Geoscience Reference
In-Depth Information
Here we define
(u
j
θ
t
)
p
−
u
j
θ
t
.
r
jθ
=
(12.33)
Their sum is the equation for the temperature fluctuation:
,j
u
j
+
u
j
θ
u
j
θ
,j
θ
,t
+
θ
,j
U
j
+
−
=
γθ
,jj
.
(12.34)
By constructing kinetic energy budgets of the wave and turbulence components
Finnigan
et al
.
(
1984
) found that kinetic energy flows from wave motions to the
turbulence and that the mechanics of this energy flow depends on the nonlinear
character of the wave field. The TKE so produced is dissipated primarily byworking
against buoyancy rather than viscous forces, which they point out is consistent
with the quasi two-dimensional character of the turbulence imposed by the strong
stability of their flow.
Finnigan
(
1988
) later generalized the approach to unsteady
waves and studied several wave-turbulence interaction events in detail.
12.3 The quasi-steady SBL
We have seen in
Chapter 10
that M-O similarity has given remarkable order to
turbulence statistics in the stable surface layer. Furthermore, acoustic sounding
(
Neff and Coulter
,
1986
;
Neff
et al
.
,
2008
) has provided evidence of quasi-steady
SBLs of both the nocturnal and the long-lived variety. These developments have
stimulated analytical work on the SBL.
12.3.1 The stable surface layer limit
The behavior of the mean vertical gradients of wind and potential temperature in
very stable conditions,
Figure 10.3
,
indicates that
kz
u
∗
∂U
∂z
z
L
,
∂U
∂z
u
L
,
φ
m
=
∼
so that
∼
(12.35)
and similarly for
∂/∂z
. This reflects the emergence of
z-less scaling
in very stable
conditions.
We can also view this stable limit through the flux Richardson number, the ratio
of the rates of buoyant destruction and shear production of TKE:
g
θ
0
θw
uw
∂
∂z
R
f
=
.
(12.36)
This can be expressed in M-O similarity terms as
(Problem 12.17)
z
L
1
φ
m
.
R
f
=
(12.37)
