Geoscience Reference
In-Depth Information
Figure 10.4 Kansas observations of the rms vertical velocity (upper) and rms
temperature (lower) for unstable conditions only, plotted in M-O coordinates.
The dashed lines are the asymptotic predictions for very unstable conditions,
Eq. (10.36)
.From
Wyngaard
(
1973
).
shows it even sooner. The shear- and buoyant-production terms in the TKE budget
are, in M-O terms,
u
3
u
3
uw
∂U
g
θ
0
θw
z
L
.
∗
∗
kz
−
∂z
=
kz
φ
m
,
=−
(10.37)
φ
m
decreases monotonically with increasing instability; the Kansas data, for exam-
ple
(Figure 10.3)
,
are fit well by the formula
φ
m
15
z/L)
−
1
/
4
on the
=
(
1
−
unstable side. At
z/L
2, where evidence of local-free-convection behavior
of
σ
w
appears, this expression says that the buoyant-production rate exceeds the
shear-production rate by a factor of nearly five.
=−
10.3.2 The asymptotic stable state
The ratio of fluctuating buoyancy and inertia forces for a velocity-field eddy of size
r
(Chapter 2)
is an “eddy Richardson number”
Ri
e
(r)
,
gθr
1
/
3
θ
0
2
/
3
,
gθr
θ
0
[
u(r)
]
2
Ri
e
(r)
=
=
(10.38)
