Geoscience Reference
In-Depth Information
10
−
3
s
−
1
. The experimental dye is
represented by
M
= 3600 tracer particles, visualized as a
line in Figure 12.7.
The solution shown in Figure 12.7 is visibly similar to
the evolution of the experimental dye line in Figure 12.3.
In particular, the progression in Figure 12.7 of the solu-
tion from a long, smooth wave in panel (b) to a backward-
breaking wave in panel (c) is strikingly similar to the
behavior of the experiment. However, the steepening of
the wave and in panel (d) the formation of the wave
train appear to happen more rapidly than they do in
Figure 12.3. By eye, Figure 12.3e corresponds better to
Figure 12.7d than Figure 12.7e, despite being separated
by 21 s. This may be due to imperfect conservation of PV
on fluid columns crossing the experimental slope or our
bottom friction coefficient
κ
may simply be too small. In
Figure 12.7 the two sides of the wave train in panel (e) do
eventually curl up on themselves, as shown in panel (f) and
as observed in our reference experiment. In Section 12.6
we will quantify this comparison over a range of Coriolis
parameters and coastal current speeds.
A much larger volume of deep water is retained on the
shelf in Figure 12.7f than in Figure 12.3, associated with
the growth and persistence of the large-amplitude shelf
wave. The streamlines reveal that the growth of this wave
leads to constriction of the along-shelf flow. This drives
water off the shelf and across the slope far upstream of
the protrusion, resulting in acceleration of the incom-
ing flow in the deeper portion of the channel. In panels
(d)-(f) the wave envelope drives an exchange across the
slope, drawing inflowing deep water onto the shelf at the
protrusion and then exporting it further downstream. At
t
= 104s all inflowing deep water makes an excursion
onto the shelf before continuing downstream. This is facil-
itated by patches of closed streamlines, corresponding
to barotropic vortices/eddies in the flow that sustain the
exchange of water across the slope.
comparison between our theory, numerical solutions, and
experiments.
of approximately 6.1
×
12.6.1. Breaking Conditions
We obtain theoretical predictions of breaking wave
properties using the nondispersive (
γ
= 0) form of the
nonlinear shelf wave equation (12.28). The dispersive
form (
γ
= 1) is formally more accurate but inhibits shock
formation, which describes wave breaking in our theoret-
ical framework. For each parameter combination
(f
,
f
)
we solve (12.28) on an azimuthal grid of
N
= 7200 points
using the method described in Section 12.4.5. We define
the solution to have formed a shock, and thus the wave to
have broken, when the gradient of the interface
r
=
R(θ
,
t)
satisfies
1
R
∂R
∂θ
max
θ
∈[
> S
max
.
(12.50)
0,2
π)
The gradient threshold
S
max
should be maximized to
ensure that interface has come as close as possible to a
discontinuity before (12.50) is satisfied. We use
S
max
=20
because the pseudospectral solution cannot reach much
larger gradients on a grid of size
N
= 7200 without
becoming subject to the Gibbs phenomenon.
In our numerical solutions we track the positions
(ρ
i
(t)
,
φ
i
(t))
for
i
=1,
...
,
M
of tracer particles that lie
initially over the center of the slope. We deem the wave-
form comprised of these particles to have broken when the
initial azimuthal ordering of the particles reverses, i.e.,
φ
i
(t)>φ
i
+1
(t)
for any
i
=1,
...
,
M
.
(12.51)
The addition
i
+ 1 is taken modulo
M
because particle
positions wrap around the annulus.
We identify wave breaking in our experimental results
by processing the images of the tank to extract the posi-
tions of the dye line in each frame. Our focus on the
initial development and breaking of the wave permits the
use of a simplified algorithm, because the dyeline may
be described as a single-valued function of azimuthal
position. Each image is first filtered to remove shades dis-
similar to that of the dye line. Then at each azimuthal
position
θ
p
for
p
=1,
...
,
P
, we search radially to locate
the midpoint of the dye line, which we denote as
R
p
.We
thereby construct a series of points along the dye line
in each frame,
(θ
p
,
R
p
(t))
, similar to our description of
the interface
R(θ
,
t)
in our nonlinear wave theory. We
omit azimuthal positions
θ
p
obscured by the clamps vis-
ible in Figure 12.2, and successive frames are compared
to remove erroneous measurements due to the dye dis-
penser. We deem the wave to have broken by applying a
condition analogous to (12.50) but using the mean gradi-
ent over a small range of
θ
to eliminate noise that arises
from the image filtering. In practice, the choice of
S
max
12.6. SHELF WAVE BREAKING
The only consistent point of comparison between our
numerical and experimental results is the formation of a
breaking lee wave behind the bump in the outer wall of the
channel. The evolution therafter varies widely across our
parameter space in
f
and
f
. When the coastal current
is weak and the background rotation is strong, the break-
ing wave rapidly rolls up into eddies, which are beginning
to form in Figures 12.3f and 12.7f. A strong coastal cur-
rent in the presence of weak background rotation will tend
to inhibit wave breaking and lead to the formation of a
wave train resembling that in Figures 12.3e and 12.7d.
The wave breaking is therefore the most natural point of
