Geoscience Reference
In-Depth Information
(a)
(b)
0.07
0.07
0.06
0.06
0.05
0.05
0.04
0.04
0.03
0.03
0.02
0.02
0.01
0.01
1
1.5
2
2.5
3
1
1.5
2
2.5
3
f
(rad / s)
f
(rad / s)
(c)
0.07
0.06
0.05
0.04
0.03
0.02
5.3
0.01
1
1.5
2
2.5
3
f
(rad / s)
Figure 12.10.
Contour plots of (a) the time of wave breaking in seconds, (b) the breaking wave length in centimeters, and (c)
the breaking wave amplitude in centimeters, extracted from experimental results. The contours have been interpolated linearly
between data points, which are indicated by crosses.
position (
θ
,
φ
i
,or
θ
p
) at which the wave has broken (
θ
B
)
and the smallest subsequent
θ>θ
B
at which the wave
envelope lies below the shelf line
r
=
R
h
. In Figures 12.8b,
12.9b, and 12.10(b) we compare
L
B
between our nonlin-
ear wave theory, numerical solutions, and experiments. We
have converted
L
B
to distances in centimeters along the
shelf line.
As with
T
B
in Section 12.6.2, we find that
L
B
has a
qualitatively similar dependence on
f
and
f
in our the-
ory and numerical solutions. A swift coastal current or
weak background rotation yields a long wave at breaking,
because water is drawn further onto the shelf by the mean
flow before the relative vorticity it acquires can steepen
and break the wave. Waves evolving under the nonlinear
wave equation break at approximately half the length of
the waves in our numerical solutions, consistent with their
shorter breaking time. The nonlinear wave theory predicts
an unusually large
L
B
for
f
=3rad
/
sand
f
=0.01rad
/
s.
In this corner of parameter space the wave speed given
by (12.28) exceeds the mean-flow speed at the protru-
sion for sufficiently small perturbations of the interface
R
about the shelf line
r
=
R
h
. This allows the wave enve-
lope to propagate further upstream, resulting in a larger
wavelength at breaking.
The breaking wavelength shows good qualitative and
quantitative agreement between our numerical solutions
and laboratory experiments. Figure 12.11b shows that
the relative error in the numerically computed breaking
lengths,
L
B
=
L
B
numerical
/L
B
experiment
−
1, is consistently
below 20%. This is surprising because the experimental
