Geoscience Reference
In-Depth Information
of
Wilson
(
2001
), for example, has a local-free-convection limit in which
K
m
behaves as
g
θ
0
Q
0
1
/
3
z
4
/
3
.
K
m
∼
u
f
z
∼
(11.12)
The
K
m
defined in
Eq. (11.10)
has not been documented extensively above the
surface layer, perhaps in part due
to
the inherently large scatter. To illustrate, con-
sider a mid-layer kinematic stress
uw
of the order of
−
u
2
0
.
05
w
∗
z
i
(which we'll see shortly is a reasonable estimate) the mean wind shear is
,sothatif
K
m
∼
∗
u
2
∂U
∂z
=
−
uw
K
m
∗
0
.
05
w
∗
z
i
,
(11.13)
0
.
3ms
−
1
,and
w
∗
=2ms
−
1
, all typical of the CBL in clear
weather over land, this gives
∂U/∂z
For
z
i
=
1500 m,
u
∗
=
10
−
4
s
−
1
- a mean wind difference of
0.6 m s
−
1
per 1000 m height. The uncertainty in a mean wind difference due to
finite averaging time scales with the rms wind fluctuation
(Chapter 2)
;thisisofthe
order of
w
∗
, which here is three times the mean wind difference over a height of
1000 m. Likewise, the error in time-averaged stress scales with
w
2
6
×
, which here is
two orders of magnitude
larger than the mean stress. Thus reliably determining the
K
m
defined in
Eq. (11.10)
in the mixed layer with conventional observations can
require unachievable averaging times.
∗
11.2.3.4 Inferences from the stress budgets
The Reynolds-stress budgets
(8.69)
provide another path to diagnosing the behavior
of
K
m
. As discussed in
Chapter 9
, their Coriolis terms are typically small in the
ABL and in steady, locally homogeneous conditions they reduce to a balance among
shear production (
S
), turbulent transport (
T
), buoyant production (
B
), and pressure
covariance (
P
)terms:
w
∂p
,
∂
∂t
uw
w
2
∂U
∂
∂z
uww
g
θ
0
θu
1
ρ
0
u
∂p
∂z
=
0
=−
∂z
−
+
−
∂x
+
(11.14)
S
T
B
P
w
∂p
.
∂
∂t
vw
w
2
∂V
∂z
∂
∂z
vww
g
θ
0
θv
1
ρ
0
v
∂p
∂z
=
=−
−
+
−
∂y
+
0
(11.15)
We'll continue to take
x
as the direction of the mean wind in the surface layer.
Wyngaard
(
1984
) found two runs from the Minnesota e
xpe
riment (
Kaimal
et al
.
,
1976
) with enough lateral mean wind shear to allow the
vw
budget to be resol
ved,
and combined four runs from AMTEX (
Lenschow
et al
.
,
1980
) to produce a
uw
