Geoscience Reference
In-Depth Information
Equating now the right-hand sides of eqs. (
2.13
) and (
2.14
) yields the desired
relation for
α
0
1
α
0
=
arctg
z
0
)
.
(2.15)
1
+
2
γ
z
p
ln(
z
p
/
Equation (
2.15
) still depends on the friction velocity
u
via the definition of
γ
:
∗
f
γ
=
.
(2.16)
2
κ
u
∗
z
p
Thus,
u
∗
has to be determined iteratively starting with a first guess for
u
∗
in
eq. (
2.16
), subsequently computing
α
0
from eq. (
2.15
), and then re-computing
u
∗
from eqs. (
2.13
)or(
2.14
). Inversely, the system of eqs. (
2.13
)to(
2.16
) can be used
to determine
z
p
if
u
∗
is known. Application of eq. (
2.12
) needs the knowledge of
three internal parameters.
The second idea, to modify the dependence of the mixing length with height,
has been proposed by Gryning et al. (
2007
). They reformulated the mixing length
l
(
L
L
in the Prandtl layer) in order to limit its growth with height and thus to
extend the validity of eq. (
2.5
) to above the surface layer. They choose
=
κ
z
=
1
l
=
1
L
L
+
1
L
M
+
1
L
U
.
(2.17)
A modified mixing length is formed in eq. (
2.17
) by introducing a length scale
for the middle part of the boundary layer,
L
M
=
55)
−
1
and
u
∗
/
f
(
−
2ln(
u
∗
/
fz
0
))
+
a length scale for the upper part,
L
U
=
(
z
i
−
z
). This results in the following wind
profile alternative to eq. (
2.5
)oreq.(
2.12
):
ln
.
u
z
z
0
+
z
L
M
−
z
z
i
z
2
L
M
∗
κ
u
(
z
)
=
(2.18)
Peña et al. (
2009
) suggest a similar approach starting from Blackadar's (
1962
)
principal approach for the mixing length,
l
κ
z
l
=
d
,
(2.19)
κ
z
η
1
+
which can be rewritten as
d
−
1
1
l
=
1
κ
z
+
(κ
z
)
,
(2.20)
d
η
and incorporated this approach into the logarithmic profile law giving
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