462 
of the irregular vorticities will be the dominating one 
at middle latitudes, resulting there in an increase of 
the westerlies. However, the arguments used are not 
convincing, and without carrying out numerical calcu- 
lations nothing certain can be said with respect to 
the resulting changes in the mean zonal flow. Pre- 
liminary calculations [15] seem to indicate that rather 
than increasing the westerlies at middle latitudes, the 
tendency under certain conditions may as well be to 
decrease the westerlies at middle latitudes and increase 
them in belts to the north and south, particularly to 
the north. 
In a barotropic atmosphere the energy equation re- 
duces to ; 
/ ty" dF = — i 17” dF + const. 
Consequently, there is an upper bound to the kinetic 
energy of the disturbances. In the case where ¢.s varied 
monotonically from one isoline of vorticity to the next 
it was found that there must be a lower bound to 
i ¢? dF which is different from zero. The same must, 
therefore, be the case for the kinetic energy of the 
irregular flow. Thus, in this case 
A= fav? ar > a, 
(a > 0). (37) 
The stability under condition (36c) therefore does not 
imply that the disturbances are entirely damped out. 
Whether the total kinetic energy of the disturbances 
will increase or decrease is usually difficult to ascertain 
and will depend upon the character of the changes in 
u. Writing 
2 / P= ans / (@ — 2) aF 
: (38) 
= -[ 2 Gr — wk) ar - sf = the)? dF, 
one finds easily [14, p. 23] by using condition (81) in 
the form 
| cr — wk) ar = 0 (39) 
that @ has to decrease where %/F is large and increase 
where %/R is small, if the kinetic energy of the dis- 
turbances is to increase. Particularly if one has to do 
with small disturbances in a zonal flow, the upper 
bound to the changes in % which can be caused by a 
transport of the initial irregular vorticities is also a 
small quantity. In order to find necessary conditions 
for real instability one has therefore to investigate 
the effect from the transport of ( + f. It was found 
earlier that « would decrease if d(f + f)/dy > O. 
By similar arguments one will find that a on the other 
hand has to increase where d(¢) + f)/dy < 0. So, the 
necessary conditions for real instability will be that 
DYNAMICS OF THE ATMOSPHERE 
to 
R is large, 
(& +f) >0 where 
(40) 
@ re Uo . 
ay et F) <0 where 5 is small. 
The trivial case with solid rotation in the mean flow, 
%/R = const, implies, of course, according to (38) 
and (39), that / 3v? dF at most can remain constant 
in time, but will decrease if some changes in @ result. 
Hitherto, the most complete mathematical treat- 
ment of barotropic waves in a basic flow which is 
unstable in the above sense has been undertaken by 
Kuo [18]. This instability is fundamentally the same 
as the one occurring when a gliding discontinuity exists. 
However, by treating the realistic case with a con- 
tinuous shear and including the variation of the Cori- 
olis parameter, two important modifications result: 
1. As results of an assumed continuous shear under 
average atmospheric conditions: 
a. All waves below approximately 300 km are stable. 
b. An intermediate wave length of maximum in- 
stability exists. 
2. As a result of the inclusion of df/dy ~ 0, the 
longest waves become stable. 
It is important to notice that the most unstable 
waves of the type discussed above are relatively long 
waves compared with the most unstable baroclinic 
waves [10; 14, p. 50]. 
How can the stability occurring for short waves 
mentioned above be understood? It was previously 
seen that provided conditions (40) were fulfilled the 
kinetic energy of the disturbances would necessarily 
increase as a result of a transport of >) + f. Therefore, 
when the shortest waves become stable this can only 
be a result of the transport of the initial irregular 
vorticities, (0 , which furthermore must tend to be the 
dominating effect for the shortest wave lengths. An 
understanding of the stabilizing influence arising from 
the transport of irregular vorticities may be obtained 
in the following way: Suppose a wavelike disturbance 
with untilted troughs and ridges to exist initially in a 
nonuniform zonal current. The instantaneous transport 
of the irregular vorticities is accomplished by a com- 
ponent wi of the mean flow and a component vo of 
the irregular flow. When small disturbances are con- 
sidered, or disturbances in which vo is essentially parallel 
to the lines (> = const, only the transport by the first 
component has to be considered. It is now obvious 
that if the angular velocity %/R varies with latitude, 
the lengths of the lines ¢’ = const have to increase 
as a result of this transport. On the other hand, the 
areas enclosed by these lines are conserved as are 
also the values of ¢’ because the effects of the transverse 
displacements of the vorticities f) + f are disregarded 
in this connection. Consequently, it follows from Stokes’ 
theorem that the velocity circulation for the disturb- 
ances taken along the closed curves v(’ = const, which 
approximately are also streamlines for v’, must remain 
