192
Equator
6 Internal Waves
90· Latitude
Poles
Fig. 6.6: Allowable frequencies and periods for internal gravity waves
Monin (1978), and many articles in professional literature. Some cases of the
frequency profile are more tractable, for example, the case when No = constant.
This case corresponds to the exponential profile of density (Fig. 6.7):
p(z) = poexp (-NJz/g) ,
(6.19)
as from Eq. (6.10) we obtain:
{ ( N2) }1/2
N(z) = -~ --:- p
= No = Const.
(6.20)
As was shown by Krauss (1972) and Phillips (1977), the assumption that N =
constant (or when the density profile is given by Eq. 6.19) is an acceptable
local approximation to the thermocline.
The fact that the vertical velocity, w, of internal waves at the sea surface and
the sea bottom should vanish, suggests the following expression for evolution
of velocity, w, in space and time (Pond and Pickard, 1983):
w{x, z, t) = a sin kz{z + h) sin{kxx - wt),
(6.21)
in which h is the water depth, and kx and kz are the wave numbers of oscillations
in the horizontal, x, and vertical, z, directions. To find kz, we again use the
fact that at z = 0, the velocity component w = O. Thus,
(6.22)