THE PERTURBATION EQUATIONS IN METEOROLOGY 
The results of hydrodynamic wave theory give numer- 
ous examples to support this statement. 
It may be remarked here in passing that there are 
certain problems in fluid motion which reduce to linear 
equations without the introduction of the assumptions 
of the perturbation theory. Consider, for instance, the 
frictionless horizontal motion of an incompressible and 
homogeneous fluid. Then the continuity equation can 
be satisfied by assuming a stream function (a, y, ¢), 
and the pressure can be eliminated from the two equa- 
tions of motion by cross differentiation resulting in the 
so-called vorticity equation which is in this case a 
nonlinear differential equation for y. However, if 
Vw = const y, 
the two quadratic terms cancel each other (more gen- 
erally if Vy = f(W), but this relation is no longer 
generally linear). The foregoing equation for y com- 
prises a great number of equations arising in mathe- 
matical physics. In the meteorological literature, ex- 
amples for this type of motion will be found, among 
others, in papers by Craig [7], Neamtan [15], and 
Rombakis [17]. 
SYSTEMS OF HYDRODYNAMIC EQUATIONS 
Only a very short review of the equations of motion 
and of continuity in the Eulerian and the Lagrangian 
systems will be given here since these equations can be 
found in all treatises on hydrodynamics. But since 
in classical hydrodynamics the fluid is as a rule regarded 
as incompressible and homogeneous, some more de- 
tailed remarks are in order concerning the modifica- 
tions which are necessary when such inhomogeneous 
and compressible fluids as the atmosphere are con- 
sidered. 
Barotropic and Baroclinic Fluids. In fluids which 
are inhomogeneous and compressible the surfaces of 
equal density and of equal pressure do not as a rule 
coincide; the stratification of the fluid is baroclinic. 
In special cases, however, the density distribution may 
be given completely by the pressure distribution; then 
the stratification of the fluid is called barotropic. A 
barotropic stratification exists, for instance, in an ideal 
gas whose temperature is the same everywhere or in 
a fluid where pressure and density are functions only 
of the elevation. An incompressible and homogeneous 
fluid is evidently barotropic and will always remain 
barotropic. A fluid whose stratification always remains 
barotropic is called autobarotropic. In general, a baro- 
tropic state will be destroyed by the motion of the 
fluid; the fluid will become baroclinic. 
The significance of the distinction between barotropy 
and baroclinity for the variation of the circulation of a 
fluid need not be discussed here since it is explained in 
the texts on dynamic meteorology. 
Tt is often convenient to introduce a ‘‘coefficient of 
barotropy”’ when the stratification is barotropic, 
q* = q*(p*). (5) 
The coefficient of barotropy T is then defined as fol- 
lows, 
403 
—e (6) 
In an atmosphere obeying the ideal-gas law with a 
constant vertical temperature gradient ¢, for instance, 
rGsF he 8 
where F is the gas constant for this atmosphere, g the 
acceleration of gvavity, assumed to be constant, and 
T the temperature. 
It is possible to generalize this terminology of baro- 
clinity and barotropy to apply to the distribution of 
other variables besides pressure and specific volume 
[4, p. 3]. 
The Physical Equation. Since in meteorological and 
oceanographic investigations the fluid medium can 
rarely be considered as homogeneous and incompress- 
ible, the density has to be regarded as one of the un- 
known variables. In this case the four equations used 
mainly in classical hydrodynamics, namely, the three 
equations of motion and the equation of continuity, 
are not sufficient to describe the fluid motion because 
the density has been added as a fifth variable to the 
three velocity components and the pressure. Itis there- 
fore necessary to have an additional equation describ- 
ing the behavior of the fluid. Such an equation, in the 
case of the atmosphere, can be derived from thermo- 
dynamic considerations. In the first place for the varia- 
bles of state, pressure p*, density g*, and temperature 
T*, an equation of state holds of the form 
E(p*, Gis; IP, A, B, C, ? -) = 0, (8) 
where A, B,C, --- indicate additional parameters char- 
acterizing the state of the fluid. In the case of the 
atmosphere the parameters A, B, etc., refer to the 
differences in the composition of the atmosphere. For 
many problems it is sufficient to use the equation of 
state for ideal gases: 
p* — Rq*T* = 0, (9) 
where F is the gas constant for air, which depends on 
the composition of the atmosphere. In the lower atmos- 
phere, F& is mainly affected by the variable amount of 
water vapor; at higher levels it depends on the separa- 
tion of the various atmospheric constituents and dis- 
sociation of some of the molecules. Consequently R& 
may vary from parcel to parcel in the atmosphere. 
While equation (8) or (9) adds an additional equation 
to the system of hydrodynamic equations it is not 
sufficient to complete the system since it involves also 
an additional variable, namely, the temperature 7™. 
In order to obtain a more complete system of equa- 
tions the first law of thermodynamics may be added: 
dh = de + p*d(1/q*), (10) 
where dh denotes the amount of heat added to the unit 
mass during an infinitely small change of state, de 
the change of internal energy per unit mass, and where 
p*d(1/q*) represents the amount of work done because 
of the expansion of the unit mass. The amount of heat 
