Geoscience Reference
In-Depth Information
9.6.1 The Ekman solution
Ekman
(
1905
) assumed there is a constant eddy viscosity
K
such that
(uw, vw)
=
K
∂
∂z
,
∂
∂z
.
If we align the
x
-axis with the geostrophic wind (which is assumed
not to depend on
z
), then
V
g
−
=
0. With the boundary conditions that the mean
wind vanishes at
z
0 and approaches the geostrophic value
U
g
at very large
z
,
his steady solution to
Eq. (9.22)
is the
Ekman spiral
=
e
−
γz
cos
γz), V
U
g
e
−
γz
sin
γz, γ
(f/
2
K)
1
/
2
.
U
=
U
g
(
1
−
=
=
(9.23)
For small
γz
we have
U
=
V
→
γzU
g
,
whereas for large
γz
(i.e., above the
boundary layer)
U
0
,
so the surface-layer wind is at 45
degrees to the wind above the boundary layer.
This Ekman solution
(9.23)
is not realistic for geophysical boundary layers for
several reasons. First, numerical experiments show that several 1
/f
times - say
on the order of one day - can be required for this neutral boundary layer to reach
a steady state from arbitrary initial conditions. Steady, horizontally homogeneous
conditions persisting that long are rare. Second, observations show that an elevated
inversion, rather than the Ekman dynamics, typically establishes the ABL depth.
Third, the mean horizontal pressure gradient (the geostrophic wind) often varies
significantly with height. Finally, the surface and the effects of stability typically
cause both
u
and
to vary significantly with height, giving
K
→
U
g
,V
→
V
g
=
u
a pronounced
vertical structure. The solution of the mean-momentum equation with
K
∼
con-
stant can be quite unphysical, as can be demonstrated in turbulent Couette flow
(Problem 9.19)
.
Only under stably stratified or baroclinic conditions do we tend to see the Ekman
wind-turning angle of 45 degrees in the ABL. The Ekman solution is a particularly
poor representation of the stress and mean wind profiles under typical convective
conditions.
=
9.6.2 The baroclinic case
Horizontal density gradients can cause the horizontal mean pressure gradient to
depend on height
z
,giving
baroclinity
that can profoundly affect ABL structure.
From the definition
(9.21)
of the geostrophic wind, we see that
1
ρ
0
f
∂U
g
∂z
∂
∂z
∂P
∂y
1
ρ
0
∂
∂y
∂P
∂z
+
1
ρ
0
∂ρ
0
∂z
1
ρ
0
∂P
∂y
=−
=−
(9.24)
fU
g
H
ρ
∂
∂y
1
ρ
0
∂P
∂z
=
−
+
,
