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this observation implies that the relative strength of the breaking, i.e. severity coefficient
s
(2.24)
, goes up at higher frequencies/smaller scales.
This is clearly illustrated in the second panel. A surrogate dimensionless severity here is
plotted as a distribution of the ratio of
R
0
(
f
)
, which stands for the energy loss, and spectral
density
P
, which is a measure of wave energy - across the frequency. The growth is
very consistent and large, about six times on average between
f
p
and 2
f
p
. Once again, we
should stress that this estimate and the trend can only be treated qualitatively, since bubble
size is not a linear proxy of the energy loss and the calibration in
Section 3.5
was only done
for a monochromatic wave.
Some interesting observations with respect to this surrogate severity can, however, be
made. Firstly, unlike in the other subplots of this figure, all the records are bundled together.
This fact signals that an increase of the severity across the spectrum is mainly a function
of the relative frequency, with respect to the spectral peak, rather than of any environ-
mental factors and forcings as in the other panels. Secondly, in the Lake George finite-
depth environment, where the downshifting is apparently restricted or even blocked by the
lower-frequency breaking, the severity of this breaking is greater than at the peak. In fact,
breaking severity at the peak is the smallest.
It is interesting to notice here that the breaking-caused directional distribution of the
dissipation is also smallest in the main direction of wave propagation (
Figure 7.9
,
Section 7.3.6
). That is, the breaking tends to make the spectral peak narrower, both in
the frequency and in the directional domains.
Below the peak, the weakest severity corresponds to the highest breaking rates (see
Figure 5.27
). This is consistent with the observation made in
Section 6.1
that, in the case of
breaking due to modulational instability, a stronger wind forcing makes the breaking more
frequent but less powerful. We should remember, however, that the lower-frequency break-
ing at Lake George is affected by the bottom proximity, i.e. influenced by a combination
of causes.
Since the larger bubbles correspond to more severe breakers, the product of the mean
bubble size
R
0
(
(
f
)
can be treated as a surrogate dissipa-
tion rate
(2.20)
at frequency
f
. Again, the surrogate dissipation must only be interpreted
qualitatively, and not even in terms of proportionality.
Here, it has to be mentioned that another method, also based on bubble size, was sug-
gested by
Garrett
et al.
(
2000
) to connect the bubble-size spectra and the dissipation. It
is based on an expression for the maximal bubble radius, for which bubble breakup is
prevented by the surface tension, and observations of the bubble spectra. The energy dis-
sipation is estimated from a dimensional energy-cascade argument. The energy brought in
by the breaking is partially spent on breakup of the bubbles and cascading the bubbles by
their size. The cascade is stopped at the bubble size whose further breakup is prevented by
the surface tension which opposes the turbulent inertia forces. At this size, the bubbles will
be accumulated and, if this radius is measured, it gives an indication of the energy initially
produced by the breaker.
f
)
and the breaking rate
b
T
(
f
)
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