Geoscience Reference
In-Depth Information
Figure 9.4
Frame of reference for the
analytical solution of Heaps (Heaps,
1972
) to density-driven flow influenced
by the Earth's rotation.
z
R
w
y
τ
s
=0
Coast
η
x
∂ρ
∂
x
= depth uniform
v(z)
h
u(z)
Continuity requires that the net transport normal to the coast is balanced by the net
input of river water R
w
per unit length of the coast:
ð
0
udz
¼
R
w
ð
9
:
12
Þ
h
and the solution is closed by boundary conditions, which are:
¼
0) the stresses t
x
¼
t
y
¼
(i) At the surface (z
0;
k
0
b
ru
b
; t
y
¼
k
0
b
rv
b
(ii) At the bottom (z
¼
h) stresses follow a linear drag law: t
x
¼
where u
b
and v
b
are the velocity components at the seabed.
The mathematical details of Heaps' analytical solution are rather intricate and
lead to algebraically complicated expressions for the velocity profile. The equa-
tions are, however, readily solved numerically (see Heaps' and ROFI modules at
the website). The essential result shown in
Fig. 9.5a
is an along-gradient compon-
ent of circulation, analogous to the estuarine circulation, plus a component of
flow perpendicular to the gradient which increases with height above the bed.
With an appropriate choice of the eddy viscosity (N
z
¼
10
3
m
2
s
1
), the
Heaps theory is seen to be in fair accord with the observed structure of the mean
sub-tidal flow in the Rhine ROFI; look at the correspondence between the Heaps
solution and the ADCP data in
Fig. 9.5a
. The persistence and the offshore extent
of the flow are illustrated in Fig
9.5b
, which shows a 16-day average of the surface
flow measured by HF radar (Souza, Simpson, et al.,
1997
) together with the
surface salinity contours. Inside the region of salinity deficit (S
2.5
<
34.5) almost all
0.15 m s
1
.
The balance between the alongshore and cross-shore components of the circulation
is set by the ratio of the frictional forces to the Coriolis force, known as the Ekman
number Ek
vectors indicate flow to the north-east with speeds up to
∼
N
z
/(fh
2
). For large friction (high Ek) the flow reverts to an estuarine
circulation. By contrast for low Ek, the flow is controlled by a geostrophic balance
and is a pure coastal current parallel to the isopycnals, in which the sea surface slopes
upwards to the coast according to:
¼
@
@
0
@
h
x
¼
ð
9
:
13
Þ
@
x
Search WWH ::
Custom Search