458 
in terms of the horizontal fields of velocity and tem- 
perature only. The conclusions arrived at are very 
much like those of Bjerknes and Holmboe [3]. The 
line of argument followed below is in some respects 
similar to that of Sutcliffe [26, 27]. (See also [8] and the 
article by J. G. Charney in this Compendium.+) 
From (2) one finds the vorticity equation in the 
vertical component to be 
a+ ve VE + By + fVn-v = 0, 
when the presumably small terms 
w(df/dz) + 6Va-v + Viw X (dv/dz)-k 
are neglected. In this equation ¢ is the vorticity and 
6 is the variation of the Coriolis parameter with lati- 
Ande, led a he defined! fiom of [ OQ = il Qa dh, 
0 0 
where Q is now supposed to be the standard density 
in some isothermal atmosphere. The term a* will 
represent the mean, in the vertical, of a with respect 
to mass. Taking this mean of the vorticity equation, 
one obtains, by virtue of the continuity equation (3) 
and the boundary conditions Qw = Oforz = 0,z = ~, 
ar* 
apt vi -VS* + Bo* 
(22) 
+ [(, — vi)- Ve — ¢*)]* = 0. 
In this equation the last term depends upon the exist- 
ence of vertical wind shears, or in consequence of the 
thermal wind equation, upon the horizontal tempera- 
ture gradients. It must therefore be this term that 
provides for the effects responsible for conversion of 
potential into kinetic energy. For a rough estimate of 
this term one may assume dv,/dz’ = 0. From the 
thermal wind equation this implies that Vax = Vpx*. 
Equation (22) now takes the form 
ow Bs * * * 
Ob = —Vr VE a Bo ms Vr: Vor, (23) 
where 
Vn V--0 + Vr, 
Va = ay, = mors Vu* Xk, (24) 
dz f 
and H = height of the homogeneous atmosphere. By 
taking the mean in the vertical of the physical equa- 
tion (4), one further obtains 
Oe * “ Ox 
OL = Vn Vx (w = (25) 
To these equations one may add 
V-vi = 0, V-vr=0, (26) 
of which the first is exactly true owing to the definition 
1. “Dynamic Forecasting by Numerical Process” by J. G. 
Charney, pp. 470-482. 
2. The assumption of isothermalcy is not necessary, but 
convenient. 
DYNAMICS OF THE ATMOSPHERE 
of vz; , and the second approximately true because of 
the identity (24). 
It may be inferred from (22) that, under the fore- 
going assumptions, vorticity in the vertical-mean mo- 
tion can vary individually only as a result of an ad- 
vection of the vorticity of the thermal wind by the 
thermal wind v,. It is also apparent that if in the 
thermal wind field there is a transport of cyclonic 
thermal wind vorticity into regions of high vorticities 
in the actual motion, these vorticities will intensify. 
This corresponds to one of the rules developed by 
Sutcliffe for the sea-level motion [26, p. 205]. Applied 
to troughs which are symmetrical with respect to meri- 
dians, this leads, as Fig. 2 illustrates, to the synopti- 
Fic. 2.—Flow pattern (solid lines) and temperature pattern 
(dashed lines) in an intensifying trough. 
cally well-known rule that troughs in the upper-air 
flow pattern intensify if troughs in the temperature 
patterns lag behind the troughs in the streamline pat- 
terns. A similar rule applies to ridges. It should be 
noted that these intensifications cannot be a result of 
the solenoids in horizontal planes, since these were 
neglected in the derivation of (2). The corresponding 
increase in the kinetic energy must have resulted from 
a conversion of potential energy. With reference to 
the discussion earlier in this section, it may therefore 
be concluded that, under the conditions mentioned 
above, the cold air must be subsiding relative to the 
warmer air, and further that the temperature changes 
due to the vertical motions can only partly compensate 
the advective changes. This rather definite knowledge 
of the three-dimensional flow structure based only upon 
a knowledge of the structure of the horizontal tem- 
perature and flow patterns is noteworthy. Clearly, noth- 
ing could in principle prevent a horizontal flow as 
illustrated in Fig. 2 from having any distribution of 
vertical velocities initially. This seeming discrepancy 
is due to the specific use which has been made of the 
conditions for quasi-geostrophic motion, which actu- 
ally may be interpreted as effecting a smoothing similar 
to the one mentioned under the study of the mean 
meridional motions. As there, the success of such a 
smoothing, and therefore of the specific use of the 
conditions for quasi-geostrophic motion, depends upon 
the stability of the noise motion which is superimposed 
upon the ‘‘geostrophically” conditioned trend. 
The question may now be raised as to what are the 
different mechanisms leading to the conditions men- 
tioned above, under which potential energy can be 
converted into kinetic energy of the disturbances. Ad- 
