THE PERTURBATION EQUATIONS IN METEOROLOGY 
where the primes refer to one fluid layer, the coordinates 
without primes to the other layer and where, as indi- 
cated, the same function applies on both sides of the 
boundary. When the components of the radius vector 
r*, or r*’ in the other fluid layer have been determined, 
a, b, c or a’, b’, c’ have to be expressed by these com- 
ponents and ¢. The results are two equations, 
INE Ol = Oh) i ee (31) 
where the functions will now be different for the 
different fluid layers. The kinematic boundary con- 
ditions require that both equations (31) represent 
the same surface at all times ¢, a requirement which VY. 
Bjerknes writes symbolically 
reso) = 0] = Fa", = O. 
The functions r* and r*’ of a, b, c, ¢ and a’, b’, c’, t, 
respectively, will contain certain arbitrary integration 
constants which have to be chosen so that (82) holds. 
At a free surface or a rigid lower boundary only one 
of the two equations in (30) and (81) exists, and this 
equation has then to be adjusted in such a manner 
that any prescribed condition concerning the motion 
at this boundary is satisfied. For instance at a rigid 
boundary, r* must be such that displacements are 
parallel to it. 
In addition the dynamic boundary condition has to 
be satisfied, namely, that the pressure at an internal 
boundary is continuous. The pressure in both layers 
is given by p*(a, 6, c, t) and p*’(a’, b’, c’, t). Then, 
after r* and r*’ have been determined as functions of 
a, b, c, t and a’, b’, c’, t, respectively, p* can be ex- 
pressed as a function of ¢ and r*, or rather of the com- 
ponents of the radius vector, and after obtaining the 
analogous expression for p*’ one has to satisfy the 
condition that 
pi, t) = p* (r,t) for 
(32) 
eels (83) 
At a free surface the right-hand side of this equation 
has to be put equal to zero or to the given external 
pressure. 
THE PERTURBATION EQUATIONS 
The two systems of hydrodynamic equations derived 
in the preceding sections can be linearized by the 
assumptions set forth on page 402. This linearization 
will now be performed, first on the Eulerian system, 
then on the Lagrangian system. From the perturbation 
equations in vector form it is more or less easy to ob- 
tain the equations in the coordinate form which is 
most suitable for any particular problem. As an example 
which is important for the study of large-scale motions 
on the earth, the perturbation equations for spherical 
polar coordinates in the Eulerian system are derived 
rather explicitly in a later section. 
The Eulerian System of Perturbation Equations. 
The undisturbed motion will be denoted by capital 
letters as stated earlier. In order to write the complete 
system of equations for the undisturbed motion it is 
therefore merely necessary to replace in the relevant 
407 
preceding equations the quantities with asterisks by 
the corresponding capital letters. Thus the equation 
of motion is obtained from (18), the equation of con- 
tinuity from (19), the physical equation from (20) 
(with the previously stated restriction that only piezo- 
tropic fluids are considered), the boundary equation 
from (25), the kinematic boundary condition from (26) 
(from (27) if an internal boundary exists), the dynamic 
boundary condition from (28), and the mixed boundary 
condition from (29) which may be used instead of 
the kinematic boundary conditions. Thus the equa- 
tions of the undisturbed state are 
ON te aay EOD) Se -(4) vP — ve, (34) 
at Q 
at + div (QV) = 0, (35) 
aQ aP 
aq Wa (F+v vP) =0, (36) 
mG. )) = 0, (37) 
ar ei ciay oaas 
apa ape VF =0, (8) 
P(r, ) — P’G,d) = 0, (39) 
5 (P - P) + V-vP - PY =, 
(40) 
ae = 2) 2. WP = B) = 0, 
It is understood that in the boundary conditions the 
coordinates have to satisfy (37). If any particular mo- 
tion whose perturbations are to be investigated is se- 
lected, one has to make sure first that this undisturbed 
motion satisfies equations (34)—(40). 
In order to derive the perturbation equations one 
has now to substitute in the equations previously men- 
tioned, namely, (18)—(20) and (25)—(29), for the quan- 
tities with asterisks the sums (2)—(4). Furthermore 
f* has to be replaced by F + f, where F refers to the 
boundary in the undisturbed position while f denotes 
the variation of the boundary due to disturbance. The 
resulting equations can be simplified, as explained on 
page 402, because 
1. The undisturbed quantities must satisfy (34)—(40), 
and 
2. Terms of second or higher order in the perturba- 
tion quantities can be neglected. 
Consider, for instance, equation (18) which may 
now be written 
OoV ov 
Fy oP Bp an WENNER OA A A Mia v:-VWw+22x<V 
1 
$2axv=-(q45) 1? +9) — ve 
The first term on the right-hand side may be expanded 
as follows, 
