Geoscience Reference
In-Depth Information
We can make another estimate of
τ
t
as follows. If the only important physical
parameters determining
τ
t
are
u
, the velocity scale of the turbulence doing the
mixing, and
d
, the depth of mixing, then
τ
t
d/u
. This says that the
time scale for turbulent mixing is of the order of the time required for turbulent
motions to traverse the depth of the tank. That is very physical.
Equating these two estimates yields
=
τ
t
(d, u)
∼
d
2
K
∼
d
u
,
τ
t
∼
so that
K
∼
ud.
(2.77)
Since we expect the dominant eddy size
to be of the order of
d
, we can write this
as
K
u
. Thus we have inferred on plausible grounds that the turbulence in our
“thought problem” has an eddy diffusivity
K
∼
∼
u
, the product of the velocity and
length scales of its dominant eddies.
Does this mean there is a formal analogy between molecular and turbulent
diffusion? No, for there are several important differences between the two:
• In molecular diffusion the time and length scales of the diffusing motion, a
microscopic process, are very small compared to those of the diffusion problem. Such
scale separation tends not to exist in turbulent diffusion.
• The most general linear relation between the flux
f
i
of a scalar constituent and its gradient
g
i
is
f
i
=−
γ
ij
g
j
, with
γ
ij
a second-rank diffusivity tensor. Since molecular diffusion is
a microscopic process one expects
γ
ij
to be isotropic - having no preferred direction -
so that
(Part III)
γ
ij
=
γδ
ij
with
γ
a scalar. Thus in molecular diffusion
f
i
=−
γ
ij
g
j
=
γg
i
. Turbulent diffusion is done by the dominant eddies, which can span the
flow and are inherently anisotropic; thus there is no reason to expect the eddy diffusivity
tensor
K
ij
to be isotropic
(Chapter 5)
.
• The eddy diffusivity is a property of the turbulent flow, but the molecular diffusivity is a
property of the fluid and the constituent being diffused.
• A diffusivity relates a mean flux and a mean gradient; it has no meaning without an
averaging process. The averaging involved in the molecular diffusivity has been done
for us in going from molecular-scale interactions to continuum fluid mechanics. Thus,
instantaneous turbulence fields have a molecular diffusivity but not an eddy diffusivity.
• As we shall discuss in
Chapter 4
and also in
Part II
, there are situations in which the eddy
diffusivity is poorly behaved (e.g., has a singularity and becomes negative).
−
γδ
ij
g
j
=−
In summary, there is a tempting, but shallow, analogy between molecular and
turbulent diffusion. Turbulent fluxes and mean gradients can show an eddy-
diffusivity-like relationship
(Chapter 4)
, but the eddy diffusivity is typically a
spatially variable flow property that can be a tensor, rather than a scalar. In a given
flow it can also depend on the diffusion geometry
(Chapter 11)
.
