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where S
M
and S
H
are stability functions which allow for the effects of stratification
on the eddy parameters. In numerical models, the eddy parameters are recalculated
from these relations at each time step using the instantaneous mean flow, density and
turbulent kinetic energy (TKE) profiles. The new N
z
and K
z
are then inserted in
the equations of motion, the transport equation and the TKE equation to step
forward in time and so on. We shall see more of how such models work in
Chapter
7
, when we use a turbulence closure model to study the evolution of dissipation and
mixing in the shelf sea water column. Such a turbulence closure scheme also lies at the
heart of the physics-primary production model which is utilised later in this topic and
is available at (
www.cambridge.org/shelfseas
).
Summary
..................................................................................
This chapter described two very different classes of motion which play important
roles in the shelf seas and are of particular importance in relation to the mixing of
heat, salt, nutrients and other properties through the water column. We first dealt
with surface wave motions which dissipate large amounts of energy and promote
mixing in the near surface layers. The orbital motions of water particles in surface
waves, the energy content of such waves and its propagation at the group velocity
were discussed in some detail as a preliminary to understanding the analogous but
more complex motions involved in internal waves propagating in a two-layer
system. Internal waves travel much more slowly than surface waves but transmit
significant amounts of energy whose dissipation may make an important contribu-
tion to internal mixing in low energy regions. The velocity field of internal waves in
a two-layer system involves patterns of convergence and divergence at the surface
which may be manifest through changes in surface roughness.
Turbulent motions are random, dispersive and lead to energy dissipation. The fluxes
of scalar properties which arise from turbulent fluctuations in the flow, contribute to
the transport of properties which is summarised in the advection-diffusion equation.
Turbulent fluxes of momentum are similarly important in the dynamics, where they
appear as the internal frictional forces. Fickian diffusion of scalar quantities provides
an idealisation involving an eddy diffusivity which is uniform and independent of
scale. Most dispersion in the ocean involves diffusivities which increase with scale, but
an important exception in shelf seas is the process of tidal shear diffusion.
In a stratified fluid, buoyancy forces act to suppress turbulence by extracting
energy from the flow through vertical mixing. Turbulence can then only be sustained
if the mean flow supplies sufficient power to outcompete the buoyancy forces, a fact
expressed in the Richardson number criterion. This is a fundamentally important
result that will be relevant to understanding constraints on the biogeochemistry of
the water column: pycnoclines are good at inhibiting vertical mixing. A more general
description of the mechanisms transporting, generating and eroding turbulent kinetic
energy (TKE) is given in the TKE equation. Turbulence closure schemes are needed
to determine the turbulent diffusion and eddy viscosity parameters. The lengthscales
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