Waves, turbulent motions and mixing - Introduction to the Physical and Biological Oceanography of Shelf Seas - page 95

Figure 4.3 Particle motions at the
surface (z l/5 in a

surface wave with amplitude A 0 ¼

¼ 0) and at z ¼

0.25

metres, period T p ¼ 6 s, wavelength

l ¼ 56.2 metres. The forward drift
results from the particle's (Lagrangian)
velocity differing slightly from the
(Eulerian) velocity at the centre point
of the orbit. The Stokes drift velocity
at the surface based on Equation (4.10)
is u St ¼ 0 : 73 cm s 1 which may be
compared with the particle's
instantaneous orbital speed of
∼ 26 cm s 1 .

u st

z =0

z =- λ /5

which is a net flow in the direction of propagation. It has a maximum of
u St max ¼ k 2 A 0 c at the surface and falls off rapidly with depth, being

< 5% of its

surface value at z l/4. The depth-integrated, time-averaged transport by the

Stokes drift is just U St ¼

¼

kA 0 c pA 0 = = 2 ¼ T p , so for waves of period T p ¼ 6 s and

1 metre, there would be a wave-induced flow of 0.52 m 2 s 1 with a
not-insignificant mean surface current of 0.12 m s 1 .

amplitude A 0 ¼

4.1.4 Energy propagation

Waves have both kinetic energy due to their orbital motions and potential energy due
to vertical displacement of particles. For deep water waves, it is readily shown using u
and w from Equation (4.9) that the average kinetic energy per unit area is given by:

ð 0
1 ð

gA 0
4

1
2

¼

u 2 w 2

T w ¼

þ Þ ð : Þ

dz 4 11

while for the average potential energy we have

ð l

gA 0
4

1
l

1
2

¼

2 dx

V w ¼ g : ð 4 : 12 Þ

0

Hence, in this case the energy of the waves is equally divided between the two
forms and the total energy is just gA 0 =

2. This energy is transmitted
by the pressure forces acting within the waves. The rate of working by the pressure

E w ¼ T w þ V w ¼