Geography Reference
In-Depth Information
ξ
2
∂
V
∂z
=
2
∂
V
∂z
=
where the eddy viscosity is now defined as K
m
and the mixing length,
ξ
2
1/2
≡
is the root mean square parcel displacement, which is a measure of average eddy
size. This result suggests that larger eddies and greater shears induce greater tur-
bulent mixing.
5.3.4 The Ekman Layer
If the flux-gradient approximation is used to represent the turbulent momentum
flux divergence terms in (5.18) and (5.19), and the value of K
m
is taken to be
constant, we obtain the equations of the classical Ekman layer:
∂
2
u
∂z
2
f
v
v
g
=
K
m
+
−
0
(5.26)
∂
2
v
∂z
2
f
u
u
g
=
K
m
−
−
0
(5.27)
where we have omitted the overbars because all fields are Reynolds averaged.
The Ekman layer equations (5.26) and (5.27) can be solved to determine the
height dependence of the departure of the wind field in the boundary layer from
geostrophic balance. In order to keep the analysis as simple as possible, we assume
that these equations apply throughout the depth of the boundary layer. The bound-
ary conditions on u and v then require that both horizontal velocity components
vanish at the ground and approach their geostrophic values far from the ground:
u
=
0,v
=
0atz
=
0
(5.28)
u
→
u
g
, v
→
v
g
as z
→∞
To solve (5.26) and (5.27), we combine them into a single equation by first mul-
tiplying (5.27) by i
1)
1/2
and then adding the result to (5.26) to obtain a
second-order equation in the complex velocity, (u
=
(
−
+
iv):
∂
2
(u
if
u
g
+
iv
g
+
iv)
K
m
−
if (u
+
iv)
=−
(5.29)
∂z
2
For simplicity, we assume that the geostrophic wind is independent of height and
that the flow is oriented so that the geostrophic wind is in the zonal direction
(v
g
=
0). Then the general solution of (5.29) is
A exp
(if /K
m
)
1/2
z
B exp
(if /K
m
)
1/2
z
(u
+
iv)
=
+
−
+
u
g
(5.30)
