THE PERTURBATION EQUATIONS IN METEOROLOGY 
to follow the future life history of an unstable per- 
turbation. In order to do so one has to find solutions 
of the complete hydrodynamic equations in their non- 
linear forms, a mathematical problem which is much 
more complicated than the solution of linear problems 
because no general mathematical procedures exist for 
its solution. It is conceivably possible that solutions 
of the nonlinear problem of the later development of 
unstable perturbations can be obtained by successive 
approximations, starting out with the perturbation as 
deseribed by the linear perturbation equations and 
obtaining the next approximation as a perturbation 
of the first approximation. However, it is doubtful if 
such a method would be simpler than a more direct 
approach which starts immediately with a considera- 
tion of the nonlinearized hydrodynamic equations. No 
matter which method of approach will be found feasi- 
ble, a quantitative study of developing perturbations 
beyond their nascent linear stages appears indispen- 
sable for the future development of the perturbation 
theory. 
Such a study may also be expected to shed additional 
light on another question connected with the perturba- 
tion theory, namely, the problem of how an unstable 
flow pattern can develop. If it is found that a given 
state of fluid motion is unstable, the implication is 
that any small disturbance will increase indefinitely in 
amplitude and lead to a breakdown of the original, 
undisturbed state. Since in a fluid system such as the 
atmosphere or the ocean such small disturbances are 
always present, it is difficult to see offhand how an 
unstable state could develop and exist for any appre- 
ciable length of time unless during a certain state of its 
development the factors which built up this state are 
more effective than those which contribute to its even- 
tual breakdown due to its inherent instability. Perhaps 
the effect of friction is responsible for delaying the 
breakdown. In the stability investigations it is always 
assumed that the undisturbed state whose instability 
one wishes to investigate is given a priori. Supplemen- 
tary investigations of the mechanism leading to its 
development are evidently in order. Again, as in the 
case of the nonlinear perturbations, it is doubtful 
whether the linear perturbation equations are the ap- 
propriate mathematical tool for such a study. 
Even though the foregoing remarks indicate that 
for a future development of instability studies the 
investigation of nonlinear problems, either by approxi- 
mation methods or by direct means, will be an im- 
portant step forward, a continuation of studies along 
the limes conducted so far will deserve an important 
place in theoretical meteorology. For a first investiga- 
tion whether a given state of motion is unstable, the 
linearized perturbation equations are evidently the 
appropriate system of equations. Since the model 
atmospheres whose stability conditions have been in- 
vestigated so far are all only more or less close ap- 
proximations to reality, it is evident that much work 
remains to be done in the study of progressively more 
realistic models. In view of the rapidly mounting diffi- 
culties of such analyses, as baroclinic wind fields, com- 
419 
pressibility, and horizontally and vertically variable 
temperature distributions are taken into: account, it is 
not likely that all fruitful problems ‘which can be 
treated by the theory of linear De vanp aii will find 
an early solution. 
It is necessary to make simplifying assumptions such 
as those discussed in the last section, namely, that 
the motion is quasi-static or quasi-gedstrophic. These 
assumptions are physically plausible and permit the 
omission of certain terms in the equations. In view 
of the complexity of the equations such procedures are 
indispensable. In these simplifications, terms are neg- 
lected which are of smaller orders of magnitude than 
others. This may give rise to errors in the subsequent 
analysis because in the reduction of the system of 
differential equations to one equation it is necessary to 
carry out differentiations in order to eliminate all un- 
known variables but one. It is conceivable that while 
of two quantities one may be considerably larger than 
the other their derivatives may be of the same order 
of magnitude. Consequently, the omission of terms in 
the original system of equations may lead to erroneous 
equations when the necessary eliminations are carried 
out. This possibility has been discussed by Queney 
[16] who pointed to this as a possible explanation of 
some contradictory results which have been obtained 
by different authors. It is not proposed to discuss this 
question here but at any rate this controversy shows 
as stated before that a great number of problems re- 
main to be settled with the aid of the perturbation 
equations. 
The problems to be studied by means of the perturba- 
tion method should include questions not only of the 
stability and instability of given fluid states, but also 
of the fluid motions caused by these perturbations. 
The solution of the linearized equations of fluid motion 
has led to many satisfactory descriptions of fluid mo- 
tions as, for instance, the theory of tidal and surface 
waves, of sound waves, and of atmospheric waves 
both where a smaller scale is involved, for instance, ~ 
in the theory of mountain waves and of billow clouds, 
and where motions of a much larger scale are con- 
sidered. 
It may finally be repeated that very little work has 
been done so far to extend the perturbation theory 
to viscous fluids. Even though friction plays presum- 
ably only a minor role in the free atmosphere its effect 
iS quite important near the ground and at very high 
levels, in the ionosphere, where the kinematic viscosity 
must reach large values. Along the same lines, pre- 
sumably more attention will have to be given to the 
problem of heat conduction in perturbation motions. 
Such problems as the land and sea breeze, or the 
monsoon circulations, require that the conduction from 
the earth’s surface into the atmosphere and the eddy 
conduction within the atmosphere be taken into ac- 
count. Among other nonadiabatie processes which have 
received little attention in connection with the per- 
turbation theory is radiation. Studies of perturbation 
motions when radiative transfer of heat occurs can 
presumably shed additional light on atmospheric circu- 
