THE PERTURBATION EQUATIONS IN METEOROLOGY 
tion. According to Taylor’s theorem, 
@(R + r) = ®(R) + r-V,@ + higher-order terms. 
Here, the operator Vr denotes space differentiation with 
respect to the undisturbed position vector R. Similarly, 
for the vector of the earth’s rotation, 
Q(R + r) = Q(R) + (:Ve)Q. 
The boundary condition (51) given above for the un- 
disturbed state remains the same for the disturbed 
state since the condition refers to the time f when the 
particles were in their undisturbed position. The fune- 
tions F(R, t) and F’(R’, ¢) in (52) are modified in the 
perturbed motion because the particles at the boundary 
whose position vectors in the undisturbed motion were 
R and R’ are now R + r and R’ + 1’, respectively. 
According to Taylor’s theorem, 
F(R-+r,t) = F(R,t) + 1r-VeF + higher-order terms, 
and an analogous relation holds for F’(R’ + 1’, ¢). 
Thus, because of (52) the kinematic boundary condi- 
tion for the disturbed motion becomes as written below 
in (58). In deriving the dynamic boundary condition 
(59), use is made of the facts that the same particles 
are considered in the undisturbed and the disturbed 
case and that in the perturbation quantities the dis- 
turbed position may be replaced by the undisturbed 
position. Thus the system of perturbation equations 
in Lagrangian form becomes 
me fe t20 (3) + [298] (Gr) 
dR Oy ee er 
4. we + 2Q xX fel Seria (54) 
q(vP) 
aap — V(r-V,S), 
q (OR OR\ (oR 
ae % oy i ee) 
or oR oR 
melee ae ce) 
(55) 
OR ér\ (dR 
laa Pea Aer) 
dR OR\ (or\ _ 
i (a) ‘ i) Ga Tar 
qd = YP, (56) 
iG, o @) =O; Fo(a’, b’, ec’) = 0, (57) 
r-VeF = r’-VeF’, R= Re (58) 
p(R, t) = p’(R’, t), R = R’. (59) 
As in the case of the Eulerian system the perturbation 
equations (54) to (59) can be obtained from the equa- 
tions (48) to (53) by forming the variation of these 
equations. 
A comparison of the three basic equations, namely, 
of motion, of continuity, and physical changes in the 
409 
Eulerian system and in the Lagrangian system, shows 
that the equation of motion is of the first order in the 
Eulerian form (41) while it is of the second order in 
the Lagrangian form (54). The equation of continuity 
in the Hulerian form (42) as well as in the Lagrangian 
form (55) is of the first order. But the physical equation 
is a first-order differential equation in the Hulerian 
system (43) while in the Lagrangian system it has 
the simple form (56) so that the perturbation density 
q can, with the aid of these equations, be eliminated 
directly from the equations of motion and continuity. 
In the Eulerian system, on the other hand, additional 
differentiations will in general be necessary in order 
to eliminate the perturbation density, so that for a 
compressible fluid the Eulerian equations are not 
simpler than the Lagrangian equations. The Eulerian 
system is simpler than the Lagrangian system only 
for an incompressible and homogeneous fluid, the case 
mostly considered in classical hydrodynamics. 
From a physical viewpoint it may be remarked that 
meteorological observations are made at a given local- 
ity and not following individual air particles so that 
the Hulerian rather than the Lagrangian method is used 
in this case. However, when trajectories or the motion 
of air masses are studied the Lagrangian method offers 
amore direct approach than the Hulerian method. It is, 
of course, always possible to make the transition from 
the results in the one system to those in the other. 
When the Lagrangian system has been used the posi- 
tions of the particles and their pressures and densities 
are given. By a differentiation the particle velocities 
can be obtained, and the distribution in space of the 
velocity, pressure, and density can then be found be- 
cause the position of the particles in space 1s known. 
When the Eulerian system is used the particle trajec- 
tories can be found from the known velocity distribu- 
tion. 
In some instances a certain simplification may be 
achieved in the work leading to the mathematical solu- 
tion depending on whether the problem is formulated 
in the Hulerian or Lagrangian system. But no general 
statements about the relative difficulty of the one or 
the other approach can be made, and it has to be seen 
by direct comparison whether the Lagrangian or the 
Eulerian method is more advantageous. 
Perturbation Equations for Special Coordinate Sys- 
tems. From the general equations given in the two 
preceding sections the special forms suitable for a spe- 
cific problem can be derived. It will depend on the 
particular fluid model to be considered, in particular 
on the boundary conditions, what simplifications can 
be made in the system of perturbation equations, and 
which type of coordinates is most suitable. As long as 
the earth can be considered as flat, a rectangular 
Cartesian coordinate system is the best choice, unless 
the motion has a circular symmetry. In the latter 
case, cylindrical polar coordinates are most appropriate. 
For large-scale disturbances on the earth its curvature 
has to be taken into account, and in this case spherical 
polar coordinates represent the most suitable frame 
of reference. 
