Geoscience Reference
In-Depth Information
where a is the volume expansion coefficient of seawater. We can think of this lower
density layer as floating on the surface with a buoyancy force of:
ag
Q
c p h 1
Nm 3
b
¼
g
¼
½
ð
6
:
6
Þ
which implies a total force on the layer per unit area of:
ag
Q
Nm 2
B
¼
bh 1 ¼
½
:
ð
6
:
7
Þ
c p
Now suppose the water column is being mixed by mechanical stirring. How much
work must be done against the buoyancy force in order to completely mix the
stratification? As the density becomes uniform, the density deficit is distributed over
the full depth h. You can think of this as being the same as moving the buoyant force
down by a distance
h/2 on average, so the work that needs to be done is Bh/2 or
ag
Qh
2c p
Jm 2
PE
¼
h
F ¼
½
:
ð
6
:
8
Þ
If the heat is supplied in a short time interval
D
t at a rate Q i then we have for the
change of
F
:
agh
2c p Q i
Jm 2
h
F ¼
PE
¼
t
½
ð
6
:
9
Þ
or as
D
t
!
0, we can write the time derivative of
F
as:
heat ¼
@F
@
agQ i
2c p
Wm 3
½
:
ð
6
:
10
Þ
t
This is the rate at which
increases for heating in the absence of stirring. The heat
flux Q i is the net flux of heat across the sea surface, as we described in Section 2.2.4 .
In order to maintain vertical homogeneity (
F
F ¼
0), the stirring must provide at least
this amount of power.
Under our present assumptions, the power must come from the tidal flow, so the
next step is to develop an appropriate representation of tidal energy input. We saw in
Section 4.4.1 that for a shear flow u(z) in the x direction, turbulent energy production
occurs at a rate given by
z per unit volume. The total rate of production over
the water column is therefore just:
t
@
u/
@
ð
0
t @
u
Wm 3
P T ¼
z dz
½
ð
6
:
11
Þ
@
h
which can be evaluated if the profiles u(z)andt(z) are known. Otherwise we can
approximate the integral under the simplifying assumptions (i) that there is a uniform
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