STABILITY PROPERTIES OF LARGE-SCALE ATMOSPHERIC DISTURBANCES 
mitting that several such mechanisms may exist, the 
discussion here will be confined to the self-exciting 
one, by means of which small disturbances may amplify 
because of an instability of the underlying basic zonal 
flow. A theoretical attack on the problem of waves in 
a baroclinic atmosphere was first undertaken in the 
Norwegian polar front school of meteorology, princi- 
pally by H. Solberg [25] (in addition, see [2, 5]). Waves 
were examined on an inclined surface of discontinuity 
separating two barotropic layers. Thus, all the baro- 
clinity was considered as concentrated in a surface of 
discontinuity. In principle, however, with regard to the 
possibility of converting potential into kinetic energy, 
there is no difference between such a basic flow and 
one with continuously distributed baroclinity. In con- 
trast to the pure, polar front waves, which may possibly 
feed also upon the kinetic energy of the basic current, 
are the baroclinic waves for a horizontally uniform 
basic flow, first examined by Charney [6]. The essential 
stability properties of these waves may be found easily 
by means of equations (23)—(26), if one neglects the 
vertical transport of potential temperature. As re- 
marked in an earlier connection, and confirmed by the 
more rigorous solutions [10; 14, pp. 46-51], this approxi- 
mation will be justified for the relatively large wave 
lengths. The adiabatic equation (3) may then be written 
* 
(B42) t+ Moe <0, 
or, by differentiation with respect to x and use of (24), 
(27) 
cy) x 9 Ov* , Or x oe 
(G4u 2 )o. Cee EO ticgg Laat 
Let it now be assumed by way of example that 
ov* sa Ovr 
dy ay 
so that the results arrived at may be said to apply 
approximately to disturbances whose scale in the y-di- 
rection is large compared with the scale in the x-direc- 
tion. It follows then from (26) that du*/da = du7/dx = 0. 
With the assumption of no horizontal shear for the 
zonal flow, u* and wr will now have to be independent 
of x and y. It is also easily understood that neither 
can they depend upon time. Equations (23) and (25) 
now reduce to 
Onn (OO! P 
(2+ i =| ae Ble 
Wee Or — (3+ uw?) vr = 0 
Suppose 
v* ~ exp [1(ua@ + wt)] 
and 
vr ~ exp [t(ux + ot)] 
459 
to be solutions of (28). By substitution into (28) one 
gets as the condition that v* and vr do not vanish 
identically, 
— 
o= Spit / (2) = wut. 
When the square root becomes imaginary, and the 
negative sign is taken, v* and v7 will increase exponen- 
tially. When the substitution for wr is made from (24), 
the criterion for stability and instability therefore be- 
(unstable) 
comes 
(2) - we (Ey St 
Di ii dz) >0 (stable). 
Formula (29) reveals the high degree of instability of 
a horizontally uniform current in a baroclinic atmos- 
phere. This result is common to all the different studies 
of baroclinic waves, which otherwise differ widely both 
with respect to the manner of formulating the problem 
of instability, and with respect to some of the con- 
clusions obtained [1; 6; 10; 14, pp. 35-51]. Below is a 
summary of some of the assumptions and conclusions. 
(29) 
CHARNEY 
Assumptions: 
1. Geostrophic approximation. 
2. Compressibility. 
3. Infinite atmosphere. 
4. Vertical stability. 
Conclusion: 
All waves, at least down to about 1200 km, are un- 
stable for a sufficiently large meridional tempera- 
ture gradient. 
Eapy 
Assumptions: 
1. Geostrophic assumption. 
2. Incompressibility. 
3. Vertical stability. 
4. Inertia effects of inhomogeneity neglected: 
Dv Dy 
@ (2% + se xv) = const (DF + sk xv). 
5. Constant Coriolis parameter: df/dy = 0. 
6. (a) Finite atmosphere bounded by two rigid walls 
at distance h apart. 
(b) Infinite atmosphere. 
Conclusions from assumption (6) are: 
(a) All waves are unstable? if and only if 
ING 7 Ox 
@ ise (« =). 
(b) No waves are unstable unless a layer of smaller 
vertical stability is underlying one of greater sta- 
bility. 
3. Compare with the result obtained on p. 457. 
