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excursion crosses the zero line (i.e. effectively dominant breaking) and those whose excur-
sion is above the zero line (i.e. short waves breaking near the crest of the dominant waves,
see e.g. Longuet-Higgins & Stewart , 1960 ; Phillips , 1963 ; Donelan et al. , 2010 ). Ochi &
Tsai ( 1983 ) found that the second type of breaking is approximately 27% of the first type.
The significance of this study is hard to overestimate. Because the model was intended
for a non-narrow-band process, it can be applied both at the spectrum peak which has
a characteristic bandwidth and at the spectral tail which does not. This is a very essen-
tial advantage as the present understanding of wave breaking indicates two-phase physics
of the phenomenon: inherent breaking of dominant waves, determined by their narrow-
banded nature, and induced breaking of short waves (see Chapters 5 , 6 and Section 7.3.4 ).
The Ochi & Tsai ( 1983 ) results point to a general steepness threshold for both phases of
wave breaking, which is consistent with the present understanding that the waves break
because they reach a limiting steepness regardless of the nature of physical processes that
led to this steepness being achieved (see comments in Section 2.9 ). The relative fraction of
the short-scale breakers of 27%, however, appears very low and in this regard the Ochi &
Tsai ( 1983 ) probability model would perhaps need further revision.
Unrealistic also is the conclusion that no breaking is expected unless the significant wave
height is greater than 4m. Mathematically, the most important result of the Ochi & Tsai
( 1983 ) model is the outcome that breaking probability depends on the shape of the wave
spectrum and that the fourth moment of the spectrum is the main parameter in this regard
(see also Srokosz ( 1986 )).
A probability model of a different kind was proposed by Snyder & Kennedy ( 1983 ).
They introduced an artificial 'breaking variable' and then considered the statistics of this
variable and its relation with the wave directional spectrum. The variable was set equal
to one inside a whitecap and zero outside, the whitecap having both horizontal and ver-
tical extents. Defining the whitecapping volume was done in terms of another variable
which had a dynamic threshold such that it would indicate a breaking. A number of further
assumptions were made with respect to the dynamic variable, one of which was that it has
a maximum at the free surface.
To clarify this reasoning in physical terms, we should simply say that the acceleration
was eventually used as the dynamic variable. If it was over some threshold limit, the point
of the wave body in space and time was regarded as breaking and the breaking variable
was set to one. Then, mapping of the whitecapping surface was done in terms of geometric
moments of this variable. Further on, the dynamic variable (acceleration) was related to
the wave spectrum, and thus the statistics of the breaking variable in terms of its moments
was connected to this spectrum.
A number of very insightful conclusions regarding breaking probability were obtained
with this model. It was found that this probability, and whitecap coverage, are a function
of the ratio between the rms of the vertical acceleration m 4 (see eq. 2.10 ) and its critical
level
in (2.60) . The probability appeared as a simple inverse function of the wave fetch,
and interestingly the cross-wave scale of whitecaps was concluded to be greater than the
down-wave scale. The presence of breaking water at some point had certain positive and
γ
 
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