Non-dissipative effects of breaking on the wave field - Breaking and Dissipation of Ocean Surface Waves - page 323

here we would like to stress the fact that breaking shapes the spectrum tail and its level, and
to concentrate on this character of the breaking rather than on the associated dissipation.
This section is relatively short as a large proportion of its content serves as a summary of
relevant outcomes and conclusions which have been discussed throughout the topic in a
different context.
With respect to the dissipation, the content of Section 8.3 is quite unique. It discusses
the issue of enhancement of the wind energy input due to wave breaking. In other words, in
a way in this sense the breaking provides a negative dissipation because it instigates higher
energy and momentum fluxes from the wind to the waves when it happens.
8.1 Spectral peak downshift due to wave breaking
In the context of the radiative transfer equation (2.61) used in spectral models to forecast
the evolution of wave fields, downshift of the spectral peak is typically attributed to the
weak resonant four-wave interactions, i.e. term S nl in RTE. This is another long-lasting tra-
dition in wave-modelling applications, analytically based on the theory initially suggested
by Hasselmann ( 1962 ) and further developed by Zakharov ( 1968 ), Krasitskii ( 1994 ) and
Zakharov (2010) for homogeneous wave fields and recently by Gramstad et al. ( 2010 )for
non-homogeneous conditions. For this mechanism to be active, the wave fields have to be
directional, with continuous distribution of wave energy along frequency and wavenumber
vectors, and exact resonance of the four waves defined by the dispersion relation (2.17)
has to occur. While the continuous spectrum is a natural appearance of the wind-generated
waves, the exact resonance is a more limiting condition. First, the real waves are always
nonlinear and non-steady, and this fact causes deviations from the linear dispersion (2.14) ,
which can be as large as of the order of 10% (see Section 5.1.2 ). Secondly, downward accel-
eration of the short waves is strongly modulated by the dominant-wave motion at a scale as
large as the magnitude of the gravitational acceleration itself (see e.g. Sections 3.2 and 3.7 ).
In the meantime, it is generally known that downshifting of the spectral energy of
nonlinear wave trains, propagating in a dispersive environment, is also an outcome of mod-
ulational instability. This dispersive environment does not have to be surface water waves.
In nonlinear optics, for example, it is the so-called Raman effect which transfers energy
from high to low frequencies and causes a continuous red shifting of the mean frequency
of pulses propagating in optical fibres (e.g. Gordon , 1986 ). It is described by the nonlin-
ear Schrödinger equation (NLS, Zakharov , 1968 ), the same as used for water waves (see
Section 4 ), which not only predicts the effect, but is also able to describe the magnitude of
the frequency shifting quantitatively.
For water waves, the downshifting which associates with the modulational instability
has also been observed and described, both experimentally (e.g. Lake et al. , 1977 ) and
theoretically (e.g. Segur et al. , 2005 ). Breaking, if it happens in the course of these growing
instabilities, significantly modifies and accelerates such downshifting (e.g. Melville , 1982 ;
Su et al. , 1982 ; Reid , 1992 ; Tulin & Waseda , 1999 ; Hwung et al. , 2009 ; Babanin et al. ,
2010a ; Hwung et al. , 2010 ).