410 
Some simple typical examples of problems to be 
treated by the perturbation method will be considered 
later. Here we shall discuss the transition from the 
vector form of the equations to the coordinate form. 
This transition presents little or no difficulty in the 
case of straight-line coordinates, such as the Cartesian 
rectangular coordinate system, but it is somewhat more 
awkward if curvilinear coordinates, such as spherical 
or cylindrical polar coordinates, are to be used. The 
reason for this difficulty is that in the case of curvi- 
linear coordinates the change in direction along the 
coordinate curves has to be taken into consideration. 
The problem of writing the perturbation equations 
in arbitrary curvilinear coordinates has been solved 
by V. Bjerknes [8, 4], with the use of Lagrange’s equa- 
tions of the second kind, but the resulting equations 
will not be reproduced here since the vector equations 
given previously permit also the derivation of the equa- 
tions in any particular coordinate system. In order to 
show by an example how the equations for the un- 
disturbed and the disturbed motion can be derived 
for a curvilinear coordinate system from the vector 
equations given above, spherical polar coordinates will 
be chosen, and the equations will be written in the 
Eulerian system. 
Depending on the amount of vector and tensor anal- 
ysis assumed to be known this derivation naturally 
becomes shorter or longer. Here only the more ele- 
mentary theorems of vector analysis will be used. 
Consider a right-hand rectangular Cartesian coordinate 
system whose z-axis, for convenience, is parallel to 
the earth’s axis. Let 3, A, and r be the colatitude, 
longitude, and distance, respectively, from the earth’s 
center. Further, let i, j, and k be the unit vectors in 
the a, y, and z directions, respectively. Then the radius 
vector 
r= ix + jy + ke 
(60) 
= irsindcos\ + jrsindsinA + krcosd. 
Further, since the radius vector r is also a function of 
r, 3, and r 
ar = (55) 20 + (5 a+ (5, ") ar (61) 
Here dr/00 is evidently a vector tangential to the curve 
of intersection between the coordinate surfaces 
\ = const and r = const, 
or as it may be called briefly a ‘direction vector” 
in the direction @. Similarly dr/d\ and or/dr are direc- 
tion vectors in the directions \ and r, respectively. 
These three direction vectors play in the polar co- 
1. It is also possible to use elliptic coordinates if the flat- 
tening of the earth is to be taken into account [20]. For most 
meteorological considerations, however, this factor can be ne- 
glected, and the sum of gravity and centrifugal force due to 
the earth’s rotation can be assumed in the direction to the 
center of the earth. 
DYNAMICS OF THE ATMOSPHERE 
ordinate system a role similar to that of i, j, k in the 
Cartesian system except that they are not unit vec- 
tors, a shortcoming which is immaterial here. By ap- 
propriate differentiations of (60) the following relations 
are obtained: 
or 
ag = i” cos 8 cos + jr cos d sinh — kr sin 3, 
or Aahte t Nae 
A —ir sin 3 sin \ + jr sin ? cos d, (62) 
OT eh ova alt 4 
57 Ls cos XT j/sin sim) ay, cos\y- 
From these three equations it is found that 
irsin a = (% cos 0 +H r-sin d) sin 8 cos 
od 
or . 
= = Ss 
jr sin 3 = Ss cos & + sr sin ») sin 3 sin X (63) 
+ = cosa, 
kr = — 3 sind + Or cos 9. 
In the following we shall for the moment consider 
only the undisturbed motion. The extension to the 
disturbed motion will be recognized easily. From (61) 
it follows by individual differentiation with respect to 
time that 
1B (Do+(2) Be w 
Here the dots indicate individual differentiation with 
respect to time. The following relations exist between 
the linear and angular velocity components, 
rO=Vs, rsmdA=V,, R=V,. (65) 
For the present the notation in (64) will be retained. 
With the foregoing relations the Coriolis term 2 X V 
can be evaluated. Since @ is in the direction of 2, 
Q = Ok. Further, V may be expressed by its compo- 
nents in the directions of x, y, and z with the aid of 
(64) and (63) and the vector product of 2 and V may be 
evaluated. In the resulting expression the unit vec- 
tors i, j, and k can be replaced again by the direction 
vectors 0R/dd, OR/dd, and dR/dr so that 
22 X V = 
—29 {sin Oo) i cos } + Ss =) r sin 9| A (66) 
ue E cot 3 + “| [= h 
