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with the universal constant a
1
= 0.5 and the non-dimensional dissipation rate for
turbulent kinetic energy /
e
= jze/u
3
. Kaimal et al. (
1972
) derive from data of the
Kansas experiment:
2
=
3
z
L
for
2
L
0
¼
1
þ
0
:
5
u
2
=
3
e
ð
3
:
87
Þ
3
=
5
z
L
for 0
L
2
1
þ
2
:
5
For lower frequencies f \ 4(
3.85
) and (
3.86
) depend on z/L
*
as well.
According to Kaimal et al. (
1972
) the shape of these spectra is similar to (
3.74
)to
(
3.76
).
The height dependence of the wave number k
max
of the maximum of the
spectrum that describes the lateral extension of turbulence elements has been
found empirically to be (Schroers et al.
1990
):
k
max
¼
0
:
0028z
0
:
27
ð
3
:
88
Þ
The integral length scale that describes the longitudinal extension of turbulence
elements is found to vary according to (Schroers et al.
1990
):
L
x
ð
z
Þ¼
112
:
3z
0
:
27
ð
3
:
89
Þ
This means that L
x
= 367 m at 80 m and 389 m at 100 m height. L
x
and 1/k
max
are related to each other. L
x
is about one third of 1/k
max
. Schroers et al. (
1990
)
further found L
x
/L
y
= 4.6 at 48 m height and L
x
/L
y
= L
x
/L
z
at 80 m, where L
y
is
the lateral and L
z
the vertical extension of the turbulence elements.
3.4 Diurnal Variation of the Wind Profile
The usual daily changes in the thermal stratification of the atmospheric boundary
layer over land influence vertical wind profiles as well. These wind profiles have
been introduced for stationary conditions in
Sects. 3.1.1.2
and
3.1.1.3
. Non-sta-
tionarity provokes additional features not covered by the stationary wind laws
which go beyond the necessary changes between the differently shaped wind
profiles under different thermal stratification. Over oceans the diurnal cycle is
practically absent due to the high heat capacity of the water. Instead, we find here
an annual cycle. See
Sect. 5.2
for further details.
The diurnal variation is considerably different for near-surface winds and winds
above a certain height which has become known as the ''reversal height'' or
''cross-over height''. Near-surface winds under clear sky conditions behave as
everyone knows from own experience: the wind freshens during daytime and
calms down at night-time. The opposite is occurring above the cross-over height:
wind speed is higher at night-time and decreases during daytime. This feature has
already
been
described
by
Hellmann
(
1915
)
and
Peppler
(
1921
)
from
the
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