402 
dynamics the linearization is achieved directly as a 
result of the far-reaching simplifying assumptions. For 
instance, the two-dimensional irrotational motion of a 
homogeneous and incompressible fluid can be dealt 
with by the methods of the theory of complex functions. 
In other cases it is necessary to adopt different pro- 
cedures. In tidal theory, for instance, it is customary 
to assume that the motion is sufficiently small so that 
terms which consist of products of the velocities and 
their derivatives can be neglected in comparison with 
terms which are of the first order with regard to the 
derivatives of the velocity components. In consequence 
of this assumption the convective terms in the equa- 
tions of motion can be neglected, and the system of 
hydrodynamic equations becomes linear, at least as 
long as an incompressible fluid is considered. The char- 
acteristic feature of this approach is that the tidal 
motion is considered as a small perturbation of an 
original undisturbed state. This method has been used 
frequently in hydrodynamics, but V. Bjerknes was 
the first to systematize this procedure and to apply 
it to atmospheric problems when the earth’s rotation 
and the compressibility and the inhomogeneity of the 
fluid system have to be taken into account. The system 
of hydrodynamic equations obtained by him is referred 
to as the atmospheric perturbation equations. The 
present article deals with their derivation and shows 
how they can be applied to various geophysical prob- 
lems. It is not the purpose of this article to deal with 
the mathematical methods of solving the resulting sys- 
tems of differential equations; that is a problem in 
mathematical physics and is treated in the appropriate 
textbooks. Neither is it the aim of this article to discuss 
some special meteorological problems. It is solely the 
method of approach which is being stressed here; for 
the solution of specific problems the reader is referred 
to the literature. Only some very simple examples are 
given as illustrations in later sections. 
Basic Assumptions of the Atmospheric Perturbation 
Theory. As stated in the preceding section the pertur- 
bation equations are derived under the assumption 
that the state of the fluid motion can be regarded as 
made up of two parts, namely an undisturbed motion 
on which is superimposed a perturbation, the sum of the 
two representing the total motion. Such a division of 
an actual state of motion is often directly suggested 
by observations. The capillary ripples on the free sur- 
face of a pond can be regarded as a perturbation pro- 
duced by a slight gust on a fluid otherwise at rest. On 
a much larger scale the nascent cyclones have been 
regarded in the wave theory of cyclones as disturbances 
in an undisturbed flow which consists of two currents 
of different densities, flowing side by side. It was, in 
fact, the wave theory of cyclones which led V. Bjerknes 
[2, 3] to the formulation of the perturbation equations 
as the tools by means of which the stability of such a 
system could be investigated. 
Since the total motion thus consists of an undisturbed 
motion on which a perturbation is superimposed, it 
follows that not only the total motion, but also the 
undisturbed motion alone must satisfy the system of 
DYNAMICS OF THE ATMOSPHERE 
hydrodynamic equations, that is, the three equations 
of motion, the equation of continuity, physical equa- 
tions, and such initial and boundary conditions as are 
appropriate for the specific problem under considera- 
tion. That the undisturbed motion by itself must satisfy 
the hydrodynamic equations is evident from the fact 
that it can exist without a superimposed perturbation. 
It is possible to select for any specific problem a suffi- 
ciently simple fluid state as the undisturbed motion 
so that little or no difficulty is encountered in satisfy- 
ing the hydrodynamic equations. For mstance, in the 
two cases mentioned above, the undisturbed state would 
appropriately be one of rest or geostrophic wind motion,. 
respectively. 
For the subsequent discussion, the variables express- 
ing the actual disturbed plus undisturbed state will be 
denoted by letters with an asterisk, the undisturbed 
state will be denoted by capital letters, and the effect 
of the perturbation by the corresponding lower-case 
letters. Thus we have the following relations: 
for the position vector r* = R +1, (1) 
for the velocity v®¥ =V-+yV, (2) 
for the pressure p* = P+ p, (3) 
g=Q+4q. (4) 
It is now assumed that the deviation from the un- 
disturbed state produced by the perturbation is so 
small that those terms in the equations which contain 
products of the perturbation quantities and their deriv- 
atives can be neglected as being small of higher order 
compared to those terms which are of the first order 
in the perturbation quantities. It is immediately ap- 
parent that with this assumption the hydrodynamic 
equations become linear with regard to the variables 
expressing the effects of the perturbation. 
This basic assumption about the smallness of the 
perturbation does not, of course, represent an equally 
satisfactory approximation to the actual state of affairs 
in every case. It is particularly appropriate in cases 
where the stability of a certain state is to be investi- 
gated. This state will then be regarded as the undis- 
turbed state and the stability investigation becomes 
the problem of determining whether or not a super- 
imposed small perturbation will grow with time. In 
such a case the initial perturbation of the undisturbed 
motion may be regarded as infinitely small since a 
deviation from this motion must start gradually. If the 
perturbation increases with time the motion is unstable, 
otherwise it is stable. The limitations of this statement 
will be discussed later. The wave theory of cyclones and 
many other types of atmospheric and oceanic motions 
pose such stability problems. In other mstances where 
the perturbation equations are applied it will depend on 
the particular circumstances how well the resulting 
solution represents reality, and it is impossible to make 
general statements about the success or failure of the 
method. However, it can at least be said that in many 
cases the linearized perturbation equations give a satis- 
factory approximation to the observed fluid motions. 
for the density 
