Waves, turbulent motions and mixing - Introduction to the Physical and Biological Oceanography of Shelf Seas - page 103

presence of internal waves (Pingree and Mardell, 1985 ) . The alternation of the
surface flow direction may also cause the modulation of the roughness of the sea
surface by changing the steepness of short surface waves. We shall see in Chapter 10
that, under favourable conditions, this effect may allow the detection of internal
waves from satellites using synthetic aperture radar (SAR) or even by the standard
marine radar aboard research vessels.

4.2.3 Energy and the group velocity of internal waves

Like surface waves, internal waves transport potential and kinetic energy as they
propagate. The potential energy density of an internal wave is found by evaluating
the average change in potential energy resulting from the vertical displacement of
water particles within the wave. For our wave travelling on the interface between two
layers, we have the potential density as:

1
2

1
4

g 0 z 2 g 0 A 0

V w ¼ ¼ ð 4 : 23 Þ

which is the same result as we had for surface waves except that g is now replaced by the
reduced gravity g 0 . As for surface waves, the kinetic and potential energy densities of
this type of internal wave are equal (see Problem 4.1) so that the total energy is just:

1
2

g 0 A 0 : E w ¼ V w þ T w ¼ V w ¼

ð : Þ

2 4 24

This energy moves at the group velocity which can be found from Equations (4.14)
and (4.18) as:

kh 1 sinh 2 kh 2 þ kh 2 sinh 2 kh 1

U g ¼ @

o 1

2 þ

k ¼

c ð 4 : 25 Þ

@ sin 2kh 1 þ sinh 2kh 2

where c is again the phase velocity. This rather unwieldy result has two simple
limiting forms. For long waves (kh ≪ 1), it becomes U g ¼

c, so that group and phase
velocities are equal, as we found for long surface waves. For short waves (kh

≫

1)

Equation (4.25) reduces to U g ¼ c/2 which is again the same result as for surface waves

when the depth is large compared with the wavelength.

4.2.4 Continuous stratification and rotation

The above description of internal waves in an ocean with two homogeneous layers
serves us well as a reference model of internal motions but, in the real ocean, the
density structure is not usually so simple. Generally we are dealing with a water
column which is continuously stratified with the degree of stability varying with
depth. In this case, the motion is not restricted to the lowest mode and can take on
more complicated forms of vertical structure in higher modes. The mth mode is