Geoscience Reference
In-Depth Information
8.2.2
Reasons for non-equilibrium: stored buoyancy and inhibited mixing
In practice, frontal positions are displaced much less than the equilibrium prediction,
typically only 2-4 km (Simpson and Bowers,
1979
; Simpson,
1981
). The discrepancy
arises for two reasons. First the adjustment time scale is much longer than the
fortnightly cycle; physically this is because the mixing during the springs phase of
the tidal cycle has to remove buoyancy accumulated during the neaps phase of the tide
as well as mixing the new input of heat. This will slow the rate of adjustment of the
front as spring tides approach, and so reduce the displacement distance of the front
which will move only as far as S (see
Fig 8.2
). Allowing for this, we see that the spring
tide position of the front is set by the condition that the average stirring power over the
spring-neap cycle, P
S
, matches the buoyancy input. We can formulate this condition
and the corresponding one for neaps by neglecting wind mixing and assuming that the
depth-mean tidal stream amplitude varies over the fortnightly cycle according to:
j^
u
j¼^
u
M2
ð
1
þ
r
SM
cos
t
Þ
ð
8
:
3
Þ
where s
2p/T
SN
is the angular frequency for the spring-neap cycle with period T
SN
and r
SM
¼ ^
¼
u
M2
is a constant for a given location.
1
The balance at spring tide
between buoyancy input and the average stirring, which defines the limit of the
retreat of the front, is then given by:
u
S2
=^
Z
T
SN
agQ
i
2c
p
¼
1
T
SN
3
dt
P
S
ð
1
þ
r
SM
cos
t
Þ
ð
8
:
4
Þ
0
3
2
r
SM
¼
P
S
1
þ
:
The furthest advance of the front at neaps will be set by the balance for minimum stirring:
agQ
i
2c
p
¼
3
P
N
ð
1
r
SM
Þ
ð
8
:
5
Þ
where P
N
is the mean stirring power at the neaps frontal position. So for a constant heating
rate and with r
SM
¼
0.3, the ratio of stirring power at the springs and neaps positions will be
þ
3
2
r
SM
P
N
P
S
¼
1
3
¼
3
:
3
:
ð
8
:
6
Þ
ð
1
r
SM
Þ
This result implies much smaller displacement than for equilibrium adjustment
(
0.5) but still exceeds the observed displacement of fronts over the fortnightly
cycle by about a factor of 2.
The second reason for over-estimating the adjustment distance lies with our assump-
tion of a constant mixing efficiency e. This does not allow for the inhibiting of mixing
D
SH
1
For much of the NW European shelf seas, r
SM
0.3
0.4.
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