460 
FJORTOFT 
Assumptions: 
1. Incompressibility. 
2. Vertical stability zero. 
3. Constant Coriolis parameter: df/dy = 0. 
4. Finite atmosphere bounded by horizontal walls 
at distance h apart. 
Conclusion: 
All waves are unstable if and only if 
L 2 x 7 o. 
h 2f? \dz 
It may be remarked that Eady’s second conclusion 
will no longer hold if the simplification mentioned in 
his fourth assumption is not made, in other words, 
even the infinite atmosphere with uniform vertical 
stability will be unstable if account is taken of the 
fact that density diminishes to zero for increasing 
heights. The appearance of a stabilizing influence from 
the vertical shear for short wave lengths in Fjgrtoft’s 
conclusion is due to effects to be discussed at the end 
of the following section. This stabilizing effect could 
not possibly appear in the other studies because there 
(w + 2ru/L)° was neglected compared with f° [14, pp. 
41-42]. 
Barotropic Disturbances 
In the case of barotropy there can be no vertical 
wind shear in the state of quasi-balance which charac- 
terizes the motions with which this article is concerned. 
Equations (18) and (21) therefore reduce to 
ID 5. 
dt 
= 0; (30) 
with the asterisks now dropped as superfluous. These 
equations represent the classical equations for con- 
servation of vorticity in a two-dimensional nondivergent 
flow of a nonviscous fluid. The fundamental investiga- 
tion of wave motions in such flows was carried out by 
Rayleigh [22]. Although in the studies of polar front 
waves the destabilizing effects of a gliding discontinuity 
were considered to be important, it has not been until 
recently that barotropic phenomena in their full and 
complex generality have been taken up for systematic 
investigations, primarily by the Chicago school of mete- 
orology. Starting with Rossby’s work on planetary 
waves [24], in which the specific importance of the 
variability of the Coriolis parameter was discovered, a 
series of papers have followed in which different baro- 
tropic phenomena have been discussed. Briefly, one 
may say that while some of them, apart from the 
modifications following from the spherical shape of 
the earth, are analogous to the classical works by Ray- 
leigh, others represent original investigations as exem- 
plified by those treating stationary solutions of the 
nonlinear vorticity equation [9, 13, 17, 20]. The mathe- 
matical solution of (80) involves, of course, all the 
difficulties connected with the solution of nonlinear 
equations. The difficulties may even be very great for 
DYNAMICS OF THE ATMOSPHERE 
the linearized equations, particularly when there is 
a variable zonal current [18]. It is, however, possible 
and also useful to obtain an understanding of several 
of the most important barotropic phenomena by direct 
physical considerations [14, pp. 15-35]. In the follow- 
ing discussion this kind of argument will be used. We 
will make the purely formal simplification of assuming 
that the horizontal motion takes place as if on a circu- 
lar disk; however, f will be kept a variable parameter. 
One of the physical principles which may be used 
for a general discussion of some barotropic phenomena 
is the conservation of total angular momentum: 
[ur dF = const. (81) 
Here # is the distance from the pole, and dF a surface 
element in the plane of motion. By introducing the 
20 
velocity circulation ¢ = [ uk dy along zonal circles, 
0 
(81) becomes equivalent t6 
[ear = const. (82) 
Another equivalent expression can easily be shown to 
be [14, p. 21] 
G= / fave(Ro ) ¥o)R? dF = const. _—(88) 
Here, fabs is represented as a function of Lagrangian 
coordinates, and is therefore independent of time be- 
cause the absolute vorticity is conserved, and Ff repre- 
sents the generally time-variable radial positions of 
the fluid particles. Equation (83) is now a condition 
which restricts all future radial positions. In Fig. 3 
Fie. 3.—Isolines for the vertical component of absolute 
vorticity. 
the irregular lines are curves of equal absolute vorticity 
for the case that £.p; varies monotonically from one 
isoline to the other. Clearly, a necessary and sufficient 
condition for having a pure zonal flow is that the lines 
