Waves, turbulent motions and mixing - Introduction to the Physical and Biological Oceanography of Shelf Seas - page 94

z / h

0

-0.2

-0.4

-0.6

-0.8

-1.0

kh =5 kh =1 kh = 0.2

(a) (b) (c)

Figure 4.2 Particle orbits for surface waves. (a) kh

1.25h; short waves with circular
orbits decreasing exponentially with depth. (b) kh ¼ 1; l ¼ 6.28h; intermediate case with
elliptical orbits decreasing with depth. (c) kh ¼ 0.2; l ¼ 31.4h; long waves with elliptical orbits.

¼ 5; l ¼

higher order approximations for waves of finite height, like the Stokes wave
(see Chapter 5 of Kinsman, 1965 ), the wave profile becomes asymmetric with sharper
crests and flatter troughs, as you may have observed is the case for real steep waves
approaching a beach. Steep waves also travel a little faster than those of small
steepness. For example, a wave with steepness of ka ¼ 2pa/l ¼ 0.1 has a phase speed

∼

in deep water which is 1% greater than that given by Equation (4.7) . Waves with

steepness ka 0.14 become unstable and break. Even for waves close to this maximum

steepness, the correction to the phase speed does not exceed 10%. Generally, for waves
of small and moderate steepness, the first order velocity potential theory does not
differ greatly from the finite amplitude theory. It therefore serves as a reasonable
model for real surface waves and is widely used as such. It is also usually much easier to
apply than the higher order theories, which are mathematically complicated.
One important finite amplitude effect in waves is the residual transport which they
induce. For strictly infinitesimal waves, the orbits are closed and there is no net drift
of particles due to the wave motion. As the wave amplitude increases, the speed of a
particle changes slightly as it moves around its orbit; for example the speed will be
greater at the highest point in the orbit than at the lowest point because of the
exponential variation in velocity with depth. These changes combine with similar
effects due to the horizontal motion to give an 'unclosed' orbit and a net forward
motion illustrated in Fig 4.3 . We can deduce the magnitude of the forward drift for
waves in deep water using the Eulerian velocities specified in Equation (4.9) and
allowing for a finite orbital amplitude. The resulting residual current, called the
Stokes drift, is given by

k 2 A 0 ce 2kz

u St ¼ ð 4 : 10 Þ