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ð
0
h
3
3N
z
h
4
8N
z
g
g
R
w
¼
udz
¼
ð
9
:
8
Þ
h
which means that the surface slope w must be related to x according to
3
8
3N
z
R
w
gh
3
¼
h
ð
:
Þ
9
9
and the velocity profile, which we will write in terms of the fractional depth z
0
¼
z/h,
becomes:
h
3
48N
z
ð
g
3
2
R
w
h
ð
9z
0
2
8z
0
3
z
0
2
u
ðÞ¼
1
Þ
1
Þ
3
2
u
0
ð
ð
9
:
10
Þ
9z
0
2
8z
0
3
z
0
2
¼
u
s
ð
1
Þ
1
Þ
density-driven
river flow
:
the density gradient x with surface velocity seaward of u
s
¼
gxh
3
/48N
z
and an average
R
w
/h due to the river discharge R
w
(in units of m
2
s
1
). The
density-driven component is landwards below z
downstream flow of u
0
¼
¼
0.42h with a maximum speed of
0.69u
s
at z
0.75h. This flow will interact with the density gradient to move lower
salinity water seaward in the surface layer and denser, more saline water landward in
the lower layer and thus induce stratification.
It should be remembered that the form of the estuarine circulation which we have
derived applies to estuaries which may be vertically well mixed or stratified, the only
requirement being that the horizontal gradient of density is independent of depth.
The velocity profile represented by Equation (
9.10
) was used by Hansen and Rattray
to derive a corresponding steady state salinity profile and from the combination of
the two profiles they established a widely used system of estuarine classification
(Hansen and Rattray,
1966
).
¼
9.2
Density-driven circulation in a ROFI: rotation
and coastal currents
......................................................................................................................
The arguments of
Section 9.1
for the density-driven circulation in an estuary can be
extended to a ROFI system where the horizontal flow is no longer restricted to one
dimension by the topography of the estuary. The response to density forcing is then
free to take a two-dimensional form. The increase in horizontal scale, previously set
by the width of the estuary, means that we must now take account of the Coriolis
forces. We will first look at what happens to the low salinity surface outflow as it
leaves the confines of the estuary, and then re-visit the estuarine exchange problem
but this time including the Coriolis term.
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