406 
and an analogous relation holds for &. Hence, the 
operation Vr*- applied to (16) results in the Lagrangian 
equation of motion 
ort or* 1 
Bs 22 x (— )| = —( =) vp* — ve. 
vr [= AF x e )| (4) v V® 
The equation of continuity may be written in the 
following form: 
or* or* or* * 
* 2 = 
: gl Ge te 
where go represents the density of the fluid particle 
at the time ¢ when a, b, and c were the coordinates of 
the particle. It is easily verified that in a rectangular 
Cartesian coordinate system (23) is identical with 
* D(x*, die a) = % 
D(a, b, ¢) Me 
(23) 
(24) 
where the Jacobian is used to express the volume change 
of the fluid particle. 
To complete the Lagrangian system of equations 
the physical equation has to be added. While for the 
Eulerian method this equation had to be differentiated 
in order to obtain quantities referring to fixed points 
no such differentiation is required in the present case 
since both the physical equation and the equations of 
motion refer to individual fluid particles. Of course, 
the parameters A, B, --- in (14) may differ from parti- 
cle to particle so that they may have to be considered 
as functions of a, b, and c, but they will not change 
with time. 
The Boundary Conditions. At rigid or internal bound- 
aries and at free surfaces the fluid has to obey certain 
conditions which exert considerable influence on the 
possible motions. We shall again first consider the 
form of these boundary conditions in the Eulerian 
system. The boundary of the fluid is given by an equa- 
tion of the form, 
fC, ) = 0. (25) 
The kinematic boundary condition expresses the fact 
that a fluid particle which forms part of the boundary 
must remain on this surface and that a particle which 
is not at a given time part of the boundary can never 
become part of it. This condition is expressed by the 
following relation, 
ae + v*-vf* = 0. 
(26) 
As is known from hydrodynamics this relation can 
also be interpreted as the condition that a fluid particle 
at the boundary has a velocity component normal to 
the boundary which must equal the velocity of the 
surface itself. 
If the boundary is a rigid surface, as for instance 
the surface of the earth, f* is independent of the time 
(neglecting such unpleasant geophysical phenomena as 
earthquakes) and (26) becomes 
v*-Vf* = 0, (26a) 
DYNAMICS OF THE ATMOSPHERE 
which states that the fluid velocity at the boundary 
must be parallel to the boundary. 
In the case of an internal boundary between two 
different fluid layers a condition similar to (26) must 
hold on the other side of the boundary, in the second 
fluid layer, where the velocity is v*’, 
af* 
— EG 7 = 
a + v*’-Vf 0. 
(27) 
In addition to the kinematic boundary conditions 
the dynamic boundary condition has to be satisfied, 
that is, the pressure across the boundary must be 
continuous, 
p*(r, t) — pa, t) = 0, 
where the prime refers again to the second fluid. At a 
free surface 
(28) 
p*(r, ¢) = const. (28a) 
This constant may be zero, for instance if the fluid 
is bounded by empty space. In some cases the constant 
may have to be replaced by a function of space and 
time, for instance when the effect of variations in 
atmospheric pressure on the ocean surface is to be 
studied. Equation (28) or, in the case of a free surface, 
(28a) represents the boundary given by (25). In many 
problems the function appearing in (25) will, in fact, 
have to be determined by (28) or (28a). Therefore, 
the pressure difference at the boundary, or in the case 
of a free surface the pressure there, may be introduced 
into equations (26) and (27). Thus, the following bound- 
ary conditions, which represent a mixture of kinematic 
and dynamic conditions [19], are obtained: 
ae — p*’) + v* -V(p* — p*’) = 0, 
5 (29) 
at (p* — p*’) + v*’-V(p* — p*’) = 0. 
It is only at a rigid boundary that the dynamic condi- 
tion need not be satisfied, and the geometric form of the 
boundary must here be known from other sources. 
Then equations (26) and (27) have to be used. 
In the Lagrangian system the form of the boundary 
conditions corresponding to those just given for the 
Eulerian system is considerably more complicated be- 
cause attention is now focused no longer on points in 
space but on individual particles on both sides at the 
boundary which may have been neighboring at the 
time ¢ = 0, but will in general be at a finite distance 
from each other at a later time. A complete discussion 
of these conditions is given by V. Bjerknes and col- 
laborators [4, pp. 63-64, 103-104]. We shall here discuss 
only some simpler forms, also given by V. Bjerknes 
[3], which are adequate for many problems of fluid 
motion. 
At the time ¢ = 0 fluid particles which are at the 
boundary must satisfy one of the following two equa- 
tions, 
fe(a,b,c)=0, — f(a’, b’,c’) = 0, (30) 
