THE PERTURBATION EQUATIONS IN METEOROLOGY 
of viscosity and friction are disregarded, and an “‘ideal,”’ 
nonviscous fluid is considered. In particular, the at- 
mospheric perturbation equations have so far almost 
exclusively been developed for and applied to such 
ideal fluids. Very often, far-reaching agreement has 
been obtained between theoretical models dealing with 
ideal fluids and observed atmospheric motions, so that 
there is considerable justification for making use of 
the substantial simplifications which arise when the 
effects of viscosity on fluid motion are neglected. 
In the case of the atmosphere it is customary to 
describe the motions relative to the rotating earth 
since the observing meteorologist participates in the 
earth’s rotation. According to the laws of Newtonian 
mechanics we may then write that 
dr* dr* 1 > 
qe t 22 Xx 5 (4) vo = VW (UG) 
Here r* is the radius vector of a fluid particle, @ the 
vector of the earth’s rotation, g* the density, p* the 
fluid pressure. It is assumed that the external forces 
acting on the fluid have a potential 6, because in the 
case of the atmosphere this external force is, as a rule, 
only gravity—composed of the gravitational attraction 
of the earth and the centrifugal acceleration due to 
the earth’s rotation—which has a potential. In some 
cases, of course, this gravity potential may have to be 
replaced or augmented by additional forces, for instance, 
in the study of tides. 
In order to apply equation (16) to problems of fluid 
dynamics one can either focus one’s attention on each 
individual particle and describe its future position, 
pressure, density, etc., or describe the distribution of 
velocity, pressure, density, etc., throughout the space 
occupied by the fluid, as functions of the time. The 
first method leads to the so-called Lagrangian equa- 
tions, the latter to the Eulerian equations, although 
both systems of equations were originally derived by 
Euler. The Eulerian system is used much more widely 
in hydrodynamics and will be described first. We intro- 
duce in (16) the velocity 
The operator d/dt represents the time differentiation of 
an individual particle. In order to have only quantities 
referring to a fixed point in the coordinate system in the 
equation, this operator may be written 
d 
_?@ *, 
Fines + v*-V. (17) 
Thus the Eulerian equation of motion becomes 
ov* * * * 1 * 
7a a Ne +20 Xvt=— ay) Ne — Ve. (18) 
In addition the fluid must satisfy the equation of con- 
tinuity, 
dq* ; ae 
apt div (q*v*) = 0, (19) 
according to which the density change anywhere in 
405 
the fluid must equal the mass convergence there. In 
order to complete the system the physical equation 
has to be added as explained in the preceding section. 
We shall assume that the fluid is piezotropic so that 
an equation of the form (14) holds. Most investiga- 
tions so far have been dealing with piezotropic fluids, 
although in some cases it has been assumed that a 
certain amount of heat energy, given as a function of 
space and time, is added to the system (for instance, 
[14]) or that heat is conducted from the earth’s sur- 
face into the atmosphere (for instance, [12]). In the 
latter case the effects of eddy viscosity are taken into 
account in order to have a consistent fluid model. 
Equation (14) does not fit into the Eulerian system 
because it refers to an individual particle, but upon 
individual differentiation with respect to time and with 
(15) and (17) it follows that 
(20) 
It should be noted that in general y will vary in space. 
Equations (18), (19), and (20) represent a complete 
system of equations which together with the boundary 
conditions (to be discussed in the next section) and 
initial conditions, determine the motion of the fluid. 
In order to apply the Lagrangian method to the 
study of the fluid motion it is necessary to identify 
each particle individually. This can be done by the 
use of three suitable parameters, a, b, and c, so that 
the state of the fluid and each of its individual parti- 
cles is described by expressing r*, p*, and g* as func- 
tions of ¢, a, b, and c. The choice of these parameters 
is free provided that they permit one to characterize 
and identify each particle uniquely. It has, for instance, 
been suggested that under certain conditions potential 
temperature, specific humidity, and potential vorticity 
would be suitable parameters [21]. As a rule the co- 
ordinates at a given time ¢ are chosen as the three 
identification parameters, and this practice will be used 
here. 
In order to write the Lagrangian equations in vector 
form we shall make use of the operation VB-A. The 
meaning of this expression is that after the scalar 
product B-A has been formed the vector operator 
V operates only on the first vector. Thus 
OB, 7 Bs OB: 
a. (=) ae (2) aes (2). 
It may be noted that 
Vr-A = A. (21) 
In the case of the Lagrangian equations the operator 
V has the components 0/da, 0/db, and 0/dc. When 
the operator with the components 0/dx, 0/dy, and 
d/dz is used in the remainder of this section it will be 
denoted by V; . It should be noted that instead of the 
total derivatives with respect to time in (16) we may 
now write the partial derivatives since a, b, and ¢ are 
independent of time. Furthermore, according to (21), 
Vr*-Vi.p = Vp, 
