Geoscience Reference
In-Depth Information
2.2.1.3 Unified Description of the Wind Profile for the Boundary Layer
For many purposes, a unified description of the wind profile for the lower part of
the ABL is desirable, which is valid beyond the surface layer. Due to the assump-
tion of the constant exchange coefficient
K
M
, the relations (
2.9
) and (
2.10
) cannot
be extended from the Ekman layer down into the Prandtl layer. Likewise, due to the
assumption of a mixing length, which grows linearly with height, the relation (
2.5
)
cannot be extended into the Ekman layer. Therefore, two approaches have been
tested to overcome this problem. The first idea is to fit the Prandtl and Ekman pro-
files in such a way together that there is a smooth transition in terms of wind speed
and wind shear between both regimes. The second idea is to modify the mixing
length in order to extrapolate the Prandtl layer wind profile into higher layers.
Etling (
2002
) has proposed the first idea by presenting a wind profile description
with a linearly increasing
K
M
below the Prandtl layer height,
z
p
, and a constant
K
M
above this height:
⎧
⎨
u
∗
/κ
ln(
z
/
z
0
) r
z
<
z
p
−
α
0
+
α
0
) r
z
=
u
(
z
)
=
u
g
(
sin
cos
z
p
(2.12)
⎩
2
√
2exp(
u
g
[1
−
−
γ
(
z
−
z
p
))
sin
α
0
cos(
γ
(
z
−
z
p
)
+
π/
4
−
α
0
)
for
z
>
z
p
z
p
)) sin
2
α
0
]
1
/
2
+
2exp(
−
2
γ
(
z
−
Equation (
2.12
) depends on five parameters: the surface roughness
z
0
,the
geostrophic wind speed
u
g
, the height of the Prandtl layer
z
p
, the friction veloc-
ity
u
α
0
.The
two variables
z
0
and
u
g
are external parameters, the other three of them are internal
parameters of the boundary layer. If a fixed value is chosen for
z
p
, then two further
equations are needed to determine
u
∗
and
, and the angle between the surface wind and the geostrophic wind
∗
α
0
. Equation (
2.12
) describes a smooth
transition of wind speed from the Prandtl layer to the Ekman layer.
Deviating from Etling (
2002
), these equations should be generated from the more
realistic physical requirement that both the wind speed as well as the wind shear are
continuous in the height
z
z
p
(Emeis et al.
2007
). Equating the first two equations
of the wind profile, equation (
2.12
)for
z
=
=
z
p
gives
u
∗
=
κ
u
g
(
−
sin
α
0
+
cos
α
0
)
,
(2.13)
ln(
z
p
/
z
0
)
and from equating the respective shear equations in the same height
z
=
z
p
, we get
a second equation for
u
∗
2
u
g
γκ
u
∗
=
z
p
sin
α
0
.
(2.14)
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