THE PERTURBATION EQUATIONS IN METEOROLOGY 
The operator 
I OR WRe lo (2E\ (2 
Nye @ (2) ) a r sin? 3 (=) is) 
This relation can for instance be verified directly from 
the form of V in Cartesian rectangular coordinates with 
the aid of (63). Further, 
vvn0(2)+a(4)+a(2). 
In order to obtain the expression V-VV which appears 
in the equations of motion the direction vectors have 
also to be differentiated with respect to 3, \, and r. 
This differentiation can be carried out with the aid 
of (62), and the resulting expressions can again be 
written as direction vectors according to (63). It will 
be found that 
V-VV = e | w: 16 +26" — i? sin cos | 
(67) 
(68) 
i (®) lw. VA +204 cote + 22 | (69) 
+ (2) lw. V)R — Or — rd’ sin “| 
The expression for the divergence in polar coordinates 
is sufficiently well known so that it need only be quoted 
here: 
div V = (70) 
eae ae ie (cot 0) + 2% 
With these Re eae the equations 
(84) to (40) can now be written in polar coordinates. 
The three component equations of (34) can be obtained 
if, after the necessary substitutions are made from 
(64) and (66)—(69), the factors of each of the direction 
vectors are equated separately. In writing down these 
equations the linear velocities have been substituted 
according to (65). Then 
me Cs) to) = 
V-V Vs (25) + Vo (oa) + (2), (71) 
and the equations of motion assume the following form: 
He Hs ao Le ae _y s *) 
pes (72) 
eed 7. = —' Ta 
20 cos 3 Vy Q Se 
oA tv. Wht 4 vay (2 
ee (73) 
+ 20(Vy cos 3 + V,sin 3) = Fan) 
2 2 
_— — Wav elepinete snr 
r (74) 
411 
The equation of continuity is 
OU + VQ + Q div V = (75) 
where 
a(sind Vs) avn 1 a(r'V,) 
d = 
NEN r sin 3 0d: rsin 3 0X 7 TT OP (76) 
and the physical equation as well as the boundary 
conditions have the same form as that given for the 
Eulerian system of perturbation equations on page 407 
except that the operator V-V is now of the form given 
by (71). 
As explained before, the perturbation equations can 
be written down directly by forming the variation 
6 of the equations of the undisturbed motion and re- 
placing the quantities 6V» , 6Vx, 6V,, 6P, and 6Q by 
the corresponding perturbation quantities. This pro- 
cedure which is more rapid than the one used in the 
two preceding sections leads to the following equa- 
tions: 
ae sae 9, Vrvs 
iP 
+ V-Vos + v-VV5 + + 
_ 2Vx» cot 
PS LRT 
i (28) + q = 
Q \rad Q? \rav)’ 
VA 
+ VeVi + ¥-WN + 2D — 
— 20 cos 2 vp (77) 
+ 
Vs Vx cot & 4 Vor Vy Cot 
r r 
+ 20(v5 cos 3 + v, sin &) 
aa (eis ta ites) 
Q \r sin 3 ON Q? \rsind an J’ 
2 
= + V-Vo, & ¥-VV- - ee 5 a2 
ea — _ I! op q (oP 
20 sin 0 vy, = ale Ale), 
of 4 V-vq + v-VQ + Qdivv + qdiv V = 
+ 
(78) 
(79) 
(80) 
The expressions v:V and div v are analogous to (71) 
and (76), respectively. The physical equation and the 
boundary conditions have again the same form as the 
analogous equations in the Eulerian system of perturba- 
tion equations. 
EXAMPLES OF THE USE OF THE 
PERTURBATION METHOD 
The present article deals with a method of handling 
theoretical problems rather than with a theory and 
interpretation of atmospheric phenomena. Therefore, 
the discussion of the following examples stresses this 
method rather than the application of the results to 
