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direction. The atmospheric stability in this analytical approach is limited to neutral
conditions.
A different approach which divides the flow field in three layers has been
developed by Sykes (
1980
). He distinguished the following layers: a very thin wall
layer, a Reynolds-stress sublayer across which the Reynolds stresses vary rapidly,
and an outer layer. The flow perturbations due to the presence of the hill are
calculated for different orders of the slope e
1/2
= H/L (e
1). The height of the
Reynolds-stress sublayer is of the order eL. For an aspect ratio of H/L = 0.1 this is
quite close to the inner layer height from (
4.9b
).
4.2.3 Modifications to the Potential Flow: Consideration
of Thermal Stability
As a preparation we rewrite Eq. (
4.1
) in terms of the fractional speed-up:
ð
4
:
18
Þ
Ds
ð
x
;
z
Þ¼
Du
pot
ð
x
;
z
Þ
u
1
ð
l
Þ
¼
u
1
ð
L
Þ
u
1
ð
l
Þ
H
L
r
L
;
z
x
L
Bradley (
1983
) studied the dependence of the fractional speedup ratio on sta-
bility. As a first approximation, Bradley assumed that Eq. (
4.18
) is still valid for
non-neutral flow as long as buoyancy forces are small compared to pressure
gradient forces. Then (
4.18
) is approximately valid but the velocities u
?
(L) and
non-neutral stratification, one obtains:
ln
z
0
W
L
L
H
L
r
L
;
z
x
Ds
ð
x
;
z
Þ¼
ð
4
:
19
Þ
L
l
l
L
ln
z
0
W
to Eq. (31) in Frank et al. (
1993
). Ds increases with increasing stability and is
reduced with unstable flow (Fig.
4.7
). This becomes also intuitively clear, because
increasing stability opposes to the vertical displacement of the streamlines over the
hill. Thus, the streamlines are squeezed together and the speed-up is increased.
Evidence from real data is depicted, e.g., in Fig. 2 of Frank et al. (
1993
).
4.2.4 Weibull Parameters over a Hill
SODAR measurements on a hill top have been evaluated in Emeis (
2001
) to derive
vertical profiles of the two Weibull parameters over a hill. The form parameter is
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