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with the convective velocity scale,
gz
i
w
.
w
∗
=
(2.37)
This means that the standard deviation of the vertical velocity component
increases with height in the unstable Prandtl layer and then stay constant above it.
2.2.3 Stable Boundary Layer
This type of ABL, which is characterized by an downward (positive) surface heat
flux (
L
0) and a stable thermal stratification of the air, is usually found at night-
time, over waters that are colder than the air above, and over ice and snow-covered
surfaces. For positive values of
z
∗
>
, the correction function for eq. (
2.26
) reads
(Businger et al.
1971
, Dyer
1974
, Högström
1988
, Lange and Focken
2006
)
/
L
∗
⎧
⎨
−
az
/
L
∗
for 0
<
z
/
L
∗
≤
0.5
m
(
z
/
L
∗
)
=
,
(2.38)
/
L
∗
+
/
L
∗
−
/
⎩
Az
B
(
z
C
D
)
for 0.5
≤
z
/
L
∗
≤
7
−
/
+
/
exp(
Dz
L
∗
)
BC
D
where
a
=
5,
A
=
1,
B
=
2
/
3,
C
=
5, and
D
=
0.35. Then the wind profile
u(z)
,
u
∗
,
and
α
can be computed again from eqs. (
2.27
)to(
2.30
), with
ϕ
(
z
/
L
∗
)
=
1
+
az/L
*
.
(2.39)
The temperature in the stable boundary layer vertically decreases less than the
adiabatic lapse rate; the potential temperature increases with height. The standard
deviations of the wind components are usually assumed to be constant with height
in the same way as described by eq. (
2.7
) for the neutral ABL (Arya
1995
).
Under suitable weather conditions (anticyclonic, sufficient large-scale pressure
gradient, clear skies), a low-level jet can form at the top of a nocturnal SBL. When
the atmosphere stabilizes rapidly after sunset, the vertical turbulent momentum flux
decreases rapidly and the wind-braking effect of the rough surface is no longer
communicated to the air at the top of the nocturnal SBL. This rapid change in the
equilibrium of forces leads to an inertial oscillation. Prominent features of this oscil-
lation are a speed-up of the wind speed in the first half of the night and a turning
wind direction throughout the night. The phenomenon ends with the restoration
of the vertical turbulent momentum flux after sunrise. Following Jacobi and Roth
(
1995
), this inertial oscillation of the wind components is described by:
∂
u
t
=−
f
(
v
g
−
v
),
(2.40)
∂
∂
v
t
=
f
(
u
g
−
u
).
(2.41)
∂
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