Geography Reference
In-Depth Information
where we have parameterized the surface momentum flux in terms of the eddy
viscosity coefficient. In applying K
m
in the Ekman layer solution, we assumed a
constant value throughout the boundary layer. Near the surface, however, the verti-
cal eddy scale is limited by the distance to the surface. Thus, a logical choice for the
mixing length is
u
∂z
.
Substituting this expression into (5.32) and taking the square root of the result
gives
(kz)
2
∂
=
kz where k is a constant. In that case K
m
=
¯
u
∂z
∂
¯
=
u
/(kz)
(5.33)
∗
Integrating with respect to z yields the
logarithmic wind profile
=
u
k
ln
z
z
0
u
¯
(5.34)
∗
¯
=
where z
0
, the
roughness length
, is a constant of integration chosen so that
u
0
=
at z
z
0
. The constant k in (5.34) is a universal constant called
von Karman's
constant
, which has an experimentally determined value of k
0.4. The roughness
length z
0
varies widely depending on the physical characteristics of the surface.
For grassy fields, typical values are in the range of 1
≈
4 cm. Although a number of
assumptions are required in the derivation of (5.34), many experimental programs
have shown that the logarithmic profile provides a generally satisfactory fit to
observed wind profiles in the surface layer.
−
5.3.6
The Modified Ekman Layer
As pointed out earlier, the Ekman layer solution is not applicable in the surface
layer. A more satisfactory representation for the planetary boundary layer can be
obtained by combining the logarithmic surface layer profile with the Ekman spiral.
In this approach the eddy viscosity coefficient is again treated as a constant, but
(5.29) is applied only to the region above the surface layer and the velocity and
shear at the bottom of the Ekman layer are matched to those at the top of the
surface layer. The resulting modified Ekman spiral provides a somewhat better fit
to observations than the classical Ekman spiral. However, observed winds in the
planetary boundary layer generally deviate substantially from the spiral pattern.
Both transience and baroclinic effects (i.e., vertical shear of the geostrophic wind in
the boundary layer) may cause deviations from the Ekman solution. However, even
in steady-state barotropic situations with near neutral static stability, the Ekman
spiral is seldom observed.
It turns out that the Ekman layer wind profile is generally unstable for a neutrally
buoyant atmosphere. The circulations that develop as a result of this instability have
horizontal and vertical scales comparable to the depth of the boundary layer. Thus,
it is not possible to parameterize them by a simple flux-gradient relationship. How-
ever, these circulations do in general transport considerable momentum vertically.
The net result is usually to decrease the angle between the boundary layer wind
