Geoscience Reference
In-Depth Information
Hill
(
1989
) has pursued the implications of M-O similarity for scalars. One
finding is that the correlation coefficient of any two scalars is
1. This is unphysical,
and he postulates that here it is necessary to include the scalar physics of the surface
itself in order to obtain realistic results.
±
10.4.1 Covariance budgets
The equations or
budgets
governing turbulent kinetic energy, temperature variance,
Reynolds stress, and temperature flux had been known for some time, but the 1968
Kansas experiment provided the first comprehensive measurements of them in the
atmospheric surface layer.
10.4.1.1 The TKE budget
In surface-layer coordinates this is, from
Eq. (8.70)
,
uw
∂U
∂
∂z
u
i
u
i
w
2
1
ρ
0
∂
∂z
pw
g
θ
0
θw
∂z
+
=−
+
−
.
(10.43)
The terms are, in order, shear production, turbulent transport, pressure transport,
buoyant production, and viscous dissipation.
In the Kansas experiment all budget terms but pressure transport were measured
over the M-O stability range (
Wyngaard and Coté
,
1971
). An unexpected finding
was the large rate of gain by pressure transport, which was inferred from the imbal-
ance of the measured terms. Up to that time pressure transport had generally been
assumed to be negligible, so the Kansas findings stimulated much discussion and
debate in the micrometeorological community. Consensus results for the unstable
side as of about 1990 are shown in
Figure 10.8
;
overall they confirm the Kansas
results. The large gain due to pressure transport approximately balances the loss to
turbulent transport; thus the sum of shear and buoyant production approximately
balances viscous dissipation, as also reported by
Bradley
et al.
(
1981a
).
The TKE budget under stable conditions suggests a much more placid picture.
Both turbulent and pressure transport are found to be negligible, so that shear
production is essentially balanced by buoyant destruction and viscous dissipation.
10.4.2 Conserved scalar variance budgets
The quasi-steady budget of potential temperature variance is, from
Eq. (8.71)
,
wθ
2
2
θw
∂
∂
∂z
∂z
+
=−
χ,
(10.44)
