Geography Reference
In-Depth Information
directed toward lower pressure. As in the mixed layer case, this is a direct result of
the three-way balance among the pressure gradient force, the Coriolis force, and
the turbulent drag.
The ideal Ekman layer discussed here is rarely, if ever, observed in the atmo-
spheric boundary layer. Partly this is because turbulent momentum fluxes are usu-
ally not simply proportional to the gradient of the mean momentum. However,
even if the flux-gradient model were correct, it still would not be proper to assume
a constant eddy viscosity coefficient, as in reality K
m
must vary rapidly with height
near the ground. Thus, the Ekman layer solution should not be carried all the way
to the surface.
5.3.5
The Surface Layer
Some of the inadequacies of the Ekman layer model can be overcome if we dis-
tinguish a
surface layer
from the remainder of the planetary boundary layer. The
surface layer, whose depth depends on stability, but is usually less than 10% of the
total boundary layer depth, is maintained entirely by vertical momentum transfer
by the turbulent eddies; it is not directly dependent on the Coriolis or pressure gra-
dient forces. Analysis is facilitated by supposing that the wind close to the surface
is directed parallel to the x axis. The kinematic turbulent momentum flux can then
be expressed in terms of a
friction velocity
, u
, which is defined as
4
∗
(u
w
)
s
u
2
∗
≡
Measurements indicate that the magnitude of the surface momentum flux is of
the order 0.1 m
2
s
−
2
. Thus the friction velocity is typically of the order 0.3ms
−
1
.
According to the scale analysis in Section 2.4, the Coriolis and pressure gradient
force terms in (5.16) have magnitudes of about 10
−
3
ms
−
2
in midlatitudes. The
momentum flux divergence in the surface layer cannot exceed this magnitude or
it would be unbalanced. Thus, it is necessary that
δ
u
2
∗
10
−
3
ms
−
2
≤
δz
10
−
2
m
2
s
−
2
, so that the change in the
vertical momentum flux in the lowest 10 m of the atmosphere is less than 10% of
the surface flux. To a first approximation it is then permissible to assume that in
the lowest several meters of the atmosphere the turbulent flux remains constant at
its surface value, so that with the aid of (5.25)
10 m, this implies that δ(u
2
∗
For δz
=
)
≤
¯
K
m
∂
u
∂z
=
u
2
(5.32)
∗
4
Thus the surface eddy stress (see footnote 3) is equal to ρ
0
u
2
.
∗
