Waves, turbulent motions and mixing - Introduction to the Physical and Biological Oceanography of Shelf Seas - page 115

diffusivity K z , a result which, at first, may seem counterintuitive. The inverse depend-
ence on the vertical diffusivity results from the fact that, for low vertical mixing, the
horizontal shear in the current is able to generate greater horizontal stretching of the
tracer, whereas increased vertical mixing counters horizontal stretching of the tracer.
The general problem of determining the diffusivity in an oscillatory flow with
period T 2 for an arbitrary mixing time T m is more difficult, but can be solved
(see (Fischer et al., 1979 )). The results show that maximum dispersion occurs for a
mixing time of T m ¼ T 2 . With this optimal matching of the vertical mixing time to the

tidal period, the shear diffusion components are

0067u T T 2 ; 0067v T T 2

K x ' 0 : K y ' 0 : ð 4 : 47 Þ

which sets an upper bound for K due to shear dispersion.
Since u T and v T are proportional to the tidal stream amplitudes, Equation (4.46)
implies an anisotropic dispersion which is strongest in the direction of the major axis of
the tidal ellipse. Tidal shear dispersion of this kind has been widely invoked in studies of
scalar transport in shelf seas. For example, in a simulation of the spreading of the 137 Cs on
the European shelf from a source in the Irish Sea, and its subsequent transport around the
north of Scotland and into the North Sea, Prandle (Prandle, 1984 ) used a combination
of tidally controlled dispersion and advection by the mean flow including the component
due to the rectification of the tide. It achieved a convincing representation of the
observed large-scale distribution of the Caesium tracer shown earlier in Fig. 3.17 .

4.4 The energetics of turbulence

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As we noted above, turbulence is necessarily dissipative, i.e. it consumes energy.
In the turbulent motions of a homogeneous fluid (r constant), energy is transferred

to smaller and smaller scales until it is eventually converted to heat at the smallest
scale by molecular friction. In order to maintain turbulence, it is necessary to supply
energy at a sufficient rate to satisfy this frictional demand.
In a stratified fluid there is an additional energy demand because turbulence acts
to bring about vertical mixing, which increases the potential energy of the fluid.
If there is insufficient energy being supplied to the turbulence to meet this demand,
the turbulent motion will diminish and the flow will revert to the laminar form. This
competition between the energy supply and the turbulence-suppressing influence of
stratification, which is formulated in the following section, leads to an important
criterion for when flow in the ocean is, and is not, turbulent.

¼

4.4.1 Buoyancy versus shear production of turbulence:

the Richardson number

In a turbulent stratified fluid, water particles with varying density differences will
be moving up and down across each horizontal plane with turbulent vertical velocity
w 0 ¼

Dz p /Dt, where z p is the height of a particle above a reference level, as shown