Geography Reference
In-Depth Information
It can be shown that
√
i
i) /
√
2. Using this relationship and applying the
boundary conditions of (5.28), we find that for the Northern Hemisphere (f > 0),
A
=
(1
+
=
0 and B
=−
u
g.
Thus
u
g
exp
−
i) z
+
u
+
iv
=−
γ (1
+
u
g
(f/2K
m
)
1/2
.
Applying the Euler formula exp(
where γ
=
isin θ and separating the real
from the imaginary part, we obtain for the Northern Hemisphere
−
iθ)
=
cos θ
−
u
g
1
e
−
γz
cos γz
, v
u
g
e
−
γz
sin γz
u
=
−
=
(5.31)
This solution is the famous Ekman spiral named for the Swedish oceanographer
V. W. Ekman, who first derived an analogous solution for the surface wind drift
current in the ocean. The structure of the solution (5.31) is best illustrated by a
hodograph as shown in Fig. 5.4, where the zonal and meridional components of
the wind are plotted as functions of height. The heavy solid curve traced out on
Fig. 5.4 connects all the points corresponding to values of u and v in (5.31) for
values of γzincreasing as one moves away from the origin along the spiral. Arrows
show the velocities for various values of γz marked at the arrow points. When
z
=
π/γ , the wind is parallel to and nearly equal to the geostrophic value. It is
conventional to designate this level as the top of the Ekman layer and to define the
layer depth as De
π/γ .
Observations indicate that the wind approaches its geostrophic value at about
1 km above the surface. Letting De
≡
10
−
4
s
−
1
, the definition
=
1 km and f
=
5m
2
s
−
1
. Referring back to (5.25) we see that for a
characteristic boundary layer velocity shear of
of γ implies that K
m
≈
10
−
3
s
−
1
, this value
of K
m
implies a mixing length of about 30 m, which is small compared to the depth
of the boundary layer, as it should be if the mixing length concept is to be useful.
Qualitatively the most striking feature of the Ekman layer solution is that, like
the mixed layer solution of Section 5.3.1, it has a boundary layer wind component
|
δ
V
/δz
|∼
5
×
Fig. 5.4
Hodograph of wind components in the Ekman spiral solution. Arrows show velocity vectors
for several levels in the Ekman layer, whereas the spiral curve traces out the velocity variation
as a function of height. Points labeled on the spiral show the values of γz, which is a
nondimensional measure of height.
