Geoscience Reference
In-Depth Information
Fig. 3.4 Dependence of the
power law exponent a on
height and surface roughness
following (
3.27
)(dashed line)
and (
3.28
)(full line)
z
z
0
a
¼
ln
1
ð
3
:
27
Þ
which equals the formulation given by Sedefian (
1980
) in the limit of neutral
stratification. Comparison with the definition of the turbulence intensity (
3.10
)
reveals that the exponent a is equal to the turbulence intensity for neutral strati-
fication. This means, that a logarithmic wind profile and a power law profile have
the same slope at a given height if the power law exponent equals the turbulence
intensity at this height. Equation (
3.27
) is plotted in Fig.
3.4
.
Equation (
3.27
) implies that the exponent a decreases with height for a given
roughness length z
0
. The height in which the slopes of the two wind profiles (
3.6
)
and (
3.22
) should be equal—this is usually the anemometer height z = z
A
—has
therefore to be specified a priori. The dependence of the power law exponent a on
height is stronger the smaller the ratio z/z
0
is (see Fig.
3.4
). Due to this fact, the
dependence of the exponent a on height is stronger for complex terrain where the
roughness length z
0
is large and it can nearly be neglected for water surfaces with
very small roughness lengths.
In order to see whether we can find an exponent a so that both the slope and the
curvature agree in a given height we must equate the formulas (
3.24
) and (
3.26
) for
the curvature of the two profiles. This yields a second relation between the
Hellmann exponent and the surface roughness length:
z
z
0
a
ð
a
1
Þ¼
ln
1
ð
3
:
28
Þ
For low heights over rough surfaces with z/z
0
\ 54.6 Eq. (
3.28
) has no solution
at all (see the full line in Figure
3.4
). For z/z
0
= 54.6 it has one solution (a = 0.5)
and for greater heights over smoother surfaces with z/z
0
[ 54.6 it has two solutions
of which we always choose the smaller one. This solution approaches the solution
of Eq. (
3.27
) asymptotically as z/z
0
tends to infinity (for very smooth surfaces such
as still water surfaces). Therefore, a power law with equal slope and curvature as
the logarithmic profile can only exist in the limit for perfectly smooth surfaces
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