Geoscience Reference
In-Depth Information
10.2.4 Uses of M-O similarity
10.2.4.1 Wind and temperature profiles
As indicated in
Eq. (10.12)
, M-O similarity predicts that the vertical gradients of
mean wind speed and mean virtual potential temperature in a locally homogeneous,
quasi-steady atmospheric surface layer are
kz
φ
m
z
,
kz
φ
h
z
.
∂U
∂z
=
u
∗
∂
∂z
=
T
∗
(10.15)
L
L
In stable conditions the functions proposed for
φ
m
and
φ
h
tend to be linear and thus
integrate easily. For example, Hogstrom's
φ
m
and
φ
h
forms,
Eqs. (10.14)
,give
ln
z
,
ln
z
,
(10.16)
u
k
4
.
8
z
L
T
k
7
.
8
z
L
U(z)
=
z
0
+
( )
=
(z
r
)
+
z
r
+
with
z
r
the reference height for the “surface” temperature.
Many different
φ
m
and
φ
h
forms have been proposed for unstable conditions,
and they tend to be more difficult to integrate analytically. Based on fits to
Hogstrom's (
1988
) extensive data,
Wilson
(
2001
) proposed
2
/
3
)
−
1
/
2
φ
m,h
=
(
1
+
γ
|
z/L
|
(10.17)
for both
φ
m
and
φ
h
. He shows these give the mean profiles
ln
z
3ln
1
,
1
2
/
3
u
k
+
+
γ
m
|
z/L
|
U(z)
=
z
0
−
1
(10.18)
2
/
3
1
+
+
γ
m
|
z
0
/L
|
ln
3ln
1
,
+
1
P
t
T
∗
k
z
z
r
−
+
γ
h
|
z/L
|
2
/
3
(z)
=
(z
r
)
+
1
(10.19)
2
/
3
1
+
+
γ
h
|
z
r
/L
|
and suggests the constants be taken as
k
=
0
.
4
,P
t
=
0
.
95
,γ
m
=
3
.
6
,γ
h
=
7
.
9.
10.2.4.2 Inferring a surface-layer property from measurements of others
In a neutral surface layer
z/L
=
0
(w
2
)
1
/
2
u
∗
σ
w
u
∗
=
1
.
2
.
(10.20)
|
|
Thus, under near-neutral conditions (i.e., at
z/L
1) one can use
σ
w
/
1
.
2asa
surrogate for
u
∗
. It is easier to measure
σ
w
than to measure
u
∗
√
−
uw
and it
requires much shorter averaging times
(Chapter 2)
.
