Geoscience Reference
In-Depth Information
where
z
r
is a reference height and
a
is the power law exponent (sometimes called
“Hellmann exponent”).
a
depends on surface roughness and the thermal stability
of the Prandtl layer. (A comparison of eqs. (
2.5
) and (
2.5a
)isdiscussedinEmeis
(
2005
).) The wind speed increases with height without a turning of the wind direc-
tion. A scale analysis gives for the height of the Prandtl layer
z
p
(Kraus
2008
):
0.00064
v
g
f
0.01
u
f
z
p
≈
≈
.
(2.6)
Putting in numbers (
u
∗
=
=
/
=
0.5 m/s,
f
0.0001 1
s
,
v
g
8 m/s) gives a typical
height of the Prandtl layer of 50 m.
In a well-mixed Prandtl layer, the temperature
T
is decreasing with height accord-
ing to the adiabatic lapse rate,
g/c
p
(
g
is gravity acceleration,
c
p
is specific heat at
constant pressure). The potential temperature,
p
)
R
/
c
p
is constant with
=
T
(
p
0
/
height (
R
is the gas constant for dry air).
The standard deviations of the three velocity components (
u
denotes the stream-
wise component,
v
crosswise,
w
vertical) are independent of height and scale with
the friction velocity (Stull
1988
,Arya
1995
):
σ
u
u
σ
v
u
σ
w
u
≈
2.5;
≈
1.9;
≈
1.3.
(2.7)
∗
∗
∗
The turbulence intensity in the streamwise direction decreases with height due to
the increase of the mean wind speed with height. By inserting the left-hand relation
from eq. (
2.7
) into eq. (
2.5
), we get (Wieringa
1973
)
σ
u
u
(
z
)
=
1
ln(
z
z
0
)
.
(2.8)
/
2.2.1.2 Ekman Layer
The Ekman layer covers the major part of the ABL above the Prandtl layer. If the
simplifying assumption is made that the height-dependent growth of the exchange
coefficient
K
M
stops at the top of the Prandtl layer and that it is vertically constant
within the Ekman layer, we can solve eqs. (
2.1
) and (
2.2
) analytically for the wind
profile in the Ekman layer:
u
(
z
)
=
u
g
(1
−
exp(
−
γ
z
) cos
γ
z
),
(2.9)
v
(
z
)
=
u
g
exp(
−
γ
z
)sin
γ
z
,
(2.10)
√
f
γ
=
/
where
(2
K
M
). Equation (
2.9
) describes the well-known “Ekman spiral.”
The height of the Ekman layer is estimated by
z
E
=
π
γ
(2.11)
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