Geoscience Reference
In-Depth Information
þ
4
:
7
1
z
z
0
z
L
z
L
[ 0
a
ð
a
1
Þ¼
ln
for
ð
3
:
36
Þ
Now, for stable stratification—in contrast to the neutral stratification above and
to unstable conditions—we have the possibility to define conditions in which
Eqs. (
3.34
) and (
3.36
) can be valid simultaneously. For such a power law profile
which has equal slope and curvature at the height z = z
A
, the following relation
between the Hellmann exponent, a and the stability parameter, z/L
*
must hold:
1
z
L
a
¼
1
1
þ
4
:
7
ð
3
:
37
Þ
In contrast to the neutral case it is possible to find an exponent a for stable
conditions, but this exponent depends on the static stability (expressed by z/L
*
)of
the flow. The possible values for a in the phase space spanned by z/z
0
and z/L
*
can
be found by either equating (
3.34
) and (
3.37
) or by equating (
3.36
) and (
3.37
):
¼
2
þ
z
z
0
1
4
:
7
ln
ð
3
:
38
Þ
z
L
Figure
3.6
illustrates the solutions from Eqs. (
3.33
)to(
3.36
), and (
3.38
). An
evaluation of (
3.38
) demonstrates that the stability of the atmosphere must increase
with increasing roughness and decreasing anemometer height in order to find a
power law profile with the same slope and curvature as the logarithmic profile. The
curved thin lines from the lower left to the upper right represent the solution of
Eqs. (
3.33
) and (
3.34
), the lines with the maximum just left of z/L
*
= 0 the
solution of Eqs. (
3.35
) and (
3.36
) (please note that the lowest line is the one for
a = 0.5, and that the lines for a = 0.3 and a = 0.7 are identical), and the thick
line marks the solution of (
3.38
). As designed, the thick curve goes through the
points where solutions from (
3.34
) and (
3.36
) are identical.
Figure
3.7
displays three examples of wind profiles for non-neutral stratification,
one for unstable conditions and a large roughness length, one which lies exactly on
the curve from Eq. (
3.38
) so that slope and curvature coincide simultaneously, and
one for very stable conditions. For a roughness length of z
0
= 0.023 m (z/
z
0
= 2,173) and a Obukhov length of L
*
= 1,500 m (z/L
*
= 0.0333) a power law
profile with a = 0.15 has equal slope and curvature at z = z
A
= 50 m as the log-
arithmic profile. At z = 100 m the two profiles only differ by 0.1 %, at 10 m by
0.9 %. This is an even better fit than the fit for the neutral wind profile with z/
z
0
= 5,000 in Fig.
3.4
. For the two profiles under unstable conditions the respective
deviations at 100 m and at 10 m are 4.5 and 89.9 %, for the two profiles under very
stable conditions these deviations are -3.5 and -14.0 %.
This extension of Sedefian's (
1980
) analysis has shown that only for certain
conditions in stably stratified boundary-layer flow is it possible to find a power law
profile that has the same slope and curvature as a logarithmic wind profile and thus
fits the logarithmic profile almost perfectly over a wide height range. In a purely
Search WWH ::
Custom Search