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stress gradient so that
@
u/
@
z
¼
constant, (ii) t
¼
0 at the surface and u
¼
0 at the
bottom. Under these conditions we have:
3
Wm
2
P
T
¼
t
b
^
u
¼
k
b
0
j^
u
j
½
ð
6
:
12
Þ
where the bottom stress t
b
is related to the depth mean velocity
u by the quadratic
^
drag law t
b
¼
k
b
j^
u
j^
u with a drag coefficient k
b
∼
0.0025 and
j^
u
j
is the modulus
0
u.
1
P
T
is the power input to the turbulent motions which are respon-
sible for stirring the water column and bringing about vertical mixing. Most of the
turbulent energy is ultimately dissipated and hence transformed to heat within the
mixed layer, but some of it is converted to potential energy in the mixing process.
We shall hypothesise that a fixed fraction e of P
T
is used in this way so that the power
available to change the water column potential energy, and hence
(magnitude) of
^
F
, is just
@ð
PE
Þ
h
@F
@
3
Wm
2
ek
b
0
j
^
¼
eP
T
¼
u
j
¼
½
:
ð
6
:
13
Þ
@
t
t
stir
The parameter e can be thought of as an 'efficiency' of mixing. Combining Equations
(
6.10
) and
(6.13)
we have for the net change in
F
due to heating and tidal stirring:
3
@F
@
agQ
i
2c
p
ek
b
0
j^
u
j
Wm
3
t
¼
½
h
ð
6
:
14
Þ
heating stirring
:
Equation (6.14)
represents the essential competition between the promotion of
stratification by surface heating and its erosion by tidal stirring. Note that the heating
term in (6.14) can be either positive (net heat supply) or negative (net heat loss, such
as in winter) as we discussed in
Section 2.2
; this is where the seasonality of the
stratification enters. The stirring term is always negative; i.e. turbulence always acts
to destroy stratification. If Q
i
and
^
u are known,
Equation (6.14)
can be integrated
forward in time to predict the evolution of stratification. The tidal stirring term can
be simplified if we assume that the tidal flow is dominated by the M
2
semi-diurnal
constituent of the form
u
M2
is the depth-mean current
amplitude of the M
2
constituent and o
M2
is the M
2
frequency. It can then be shown
that, averaged over one or more whole tidal cycles:
u
^
¼ ^
u
M2
sin o
M2
t, where
^
4
3p
^
3
u
3
M2
j^
u
j
¼
ð
6
:
15
Þ
so that
Equation (6.14)
becomes:
1
Alternatively, if we consider the motion relative to the water, the stress exerted by the seabed moves
backwards at the near-bed velocity u
b
so the work done on the water is t
b
u
b
¼
k
b
0
u
2
j
u
b
j
which
gives an estimate comparable to that from Equation (
6.12
) but modified by a factor u
b
=
u. Other profiles
of u and t all lead to the u
3
dependence but with different numerical factors.
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