HYDRODYNAMIC INSTABILITY 445 
i ~ j. In (66) the K,’s are the components of the 
transverse vector 
W 
K = Bue eeevi 
C? u—C 
any ev (1 
(67) 
C2 C2 : 
Equation (66) has meaning only if A ~ 0, that is, 
when the orbital circular frequency » = u(e — u) of 
the perturbation is different from the circular fre- 
quencies ys and yp for inertial oscillations. When » = 
vs (or vp), and this may occur at certain points of the 
transverse plane, the amplitude relations deduced from 
(62) are compatible only if 
3 2 0 
at the above-mentioned points. 
Continuing the analysis, we substitute( 66) into (58) 
and (61) and obtain the amplitudes £ and o as func- 
tions of @, that is, 
UES iC 
ii u(u — =) _ Ve 
_ apo fos _ _ 8s) 
coro dui (4 nS ) u-—ec’ (68) 
py ee -K,2) _ 8a 
© dx? \da! 4 Cr" 
Finally we replace £’, £, and o in (60) by their values in 
(66) and (68); there results an equation in the partial 
derivatives of amplitude o of the perturbation pres- 
sure p [31, 42], 
19 7 OO) [1 8 (ygiig 
sam (54 ) [Emo i) 
J Orig (1 cairn e)) 2 | =i 
“7 (u— ©)? @ 
We observe at once that equation (69) is of elliptic 
or hyperbolic type depending on whether A > or < 0. 
The determinant A = || A:;|| = vo —(an + ay)vo + 
1 G2 — (a2) is identical with the first member of the 
equation for circular frequencies of inertial oscillation; 
therefore, we have || A;; || = (vs — v5)(va — vs). Since 
|v| <|vpv|<]|vs|and vg > O in the atmosphere, 
A > or < 0 as vj > or < 0; there follows the proposi- 
tion [42]: the partial differential equation in the perturba- 
tion pressure is of elliptic or hyperbolic type depending 
on whether the geostrophic motion is stable or unstable. 
The amplitude @ which solves the present problem 
is that solution of the second order equation (69) 
which satisfies the boundary conditions [42] 
Aas 2 ~K,e) a) 
for «x =0(orz = 0), and 
sai (22 - K,o) —5=0 
(69) 
(70) 
Ox? \ da? 
for P(x, %2) + a(2', 2°) exp \/—1u(a — ct) 
= const. 
It is probable (1) that the existence of a solution o of 
(69) satisfying (70) implies a dispersion equation among 
the quantities v?, vi, and N, characteristic of the 
simple motion, on one hand, and the wave length L 
and the speed of propagation c of the perturbation on 
the other hand; and (2) that the condition as to 
whether the dispersion equation has imaginary roots 
c depends on the sign of y?v? — N’, that is, the hydro- 
dynamic stability. However, hydrodynamic instability 
is only a sufficient condition for dynamic instability 
of the perturbation, since selective instability of a 
perturbation is possible even in the case of a stable 
“simple motion.” 
Clearly the problem of these perturbations is so 
difficult that drastic assumptions are inevitable if one 
really wishes to undertake its solution. Generally the 
discussion is confined to homogeneous (S = constant), 
incompressible (C = «), or isothermal (C = con- 
stant) media; vertical accelerations are neglected 
(quasi-static hypothesis); the basic flow u is assumed 
to be constant or at most to be linearly dependent on 
altitude z [3], or it is assumed that the current u con- 
sists of two constant, parallel, adjacent flows; some- 
times the meridional component of the perturbation 
velocity is even assumed to be geostrophic [8]. The 
effect of these simplifications is to make the partial 
differential equation (69) degenerate into an ordinary 
differential equation, integrable by means of simple 
functions. 
Four fundamental factors are involved in the general 
equations (62) and (64): (1) gravity g (gravity waves), 
(2) compressibility C~” (sound waves), (3) the earth’s 
rotation (inertial waves) and the variation of the 
Coriolis parameter with latitude (planetary waves [28]) 
and (4) the zonal current w and its vertical and merid- 
ional gradients (shear waves). Examination, employ- 
ing simple media, of each of these factors separately 
(the others being assumed to be zero) demonstrates 
that the atmospheric perturbations (c = 10 m sec _, 
L = 10° m) can be compared to inertial waves whose 
instability results from the gradient of w [2, 31]. 
On the whole, the qualitative results of the perturba- 
tion theory have been obtained under assumptions 
which are never even approximately realized in the 
atmosphere. To be convinced of this it is sufficient to 
refer to recently published vertical cross sections of 
the “jet stream” (22-24, 32]. The perturbations which 
interest us are those of a zonal baroclinic (V + 0) 
current whose intensity w depends not only on altitude 
z but also on latitude y. The problem of atmospheric 
perturbations is thus one in partial differential equations 
and not a problem of an ordinary differential equation. 
The synoptic weather charts show that the propaga- 
tion velocity c and the wave length LZ are such that 
products and squares of the ratios u/C, ¢/C, and 
(u — c)/C, as well as the square of the ratio 
[2r(u — c)/L\/f, are negligible compared to unity 
[3, 4]. One is then justified in making these approxi- 
mations, but they must be introduced nof into the 
initial equations (58) to (61), which must retain all 
their generality, but rather into the final equation (69). 
