Energy dissipation across the wave spectrum - Breaking and Dissipation of Ocean Surface Waves - page 254

hand side. The energy transfer across the spectrum due to the nonlinear term S nl becomes
essential at the scales of thousands and tens of thousands of wave periods ( Hasselmann ,
1962 ; Zakharov , 1968 ). In our case of a stationary fully developed constant-depth wave
environment, the full derivative is zero and the right-hand-side terms of (2.61) are balanced.
At medium scales of hundreds of wavelengths, the wave fields are usually assumed to
be stationary and homogeneous. Time/space series of surface elevations in such waves are
used to obtain statistically reliable estimates of wave spectra in experiments, and spectra of
this averaging scale are used in wave forecast and research spectral models. Such models
have been reasonably successful and this, to some extent, justifies the assumption.
Indeed, the small advective terms and S nl are not capable of bringing about significant
changes to the wave spectrum at such time scales (unless changes to the spectrum are sud-
den and abrupt; see e.g. Young & van Vledder , 1993 ). For the dissipation term controlled
by wave breaking, however, the scale of hundreds of waves is not small. For example,
laboratory experiments on unsteady deep-water breaking by Rapp & Melville ( 1990 ) and
Babanin et al. ( 2010a ) show that the breaking is a rapid process of the same order of magni-
tude in time as the wave period, tens of periods at most, and may cause a major energy loss
from the wave group where the breaking occurs. If, hypothetically, within the measurement
time-span of hundreds of waves, each wave breaks even once, changes to the spectrum will
be very significant. Such dramatic losses of energy, however, are not observed at these time
scales since the breaking rates are typically not very large ( Holthuijsen & Herbers , 1986 ;
Babanin et al. , 2001 , among others) and, at this time scale, the wind input is apparently
capable of restoring the mean spectrum after breaking. This also means that, at this time
scale, energy input by the wind is a slower process than the energy loss from breaking, as
the energy is input to every wave in the field, whereas it is only lost from a small fraction
of the breaking waves.
There are points of view that the spectral models based on the medium-scale averaging
of the wave fields may have reached their limit in the accuracy with which they can simulate
realistic wave generation and growth conditions (e.g. Liu et al. , 1995 ). This can be, in part,
due to the fact that they average out variations of the wave field at scales of several waves
(wave group scale). It is this shorter-scale group structure which plays a major role in
intermittent wave breaking ( Donelan et al. , 1972 ; Holthuijsen & Herbers , 1986 ; Babanin ,
1995 ; Banner et al. , 2000 ; Babanin et al. , 2007b , 2010a ) and modulation of the wind stress
( Skafel & Donelan , 1997 ). There is modulation of the surface roughness at even shorter
scales of dominant waves ( Hara & Belcher , 2002 ; Kudryavtsev & Makin , 2002 ) and over
breaking waves ( Babanin et al. , 2007b ), and disregarding this effect in models of wave
growth can lead to underestimation of the growth rate parameter by a factor of 2-3 when
compared to measured values (see Donelan et al. , 2006 , for a discussion). The spectral
equation (2.61) is not designed for applications at such time scales of individual waves and
wave groups.
What happens at the scales of dozens of waves or a hundred waves? In the case of
ordinary weak-in-the-mean breaking conditions, there should not be much variation in
properties at this scale. In the case investigated here, the strongly forced and frequently