444 
of the pressure field. The motion then set up loses its 
inertial character. In short a theory of cyclogenesis 
cannot be established without consideration of a pres- 
sure perturbation [81, 42]. 
Hydrodynamic Instability and the Theory of Atmos- 
pheric Perturbations 
Tn order to simplify the equations, we will designate 
by x’ the horizontal meridional coordinate y and by 
x the vertical coordinate z; hence the Hulerian variables 
x, y, 2, and t will here be written x, 2’, x’, and ¢. In this 
systan of notation we have: w, = =, oy, = 2w COS e 
= dW/da’ and 2v, = f = 2 sin d = —OW/dx, 
where W = o X R represents the linear velocity due to 
the earth’s rotation. Equation (1) of zonal “simple 
motion” then assumes the form [42] 
SVP = OVIL = uVW — Ve, (56) 
while the equilibrium condition (2) can be written 
vuX VW = VUXVW= VOXVI= VSXVP, (57) 
where u = u(a’, x’) represents the intensity of the 
westerly current, U = U(a’, a”) = W + uw the absolute 
zonal current, and S = Sa x) the specific volume 
of air. 
This being the case, the equations of adiabatic ‘‘small 
motions’? [2] which can be superimposed on simple 
zonal motion (56) can be written [42] 
oe 2 oe pS w (58) 
Ou" Ox 
Oe a Oe Gee OE a GD) 
Ox" (chi Ox" 
ASets Oz av’ 
ye Sirs cata chee 
Soone 
I ease 1 
Ds +5 ip = AER 0, (61) 
where z = 1, 2; p designates the perturbation pressure; 
s, the corresponding perturbation of specific vol- 
ume; C, Laplace’s speed of sound, 
= (c,/¢,)PS = (¢,/¢)RT & 10° m’ sec [2]; 
Cy , the specific heat at constant volume; T, the a absolute 
air temperature; v,, the zonal component; %y = v', the 
meridional component; and v, = v, the vertical conn 
ponent of the perturbation salosiiay v relative to the 
earth. We have also substituted D = 0/dt + wd/dz. 
The operator D represents the local derivative with 
respect to time ¢ in a reference system embedded in 
the zonal current u. By eliminating v, and s among the 
Hulerian equations (58), (59), and the equation of the 
adiabatic process (61), we obtain the equations of 
transverse perturbation motion (v’, v) as a function 
of perturbation pressure p; these are 
S Ge | ei _ gp 22 
Dye =k Gy = = =) == - 
Ge C? dx? 0x’ = 0 On’ 
(62) 
where, (57), 
DYNAMICS OF THE ATMOSPHERE 
aW aU _ SeP a0 
0x’ Ox? = © 0x OX 
_oWaU _ SeP 60 _ 
ax? Oxi dai dni * 
are simply the coefficients (12) of Kleinschmidt’s quad- 
ratic form Q (13). 
We next eliminate v, and s between the equation 
(58) of zonal perturbation motion and the equations 
of eae (60) and the adiabatic process (61), 
ay = 
(63) 
S oP aU av’ 
T @ C2 xi Do Oat Ox 
Dy a oe 
Deep 
+5( ap) 
System (62) and (64) of three equations containing 
partial derivatives of three unknown functions v’, v’, 
and p with respect to the independent variables x, 2’, 
az, and t, must be integrated under the following condi- 
tions at the external boundaries: 
(a) at the surface of the globe: | 
9 
£=2= 
(b) at the free surface: 
P@, x) + p@,2,2,#) = 
Dp +(8P/dx’)v* = 0. 
We observe that setting p = O in the differential 
system (62) reduces it to the differential system (48) 
for inertial oscillations of a zonal geostrophic current 
(w, = 0) and that the characteristic determinant 
D* = (au ar as) D” +f 3199 — (a2) 
of (62) is formally identical with the left side (49) of 
Solberg’s equation of circular frequencies. This cir- 
cumstance demonstrates the influence of the nature of 
the inertial oscillations or, what amounts to the same 
thing, the influence of the nature of the hydrodynamic 
equilibrium wpon the solution of the differential system 
(62) and (64) for transverse perturbation motion. 
After having thus stated in all its generality the 
problem in mathematical analysis posed by the theory 
of perturbations, we make the customary restriction 
to a sinusoidal disturbance which is propagated by 
plane waves with constant velocity c parallel to the 
zonal axis x. In this case the quantities v,, v', s, and 
p are proportional to exp [1/ —1yu(« — cé)], with p = 
2n/L, where L is the wave length. We designate by &, 
£" o, and » the amplitudes of these quantities, which are 
tomotlorns of the transverse coordinates « and x. The 
differential system (62) gives the amplitudes &’ directly 
as functions of 3; we find [42] 
ij | O® 
SOV ah @) OVA a SI Ks), os 
@, 7 = 1, 2) 
where A‘? is the normalized algebraic minor of the 
element A;; = —pn’ (wu — ¢) yz + Gi; of the determinant 
A = ||A,;||, with y; = 1 when 7 = j and = 0 when 
const; 
