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þ
4
:
7
1
u
ð
z
Þ
1
þ
4
:
7
z
L
o
u
oz
¼
ln
z
z
0
z
L
z
L
[ 0
for
ð
3
:
30
Þ
z
where the first Eq. (
3.29
) is valid for unstable stratification and the second Eq.
(
3.30
) for stable stratification. The curvature of the diabatic wind profile (
3.16
)is
given by:
W
1
u
ð
z
Þ
z
2
1
þ
x
o
x
o
2
u
oz
2
¼
ln
z
z
0
z
L
z
L
\0
oz
for
ð
3
:
31
Þ
x
þ
4
:
7
1
u
ð
z
Þ
z
2
2
u
oz
2
¼
ln
o
z
z
0
z
L
z
L
[ 0
for
ð
3
:
32
Þ
where, once again, the first Eq. (
3.31
) is valid for unstable stratification and the
second Eq. (
3.32
) for stable stratification. The expression (z/x) qx/qz in (
3.31
)
equals -3.75 z/L
*
(1/x
4
), where x has been defined after (
3.15
). Slope and cur-
vature of the power law (
3.22
) do not depend explicitly on stratification and thus
remain unchanged. Looking for equal slopes in non-neutrally stratified flow now
requires the investigation of the possible identity of (
3.29
)/(
3.30
) and (
3.25
). We
get:
W
1
1
x
z
z
0
z
L
z
L
\0
a
¼
ln
for
ð
3
:
33
Þ
þ
4
:
7
1
z
z
0
z
L
z
L
z
L
[ 0
a
¼
ln
1
þ
4
:
7
for
ð
3
:
34
Þ
which are exactly the equations found by Sedefian (
1980
) From (
3.33
) and (
3.34
)it
is obvious that a is smaller with unstable stratification than with neutral, but is
larger with stable stratification, because x and the expression in brackets con-
taining z/L
*
are both larger than unity.
We had seen from Fig.
3.5
that the neutral logarithmic profile is always steeper
(in the manner we have plotted the Figure, steeper means that wind speed is
increasing less with height) than a power law profile fitted to it at the height
z = z
A
. As the logarithmic profile for unstable stratification is even steeper than the
one for neutral stratification we do not expect a match with the power law profile
for unstable stratification. But for stable stratification, the slope of the logarithmic
profile is smaller than for neutral conditions and a fit may become possible. We
therefore equate the curvatures from Eqs. (
3.26
) and (
3.31
)/(
3.32
) yielding:
W
1
1
3
:
75
z
L
1
x
4
z
z
0
z
L
z
L
\0
a
ð
a
1
Þ¼
ln
for
ð
3
:
35
Þ
x
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