STABILITY PROPERTIES OF LARGE-SCALE ATMOSPHERIC DISTURBANCES 
of equal vorticity have a zonal distribution. The motion 
corresponding to the vorticity distribution in Fig. 3 
is therefore certainly not zonal. It may now be inferred 
from (33) that neither can it be so at any later time. 
The reason for this is that by varying the positions of 
the fluid particles, G assumes an extreme value if and 
only if the lines of equal vorticity are zonal. In the 
present case this extremum would be either an absolute 
minimum or maximum. Therefore, the isolines of fabs 
could not possibly become zonal unless at the same 
time G varied in contradiction to (83). Thus, at all 
times one must have 
[srar2m,  (m>0), (34) 
where a prime again has been used to indicate devia- 
tions from zonal means. With possible exceptions for 
some singular cases, it can be shown that (84) must be 
true for a quite general distribution of vorticities. 
A problem of considerable interest is involved in 
the question as to what will happen in a barotropic 
atmosphere left with a certain distribution of vortic- 
ities that do not correspond to a motion which is sta- 
tionary, either absolutely or with respect to a co- 
ordinate system rotating at a constant speed. Will 
the structure of the subsequent motion tend to approach 
a more or less definite limit, or will different structures 
be repeated more or less periodically? Synoptic experi- 
ence is probably most in favor of the first point of 
view, though there have been attempts to interpret 
some cycles in the general circulation on a strictly 
barotropic basis. In favor of the first point of view one 
can at least say with certainty that theoretically no 
oscillations in the strict sense are possible. This is 
because in a purely oscillatory motion the fluid particles 
would simultaneously have to reassume earlier posi- 
tions, but with different velocities. However, this is 
impossible because, according to the conservation of 
vorticity, the positions of the fluid particles determine 
uniquely the vorticity distribution which in turn de- 
termines the velocities. 
From the condition | cP dF + / ft, dF = const, 
there is also an upper limit to / ¢?dF, so that (34) 
may be extended to 
M= pe dF =m, (m> 0). (35) 
Returning now to the kind of vorticity distributions 
which are illustrated in Fig. 3, it will be assumed as a 
particular case that the vorticity distribution deviates 
only slightly from a zonal one. The corresponding flow 
is a zonal flow upon which there are superimposed 
small disturbances. The time-invariant value of G will 
now be but slightly different from a maximum or 
minimum value, which implies that no change in the 
vorticity pattern from the nearly zonal one to one 
characterized as in Fig. 3 can take place without neces- 
sarily changing the value of G. The proof for this is 
461 
simply the reversal of the one stating that it was not 
possible to come arbitrarily near to a zonal distribu- 
tion. The corresponding zonal flow is, therefore, stable 
in this case. Let & + f and ¢% be the initial vorticities 
in the mean zonal flow and.in the disturbances. An 
expression and criterion for the present stability is 
M = [scar > m, 
(m > 0), (86a) 
M-—0 when fs dF — 0, (866) 
if 
€) + f varies monotonically with latitude. (36c) 
It should be mentioned that the result which has 
been found that polar anticyclones are always unstable 
in a barotropic atmosphere [23] cannot be true if (36) 
is true. That is because anticyclones may exist for 
which condition (36c) still may be true. 
Suppose now again that the absolute vorticity for 
the mean zonal flow varies monotonically with latitude 
and consider the possible changes in the mean zonal 
flow which will result from the meridional exchange 
of air. It may be assumed, in accordance with the 
most frequent conditions, that d(> + f)/dy > 0. One 
has 
e c 1 
i =e see 
where F is the area north of the latitude circle which is 
being considered. Since equal areas of fluid are going 
in and out of a latitude circle, % will have to decrease 
or increase according as the vorticities leaving the circle 
are replaced by vorticities of lower or higher magni- 
tudes, respectively. Now, each fluid particle is assigned 
a certain absolute vorticity € + f+ ¢0 which is moving 
with the particle. The effect of this transport may be 
considered as a separate effect from those of the trans- 
port of & + f and of ¢. As to the first effect it is 
easily understood that because f + f is increasing 
northwards, air streaming out of a zonal circle will 
have to be replaced by air with lower value of the 
absolute vorticity. This effect, when considered alone, 
will therefore amount to a decrease in ¢ at all latitudes. 
This is in conflict with the principle of the conservation 
of total angular momentum, equation (32). It can, 
therefore, immediately be inferred that the effect from 
the transport of the initial irregular vorticities must 
be to compensate exactly this loss in angular momen- 
tum, that is, to create in the mean a compensating 
westerly flow [14, p. 28]. In consequence of this it 
may be concluded that, in the mean at least, the air 
with positive (> has to be transported to the north, 
and that with negative ¢> to the south. In the special 
case of a stationary flow pattern, as for instance for 
the Rossby waves, the two effects compensate each 
other exactly at each latitude. In other cases, however, 
one can only expect that the compensation is accom- 
plished after integration over all latitudes. It has been 
assumed by Kuo [19] that the effect of the transport 
