Waves, turbulent motions and mixing - Introduction to the Physical and Biological Oceanography of Shelf Seas - page 101

z / h

0

-0.2

-0.4

-0.6

-0.8

-1.0

kh =5 kh =1 kh = 0.5

(a) (b) (c)

Figure 4.7 Particle orbits for sinusoidal progressive internal waves in a two-layer system
derived from the velocity potential (Equation 4.20) . (a) kh ¼ 5; l ¼ 1.25h; short waves with
circular orbits decreasing exponentially in both layers from a maximum at the interface.
(b) kh ¼ 1; l ¼ 6.28h; intermediate case with elliptical orbits becoming weaker and increasingly
flat towards the boundaries. (c) kh ¼ 0.5; l ¼ 12.5h; long waves with elliptical orbits;
amplitude of horizontal movement is practically uniform in each of the two layers.

which describes elliptical motion with the particles revolving clockwise about their
mean position. The ratio of the amplitudes of vertical to horizontal motion is therefore:
j w j

j ¼

tanh k ð h þ z Þ ð 4 : 21 Þ

j u

which is the same as for the case of a surface wave in an un-stratified ocean.
If the depth is large relative to the wavelength (kh

1), the ratio of w and u
components tends to unity and the motion becomes circular in the interior of the layer
and dies out towards the boundary, as shown in Fig. 4.7a . Notice here that the bottom
layer in an internal wave is behaving in a similar manner to a surface wave. In the
upper layer, the particles revolve in the opposite, anticlockwise, direction and again
the motion diminishes in amplitude as we move from the interface towards the upper
boundary. At the interface, where the sense of the orbits changes abruptly, there is
node (a zero) in the horizontal velocity with a large velocity difference (shear) across
the interface. This shear, which increases with the wave amplitude, can result in
instability of the wave and the generation of turbulence, as we shall see in Section 4.4 .
For longer waves (kh

≫

1) the particle orbits, shown in Fig. 4.7b , c , become
flattened as they do for surface waves. The vertical velocity is zero at both top and
bottom boundaries and increases linearly with distance from the boundaries to a
maximum on the interface, while the horizontal velocity in each of the two layers
becomes depth uniform with a 180 phase shift between the layers.
The particle orbits shown here are for small (strictly infinitesimal) amplitude waves
for which the orbits are closed. For finite amplitude internal waves, just as for surface