May 4, 19 16] 



NATURE 



21 1 



The Rigime of the Stratosphere. 

 But our new point of view only shows our problem 

 removed one step further; we have now to begin 

 again and imagine for ourselves what is the rigime 

 of pressure and winds in the stratosphere until the 

 enterprise of meteorologists completes our knowledge 

 of what it actually is. The problem is, at any rate, 

 much simplified, because convection is avoided ; we 

 deal with an atmosphere which, being nearly iso- 

 thermal, is inherently stable ; density goes directly 

 with pressure, layer lies on layer like a light liquid on 

 a heavy one ; temperatures are uniform, or very nearly 

 so, in the vertical direction, and therefore isotherms 

 are also isobars, and winds are proportional every- 

 where to pressure-differences — that is, to temperature- 

 differences. Outside the equatorial region the rotation 

 of the earth secures that air always moves along the 

 lines of pressure, keeping high pressure or low tem- 

 perature on the right. So the general idea is simple, 

 but whether the streams of air are long, straight 

 currents or centrical whirls we do not yet know. 



Numerical Calculations. 



Speculations of a qualitative character are apt to 

 lead the speculator into serious error; the real test 

 of any physical theory is its quantitative application. 



It will be of great advantage to the further develop- 

 ment of our ideas if we can trust implicitly to the 

 hypothesis of pressure balanced by motion (let us 

 call it the principle of strophic balance) as the founda- 

 tion of the structure of the atmosphere, and that 

 hypothesis will be confirmed in the orthodox scientific 

 manner if the quantitative conclusions to be drawn 

 from it are verified by observation. I propose to 

 ask your attention to some applications of that hypo- 

 thesis which can be tested numerically. 



From this point of view the theory of strophic 

 balance has the great advantage of giving a definite 

 relation between wind velocity, pressure, and tem- 

 perature, and therefore brings the relations between 

 all these quantities within the region of arithmetical 

 computation. 



Let us consider some of these relations. We 

 require a number of symbols for the meteorological 

 quantities : — 



/> represents the atmospheric pressure 

 6 „ „ „ temperature 



p „ „ „ density 



/ „ „ horizontal distance 



h „ „ vertical height 



„ horizontal pressure gradient 



„ „ temperature gradient 



„ velocity of the wind 

 R=/>l(p6) „ „ constant of the gas equation. 



Certain geodesic quantities also come in, viz. : — 



E, the radius of the earth. 



g, the acceleration of gravity. 



r, the angular radius of a small circle on the earth's 

 surface which indicates the path of air in a cyclone. 



A, the latitude of the plaqe of observation. 



<u, the angular velocity of the earth's rotation. 



We require also some convention as to the positive 

 and negative of v. 



V positive represents the winds when the pressure^ 

 difference Ap represents higher pressure on the right 

 of the path. 



The fundamental relation between the velocity of 

 the wind at any level and the pressure-gradient there 

 is : — 



(=1) •■ 



(40 .. 



^ = -jf = 2a)Vp sm X ± — p cot r 

 NO. 2427, VOL. 97] 



(F) 



The two terms which make up the right-hand side 

 of this equation are of different importance in different 

 places and circumstances ; for example, if the air is 

 moving in a great circle; r is 90° and cot r is zero; 

 the first term alone remains. On the other hand, at 

 the equator the latitude A=o, sin A is zero, and the 

 second term alone remains. Away from the equatorial 

 region the second term is relatively unimportant 

 unless the velocity v is great. In temperate and 

 polar latitudes the path of the air differs little from a 

 great circle except in rare cases near the centre of 

 deep depressions ; consequently the first term may be 

 regarded as the dominant term in these regions. 



We call the wind computed according to the first 

 term the geostrophic wind, and regard it as generally 

 representing the actual wind of temperate and f)olar 

 regions. 



We call the wind computed according to the second 

 term the cyclostrophic wind, and regard it as repre- 

 senting the actual wind (in so far as there is any 

 regular or persistent wind at all) in the equatorial 

 regions. It represents the wind of tropical hurricanes, 

 and winds of the same character may also occur 

 locally in temperate regions as tornados and other 

 revolving storms. 



Thus we have the following auxiliary equations : — 



Horizontal gradient of pres-") 



sure J 



Horizontal gradient of tem-\ 



perature . . . .J 

 Winds of temperate and polar"! 



regions — geostrophic winds/ 

 Winds of equatorial regions — ) 



cyclostrophic winds . .J 



The measurement of pressure-} ^= -^p 



dl 



d6 

 ^=dl 

 s =20)t'p sin X 



s =p cot r 

 E 



Lrfyi 



(I) 



(2) 

 (3) 



gaseous laws (assumed) j, n /, , ^ 



dry air). . . .|/=Rp^ (4) 



_ i^d6 ^E 

 ddh cot r 



The 

 for 



From these by simple manipulation I have deduced 

 the following : — 



For change of pressure gra-) ds _ Iq s\ 

 dient with height . •Sdh~^^\6~p) 



For change of wind velocity"! . 

 with height — J- ~ = 



geostrophic winds . .J '*'* 



cyclostrophic winds. X — 

 y d/t 



Deductions from the Theory of Equivalence of 

 Pressure-distribution and Wind. 

 These equations serve to explain the following facts 

 established by observation * : — 



I. Light winds in the central region of an anti- 

 cyclone. 



It follows from the fundamental equation F when 

 the negative sign is taken, as it must be for an anti- 

 cyclone, that the values of v will be given by the roots 

 of a quadratic equation, which will be impossible if 



V is greater than "^^'^ . This, for a circle of 



70 iniles' diameter, only allows a velocity of about 

 4 metres per second. 



This is confirmed in practice, and furnishes a 



* The folIo«ang references may be given for the statements enumerated 

 here :— (i) Barometric Gradient and Wind Force. Report by Ernest Gold. 

 M.O. Publication No. 190. (2) Shaw. Journal of the Scottish Met. Soc., 

 vol. xvi., p. 167, 19x3. (3) Shaw, Q.J. Roy. Met. Soc., vol. xl., p. m, 

 1914. (4) The Free Atmosphere of the British Isles. Report by W. H 

 Dines, F.R.S. M.O. Publication, No. 202. C. J. P. Cave, The Structure 

 of the Atmosphere in Clear Weather. (Cambridge University Press.) 

 E. Gold, The International^ Kite and Balloon Ascents. Geophysical 

 Memoirs, No. 5. M.O. Publication 210^. The computations of equations 

 B and C are not yet published ; the direction of the wind is regarded as 

 not being subject to change with height. (5) Shaw, Principia .Atmospherica. 

 Proc. R.S.E., vol. xxxiv., p. 77, 1914. 



