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the three is initial breaking of the wave; the evolution
therafter varies widely, impeding quantitative compari-
son. In our long-wave theory a stronger coastal current
(
f
) results in a larger PV flux across the slope, leading
to longer shelf waves with larger amplitudes that break
more rapidly. Stronger rotation
f
produces a larger rela-
tive vorticity in the wave envelope, resulting in shelf waves
that break more rapidly at smaller wavelengths and ampli-
tudes. Our numerical and experimental results exhibit
similar patterns of breaking time (
T
B
) and wavelength at
breaking (
L
B
), but the amplitude at breaking (
A
B
)may
increase or decrease with
f
, depending on the size of
f
.
We attribute this to competing tendencies for the wave to
transport more water across the slope, but also to break
more rapidly, when the relative vorticity in the wave enve-
lope is larger. Figure 12.11 compares the properties of
breaking in our numerical solutions and laboratory exper-
iments. While
L
B
is consistently within around 20% error,
our QG model underpredicts
T
B
and overpredicts
A
B
,
in each case by a factor of
3
on average. This is again
consistent with imperfect conservation of PV or stronger
bottom friction in the rotating tank.
The large amplitude of our coastal shelf waves distin-
guishes our experiments from previous studies of topo-
graphic Rossby wave generation, and the development of
a nonlinear wave theory aids in interpreting our results
(see Section 12.4). Previous investigations of Rossby
waves over continental slopes, such as those of
Caldwell
et al.
[1972],
Caldwell and Eide
[1976], and
Cohen et al.
[2010], have shown very good agreement with the cor-
responding theoretical dispersion relations.
Pierini et al.
[2002] found that numerical solutions of the barotropic
shallow-water equations closely reproduced Rossby nor-
mal modes on an experimental slope between two walls.
Our QG numerical solutions accurately describe the evo-
lution of our experiments, as shown in Section 12.6. This
indicates that flow in the rotating tank adheres closely to
columnar motion, despite the formal requirements for the
shallow-water QG approximation being poorly satisfied
[see
Williams et al.
, 2010].
Solutions of the nonlinear shelf wave equation in
Section 12.4 differ substantially from our experimental
results. Our nonlinear wave theory cannot capture the
overturning of the wave and the resulting break-up into
eddies, due to the assumption of a single-valued inter-
face
r
=
R(θ
,
t)
. The theory should describe the flow accu-
rately as long as the azimuthal gradients remain
and 12.7 indicates that some dispersion takes place but
that it is not sufficient to prevent breaking. This phe-
nomenon is the subject of ongoing research.
Previous laboratory investigations of coastal currents
have focused on buoyant gravity currents [
Whitehead
and Chapman
, 1986;
Cenedese and Linden
, 2002;
Folkard
and Davies
, 2001;
Wolfe and Cenedese
, 2006;
Sutherland
and Cenedese
, 2009] or turbulent jets [
Cenedese et al.
,
2005]. Our experiments employ a barotropic coastal cur-
rent generated via a rapid change in the tank's rotation
rate. Such a configuration has received little attention in
previous laboratory studies, though
Pierini et al.
[2002]
generated a barotropic shoreward mean flow using a
wide, slow-moving plunger. Our use of dye to visualize
our experimental results implicitly describes the evolution
of the coastal current as a wave. This may be insuffi-
cient for more realistic bathymetry like the coastal trough
of
Sutherland and Cenedese
[2009], though their trough-
crossing current resembles a trapped topographic Rossby
wave [
Kaoullas and Johnson
, 2012]. The wavelike descrip-
tion of the current breaks down when eddies form, as in
the experiments of
Folkard and Davies
[2001] and
Wolfe
and Cenedese
[2006], and in the long-term evolution of our
experimental and numerical coastal currents.
Our results show that retrograde coastal current flow
past a continental headland shifts the PV front shoreward
from the continental slope. The resulting large-amplitude
lee wave breaks and subsequently tends to form eddies that
exchange water between the coastal waters and the deep
ocean. However, if the velocity of the current is sufficiently
large, then the wave and any eddies are simply swept away
downstream. Our parameter sweep in Section 12.6 yields
insight into the eddies formed by the wave breaking and
the resulting exchange of water across the shelf break.
For the purpose of comparison with the real ocean,
f
measures the velocity of the coastal current, and
f
mea-
sures the PV jump between the continental shelf and the
open-ocean. The lee wave is longest, resulting in the largest
eddies, when the coastal current is strong and the PV jump
is small. We measure the penetration of open-ocean water
onto the experimental shelf using the amplitude of the lee
wave at breaking. In general, water is transported further
onto the shelf when the coastal current is strong and the
PV jump is large, but the complete dependence on
f
and
f
is somewhat more complicated (see Section 12.6.4).
There is a developed body of theory that describes the
evolution of coastal currents in terms of shelf waves [e.g.,
Mysak
, 1980b;
Johnson and Clarke
, 2001]. Predictive mod-
els of coastal shelves require adaptation for the sharply
varying bathymetry and must employ very high resolution
to capture the shelf processes [e.g.,
Haidvogel et al.
, 2008].
As a result, there remains scope for laboratory experi-
ments to inform oceanographic research on the subject
of continental shelf dynamics and to evaluate numerical
(μ
1
/
2
)
,
which is certainly the case during the initial generation
of the wave. The dispersive terms in (12.28) may be
expected to inhibit wave steepening and thereby maintain
a small amplitude-to-wavelength ratio, as in Figure 12.4.
It is therefore unclear why the numerical and experimen-
tal shelf waves develop gradients that are
O
O
(
1
)
within
−
10
50s. The formation of a wave train in Figures 12.3
