Geoscience Reference
In-Depth Information
inversion can damp upward motions quite effectively. Passengers in an airplane
taking off from a large metropolitan area can often see the dark smudge of pollu-
tants trapped in the boundary layer below, a sign of such an inversion. In Denver,
Colorado, this is known as the “brown cloud.”
In unstable stratification,
∂/∂z
negative, we expect
wθ
, the vertical flux of
temperature, to be positive, so the buoyancy term represents a rate of gain in the
TKE budget,
Eq. (8.70)
. In stable stratification we expect a negative flux and cor-
respondingly a rate of TKE loss to buoyancy. Labeling this TKE budget term as
buoyant production
might be confusing, since that might be taken to imply an
energy source term, so where possible we will refer to it under stable conditions as
buoyant destruction.
It represents energy conversion
(Problem 9.12)
.
The TKE
equation (6.54)
for a horizontally homogeneous resolvable-scale field
with buoyancy effects becomes
u
i
u
i
,t
=−
p
r
u
r
3
,
3
−
u
i
u
i
u
3
,
3
−
u
i
u
i
u
3
,
3
+
1
2
1
ρ
0
1
2
g
θ
0
θ
r
w
r
.
(9.8)
U
1
,
3
u
r
1
u
r
3
−
I
−
If the spatial resolution is sufficient to give
θ
r
w
r
θw
, then the buoyancy term
here is well resolved. This is the case if
.
The corresponding TKE equation for the SFS field is
(6.56)
generalized to include
buoyancy effects:
2
u
i
u
i
,t
p
s
u
3
,
3
+
2
u
i
u
i
u
3
,
3
1
1
ρ
0
1
U
1
,
3
u
1
u
3
−
=−
I
−
(9.9)
u
i
u
i
u
r
3
g
θ
0
θ
s
w
s
−
,
3
+
−
.
If the filter cutoff separating the resolvable and subfilter-scale fields lies well beyond
the energy-containing range, i.e., if
, the subfilter-scale field contains
negligible variance and flux. Then as we saw in
Chapter 6
this TKE equation
reduces to
u
i
u
i
,t
1
2
I
−
,
(9.10)
a statement of the energy balance of all eddies beyond the energy-containing range.
This was central to Kolmogorov's thinking.
9.3.3 The convective ABL
When a parcel of buoyant air rises from the warmer or moister surface its buoyancy
lasts as long as its potential temperature excess
θ
does. Distortion of the parcel by
