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long-wave theory, which implicitly assumes a small ratio
of cross-slope to along-slope transport. We have recalcu-
lated Figue 12.8c using the full dispersive wave equation
(12.28), with a much smaller gradient
S
max
in (12.50), but
this still fails to capture the pattern shown in Figure 12.9c.
The pattern of
A
B
in Figure 12.10c is qualitatively simi-
lar to Figure 12.9c, albeit somewhat distorted because
A
B
is particularly sensitive to artifacts in the filtered exper-
imental images. In Figure 12.11c we plot the relative
error in the numerically computed amplitudes at breaking,
A
B
=
A
B
numerical
/A
B
experiment
- 1. The wave ampli-
tude is typically around 1-2cm larger in our numerical
solutions than in our experiments. We attribute this to
our assumption of horizontally nondivergent flow under
the QG approximation, which neglects depth changes in
the mass conservation equation. Our numerical solutions
therefore overestimate the volume of fluid drawn onto the
shelf in the lee of the protrusion in the outer wall.
To interpret the behavior of our experimental shelf
waves, we introduced in Section 12.3 a QG shallow-water
model. The model assumes nondivergent, geostrophically
balanced barotropic flow beneath a rigid lid with small
variations in depth. We applied no-flux boundary con-
ditions for the channel walls and prescribed initial con-
ditions that conserve the absolute vorticity of each fluid
column and the total kinetic energy during the initial
change in the tank's rate of rotation. Some discrepancy is
to be expected between this model and our experiments, in
which the surface height may vary across the channel and
the shelf occupies a quarter of the water depth. However,
QG theory has been shown to describe rapidly rotating
laboratory flows far outside the formal range of validity
[
Williams et al.
, 2010] and retains the vortical dynamics
required to capture the evolution of shelf waves [
Johnson
and Clarke
, 2001].
In Section 12.4 we derived a nonlinear wave theory to
provide an intuitive description of the shelf wave evo-
lution. Our theory follows that of
Haynes et al.
[1993]
and
Clarke and Johnson
[1999] for a straight channel,
approximating the slope as a discontinuity in depth and
posing an asymptotic expansion in a small parameter that
measures the ratio of the wave amplitude to the wave-
length. The theory does not require the wave amplitude
to be small, however: Making this additional assump-
tion yields a Korteweg-de Vries-like equation [
Johnson
and Clarke
, 1999]. Our nonlinear wave equation (12.28)
improves upon the derivations of
Stewart
[2010] and
Stew-
art et al.
[2011] via the addition of an evolution equa-
tion (12.30) for the stream function on the outer wall.
Figure 12.4 shows that the evolution of the first-order
nonlinear wave equation initially resembles our experi-
mentally generated waves, but dispersion prevents these
solutions from capturing the wave breaking. By contrast,
solutions of the nondispersive zero-order wave equation
will always break but thereafter form a persistent shock
that does not resemble the experimental flows.
To resolve the disparity between our experiments and
our long-wave theory, in Section 12.5 we developed a
numerical scheme that solves the QG model equations in
a wall-following coordinate system. We advect an array of
tracer particles that delineates the wave envelope, mimick-
ing the dye line in our laboratory experiments. Figure 12.7
shows that the numerical shelf wave is qualitatively simi-
lar to our laboratory experiments, but the evolution occurs
more rapidly. This may be due to imperfect conservation
of PV or stronger bottom friction in the rotating tank. The
numerical solution shows that regions of closed stream-
lines appear as the shelf wave curls up, forming eddies that
transport water across the slope.
In Section 12.6 we compared shelf waves formed in
our long-wave theory, numerical solutions, and laboratory
experiments. The clearest point of comparison between
12.7. SUMMARY AND DISCUSSION
The large-scale flow of coastal currents is dominated
by the interaction of the current with the strong topo-
graphic PV gradient at the continental shelf break, which
results in complex behavior even in a strongly barotropic
fluid. This has motivated previous experimental studies
ranging from generation of isolated topographic Rossby
waves [
Ibbetson and Phillips
, 1967;
Caldwell et al.
, 1972;
Cohen et al.
, 2010] to turbulent coastal currents [
Cenedese
et al.
, 2005;
Sutherland and Cenedese
, 2009]. This chap-
ter has focused on the flow of a retrograde coastal current
past a headland that protrudes out onto the continental
shelf. Examples of such a configuration include the flow
of the Agulhas current through the Mozambique Channel
[
Bryden et al.
, 2005;
Beal et al.
, 2006, 2011] and the Gulf
Stream approach to Cape Hatteras [
Stommel
, 1972;
Johns
and Watts
, 1986;
Pickart
, 1995].
In Section 12.2 we introduced our laboratory experi-
ments whose setup is sketched in Figure 12.1 and illus-
trated in Figure 12.2. We constructed an annular channel
in a rotating tank with a narrow slope leading to a raised
shelf around the outer rim. The shelf was constricted over
an azimuthal length
b
by a protrusion in the outer
wall, representing a continental headland. We generated
a retrograde azimuthal mean flow, representing a coastal
current, by changing the rate of the tank's rotation. This
resulted in the formation of a large-amplitude Rossby
shelf wave in the lee of the protrusion, which we visualized
using a dye line that was positioned initially along the cen-
ter of the PV front. Figure 12.3 characterizes the evolution
of the flow, which develops a long, large-amplitude shelf
wave that breaks and overturns. Thereafter the wave may
unravel and form a wave train, as shown in Figure 12.3e,
or may instead roll up into eddies.
∼
