by these studies based upon actual observations. JULY, 190(5. MONTHLY WEATHER REVIEW. 307 STUDIES ON THE THERMODYNAMICS OF THE ATMOS- PHERE. By Prof. FRANK 11. BIUELOW. VI. THE WATERSPOUT SEEN OFF COTTAGE CITY, MASS., IN VINEYARD SOUND, ON AUGUST 19, 1896. 1 THE SOURCES OF THE DATA USED IN THE DISCUSSION. This waterspout has an especial scientific interest for mete- orologists because it was seen under circumstances remarkably advantageous for making observations and photographs, from which it is possible to compute, with much accuracy, the dimensions of the tube, and thus facilitate the application of the mathematical theory of vortices. A series of papers and letters from various persons who saw the phenomenon, and a very complete set of photographs, were secured at the time by the Editor, which he has courteously placed at my disposal for incorporation in this paper, and they will be found inserted in the following pages. I have myself been familiar with that part of the Massachusetts coast, and have therefore been interested to study the facts as thoroughly as possible, as a preliminary to the discussion of this type of vortex motion. I accordingly visited Cottage City the follow- ing September, and was conducted by Mr. Chamberlain to the spot where he placed his camera for making his photographs. There I made a sufficiently accurate survey of the linear dis- tances between that spot and the telegraph poles shown in his pictures to determine the scale of distances for all objects. Furthermore, by collecting and collating all the data relative to the positions of the waterspout and the schooner seen in the several photographs, I am able to plot them on the Coast and Geodetic Survey Chart No. 112, in such a way as to recon- cile nearly all of the statements made regarding the distances and progress of the two objects, respectively. The photo- graphs taken from such distances as Vineyard Haven and Fal- mouth Heights give an excellent view of the whole cumulo- nimbus cloud from which the tube descended, and its connec- tion with the thunderstorm which preceded it. All these data will enable us to discuss the subject of. tornado and waterspout formation with considerable fulness, and with the conviction that confidence may be placed in the comparison of the obser- vations with computations. There is every reason to believe that the photographs are perfectly genuine, and free from touches to add to their artistic beauty at the expense of scien- tific accuracy. Certain preliminary computations were made in 18!)7, the result of which was published in the International Cloud Report, page 633, Report of the Chief of the Weather Bureau 1898-99, Volume II; this was republished in the MONTHLY WEATHER REVIEW.* My purpose then was to illustrate the application of certain formulas, and it was my intention at that time to complete the study as soon as my other duties per- mitted. In these present papers I shall begin with the descrip- tive accounts of the waterspout, then pass to a discussion of the facts as shown by these reports and the photographs, and finally consider the dynamic motions and the thermodynamic conditions present in the atmosphere near Cottage City on that occasion. r.ETTF.RS AND REPORTS OF OBSERVERS. The following letters, reports, and observations have been furnished by the several authors. It will be instructive to refer to fig. 25 while reading these papers. (A) IC.XTHVI.T HiCIM THE DAILY JOURNAL OF U. S. WEATHER BUREAU STATION, VlNE- YARD HAVEN, MASS., W. \V. NEIFERT, OHSERVER. Auyiist W, 7,s'.%'. Partly cloudy weather during the morning, with gentle northerly wind. Three magnificent waterspouts were observed in Vineyard Sound to-day, in northerly direction from station, about ten miles distant. During the entire afternoon the weather was partly cloudy and sultry, with groat masses of cumulus clouds in the north and northeast. At 1.1:45 the first display was observed. At first a long spiral 1 Xo. V of the series (" The Horizontal Convection In Cyclones") will follow later. 'May, 1902. Vol. XXX, pp. 257, 258. column seemed to fall from the clouds, about the thickness of a man's body, but this gradually increased in size as the cloud lowered, and when it reached the water it was as thick as a large sized cask, and changed in color from a rich gray to a black, and assumed a funnel shape at the base of the clouds. The cloud seemed of a yeasty white where the column came in contact with it, and looked as though the water was hauled up to it. The area of contact appeared small. The spout was very straight and almost perpendicular, kicking up a great sea as it traveled. When it disappeared it began to do so at the base and rapidly reached the top, having the appearance of clouds, and finally cleared away, like steam from an engine, at 12:58 p. m., leav- ing a clear sky for a background and the original clouds above. At 1 p. m. it formed the second time, which was really the most interesting spectacle of all. From a mass of inky clouds it reached down, finger- like, to almost the ocean's surface. Below it the water was stirred to an angry whirlpool, the foam reaching up perhaps a hundred feet. It appeared as though great volumes of water were traveling up to the cloud by an endless screw, when suddenly, at 1:18 p. m., the long arm disappeared in a manner similar to the first. At 1:20 it formed for a third time and scarcely reached the water, but had a decided funnel shape, lasting about five minutes, when it slowly withdrew into the blackness above and the surface of the ocean became quiet. There was a sprinkle of rain from 12:54 to 1 p. m., amounting to a trace. During the display the wind at the station was six milt- s per hour from the northwest; tem- perature 72, with a fall to 56.5 during the thunderstorm which fol- lowed, passing ov<*r the station from northwest to south. Thunder was first heard at 1:45 p. m.; loudest at 3:04 p. m.; last at 3:45 p. m. Heavy downpour of rain from 3:04 to 3:15 p. m., then continued light rain until 3:30 p. m. Amount, 0.38 inch. The summer residents were stricken with fear at the approach of the dark clouds over the sound, and viewed the waterspout with mingled feelings of awe and interest. It was a sight long to be remembered, and when the weather cleared, about 4 p. m., each expressed himself as being most fortunate in having es- caped some dreadful calamity. No noise was heard here, but the sehooner-yacht Avalon of Boston was very near the spout aud those on this vessel reported plainly hearing the noise and the wind blowing around the vortex with wonderful rapidity; to them the spout appeared to be one hundred feet in diameter. The three spouts moved gracefully to the eastward. This is the first display of this phenomenon witnessed here for 27 years. Mariners hero who have circled the globe a number of times, and have seen dozens of waterspouts, declare it to be the most perfect specimen they ever observed. (B) LETTER FROM MR. NKIFKRT TO MR. A. J. HKNRV. DATKD VINEYARD HAVEX, MASS., DKCKMIIKII 19, 1896. When I first saw the waterspout it was in the vicinity of Black Buoy No. 13, on the east end of L'Hommedieu Shoal. Can not say now ex- actly, but in that general direction. Oould just sec base off East Chop. When the photographic view was taken here it was about where the red dots surround sounding marked 8i. It appeared nearer then, but I pre- sume this was caused by its base being hidden by the "highlands". The view from here was taken from on board of a yacht which lay at the red dot between the two wharfs or the head of the harbor under the figure 3 of the sounding marked 13. * It may not be so far, but that is as I remembered it. There was so much confusion, women and children cry- ing, that I was not very observant until it was over. Coolidge was just north of the head of the wharf in Cottage City, and the "spout "was in an east-northeast direction from him. His views were made from the same position, and only time enough elapsed be- tween them to change the plates. () EXTRACT FROM THE DAILY JOURNAL OF THK U. S. WEATHER BUREAU STATION, NANTUCKET, MASS., MAX WACXEK, OIISEKVEK. August 19, 1896. Clear weather all day, except in the afternoon, when light rain began at 2:40 p. m. and ended at 4 p. m. Total amount 0.03 inch. Cooler, with rising barometer. Mr. Wagner went to Cottage City in the morning to check up; from there he observed the big waterspout that formed in Nantucket Sound. An ordinary thundershower was pass- ing across the sound when, about 12:40 p. m., a huge black tongue shot down from an alto-cumulus cloud that floated a half mile high at the northern edge of the shower, and after rising and falling a number of times, finally joined a shorter tongue that seemed to leap out of the water to meet it. Twice the column parted for a moment, but joined again instantly. There was no apparent motion of the waterspout for- ward, and the phenomenon lasted for half an hour. It was pronounced by many sea captains who witnessed it the finest waterspout they had ever seen. No damage was done by the spout, but a small catboat which arrived at night reported being becalmed near the spout, the crew being badly scared. (D) EXTRACT FROM THK DAILY JOURNAL, U. S. WEATHER BUREAU STATION, WOODS HOLE, MASS., J. D. BLAODKX, OBSERVER. August 19, 1896. Three waterspouts were reported in the Vineyard 3 On chart not reproduced. EDITOR. 3UH MONTHLY WEATHER REVIEW. JULY, 1906 EAST FALMOUTH HAMLIN PT. GUMrllMG PT. SUCCOHESSE.T PT FALMOUTH >*EIGIHT QUAMOUISSET 5UCCONESSET -SHOAL LIGHT-SHIP * nOriAMESSET VIMEYAR.D HA EDQARTO R LISHT SCALE OF MILES -3 KILOMETERS KATA.MA BAY FIG. 25. Location of waterspout seen In Vineyard Sound, August 19, 1896. (Reduced from United States Coast and Geodetic Survey chart No. 112.) JULY, 1906. MONTHLY WEATHER KEVIEW. 309 being generally accompanied with flashes of lightning and a sulphurous smell showing the activity of the electrical principle in the air. FIG. 26. Diagram of the survey between site of Chamberlain's camera and four telegraph polos shown in his photograph, 2d A (fig. 27). Sound and one in Buzzards Bay between 12:35 and 2 p. m. spout was photographed with excellent results. One water- COPY OF \ CIRCULAR ACCOMPANYING COPYRIGHT PHOTOGRAPHS. BERLAIN, OP COTTAGE CITY, MASS. BY J. N. About 12:45 noon, August 19, 1896, we were startled by the cry of " A waterspout ! " and wit h our assistants started with the camera to the park in front of Doctor Tucker's residence, where we could see, a little north of the direction of Nantucket, very dark and angry clouds, out of which a funnel-shaped cloud of various colors, with a pointed streak, issued downward until it touched the water. We obtained two photographs of this, showing a slight difference. [One of these views is reproduced as fig. 27.] After about twelve minutes it gradually and completely vanished. Very soon a second one appeared, more curved than the first, with a long sharp streak from the same clouds and slowly extend- ing downwards to a point about one hundre'd feet from the surface of the ocean. In a few moments this changed to a smaller streak with a different curve bending to the south, while the former bent to the north. Both of these we photographed [figs. 34, 35]. The height of this, which Professor Dwight of Vassar College says was a genuine waterspout, was about a mile. The cloud-burst disturbed the water in the sound for several hundred yards until it looked like a boiling pool. We could trace through the camera the spiral motion of the water as it was drawn into the clouds, every moment augmenting their portentous darkness. The cloud from above and the spray from below were drawn together by suction, and condensed torrents of water poured down a few hours later, which was found by persons in different places on the island to be salt, and proves that it was can led up to a height and scattered round as solid bodies are by tornadoes on land. The Greeks applied the term "prester" to the waterspout, which signifies aflery fluid, its appearance (F) A REPORT TO THE EDITOR BY PROF. W. B. DWIGHT, VASSAR COLLEGE, POUOH- KEEPSIK, N. Y. DATED MARCH 22, 1897. I now inclose such statements as I am able to make without the few memoranda, noted on the spot and since lost, of the waterspout of last summer at Cottage City. The basis of my estimates of the height of the waterspout is rather hypothetical, but I submit them for what they may be worth. I have endeavored to assume my units of measurement so as to be below rather than above the fact, in order that the estimate might not seem to be made in a spirit of exaggeration. Thus, I am inclined to think that the dis- tance of the schooner, in the photographs, from the shore is nearer three than four miles, which would make the spout higher than my estimate. One reason for my thinking so is that there is a buoy, the three-mile buoy, so called, not far from the position of the schooner and in front of her, about three miles from the shore and marking the channel. She was likely aiming for that buoy and then .would not be very far from it. On the other hand, the state of the wind might lead her to go as much as half a mile or more outside (to the east) of it. I presume that the opinion of the seamen at Cottage City on the distance of the schooner could be easily obtained and would be of value. I think that I obtained such an opinion, but it is lost with my other memoranda and I can not now recall it. I have searched for the three-mile buoy in the photographs, but it is a very small object and I cannot identify it. Some statements as to the waterspout in Nantucket Sound (sometimes called Vineyard Sound) easterly from Cottage City, Marthas V-neyard, Mass., at noun of August 19, 1896, made from personal observations by William B. [Height, of Vassar College, Poughkeepste, N. Y. (resident vn the summer at Cottage City). My summer cottage is situated close to the beach at Cottage City, with unobstructed view of the ocean. I was standing upon my private wharf, nearly in front of the cottage, when the waterspout of August 19, 1896, began; I saw it at the outset and was among the first to call general atten- tion to it in our part of the town. I watched it closely, with the assist- ance of a good field-glass till the close of the phenomenon, but I had no proper instrument at my command for taking the altitude. Excellent photographs were taken by Mr. Coolidge and Mr. Chamber- lain. I am able to testify to their general correctness as corresponding with personal observation. Mr. Coolidge's are most artistic views of the whole scene and scenery. Three of Mr. Chamberlain's present with accu- racy three consecutive views of the waterspout in its phases, changes, and progress taken from one and the same spot. They were taken with total disregard of the foreground and the sole aim of getting the best views of the spout itself. These, facts give these three views a marked scien- tific value, and these photographs will repay a careful scientific study. Like all of the three phenomena of this kind which I have personally observed (and this is the second which I have seen from Cottage City), the funnel of the tornado is constantly changing its form, length, and other dimensions; and occasionally, or at intervals, it may entirely dis- appear in its cloud, only to reappear again in full force. This one had several such successive appearances with intervals of total disappear- ance. Hence the photographer, the newspapers, and the spectators gen- erally described the appearance of several waterspouts on this occasion. I consider this an unscientific and unfortunate mode of describing this phenomenon, chiefly for these two reasons. 1. There was only one great but entirely distinct and Individual cloud concerned in the phenomenon from beginning to end, and in fact only one particular spot in that cloud. This not only follows from my own observation but is demonstrable from a study of Chamberlain's three photographs, as I propose later to show. This cloud and its point of vortex movement sustained constantly throughout the waterspout phe- nomenon, three quarters of an hour (or more), the same relation to the furious squall of lightning, thunder, rain, and hail, going on about a mile to the southeast of the waterspout, i. e , a mile from the thunderstorm to the edge of the waterspout. This squall is clearly visible In Cham- berlain's first photograph of the three mentioned. 2. From the point mentioned in the tornado cloud, (as I will designate it in distinction from the squall cloud), a watei spout funnel would de- scend to the ocean, and move along its surface in an easterly direction, with its cloud; after a while it would thin out, or break into pieces, and nearly or quite disappear. For the most part, however, the location of its minimized force in the cloud remained marked clearly by a downward bulging of that part of the cloud, with indications often of rotary move- ment at the spot. Once, however, the spot where this tendency to vortex motion still existed was for a few minutes lost to view; but soon the vortex movement visibly returned somewhere along the line between the cloud and the ocean, from the point of the cloud which was affected. It generally appeared first at the cloud, but once the vortex movement at the ocean's surface was practically simultaneous with that at the cloud; then another column or spout was completely formed, but as the cloud had been moving eastward during the interval, the spout would of course be seen in a position somewhat to the eastward of its former place; and so this disappearance and reappearance was several times 310 MONTHLY WEATHER REVIEW. JULY, 1906 repeated. Those of the more intelligent observers who insist that there wore "several waterspouts" ou this occasion base their statement on these two arguments: (1), that there were successive spouts seen; (2), that no two of the spouts were in the same place. On tho contrary, I hold that my preceding remarks, and the further facts to which I shall call attention later, show that this was tho same phe- nomenon, that is, the same center of vortex action, throughout, and that its different appearances were not different waterspouts, but simply dif- ferent and varying phases of one and the same phenomenon. As to the second point, the difference in position, I contend that the differences in position were only those which a waterspout drawing itself up into its cloud, and then coming down again, must necessarily take in conse- quence of tho constant southeastward progress of the storm. It could not come down in the same place any more than a circus rider can when he leaps up from the back of a running horse und comes down again several feet ahead of his former position. The successive phases of this waterspout, In their positions, follow strictly the eastward move- ment of the tornado cloud, and inspection of the three photographs of Chamberlain's sot shows that a lino between the first and last phase of his three would pass through the position of the intervening one. This may seem a matter of little consequence in terminology; but it is of importance in view of the fact that the expressions "two water- spouts", "several waterspouts", etc., are positively needed for cases where two or more entirely independent phenomena of the kind are in sight at the same time, or nearly so; as when a friend of mine once saw eleven waterspouts on the ocean simultaneously. I will now give a brief description of the successive phases of the waterspout as I observed it. I was standing on my own private boat wharf, which is on the sea- shore at the extreme southeast point of Cottage City, one-half mile ex- actly south of tho "Oak Bluffs" or main wharf, the wharf shown in Coolidge's photograph, No. 7933, fig. 28, a little after half-past twelve. In the excitement of the occurrence I failed to note the exact time An exclamation from a friend standing near mo drew my attention to the waterspout, which had just formed in the rear of a black thundersquall which we had been watching to the southeast, the wind being from the northwest. The waterspout being a mile or more in the rear of the squall and separated from it by a clear interval, was a little north of oast from my position of observation; it appeared to be somewhat nearer than the Sueconesset light-ship (on Succonesset Shoals), which is nine (9) miles easterly from Cottage City; at the same time it was evidently nearly as far. I had several interviews subsequently with captains of the local fishing catboats, all men of lifelong experience as coasters, with refer- ence to the probable distance of the spout. They all estimated It as from eight to ten miles away; no one gave a less estimate than eight miles. All but one of the captains had seen it only from Cottage City. One captain, however, told me that he was sailing to Cottage City from Cape Poge, a point seven miles to the southeast of Cottage City, and saw the waterspout when he was off that cape, and that it was certainly nearer to Cottage City than the Succonesset light-ship (which from his position would be much in the same direction); ho said it was, in his judgment, about one mile nearer to Cottage City than the light-ship; this is excellent testimony on this point, and I think wo may safely set the distance of the waterspout from the Cottage City wharf as having been just about eight miles. At this first phase the waterspout was very tall and very thin; in fact it presented very much the same appearance as in No. 3 of Chamberlain's set, fig. 35, though in a much more northwesterly position; at its base was a spherical mound of up-whirling water and spray several times wider than the main portion of the column, a white dot of foaming water appearing at tho center of this mound at the ocean's surface; the column was sinuous and moderately expanded as it joined the cloud. The tor- nado cloud had a broad, flat, angry looking under surface, little tufts of mist or rain descending from it here and there; it extended at least a mile to tho oast and southeast, joining the thundersquall in the latter course. Toward the north and west it was much less extensive, and in fact more than one-half of the sky over Cottage City was in bright sunlight. At times it appeared as if streaks of rain were descending from the tornado cloud to the ocean all around the waterspout in all its successive phases. This first phase is not xhown in any of the professional photographs, though probably some amateur's camera may have caught It. Compara- tively few persons saw it, that is only those who happened to be at the beach; the morning bathing hour was mostly over; the professional pho- tographers were In their offices inland; it took time to get word to them and for them to bring out their instruments and get them placed in good positions. Meanwhile, this first phase faded away, and that one of tho views of the photographers which is generally called the "first water- spout " is not at all the first, but the second phase, and a much larger and grander one. The second phase, which appeared at about a quarter to one o'clock, was by far the grandest one of all. It is the " first" one of the photographers, the one shown by Mr. Coolidge's photograph numbered 7933, fig. 28. It began by the formation of a broad funnel on the under side of the tornado cloud, which then became a very broad black tube. This rapidly stretched down to the ocean, where it raised a large mound of whirling, foaming, rising water at the center, and of spray around its margin. The white center of upward rushing water was usually clearly visible to the naked eye, and through a Bold glass was very marked. At other times it was completely obscured by the surrounding mist and spray, and was never relatively large to tho view, because so thoroughly enveloped. * * * There was no white water vi.-iblo at any time in the tube proper, above the mound. This phase lasted probably about fifteen minutes, during which it varied in form from a slender, even cylinder, to a massive, imposing conical tube, a^ it swept on slowly and majestically to the southeast. From this phase we are enabled, through Mr. Chamberlain's valuable set of photographs, to trace the forward progress of tho tornado cloud visibly, and the relations of this and tho succeeding phase to each other, since these three views were taken by the same camera and lens, and from exactly the same point. (I have established this point, as it is easy for any one to do, on the spot, by taking a position on the west mar- gin of the main portion of the Ocean Park adjoining the Oak Bluffs dock, where the relative positions of the telegraph poles, and their several arms and wires can be made to coincide exactly with their relative posi- tions in tho photographs. It will bo noted that these positions arc ex- actly the same in the three views, though in the last one the camera was revolved more to the southeast than in tho others. This point of obser- vation proves to have been at the center of the convex western edge of the main body of the park, just cast of the carriage road which extends in a curve from one point of the Sea View avenue to another point of the same avenue around the west edge of this main portion of the park. It is called Ocean avenue. The point whore the camera stood is just east of Ocean avenue, where a straight lino running through the east end of Fisk avenue would strike it.) Now, in examining Chamberlain's first view, fig. 27, where the grandest phase is seen, the waterspout is shown a little north of the two central telegraph posts of the view, while a schooner is seen sailing southeast about three or four miles from the shore, some little distance to tho north of the waterspout. Also, a little northerly of the waterspout, about a third of the apparent distance (in the view) between it and the schooner, the masts of a vessel at anchor appear ou the horizon. This is apparently the Succonesset light-ship, nine miles away. It is cer- tainly about the position of that light-ship, and resembles closely its familiar appearance, as seen from Cottage City. This would seem to locate exactly the direction of this phase of tho phenomenon from Cot- tage City. As my charts of Nantucket Sound (or Vineyard Sound) are at Cottage City, and my notes made on the spot are lost, I can not at this time state the bearing of the light-ship from tho point mentioned as tho position of Chamberlain's camera; but it can easily be settled by reference to such a chart. Mr. Coolidge's photograph No. 7934, fig. 32, seems to show a later form of this phase, the column having grown thicker and more evenly conical since Chamberlain's first and earlier view was taken. This ap- pears from the fact that the schooner is now much nearer to the line of direction of the waterspout, in which direction she was sailing faster, apparently, than the spout was moving in the same direction. It is true that Coolidge's standpoint was evidently to the north of Chamberlain's, being near the steamboat dock, and quite near the shore, while Cham- berlain's position was about five hundred feet from the shore. But when all allowances for difference in position have been made, there seems to be still quite a margin which eau only be explained by the fact that the vessel had actually had time to sail some distance southeasterly. When this grandest phase disappeared the spout was for a few minutes totally absorbed into the cloud. Then, while I was closely watching it for further developments, a whirling funnel began to bulge down from the tornado cloud, while at the same moment, and before the funnel was more than a mere projecting knob on the cloud, the water on the surface of the sound just beneath began to boil furiously, and to rise up in a whirling mound, indicating a line of vortex motion already established the entire distance between the cloud and the sea. Next, while the upper tube began to extend downward, a central portion of the tube was formed, entirely independent of the upper and lower portions, as clear spaces existed between. This Is finely shown in Chamberlain's second photograph, fig. 34, where the three portions are distinctly seen before their union. Many per- sons on seeing his three views take tho one which shows the waterspout as a very slim tortuous tube, fig. 35, as the first exhibition of a waterspout, of which the one in three portions is but the breaking up phase. That this is not the case is indicated by my own most positive observation of the triple formation of the spout, to which I called tho attention of by- standers at the time. But, further, the photographs themselves prove the incorrectness of that idea; for it will be noted that in the view of the spout having the triple structure, fig. 34, the column has just passed from its second phase, where it appeared a little to tho north of the pair of central telegraph posts, to a position a little to the south of the more northerly of these posts, while the schooner has passed to a central position between them. In the other view, fig. 35, the column has come near to the more southerly of the posts, while the schooner has passed considerably to the south of both, and lias been closely approached by a tug towing three barges, which was only just coming into view in the other photograph. I need only to remark further, therefore, in this connection, that the JULY, 1906. MONTHLY WEATHER REVIEW. 311 third of Chamberlain's set of photographs, fig. 35, represents the com- pleted form of the waterspout shown in the second view, and the third and last actual phase of the entire phenomenon. As this phase dis- appeared at about 1 :25 p. m., the entire occurrence covered a little over three-quarters of an hour. If one were disposed to form some estimate of the rate of progress of the waterspout, from that of the schooner, the following points would deserve consideration : 1. Though the wind was violent in the vicinity of the squall, there was but a moderate wind on the shore and in the vicinity of the schooner. This is not a matter of my recollection alone, for the photographs show it; the sea near the shore is little disturbed, while the schooner carries all sail except topsails, and has no reefs. If we may estimate the probable length of her hull as 75 feet (a moderate estimate), she must have gone somewhat over half a mile during three-quarters of an hour. Her course being somewhat oblique to the line of vision, and veering away from the observer, is really longer than it measures on the photograph. The tide runs with great force, and may have worked against the schooner. 2. The waterspout was certainly twice as far away as the schooner. Supposing, as is very likely from evidence elsewhere offered, that it was just about twice as far, and moving, as appears to have been the case-, in the same course, the tornado cloud and the waterspout must have progressed considerably less than twice as fast as the schooner, since the latter, starting from a spot considerably to the north of the spout passed to the south of it. Toward the close of this phenomenon the eastern half of the sky be- came quite black with clouds, while the entire western half, where the Cottage City observers were, was brilliant with sunlight, which at this hour glanced easterly beneath the blackness. The chromatic effects were of an indescribably rare and beautiful kind. The surface of the sound for several miles out was lighted up with weird hues of bright blue, green, yellow, and gray, in patches, according to the nature of the variable weedy and sandy bottom, greatly Intensified by the solemn, black storm clouds and waterspout overhead. Thousands of spectators, crowding the beach, gazed on the sight with mingled admiration and awe. I append some estimates of the probable dimensions of the waterspout founded on its apparent distance, as ascertained by investigation, and upon measurements of the photographs, using as a unit the hull of the schooner in view, estimated as 75 feet long and four miles from shore. I am inclined to think that the schooner's distance was nearer three than four miles, and that the estimates should be increased proportion- ally. By my estimate the waterspout would have an altitude of about half a mile. The conditions of the photographs of Chamberlain's series might afford another basis for an estimate. By ascertaining the distance from the spot where the camera stood to the higher one of the central pair of telegraph posts (as the other stands on lower ground), which distance is not far from 500 feet, and the height of the telegraph post, and assuming the probable distance of the water- spout as eight miles, triangles could be constructed from which the height might be calculated, provided that the telegraph posts were not so near to the camera as to be disproportionately magnified. I suppose corrections might be applied for such irregularity if the power of the lens were known. Rough estimates as to dimensions of the icaterspoui seen from Cottciije City, Mass., Augitst I'.i, 1896. (1) Estimates founded on photograph No. 7934, fig. 32, taken by Coolidge, and based on the supposition that the waterspout was eight miles from Cottage City, and that the schooner visible to the north of it is 75 feet long and four miles from Cottage City. In this photograph, the schooner's hull (not including the bowsprit), estimated at 75 feet, measures one-tenth inch. The waterspout being twice as far distant should therefore measure 150 feet for every one-tenth inch of dimension. Some correction should be made, in strict calculation, for the tendency of the camera to magnify near objects more than distant ones; but this is comparatively slight for two objects both of which are distant, and may be disregarded in a rough estimate. Using this unit of measurement, 1-10 inch on the photograph corres- ponds to 75 feet at the schooner, or 150 feet at the spout: Feet. Height from surface of ocean to lower edge of cloud (= 1.917 inches) 2874 Mound of spray at base of the spout: Height 600 Breadth 750 Breadth of the mass of foaming water in this mound . 150 Tube or funnel proper, above the base mound: Breadth (diameter) just above the mound 150 Breadth about the middle 300 Breadth at extreme top where it joins the cloud . . . 600 (2) Estimates founded on a photograph by Chamberlain, fig. 27, and called by him the " first " waterspout. It shows the spout of larger size and grander appearance than the later views. These estimates are based on the assumed length of the schooner in sight, and its distance, assisted by known facts as to the position of the ship channel, and on the supposed distance of the waterspout, judging from reports of cap- tains of fishing boats who got its range. This is really the second and not the first stage or appearance of the spout, as witnessed by myself from its very beginning. In this photograph the hull of the schooner (75 feet) measures 13-100 of an inch. Its estimated and probable distance is not over four miles, that of the waterspout eight miles. Hence the unit of measurement, 13- 100 inch, would cover 150 feet at the waterspout. The dimensions would then be as follows: Feet. Height from ocean to cloud 2600 Height of basal mound of spray 300 Width (diameter) of basal mound of spray 600 Tube proper: Width at lowest point 150 Width at middle point 100 Width at top 375 Remarks. This photograph represents, as I can testify from personal observation, the same spout, or phase of the spout, as Coolidge's No. 7934; but at a different minute of its existence, since the form is consid- erably different. The spout was changing constantly in length and width within certain limits, but was throughout the largest of the phases. The difference in the apparent length of the schooner's hull from that in Coolidge's photograph is due probably to a stronger lens. (G) LETTER OF E. U. HANKS, TO THK EDITOR. DATKD COTTAGK CITY, MASS., OCTOUKR 26, 1896. Your letter of October 20 is at hand. I am sorry to say that I do not know of any scientific observations of the waterspout seen in Vineyard Sound, August 19. I had an excellent view of it throughout its entire duration, a portion of the time through a six-inch astronomical telescope. It occurred August 19, 1896, at about 1 o'clock p. m. It had been a calm summer day, with but few clouds, temperature about 70, with but little variation before and after the phenomenon. It has been stated that there were two or three waterspouts ; this, I think, is hardly correct, as no one saw more than one at the same time, the so-called different ones being different forms or reformations of the same spout. Its beginning was, from my point of view at Cottage City, about six miles distant, in a line toward Cotuit; its ending, about eight miles, in a line toward Hyannis. It had a steady progressive movement and was inclined forward in the direction of advance. I estimate its forward movement at about eight miles in the thirty-five minutes it continued. During the time of the waterspout, showers, with lightning, could be seen preceding and follow- ing it in its course; about an hour afterward Cottage City was visited by a tempestuous downpour of rain. Through my telescope the column seemed to be surrounded by a dense vapor, which radiated like smoke from its edges, and, condensing, fell in torrents of rain for a distance in either direction about equal to the diameter of the column. At first the edges of the column were quite well defined, later it grew much larger in diameter and more diffuse, its height remaining the same throughout. While I could not penetrate with my telescope the enveloping mist so as to see if there was a solid or tubular mass of water either ascending or descending the inner part of the spout, nor detect a whirling or spiral movement, yet the funnel shape at the top and general appearance indi- cated that character. Based upon estimates of the most careful obser- vers, its probable size was from KlO to 300 feet in diameter at different periods, and 4000 to 5000 feet high. Where the column joined the sea there was a great churning and splashing of the water, which extended as white mist for 200, or more, feet upward and outward; this was more pronounced toward the last. When the spout finally disappeared it grew slender and broke about midway of its height, the lower portion drop- ping into the sea and the upper dissipating into the cloud. (II) EXTRACT FROM THE REPORT OF THE CLIMATE AND CROP SERVICE, NKW ENKLASD SECTION, AUGUST, 1896, BY J. WARREN SMITH, SECTION DIRECTOR. On August 19 three well-defined and magnificent waterspouts were observed in Vineyard Sound, between the eastern edge of Marthas Vine- yard and the mainland, about off Succonesset Shoal. Mr. W. W. Neifert, the Weather Bureau observer at Vineyard Haven, writes : " During the entire forenoon the weather was partly cloudy and sultry, with great masses of cumulus clouds in the north and northeast ". | The remainder of this quotation is practically identical with (A) above. EDITOR.] We have reports of the phenomenon from Mr. E. H. Garrett, Vho ob- served it from the coast between Hyannis and Oysterville; from John B. Garrett, who saw it from Falmouth Heights, and from Dr. S. W. Ab- bott, Secretary of the State Board of Health, who was in West Falmouth Harbor at the time. Mr. E. H. Garrett says: "We were out on the beach and saw an odd look- ing cloud in the sky. It seemed to have a curious appendage at first, which one of the party described as looking like ' an icicle.' We turned to go home, when one of the group looking back saw the ' icicle ' chang- ing, and we all watched. It grew larger, then looked like a long, thin, gray veil of mist and as it descended the water from the Sound began to rise. I watched it carefully and should say It was over 300 and nearer 500 feet high, and in comparison with the measurement of schooners MONTHLY WEATHER REVIEW. JULY, 1906 lying near it, it certainly could not have been less than 125 feet in di- ameter". Mr. John B. Garrett saw it from Falmouth Heights in an east-south- east direction, and its distance was estimated to be from six to ten miles. He says: "In form it was much like a short section of rubber pipe, flexible, and of the color of a heavy watery cloud. It was tele- scopic, the upper end of the column vanishing in the small end of a fun- nel-shaped cloud somewhat larger than itself. " Shortly before it broke and disappeared, the main column drew up- ward, disclosing at its lower end a smaller column or tube within the main one. There was also visible for a time, as it broke, a distiuct spiral and rotary motion, extending about one-third the length of the column from its upper end. " During the whole appearance the water at its base, considerably wider than the column, was churned into a seething mass and raised to a great height. " If the estimate of the distance from Falmouth Heights be approxi- mately correct, your previous correspondent'^ estimate of the diameter of the waterspout, 125 feet, must be within rather than beyond the actual; and, assuming this as correct, the height of the column can not have been less than 750 feet. The height to which the spray was thrown was decidedly greater than the width of the column, and must, there- fore, be estimated above 125 feet ". Doctor Abbott estimates the height of the waterspout to have been 3000 or 4000 feet, judging from its appearance above a distant hill, and the " probable distance away of the phenomenon ". He says: " From all that I can learn, the waterspout was about 25 miles distant". But it could not have been that distance away from him, and yet have been seen in an " east-southeast " direction from Falmouth Heights, and in a "northerly" direction from Vineyard Haven. Still from his point of view, at ten miles distance, it must have been over 10UO feet in height. He writes that " the waterspout was soon followed by marked atmos- pheric disturbances. Thunder, lightning, hail, and rain in abundance fell within an hour or more. A dense, dark cloud formed in the north- west, followed by a squall from the southwest, and the wind shifted in a short time from northeast to southeast, and then by southwest to northwest. The thermometer at 2:0u p. m. indicated 56, a very low reading for a place where it has varied but little from 70 all summer ". (I) COPY OF LETTER OF REV. CRANDALI. J. NORTH, OF NEW HAVEN, CONN., IN THE CHRISTIAN ADVOCATE FOB SEPTEMBER 24, 1896. Thousands of summer residents of Marthas Vineyard, Nantucket, and the adjacent Massachusetts coast were treated to a spectacle of remark- able grandeur one day in August, last. Guests at the hotels and occu- pants of cottages at the various resorts were just rising from dinner when the cry was raised, "A waterspout, a waterspout !" The scene presented to view was such as not one in a thousand had ever wit- nessed before or would ever see again. A large mass of heavy black cloud hung high above the ocean between Nantucket and Cape Cod. Suddenly it was seen to project a circular column of its own dense vapor perpendicularly downward, rapidly but not precipitantly, until sea and cloud were connected by a cylinder one or two hundred feet in diameter, straight as a pine tree, and at least a mile high. It was a waterspout indeed, of most unusual proportions and indescribable beauty. The sea was perfectly calm, the air almost motionless, the sun shining brightly, light summer clouds hangiug here and there over the deep blue sky; and in strange contrast with all the rest, was this lofty ihass of black vapor with its absolutely perpendicular support. To add to the weird effect occasional livid streaks of forked lightning shot athwart the black monster cloud above. The column was only slightly funnel-shaped just where it joined the cloud, and was of equal diameter the remainder of its length. At its ba*e th sea was lashed into a mass of white foam and spray that mounted upward as high as the masts of a large schooner. From Cottage City it seemed about six miles distant, but careful obser- vation through a glass from the writer's view-point showed that it was nearly in line with the light-ship off Hyannis Harbor, and still farther distant, its foot resting upon the sea beyond the horizon line. It must have been twenty or twenty-five miles away, but such was its magnitude that it seemed not more than one quarter of that distance. It moved slowly eastward, and continued with little change in form for seventeen minutes. Then it gradually attenuated till it looked like a dark nbbon hanging out of the cloud, and at length disappeared. The lashing of the water into foam and spray where its base had rested con- tinued unabated, which was evidence that the waterspout was still there, though now invisible, and that it might be expected to reappear. Surely enough, after an interval of about ten minutes, the cylindrical form of black vapor began to push its way downward again from the cloud and continued until it stood again upon the white mass of foam and spray mounting up from the sea surface. This time its top was more funnel- shaped and curved to the eastward. It continued eight minutes and disappeared. The projecting of the visible vapor downward caused the illusion that its origin was from the cloud rather than from the sea, and many supposed that it was a cloud-burst rather than a waterspout; but this is disproved by the continuance of the agitation of the sea surface during the interval between the disappearance of the first column of visible vapor and the formation of the second. Also the descent of the column was too slow for a mass of water falling from a cloud-burst, as was clearly apparent a little later, when a real cloud-burst occurred upon the mainland opposite, in full view from our point of observation. The apparent formation from top downward was due to the fact that the atmosphere became more rarefied by the swifter gyrations of the whirlwind at the higher altitude, causing the invisible vapor carried up from the sea surface to condense and become visible at the highest level first; then its visibility gradually extended downward as the velocity of the gyrations below increased. The whirlwind lashed the sea into foam and spray and vapor, and stood it up in an invisible column ; but it turned into cloud at the top first, then downward its entire length, until there it stood for many minutes before the wondering gaze of thousands, a veritable " pillar of cloud by day". The old sea captains of Marthas Vineyard said that this waterspout exceeded in size and grandeur anything of the kind they had seen dur- ing all of their seafaring experience. Enterprising photographers se- cured several good photographs of the remarkable phenomenon. (J) COPY OF DESCRIPTION BY DR. F. C. V. II. VOM SAAL; APPARENTLY COMPILED FROM OBSERVATIONS AT COTTAUE ClTY, AND PUBLISHED IN TUE SCIENTIFIC AMERI- CAN, NEW YORK, SSEPTEMIIER 26, 1896. About 12:30 p. m., August 19, 1896, one of the very dark clouds hover- ing over Vineyard Sound, between the mainland and Cottage City, was seen to send out a downward and sharply pointed streak of cloud matter, whose funnel-shaped basis above was not at all times visible. After a duration of about fifteen minutes it broke and completely vanished The apparition quickly emptied of their summer residents all the cottages along the bound and adjacent islands, Nantucket included. No photo- graphs were taken of this "first spout, to my knowledge. Shortly afterward a long tongue emanated from the same clouds, and was slowly pushed downward to a point about 100 feet from the surface of the ocean. Its height was certainly a mile, and the band-like shape gradually increased in width. With a glass, slow gyratory movements could be detected, also longitudinal stripes caused by falling water. This cloud-burst 4 made the water below, over a surface of many hundred yards, look like a boiling pool. The jumping spray from this was also caught and drawn upward into the whirl toward the downpouring col- umn. This latter, now of lighter color, being struck by the sun, was gradually withdrawn upward, evidently thinning and broadening toward its base. With a glass, mists could still be seen falling into the snow- white foaming area below. The duration of this second and most perfect phenomenon of i he day - there were three in all was about half an hour. About twenty minutes after its disappearance a third began to form, gradually coming downward from the same clouds, though from a spot a little farther north; but it hardly reached completion. It is very im- portant to note that, in this third case, the ocean below was entirely quiet for a time, being only disturbed later on, when the same process of condensation, mentioned above, caused a similar downpouring, espe- cially noticeable in the period of retraction. It was soon apparent that the agency causing the spouts had spent its energy; the column was eUdently thinner in substance and its formation slower and hesitating. It stopped midway, sending only an attenuated end farther, to be with- drawn upward soon after. During almost all of the time since the appearance of the first spout there was a heavy rainstorm accompanied by flashes of lightning from the northern and darkest portion of the long motionless stratum of clouds above mentioned. Cottage City, which had been in sunshine until then, was visited by a drenching rain some hours later. The long duration of the phenomena just described enabled the writer to form a somewhat different opinion of the nature of such water- spouts from what is commonly held. True, I must fall back upon the old (or rather older) explanation, that such whirls are caused by two winds striking each otherat an obtuse angle. The greatest rotary velocity must be placed at the spot, about 100 feet above the ocean, toward which the cloud matter from above and the spray from below were drawn. As condensation was continually transforming this cloud matter into water, it stands to reason that by far greater quantities of it were drawn down than was apparent to the eye. But the spout is from above and not from below, as a glance at- the cut conclusively proves. This also definitely settles the question as to what part the ocean takes in the constitution of the column, which is prac- tically none. The "boiling as if in a caldron" is not caused by the action of the circling wind, but by the great quantities of falling water. Nor is there a whirlpool action in, nor rising from, the body proper of the ocean. The way the spray, caught and drawn up, looked at times, easily explained to me how this delusion originated. The surprising tranqutlity of the clouds shows that such currents of wind need not be of great height, ht least not at their borders, where alone such whirls can take place. That the spouts scarcely shifted their position is proof that the velocity of the concurrent winds was almost 4 Of course this was not the " cloud-burst " of technical meteorology, for that is simply an unusual excessive rainfall. EDITOR. JULY, 1906. MONTHLY WEATHER REVIEW. 313 equal. It is certain that this velocity can not have been great. Several small vessels in close proximity at the time report that there were a great noise and gusts of wind in the immediate vicinity of the display, while beyond this there was almost a dead calm (Boston Globe, September 1). This latter statement, however, seems to be somewhat exaggerated. The following is a list of the photographs and the times when they were taken: SECOND APPEARANCE. THE PHOTOGRAPHS. We have to acknowledge our debt to the photographers who happened to be in the neighborhood of Cottage City on Au- gust 19, 1896, for an admirable series of pictures which cover the most important features of the phenomenon. Messrs. Bald- win Coolidge, 146 Tremont street, Boston, Mass; J. N. Cham- berlain, Cottage City; F. W. Ward, 16 Adams street, Burling- ton, Vt. ; Dodge, of Bangor, Me.; and E. K. Hallet, through Mr. Coolidge have placed their photographs in the care of the Weather Bureau for study, and our thanks are hereby extended to these gentlemen for their courteous contributions to the available scientific data that have come into our posses- sion. Both Coolidge and Chamberlain were stationed on the bluff at Cottage City, and had a clear view over the ocean to the waterspout; Mr. Ward stood near the foot of Hope avenue, in Falinouth Heights; Mr. Dodge was at the head of Vineyard Haven Harbor and saw the spout across the headland near East Chop light; Mr. Hallet was on the high ground west of Cottage City. These locations are shown on the chart, fig. 25. Some other photographs were taken on a small scale which contributed somewhat to the information contained in those reproduced in this memoir. The first appearance of a waterspout began at 12:45 p. m. and ended at 12:58 p. m. ; no photographs were made of this phenomenon, as it required time to bring the cameras into operation. Mr. Coolidge was half a mile away from his studio, at home for dinner, when this spout appeared; he started to secure his instrument, when unfortunately the spout disap- peared. He was ready on the bluff for the second appearance, which began at 1:00 p. m. and ended at 1:18 p. m. He used a rapid symmetrical Ross lens, with a focal length 14| inches from the diaphragm to the ground glass, or 13| inches from the back of the lens to the ground glass. Mr. Chamberlain brought his camera from his studio to the edge of the park, about 200 yards from the water, and his pictures, therefore, include a foreground showing several telegraph poles. The measured distances between these objects give the scale of the photograph, which becomes more valuable on this ac- count He used a large camera, No. 5 euroscope, with equiva- lent focus of 17 inches from the optical center to the sensitive plate, and a lens of 3-J- inches diameter. These sets of photo- graphs by Coolidge and Chamberlain both show a schooner which was sailing southeastward, and the positions of the schooner relative to the waterspout in its successive positions are very useful in determining the time intervals between the successive pictures. One of Chamberlain's, fig. 27, also shows the Succonesset Shoal light-ship, together with the waterspout near it, and this is important in identifying the direction of the sight lines from Cottage City. Mr. Ward's picture was taken with an Anthony kodak triad camera, 4| inches focal length, and the plate is 4 by 5 inches; this was enlarged by Coolidge to the 8 by 10 size. It shows the cloud formation and is most instructive as to general meteorological conditions; it also shows the curvature of the vortex tube at right angles to the view from Cottage City, where it seemed nearly straight, as seen in perspective during the second appearance. Mr. Dodge caught a distant view of the spout, and his picture also shows the great cumulo-nimbus cloud from which it descended ; his sight line passed just to the south of the East Chop light- house, and this distinctly identifies the direction of the spout at that time. Ballet's picture was taken with a small camera, but shows the large cumulo-nimbus cloud so well that I have taken it as the basis of the thermodynamic computations. Fig. No. Serial No. of photograph. 27 28 1 2 29 3 30 4 31 5 82 6 33 7 ase. Photographer. Moment of exposure. Photogra- pher's numeration of negative. 2d A Chamberlain 1:02 p. in. 2d B 1-OJ p m 7933 M C 2d I) Hs.Het 1:08 p. m. 1-12 p m 2d E 2d F Ward Cooli.lge 1:14 p.m. 7934 2d G Coolidge 1*17 p- ni 7936 THIRD APPEARANCE. 34 8 3d A Chamberlain 1:20 p.m. 35 9 3d B Chamberlain 1:24 p.m. 36 10 3d C Coolidge 7935 Notes on the photographs. No. 1. See fig. 27. Chamberlain, %d A, at 1:02 p. m., showing the waterspout 5.75 miles away, the lower face of the cloud in great detail, the foreground, the schooner about two miles out, and Succonesset Shoal light-ship about eight miles distant; the latter can be seen on the horizon about one third of the apparent distance from the spout to the schooner. No. 2. See tig. 28. Coolidge, 2d B, at 1:03 p. m., includes the Marthas Vineyard steamer, the spout and cloud in nearly the same condition as shown by 2d A. No. 3. See fig. 29. Hallet- Coulidge, 2d C, at 1:08 p. m. This picture is attributed to E. K. Hallet, photographer, and is copyrighted by Baldwin Coolidge, Boston, Mass., 1897. It seems to be somewhat later than 2d B because the vortex is leaning more toward the south, in accordance with the drift of the cloud stratum, which is brought out more positively in the third appearance, 1:20 to 1:25 p. m. ; it also gives us the dimension of the upper cloud which is not seen in the pictures 2d A, 2d B. Such small-scale photographs of the whole cloud region serve admirably to supplement the details to be found only on the large-scale pictures, and should always be made if possible by those having kodaks at hand. No. 4. See fig. 30. Dodge, 3d D, at 1:12 p. m., is chiefly of importance in locating the line from the head of Vineyard Haven Harbor to the waterspout. The curvature toward the southwest in the center begins to be seen from that angle I estimate that this was taken at 1:12 p. m., though there may be some doubt about the exact minute. No. 5. See fig. 31. Ward-Coolidge, 2d E, probably at 1:14 p. m., taken by F. W. Ward, enlarged and copyrighted by Bald- win Coolidge. The curvature of the tube is now fully seen from Falmouth Heights, where this plate was taken, this sight line being nearly at right angles to those from Cottage City. The vortex column appeared vertical at Cottage City, but strongly curved at Falmouth Heights with convexity toward the southwest. In the third appearance the convexity is seen nearly broadside on at Cottage City, and this indicates some change in the drift of the lower surface of the cloud relative to the layer of air at the water. This photograph gives the horizontal extent of the cloud. The Hallet photograph, fig. 29, 2d C, shows the precipitation in the thunderstorm preced- ing the waterspout by about one mile. No. 6. See fig. 32. Coolidge, 2d F, at 1:15 p. m., shows the enlargement of the tube before breaking up, the spray being cast out from all parts of the tube, especially at the top, thus causing the conical form. No. 7. See fig. 33. Coolidge, 2d G, at 1:17 p. m., gives the phenomenon at the breaking up of the second appearance, and it locates the schooner well up to the place of the vortex. There are three photographs of the third appearance. 314 MONTHLY WEATHEB REVIEW. JULY, 1906 No. 8. See tig. 34. Chamberlain, Sd A, at 1:20 p. m., shows the top of the vortex advanced toward the south relative to the base, indicating- the drift in the cloud stratum. The schooner has moved beyond the base of the waterspout and is between the two telegraph poles; a tow of barges is just com- ing into view on the extreme right of the photograph. No. 9. See fig. 35. Chamberlain, 3d 11, at 1:24 p. m., is simi- lar to the preceding, but the base of the spout has moved toward the southeast; the schooner and the barges are ap- proaching each other. No. 10. See fig. 36. Coolidge, 3d C, at 1:27 p. m., is a later phase of the third appearance, with the schooner and head of the tow nearly in the same line. The schooner is about two and one-half and the barges about three miles distant from Cottage City. POSITION OF THE WATERSPOUT IN THK SOUND. It will be seen that from the foregoing notes, the photo- graphs, and the chart we have considerable data with which to find the position of the waterspout in Vineyard Sound. It will be best first to fix our attention upon the first part of the second appearance as shown in the photograph, Chamberlain, 2d A, fig. 27, taken at about 1:02 p. m. My own personal survey of the ground gives the following distances approxi- mately, as plotted in fig. 26. The telegraph poles are marked 1, 2, 3, 4, and Chamberlain's camera is marked 5. We have the distances, 5-1=450 feet, 5-4=504 feet, 1-2=72 feet, 2-3=120 feet, 3-4=66 feet, 4-1 = 132 feet. The sight line to the water- spout is laid down, also that to the schooner; the angular distance between them is 5.44. On photograph, Chamberlain, 2d A, fig 27, is also shown the Succonesset Shoal light-ship, which appears as a dot on the horizon about one-third the dis- tance from the foot of the spout to the schooner. This enables us to orient the entire drawing with great accuracy. These lines are now transferred to the chart of Vineyard Sound, pub- lished by the U. S. Coast and Geodetic Survey as No. 112, August 1901, of which a portion is reproduced as tig. 25. On photograph, Dodge, 2d D, fig. 30, taken from the head of Vineyard Haven Harbor, the spout is shown just to the south of East Chop light-house, and that line is added to the chart. The spout was also seen from Woods Hole at the head of Little Harbor, and a measurement of the line as described gives mag- netic declination S. 75 E., which is also drawn. It was seen from Edgartown, 10 east of true north, by one report, this line being indicated on the chart, fig. 25. Dr. George Faulkner's family saw the second appearance from their residence near the water in the town of Falmouth, and their sight line passed just south of the steamboat wharf at Falmouth Heights. This enables us to fix another line as shown on the chart. These lines all converge quite accurately to a point a little south of east of L'Hornmedieu Shoal, where other observers also placed it by estimate, and I have accord- ingly cut off the Chamberlain line of sight at that point on my chart, fig. 25. This makes the distance from Chamberlain to the waterspout 5.75 miles, and to the Succonesset Shoal light- ship eight miles; as the schooner was in the usual inside chan- nel it was about two miles distant, as shown on this same chart. It is instructive to note that some spectators imagined the spout to have been more than twenty miles from Cottage City. It is of great importance to be able to accurately convert the distances shown on the photographs into angles, because the angles, combined with the length of the sight line, give the corresponding linear dimensions at the spout and at the schooner. We have to measure the linear distance on this photograph from the middle of the schooner to the middle of the waterspout, which is 48 millimeters on fig. 27, Chamber- lain, 2d A; at the same time the angular distance between the sight lines from the camera to these two objects is found from the survey to be 5.44. This was determined by plotting the lines of the survey on a large scale, and testing the result by numerous checks on the other distances measured on the pho- tographs. Hence, 1 millimeter = 6' 48" of angle. This is th<- fundamental dimension, and it leads to 1 millimeter = 60 feet = 18.3 meters at the waterspout. DIMENSIONS AS MKASCHEI) ON I'HOTOdKAl'H 2l> A, FIG. 27. By this process we obtain the absolute dimensions given in the accompanying table. Distance from middle of schooner to middle of waterspout on the horizon, measured on the photograph, 48 mm. Angular distance subtended by the sight lines at th< camera, as determined by the local survey, 5.44. Hence,! mm. subtends 5.44 H- 48= 0.1133= 6.80' = 6'48". I'Y. i. Heten, 3014 10 444 >0 888 60 00 tV 7 19111 *:t <; 240 720 '2W '111 4"'0 144 t.H .S'l 840 256 O'i 3GOO lu -7 8 Approximate height of the tp uf thr Homl (from '.id <.', fig. 29) .. . 1GOOO 4876.8 The distance moved by the waterspout from the beginning of the first appearance at 12:45 p. in. to the end of the third appearance at 1:28 p. m. can be found as follows: The positions of the schooner and the waterspout at the time of taking Chamberlain's three photographs are shown on the chart (see fig. 25), as nearly as can be determined; 2d A, at 1 :02 p. m. ; 3d A, at 1:20 p. m. ; 3d B, at 1:24 p. m. In the inter- val, 1:02 to 1:24 p. m., 22 minutes, the schooner moved about 0.65 mile. This is at the rate of 1.7 miles per hour. The schooner was sailing nearly east-southeast, and the sails were set to catch a wind from the northwest. The wind was very light at the time, as stated by several observers, and as is shown on the photographs by the smoothness of the water. In the interval, 12:45 to 1:28 p. m., 43 minutes, the vessel passed over the distance 1.27 miles. Similarly, the waterspout passed over the distance 0.4 mile in the interval, 1:02 to 1:24 p. m., and over the distance 0.78 mile, or 4018 feet, in the interval, 12:45 to 1:28 p. m., while the whole phenomenon was in evidence. This is at the rate of 1.10 miles per hour. It is instructive to compare these results with the estimated dimensions and distances as reported by different spectators. Mr. Hanes estimated the eastward progress as 2 miles, diame- ter from 100 to 300 feet, height 4000 to 5000 feet. Mr. North made the distance of the waterspout from Cottage City 20 miles, or more, supposing that the foot of the vortex \vas be- yond the horizon, and that from his view-point the base of the tube was 20 feet above the sea level; he made its eastward move- ment about equal to its own height before it disappeared, which is nearly correct, and called this one mile. Mr. Coolidge. October 19, 1896, estimated the height of the spout at from 6000 to 10,000 feet, or 21 to 28 times its diameter, and the latter at 300 to 375 feet and the distance 8 miles. Mr. Cool- idge, September 1, 1897, made it 400 to 600 feet in diameter at its mid-height, from 4000 to 6000 feet, or perhaps 10,000 feet high, and 5 miles distant. The observers on the yacht Avalon, which was very near the waterspout, made the diame- ter 100 feet. Mr. E. H. Garrett estimated over 300 to nearly 500 feet high, and 125 feet in diameter; Mr. John B. Garrett: 6 miles distant, height, 750 feet; diameter, 125 feet; height of cascade, 125 feet; Mr.Abbott; height, 3000 to 4000 feet. [The treatment of this waterspout will be continued in Sec- tions VII, VIII, and IX.] I o .0 cS 3 O > 5 1C CD 2 "3 o o o. - a 8 M a o a s O) _e \* oT a P i ~ CS T3 a fi a T-l I d I E = Oi ' W o I a a 1 E o I *w 6 X! s i 5 G, a, xi J-H I a i S 0! oo I 4-3 5 1 o T3 Lt IS *J M a CO I E AUGUST, 1906. MONTHLY WEATHER REVIEW. 360 STUDIES ON THE THERMODYNAMICS OF THE ATMOS- PHERE. By Prof. FRANK H. BIOELOW. VII THE METEOROLOGICAL CONDITIONS ASSOCIATED WITH THE COTTAGE CITY WATERSPOUT. The data that have been collected regarding the meteoro- logical conditions prevailing at the time of the Cottage City waterspout are sufficiently extensive and accurate to enable us to study carefully the causes that produced the phenomenon, and to derive several important results regarding the forma- tion of lofty cumulo-nimbus clouds and the dynamic actions going on within them. In this instance we can compute approximately the forces producing the ascension of the buoyant vapor in the cloud, the formation of hail and the energy working in the vortex at the base of the cloud which developed as the waterspout. I shall proceed to give these facts in detail, as this computation may serve as a type to be followed in discussing other cases of similar local atmospheric action. Besides the formation and dissipation of the tube, noted in the several reports and shown in figures 27-36, there are special features to which attention must be directed: (1) In the third appearance the vortex tube shows a gradual tapering of the form from the cloud to the sea level, but in the second arjpearance the tube seems to have about the same diameter from the cloud to the sea level. It is necessary to account for this divergence in the type. (2) The photographs show a peculiar set of boundary curves in the cloud level which depend upon certain dynamic forces that we shall attempt to discover. (3) At the foot of the tube, near the sea level, there was a great commotion of the waters, with a white nucleus just under the tube, and finally a beautiful cascade of imposing dimensions surrounding it. These are topics of especial interest besides those usually considered in discussing such vortices. The meteorological conditions are given quite fully by the regular observations of the neighboring Weather Bureau sta- tions, Nantucket being a station of the first order and having a continuous barograph and thermograph record ; Woods Hole, a station of the second order, with complete daily evening observations; and Vineyard Haven, a station of the third order, with daily temperature, wind, and cloud reports. The daily weather map of 8 a. m., August 19, 1896, exhibits the general conditions for the United States, and from it can be obtained the local conditions prevailing at that hour, at least approxi- mately. The physical appearance of the waterspout has been described fully in the reports already given, and there is also a series of notes of which further use will be made in the proper places. We shall endeavor in this Section, VII, to discuss the scientific problems which are naturally suggested by these data, with the view of illustrating typical methods of treating waterspout and tornado phenomena whenever these Clear during the forenoon, partly cloudy during the afternoon. Thun- derstorm: thunder first heard, 1:58 p. m.; loudest, 3:02 p. m.; last, 3:50 p. m. Storm came from the northwest and moved toward the southeast; temperature before the storm 66, after 67; direction of the wind before the storm northwest, after, northwest; during the storm the wind shifted to the northeast. Rain began 2:55 p. m.; ended 3:20 p. m.; amount 0:33 inch. Maximum wind velocity 38 miles per hour from the northwest, at 3:00 p. m. A few hailstones fell about 3:10 p. m., and quite a heavy fall of hail was reported a few miles north of this office. The weather map of August 19, 1896, is represented as fig. 37. In Section A of Table 50 the meteorological data are given for Woods Hole, Vineyard Haven, and Nantucket on August 19, 1896. They are extracted from Forms 1001-Met'l. of Woods Hole and Nantucket, and Form 1004-Met'l. of Vine- yard Haven. The notation is as follows: B= barometric pres- sure; <=dry-bulb thermometer; 1.1=0.0130. ( argument, j B L 0.0128 0.0135 Resulting ratio of |L. -"l "! This ratio should have come out -J- = 0.0130 if the trial val- ues of t l had been correct, and interpolation shows that in order to make it so the value of <, should be 9.3. Repeat the computation, *, 9.3 e 8.72 Smithsonian Table 43. I a . 1.3509 Cloud Report Table 95. (7 a .9022 As above. (\ nisn M - !! - .4487 B 1 672 Cloud Report Table 96. This completes the check, and gives ,= 672, < l =9.3,e 1 =8.72 at the base of the cloud. We next compute the height at which the visible cloud base floats above the sea level. For Table 91 we have the formula of barometric reductions and the corresponding height, 365 MONTHLY WEATHEB REVIEW. AUGUST, 1906 log B a log J5,= 7; 763.27 7?, 672 +m /3m ym. Logarithm. 2.88268 2.82737 m /3m fm, .05531 Since Table 91 is constructed to find m through the argu- ments, height = H and mean temperature T m = 0, in order to obtain H by the inverse process we must compute m itself fromm /3m ym=. 05531, by taking m =.05531+ /3m + rm. By Table 92, with the arguments (7? =763 and e =10.9) we have: I=.378 = 0054 and For argument For argument II=/3 (forassumed//=1100) .0048 So that+/3m (for m=.055) ....................... +0.00026 By Table 93, + r m (for =4.57 B 672 t. 9.3 e,=8.72 m ff. B l -B 128 i_ _9.3 1728 meters. e'-e,-4.15 m ^TO ym .00401 The corrections for /3m ym can be neglected. j 74 meters. Table 91, argument (0=0, m=. 00401) ' ' ' | 243 feet. The pressure gradient in the ^-stage is, (G. B ) = 6.76 By observation. (G. B ) c = 6.70 Table 147, V, of the Cloud Report. (D) THE (5-8TAGE. OR FROZEN PROCESS. The 5-stage extends to the visible tops of the clouds, and we may, therefore, take such a temperature as will, through B. .0130. Divide B l B l etc. by 17.28, or the height in units of 100 meters. the intermediate computation, produce a height which agrees (G.B t ) 7.40 (G.tjo .538 (.e,) .240 By observation. with the height measured on the photograph. A preliminary (G.BJc 7.40 (G.tjc .540 (G.e^) c .260 Table 147, IV, for trial of 11C. for the temperature gives a height that is somewhat lower than the apex of the cloud, and a second trial of 12C. seemed to be about right, so that it has been In order to compare this with the normal conditions of the adopted for the numerical example, atmosphere in August, we again take the Nantucket data, For the constant in the 5-stage, we have, Barometry Report, page 707, where (B, t, e) are given on the t n 3500-foot plane and the 10000-foot plane. B l The depth of the o-stage = H a = 1078 meters = 3537 feet. e n The depth of the /3-stage = Hp = 1728 meters = 5669 feet. ^ Height of top of /3-stage above sea level H l 2806 meters = 9206 feet. -^ By interpolation to the given heights, 1078 meters=3537 feet and 2806 meters=9206 feet, we obtain in metric measures, e from page 707 of the Barometry Report, the data. mm. C. mm. B 1 548.6 t 1 9.22 e 1 4.95 B, 671.6 t, 15.72 e, 11.25 0C. ) 539. v brought from the top of the j'-stage. 4.57 j 534.4 ' B, 123.0 t l t, 6.50 e 1 e, 6.30 II, Assume t n Divide by 17.28, the height of the /3-stage in units of 100 meters. Ills (G.B^B7.11 (G.t^B.BlG (G.e^B -364 These sets of gradients for the /3-stage will also require ''-Constant further discussion. (c) THE 7--8TAGE, OR FREEZING PROCESS. In the freezing stage there is no change in the temperature, which is =0, nor in the vapor tension, which is e =4.57 mm., and we have only to compute the variation in the pressure B . Constant 110 + III0 .4163 ] .0086 .0130 Arguments. 1.3664 TablelOl, (<-0,- .0130 V. \ B I - .4273 Table 98, (s u e n = 534.4, e - =.0130 V + .0134 Table 102, (t=0 , e " =.0086). \ B \\~ e n ' .9525 12 C. L o e B 544. 4.57 0.0130 e 11 1.64 mm. Table 103, Is Constant .9525 Argument. t= 12C. 1.3556 Table 101, (<=-12,-? =.0130 \ \ R I brought from the /3-stage. .4031 This is the constant to determine B u . Assume B n 415.0 419.0 e 11 1.6 1.6 B"e n 413.4 417.4 367 MONTHLY WEATHER REVIEW. AUGUST, 1906 e" B u e" .0040 .0040 II a -.4098 -.4105 + .0066 .0066 Table 98, argument. Table 102, arguments. -.4032 -.4039 Interpolation indicates 7/ u =414.5. Assume B 11 414.5 e" 1.64 Be" 412.9 .0040 Arguments. II S - .4097 Table 98/7?" e"=412.9,^=.0130j. +.0066 Table 102 <=- 12, \ .4031 This checks the constant for the pres- sure 7? n =414.5 mm. The depth of the - . 072 ) - . 078 - .319 - .302 - . 00170 - . 00169 (G) Observed, (G) Cloud. I [Lower 102.8 2880 539.0 4.57 4.05 9,449 21.22 32.0 0.180 I 6 74 6.76 0.10 243 .082 (G) Observed. Upper .... 100.2 2806 544. 4. 57 3.95 9,206 21.42 32.0 0.1HO Gradients. C radients C -7.10 .638 - .210 C - .089 - .291 - .00288 (G) Observed. /3-stage Range .... 61.7 1728 < - 7.60 \ -7.11 - .610 - .260 - .376 - .361 2.43 5,669 ] - . 091 ( - . 086 - .294 - .207 .00312 - . 00137 (G) Cloud. (G) Barometry. [Lower 38.5 1078 672.0 9. 3 8. 72 1,52 3,537 26.46 48.7 0.343 f Upper.... a-stage -[Range.... 38.5 C ( - 8. 16 1078 \ - 8. 10 ( - s. 21 rradients - 0. 963 - 0.950 - 0.376 -0.201 - 0.192 - 0.296 1.52 3,537 r C - 0.101 I - 0.101 < - 0. 098 radients - 0. 531 - 0. 522 - 0. 206 ' - 0. 00216 - 0. 00230 - 0. 00355 (G) Observed. (G) Cloud. (G) Barometry. I Sea level.. 763.27 19.72 10.92 30.05 67.5 0.430 (2) In the temperature of the a-stage the normal gradient is 0.206 F. per 100 feet, and the waterspout gradient is 0.531, which is only a little short of the true adiabatic gradient. The normal temperature fall from the sea level is 7.3, or a change from 67.5 to 60.2, at the 3537-foot level. This change from the normal temperature fall (which is small on the Atlantic coast in summer because of the southward bend- ing of the isotherms over the ocean areas) to the adiabatic convectional gradient is a very striking fact. The latter gives a fall of 18.8 F., or a fall from 67.5 to 48.7 F., instead of to the normal temperature 60.2 F., showing that something has occurred to suddenly reduce the normal temperature at the 3637- foot level by the amount 11.5 F. The cause of this will be explained in the following para- graph. (3) The normal gradient of the vapor pressure is 0.00355 inch per 100 feet, for Nantucket in August, and this gives a total fall of 0.126 inch, or a change from 0.430 at sea level to 0.304 at the cloud base. The abnormal or convectional gra- dient prevailing at the waterspout is 0.00246, which makes a change of 0.087, or a diminution of the normal vapor pres- sure 0.430 to 0.343. Thus, we have a gain of 0.039 in the vapor tension for the waterspout. The result for the a-stage is a total decrease of the pressure by 0.110 inch, a total de- crease in the temperature by 11.5 and a total increase in the vapor tension by 0.039 inch. This must be interpreted to mean that the air is rising from the sea level, carrying with it aqueous vapor into levels of temperature about 11.5 F. lower than that which prevails in the normal August weather. (4) The depth of the a-stage is 3537 feet, of the /3-stage 5669 feet, of the ^-stage 243 feet, and of the 3-stage 6765 feet, to the assumed top of the cumulo-nimbus cloud. In the ,3-stage the pressure gradient changes from the normal rate 0.085, to the convection rate 0.089 per 100 feet. This is equivalent to a fall of 4.82 inches in the normal state, from 26.58 to 21.76 inches; and to a fall of 5.04 in the convection cloud, from 26.46 to 21.42 inches. There is, therefore, a total fall in pressure of 0.227 inch, ( .004 x 56.7 = 0.227), induced by the change from the normal to the waterspout conditions in the /3-stage of the cloud. We have thus a dif- ferential gradient per 100 feet in the a-stage of 0.0031, and in the /3-stage of 0.0041 inch in favor of a vertical current. The excess in the /3-stage over the a-stage of 0.0010 inch may be taken as the effective gradient due to the additional latent heat produced by the condensation of the aqueous vapor to liquid water. This is only one-third the amount of the gradient due to the cause which produces the general uplift of the air that feeds the cloud. This criterion also proves that there is a more efficient cause for the vertical pressure gradient than the condensation of the aqueous vapor, which has been so generally considered by meteorologists to be the true source of the energy that drives cyclones, following Espy's sugges- tion of fifty years ago. (5) The temperature gradient in the /S-stage changes, de- rived for the normal, is -0.207 F. per 100 feet, and 0.294 F. per 100 feet in the cloud. This amounts to a fall of .-11.7 F. in 5667 for the normal state, and 16.7 F. for the convec- tional state, carrying the normal temperature from 60.2 to 48.5 F. in the /3-stratum, and from 48.7 to 32.0 F. in the actual /3-stage of the waterspout cloud. This is equivalent to a gradient excess in the a-stage of 0.325 of the convection over the normal gradient, and in the /3-stage is one-fourth that in the a-stage. In the case of the pressure the excess in the /?-stage is four-thirds that in the a-stage. The tempera- ture difference of gradient diminishes rapidly in proportion to the height up to the f-stage. In that stage and the 3-stage there is not much difference between the normal and the com- puted convectional gradients. This indicates that the effec- tive cause of a vertical gradient is about exhausted at the height where the isotherms of the is located, or, in other words that the vertical convectional action is properly confined in the waterspout cloud to within about two miles of the ground, and is most active in the lower portion of the cloud. In cyclones this vertical convection is usually limited to within two or three miles of the ground, though the accompanying dynamic action may penetrate into the upper strata as high as three or four miles; in hurricanes the penetration reaches to six or seven miles at least. (6) In the /9-stage the gradient of the vapor tension changes from the normal 0.00437 per 100 feet to 0.00288; the total fall in 5669 feet amounts to 0.248 inch, ( 0.00437 x 56.7 = 0.248), in the normal, to 0.163 in the cloud convection, making the fall which should be from 0.304 to 0.056 in the normal state, from 0.343 to 0.180 in the cloud. The differ- ence of gradient is +0.00109 per 100 feet in the a-stage, and + 0.00149 in the /3-stage, showing that large quantities of vapor are carried upward from the sea level in both stages, but that there is a condensation of aqueous vapor to water equiva- lent to -f- 0.00040 inch per 100 feet. We can not carry out this comparison between the normal and the convectional gra- dients in the ^--stage and the 3-stage, but the evidence is that they have become practically identical. 369 MONTHLY WEATHER REVIEW. AUGUST, 1906 (7) It is desirable to compare these vertical gradients with the commonly observed horizontal gradients. We have in the pressure 0.34 inch in 9206 feet. This is equivalent to 13.47 inches in 364,525 feet, or 111,111 meters, or one de- gree in the standard latitude of forty-five degrees. Now, on the weather maps, 0.70 inch in five degrees, or 0.14 inch in one degree, is the average horizontal gradient in a highly developed cyclone. Hence, in the convectional cloud forma- tion the vertical gradient is about one hundred times as large as in such horizontal motions. In the temperature the fall of 16.5 in 9206 feet is equivalent to 654 in 111,111 meters, or one degree, and this too is about one hundred times the horizontal gradients which are found on the weather maps. This indicates that the scale of operations on the horizontal plane is only one hundredth that which occurs in the vertical direction in convectional clouds. The linear dimensions of cyclones are usually about one hundred on the horizontal to one in the vertical direction, and these two facts, taken together, show how much less force is required to drive a horizontal than a vertical current. THE CAUSE OF THE FORMATION OF THE WATERSPOUT CLOUD, AND THE VERTICAL CONVECTIONAL VELOCITY. (1) Vertical convection due to surface heating. We now reach the important question, what was the physical cause of the formation of the cloud, and the vertical convection within it that was the immediate condition of the generation of the vortex tube extending from the base to the ocean? Fortun- ately, one answer to this question is entirely excluded from our consideration. The disturbance of the normal stratifica- tion which produces an abnormal system of gradients and the corresponding vertical currents, may be due to two causes, (1) the surface layers may be overheated relatively to the upper layers, or (2) the upper layers may be undercooled re- latively to the surface layers. Either cause would be equally efficient, and it is only a question of which one is actually operating in the case of this waterspout. Overheating the surface layers is due to a perfectly definite physical process, namely, as follows. The effective solar radiation falling upon the earth's atmosphere consists of short wave lengths from 0.30 fi to 2.00/i. (Compare figs. 3 and 4, Tables 1 and 2 of my paper on " Solar and terrestrial physical processes," MONTHLY WEATHER REVIEW, December, 1902, Vol. XXX, pp. 562-564.) These short waves, whatever may be the true effective solar temperature at which they are produced, penetrate to the sur- face of the earth with two sources of depletion, the first, by scattering in the upper atmosphere which cuts out a large percentage of them, and produces the strong glare that is characteristic of the higher layers; the second, by absorption in the aqueous vapor at certain wave lengths, which causes the observed depressions or cold bands in the energy spec- trum. There is good reason for believing that the upper as well as the lower atmosphere is heated by the passage of the remainder of the short waves by only a very slight amount, and that this is practically negligible in general discussions. But these short rays falling upon the surface of the earth are readily absorbed, and this absorption powerfully raises the temperature of the land and ocean areas. That practically ends the history of the incoming solar radiation. The terrestrial radiation, on the other hand, is of an entirely different character, and it has a very different effect upon the earth's atmosphere. The heat radiations at terrestrial tem- peratures, where the absolute temperature ranges from r=200 to T=325 , have wave lengths extending from %L to 40/i, and, thus the outgoing wave lengths begin where the in- coming lengths end. Many of these long waves, in radiating from the surface of the earth, are quite readily absorbed by the atmosphere, and the heat percolates from the lower through the upper strata by a process of slow conduction and convection. The aqueous vapor certainly absorbs many waves, as from 4/j to 8,'t, and possibly most of the waves beyond 12/j.. It was shown very distinctly in my International Cloud Report that the surface temperatures do not diminish with the height at an adiabatic rate, but much more slowly, as is indicated on charts 78 and 79. In Table 162 of the same report is given the number of calories per kilogram required to convert an adiabatic atmosphere into the actual atmosphere as observed. It shows an increase from the ground until the number is about 9.5 calories at the 13,000-meter level in summer and 11.0 calories in winter. This amount of heat may be taken to represent the effect of the outward flowing flux, which, like a slow conduction, keeps the upper atmosphere warmer than it would be if the outward-going waves had the same length as the inward-coming waves. This difference in wave length is the most important factor in the economy of the earth's atmospheric temperatures. We may now fix our attention more closely upon the changes in the surface temperatures as measured in the normal diurnal and annual periods, and in the local variations of all kinds upon the average conditions. The change in the transparency of the atmosphere due to cloudiness, the difference of altitude of the sun, the character of the surface, whether water area, moist ground, or dry desert, all determine the effective temperature of the surface at any given time. These react upon the cor- responding outgoing radiation, which first heats up the lowest strata of the atmosphere, or cools it, according to the pre- vailing conditions. Strong surface heating by day and cool- ing by night is, therefore, the regimen to which these layers are subject, and the integral effect of this action in its pas- sage through the upper layers, finally builds up the observed normal gradients of temperature which by no means produce an abiabatic rate of stratification. Temporary convectional currents upward by day, downward by night, upward in some local areas, downward in other areas, constitute the common types of motion due to these causes. The formation of the lower cumulus clouds with moderate convection, of thunder- storms in strong convection, and of desert sand vortices are typical examples of purely surface overheating with vertical convection. But there is an entirely different class of vertical convection, to which sufficient attention has not been paid in meteorological investigations. (2) Vertical convection due to the overflow of cold upon warm cur- rents of air. It is evident that the vertical convection in the cumulo-nimbus cloud of the Cottage City waterspout could by no possibility have been due to the overheating of the surface by the incoming solar radiation. The phenomenon occurred over the ocean and all the meteorological data of Table 50 show that there was actually no superheating effect near the surface at that time. The prevailing temperature was 67.5 F., while the normal for the month of August at Nantucket was 67.7 F. ; and the humidity was 64 per cent, while the normal was 84.3 per cent. In fact, the 19th of August, 1896, was the driest day of the entire month, and the powerful vertical convection then taking place could not have been due to the solar radiation act- ing on the surface conditions. We must, therefore, look for another efficient principle capable of producing the powerful effects shown in the showers preceding the family of water- spouts and the thunderstorm with downfall of hail following them. This we can readily discover by referring to the weather chart of the date, August 19, 1896, tig. 37. This map shows that a well-defined area of high pressure was just pushing its southeastern front over Vineyard Sound, that the winds were from the northwest, and that there was a fall in temperature of about 15 along the coast line, due to the advance of this cold area. We have, therefore, merely to assume, in accordance with the general fact, that the upper strata are moving eastward in advance of the lower, and that this cold air from the high area was blown forward over Vine- AUGUST, 1906. MONTHLY WEATHER REVIEW. 370 yard Sound earlier in the strata a mile or two high than at the surface. A sheet of cold air overran the low, warm, and quiet strata about midday, while the cold air followed at the surface a few hours later, and in these facts we have the exact conditions required to produce the observed powerful convection. Such abnormal cooling of the higher strata is as efficient in producing vertical convectional gradients as a superheating of the surface would be, and the evidence that this was the actual case is so good as to render it a practical proof of this circumstance. The upper strata were cooled suddenly by 12 to 15 F., and this brought showers, the waterspouts, and the thunderstorms in close suc- cession. These were followed later by cooler conditions at the surface, giving a temperature fall from the maximum of the day, 72, to the minimum, 56.5, at Vineyard Haven. Under these conditions all the observed facts find so natural and satisfactory an explanation that no further remarks seem to be needed to enforce the theory. But, it should be noted that this overflow of relatively cold layers of air at a moderate elevation upon the warm surface layers, this forereaching and temporary stratification causes an abnormal system of gradients which produce the vertical cur- rents required to set up the motions that tend to reestablish the normal equilibrium of the atmosphere. This local disturb- ance of the average gradients, due to the fact that the cold upper air, under certain configurations of the lower currents, is drifted forward upon them, is the primary cause of most of the phenomena classified as thunderstorms, tornadoes, cyclones, and hurricanes. In short, all these violent local disturbances of the lower air are largely due to this cause, and this is the true source of the energy expended, though it has been attributed by one school of meterologists to the latent heat of condensation, and by the other school to the eddies established by differen- tial horizontal velocities. These two latter sources of energy need not be excluded from consideration, for they contribute their quota to the total energy of circulation, but the first cause is the abnormal stratification of the air at moderate elevations. Thus the groups of thunderstorms which frequent the southeastern quadrants of the cyclone are due to the over- flowing of the cold northern current upon the warm current from the south. Tornadoes have the same origin and their location shows that they are due to this cause. The cyclone itself is generated by warm currents of air from the Tropics underrunning the cold sheet which rotates above the surface of the earth, in the hemispherical whirl north of latitude 35. The reason for the outflow of warm currents from the Tropics has been indicated in the International Cloud Report, chapters 8 and 10; also there will be found in the MONTHLY WEATHEB REVIEW for January and February, 1903, and in the preceding papers of this series, further illustrations and re- marks on this theory. The West Indian hurricanes in a simi- lar way are produced in the late summer and autumn by the overflow of the cool upper sheet from the North American Continent upon the warm tropical lower strata, because this sheet is then increasing in size with the southward retreat of the sun. The withdrawal of the sun to the south in fact brings the thermal equator of the higher strata toward the geographical equator earlier than that corresponding to the lower strata. Hence, relatively cold air from the temperate zones at considerable heights, begins to overlay the tropical warm and moist lower strata, and this induces the long continued vertical convection, localized in the hurricane vor- tex, which in its progressive movement may traverse thou- sands of miles along its parabolic track. The form of the track is due to the influence of the general circulation local- ized in centers of action, which builds the south Atlantic high area on the ocean and is manifested in the trade winds, so that the hurricanes usually gyrate along the edge of this special configuration. The power which is expended for days in succession in a hurricane is due to the fact that the wide expanse of the upper cold sheet covers the temperate zones and overlaps the Tropics at moderate heights. As long as this contrast of temperature, due to abnormal stratification, continues, there is a sufficient source of energy in the resulting thermal engine to produce powerful vertical convection currents, and to sustain the most violent hurricanes, in which the vortex has a depth of several miles in a vertical direction. This theory seems to harmonize completely with what is known about the meteor- ology of the lower air, and to be such a satisfactory escape from the difficulties of (1) the condensation theory and (2) the dynamic eddy theory, which have always encountered both practical and theoretical objections, that we may expect to find confirmation of it in the future development of the mechanics of the atmosphere. OCTOBER, 1906. MONTHLY WEATHER REVIEW. 470 STUDIES ON THE THERMODYNAMICS OF THE ATMOS- PHERE. liy I'rof. FRANK II. HIGEI.OW. VIII. THE METEOROLOGICAL CONDITIONS ASSOCIATED WITH THE COTTAGE CITY WATERSPOUT Continued. RELATIONS BETWEEN WIND VELOCITIES AND ATMOSPHEKIC PRESSURES. In meteorology there are various relations depending on the influence of wind velocities upon pressure, which must be considered in addition to the usual static barometric pressure used in the construction of synoptic weather maps, and in the determination of heights. Those reductions assume that the air is calm, and that the difference of pressure depending on wind velocity may be neglected. On the other hand, in torna- does, waterspouts, hurricanes, and strongly developed cyclones the velocity of the wind gives rise to variations of pressure from point to point. In theoretical meteorology, and in the practical calculation of the effects of high winds upon build- ings and other structures, as when a tornado passes over a city or thru a forest, it is very important to have a definite knowl- edge of the relations between these two phenomena. The literature of this subject is very extensive, but an attempt will be made to bring together in suitable form for reference the facts likely to be of value to meteorologists, engineers, and architects. FORMULAS FOE WIND VELOCITIES AND PRESSURE GRADIENTS. Table 52, formulas 1-13, contains the development of the velocity-pressure function from the primary equation (203), page 505, Cloud Report, Vol. II, 1898-1899. By means of the auxiliaries on the side of the table the final equation 13 is found, TABLE 52. Formulas for wind velocities and pressure gradients. Auxiliaries. (1) (2) (3) (4) (5) (6) (7) OP = i u + vO v + wdw -f- gdz. P P T = udu + vdv + wdw -\- gdz. 2 ' /'o T * P = - p = P\\ 1 T n P Po P T n gdz). i T r i -logp= j T 5 (5 2 -?o 2 ) + ( z -<0 log n Po -log,, p = log p-log p = -l^ 9 dP _dp _9B ~r^ == Y~~ : ~~B" /) m = 13595.8 kilograms. /> = 1.29305 kilograms. I = 7991.04 meters. g = 9.806 meters. Natural logarithms. Common logarithms. J B = 0.001742 B q l . Integrate the velocity term alone when the gravity term is omitted. (8) (9) (10) (11) (12) (13) (14) Standard. (15) Other density. = 0.001742. q 2 = 574.06 q = 23.06 29,1 For 9o = 0. B and AB in meters. q = velocity in meters per second. TABLE 53. Barometric gradient sustaining an eastward velocity only. Auxiliaries. D= 111 111 000 in millimeters. ;/= 9.806 meters. 2 n = 0.000 1458. 5=0.1572 " f^usiny. . ,. 760 T l> m millimeters. 17? 7? = .._ _ -v sin . v in meters per second. 17.7.J ./ AR= In v sin n J. = 0.02589 B T D*iu 1 lc. 0.02859 0.03366 The weight of the body to be just sustained, a = specific weight. (25) Weight W a = f- * ' P x 1000 kilograms = 4188.8 r 3 p = 4188.8 r" ^ /'l = * ff ^ ^ x 1000 kilograms = 523.6 Z* 3 /. = 523.6 Z) 8 ?J? "8 ^) (26) Specific weight ? = specific weight of body = /Oj specific weight of water ,_, p density of air above surface B T /> ~ density of air at surface ~ B T' (28) For equilibrium J p A = W g = 37.200 E - r 1 . w>. k = 4188. 8 r'. Pv> . (29) = 9.300 B T D\ii?.k= 523.6 Z>'./<,,, (30) = 0.10356 ^ ^ r 1 . w>.k= 4188.8 r 8 . />,. (31) = 0.02589 ^- ^ Z>'. w'. k = 523.6 Z>". Pw . OCTOBER, 1906. (32) (33) (34) (35) MONTHLY WEATHER REVIEW. TABLE 56. Velocities. 472 _ ^l 8 ^ 8 T r jl" 37.20 B"k' 523.6 T.^, 9T300 ~F' 2 _ 4188.8 B 9 T^ rpy ~~ (U0356 7i 2; A iv* = 523.6 w = w = 10.610 7.503 201.12 1 k Logs. [1.02578]. [0.87526]. [2.30345]. [2.15294]. VI ' /", VI Dp w I k VJ T 4 . / o ' ) 1 0.02589 5 T. fc ' o Units. Ceutimeter-gram-second. (C. G. S. ) Taking B, r, and Z> in centimeters, />, in grams, the velocities then become, -= 10.610^/^^4 (36) Velocity in meters per second. (37) (38) (39) w= 7.503 to = 20.112 J" r^J-20.112 = 14.221. l~'~ Application of formula (37); w in meters per second. w = 7.503 / j Rain Pu , = l. Hail p., = 0.917. 1 Friction factor. Height in meters. B T D JD D /Ti k *? 273 283 293 qnq OUO 313 *f fw 0. CHI. 76.00 67.51 60.25 53.28 47.34 42.16 37.37 33.20 29.59 in. p. s. 14.22 15.09 16.01 16.99 18.02 19.12 20.28 21.52 22.83 m.p.s. 14.48 15.36 16.30 17.29 18.35 19.47 20.65 21.91 23.24 m.p.s. 14.73 15.63 16.58 17.60 18.67 19.81 21.01 22.29 23.65 m.p. s. 14.98 15.90 16.87 17.89 18.98 20.14 21.37 22.67 24.06 m.p. s. 15.23 16.16 17.14 18. 19* 19. 29) 20.47 21.72 23.06 23.44 cm. 1.00 0.90 1.00 0.95 0.89 0.84 0.78 0.71 0.63 0.55 0.45 0.32 0.22 0.10 0.03 cm. 10 9 8 7 6 5 4 3 2 1 3.03 2.87 2.71 2.53 2.35 2.14 1.90 1.66 1.35 0.96 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 1.00 0.95 0.91 0.88 0.8S 0.82 0.78 0.77 0.75 0.73 1000 2000 0. 80 0.70 Large drops 0. 60 0.50 40 Common drops. 0.30 0.20 0.10 Fine drops.... ' <* 1.0.001 3000 4000 5000 6000 7000 8000 TABLE 57. Conversion factors for units of length, mass, and pressure. Meter-kilogram-seconds. Decimeter-gram-seconds. C'entimeter-gram-seconds. Foot-pound-seconds. Inch-grain-seconds. Number. Logarithm. Number. Logarithm. Number. Logarithm. Number. Logarithm. Number. Logarithm. Length, 1 meter. 0.00000 10 1.00000 100 2.00000 3.2808 0.51599 39.37 1.59517 /;. 0.1 9.00000 1 decimeter. 0.00000 10 1.00000 0.32808 9.51599 3.937 0.59517 0.01 8.00000 0.1 9.00000 1 cm. 0.00000 0. 032808 8.51599 0.3937 9. 59517 0. 3048 9. 48402 3.048 0. 48402 30.48 1. 48402 1 foot. 0.00000 12.0 1.07918 0. 0254 8. 40484 0.254 9. 40484 2.54 0. 40484 0. 08333 8.92082 1 inch. 0.00000 Mass, 1 kilogram. 0.00000 1000 3.00000 1000 3.00000 2. 2046 0. 34333 15432. 4 4. 18843 If. 0.001 7.00000 1 gram. 0.00000 1 0.00000 0. 0022046 7.34333 15.4324 1. 18843 0.001 7. 00000 1 0.00000 1 gram. 0.00000 0.0022046 7.34333 15.4324 1.18843 0. 453593 9. 65667 453. 593 2. 65667 453. 593 2.65667 1 pound. 0. OO(KX) 70(10 3. 84510 0. 000064799 5.81157 0. 064799 8.81157 0.064799 8.81157 0. 00014286 6. 15490 1 grain. 0.00000 Pressure, 1 kilo. /m.< 0.00000 in 1.00000 0.1 9.00000 0. 20481 9.31135 9. 9562 0.99809 , _.v 0.1 9.00000 1 gram/dm. - o.oooon 0.01 8.00000 0. 020481 8.31135 0. 99562 9. 99809 ' /, 10 1. 00000 100 2. 0001 III 1 gram/cm. 2 0.00000 2.0481 0.31185 99.5620 1. 99809 4.3323 0.63363 43. 823 1. 63363 0. 18323 9. 68363 llb./ft" 0.00000 13.6110 1. 63671 0.10044 9.00189 1.0044 0.00189 0.010044 8.00189 0.020571 8.00000 1 gr. /in. - 0.00000 473 MONTHLY WEATHER REVIEW. TABLE 58. Conversion factors for units of dMtmcr,, time, and velocity. OCTOBER, 1906 1 llil^. Mi t' r srroml. Number. Logarithm. Number. Logarithm. Number. Logarithm. Number. Logarithm. DistatK-e, & 1 meter 1000. 1609. :i!5 0. 304794 0.00000 3.00000 a 20664 9.48400 0.001 1 kilometer. i ww.-n.-. 0. 00030479 7.00000 0.00000 0.20664 (1. 48400 0.00062188 0. 62138 1 mile. 0.0001894 i;. 7'.c:iti :i. 7!i:i:;n 0. OOIKK) 0.27737 8. MM 8280.9 5280. i r.H.t. 0.51000 1.51600 3. 72263 0.00000 Time. T. 1 second 3600 3600 1 0.00000 3.55630 3.55630 0.00000 0. 0002778 1 hour. 1 0. 0002778 6.44370 0.00000 0.00000 6. 44370 0. 0002778 1 1 hour. 0. 0002778 i;. 41:170 0.00000 0.00000 6. 44:(70 i MM 3600 1 second. 0.00000 3.55630 8.H6SO 0.00000 Velocity, '- 1 m. /sec. 0. 2778 0. 4470 0.3048 0.00000 9.44:i7" 9. 65034 9.48400 3.600 1 kilo. /hour. 1.6093 1.0973 0.55630 0.00000 0.2(16111 0.04030 2.2369 0. 6215 1 mile/hour. 0.6818 0. 34966 9. 79335 0. 00000 9. 83367 3.2809 0.9113 1.4666 1 ft. /sec. 0. 51600 t.Msm 0, 16633 0.00000 TABLE 59. Conversion factors for pressures and velocities. Pounds / foot 2 . Miles / hour. Ap = 1 cv* Kilogr. / met. 2 . Ap = 4.8823 cr* Feet / second. = (0.6818)" ei', 1 0.4649 [9.66734] log Meters / second. (2.2369)' cr/ 5.004 [0.69932] log = 4.882 x (0.6818)' cv s * = 4.882 (2.2369)' cr s 2 2.270 24.43 [0.35597] log [1.38795] log Grams / cm.*. Ap = 4.8823 X 0.1 cr, 2 0.2270 [9.35597] log Grains / inch 2 . Jp = 4.8823 x 9.9562 cv* = 48.61 (0.6818)' cv* 48.61 22.CO [1.68674] log [1.35408] log Grams / dm.*. Ap = 4.8823 x 10 cr,, 2 = 48.823 (0.6818)' cv* = 48.823 (2.2369)' cr 3 ' 48.823 22.697 [1.68863] log [1.35597] log 0.4882 (0.6818) 2 cv* = 0.4882 (2.2369) 2 cr s 2 2.443 [0.38795] log = 48.61 (2.2369)' cv* 243.3 [2.38606] log 48.82? 244.32 [2.38795] log Kilometers/ hour. (0.6215) a cu 4 * 0.3861 [9.58672] log 4.882 (0.6215)' cr' 1.885 [0.27535] log 0.4882 (0.6215)' cr,' 0.1885 [9.27535] log 48.61 (0.6215)' cr, 1 1.877 [0.27346] log 48.823 (0.615)" m* 18.852 [1.27535] log c represents the other terms in the general formula for J p. The numbers inclosed in brackets are logarithms of the factors accurately computed. TABLE 60. Resistance to a solid moving in a jluid. Newton's theorem and the coefficient k. p = density, h = height, w = vertical velocity, A = Area. The resistance between the solid and fluid is equal to the pres- sure due to the weight of a column of the fluid p h A = p N , 10* 10* where w 1 = Zgh,tindh= ~ , so that p x = p-^-. A (21). By observations this requires a coefficient k= -* . PN On account of viscosity and other forces the more complete formula is p = aw + bio* + cw* + . . . . For air blowing against a plate normally there is an excess of pressure -f Jp, on the front side, and a defect of pressure Jp 2 on the back side of the plate. Take the static pressure of the air on the body = p. (42) Wind pressure = p t p 2 = Jj>= + J/^ TABLE 61. Differential coefficients. n From (24) J/> = c. k fpiir . c= constant. (43) (40) Front side pressure = p^ = p + Jp l= p -\- k l f> A w > (41) Back side pressure = /> 2 = p ^P,=P &, p - A (45) (46). fc= coefficient of resistance = 1.30. = 77 die. Increase J k = + 0.1 ; increase J/>= 1.1%. = 0.13 dli mm .' Inc. JB = + 1 """ ; increase J/-= 0.13%". -- 3 dT=-0.37dT. Decrease J 7'= -1; 100 di, 1.3 100 . . 100 ncrease J]> = 100 dlr OCTOBER, 1906. MONTHLY WEATHER REVIEW. 474 ck 100 d B (47) (48) ck T tt> 2 , B 760 1 OIK/7' 27:: (49) From (24) w 2 = 100 x 2 10 d w _ J > C . A" T 1 Ji e.k WQ dw = a^p w 2 J p (60) (51) (52) (63) (54) (55) Maxwell. //= 0.000 000 0256(461' 1 % error in J ;) = i % error in w. 1 rfT 100 = 0.18 dl 7 . 2 273 1 error in T = 0.18 % error in w. 1 f//y 100 = 0.07 dB. 2 760 !" error in B = 0-07 % error in w. 1 rf ~2 L3 0.1 error in k = 3.8 % error in. TABLE f 2. Coefficient of viscosity for air, /t. , pound 1 foot* ' foot The pressure in pounds required to slide 1 square foot of air at the rate of 1 foot per second parallel to a layer 1 foot distant, when the temperature in Fahrenheit degrees is t. Maxwell. //=0.000 1878 (1 + .00275 1), G. G. S. units. O. E. Meyer. 0.000 1727. Compare Basset's Hydrodynamics, Vol. II, p. 251. Coefficient of resistance for air, k. 56) Poncelet and Unwin. Coefficient k 1.85 A section of the current ~~ a section of the body _ a, section of contracted current a ~ section of the body For circular plates. Morin. For plates 1 foot square. Thibault. For plates 0.3 to 0.5 meter square, velocity 2 to 8 meters per second. Dvichemin. Reduction of rectilinear motion to circular motion. p = pressure for rectilinear motion with velocity i'. p c = pressure for circular motion with velocity u. A = area of greatest section of the body. I* = density of the fluid. B = arm of rotation at the center of A. x = distance of center of figure of A from the center of gravity of the half section of A on side of axis. i = angle of incidence of air striking the front of A. ft = i */ A si 11 * = thickness of the flowing stream on A. (57) 1.30 1.36 1.83 For plane surfaces ( 1.25 ( 1.30 Mariotte. Plates with low velocities. 1.74 Weissbach. Fixed plates in a moving current of air. 1.25 ( 1.19 Woltinann. Experiments at Hamburg, 1785-1790 ( 1.49 ( 1.34 Munche. From experiments of Woltmann, de Borda, Hutton. 1.30 Dubuat. Grashof's discussion. 1.43 565 Didion. k= 1.318+ r^- . 1.32 w~ Hutton. Sphere for velocities under 90 meters per second, using a ballistic pendulum. Nordmark. Double conical bodies. Cylindrical bodies. Spherical bodies. Piobert. ] In joint experiments with plates 0.5 to 1.0 Morin. meter square, velocities to 9 meters Didion. ) per second. Poncelet. Recommends as the result of discussing all data available to him. 1.40 0.67 0.91 0.83 1.357 1.30 1.39 1.49 1.64 1.24 1.43 1.85 1.15 1.23 1.90 1.84 De Borda. Whirling machine, plates 0.10 to 0.03 square meter, velocity 3 to 4 meters per second. Hutton. Using Robbin's whirling machine, plate 0.01 to 0.02 square meter, and velocities up to 6 meters per second. Rouse. Whirling machine. Pressure in pounds per foot 2 ,/> = 0.00492 if. Velocity in miles per hour. Beaufoy. Whirling machine, plate 1 foot square, ve- locity 2 meters per second, and for a larger plate. Prechtel. Square plate rotatiugaroundoneedgeasaxis Rouse. And the Royal Meteorological Society. Hagen. Whirling machine with disks which were square, circular, and triangular, the av- erage circumference being 0.50 meter, and velocities between 1.5 and 5.5 feet per second. For C= circumference of the area A = 0.50 meter. (58) Jp= (0.00707 + 0.0001125 C)A v* (gram- decimeter second). Ap= (0.0707 + 0.01125 C) A t; 2 (kilogram- meter second). Jp = (0.0028934+ 0.0001403 G) A u 2 (uin miles per hour, p in pounds, C in feet, and A in square feet). From these results, k = 1.135+0.1805 C = 1.135 + 0.090. 1.225 Thiessen Whirling apparatus with cylindrical bars, and D = diameter 2 to 3 millimeters, L = Schellbach. length 0.3 to 1.0 meter'. (59) Ap = (7.25 u + 0.486 D v* + 0.0000698 Z> 2 u s ) 10 L, (C. G. S). J p = (0.0000725 i' + 0.0486 D u 2 + 0.0698 D 2 u s ) L, (M. K. S). For B 760 millimeters, and t = 20 C, k = 0.00118 JL + 0.7914 + 1.1366 Dv Dv (for cylinders). k= 0.00154 A + 1.008 + 1.448 Dv, Dv (for squares). 475 MONTHLY WEATHER REVIEW. OCTOBER, 1906 Thiebault. Langley. Nipher. Dines. k = 0.546 1 + 1.008 + 0.0040 v v (for D = 0.00275 m). k for average velocities. 1.300 Thin plates on a whirling machine. Square plate, 4=0.026 square meter. 1.525 Square plate, 4=0.10304 square meter. 1.784 Rectangular plate, 4=0.10304, long side 1.900 radial. Rectangular plate, 4=0.10304, short side 1.677 radial. Square plate, 4 = (0.323) 2 , radius= 1.370. 1.784 Square plate, 4=(0.227) 2 , radius=0.966. 1.784 Square plate, 4 = (0.161) 2 , radius=0.685. 1.784 Whirling machine, square plate with veloci- ties 4 to 11 meters per second. 1.31 Railroad car direct wind pressures. 1.37 Stokes. route FIG. 38. Special form of whirling apparatus used by Dines. W= weight of adjustable piece. a> = angular velocity of rotation of W about C. The piece P B W is rigid and by its rotation about B assumes a position of equilibrium. For equilibrium. (60) (61) Px=k 23.68 (r+a;)'< J 4.;r=7r cos (0.116) 2 for air. The maximum velocity of a falling body be- comes permanent when, p w = density of the sphere. /i = density of the fluid thru which it falls. g = acceleration of gravity in inches, 386 inches. iu = velocity inch/second: 1 inch/second =0.0254 meter/second. For small drops of water at small velocities the viscous resistance of the air is far larger than the impact resistance, as com- puted by the Newtonian theorem. For r = 0.0005 inch, water /> u , = 1, air in the lower clouds, p =0.001, we find w m = 1.593 inch/second = 0.133 foot/second. Recknagel. Pressure at the center of the front of a plane plate and at the apex of a solid of revolu- tion. = pressure in still air surrounding plate, kilogram/meter 2 . -= mass of 1 cubic meter of air. 9 9 p m = = v a = velocity of the air relative to center of plate, meter/second. Q -^ = Tc = 1.41 = ratio of specific heats. " k 1 = 0.2908. =0.1754 Jp^|= 0.8564 Jp pound/foot 2 . (65) Pi = P (1 + J, 1 ,.2 IS. IS] * *Vp, 7?i v 2 for low velocities. 2 for any velocities. p, = p + The pressure diminishes from the center to the edge of the plate. Schreiber. A discussion of the distribution of the pres- sure over a flat plate is given on pages 36- 38, of Studien.iiber Luf tbewegungen, von Paul Schreiber, Abh. d. Kon. Sachs, me- teorol. Inst. Heft 3, 1898. Nipher. A complete experiment of the distribution of pressure over a plate is given in his paper, "A method of measuring the pres- sure at any point on a structure, due to wind blowing against that structure," by Francis E. Nipher, Transactions of the Academy of Science, St. Louis, Mo., Vol. Ill, No. L This agrees with the formula, given by Hann on page 11, tlber die tiigliche Drehung der mittleren Windrichtung, etc. Wien, 1902. OCTOBER, 1906. MONTHLY WEATHER REVIEW. 476 Table 53, formulas 14, 15, deduces the barometric gradient which, acting along the meridian from south to north, will jUst sustain the wind velocity v directed due eastward. This is the formula for determining the relation between the east- ward drift of the atmosphere in the upper strata and the nor- mal gradient which is required to sustain it. This is also found on page 11 of Hann's paper and on page 472 of his Lehrbuch der Meteorologie. Table 54, formulas 16-21, contains the deduction of the Newtonian theorem for the pressure exerted by wind velocity on a body. The general equation becomes (19) p = f> W + !>h. In case there is equilibrium between the weight represented by p h and the pressure exerted by a vertical velocity u 2 , so that p = 0, we have (20) ^=A. which is the law of the velocity for a freely falling body. Hence, the pressure exerted by the first term, p ~=p y , is the Newtonian pressure. From observations it is found that this pressure must be multiplied by some factor k, to reduce it to the actual pressure which is exerted upon a rigid body of sensible dimensions. When such a body moves thru a still fluid, or when a moving fluid passes a fixt body, the stream lines of the fluid are de- flected in passing the body, making an excess of pressure on the front side, + dp v and a defect of pressure on the back side, Jp. 2 , so that the total resultant pressure is This is not equal to the Newtonian pressure, but differs from it by some factor, k = Preservation) = P. . The deflection p (rsewton) p^ of the stream lines causes vortices and hydrodynamic pres- sures of a complicated kind, which are integrated in the total excess of pressure of the positive and the negative types. Many experiments have been made to determine the relations be- tween the front and the back pressure, but they depend largely upon the shape and size of the body and the density of the fhiid. The existence of a diminished pressure and consequent inflow, or so-called " suction ", on the leeward side of bodies exposed to the wind has been generally recognized, 1 but the experiments of Mr. Irminger, a Danish engineer, made to de- termine the amount of such suctions, shows it to be present to an unexpected extent. His measurements were made by the use of hollow plates and models of thin sheet iron exposed in an air duct 4| by 9 inches cross section, at various angles and positions, to velocities ranging from 16 to 32 miles per hour. We quote the data from Julius Baier's paper on wind pressures in the St. Louis tornado, American Society of Civil Engineers, Vol. XXXVII, 1897, No. 805. (See Table 63.) This shows that the percentages vary widely with the shape of the body, and its exposure to the direction of the wind, and that the lee suction is often much in excess of the front pressure. For spheres the front pressure is 28 per cent and the back suction 72 per cent of the total pressure p. The total pressure on a sphere is only 57 per cent of that on a thin plane having each side equal to the diameter of the sphere. The subject is very complex in application to special cases. 'See Abbe in the Monthly Weather Review, November, 1886, Vol. XIV, p. 332, and his publication of observations on the pressure and suction around the Weather Bureau station at Mount Washington, N. H., in his Meteorological Apparatus and Methods, pp. 142-144. See also the results of experiments on chimneys and cowls, Proc. Am. Acad. Arts and Sciences, Boston, 1848 ; Journal Franklin Institute, Philadel- phia, 1842. TABLE 63. Percentage of front and backpressures (Irminger's results). PRESSURE IN A HORIZONTAL DIRECTION PRESSURE ON THE WINDWARD SIDE $TOTAL PRESSURE SUCTION ON THE LEEWARD SIDE .95p 67 43 V% J - 0.79p 24 76 O 0-57P 28 72 0.25p 18 82 jAf 0.59p 68 42 <] 0.42p 14 86 > 0.71p 63 37 The wind velocities are usually taken in the United States by the Robinson anemometer, but the indicated velocities must be reduced about 20 per cent, in order to obtain true values of the velocity to enter into the formula. The reduction factor from miles per hour to meters per second is as follows: 1 mile per hour = 0.4470 meter per second. MARVIN'S CORRECTION TO OBSERVED WIND VELOCITIES. The velocity of the wind is very generally measured by some form of the Eobinson cup anemometer. From early experiments it was found that the distance passed over by the center of one of the revolving cups would, if multiplied by three, give the velocity of the wind, and the wheels and recording dials of the instrument were geared to read wind velocities directly by taking into account this factor of reduction. Later experiments have shown that with anemometers of the size commonly used this ratio is erroneous, and the indicated velocities are about 20 per cent too great, but no change has been made in the recording device and the " wind observations published by the various meteorological institu- tions at the present time have only a relative, but not absolute, value. It is very probable that many experiments on the relations of wind velocities to wind pressures have been made in which this anemometer correction has not been properly applied ". Professor Marvin has determined the correction to be applied to the readings of the standard form of anemometer used by the Weather Bureau. It is expressed by a logarithmic formula from which the follow- ing table taken from his report on wind pressures is computed: TABLE 64. Corrected wind velocities as indicated by a Robinson anemometer, in miles per hour. Indicated velocity. 1 2 3 4 5 6 7 8 9 6.1 6.0 6.9 7.8 8.7 10 9.6 10.4 11.3 12.1 12.9 13.8 14.6 15.4 16.2 17.0 20 17.8 18.6 19.4 20.2 21.0 21.8 22.6 23.4 24.2 24.9 30 25.7 26.5 27.3 28.0 28.8 29.6 30.3 31.1 31.8 32.6 40 33.3 34.1 34.8 35.6 36.3 37.1 37.8 38.5 39.3 40.0 50 . .. . 40.8 41.5 42.2 43.0 43.7 44.4 45.1 45.9 46.6 47.8 60 48.0 48 7 49.4 50.2 50.9 51.6 52.3 53.0 53.8 54.5 70 55.2 55 9 56.6 57.3 58.0 68.7 59.4 60.1 60.8 61.5 80 62.2 62.9 63.6 64.3 65.0 65.8 66. 1 67.1 67.8 68.5 69 2 Total velocity = number on side and number on top of same column. Thus, for 41 the corrected velocity is 34.1; for 57 the corrected velocity is 45.9. Table 55, formulas 22-31, contains the formulas for the vertical velocity that just sustains a freely falling body. The successive steps are clearly indicated. In formula 24 the co- efficients are computed for the value of k = 1.3 employed by Professor Schreiber. Since the weight of a body, W g =. ^xr'p x 1000 kilograms, must be put equal to Jp A = 4 B '-. According to Irininger's tests the coeffi- cient of -r should be 0.57 instead of 0.50. The specific weight of the body =/< = ''"> where the specific weight of Pi water is the unit. The density of the air at any point is li T found by the usual formula p = p a -& ," . Formulas 28-31 ** * contain the several relations found by putting (24) in combi- nation with (25). Velocities. Formulas 32-35 of Table 56 contain the resulting velocities in the kilogram-meter-second system (M. K. S.). In the cen- timeter-gram-second (C. G. S.) system for the barometric pres- sure B, the radius r, and the diameter D in centimeters, for the density f> in grams, and the velocity in meters per second, we have the values of the velocity w, as given in formulas 36-39. Formula 36 gives the velocity in meters per second in terms of the absolute temperature T, the barometric pres- sure B in centimeters, the radius of sphere in centimeters, the specific weight of the body in grams. The coefficient k must be assigned from experimental data. Formula 37 employs the same data as 36 except that the diameter is used instead of the radius. Hence, 10.61 = 7.503 x v/2 and 20.112 = 14.221 x v/2. T* /TT In 38 and 39 the substitution D -? is made, in case one o P prefers to use the densities rather than the pressures and temperatures. We have taken formula 37 for development and application to the formation of hail. IT i' Formula 37, 10 = 7.503 A / Dp,- The first section of the \B w k table contains w = 7.503 /_ for the arguments T and B. T begins at 273, since in hail formation the temperature does not fall below C., and it extends to T= 313, which is the extreme of summer heat at the surface. Approximate values of B are taken at the heights H as indicated, so that H or B may be used in practical applications. If D = 1 cen- timeter, p w = 1 for water, and k = 1, the velocity in meters per second is found in the body of the table. In application to water drops of various sizes, large drops 0.70 to 0.50 cm., common drops 0.40 to 0.20 cm., fine drops 0.100 to 0.001 cm., the subjoined multiplying factor x/J9 must be used. In the case of hail, for p w = 0.917, the tabular factor ranges from D = 10 cm. to D = 1 cm. giving the several values of It is shown in the following tables that the coefficient k is equal to about 1.30 for bodies of considerable dimensions, such as engineers naturally employ, plates, disks, spheres, cubes, and parallelepipeds placed on whirling machines for tests. These can hardly be true for water drops which are not rigid, but very flexible in their form while falling, since they undergo periodic changes in shape when in motion, like oil drops rising in water. It is difficult to assign values to k for fluids, and on that account the factor has been kept separate from the first section of the table. My opinion is that k = 1.1 nearly for water, but that on the solidification into hail, the value of k approaches k = 1.30, in consequence of its rigid shape in the solid state, which is of dimensions comparable with those used in the physical experiments. These tables can be readily used in numerous combinations, by selecting the suitable sets of factors to be multiplied to- gether. The effect of k is to diminish the required velocity w in meters per second, since the form of the body generates a term which is itself equivalent to an addition to the actual wind velocity. The required velocity increases with the di- mensions of the body, and with the increase of density of bodies of the same dimensions. Convention fartorx. Tables 57, 58, and 59 contain the conversion factors between several systems of vinits which are found in the papers relating to the velocities and pressures. Table 57 gives the units of length L, the units of mass M, the pressure p = , . Table 58 gives the distance S traversed in the unit of time, the unit of t< time T, the velocity V= l . Table 59 gives various combina- tions between the pressure and the velocities for several sys- tems of units. For example, the pressure in pounds per square foot with the velocity in miles per hour, becomes the pressure in grams per square centimeter with the velocity in meters per second, by multiplying with the factor 2.443 (logarithm = 0.38795). The inverse reduction is performed by multiplying with ~- = 0.4093 (logarithm = 9.61205). Resistance to a solid moving in a fluid. ' The problems in hydrodynamics relating to solids moving in a fluid, or to a fixt solid in a moving fluid, are very numer- ous and their discussion can be found in several treatises. In Table 60, formulas 40 to 42, the Newtonian theorem is re- sumed in connection with the factor k. But it was shown that there is a factor k l for the front side of a plate or solid of any form, and a factor A" a for the back side of this body. The wind pressure is made up of two parts Ap = J/>, + Jjo s , and the factor k of two parts k = fc, + k f These must be de- termined for special objects, and their values can not be assigned from general considerations. Differential coefficients. The variations of pressure dp depend upon changes in the barometer height J B, the temperature J T, the coefficient Jjfc, and the velocity J w. Table 61, formulas 43 to 54, contains the differential coefficients, and the corresponding changes of the terms in percentages. Thus -(- 0.1 J fc = 7.7 %Jp. + 1" JB= -1 JT= 1 % error in A p 0.5 % error in n: 1 error in T = 0. 18% error in u: jmm error j n ft _ 0.07% error in n: 0.1 error in k =3.8 % error in ic. These values of the ratios are convenient in estimating the mutual changes which take place among these quantities. The resistance coefficient k. It is evident that an error in the value of the resistance co- efficient k is much more efficient in producing an error in the wind velocity as computed than are similar errors in the de- termination of the barometric pressure and temperature. It is, therefore, important to discuss the coefficient k with much care, and to pass in review the results of the experiments which have been executed for the purpose of establishing its value under different conditions. These involve bodies of different sizes and shapes, carried thru fluids, such as air and water, with variable velocities. The experiments extend thru the past century, and many of them bave been executed with all possible care as to details, in order to secure scientific precision. Summaries of these studies may be found in Abbe's 2 'Treatise on Meteorological Apparatus and Methods, by Cleveland Abbe, Appendix 46, Annual Report Chief Signal Officer, 1887, pages 218- 240. Mechanics of the Earth's Atmosphere, C. Abbe, Translation of Hagen's paper, Smith. Misc. Coll., 843, 1891. OCTOBER, 1906. MONTHLY WEATHEK REVIEW. 47H and in Schreiber's 3 compilations to which I am chiefly indebted for the accompanying data. The original papers are enumer- ated therein, and the bibliography need not be repeated in this place. Table 62, formulas 55 to 65, contains the summary of values of k, together with a brief statement regarding the nature of the experiments and the fundamental formulas of the instru- mental work. Formula 55 gives Maxwell's value of the co- efficient of viscosity in British units and in C. G. S. units, with reference to Basset's Hydrodynamics. Formula 56 gives Poncelet and Unwin's equation, which takes account of the contraction and expansion of the stream lines in passing by a body. Formula 57 gives the equation for transforming the pressure on a body moving rectilinearly into that encountered by it when carried on a whirling machine. The group of equa- tions under formula 58 contains Hagen's restilts in (C. G. S.,) (K. M. S.), and (P. F. S. ) units, respectively. It is to be ob- served that a Considerable factor depends upon the circum- ference of the plates, and that the value of k = 1.104, when the circumference is very small, as in raindrops. Formula 59 contains the result of Thiessen's and Schellbach's experiments on long cylindrical rods whirled by a machine, and it shows that there is a complex function depending upon the first and second powers of the velocity which is involved in the coeffi- cient of resistance. Formula 60 gives the equation for equi- librium in Dines's machine, which has a special device for determining the pressure (fig. 38). The rectangular arm I' K W is rigid and rocks upon the axis B; the arm B W is stayed between two stops, with electric contact, so that the length of the working arm for W can be accurately adjusted to the whirling pressure P on the plate A. Formula 61 gives the equation for k and the accompanying table of values for / and I: / is the coefficient required to find the pressure in pounds per square foot from the velocity in miles per hour, and it averages ). = 0.00355, which appears in formula 62, Jp = 0.00355 D 2 . Professor Marvin has established the value for the Weather Bureau />. = 0.00400, from which Jp = 0.00400u 2 . Professor Nipher has determined the value A = 0.00251 on the windward side alone, so that from Irminger's experiments we are safe in taking the total value as that given by Dines or Marvin for general conditions. The result of Stokes's investi- gation on the resistance of any moving body immersed in a fluid shows that it consists of two parts, the first due to vis- cosity proportional to the velocity, and the second due to the gyratory motions which are generated in the fluid under the existing conditions, proportional to the square of the velocity. Formula 63 gives Stokes's equation and the value of and the fourth w= 15^/25, which is the required velocity in meters per second. TABLE 66. Bigelow's formula for rain. w 7.503 For rain in lower atmosphere, take p^ := 1.0 for water, and ft 1. Take 7.503 - /-= = 15.0 m. p. s. approximately. I. FOR FINE DROPS, 0.01 to 0.20 ram. z>.. ^5..V5 9* r * mm. cm. m. p. s. m.p.s. 0. 01 = 0. 001 0.032 0.48 0.0032 0.02 = 0.002 0.045 0.67 0. 0127 0. 03 = 0. 003 0.055 0.83 0.029 0.05 = 0.005 0.071 1.07 0.079 0.10 = 0.010 0.100 1.50 0.32 0. 20 = 0. 020 0.142 2.13 1.27 Bmif \/l> w=15 \/ D *=.., YP mm. cm. 0. 30 = 0. 030 0.173 m.p.s. 2.60 m.p.s. 2.73 0.40 = 0.040 0.200 3.00 3.15 0. 50 = 0. 050 0.223 3.35 3.53 III. FOB LARGE DKOPS, i.OO to 5.50 mm. ,. 2r Si w = 15 \/ D Ob n. 1 k mm. 1.00 = cm. 0.100 0.316 4.74 4.40 0.98 .1 1.50 = 0.150 0.387 5.81 6.70 0.94 .1 2.00 = 0.200 0.447 6.71 5.90 0.88 .3 2.50 = 0.250 0.500 7.50 6.40 0.85 .4 3. 00 = 0.300 0.548 8.22 6.90 0.84 .4 3.50 = 0.350 0.592 8.88 7.40 0.83 .6 4.00 = 0.400 0.632 9.48 7.70 0.81 .5 4. 50 = 0.450 0.671 10.07 8.00 0.79 .6 5.00 = 0.500 0.707 10.61 8.00 0.76 1.8 5.50 = 0.550 0.742 11.13 8.00 0.72 1.9 For fine drops, I, w ranges from 0.48 to 2.13. For common drops, II, w ranges from 2.60 to 3.35. For large drops, III, w ranges from 4.74 to 11.13. These results are obtained by applying the same law for all drops, from the finest to the largest which occur in the atmos- phere. This must, however, be modified for fine drops, I, and for large drops, III, in order to conform to the experimental observations. When the drops are very fine the viscous resistance of the air becomes the prevailing force that holds them from falling, and by formula 64, Table 62, the maximum falling velocity is permanent for 9 By taking (/>, f >) = 1.00 0.00129 = 1.00, this becomes 2 or 2 w m = ~~g > which is the formula employed by Lenard for fine drops. In this g = 981 cm., /t = 0.0001 2, by formula 55, Table 62, and r is taken in centimeters. The result in the fifth column ranges for fine drops from 0.0032 to 1.27 m. p. s., and they are much smaller than those given by the general formula. In the case of common drops, Lenard uses the formula w 2 = r, where g = 981 cm., p = 0.00129, and r = 0.153, a constant derived by experiment. The results range from 2.73 to 3.53, and are in close agreement with the general formula, where k = 1.0. This shows that the formula for impact re- sistance as distinguished from viscous resistance begins to be applicable for drops whose diameters are about 0.25 mm. The fact that k = 1.0 indicates that the common drops do not expe- rience any deformation relatively to the passing stream lines, the surface tension being strong enough to simply adjust the shape of the drop to the curvature required for avoiding any resistance due to the shape of the body, except that tangential to the surface of the drops. When the drops increase in size beyond 0.50 mm. a deformation sets in which it is important to describe more fully. Professor Lenard 's experiments on large drops, with diame- ters 1.00 to 5.50 mm., were conducted by means of a machine which produced a vertical current whose velocity could be regulated and measured. In the midst of this the water drops falling from above were made to float, and their sizes when in equilibrium were studied. The resulting velocities for corres- ponding diameters are given in column 5 of section III, and they range from 4.40 to 8.00 m. p. s. It was seldom that larger drops than 5.50 mm. survived without breaking up, and the 512 MONTHLY WEATHER REVIEW. NOVEMBER, 1906 maximum current was 8 m. p. s. If we divide this experi- mental value of the vertical velocity by the computed velocity in column 4, we find the ratios / in column 6, section III. We take out the corresponding values of which are placed in the last column of Table 66, where they range from 1.1 to 1.9. We may, therefore, render the general for- mula for w applicable to raindrops of different sizes by taking =1.0 for common drops, and gradually increasing its value for large drops from =1.1 to =1.9, as indicated in the table. In his experiments on the deformation of raindrops in a vertical current of air, which could be well observed, Lenard found that the first effect of the current on the shape of the drop was to flatten it so that the axis parallel to the direction of the current was shortened. A further increase of velocity produced an increase of surface friction along the meridians of the drop, which also set up oscillations, and gradually produced vortex ring motion around a circle in the outer portion of the drop, lying in a plane perpendicular to the motion of the current. This vortex ring then separated into a corona of beads, the ring breaking up into smaller drops which became individuals, and broke up the large drop into fine drops. The fine drops began to increase in size by means of two processes, (1) collisions of drops in the current, (2) the attraction of drops by means of the electric charges which always accompany the aqueous vapor in its various stages of ionization. Lenard made counts for the number of drops of different sizes occurring in several rains and found that the number of fine drops is greatly in excess of that for large drops. The series of assorted sizes does not change regularly from the smallest to the largest, but they accumulate in groups, some sizes being entirely wanting, tho the number of large drops in any group is not so great as in the groups of small diameters. There is a continual inter- change in the sizes and numbers in each group because of the growth, deformation, and separation of large into small drops. It is evident that the true physical values of the surface ten- sion in drops can be obtained by computations on such data as that found in this manner. In the quiet air of the Tropics where the aqueous vapor content is great the drops may some- times grow to a diameter of 7.0 or 8.0 mm., tho that is not common. The time of oscillation in the process of deforma- tion and disintegration is apparently two or three seconds. The subject of rain invites to more exact experimental re- search than has yet been bestowed upon it. THE PROBABLE VERTICAL VELOCITY IN THE CLOUD. It is desirable to obtain some idea of the vertical velocity within the cloud itself for the purpose of judging of the validity of certain theories which have been proposed to ac- count for the formation of heavy hailstones. The formula to be employed is w 1 = 574.06 -g A B, in the adopted system (M. K. S.) where B and J B are in meters, tho, since they occur in the ratio =-, they can be taken in millimeters; w is the velocity in meters per second. Referring to the data for the waterspout in Table 51, we find the quan- tities available for the discussion. It is evident that there is no difficulty regarding Tor B, but that the value to be assigned to A B = B B is very uncertain. At first I take B as the static pressure in the normal system of gradients as obtained from the Barometry Report, and B the pressure computed as above from the observed cloud conditions. a-stage. (See Table 51. Sura'raary of data.) The pressure at sea level is 763.27 mm. The gradient in the static state is 8.24mm/100m. The height is 10.78 x 100 meters. The pressure fall is 88.83 mm. B = pressure at cloud base in static state 674.44 mm. The gradient in the convection state is 8.46 mm/100 m. The pressure fall for 10.78 x 100 meters is 91.20 mm. mm. mm. B = pressure at the cloud base in con- vection state (763.27 91.20). . . 672.07 (/? B) a = difference of pressure at cloud base 2.37 r= temperature (absolute) at base of cloud (273 + 9.3) 282.3 w a = vertical velocity at the top of the a-stage 23.91 m.p. s /J-stage. The gradient in the static state is 7. 11 mni/100 m. The gradient in the convection state is 7.40 mm/100 m. The height is 17.28 x 100 meters. (B t B) ft = (7.407.11) x 17.28=0.29x17.28= (B t B) a = (8.46-8.24) X 10.78=0.22 x 10.78= 5.01 2.37 mm. mm. (B B) = total change in the pressure 7.38 mm. T = temperature at the top of the /J-stage 273 B = pressure at the top of the /9-stage 544 mm. WQ by the formula 46.11 m.p.s. It is seen from the preceding computation that we have found a vertical velocity of w a = 23.91 meters per second at the top of the -stage, and Wp = 46.11 meters per second at the top of the /J-stage. w$ = small. The velocity is evidently small at the top of the cloud. There are, however, a number of reasons for thinking that these large values of w are erroneous, and must be greatly diminished. Since the discussion may be of interest, altho no satisfactory decision is reached as the result of it, the fol- lowing circumstances should be considered. (1) The pressures as computed for the cloud levels have been directly compared with the static pressures as determined from the gradients derived from the Barometry Report, and therefore any error in either of these steps must be allowed for in the comparison. It is noted that while there are no obvious errors in the work, we are yet dealing with small quantities, (B B) a = 2.37 mm. = 0.093 inch, and (B ff) f = 5.01 mm. = 0.197 inch, and that it will require great precision in the meteorological data to make these figures perfectly reliable. (2) It has been practically assumed that these two types of pressure are in action simultaneously on the same plane icithin the cloud itself. As a matter of fact the congestive circula- tions producing a cumulo-nimbus cloud are very complex, and they involve masses of air outside the cloud limit. In the case of the Cottage City waterspout the cold air of the anticyclone flows over the ocean strata, and immediately sets up a series of currents of which the cloud itself is one effect. This indicates that the normal static pressure has already been disturbed, and, therefore, the vertical current begins to move before such wide variations in (B o B) as 2.37 or 5.11 mm. can occur. In fact the current tends to fill up the pressure difference as soon as it begins to diverge from the normal state, and if it were possible to trace this variation exactly, or if, conversely, we could accurately measure the NOVEMBER, 1906. MONTHLY WEATHER REVIEW. 513 velocity at the several points, then the problem could be finally resolved. Unfortunately neither of these measurements can be made and therefore we are limited to general discussions. It is my impression, however, that these facts indicate that the pressure difference will seldom be as large as 1 millimeter or 0.040 inch, which implies a vertical velocity of only 15 meters per second. There are reasons for thinking that this is the maximum value of the vertical velocity, and that it seldom can occur in nature. Probably the vertical velocity is usually some- thing like 5 meters per second or even less, in the midst of such a cloud, but this is merely an opinion. (3) It is very probable that the vertical velocity increases to a maximum within the cloud at about the level of the top of the j?-stage, and that it is small at the top of the cloud, also at the sea level except in the midst of the vortex. The appear- ance of the cloud, as usually observed, shows that there is a rapid vertical growth in the central mass from the base upward, and that a sort of boiling with overflow to the sides takes place, except as disturbed by penetrating into other moving strata. Since several theories of hail formation depend upon a very strong vertical current, it has been important to point out the fact that the vertical velocity does not probably exceed 15 meters per second, and that evidence implies that this is generally too high. We will consider briefly the theories proposed for the formation of large hailstones, and then make such suggestions as seem warranted by the conditions deter- mined in this special cloud. APPROXIMATE POSITION OF THE ISOTHERMS AND ISOBARS IN THE COTTAGE CITY WATERSPOUT CLOUD. The practical difficulty of solving this problem lies in the fact that the data are wanting with which to compute the value of A B in the formula, 6~AB. B It is important to approach the true value at least approxi- mately, if possible, and for that purpose I have made the fol- lowing trial computations, shown in Table 67. The data for the B, t, R.H. as selected for the three hours, 4 p. m., 1 p. m., and 10 a. m., are indicated in English measures, and trans- formed into metric measures. The mean temperatures, 0, of the air column in the a, /?, ?-, a stages are determined as fol- lows: Plot the values of t, 58, 67.5 and 65 on the sea level; compute from the gradients of Table 51, summary of data, the heights at which 58 and 65 occur over the 1 p. m. and 10 a. m. columns, and draw the isotherms; plot the temperature 48.7 at the height of the bottom of the ,9-stage, 32 at bottom and top of the f-stage, and 10.4 at top of the cloud, as indi- cated on fig. 39. Then, I have drawn the isothermal slopes by judgment, admitting that they may need modification to be true to nature, tho there is no criterion now available. The temperature was determined at each 500-foot level in the three columns, and for the intermediate 250-foot points by taking the means of successive pairs of values. This gives 6 pairs in the a-stage, 12 in the /3-stage, and 13 in the <5-stage. The mean of the several a, /5, y, '' groups gives the values of 8 for the several stages as shown in Table 67. The values of the vapor tension, e, were computed, taking the relative humidity, R. H., given for the a-stage, and 100 per cent in the other stages. They may not be exactly correct, but they are sufficient for a close barometric reduction by the formula, log B = log B o m + jim + fni, the -fm term being negligible in the latitude of Cottage City. For the purpose of comparing the resulting pressures for the several stages found by the static formula just given and those found by the thermodynamic formulas as given in Table 51, summary of data, we select for 1 p. m. the following figures: Stages. Static method. (top) (bottom) 414.56 539.02 544.02 672.30 763.27 Thermodynamic method. m m. 414.50 539.00 544.00 672.00 763.27 This shows that these two groups of formulas and the de- pendent tables work together in entire harmony, considering the very different ways of using the temperature terms, since they are involved by means of their diverse functions. The resulting isobars for the stages are indicated on fig. 39, as- sumed approximate position of the isotherms and isobars. It is noted that from the beginning to the end of the disturbance, from 10 a. m. to 4 p. m., the isobars in the cloud region change by about 1 mm. of mercury. In the immediate neighborhood of the waterspout, 1 p. m., there may have been a series of abrupt rises and falls of the pressure, such as are usually found on the barograph traces in thunderstorms; the extent of these I can not determine for this case, but the important fact is that the range of J B must have been about 1 mm. in the cloud, because it is hardly probable that it should have exceeded for any short interval the total change between the 10 a. m. and 4 p. m. extremes. By entering 1 mm. in the for- mula, we find w = 15.52 meters per second. of J6000 /2OOO //OOP fooo 8000 6000 3000 '/OOP zaoo of vouch* neters Z880 30 /5 30 ZO" FIG. 39. Approximate position of isotherms and isobars. This agrees with my previous estimate of the vertical veloc- ity. It is evident that in the other computation, where the mean gradient for the month was used, as derived from the Barometry Report, namely, 8.24 for the a-stage and 7.11 for the /S-stage, Table 51, we assumed that the conditions for the waterspout were the same as those of the mean of August 514 MONTHLY WEATHER REVIEW. TABLE Gl. Computation of the isobars at three hours, August 19, 1S96. NOVEMBER, 1906 4 p. m. 1 p. m. 10 a. m. 5=30.10 <=58.0 R. 5=60% # a =51.4F. 00=36.7 P. #a =19.5F. = 764.54 mm. = 14.44 C. e= 7.33 mm. = 10.78 C. = 2.61 C. = -6.94C. 5=30.05 <=67.5 JR. H. = 64% # a =58.8F. #p=39.9F. # { =21.5F = 763.27 mm. = 19.72 C. e= 10.92 mm. = 14.89 C. = 4.39 C. = 5.8 S C. 5=30.03 = t=65.Q = /?.//. = 67% a =57.0F. = 0ft =41.7 F. = #4 =23.9F. = 762.76 mm. 18.33 C. e= 10.49 mm. 13.89 C. 5.39 C. 4.50 C. (a) 5= 1078 #=10.78 e= 7.33 log 5 = 2.88340 m = .05625 /3m =+ 18 5= 1078 #=14.89 e=10.92 log B a = 2.88268 m = .05546 /3m =+ 28 5= 1078 #=13.89 e=10.49 log /;= 2.88239 m = .05561 /3m =+ 26 log 5=2.82733 B= 671.94 log 5=2.82750 B= 672.30 log 5=2.82704 5= 671.49 (/3) 5= 1728 6= 2.61 <= 7.22 o 7 KB log 5 = 2.82733 m = .09286 /3m= + 32 H= 1728 8= 4.39 <= 9.44 c H HO log 5 = 2.82750 m = .09226 /3m =+ 38 H= 1728 #= 5.39 10.17 p q 94. log # =2.82704 m = - .09193 ftm = + 39 log =2.73479 B= 542.99 log B =2.73562 B= 544.02 log 5 =2.73550 5= 543.88 (r) 5=74 J 5= 5.00 5= 537.99 A B= - 5.00 5= 539.02 . J 5= 5.00 5= 538.88 (.5)5 = 2062 6= 6.94 e=4.57 log 5 =2.73077 m = .11477 /3m = + 28 5=2062 #=5.83 e=4.57 log 5 =2.73161 m = .11430 /3m = + 28 5=2062 #=4.50 e=4.57 log 5. =2.73149 m = - .11373 /3m =+ 28 log B= 2.61628 B= 413.31 log 5=2.61759 5= 414.56 log 5=2.61804 5= 414.99 taken day and night. But the waterspout occurred at midday, and the gradient was probably nearer the adiabatic rate, 8.46 for the a-stage and 7.40 for the /3-stage, than was supposed. The practical difficulty of successfully treating this part of the discussion is a great barrier to concluding the analysis of the dynamic conditions as derived from the thermodynamic state of the cloud, and additional observational data are much needed under similar thunderstorm actions. Finally, I adopt as the most probable maximum velocity of the vertical current w = 15 meters per second. THE BUILDING OF HAIL. The literature of the discussion of the many problems con- nected with the making of hailstones in the air is very exten- sive, but the following references are sufficient to place the subject before the reader in its most recent phases: Recent Advances in Meteorology, William Ferrel. Appendix 71, Annual Report of the Chief Signal Officer for 1885, Part 2. Pp. 302-315. Lehrbuch der Meteorologie, J. Hann. 1901. Pp. 682-699. Die Bildung des Hagels, Wilh. Trabert, Meteorol. Zeitschr., October, 1899. Pp. 433-447. Beitrage zur Hageltheorie, P. Schreiber. Meteorol. Zeitschr., February, 1901. Pp. 58-70. Hailstones, F. W. Very. Transactions of the Academy of Science and Art, Pitts burg; lecture January 5, 1904. Doctor Trabert's paper contains many references to other papers on hail. The physical structure of hailstones is about as follows: 1. Central nucleus of opaque snow or snowy ice, consisting of snowflakes and ice crystals mixed with air bubbles from 0.1 to 0.5 inch in diameter. 2. Layer of clear ice, or pellucid material containing incisted air cells and liquid in radiating inclosures. It is 0.1 to 0.2 inch thick and terminates in a sharply defined spherical boundary, or else in a more irregular boundary to which adhere mammillary masses of soft snow, which may be 0.1 inch in thickness. The clear layer itself is built up of a col- lection of many small drops, which are instantly stiffened, as in undercooled water, which is 5 to 10 below the freezing temperature, when water falls below zero without freezing and then sets with a shock. The ice crystals are mixed in a motley array. 3. A series of opaque and clear layers succeed each other, as NOVEMBER, 1906. MONTHLY WEATHER REVIEW. 515 many as fourteen having been counted, the last layer usually being opaque and adhering to a very thin layer of clear ice. The diameters of hailstones vary from 0.5 inch to 4 inches; a frequent size is 1 inch to 2 inches. The shapes of hailstones may be divided into two classes: (1) Those which have a thick base and pointed top, such as conical with a flat base, pyramidal with a round base, pear shape with a concave base, and mush- room shape with the table downward. In these the accretions are chiefly on the lower side of the body, as if gained in falling thru layers of snow and water drops to the ground. (2) Those which are regularly disposed about a center, as if they grew regularly from all sides, and are spherical, ellipsoidal, lens- form and hemispherical, the ellipsoidal being most common. The stones frequently indicate some effects as from a rotary motion about an axis, so that they grow in a certain plane more readily than in other places. The clear ice forms chiefly on the large plane like a tooth-shaped disk, the forms being many and irregular, depending on the accumulation of hex- agonal ice crystals. The order of construction from the center outward, if there were no repetition of the layers, would probably be: 1. Center = snow belonging to the 5-stage in the cloud. 2. Snow and ice mixed = the undercooled crystals of the 3. Clear ice = the gradual cooling ice of the /?-stage in con- tact with a cold nucleus. If there are repetitions, it follows that these stages are mixt in the internal circulation of the cloud, and that they are brought in contact with the nucleus in succession by falling, or by some other mechanical process. The undercooling of the f-stage and 'J-stage must be due to sudden transitions of the stone from one layer to another having different temperatures. The gradual cooling must be due to the contact of saturated water drops in the ;?-stage with the nucleus which has passed out of the 5-stage and the ?--stage into the /5-stage. The irregu- larities merely record the congested state of the air and a mixt condition of the <5-?--/?-stages. Hailstones are usually of aboiit one kind in the same storm, but they change their type from storm to storm. Hail is formed at the rear of the rising column of warm air, at the place of marked changes in the isotherms, when the barometer is beginning to rise rapidly and the wind shifts from the south to the northwest. This is the locus of the contact of two counter currents of air having very different temperatures, and hail forma- tion is one of the results of the rapid progress of the warm and cold layer* toward thermal equilibrium. The energy difference, which marks the departure from the normal equilibrium, does not lie in a vertical direction so much as in a horizontal direction. There is a rapid rise of warm air with condensation of the vapor, as in a thunderstorm, and the lightning usually occurs at about the time of hailfall, but it is sometimes earlier and at other times later, and not necessarily simultaneous. On mountains it is said that there is always lightning with hail and in the valleys sometimes lightning with hail. On moun- tains there are observed to occur simultaneously lightning, undercooled drops, and snow crystals. In falling thru the warm -stage the outer opaque coating of the hailstone becomes covered with liquid water and in this condition the stone falls to the ground. THEORIES OF THE FORMATION OF HAILSTONES. There are many theories regarding the mode of formation of hailstones, in each of which there is probably an element of truth. None of them can be said to be entirely satisfactory, and yet it is very likely that nearly all of the assigned natural causes and effects generally operate in producing the phenom- ena. The two principal facts to be accounted for are, (1) the presence of the cold which causes the sudden stiffening of the water drop at undercooled temperatures, and (2) the alterna- tion of snow and water materials in the successive layers. In the sudden cooling of the water drops there is evolved a con- siderable amount of latent heat, and the cold must be present to such an extent as to overcome the restraining effect of this latent heat, and yet produce cooling as by a shock of the molecular material in the water. The theories may be briefly summarized as follows: 1. The oscillation theory. It is assumed that two cloud layers of different temperatures are superposed, and that a hailstone oscillates up and down between them under an electrical attraction and repulsion, one cloud being charged with nega- tive electricity, and the other with positive electricity. This theory is now considered unnatural and arbitrary, and it cer- tainly is not true, because no electrical forces exist in clouds capable of thus moving heavy stones up and down in the presence of gravity. 2. The orbital theory. Professor Ferrel postulated a verti- cal orbit in the cloud, in connection with an internal vortex tube having a vertical axis, and supposed that the stones past around this thru considerable changes in altitude, and thru masses of different structure. Such a flow of air inside the cloud is very improbable, and there is no evidence that the cloud thus rotates. A modification of this view is found in the horizontal roll which very frequently exists on the back side of the warm ascending current, at the place of the most active mixing with the cold column. It is very likely that this does often develop in thunderstorm clouds, and indeed, there may be several such rolls on horizontal axes, and their action may well produce certain effects upon the construction of hailstones of different types; the effects are confined to merely differential variations of the typical structures. The vertical component can hardly lift the stones, except those of the smaller sizes. If the upward current on one side retards a freely falling stone, on the other side of the roll it would accelerate its fall, and so discharge it from the local action in the cloud by this impulse. 3. The upward current theory. The sustaining force of a strong upward current of air in the midst of the cloud, whereby a hailstone is held aloft for a considerable time while it receives accretions from the contents of the ascending stream, acting especially on the under side, is, doubtless, the most important theory to be examined. The growth on the underside of a hail- stone can be accounted for either by falling from a consider- able height thru the cloud, or by being sustained at a given height by an upward flowing current. There are several diffi- culties if not objections to this theory, when taken as the single cause of the formation. (1) The condensation products carried in the vertical current do not seem sufficient to produce the largest stones. Let W = grams of water in 1 cu. meter. W = grams of water in 1 cu. centimeter. A stone of section nr* falling thru a height dh will gain W 10" . -r' . dh grams, which is equal to a volume increase of 4 Trr 2 . dr. Hence, dr= W 4xTO~ 6 dh, and r } r l = W 4 000 000 (h t h,). If the height of fall is 2 kilometers (200,000 cm.), and W== 4 grams, at freezing temperatures, then J r = 0.2 cm. = 2 mm. This is Trabert's argument, and he thinks it does not fully account for the large stones which are found weighing as much as 250 to 1000 grams. This view conceives the stones to form in the 3 and ?--stage as ordinarily stratified in a quiet cloud, but I think that suitable modification can be indicated, which will to some extent avoid the difficulty. (2) The stream lines around the stone will doubtless carry off some of the particles of water without their touching the 516 MONTHLY WEATHER REVIEW. NOVEMBER, 1906 stone itself, and this will tend to diminish the quantity that is actually deposited, thus strengthening the former objection that the total quantity of deposit is insufficient to produce the mass of the observed hailstones. (3) It is not easy to account for the concentric layers on the upward current theory, or the downward fall theory in a simple cloud. (4) In seeking to maintain this current theory of accretion, Professor -Schreiber, it seems to me, has assumed excessive heights and improbable velocities in the ascending currents. Thus, he makes two assumptions: (a) that the vertical velocity increases steadily, at the rate of 3 meters per second per 1000 meters of altitude, as in the second column of the following table; (6) that the vertical velocity increases at the rate of 7.5 meters per second per 1000 meters, up to 20,000 meters of altitude, and then diminishes at the same rate, down to at 40,000 meters, as in the third column of the following table. Schreiber'a assumed vertical velocities. Height in meters. () () 30000 m. p. s. 90 m. p. s. 78 20000 60 150 10000 30 75 5000 15 37 5 4000 12 30.0 3000 9 22.5 2000 6 15.0 1000 3 7 5 . .. .. He makes two other assumptions for trial, (1) that the vertical current has no limit in height, and the same velocity and density thruout, and (2) that the velocity is the same thruout, but that the density diminishes with the height. He dis- cusses the sorting velocities which separate the stones of dif- ferent diameters, those larger than the critical velocity falling to the ground, and those smaller rising in the current and growing to larger size in preparation for a fall. It may be remarked, generally, that cloud heights above 10,000 meters are rarely measured, and that the vertical velocities are a maximum within the cloud, probably at the height of the p-stage, rather than at the top, somewhat as assumed in his fourth trial (p. 62), but by no means at such large values of the current. Schreiber asserts that hail forms at the top of such lofty clouds as 30,000 meters and in vertical currents of 100 meters per second, which it seems to me is impossible in view of the fact that such clouds do not exist, and that by adiabatic laws the ^-stage is seldom higher than 6000 meters in the most favorable summer conditions. In this connection refer to my discussion of the heights of the several stages, Cloud Report, Annual Report Weather Bureau, 1898-1899, pages 720-723, and chart 74. There can be little doubt that we must confine the formation of hail to the region 3000-7000 meters above the ground, and usually to the middle height, most frequently near the 5000-meter level. It is noted that Schreiber assigns a vertical velocity of 15 m. p. s. at the 5000-meter level, and that this agrees with the maximum ver- tical velocity which it seems probable can be developed in an ordinary summer cloud. This is not strong enough to sustain a hailstone of 1 cm. diameter, which is a small specimen, as a velocity of 20 m. p. s. is required for that purpose, while 40 m. p. s. is required to sustain a large stone 4 cm. in diameter. 4. The electrical attraction theory. In order to escape from such difficulties as those just enumerated in the vertical cur- rent theory, Trabert advocates the theory that the sudden accumulation of drops on the nucleus at undercooled tem- peratures is due to the electrical charging of the nuclei at the instant of a lightning flash, the surface charges having the power to attract water drops to the charged surface. The drops of a jet of water are thus suddenly drawn together by an electrified piece of wax placed near it. The drops fly together when changes take place in the electric field sur- rounding them. Some observers say that there is no hail without the electric phenomena. Each layer of ice is made suddenly by electric impulses which follow in succession for the several layers. The deposit of a layer brings the under- cooled temperature up to the freezing point. The escaping heat of the undercooled mass in condensing makes a water layer on the outside which changes to ice. There is a con- siderable quantity of latent heat evolved in the process of water and ice formation. Hail weather and lightning weather are alike in kind and different in intensity. Thunderstorms are associated with the horizontal roll due to overturning, and hailstones with the vertical vortex due to excessive convection currents. It has been suggested that the heat of the convec- tion process is transformed into electricity and that the required cooling is produced in that way, but of this there is little evi- dence. The cooling is also referred to sudden expansion in the air, but this would produce so great changes in the ba- rometer that it would be readily detected. This electrical theory ought to play a part in the formation of hail at times, but it is hardly demonstrable that hail does not fall without lightning, and certainly it is not shown that a flash of lightning occurs at the time of the deposit of the several stratified layers. A hailstorm often lasts many minutes, and during that time there must be, on this theory, such an incessant recurrence of lightning to match the numerous lay- ers of ice that it would be a very conspicuous event. There are many instances known in which the lightning seems to have really followed the hail by many seconds, but it should evidently precede it, if the time allowance for the fall from the cloud to the ground is subtracted from the instant when the hail is seen to fall upon the ground. F. W. Very writes: Severe hailstorms are almost universally accompanied by thunder and lightning, but the electrical display is apt to lag, and even to attain Its greatest development as the storm advances to its close. The rising air carries up the low surface potential on the front side of the hail squall, and the descending air brings down the high potential of the upper levels, so that there is an increase in the difference of potential at the hail level which causes horizontal flashes in the cloud, for the greater part. The earth's negative charge is carried aloft to the hail- stones, which are often negatively charged. Snow which forms in the high levels is usually positively charged. Reversals of the ordinary disposition of the electric potential have been noted as the effect of snow, hail, and water inductions brought to the surface of the ground. 5. The stratification theory. After the foregoing examination of the theories that have been heretofore proposed for the ex- planation of the growth of hailstones, I proceed to examine the subject from a new point of view, which seems to me to offer certain advantages over the other theories, and to em- body the best points of them all. This I call the stratification theory. It happens that a hailstorm cloud, which is merely an intense form of thunderstorm cloud, really consists of two component portions separated from each other by isothermal surfaces inclined forward from the vertical. On the front side the air is much warmer than on the back side, and along the line of separation the contour is strongly stratified by the mutual interpenetration from opposite directions of layers having different temperatures. Fig. 40, " Stratification of the /? and '5-stages in a thunder- storm cloud with hail ", roughly illustrates the idea. Such storms begin in consequence of the transportation of cold air into a region of warm air, and in many cases the difference of temperature amounts to as much as 20 F. The tendency for NOVEMBER, 1906. MONTHLY-jWEATHEE REVIEW. 517 such masses of air at different temperatures is to mix inti- mately and irregularly in order to restore the thermal equili- brium as rapidly as possible. The cold air is carried forward in the high levels, and like a sheet overflows the warmer lower layers, as is indicated by the first formation of clouds of the cirrus type, which later change into alto-cumulus and alto- stratus types. FIG. 40. Stratification of ,3- and J-stage in a cloud with hail. The body of warm air tends to rise and interpenetrate the cold air in a congested circulation including numerous minor whirls and small vortices. On the western side of the column of rising warm air the tendency to stratification of the warm and cold layers in horizontal directions is very pronounced, the sheets of different temperatures penetrating strongly at a series of intervals in elevation, so that they lie over each other on a given vertical in succession which may be repeated many times. The boundary between the ,3-stage and the