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Yuan et al. ( 2008 ) continued the probability-function limiting-value approach by using a
modification of the kinematic criterion (2.49) , that is by assuming that the waves at the tail
of the probability distribution cannot exist if their surface orbital velocity exceeds the wave
phase speed, which is the limiting velocity of the Stokes wave. An essential new ingredient
of their model is the surface wind-drift velocity, which modifies (increases) the speed of
water particles at the wave crest and thus promotes breaking.
Yuan et al. ( 2009 ) combined this probability model with further empirical and analytical
argument and applied it to investigating oceanic properties related to breaking, such as
energy loss, both potential and kinematic, whitecap coverage (see also Section 3.1 ) and
breaking entrainment. They continued on and compared expressions obtained with field
observations, to find satisfactory agreements.
A different class of probability models was developed over the years to target the prob-
ability of breaking occurrence, rather than breaking strength as in the studies described
above (e.g. Longuet-Higgins , 1975b , 1983 ; Houmb & Overvik , 1976 ; Nath & Ramsey ,
1976 ; Tayfun , 1981 ; Huang et al. , 1983 , 1984 ; Ochi & Tsai , 1983 ; Snyder & Kennedy ,
1983 ; Papadimitrakis & Huang , 1988 ; Papadimitrakis , 2005 ). They usually employ com-
parison of probability functions of some properties of the wave systems, sometimes
ad hoc properties, and refer to empirical criteria for such comparisons, rather than to
physics explicitly. Since the wave measurements are most often conducted as time series
of surface elevations, the majority of these probability models employed joint probability
distributions of wave height H , and wave period T rather than wavelength or wavenumber.
As an example of such a model, which does not have a narrow-band spectrum limitation,
we will describe the probability model of Ochi & Tsai ( 1983 ). They used T 2 which can
be converted into wavelength or wavenumber by means of a linear dispersion relationship
(see 2.17 , 2.55 ), and therefore the probability of occurrence of waves exceeding the Stokes
limit (2.47) , (2.55) or other limiting steepness can be estimated. Ochi & Tsai ( 1983 ) argued
that, based on their laboratory observations, the Stokes limiting-steepness criterion (2.55)
is too high and the waves will actually break if
020 gT 2
H
0
.
.
(3.46)
They concluded that (2.55) is applicable to what they called regular waves, and the irregular
(or spectral waves) have to obey the limit (3.46) . From what we know about wave breaking
now, this difference can perhaps be explained in physical terms.
If the waves break due to modulational instability, they do not follow the simple reg-
ular/irregular notion because shortly before breaking onset their steepness reaches the
Stokes limit (2.47) whereas their wavelength shrinks and the period is reduced accord-
ingly. The reduction of the period as observed by Babanin et al. ( 2007a , 2009a , 2010a )is
some 10-15%, which is in perfect accord with the difference between (2.55) and (3.46)
observed by Ochi & Tsai ( 1983 ).
Ochi & Tsai ( 1983 ) continued on to derive the prediction formula (3.46) by using a joint
probability distribution of wave excursion and the associated time interval of a non-narrow-
band random process. They subdivided the wave breakings into two types, those whose
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