Geoscience Reference
In-Depth Information
Figure 9.1
(a) Frame of reference used
in the calculation of the residual flow
using Equation (
9.10
). (b) The two
components of the residual flow
described by Equation (
9.10
).
z
(a)
τ
s
=0
Se
a su
rface
η
x
Mean level
z
=0
∂
∂ρ
= depth uniform
O
CE
AN
RIV
ER
u
=0
at
z
=
-
h
Density driven circulation
(b)
River flow
0
0
-0.5
−
0.5
-1.0
−
1.0
-1
0
u
/|
u
s
|
-2
0
-0.5
0.5
1
-1
u
/
u
0
makes the problem much more difficult than its heating-stirring counterpart and
requires that we examine the spreading process in some detail. This we do, initially,
by developing a simple model of the density-driven circulation and its interaction
with tidal flow in a narrow estuary where, because of the smaller scale, we can neglect
the effects of rotation.
9.1.1
The estuarine exchange flow
The flow of freshwater from a river into an estuary sets up and maintains horizontal
gradients of salinity and hence of density. The density gradient implies a pressure gradient
which drives a circulation. In a narrow estuary, where rotation is not important, we can
of the classical calculation by Hansen and Rattray (Hansen and Rattray,
1966
).
In this simple model of a narrow estuary, we assume that the horizontal density
gradient is independent of depth and directed in the x direction which is parallel to
the axis of the estuary, as illustrated in
Fig. 9.1a
. Note that this important assump-
tion applies, not only in the case of a vertically mixed estuary, but also to estuaries
with stratification if the vertical structure is uniform along the estuary.
We assume further that rotation can be neglected (Coriolis forces
0) and that the
flow is in steady state. Under these conditions, the momentum
Equation (3.13)
simplifies to a balance between the pressure gradient and the stress divergence, i.e.
¼
@
x
þ
@
p
t
x
@
z
¼
0
:
ð
9
:
1
Þ
@
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