Energy dissipation across the wave spectrum - Breaking and Dissipation of Ocean Surface Waves - page 255

breaking dominant waves come in alternating breaking and non-breaking trains from 4 to
120 dominant waves long. The breaking waves are, on average, significantly higher and
steeper than those not breaking ( Holthuijsen & Herbers , 1986 ). It is expected therefore
that there will be a noticeable difference between the spectra calculated over breaking
wave-train segments and the spectra over non-breaking segments.
Since breaking is the only major process to contribute to the rapid dissipation at this
time scale (the bottom friction is relatively small and also relatively constant across the
breaking/non-breaking segments), the difference can be attributed to dissipation due to
the breaking of the dominant waves in the spectrum (as the spectrogram method is able
to detect the breaking of dominant waves only - see Section 3.5 ). This difference will
constitute a non-zero term on the right-hand side of (2.61) . The main assumption, which
was discussed and supported in Young & Babanin ( 2006a ), is that the difference can be
attributed to the partial derivative only and the advective term is small. An assumption
regarding the advective term was needed as it was not possible to directly estimate its
value. The waves were measured using a spatial array of wave probes with the largest
separation of 30 cm between the probes ( Young et al. , 2005 ). At such distances, the spectral
difference along the wave fetch (between the probes) was negligible and could not be
detected with a reasonable degree of confidence. Thus, to determine the spectral energy loss
due to dominant breaking, it was assumed that it should be sufficient to measure differences
between spectra of breaking and non-breaking waves based on measurements of time series
at a point.
This difference will be a lower-bound estimate of the dissipation due to breaking. The
approach treats the segments of breaking waves as a sequence of incipient breakers. In
fact, waves breaking at the measurement point already exhibit some whitecapping and
therefore they have already lost some energy prior to arriving at the measurement point.
The broken waves in the non-breaking sequence are already gaining energy from the
wind, but this energy is still not sufficient, on average, to bring them up to the breaking
point. Therefore, the energy difference between the breaking-onset and the just-broken
waves should in fact be somewhat larger than the one actually measured by this seg-
menting method. Also, as pointed out by Ardhuin et al. ( 2010 ), it should be kept in mind
that, since the spectra are different, nonlinear interactions must be different too, and this
fact, apart from the dissipation alone, can contribute to the observed spectrum variation
as well.
Additionally, it is instructive to observe that the breaking waves are at the same time
receiving energy from the wind. This means that the wind-input rate is still slower com-
pared to the dissipation, but wave-growth rates and breaking-dissipation rates are now
comparable and differ only by a factor of 2-3. This interesting observation indirectly sup-
ports the conclusion made in Donelan et al. ( 2006 ) that the wave-growth rates depend on
the wave steepness.
To summarise the segmenting approach described here, we would mention again that,
in the wave record with 60% dominant breaking rate, the trains of dominant breakers
are treated as sequences of incipient breakers and the trains of non-breaking waves as