Waves, turbulent motions and mixing - Introduction to the Physical and Biological Oceanography of Shelf Seas - page 104

characterised by having m nodes in the vertical structure of the horizontal velocity.
Hence the simple wave we have described above is referred to as a mode 1 internal
wave, with the node at the interface between the layers. Each mode has its own
propagation speed which generally decreases with increasing mode number.
The form of all the modes and their speed of propagation can be found from
the observed density structure which is used to calculate the buoyancy frequency
(Equation 3.37) . Normal mode analysis solves an eigenvalue problem for the modes
using well-established methods (Phillips, 1966 ) which are now readily applied using
freely available M atlab codes (Klink, 1999 ). In many cases in the ocean, it seems
that the lower modes are dominant with much of the energy contained in the lowest
mode. It is, therefore, usually unnecessary to consider more than the first few modes
in analysing observations of internal waves.
The possible frequencies of internal waves are limited in two ways. No solution is
possible for an internal mode with a frequency o

N max where N max is the largest
value of the buoyancy frequency in the water column density profile (see 3.4.2). At
the same time, there is a low frequency limit which is set by the Earth's rotation,
whose influence we have not yet considered. When Coriolis forces are included in the
equations of motion, it can be shown (Kundu, 1990 ) that wave type solutions are
only possible for frequencies greater than the inertial frequency f. Internal waves on
a rotating Earth are, therefore, limited to the frequency range N max >

>

f with a
corresponding wavelength set by the phase velocity of the mode involved. Rotation
also affects energy distribution in internal waves. When we include the Coriolis forces
in the dynamical equations (Gill, 1982 ), the ratio of potential and kinetic becomes

o >

o 2 f 2 V w

T w ¼

f 2 :

ð : Þ

4 26

o 2 þ

For o ≫ f the ratio is unity as we found for non-rotating waves, but as o ! f it

diminishes and the potential energy becomes zero at o f. In this limit vertical

motion is suppressed and all energy is in the horizontal motion which has the form of
inertial oscillations.

¼

4.2.5 The importance of internal waves

Generally the energy content of internal waves is small compared with that of surface
waves and the energy is propagated around the shelf seas much less rapidly. Internal
wave energy fluxes are, therefore, very much less than those involved in surface waves
and it is reasonable to ask whether these motions need to be considered as an important
component of the shelf sea system. The answer, as we shall see in Chapters 7 and 10 ,is
that they play a significant role in the interior of the stratified shelf seas, a region
which is remote from both the surface and bottom boundaries where almost all of the
mechanical energy is input to the water column. Consequently, very little energy
reaches from the boundaries to the stable, relatively tranquil interior and the flux of