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is the norm and so the relevant criterion for whether turbulence will develop is
generally set by the energetics considerations embodied in the Richardson number.
4.4.2
The turbulent kinetic energy equation and dissipation
In the previous section we focused on the competition between the production of
turbulence in a shear flow and its conversion to potential energy in mixing. We now
proceed to take a more general view of the energetics which combines all the
important processes influencing the budget of turbulent energy. We define the
turbulent kinetic energy (TKE) as:
1
2
q
2
1
2
ð
u
0
2
v
0
2
w
0
2
E
T
¼
¼
þ
þ
Þ:
ð
4
:
57
Þ
E
T
is a scalar property which is subject to change by advection and diffusion as well
as being produced and dissipated in the fluid motion. If we assume that conditions
are horizontally uniform and that there is no vertical advection (W
¼
0), then the
evolution of
E
T
will be described by:
w
0
E
0
T
Þ
@
0
w
0
0
@
E
T
@
t
¼
@ð
1
0
t
x
@
U
@
t
y
@
V
g
z
z
þ
e
@
z
ð
4
:
58
Þ
diffusion
production
mixing dissipation
ð
Þ
ð
Þ
P
B
E
0
T
¼
E
T
E
T
and e is the rate of energy conversion to heat. The production
term is a generalised form of
Equation (4.52)
for flow in x and y directions.
The mixing or “buoyancy production” term represents work done against, or by,
buoyancy forces as expressed in
Equation (4.49)
. For a stable stratification it repre-
sents a loss of TKE to potential energy. For an unstable situation, as in convection,
this term is a positive input of TKE fuelled by the loss of potential energy.
The diffusion term i
s
the divergence of the vertical diffusive energy flux which can
be expressed as
where
z where K
q
is the eddy diffusivity for TKE. Writing the
production and mixing terms in similar form (
Equations 4.53
,
4.49
), we can express
the TKE equation as:
K
q
@
E
T
=@
!
2
þ
@
2
@
E
T
@
t
¼
@
K
q
@
E
T
@
@
U
@
U
@
K
z
g
0
@
@
þ
N
z
þ
z
e
:
ð
4
:
59
Þ
@
z
z
z
z
In many situations, the diffusion of TKE is small relative to the leading terms. If we
neglect it and assume a steady state, we have:
!
2
þ
@
2
@
U
@
U
@
K
z
g
0
@
@
N
z
þ
z
e
¼
0
ð
4
:
60
Þ
z
z
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