THE PERTURBATION EQUATIONS IN METEOROLOGY 
By B. HAURWITZ 
New York University 
INTRODUCTION 
Before entering into a discussion of the systems of 
hydrodynamic equations suitable for the investigation 
of atmospheric dynamics, it is appropriate to make 
some general remarks on the typical difficulties of 
investigations in theoretical meteorology and on the 
general principles on which the formulation of the 
perturbation equations is based. Such a discussion nat- 
urally mcludes an enumeration of the types of prob- 
lems where the application of perturbation methods 
is particularly suitable. 
The Problem of Dynamic Meteorology. Meteorology 
concerns itself with the exploration and the study of 
the gaseous envelope of the earth. Dynamic meteorology 
in particular aims at a quantitative explanation of the 
observed variations and motions of the atmosphere 
on the basis of physical “laws.” If a complete quanti- 
tative explanation has been accomplished, a quantita- 
tive weather forecast will be a by-product. Conversely, 
quantitative forecasts of the future state of the at- 
mosphere from a given set of initial conditions may 
be regarded as the ultimate goal of dynamic meteorology 
because such forecasts require a thorough understand- 
ing of the laws governing the atmosphere. 
The peculiar difficulties in meteorology, as in the 
other earth sciences, in comparison to physics, are 
that very many different factors act simultaneously on 
each phenomenon and that it is impossible to perform 
controlled experiments. In order to study the atmos- 
phere the meteorologist has to consider a fluid shell 
surrounding a sphere. This fluid is neither homogeneous 
nor incompressible, a circumstance which complicates 
the purely hydrodynamical problem. Moreover, one 
of the constituents which make up the fluid may change 
from the gaseous to the liquid or solid state at tem- 
peratures and pressures regularly occurring in the at- 
mosphere. This versatility of water vapor would be 
bad enough by itself, but the complications increase 
further because the fluid shell receives radiant energy 
in amounts which vary with time and location on the 
sphere and with the state of the water in the atmos- 
phere. Also, the properties of the lower boundary of 
the fluid shell are by no means uniform, the greatest 
differences being those between water and land which 
react differently to the incoming solar radiation and 
which have a different frictional effect on the motions 
of the gaseous layer. Further, the properties of the 
earth’s surface are by no means independent of the 
atmosphere, but depend rather strongly on its state; 
the ocean surface, for instance, may be made rougher 
by wind-generated waves, and the radiative properties 
of the land surface may be modified drastically by a 
snow cover. Nor is the land surface itself of uniform 
elevation, but rather it presents formidable obstacles 
to the air motion. It would be easy to continue this 
account of the complications which the atmosphere 
presents to the meteorologists, and to dwell on such 
matters as the fact that one of the gases (oxygen) 
appears in one layer in the triatomic state, in another 
in the monatomic state, with importantly different 
absorptive properties, or to discuss the electromag- 
netic effects in the high atmosphere. However, it may 
suffice here to state that finally this complex fluid 
system, bounded by an inhomogeneous surface and 
subjected to unequal influx of energy, is surrounding a 
rotating globe. 
At that it must be admitted that the problems of 
astrophysics are presumably more complicated. Such 
phenomena as sunspots or chromospheric eruptions 
are subject to additional influences, for instance, to a 
strong electromagnetic field, or to radiation pressure, 
influences which the meteorologist can mercifully neg- 
lect, at least in the lower atmosphere. However, the 
solar physicist has the advantage of a really detached 
viewpoint. He is not concerned with the detailed be- 
havior of the individual sunspot or the individual 
chromospheric eruption and their detailed structures. 
The meteorologist, on the other hand, attempts to 
explain the detailed structure of a cyclone or the mo- 
tion of an individual hurricane. It is likely that meteor- 
ology would appear to be in a much more satisfactory 
state if we could apply our knowledge to observations 
of the earth’s atmosphere as viewed from Mars. 
As stated before, for the solution of its rather for- 
midable problem, dynamic meteorology uses the laws 
of physics which in the case of the lower atmosphere 
means mainly the theorems of thermodynamics and 
hydrodynamics. Because of the complexity of the prob- 
lems it is always necessary to make restrictive simplify- 
ing assumptions, in other words to consider models 
which include only some of the features of the real 
atmosphere. Even with these assumptions the neces- 
sary mathematical analysis offers as a rule considerable 
difficulties. One of the most frequently recurring diffi- 
culties arises out of the fact that the hydrodynamic 
equations which describe the motion of any fluid are 
nonlinear. Very little is known about the solution of 
nonlinear differential equations. Progress in this direc- 
tion may presumably be expected eventually from work 
with high-speed computers because the study of in- 
dividual nonlinear problems will point the direction 
in which a theory of nonlinear differential equations 
should be developed. In the absence of such a general 
theory it has long been the practice in hydrodynamics 
to linearize the basically nonlinear equations of fluid 
motion. In some types of problems of classical hydro- 
401 
