Geoscience Reference
In-Depth Information
• The correlation coefficient between
w
and
c
t
is only 0.26, while that between
w
and
c
b
is 0.60.
• The bottom-up flux is carried 70% by updrafts, 30% by downdrafts. The top-down flux
has nearly the same distribution.
These joint densities also say someth
ing
about the “efficiency” of the transport
process. To illustrate, consider a positive
wc
. The first and third quadrants contribute
to it, while the second and fourth contribute to a flux of the opposite sign. Let us
call these the “forward” and “back” fluxes, respectively. A “transport efficiency” is
net flux
forward flux
=
forward flux
back flux
forward flux
+
e
t
=
.
(13.34)
The top-down process, with
ρ
=
0
.
26, has
e
t
=
0
.
6; the bottom-up process,
with
ρ
0
.
9. This shows again the unusual diffusive properties of
convective turbulence. Convective clouds could have similar diffusive properties.
Numerical models of the atmosphere and ocean need concise but reliable sub-
models of unresolved turbulence. The traditional way to develop such submodels
is to study the process in detail experimentally, to interpret its basic physics, and
then to express this physics through an approximate but concise equation or set
of equations. This process, called
parameterization
in meteorology, can be quite
difficult because the detailed, reliable turbulence data that it requires often do not
exist. As we saw in this and earlier chapters, DNS and LES can be used to provide
turbulence fields for this purpose.
=
0
.
6, has
e
t
=
13.4 The evolution equation for the probability density
In
Parts I
and
II
we manipulated the fluid equations to produce equations for the
evolution of certain statistics of the turbulent fluctuations of velocity and advected
scalar constituents. Until fairly recently the probability density of velocity or an
advected scalar was not among the statistics for which one could derive an evolution
equation. That situation changed with a remarkable paper by Lundgren in 1967.
We'll briefly sketch his derivation here.
†
If
β(v
i
;
x
i
,t)
is the probability density of the velocity at point
x
i
and time
t
,
β(v
i
;
x
i
,t)dv
i
is the probability that the velocity at
(x
i
,t)
is within
dv
i
of
v
i
.If
we define
β
α
(v
i
;
x
i
,t)
to be a “single-realization” probability density of velocity -
one that produces
v
i
(x
i
,t)
u
i
in realization
α
,
‡
then
=
β
α
(v
i
;
x
i
,t)
=
δ(u
i
−
v
i
),
(13.35)
†
We have used tensor notation rather than Lundgren's vector notation.
‡
This is an elaboration of the interpretation offered by
Lundgren
(
1967
).
