Geoscience Reference
In-Depth Information
g
g
θ
0
∂
∂z
θ
0
θw
u
i
u
j
∂U
i
rate of buoyant destruction of TKE
rate of shear production of TKE
=
R
f
=
,
i
=
∂
∂z
2
.
(9.15)
∂x
j
As we'll discuss in
Chapter 12
, there is evidence that when these Richardson
numbers exceed about 0.2 to 0.3 all turbulence can be extinguished.
9.4 Average vs. instantaneous structure
Perhaps you remember lying on your back outdoors on a summer day and watching
cumulus clouds drifting overhead. Their three-dimensional, cauliflower-like turbu-
lent structure seems “frozen” because the large-eddy time scale
/u
is so large.
You need to watch even a modest cloud with
1ms
−
1
on
the order of 20 minutes - longer than most of us can - to see it change. As a
result we are probably more aware of instantaneous cloud properties than average
ones.
Most numerical models of turbulent flows predict average properties, either the
ensemble average or a spatial average over a numerical grid cell (
Chapter 1
). The
Gaussian-plume models routinely used for effluent concentrations downwind of
sources, for example, are based on G. I. Taylor's ensemble-mean solution (
Chap-
must test their predictions against short-term time averages, and so the model
evaluations tend to be plagued by scatter. It can be difficult to apportion the
scatter between inadequate averaging time and model physics. This is illustrated
in
Figure 9.8
.
∼
∼
1000 m and
u
9.5 Quasi-steadiness and local homogeneity
The ABL over land is nonstationary due to the diurnal cycle; it also tends to be
inhomogeneous in the horizontal due to spatial variations in synoptic conditions
and surface properties. But if its time and length scales are small compared to those
of the external variations, we can often use a steady, horizontally homogeneous
model of ABL turbulence.
We'll illustrate with time changes.
Figure 9.5
shows the daily cycle of surface
temperature flux
Q
0
measured in the Kansas experiment. We expect that the tur-
bulence can be quasi-steady if the time scale of these changes,
Q
0
(∂Q
0
/∂t)
−
1
,is
much larger than the large-eddy turnover time
h/u.
This is the case if
1
Q
0
∂Q
0
∂t
h
u
1
(
quasi-steadiness
).
(9.16)
