Geoscience Reference
In-Depth Information
u
3
M2
3ph
@F
@
agQ
i
2c
p
4ek
b
0
^
t
¼
:
ð
6
:
16
Þ
Stratification will only develop if
@F
/
@
t
>
0, i.e. if
u
3
M2
3ph
agQ
i
2c
p
>
4ek
b
0
^
ð
:
Þ
6
17
which we can re-arrange to give:
>
3pag
8c
p
0
k
b
e
Q
i
h
^
1
:
ð
6
:
18
Þ
u
3
M2
This is the condition for the development of seasonal stratification. If it is not satisfied, the
water column will remain vertically mixed. When
0 there is an exact balance
between heating and stirring and the left side of
Equation (6.18)
is unity. This condition
marks the position of tidal mixing fronts which are the transitional zones between regions
which develop seasonal stratification and those which remain permanently mixed. Note
that all the terms outside the bracket expression in
Equation (6.18)
are essentially
constants (e by hypothesis) so that the control of stratification rests with the term
Q
i
h
@F
/
@
t
¼
u
3
M2
. For a particular region of the shelf seas, Q
i
may be regarded as approximately
spatially uniform, so that the positions of the tidal mixing fronts are determined simply
by the parameter h
=^
u
3
M2
(Simpson and Hunter,
1974
). Because of the large variations in
=^
u
3
M2
encountered in many shelf sea areas extends over four
or more decades. In general, maps of this parameter are frequently presented in the form
of log
10
(h/u
3
), with u being some measure of the strength of the tidal currents (see
Box 6.1
).
u
3
M2
,therangeofh
depth and
^
=^
3
Box 6.1 Different options for
in
/
u
h
u
^
So far, we have utilized the depth mean M
2
amplitude,
u
M2
, but other measures of the
tidal current speed may be used. For instance, the surface amplitude of the current at
spring tides, u
s
, is convenient as it is given in tidal atlases. Typically the surface
current is 15% greater than the depth mean, and for the northwest European shelf
the ratio of amplitude at spring tides to the M
2
amplitude is close to 9/7 so that:
^
^
u
s
¼
9
=
1
:
15
u
M2
¼
1
:
47
u
M2
7
which means that:
¼
u
s
u
3
M2
47
3
u
3
M2
log
10
h
=
log
10
h
=^
log
10
ð
1
:
Þ¼
log
10
h
=^
0
:
50
:
Another opt
ion
is to use the average over the tidal cycle of the cubed depth mean
current, i.e.
^
u
3
which is a quantity readily available from numerical models. This
option implies a further adjustment by log
10
jj
ðÞ
4
=
so that
3
¼
þ
þ
h
u
3
M2
u
s
=
^
=^
:
¼
=
:
:
log
10
u
3
log
10
h
0
37
log
10
h
0
87
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