THE NEAR-SURFACE LAYER OF THE OCEAN
,
M
g
M
g
u
v
K
fv fv
K
fu
fu
z
z
z
z
w
w
w
w
§
·
§
·
¨
¸
¨
¸
w
w
w
w
©
¹
©
¹
.
(1.134)
The classical Ekman theory assumes that K M does not vary with z, which is a
strong constraint.
The boundary conditions for the atmospheric boundary layer:
0
u v
at z = 0,
(1.135)
and
g
u u
|
and
0
v
as z o f .
(1.136)
Condition (1.136) means (without loss of generality) that the geostrophic
wind vector G
G
is directed along the x-axis. In this case the magnitude of the
geostrophic wind,
1/ 2
2
2
g
g
g
G u v
u
.
(1.137)
When the atmospheric boundary layer is in contact with a moving ocean
surface, the atmospheric surface velocity is not exactly zero as stated in
(1.135). It is, however, generally small relative to atmospheric geostrophic
velocities, so (1.135) can be taken as a fairly good approximation. As stated
in equation (1.136), as z increases indefinitely, the velocity will approach the
geostrophic velocity, although it is only necessary to state that the velocity is
bounded as z o f .
Solutions to equations (1.134) with boundary conditions (1.135) and
(1.136) are
1
E z
E
u G
e cos
z
J
J
ª
º
¬
¼ ,
(1.138)
sin
E z
E
v Ge
z
J
J
,
(1.139)
where
1/ 2
2 /
E
m
K f
J
. The velocity vectors described by solution (1.138)
(1.139) are shown in Figure 1-16a as a function of height. The tip of the
vectors traces out a spiral, which is known as the Ekman spiral.
For constant K M , the geostrophic wind vector is aligned 45
o (counter)
clockwise from the surface stress vector in the northern (southern)
56