Geoscience Reference
In-Depth Information
Multiplying by
c
r
, ensemble averaging, scaling, and rearranging as we did with
the TKE budgets gives the equation for the evolution of the variance of
c
r
:
∂c
r
c
r
∂t
=
2
Sc
r
−
I
c
=
0
.
(6.46)
2
c
r
u
j
c
,j
−
2
c
s
u
j
c
,j
.
I
c
=
(6.47)
The mean-gradient production and molecular destruction terms of the scalar vari-
ance budget,
Eq. (5.7)
, are the product of a scalar flux and a scalar gradient.
I
c
has
this form as well.
The same process gives the equation for the evolution of
c
s
c
s
:
∂c
s
c
s
∂t
=
I
c
−
χ
c
=
0
.
(6.48)
The interpretation here is analogous to that for TKE: the variance of the large-scale
part of the scalar is kept in equilibrium by the balance between the mean rate of
production by the forcing term and the mean rate of loss by transfer to smaller
scales. The variance in those smaller scales is kept in equilibrium as well by the
balance between the mean rate of transfer from the larger scales and the mean rate
of molecular destruction. Again the mean rate of spectral transfer of variance - the
cascade rate
- is the magnitude of the rate of molecular destruction of variance,
here
χ
c
.
6.4 Application to flows homogeneous in two dimensions
We saw in
Chapter 5
that the fluctuating velocity satisfies
u
i,t
+
U
j
u
i,j
+
u
j
U
i,j
+
u
i
u
j
−
u
i
u
j
,j
1
ρ
p
,i
+
νu
i,jj
.
=−
(6.49)
We'll use this equation in a boundary layer over a homogeneous surface. We choose
x
1
to be the mean-flow direction and assume that the inhomogeneity is confined to
the
x
3
or
z
(vertical) direction, so we have
U
i
=
U
1
(x
3
)δ
i
1
,
U
i,j
=
U
1
,
3
δ
i
1
δ
j
3
,
(6.50)
(u
i
u
j
)
,j
=
(u
i
u
3
)
,
3
,
u,
=
u(x
3
), (x
3
).
