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C
d
a
W
2
u
s
Wm
2
P
W
¼
t
s
u
s
¼
½
ð
:
Þ
6
20
where we have assumed that the stress is related to wind speed W by a quadratic drag
law with a drag coefficient C
d
(
1.3 kg m
3
)is
the density of air. If we further assume that surface water velocity is a fixed fraction
of the wind speed (u
s
¼
0.0012, but see
Section 2.4
) and r
a
(
g
s
W, with g
s
typically about 0.02), we then have for the
stirring power input, per unit area, from the wind:
C
d
a
W
2
P
W
¼
s
W
C
d
s
a
W
3
¼
ð
6
:
21
Þ
k
s
a
W
3
¼
where k
s
¼
C
d
g
s
. Inserting P
w
into
Equation (6.14)
gives:
3
@F
@
agQ
i
2c
p
ek
b
0
j^
u
j
e
s
k
s
a
W
3
h
t
¼
ð
6
:
22
Þ
h
where e
s
is an efficiency of wind mixing analogous to the tidal stirring efficiency e.
The condition for the frontal transition (
@F
/
@
t
¼
0) then becomes:
3
agQ
i
2c
p
¼
ek
b
0
j^
u
j
e
s
k
s
a
W
3
h
þ
:
ð
6
:
23
Þ
h
This relation gives a rather better fit than Equation (6.18) to observed frontal
positions (Bowers and Simpson,
1987
) although the good fit of tidal stirring alone
does not leave a great deal of room for improvement. The influence of wind stirring is
seen more convincingly when we use
Equation (6.22)
to predict the evolution of
stratification over the shelf and compare with observations. Integrating forward in
time from the vernal equinox, when Q
i
becomes positive, up to time t
1
, we have
t
1
ð
ð
t
1
0
j^
t
1
ð
ek
b
0
h
e
s
k
s
a
h
ag
2c
p
3
dt
W
3
dt
F
1
¼
Q
i
dt
u
j
ð
6
:
24
Þ
0
0
!
t
1
t
1
u
3
M2
h
agQ
1
2c
p
4ek
b
0
3p
^
e
s
k
s
a
W
3
h
where Q
1
is the total heat input up to time t
1
. The first term on the right of
Equation
(6.24)
represents the stratification that would occur in the absence of stirring. The second
term is the accumulated tidal stirring approximated, as before, by the cube of the M
2
tidal stream amplitude, while the third term is the equivalent wind stirring up to time t
1
.
For a region where Q
1
may be assumed uniform, and amalgamating the constant
quantities, we can write
Equation (6.24)
in the form:
!
:
u
3
M2
h
c
1
^
c
2
W
3
h
F
1
¼
c
0
ð
6
:
25
Þ
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