Geoscience Reference
In-Depth Information
The second, third, and fourth terms on the left sides of
Eqs. (5.4)
and
(5.5)
represent the three advection processes that produce fluctuations: advection of the
fluctuating quantity by the mean velocity, advection of the mean quantity by the
fluctuating velocity, and the fluctuating part of advection of the fluctuating quantity
by the fluctuating velocity. The other terms in
Eqs. (5.4)
and
(5.5)
are the fluctuating
counterparts of those in the full equations. Like their parent equations,
Eqs. (5.4)
and
(5.5)
are nonlinear since the fourth terms on their left sides involve products of
the fluctuating quantities.
5.3 Example: The scalar variance equation
5.3.1 Derivation and interpretation
We'll demonstrate the derivation process with the scalar variance
c
2
. Multiplying
the
c
-
equation (5.4)
by 2
c
, ensemble averaging, and using the averaging rules in
Chapter 2
yields the following results, term by term:
∂c
2
∂t
,
2
c
∂c
∂t
=
U
j
∂c
2
∂c
∂x
j
=
2
cU
j
∂x
j
,
2
cu
j
∂C
2
cu
j
∂C
∂x
j
=
∂x
j
,
(5.6)
∂x
j
u
j
c
u
j
c
=
∂c
2
u
j
∂x
j
∂
2
c
−
,
∂
2
c
2
∂x
j
∂x
j
−
∂
2
c
∂x
j
∂x
j
=
∂c
∂x
j
∂c
∂x
j
.
2
cγ
γ
2
γ
The resulting conservation equation for scalar variance, with the meaning of its
terms indicated, is
∂c
2
∂t
U
j
∂c
2
=−
(
mean advection
)
∂x
j
2
u
j
c
∂C
−
(
mean-gradient production
)
∂x
j
∂c
2
u
j
∂x
j
−
(
turbulent transport
)
