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limited by frictional forces at the seabed which come into balance with the applied
surface stress.
Two-layer model of upwelling and downwelling (the baroclinic response)
Our one-layer model assumes that the density is uniform and that the currents are
barotropic. In order to see how the Ekman transport drives upwelling and down-
welling motions near the coastal boundary, which will be important in addressing
exchange between the ocean and the shelf in
Chapter 10
, we require a similar, slightly
more elaborate model with two layers; a lighter warmer layer overlying a cold denser
layer, as illustrated in
Fig. 3.10b
. Each layer has depth-uniform velocity and density
and the interface between them is horizontal, at time t
0, when the wind stress is
switched on. The linearised dynamical equations for the top and bottom layers, again
assuming the alongshore terms involving
¼
@
/
@
y are zero, take the form:
upper layer :
@
u
1
@
g
@
@
@
v
1
@
t
w
0
h
1
t
¼
fv
1
x
;
t
¼
fu
1
þ
ð
3
:
52
Þ
@
u
2
@
g
@
@
g
0
@
B
@
v
2
@
lower layer :
t
¼
fv
2
x
x
;
t
¼
fu
2
@
where B is the upward displacement of the interface between the layers and g
0
¼
(
D
r/r
0
)g
is the reduced gravity associated with the density difference
r between the layers.
The surface pressure gradient can be eliminated by taking the difference between the
corresponding equations for layers 1 and 2 to give:
D
@~
u
g
0
@
B
@~
v
t
w
0
h
1
t
¼
f
v
~
þ
x
;
t
¼
f
~
u
þ
ð
3
:
53
Þ
@
@
@
~
where
u
~
¼
u
1
u
2
and
v
¼
v
1
v
2
. The continuity equations for the two layers can be
written as:
@
B
h
1
@
u
1
@
t
þ
x
¼
0
@
ð
3
:
54
Þ
@
B
h
2
@
u
2
@
t
þ
x
¼
0
@
which can be combined to give:
@
t
þ
@
~
h
1
þ
h
2
h
1
h
2
B
u
x
¼
0
ð
3
:
55
Þ
@
@
The solution to
Equations (3.53)
and
(3.55)
is then (see Gill,
1982
, p. 404):
s
g
0
h
1
h
2
h
1
þ
c
0
f
¼
t
w
1
f
e
x
=
Ro
0
Ro
0
¼
u
~
¼
0
h
1
ð
1
Þ;
f
h
2
ð
3
:
56
Þ
t
w
t
w
fRo
0
0
h
1
te
x
=
Ro
0
0
0
g
0
h
1
te
x
=
R
0
0
v
~
¼
;
B
¼
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