Geoscience Reference
In-Depth Information
Thus, both breaking probability and breaking severity become functions of the wind for-
cing, with opposite trends, and this fact will affect breaking dissipation
(2.25)
and
(2.33)
in a way which is hard to predict at this stage:
=
b
T
(
/
),
b
T
U
c
(2.37)
=
(
/
).
s
s
U
c
Here,
U
is some characteristic wind speed, and therefore the ratio
U
c
describes the wind
forcing, that is, how fast the wind is with respect to the phase speed of the waves
c
.Inany
case, the notion of roughly constant breaking severity is absolutely inapplicable in such a
scenario.
The fourth reason to be highlighted is the spectral nature of breaking severity in real
wave fields. Except for the case of pure swell which does not break anyway, ocean waves
have a continuous spectrum extending to higher and lower frequencies, and often with
multiple peaks. In the realistic spectral environment, it is quite rare that a physical process
is limited to a particular frequency or wavelength scale and does not have an impact across
the spectrum. This is certainly true with respect to such strongly nonlinear processes as
those that determine breaking severity.
An example of the spectral distribution of breaking severity is demonstrated in
Figure 6.1
.In
Figure 6.1
a, spectra of the time series of
Figure 6.2
are plotted: the solid line
is the pre-breaking spectrum and the dashed line is the after-breaking spectrum.
Figure 6.1
b
is the ratio of the two spectra.
Even though the laboratory waves were generated monochromatically and sinusoidally,
they are now strongly nonlinear and the first, second and third harmonics are clearly visible
in their spectra, indicated by vertical solid lines (again, solid lines correspond to the pre-
breaking state and dashed lines are close to the after-breaking state). The breaking severity
effect (the ratio in
Figure 6.1
b) is distinctly spectral. While it is the main wave that is break-
ing (around a frequency of 2 Hz at pre-breaking), the energy is lost from all the harmonics
as well. In both absolute and relative terms, most of the loss has come from the peak which
was reduced by a factor of 5. With the exception of the second harmonic (reduced and
shifted to 3.6 Hz), the other harmonics have almost completely disappeared. For frequen-
cies above the spectral peak, the average ratio is approximately 1.7. Across the entire range
of relevant frequencies, from the peak up to 11 Hz, the average ratio is 1.8 which translates
into a total spectral loss of
s
spectral
=
/
45%. Here, spectral severity is defined as
16
∞
0
16
∞
0
s
spectral
16
∞
0
E
s
spectral
=
F
before
(
)
−
F
after
(
)
=
F
before
(
)
f
df
f
df
f
df
(2.38)
where
E
s
spectral
are
wave frequency spectra before and after the breaking event. The factor of 16 is introduced
for consistency, as, by definition the significant wave height
H
s
is
is the energy loss across the entire spectrum and
F
before
(
f
)
and
F
after
(
f
)
∞
4
√
m
0
=
H
s
=
4
F
(
f
)
df
.
(2.39)
0
This way, the energy in
(2.38)
is expressed in terms of wave height, as in
(2.23)
-
(2.32)
.
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