Geoscience Reference
In-Depth Information
∂
2
c
2
∂x
j
∂x
j
+
γ
(
molecular diffusion
)
∂c
∂x
j
∂c
∂x
j
−
2
γ
(
molecular destruction
).
(5.7)
As indicated, the molecular term represents two effects.
The combination of the local time change plus mean advection is the time deriva-
tive following the mean motion. Thus it differs from the substantial derivative, also
called the total time derivative, which follows the instantaneous motion:
∂
∂t
+
U
i
∂
∂x
i
,
≡
time derivative following mean motion
∂
∂t
+˜
∂
∂x
i
time derivative following instantaneous motion
≡
u
i
D
Dt
=
∂
∂t
+
(U
i
+
u
i
)
∂
∂x
i
.
=
(5.8)
We can interpret the
mean-gradient-production
term as follows. A displacement
d
j
in the presence of a mean gradient
∂C/∂x
j
produces a fluctuation
c
d
j
∂C/∂x
j
,
so
u
j
∂C/∂x
j
is the rate of production of
c
fluc
tuations. Multiplying by 2
c
and
averaging then gives the rate of production of
c
2
as 2
u
j
c∂C/∂x
j
.
The third term on the right side of
Eq. (5.7)
is the divergence of
c
2
u
j
, the turbulent
flux of squared scalar fluctuation. We call this
turbulent transport
of scalar variance.
A term of this type appears in any second-moment equation.
The fourth term, molecular diffusion, is also a divergence. The sum of these three
divergence terms is
=
mean advection
+
turbulent transport
+
molecular diffusion
U
j
c
2
.
γ
∂c
2
∂x
j
∂
∂x
j
=−
+
c
2
u
j
−
(5.9)
With the divergence theorem we can express the volume integral of this divergence
as an integral over the surface that bounds the volume:
U
j
c
2
dV
U
n
c
2
dA,
(5.10)
γ
∂c
2
∂x
j
γ
∂c
2
∂x
n
∂
∂x
j
+
c
2
u
j
−
=
+
c
2
u
n
−
V
A
where n is the outward normal to the surface element
dA
. If we integrate over the
entire flow volume, so the velocity field vanishes on this bounding surface, the
