446 
To this end we must calculate expressions for A” 
and K, , substitute them into equations (69) and (70), 
carry out the derivations involving y and z, and finally 
estimate the order of magnitude of the terms in these 
equations, taking into account the above-mentioned 
approximations. By neglecting terms of order 10” * 
compared with those of order 10”, we would obtain 
the approximate differential system of atmospheric per- 
turbations, which would then have to be integrated. 
Unfortunately, it is impossible at present to make 
this choice of terms, for we have not drawn up tables 
giving the orders of magnitude of the meteorological 
elements and their first and second derivatives with 
respect to space and time as functions of altitude and 
latitude. It seems urgent to us to prepare such tables 
with the help of the sufficiently extensive aerological 
data which is already at our disposal. 
Meanwhile, using Charney’s theory of meteorologi- 
cal approximations [4], we can deduce from general 
equation (69) the approximate partial differential equa- 
tion of the perturbation pressure associated with quasi- 
static, quasi-geostrophic long waves. There is obtained 
[43] 
ve OO elie) |? Ose. ry ae 2 
f2 ay? | dz Soa ia the Ge «ie 
2 2 42 2 
_ Svs pig Ue 1 ps0 U Ou 
{ ie (1 uu -) ee uw “EF Oy’ als oz? a) 
PR 1 7 =| LOS Wt aU) 
(f+ Te eS Pa State te fo 
where 7’ designates the absolute air temperature and 
where we = (df/dy)/u" is the critical speed introduced 
by Bjerknes and Holmboe [1], that is, the intensity of 
zonal flow which corresponds to stationary, planetary 
waves of wave length L = 27/u (Rossby [28]). Equa- 
tions (66) and (68) reduce to 
f=n7=V- io, 
(72) 
2 
pe ie as Va 
Vs 0z\u— ce 
_ _ Sam _ 8 de 
Hquation (71) is identical with that of Charney [4] 
when 7’ depends linearly on z (that is, 0(T'v2)/dz = 0, 
see eq. (86)) and w is a function of z only. Dividing 
the first term of (71) by v;, and letting »? approach 
infinity, one obtains Kuo’s equation [19]. 
Equation (71) is to be integrated with suitable 
boundary conditions. The boundary conditions other 
than those referring to the earth’s surface (¢ = 0 
for z = 0) must be carefully chosen, for on them depend 
the form of the dispersion equation and the expression 
jor the criterion of selective instability. The boundary 
conditions being linear in the unknown function and its 
DYNAMICS OF THE ATMOSPHERE 
first derivatives, the mtegration of equation (71) is a 
problem of the classic “mixed” type. This integration 
can be achieved only with computing machines. 
It should be observed that in the case considered 
here the ‘‘stmple motion” is a weakly baroclinic zonal 
current u(N & 10") in which the isentropic surfaces 
have a slope of the order of 10 *. Under these condi- 
tions equation (71) is always of elliptic type; for this 
reason any hydrodynamic instability is excluded and 
only the selective instability is possible. The constant 
c is then complex (¢c = ¢, + »/—1¢;), as is the function 
® (thatis,® = wo + 4/—19”), so that the perturbation 
pressure assumes the form 
p = ela’ (y, 2) cos u(x — ¢,t) 
(74) 
— w(y, z) sin n(@ — c,t)). 
Here the real functions a’ and g’’ satisfy a system of 
partial differential equations of elliptic type which is 
easy to deduce from (71). In this case the form of the 
perturbation pressure p demonstrates the phase lag, 
both in altitude and latitude, which the troughs and 
ridges of unstable perturbations undergo. The merid- 
ional tilt of the axes of troughs and ridges is an es- 
sential characteristic of atmospheric perturbations. 
Their tilt relative to the meridians assures the trans- 
port of zonal momentum and kinetic energy along 
them. 
The Stability of Permanent, Horizontal, Isobaric 
Motion 
When the air is in permanent, horizontal, isobaric’ 
motion, the atmosphere is said to be in a state of 
hydrodynamic equilibrium. The dynamic method which 
we employed (pp. 434-437) for the case of rectilmear 
isobars can be applied to the case of curvilmear isobars, 
provided we use a system of orthogonal curvilimear 
coordinates c', «, o, fixed relative to the earth and 
having one (oc) of the three families of coordinate 
lines coincident with the lines of intersection between 
the isobaric surfaces and the equipotential surfaces of 
the gravity field. Let Oxyz be a right-handed, rec- 
tangular Cartesian coordinate system of origin O, in 
which the axes are tangent to the coordinate limes 
passing through O; axis Ox points in the direction of 
the isobaric motion. We designate by u = u(c, o) the 
velocity of air along the isobars obtained by varying 
o (gradient wind), by R, the radius of geodesic curva- 
ture, and by R, the radius of normal curvature at O of 
isobar o’ considered as a line in the surface o = o4 
passing through O. By then adopting without change 
the reasoning presented on pages 434-437, we find 
that the equilibrium between gravity, the pressure 
gradient, the centrifugal force, and the Coriolis force 
at point O, for a transverse displacement (7, , rz) from 
this point, is stable or unstable depending on whether 
[41] 
Q(t, , Te) = Gay ty Tp dye Ty Te a::72 > or < 0, (75) 
