THE PERTURBATION EQUATIONS IN METEOROLOGY 
write (101) in the form 
€C dD 2 
where Z stands for the operator acting on D on the 
left-hand side of (101) when T is replaced by 7% . If 
we put 
2 
€ € 
D=D+(s)r+(5) eto, 
comparison of the coefficients of e/7’) shows that 
U@s) = 0, (104) 
(Dy) = io ws «’Dr), afte. (105) 
dc 
The differential equation (104) for the zero approxima- 
tion can readily be solved since it is homogeneous and 
has constant coefficients. It is, of course, the equation 
obtained from (101) if for the variable coefficient T 
in this equation its constant value 7) at the interface 
is blithely substituted. The correction terms Dy, etc., 
can also be computed by elementary means since they 
involve the solution of a second-order linear and in- 
homogeneous equation with constant coefficients. It 
can be shown that the series for D converges, and for 
practical purposes when relatively short waves such 
as billow clouds are studied the zero approximation is 
sufficiently accurate. According to (104), 
Dy = ¢*"(Kie™ + Kee “’), (106) 
where AK, and K, are integration constants and 
ry g/R —« 
Me Tayo (107) 
2 22 bs 2 
NW = 4) 2  a@a/\ +  — &)Rg/o 
i +a RT, . (108) 
In order to distinguish the upper and lower layer, 
primes will be added to the appropriate letters to 
indicate that these quantities refer to the upper layer. 
After Dj has been determined for the lower layer, 
Co is found from (102). Since the vertical displace- 
ment must vanish at the lower rigid boundary where 
c = —h, it follows that a relation must exist between 
the two constants K, and K,. With the introduction 
of a suitable new constant 
—uej2 sinh N(c + h) 
Co = K Paar eA 
0 é Qa2(a!a? — 9°) (109) 
and 
Do = Ket? [ee cosh Nie + h) 
a? (241 + g) (% — g) (110) 
(c'n/2— g) sinh N(e + “| 
o (4+ g)(% — g) ; 
where x; = a (N — p/2) and 7% = o(N + p/2). 
At the top of the upper layer (h’. = To/e’) the per- 
turbation pressure must vanish (Do(h’) = 0), so that 
415 
with the introduction of a new integration constant K’, 
D) = K'e**” sinh N'(c — h’), (111) 
and consequently, 
ch = Ror =e — g) sinh Nic Te h’) 
Q'(ota' — @?) aly 
o N’ cosh N'(c — | 
TEP=a) | 
At the interface (¢ = 0) the perturbation pressures 
and the vertical displacements must be continuous. 
Thus 
Do(0) = Do(O) and Cy(0) = Co(0). 
These conditions represent two linear and homogeneous 
equations for K and K’, and a nontrivial solution 
exists only if the determinant of the foregoing system 
vanishes. This condition leads to the equation of fre- 
quency 
Qn a N coth Nh + ou/2 — g 
[o?(N — n/2) + gllo?(N + u/2) — gl 
as Qs 
ou’/2 — g + oN’ coth N’h”’ 
which gives the relation between wave velocity, wave 
length, and the parameters of the fluid system. Since 
ois contained in the abbreviations N and N’, the equa- 
tion is transcendental. An approximation which is satis- 
factory for small-scale oscillations can be obtaimed by 
putting the hyperbolic cotangents equal to one which 
is strictly correct only for infinitely deep fluid layers. 
Then (113) changes into an algebraic equation which 
can be evaluated [9]. After an approximation has been 
obtained it can be used to substitute more accurate 
values for the hyperbolic cotangents and a new value 
for o can be computed if necessary. Furthermore, the 
correction terms D,, etc., can be computed, although it 
is found that the zero approximation gives a sufficiently 
good quantitative result, for instance in the theory of 
billow clouds. 
Long Waves on a Rotating Plane. As an example 
of the application of perturbation equations when the 
earth’s rotation is taken into account, the so-called 
trough formula given by Rossby [18] may be derived. 
The earth will be regarded as a flat, horizontal plane 
and the atmosphere as incompressible and homogene- 
ous. The motion, disturbed as well as undisturbed, will 
be assumed as horizontal. If the atmosphere has a 
horizontal velocity U in the x-direction, 
(113) 
X =a-+ Ut, VY = fo, Boz & 
It follows from (48) that for the undisturbed motion, 
IvoP, 
0 = — Q da’ 
1 oP 
2 = es 
20, U 0 ab” 
1 oP 
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