Waves, turbulent motions and mixing - Introduction to the Physical and Biological Oceanography of Shelf Seas - page 96

force on a plane normal to the direction of wave in a depth element dz is just pudz per
unit width, so that the average total rate of working is given by the integral:

ð 0 ð 0

¼

gA 0
2

c
2 ¼ E w U g

goA 0 sin 2 e 2kz dz

P w ¼ pudz ¼ ð kx ot Þ ð 4 : 13 Þ

1 1

where u and p are taken from Equations (4.6) and (4.9) and the overbar denotes the
time average. Energy is, therefore, transmitted at a rate which is the product of the
energy density

c/2, termed the group velocity (because it is also
the speed at which wave groups travel). The result that energy flux is the product of
energy density and group velocity has been obtained here for a sinusoidal deep water
wave. It is applicable more generally, to a wide range of wave motions, with the
group velocity given by:

E w and a velocity U g ¼

U g ¼ @

o

k :

ð 4 : 14 Þ

@

For the surface waves being considered, we have from Equation (4.7) that:

gk so that U g ¼ @

o g

2o ¼

gT p
4p ¼

c
2

o 2 ¼

k ¼

ð 4 : 15 Þ

@

which is in accord with the result of (4.13).

4.1.5 Wave breaking and near-surface processes

Energy input to the wave field by the wind is propagated around the shelf seas in the way
described in the last section. Some of it ends up being dissipated at the coast in wave
breaking on beaches and through bottom friction in shallow water. A large fraction of
the energy, however, is dissipated in wave breaking in deep water out in the open sea in a
process termed 'whitecapping'. In, and close to, areas where waves are being actively
generated by the wind, the waves are sufficiently steep so as to become unstable and
break. Wave breaking may also be induced by the interaction between the local gener-
ated wind waves and long swell waves arriving from distant storms in the deep ocean.
As illustrated in the aerial photograph of the sea surface in Fig. 4.4 , the whitecaps
which result from wave breaking are highly turbulent patches. Within these patches,
the energy of the breaking wave is transformed into turbulent motions which stir the
near surface layers of the ocean and dissipate to heat. This energy input has an
important role in driving near surface mixing and hence promoting exchange
between the ocean and atmosphere. Its influence, however, is restricted to a relatively
shallow layer just below the surface, much shallower than the depth to which
significant orbital velocities extend (

l/2). Most of the energy is thought to be
dissipated within a length scale set by the wave height parameter H 1/3 which is the
average height of the highest one third of the waves.
Breaking waves also act to enhance ocean-atmosphere exchange through the forma-
tion of bubbles inwhitecaps. The motion in breaking waves is so violent that the integrity

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