THE PERTURBATION EQUATIONS IN METEOROLOGY 
By cross differentiation of equations (122) and (124) 
it follows that 
BO Oni / oe ve OX 1 dx 
(B42) (smo %) + 1 ox! 
Ox. 
2 ~=0. 
4 (QQ +) 5 0 
If it is assumed that the perturbations are waves of 
the frequency 6 and the wave number m, 
x = Cf(d) exp [7(8t + md)], (127) 
where f(#) must satisfy the equation, 
ea. df 
sin 3 dd (sin e = 
- (128) 
¢ m 
+| 20496 +m — 2 ]i=0. 
It is known that this equation has solutions which 
are regular at the poles only if 
2(2 + x)(B + nm) = n(n + 1), 
where n is an integer. The last relation represents the 
frequency equation, and f(#) is then an associated 
Legendre function, P”(cos 3). From (129) the period 
of the oscillation can be obtained as a function of 
the earth’s rotation, zonal wind velocity (expressed 
by the angular velocity «), wave number m, and 
the integer n. This last number determines the merid- 
ional extent of the oscillation since 
a2, O83) a 
~d(cos 8)" = P*.(cos 0). 
(129) 
f@) = C sin” 3 
Thus f(#) vanishes at the poles and at n — m circles 
of latitude which are symmetrical to the equator. The 
equator itself is a nodal parallel if m — mis odd. 
Quasi-static and Quasi-geostrophic Approximations. 
In a discussion of the simplifications made to obtain 
tractable models of atmospheric motions it is appropri- 
ate to consider the hypothesis of ‘‘quasi-static’? mo- 
tion. This hypothesis is based on the assumption that, 
in each vertical, equilibrium exists not only before 
the start of the motion, but also during motion, al- 
though the equilibrium may change with time and 
from one vertical to another. This assumption is, of 
course, the basis of all the evaluations of upper-air 
soundings and oceanographic soundings which are per- 
formed with the aid of integrals of the hydrostatic 
equation. The quasi-static hypothesis implies that the 
vertical accelerations of the motion can be neglected 
compared to the acceleration of gravity. Since the 
latter is in general much larger than the former, the 
quasi-static hypothesis appears quite plausible, but it 
is impossible to give an a priori justification for it as 
Solberg [20], especially, has emphasized. Its most satis- 
factory justification is to be found in the fact that it 
gives results which in many instances are in agree- 
ment with the observations. 
Historically, the quasi-static assumption was first 
417 
introduced in the theory of tides. It is also successfully 
used more generally in the theory of ‘‘long”’ waves, that 
is of waves whose length is large compared to the depth 
of the fluid layer. The notation ‘‘quasi-static,’’ which 
characterizes the special dynamic nature of the fluid 
motion more clearly than the expressions ‘“‘tidal’’ or 
‘dong’? waves, was introduced by V. Bjerknes [1] when 
he generalized the quasi-static treatment from incom- 
pressible and homogeneous fluid layers to autobaro- 
tropic layers. 
In order to illustrate the method, consider a fluid 
layer which is at rest in the undisturbed case. It will 
also be supposed that the coordinate system is non- 
rotating and that all motions take place in a vertical 
a, c plane. The undisturbed pressure and density de- 
pend then only on the c coordinate and satisfy the 
hydrostatic equation 
oP 
=—-—_. 130 
gQ a (130) 
The horizontal equation of motion may be written, 
according to (54), 
a @ 
ee) = 0. 
Q 
Because the vertical acceleration of motion may be 
neglected and because of (56) and of (130), the equa- 
tion for the vertical component becomes 
) p\ _ Gt = Wyo 
5 + 5) = an 
where I’ and y are the coefficients of barotropy and 
piezotropy, respectively. In the last equation the un- 
disturbed pressure P may be introduced for the height 
coordinate c with the aid of (130). Then 
d \ C= app 
i (w+ 3)- = 
The equation of continuity 
(131) 
(132) 
(133) 
qd Cin, Ce 
peel peti se pele eG) 
Q v 0a a dc 
may be written with the aid of the physical equation 
and of the hydrostatic equation (130), 
1 0x Oz Yp 
Se eae ee ee 134 
Qaa Pt @ is? 
By combination of (133) and (134) it follows that 
GhP oy) 
Se ah ey 135 
AG) tele is) 
This equation relates the perturbation pressure’ to the 
horizontal divergence. It expresses the pressure as an 
effect of mass transport. If (131) is differentiated with 
respect to P and equation (134) is used, one obtains 
the relation: 
a = a (aon ae a 
136 
dt? \aP Q 0a G36) 
