Energy dissipation across the wave spectrum - Breaking and Dissipation of Ocean Surface Waves - page 251

a typical ocean spectrum, e.g. of JONSWAP form ( eq. 2.7 , Hasselmann et al. , 1973 )or
Donelan et al. ( 1985 )form.
The real ocean spectra are continuous and what is the primary component and what are
the sidebands in such full-spectrum conditions have to be clarified, as far as the modula-
tional instability is concerned. Studies of this kind are available (e.g. Onorato et al. , 2001 ;
Janssen , 2003 ) and they demonstrate that the modulational-instability analogy should be
applicable at the peak of typical narrow-banded wave spectra. Given the fact that the fre-
quency band of this instability, relative to the primary wave, is quite limited (see e.g. Yuen &
Lake , 1982 ; Tulin &Waseda , 1999 ), the high-frequency components within the continuous
spectrum cannot be expected to participate in the nonlinear-interaction dynamics coupled
with the dominant waves anyway.
Another conclusion of Tulin & Waseda ( 1999 ) and Meza et al. ( 2000 ), which poten-
tially distinguishes instability breaking from focusing breaking, is the behaviour of low
scales in the spectrum (wave components longer than those breaking), in response to
the breaking. In both cases, spectral downshifting is observed, i.e. these components in
fact gain energy as a result. That is, not all the energy lost is gone from the wave sys-
tem, at least some part of it is transferred across the spectrum. This, again, highlights
the potential of the dissipation term S ds to participate, along with the nonlinear interac-
tions described by S in in (2.61) , in the evolution of wind-generated wave fields towards
larger waves.
Quantities of the energy gain, however, are very different. In the modulational-breaking
scenario, the lower sideband grew on average approximately six times, whereas in linear-
superposition breaking, the mean low-frequency gain was some 10%. If applied to the
natural wind-generated spectra, however, it should be kept in mind that the energy drops
away below the spectral peak very fast (see, for example, JONSWAP parameterisation
(2.7) ). Therefore, in field conditions such gains may turn out to be quite subtle and difficult
to detect, particularly if the swell is also present, which is unrelated to breaking and is a
usual occurrence in the open ocean.
Thus, the spectral output of breaking, if detected, can help to distinguish modulational
breaking from linear-focusing breaking. What are other possible mechanisms leading to
the limiting steepness (2.47) , (2.51) and (2.52) and therefore to the breaking?
Babanin et al. ( 2011a ) broadly placed such processes into two groups: instability mech-
anisms and superposition mechanisms. The modulational instability of Tulin & Waseda
( 1999 ) and the linear focusing of Meza et al. ( 2000 ) is only one example for each of
the groups. Within these two phenomenological groups, further subdivisions are obvious,
which may and perhaps will also correspond to different dynamics in terms of the dissi-
pation outcome. For example, the classical Benjamin-Feir/McLean instability ( Zakharov ,
1966 , 1967 ; Benjamin & Feir , 1967 ; McLean , 1982 ) does result in very tall waves rising
within wave groups, provided that the average wave steepness is large enough, but once
an individual-wave steepness is higher than 42, the wave-crest instability (also called

second-type instability) becomes important ( Longuet-Higgins & Dommermuth , 1997 ;
Longuet-Higgins & Tanaka , 1997 ).

= 0 .