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wave field, but unless the waves are swell or forced by very light winds, they normally do
exhibit some breaking (e.g. Rogers et al. , 2003 ). The spectrum drops below this limit in
its front face which signifies the obvious knowledge that waves, which are longer than the
dominant ones in the wind-wave spectrum, do not break.
Therefore, wave spectra exist between these two lines. The dotted line indicates a 5%
breaking rate. As is known, at typical moderate deep-water conditions breaking rates are
of the order of a few percent (e.g. Babanin et al. , 2001 , their Figure 13). Thus, in a typical
wave spectrum, the spectral density corresponding to the small/short waves is somewhat
above the dimensional threshold (5.37) , approximately at the position indicated by the
dotted line in Figure 5.29 . This means that inherent breaking in trains of such waves is
active, and the rate of such inherent breaking is determined by the degree of excess of the
real spectrum above the dashed threshold level at each frequency. The induced breaking,
however, alters such simply derived breaking rates and in fact tends to dominate at smaller
scales away from the spectral peak.
Let us try to apply the threshold value (5.37) to experimental data in order to test the
breaking probabilities in different frequency bins in dimensional terms. Indeed, while the
overall scatter in Figure 5.28 is prohibitive, data points for the spectral-peak bin (asterisks)
exhibit a reasonable correlation as a function of the saturation spectrum. In Figure 5.30 ,
the Lake George data, in a search of the dependence of b T (
f
)
on wave spectrum F
(
f
)
,are
separated into narrow spectral bins
f i
=
f p +
0
.
2 if p ±
0
.
1 f p
(5.38)
with i
4, i.e. starting from the spectral peak. Only records with breaking rates in
excess of 2% across all the frequencies were chosen to avoid bias due to zero-breaking con-
tributions when the rates are low. A riding wave removal (RWR) procedure ( Schulz , 2009 )
was used to identify the periods of the breaking waves. The standard zero-crossing analysis
becomes naturally noisier towards higher frequencies when the riding shorter waves may
not necessarily cross the mean level. The RWR technique works, once the bubble detection
signals a breaking, by finding the shortest riding waves first, and then removing all of them
from the signal before re-processing the signal to look for the next scale of largest riding
waves.
At the spectral peak (first panel), consistent with the two-phase behaviour of the break-
ing/dissipation discussed above, dependence in terms of the wave spectrum is linear:
=
0
,
1
,...
b T
2
(
F
F threshold ).
(5.39)
If, however, this dependence, as shown with solid lines in subsequent subplots, is applied to
the breaking rates at higher frequencies, it exhibits a progressively larger underestimation.
Such a result is consistent with our expectations that follow from the documented cumu-
lative behaviour. Inherent (linear) dependence of the wave-breaking rates on the spectrum
excess should be present at each frequency. However, at every next frequency away from
the spectrum peak, the contribution of the induced breaking (due to waves breaking at
lower frequencies) has to become progressively larger.
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