Geoscience Reference
In-Depth Information
where we have introduced another, this time inverse, length scale, c which depends
on the Coriolis parameter and the turbulent viscosity, K
M
:
r
f
2K
M
c
¼
ð
3
:
51
Þ
Usually, the top height of the Ekman layer z
g
is estimated from this inverse
length scale by:
z
g
¼
p
c
ð
3
:
52
Þ
Equation (
3.50
) can be mathematically simplified if the height z is small
compared to the length scale 1/c. Then the cosine-function in (
3.50
) is close to
unity and we get:
1
2e
cz
þ
e
2cz
u
2
ð
z
Þ¼
u
g
ð
3
:
53
Þ
and after taking the square root we end up with:
u
ð
z
Þ¼
u
g
1
e
cz
ð
Þ
ð
3
:
54
Þ
The simplified Eq. (
3.54
) describes an exponential approach with height of the
wind speed u(z) from lower wind speed values within the Ekman layer to the
geostrophic wind speed u
g
above the Ekman layer, while the full Eq. (
3.50
)
describes this approach as well, but including a small oscillation of the wind speed
around the geostrophic value near the top of the Ekman layer. The full Eq. (
3.50
)is
usually preferable, because the simplified Eq. (
3.54
) gives wind speed values
which are—compared to (
3.50
)—by 1/H2 too low close to the ground. Equation
(
3.54
) is introduced here, because it has been used in some examples shown in
Sects. 3.4.1
and
4.2.4
. Generally, neither Eq. (
3.50
) nor (
3.54
) should be extrap-
olated down into the surface layer. Profile relations which are valid over the
surface layer and the Ekman layer are derived in the next section.
The vertical profile of the standard deviations of the wind components as the
major turbulence parameter has already been given above in Eq. (
3.19
).
3.2.4 Unified Description of the Wind Profile
for the Boundary Layer
For many purposes, especially in those situations where the hub height is close to the
top of the surface layer and the rotor area of a wind turbine cuts through the surface
layer and the Ekman layer above, a unified description of the wind profile for the
entire lower part of the ABL is desirable, which is valid in both layers. Due to the
assumption of the constant exchange coefficient K
M
in the Ekman layer, the relations
(
3.40
)-(
3.45
) and (
3.50
) cannot be extended from the Ekman layer down into the
Prandtl layer. Likewise, due to the assumption of a mixing length which grows
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