Geoscience Reference
In-Depth Information
θ
with
β
x
and
β
y
the components of the downslope angle
β
and
the deviation
from
θ
0
.
If we define themeanwind direction as
α
tan
−
1
(V /U)
, then from the evolution
equations (12.20)
for
U
and
V
we can derive that for
α
(Problem 12.18)
:
=
S
2
V
∂uw
+
f
G
S
1
θ
S
2
(Vβ
x
−
Uβ
y
).
(12.21)
dα
dt
=
1
∂z
−
U
∂vw
g
θ
0
cos
γ
−
+
∂z
Here
γ
is the angle between the geostrophic and mean winds. In the Minnesota
experiment the terrain-slope term in
Eq. (12.21)
had become important at
z
4m
by the mid-stages of each of the runs in
Figure 12.2
.
In five of the seven runs its
magnitude reached
f
, which in mid-latitudes corresponds to a rate of turning of
the mean wind direction of 20 degrees per hour
(Problem 12.19)
.
Given that few land surfaces are as level as the Minnesota site, we conclude
that drainage forces can be very important in the nocturnal ABL; they can foster
slope-driven local flows that evolve in time and interact in complicated ways.
=
12.2.4 An approach to treating gravity waves
The nocturnal ABL is also a rich medium for the growth and propagation of internal
gravity waves, which interact with turbulence.
Coulter
(
1990
) has reported acoustic
sounder measurements in which Kelvin-Helmholtz waves and instabilities above
the nocturnal SBL induced a factor of two to four modulation in rms vertical velocity
and other turbulence quantities within the boundary layer.
Figure 12.6
shows an
acoustic sounder record from a late-night SBL with its characteristic evidence of
strong gravity waves.
Finnigan
et al
.
(
1984
) have used the
Reynolds and Hussein
(
1972
) three-part
decomposition of flow variables, mean + turbulence + wave, to isolate the wave-
induced and turbulent motions in the nocturnal ABL and study their interactions.
That decomposition generalizes the ensemble-mean plus fluctuation decomposition
of velocity, pressure, and conserved scalar fields to
u
i
u
i
,
p
w
p
t
,
c
w
c
t
.
u
i
=
˜
U
i
+
+
p
˜
=
P
+
+
c
˜
=
C
+
+
(12.22)
As usual the tilde indicates the full variable and a capital denotes the ensemble-
mean value, but now the superscripts w and t indicate the wave and turbulent parts
of the fluctuation about the ensemble mean.
