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scales up to a maximum scale which is limited either by stratification or the proximity
of the boundaries. In stratified conditions, buoyancy forces oppose the vertical
extension of eddies beyond a point where their turbulent kinetic energy is all con-
verted to potential energy in the overturning motion. On dimensional grounds this
limiting scale should be set by the Ozmidov length (Ozmidov,
1965
):
1
=
2
e
N
3
L
o
¼
ð
4
:
69
Þ
where N is the buoyancy frequency. For strongly stratified conditions (N
2
10
3
s
2
)
∼
10
6
Wkg
1
) L
o
is only a fraction of a
metre. Where stratification is very weak, L
o
may increase to tens of metres and
exceed the water depth. A closely related scale to the Ozmidov length is the
Thorpe scale L
T
which can be determined from observations by re-ordering
particles in the density profile to remove instabilities in the water column and
calculating the r.m.s. displacement of particles (Thorpe,
2005
). Since it has been
shown (Dillon,
1982
) that L
o
∼
where turbulence is relatively weak (e
∼
0.8L
T
, profiles of L
T
and N
2
derived from high
resolution observations of density can provide estimates of the dissipation rate
using
Equation (4.69)
.
4.4.4
Turbulence closure
We have noted already that, in order to make progress in solving the equations
of motion and diffusion, it is necessary to represent the turbulent stresses and the
fluxes of scalars in terms of the mean properties of the flow. This requirement for
turbulence closure can be met by a wide variety of assumptions and parameterisa-
tions. Frequently the approach is based on the notion of an eddy viscosity or eddy
diffusivity which relates the stresses and fluxes to the relevant gradients of velocity or
scalar concentration (see earlier in
Sections 4.3.4
and
4.3.5
). The simplest closure is
then to assume that the eddy parameters are constants which may be adjusted to
match observations. Such simple closure has obvious advantages in simplifying the
maths if we are seeking analytical solutions.
We have seen, however, that turbulence is strongly influenced by buoyancy forces
in a way which is encapsulated in the Richardson number, Ri. As a first step in
putting closure on a sounder footing, it seems desirable to make the eddy parameters
N
z
and K
z
depend on Ri. Parameterisations of this kind (e.g. Pacanowski and
Philander,
1981
) are generally too difficult to implement in analytical studies but
have been widely used in numerical models. A more rigorous general approach to
the eddy parameters in terms of the product of the r.m.s turbulent velocity q and a
characteristic length scale L of the turbulence:
N
z
¼
S
M
qL
;
K
z
¼
S
H
qL
ð
4
:
70
Þ
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