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the eddy viscosity N
z
and diffusivity K
z
. These parameters, which depend on the
amount of turbulence in the flow, may be obtained from a turbulence closure scheme
(see
Box 7.1
) which relates them to the product of the intensity of the turbulence
q (m s
1
) and a length scale L (metres):
N
z
¼
S
M
qL
;
K
z
¼
S
H
qL
:
ð
7
:
7
Þ
We will refer to our new model as the 'turbulence closure' or TC model. The
turbulent intensity q is the root-mean-square speed of the turbulent motions, with
the turbulent kinetic energy E
T
¼
2
q
2
. The evolution of turbulent kinetic energy
(TKE) is determined by the TKE equation which we established in
Chapter 4
(see
Section 4.4.2
for details of the individual terms):
1
w
0
E
0
T
Þ
@
@
E
T
@
t
¼
@ð
1
0
t
x
@
U
@
t
y
@
V
g
0
w
0
0
z
x
þ
e
:
ð
7
:
8
Þ
@
x
The stability functions S
M
and S
H
represent the inhibiting effect of stratification on
mixing, i.e. simulating the response of mixing to stability seen in Turner's tank
experiment (
Section 6.1.1
). Both S
M
and S
H
depend on the gradient Richardson
number (Equation 4.56) which, in turn, is derived from the velocity and density
structure. For our TC model, we shall adopt a relatively simple formulation of the
Box 7.1 Turbulence closure schemes
In order to set up a system of equations which unambiguously describes the evolution
of the structure and flow of the water column, we need to add to the basic equations
(i.e. dynamics, continuity, advection-diffusion) a prescription for relating the eddy
diffusivity K
z
and eddy viscosity N
z
to other parameters of the system and so 'close'
the problem. Closure schemes range from the specification of constant values for K
z
and N
z
, through their prescription as simple functions of the gradient Richardson
number Ri, to more elaborate formulations based on one or two equations for
turbulent quantities. Most of the latter involve an equation for the evolution of
the turbulent kinetic energy q
2
(like our
Equation 7.8
) and a second equation for
another turbulence parameter. Two apparently different and widely used approaches
have evolved; the Mellor-Yamada (MY2.5) scheme which employs an equation for
q
2
L and the k
e approach which incorporates an equation for evolution of the
dissipation e. The two approaches, which have generated a large and sometimes
arcane literature, are distinguished by different notations and apparently differing
assumptions. However, it has been shown that, in physical terms, the two are
essentially the same and in practice lead to closely similar results (Burchard et al.,
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