Geoscience Reference
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u
, and the turbulent scalar scale as
c
, an estimate for the magnitude of the rate of
change of mean concentration following the mean motion is
D
m
Dt
m
C
∂cu
i
cu
L
p
.
=−
∂x
i
∼
(3.20)
Here we expect
c
∼
C
, in order of magnitude, so that
D
m
Dt
m
C
Cu
L
p
∼
C
τ
turb
,
∼
(3.21)
with
τ
turb
∼
L
p
/u
the time scale of this turbulent diffusion process.
Equation (3.21)
says we can interpret
τ
turb
as the eddy-traverse time across the mean plume. If
L
p
is
1mand
u
is 1m s
−
1
,
τ
turb
is 1 s, strikingly less than our estimate of 10
5
sfor
τ
molec
.
Put another way, if (in analogywith themolecular diffusion result
τ
molec
∼
d
2
/γ
)
L
p
/K
, with
K
a
turbulent
diffusivity, we find that
K
we write
τ
turb
∼
∼
uL
p
,which
in our example is five orders of magnitude larger than
γ
.
Let's summarize. Molecular diffusion is a microscopic physical process that
acts to remove macroscopic
c
-anomalies through the collective effect of molecu-
lar collisions. It seems well described by
intermingling
, a dictionary definition of
diffusion. Turbulent diffusion, viewed most broadly, involves three processes. Two
are physical - turbulent mixing by the chaotic, random deformation and advection
of
˜
c
-anomalies by turbulence; and molecular diffusion, which acts most strongly
on the smallest of these
˜
c
-anomalies. The all-important third, and new, process is
ensemble averaging, which produces a virtual concentration field
C
that is much
smoother, broader, and has smaller maximum concentrations than the turbulent
˜
c
field in any realization. As a result the concentrations observed along paths in the
instantaneous and ensemble-average plumes can differ strikingly, as shown in the
lower panel of
Figure 3.3
.
˜
3.3.2 Enhancement of molecular diffusion by turbulence
Figure 3.4
depicts the distortion of a blob of conserved scalar in a turbulent flow
in the process we call
turbulent mixing
. If we think of the distorted blob as a sheet
having a surface area and a thickness, then as the distortion increases the surface
area it decreases the thickness. The increased area and the larger scalar gradients on
it greatly increase the total rate of molecular diffusion of the scalar out of the blob.
This suggests that we examine the equation for the evolution of the gradient
g
i
˜
of a conserved scalar
(Problem 1.11)
:
∂
2
D
g
i
Dt
=
˜
∂
g
i
∂t
+˜
˜
u
j
∂
g
i
∂x
j
=−
˜
∂
u
j
∂x
i
˜
˜
g
i
∂x
j
∂x
j
.
˜
g
j
+
γ
(3.22)
