416 
The equation of continuity is satisfied for this type 
of motion since the density Q is constant. In the third 
equation above, the term with Q, has been omitted 
since it is small compared to g. In the following per- 
turbation equations ©, and Q, will be set equal to zero 
as is customary when motion on a rotating plane is 
considered. Further, only the variation of ©, in the 
y-direction will be taken into account, a restriction 
which is justified if the undisturbed motion is pre- 
dominantly zonal. Then, according to (54) and (55), 
the perturbation equations become 
2 
Os. oY + 20. Sh + AE = 0, (114) 
dy 4(20,) 
= © | 90, ot + 20. oH $y 
(115) 
1 op _ 
Ooben 
ey st = (116) 
In the last equation the variation of the height of the 
undisturbed fluid in the y-direction has not been taken 
into account. This height variation provides the meridi- 
onal pressure gradient required by the undisturbed 
zonal current. It is equal to the small inclination of the 
isobaric surfaces and therefore presumably of little 
effect on the perturbation motion. 
It may be assumed that 
_ 9(20,) 
ab 
The foregoing system of equations has coefficients which 
depend on b. It can therefore be satisfied if the unknown 
variables x, y, and p are assumed to be unknown funce- 
tions of 6 and have the periodicity factor 
exp {zala — (o — U)t]}. 
Instead of adopting this procedure we may directly 
eliminate all unknown variables but one by differentia- 
tions, a method which could also have been used in 
the preceding examples. The continuity equation (116) 
can be satisfied by a function y(a, b, ¢) analogous to 
the stream function such that 
_ oy Lee hoy 
~ ab’ aa 
By cross differentiation of (114) and (115) the vorticity 
equation is obtained, 
= const. 
ay 
00) +4 a2) = =0, (17) 
where 
Fede wear 
Te a 
Equation (117) is satisfied by the following expression 
v = A exp {zala — (o — U)t] + 26b}, (118) 
DYNAMICS OF THE ATMOSPHERE 
where A is an arbitrary constant, provided that 
Uy] 
g=U= oh (119) 
The last relation is the trough formula which relates 
the speed of long waves in a zonal current to their wave 
length and width and to the parameters of the system. 
The formula has been discussed and applied widely in 
meteorological work. 
Long Waves on a Rotating Globe. As an example 
of the application of the perturbation equations to 
motion on a sphere the same problem as in the previous 
section will be considered for a spherical fluid layer 
[11]. It will be assumed that the angular velocity of 
the undisturbed current, x, is constant and that it is 
in the direction of the geographic longitude i, so that 
the linear zonal velocity 
Vi = cE sin 3, (120) 
where H is the earth’s radius, and # is the colatitude. 
Since vertical motion is neglected, we may substitute 
in the equations for polar coordinates on page 411 
the earth’s radius # instead of r because the vertical 
dimensions of the fluid layer are small compared to EH. 
The following relations are thus obtained for the un- 
disturbed motion. From (72) 
1/ oP 
(x + 2) cos 3 Vy = alae 
From (73) it follows that the undisturbed pressure 
field is independent of \. The two terms on the left- 
hand side of (74) are so much smaller than the accelera- 
tion of gravity that hydrostatic equilibrium may be 
assumed, as in the preceding example. The equation 
of continuity is evidently satisfied by the assumed 
undisturbed motion. 
As in the plane case the perturbation motion may 
be purely horizontal. Then the two equations for the 
horizontal motion are, according to (77) and (78), if 
(120) is noted, 
(121) 
= oe he fd egal OP 
+ «— — 2k + Q) cosdy = 0 (el (122) 
. + = was 2(« + Q) cos & vy 
Ct (123) 
3 ala dp ) 
~ @Q\Esind oan] ° 
Since static equilibrium is assumed, the third equation 
of motion need not be considered. 
Tn analogy to (116) the small effects of the meridional 
variation of the height of the free surface may be 
disregarded so that the equation of continuity becomes, 
according to (80) and (76), 
A(sind vy) , On] _ 
|| ag + =| = 0) (124) 
| 1 
E sin 3 
Equation (124) may be satisfied by a stream function 
x so that 
Ox Ox. 
= eee 125 
E sind dn’ (es) 
