Geoscience Reference
In-Depth Information
c
final
c
initial
Figure 2.4 Molecular diffusion of dye in a tank of motionless fluid. The top panel
shows the initial state; the bottom panel shows the evolving vertical profile of dye
concentration during the diffusion process.
the response time
τ
m
must depend on them alone:
τ
m
=
τ
m
(d, γ ).
It follows on
dimensional grounds that the only possible form is
τ
m
∼
d
2
/γ
.If
d
=1mand
10
−
5
m
2
s
−
1
, for example, then the time scale for molecular diffusion is 10
5
s,
about one day. Molecular diffusion can be very slow.
Figure 2.5
shows a turbulent version of this problem, with the turbulence driven
by bottom heating, say. The concentration field is now very complicated, every
term in the conserved scalar
equation (1.31)
being active. How can we deduce from
that equation what we know from experience - that turbulence can mix much faster
than molecular diffusion?
As sketched in
Figure 2.5
,
we expect that the ensemble-averaged concen-
tration
C(z,t)
- the average of
γ
=
c(z, t)
over many realizations of the same
