Wave-breaking probability - Breaking and Dissipation of Ocean Surface Waves - page 199

2). While both the options reproduce the observations well, when there is a substantial
amount of breaking, the second option clearly fails the no-breaking case of f c =

p =

0 . 55 f p in

the top left panel.
Thus, Filipot et al. ( 2010 ) introduced an alternative method for quantifying breaking
probabilities across the spectrum, which does not rely on the empirical breaking-threshold
value, but employs other empirical parameters. The method was shown to work both for
deep-water and finite-depth waves and to approximate the average probability distribution
of breaking-wave height reasonably well both for inherent-breaking and induced-breaking
scales in the wave spectrum. The latter gives it an advantage compared to the probability
formulations which have to rely on the cumulative integral as in (5.40) - (5.41) , but there
are indications that, with respect to the average distribution at each scale, BWHD may still
vary in a different way depending on whether the nature of the breaking is spontaneous or
induced (i.e. Figure 5.31 ).

5.3.3 Breaking in directional wave fields

In this subsection, the effects of directionality on breaking occurrence and probability are
discussed. The main question to answer is whether the modulational instability, which
is often regarded as a two-dimensional phenomenon, is still active in directional wave
fields. The other question is whether it is more frequent than the linear-focusing, includ-
ing directional-focusing breaking. The possibility of modulational instability is discussed
in terms of the directional spectrum and the directional modulational index defined in
this subsection. Lower limits for instability existence in these terms are identified and
parameterisation of the separation of the two breaking types is proposed.
Waves on theocean surface aredirectional, i.e. three-dimensional, and this is their principal
feature. Apart from the case of pure swell, which appears unidirectional and evenmonochro-
matic but due to its low steepness has little relevance as far as wave breaking is concerned,
the oceanic wind-generated andwind-forcedwaves are characterised not only by continuous
distribution of their energy along temporal/spatial (or frequency/wavenumber) scales, but
also along directions (5.33) - (5.34) . Investigations of the directional wave fields present

apparent technical difficulties both experimentally (see Mitsuyasu et al. , 1975 ; Hasselmann
et al. , 1980 ; Donelan et al. , 1985 ; Babanin & Soloviev , 1987 , 1998b ) and numerically (e.g.
Ducrozet et al. , 2010 ).
This is obviously true with respect to all other wave-related scale-distributed proper-
ties and characteristics, including the wave-breaking probability, severity and the dissipa-
tion term. These functions are spectra, i.e. they describe the distribution of the respective
properties along frequencies/wavenumbers, and should in principle describe their distribu-
tion along directions too. That is, the breaking probability in (2.4) , technically speaking,
should be:

θ

n ( f ,θ) T n ( f ,θ)

b T ( f ,θ) = = (5.48)

t N ( f ,θ)