STABILITY PROPERTIES OF LARGE-SCALE ATMOSPHERIC DISTURBANCES 
In this article only adiabatic processes will be con- 
sidered. The physical equation is therefore 
— = —v-Vx. (4) 
From the foregoing equations one obtains by separa- 
tion into the mean and irregular flows: 
: + 9-V¥ = —Vy — fk X ¥ — xg —V'-yv’, (5a) 
V-Qv = 0, (5b) 
OE Se ys = vi yx! (5c) 
at é 
and 
/ 
= + yVy! = —Vy' — fk Xv’ 
(6a) 
— x'g — v-Vv! — v’-Vv + v’- vv’, 
V-Qv’ — 0, (6b) 
= = —v-Vx' — v'-Vx + v'-Vx!. (6c) 
These equations reveal the coupling which must exist 
between the two components of the flow, as both com- 
ponents occur in each set of equations. 
The Circular Vortex 
According to the definitions above, the orderly flow 
may be considered as a pure zonal flow with velocity 
component wi, and a meridional flow with velocity 
Vm = dj + wk, which is identically the same in all 
meridional planes. The meridional component of (5) 
may be written 
The approximate balance existing in the large-scale 
relative motion in the atmosphere gives, when V7 is 
eliminated from this equation: 
Ou Ox 
== 0 (7) 
This is the so-called thermal wind equation applied 
to the mean flow. 
One may study the axially symmetric meridional 
motions in a qualitative way by the method of veloc- 
ity circulation used primarily by V. Bjerknes [4] and 
Hgiland [16]. If Vy is eliminated from the foregoing 
equation by taking the circulation along some arbi- 
trary closed curve in a meridional plane, and the result- 
ing equation differentiated once with respect to time, 
one gets 
dt dt ae 
455 
The expression for d%/dt is obtained from the zonal 
component of (5a): 
ot = — Fn (Vit — fi) — vel (9) 
Substituting this expression for du%/dt in (8) and like- 
wise for 0x/dt from (5c), one gets 
he an, Se (Cope ne 
“¢ vais ‘or = $ i [fVaj — f jj + Vug] -6r 
(10) 
ap a-wle-o1 a § iv -vilh-on 
A study of the first integral on the right-hand side of 
this equation leads to the conditions which must exist 
if in a pure, axially symmetric motion the meridional 
circulations should accelerate or decelerate, in other 
words to the now well-known stability criteria for a 
circular vortex for vortex-ring perturbations. It will 
be assumed in this article that all orderly flows which 
are treated are stable in this sense. Most likely this is 
usually the case in the atmosphere. 
The remaining terms on the right-hand side of (10) 
represent the effects upon the acceleration of the meridi- 
onal circulations which are due to the disturbances. 
The character of the resulting forced circulations will 
now also depend essentially upon the stability prop- 
erties of the circular vortex for vortex-ring perturba- 
tions. Briefly, one may say that the presence of effects 
changing the fields of mass and velocity in the orderly 
zonal flow, other than effects of the meridional circu- 
lations themselves, will generally tend steadily to de- 
stroy the balance in the meridional motions. Because 
of the stability of the circular vortex the resulting 
added meridional circulations will act to restore the 
equilibrium, an equilibrium which will, however, be 
different from the original one. If the stability is large 
enough, the whole development may be thought of as 
one which goes through different equilibrium stages 
by smoothing out over sufficiently large periods the 
relatively high frequency oscillations superimposed 
upon this trend. The simplifications following from 
such a procedure are essentially the same as those 
introduced by the systematic use of the condition of 
quasi-geostrophic motion [7, 12]. With the simplifica- 
tions above, A. Eliassen [11] has studied the forced 
meridional circulations produced by given sources of 
heat and angular momentum. 
To see in a qualitative fashion how the irregular flow 
affects the mean meridional circulations one may use 
the simplifications mentioned above in connection with 
the circulation integrals in (10). One then obtains 
$ Sn Lf Vij — fH) + Viel ar 
= =f FH 61 _ § iv vui-on 
It will now be assumed that Vu and 0x/dy are small 
enough to be neglected where they occur in (9) and 
(11). It is further assumed that @x/dz = 0, which is 
(11) 
