Geoscience Reference
In-Depth Information
mean advection and turbulent transport contributions to the surface integral vanish.
We conclude that they move scalar variance from one point in the flow to another.
We can rewrite the molecular diffusion term as
γ
∂c
2
∂x
j
−
,
∂
2
c
2
∂x
j
∂x
j
=
∂
∂x
j
∂
∂x
j
γ
∂c
2
∂x
j
γ
=−
(5.11)
which is the negative divergence of the molecular flux of
c
2
, or molecular diffusion.
The second m
ole
cular term, being negative definite, represents the rate of molecular
destruction of
c
2
. We will label it
χ
c
:
∂c
∂x
j
∂c
∂x
j
.
χ
c
=
2
γ
(5.12)
The factor 2 is not always used in its definition, which can be a source of confusion.
We can rewrite
χ
c
as
2
γ
∂c
∂x
j
∂c
∂x
j
χ
c
=
=−
2
(
fluctuating molecular flux of
c)
·
(
fluctuating gradient of
c),
(5.13)
which has the same form as the mean-gradient production term, the scalar (dot)
product of flux and gradient.
In steady conditions the integral of
Eq. (5.7)
over the flow volume therefore
reduces to
∂
∂t
2
u
j
c
∂C
c
2
dV
=
0
=−
∂x
j
dV
−
χ
c
dV,
(5.14)
V
V
V
so it follows that
2
u
j
c
∂C
−
∂x
j
dV
=
χ
c
dV >
0
.
(5.15)
V
V
Equation (5.14)
says that in st
ead
y conditions the flow-integrated rates of production
and molecular destruction of
c
2
are in balance.
Equation (5.15)
is the consequence:
in steady conditions the flow-integrated rate of production is positive definite. The
eddy-diffusivity closure
K
∂C
cu
j
=−
∂x
j
,K
≥
0
,
(5.16)
which can be useful for simple estimates, satisfies this constraint by making the
local
mean-gradient production term positive definite:
2
K
∂C
∂x
j
2
u
j
c
∂C
∂C
∂x
j
−
∂x
j
=
≥
0
.
(5.17)
