456 
necessary to insure the stability of the circular vortex 
in the present case. Equations (9) and (11) now reduce 
@ .. <=; 
J Ot = fo -, v’-vw, 
Z 
= $v wigs — § iv-vuly 
Let it now be assumed that the path of integration 
consists of the sides of a ‘rectangle’? bounded below 
by the earth’s surface and with the top at about 
tropopause height (Fig. 1). The direction of integra- 
(12) 
(18) 
i 
Fie. 1.—Idealized meridional circulation. 
tion is indicated by arrows. With subscripts from 1 to 
4 denoting average values along the sides of the rec- 
tangle correspondingly labelled, (18) may be written 
f (is — 0;)B + Te (W. — Ws)H (14) 
= g(v’- ys — v’-yxt)H + f(v’-vus — v’-Vut)B. 
Here, B and 4 are the lengths of the sides of the rec- 
tangle. Suppose further that the rectangular-shaped 
boundary is a streamline in a corresponding simple 
cellular meridional motion, in which 9 and W are 
derived from a stream function 
Wa ~ sin sin 5 - 
This implies that for the present the atmosphere is 
treated as incompressible. It may be anticipated, how- 
ever, that the results to be obtained will also roughly 
apply to cellular motions whose kinematics are essen- 
tially the same as in this most simple cellular motion. 
By deriving + and W from (15) and forming the 
averages 0; , 03, W., and ™,, one obtains 
(Ws — M,)B = (03 — 01)H. (16) 
If one does not think of the velocities in this formula 
as averages along the sides of the rectangular boundary, 
but rather as averages over the whole region of the 
velocities in the ascending and descending motions, 
and in the north and south motions, the formula simply 
(15) 
DYNAMICS OF THE ATMOSPHERE 
states the well-known continuity principle that the 
ratio between the magnitude of the vertical and hori- 
zontal velocities is proportional to the ratio between 
the vertical and horizontal scale of the motion. By a 
suitable choice of the integration curve in the circula- 
tion integrals given above, one could therefore also 
apply (16) to cellular motions of a much more general 
character than the one determined from (15). In view 
of (16), (14) may now be written 
- 2) 
f(s — v1) (i +9 =S) 
(17) 
= fglv’- yx — v’-yx4) * + f'(v’-Vus — v’-Vvul). 
Taking likewise an average of (12) along the horizontal 
sides of the rectangle, one obtains by subtraction 
(o) i — = pep, SST ee ay 
at (a3 — th) = fs = ty) = (v’- Vus — v’-yui). 
Substituting here for f(0; — %,) from (17), one obtains 
[= -Vui — Vv’: a 
ox H? H 
i 4 (18) 
i 0 wz SS 
AF Pea REE | Serve — vw |. 
2 SS 
arg dz B? 
This formula now determines, as a function of two 
terms, the time rate of change in the vertical wind 
shear of the mean zonal flow: The first term involves 
the wind shear that would directly result from the 
dynamic effects of the irregular motion; the second, 
the mean meridional temperature gradient resulting 
directly from the thermal effects of the disturbances. 
However, it is seen that only fractions of these quanti- 
ties are effective in building up the resulting shear 
since they are multiplied by factors smaller than unity. 
This is clearly a result of the interference with the 
effects resulting from the forced meridional circula- 
tions, and could have been obtained by more direct 
considerations. The formula given above, however, may 
serve as a rough indicator of how this interference 
depends upon the stability of the circular vortex and 
the horizontal and vertical scales of the motion. In 
the present discussion it will only be pointed out how 
the dynamically conditioned increase in the wind shear 
becomes more and more compensated when the verti- 
cal stability goes to zero, or when H/B becomes smaller, 
while at the same time the thermally conditioned in- 
crease in wind shear becomes correspondingly more 
important. The relative importance, in regard to the 
general circulation, of the dynamic effects of the dis- 
turbances on the one hand, and the thermal effects on 
the other, naturally depends also upon the relative 
magnitudes of the terms v’-Vu’ and v’-Vx' and their 
distribution. This is intimately connected with the 
problems to be treated in the following sections. 
