Wave-breaking probability - Breaking and Dissipation of Ocean Surface Waves - page 195

The (5.41) -estimated data points are shown in Figure 5.30 with circles. With the amount
of uncertainty involved, they agree remarkably well with the measurements and with the
local-in-frequency-space dependences which fit these measurements, and therefore provide
a very strong corroboration of the cumulative effect, which in fact was estimated here in a
quite simple way, defined by (5.41) .
Other mathematical and physical expressions for the cumulative terms are possible of
course (e.g. Donelan , 2001 ; Ardhuin et al. , 2010 ), but one way or another these breaking-
probability functions/parameterisations have to include integral functionals, rather than be
a function of spectral density or other distributed property local in frequency/wavenumber
space. The cumulative term has a principal importance, particularly as the contribution of
the inherent breaking becomes so small at shorter scales that it renders little connection
between b T ( f ) and wave spectrum F ( f ) , to an extent that the inherent breaking can in

fact be neglected.
The principal question in this regard is what is the relative frequency (with respect to
the spectrum peak) at which the inherent-breaking and induced-breaking terms are equal?
Then, even though both terms can exhibit their influence, below this frequency the inher-
ent term will dominate, and above, the induced term will prevail. Once again we should
remember that the question about breaking has not necessarily the same answer as the
question about inherent dissipation and induced dissipation in (5.40) being equal.
There is not much direct quantitative guidance on this topic, apart from an experimen-
tal investigation by Filipot (2010, personal communication). He demonstrated that the
breaking of short waves with frequencies in excess of 3 f p is locked in phase with the
dominant-wave crests. Qualitatively, such an effect should theoretically be expected for
wave breaking due to the modulated effects of the underlying long waves which make the
short waves steeper close to the dominant crests (e.g. Longuet-Higgins & Stewart , 1960 ;
Phillips , 1963 ; Donelan et al. , 2010 ).
Indirectly, discussion of this transition can be reduced to the question about transi-
tion from the f − 4 to f − 5 tail of the wave spectrum (see Section 8.2 ). Tsagareli ( 2009 ),
Tsagareli et al. ( 2010 ) and Babanin et al. ( 2010c ), based on the physical constraint of
the total wind-to-wave momentum flux being equal to the integral of the wind-input spec-
tral term S in ( in (2.61) , argued that such a transition must necessarily exist. The f − 4

behaviour of the equilibrium interval is consistent both with observations (e.g. Donelan
et al. , 1985 ) and theory (e.g. Pushkarev et al. , 2003 ). If, however, extended to high frequen-
cies, Tsagareli et al. ( 2010 ) found that such a spectrum cannot satisfy the above-mentioned
principal constraint.
Thus, the Phillips ( 1958 ) f − 5 equilibrium interval has to be invoked at higher frequen-
cies. The existence of such an interval has also been confirmed by experiments
(i.e. JONSWAP, Hasselmann et al. , 1973 ), as well as the presence of both the subintervals
in the single spectrum and the transition between them (e.g. Forristall , 1981 ;
Evans & Kibblewhite , 1990 ; Kahma & Calkoen , 1992 ; Babanin & Soloviev , 1998b ; Resio
et al. , 2004 ). Because of the cumulative breaking behaviour, significant wave breaking
is predicted at these frequencies and thus the original Phillips concept can be applied.

f )