STABILITY PROPERTIES OF LARGE-SCALE ATMOSPHERIC DISTURBANCES 
Baroclinic Disturbances 
One way of classifying atmospheric disturbances is 
with respect to the sources of energy which are at their 
disposal. The energy equation, when integrated over 
an isolated volume 7 of the atmosphere, is obtained 
from (2) and (3) and becomes 
[ass dr =) Qxugz dr + const. 
Substituting here v = wi + Vv, + v’, one arrives at 
[aw dr =) Qugz dr — [oe dr + const, 
having neglected the relatively much smaller kinetic 
energy contained in the mean meridional circulations. 
Applying the same approximation, the time rate of 
change of this equation becomes, in view of (5) and (6), 
@ | Gav? ar = | Qeu'gar 
— fat wera — fo wu’ ar 
oy Oz 
Consequently, one may classify disturbances into three 
categories according to whether the main source of 
energy is potential, kinetic, or both. In this section 
the first one will be considered. 
Clearly, it is the correlation between the fluctua- 
tions in temperature and vertical velocity which will 
be decisive in determining whether potential energy 
shall be a source or sink for the disturbances. It is in 
accordance with synoptic experience that in most cases 
cold air masses sink relative to the warmer ones. It is 
therefore to be expected that potential energy is, at 
least partly, an important factor in creating and main- 
taining the large-scale disturbances. In order to under- 
stand how a positive correlation between the fluctua- 
tions of temperature and vertical velocity may be 
brought about, one may write 
So KV 
(19) 
/ 
iS 
CK > O). 
Here, 1 is a kind of mixing length, and a positive cor- 
relation has been assumed between | and v’. Conse- 
quently, one has 
| exv'g dr = —K | QWw!-vijw'g dr 
ay He Doni ripe i 2 Ox 
K J Qu'w'  g ar K | Qu? gar. 
Having assumed vertical stability, it is therefore seen 
that as requirements for potential energy to be fed 
into the disturbances (2) horizontal temperature gradi- 
ents must exist, and (77) the slope of the streamlines in 
the meridional planes, w’/v’, must be of the same sign 
as the slope (—0dx/dy)/(0%/dz) of the isentropic sur- 
faces of the mean flow, but have a smaller magnitude. 
This last requirement may also be stated as saying 
that in the identity v’-Vz = v'dx/dy + w'dx/dz there 
must be a tendency for the vertical transport of entropy 
457 
to compensate the horizontal transport. The latter, 
however, has to be the dominating effect, so that ap- 
proximately v’-Vx = v/dx/dy. 
For reasons of continuity it is to be expected that 
for decreasing horizontal dimensions of the disturb- 
ances the magnitude of the vertical velocities will in- 
crease relative to the horizontal velocities. It would 
therefore not be surprising if, for sufficiently small 
horizontal dimensions of the disturbances, w!/v’ would 
exceed the values for which a conversion of potential 
energy into kinetic energy of the disturbances could 
take place. This effect will now be studied in more 
detail. To obtain results comparable with others which 
will be referred to at the end of this section, an in- 
compressible atmosphere will be considered, bounded 
by two horizontal rigid planes at distance h apart. 
Also it will be assumed that 
By eliminating V,y from the horizontal component of 
(2), one now obtains 
0 0 \ Ov Ou ow 
(4u2)” gu cae ae 
(20) 
Let it be assumed that instantaneously v and its de- 
rivatives may be obtained from the identity 
i he or eee DN 
v = cons ome TW Wo ; 
representing instantaneously a simple wave propagat- 
ing with a speed given by the value of wu halfway be- 
tween the boundaries. By substituting into (20), one 
obtains 
is oe -4)o = 28 
L? dz 2 noe OR 4 
Applying the boundary condition w = 0 for z = 0, 
one obtains by integration between z = 0 and the 
height h/2 where w reaches its maximum value: 
74 4 P) 
Wan. 7h du 
v iG Ge} 
Applying now the condition (27) above to wz=n/2/2, 
one obtains 
rh du 
2L?f dz 
al Ox/ dy 
ax/dz? 
or, by substitution from the thermal wind relationship 
du/dz = —(g/f)(dx/dy): 
ib a) 
mi > 2p? 22) 
This now constitutes approximately the restriction on 
the horizontal scale of the disturbances if potential 
energy is to be converted into kinetic energy of the 
disturbances. 
It will now be shown how one may express the con- 
ditions for a positive correlation between x’ and w’ 
(21) 
