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Wind stress
(a)
(b )
Wind stress
τ
W
τ
W
z
z
V
(
x,t
)
h
1
η
h
h
2
y
y
x
x
Ro
Ro
Figure 3.10
(a) Response of a homogeneous ocean in the northern hemisphere to a coast-
parallel wind stress; (b) Upwelling, in a two-layer ocean in the northern hemisphere, forced
by coast-parallel wind stress.
@^
g
@
@
@
v
@
u
@
t
w
0
h
;
t
¼
f
v
^
x
;
t
¼
f
u
^
þ
ð
3
:
49
Þ
where
v are the depth average velocities. The corresponding, vertically averaged
version of the continuity
Equation (3.3)
is just:
u
^
; ^
@
@
h
@^
u
t
¼
x
:
ð
3
:
50
Þ
@
For a wind stress t
w
switched on at t
0, the solution of
Equations (3.49)
and
(3.50)
takes the form (e.g. see Gill,
1982
, p. 396):
¼
p
gh
t
w
e
x
=
Ro
u
^
¼
0
h
ð
1
Þ;
Ro
¼
f
f
ð
3
:
51
Þ
t
w
t
w
0
h
te
x
=
Ro
0
c
te
x
=
Ro
v
^
¼
;
¼
:
The Ekman transport, initially h
far from the coast, starts to decrease at
an offshore distance determined by the parameter R
o
and becomes zero at the coast
(x
^
u
¼
t
w
=ð
0
f
Þ
0). Ro is an important length scale. It is the Rossby radius of deformation, and
controls the extent of boundary influence. It can be thought of as the distance
travelled in a time of 1/f
¼
g
p
.
This makes intuitive sense, as the wave speed is the fastest that any signal, for
instance a change in wind forcing, can be transmitted through the ocean. In mid-
latitude shelf seas Ro is typically
¼
T
I
/2p by a long wave at the phase speed of c
¼
250 km.
While the shoreward transport is steady, the alongshore current and the surface
elevation grow linearly in time to absorb the onshore flow which is converging at the
coast. The pressure gradient due to the surface slope is also increasing with time and
remains in geostrophic balance with the increasing alongshore flow.
This solution is a good model of what happens to the sea surface elevation after the
onset of a strong wind stress and a storm surge develops at the coast. Of course, sea
level does not go on rising indefinitely because the coastal current is eventually
∼
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