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Peña et al. (
2010a
) suggest a similar approach for the mixing length starting
from Blackadar's (
1962
) principal approach for the mixing length, l
jz
l
¼
ð
3
:
62
Þ
d
jz
g
1
þ
which can be rewritten as
jz
þ
j
ð
d
1
1
l
¼
1
ð
3
:
63
Þ
g
d
Incorporating this approach into the logarithmic profile law (
3.6
) gives:
!
d
d
z
z
i
u
ð
z
Þ¼
u
j
z
z
0
þ
1
d
jz
g
1
1
þ
d
z
z
i
jz
g
ln
ð
3
:
64
Þ
For neutral stability and d = 1, Peña et al. (
2010a
) find for the limiting value of
the length scale in the upper part of the boundary layer g = 39 m; for d = 1.25
they give g = 37 m. The only necessary parameter in (
3.64
) from above the
surface layer is the height of the boundary layer, z
i
. A summarizing paper com-
paring the different approaches (
3.6
), (
3.61
) and (
3.64
) for neutral stratification and
homogeneous terrain has been written by Peña et al. (
2010a
).
With the correction functions for non-neutral thermal stability (
3.15
) and (
3.21
),
the unified vertical wind profile (
3.55
) becomes:
8
<
u
=
j ln
ð
z
=
z
0
Þ
W
m
ð
z
=
L
Þ
ð
Þ
for z\z
p
u
g
ð
sin a
0
þ
cos a
0
Þ
for z
¼
z
p
u
g
1
2
2
p
e
c
ð
z
z
p
Þ
u
ð
z
Þ¼
ð
3
:
65
Þ
:
sin a
0
cos
ð
c
ð
z
z
p
Þþ
p
=
4
a
0
Þ
for z [ z
p
þ
2e
2c
ð
z
z
p
Þ
sin
2
a
0
1
=
2
In the non-neutral case the equations for the friction velocity and the wind
turning angle (
3.56
)-(
3.58
) take the following forms, which now involve correc-
tion functions for the thermal stability of the atmosphere:
u
¼
ju
g
ð
sin a
0
þ
cos a
0
Þ
ln
ð
z
p
=
z
0
Þ
W
m
ð
z
p
=
L
Þ
ð
3
:
66
Þ
cjz
p
sin a
0
u
ð
z
p
=
L
Þ
u
¼
2 u
g
ð
3
:
67
Þ
1
a
0
¼
arctg
ð
3
:
68
Þ
2cz
p
u
ð
z
p
=
L
Þ
1
þ
ln
ð
z
p
=
z
0
Þ
W
m
ð
z
p
=
L
Þ
u
*
and a must be determined by the same iterative procedure as described after
(
3.59
). c still has the form given in (
3.59
), b is set to 16 following Högström
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