Geography Reference
In-Depth Information
v
2
)
1/2
, and the
subscript s denotes surface values (referred to the standard anemometer height).
Observations show that C
d
is of order 1.5
(u
2
where C
d
is a nondimensional
drag coefficient
,
|
V
|=
+
10
-
3
over oceans, but may be several
×
times as large over rough ground.
The approximate planetary boundary layer equations (5.18) and (5.19) can then
be integrated from the surface to the top of the boundary layer at z
=
h to give
u
w
f
¯
v
g
=−
s
h
C
d
V
¯
u
h
v
−¯
=
(5.20)
v
w
f
¯
u
g
=−
s
h
C
d
V
¯
v
h
−
u
−¯
=
(5.21)
Without loss of generality we can choose axes such that v
g
=
0. Then (5.20) and
(5.21) can be rewritten as
κ
s
V
¯
κ
s
V
¯
v
¯
=
u
;
u
¯
=¯
u
g
−
v
;
(5.22)
C
d
(fh) . Thus, in the mixed layer the wind speed is less than
the geostrophic speed and there is a component of motion directed toward lower
pressure (i.e., to the left of the geostrophic wind in the Northern Hemisphere and
to the right in the Southern Hemisphere) whose magnitude depends on κ
s
.For
example, if
where κ
s
≡
10 m s
-
1
0.05 m
-
1
8.28ms
-
1
,
u
g
=
¯
and κ
s
=
s, then
u
¯
=
v
¯
=
3.77
ms
-
1
, and
V
=
9.10 m s
-
1
at all heights within this idealized slab mixed layer.
It is the work done by the flow toward lower pressure that balances the frictional
dissipation at the surface. Because boundary layer turbulence tends to reduce wind
speeds, the turbulent momentum flux terms are often referred to as boundary layer
friction
. It should be kept in mind, however, that the forces involved are due to
turbulence, not molecular viscosity.
Qualitatively, the cross isobar flow in the boundary layer can be understood as
a direct result of the three-way balance among the pressure gradient force, the
Coriolis force, and turbulent drag:
V
V
1
ρ
0
∇
¯
C
d
h
f
k
×
V
=−
p
−
(5.23)
This balance is illustrated in Fig. 5.3. Because the Coriolis force is always normal to
the velocity and the turbulent drag is a retarding force, their sum can exactly balance
the pressure gradient force only if the wind is directed toward lower pressure.
Furthermore, it is easy to see that as the turbulent drag becomes increasingly
dominant, the cross isobar angle must increase.
5.3.2
The Flux-Gradient Theory
In neutrally or stably stratified boundary layers, the wind speed and direction vary
significantly with height. The simple slab model is no longer appropriate; some
