Wave-breaking probability - Breaking and Dissipation of Ocean Surface Waves - page 206

and some of these detected deep-water wind-generated three-dimensional waves have an
enormous (by field standards) steepness up to

= 0 . 27, the distribution is clearly biased

towards higher frequencies. Waves with > 0 . 17 are all shorter than those of the peak fre-

quency f p = 25 Hz, and the higher the individual steepness, the higher is the individual

frequency. Since on average the highest waves are observed at the peak frequency, and
the peak is very sharp, the only plausible explanation for this observation is that these
abnormally steep, but rare waves are those in transition towards or just after the incipi-
ent breaking. As deep-water waves below the peak do not break, and if the near-breakers
shorten, distribution of the incipient breakers has to be characterised by higher-than-peak
frequencies at abnormally high steepnesses, as it is. The dispersive focusing or directional
focusing, or other linear and quasi-linear processes cannot be attributed to the shortening
of the wavelength prior to breaking. Thus, the very existence of such abnormally high and
shrunken waves indicates that the modulational instability mechanism is most likely still
active in these directional field conditions (in this regard, also see steepnesses of individual
waves in Figure 8.6 ).
Finally, while discussing what implications for the wave-breaking process in general
and wave-breaking probability in particular may be brought about by the fact that the
wave fields are directional, the directional focusing has to be more explicitly mentioned
( Fochesato et al. , 2007 ). This is a truly directional phenomenon which signifies linear
superposition of waves/wave-packets of approximately the same carrier frequency con-
verging at an angle with respect to each other.
If the carrier waves are steep enough, the superposition can lead to a very high crest
(i.e. double the height of the waves involved in the superposition, in which case nonlinear
effects may start playing a role (e.g. Brown & Jensen , 2001 )), reaching the steepness limit
(2.47) and ultimately breaking. Depending on the angle and the length of the converging
crests, the breaking can have different severities and spatial/temporal extents.
How frequent is such breaking? In order to achieve the limit (2.47) , there has to be
a superposition of either multiple wave crests, which should be quite a rare event, or,
for example, superposition of only two crests with half-the-limit steepness of Hk

0 .

/ 4 =

limiting / 22; such crests are themselves quite rare events to begin with (see

Figures 5.35 and 8.6 ).
In the experiments by Babanin et al. ( 2011a ), dedicated specifically to separating
modulational-instability and directional-focusing breaking, where such initially steep
crests were intentionally mechanically produced, it was noticed that the linear focusing
only exists at average steepness ak

2 = 0 .

24. This signifies superposition of two waves
leading to the breaking at limiting steepness (2.47) . Thus, superposition of three waves
is unlikely, at least it did not happen in the course of the records which encompassed
some 500-700 waves in the experiment, whereas breaking did happen. This is an impor-
tant observation, since in typical conditions, with much less steep waves and dispersive
rather than directional focusing, i.e. focusing of waves of different lengths and there-
fore heights, superpositions of even greater numbers of waves would be required. Such
an observation indicates that the probability of linear superposition, which would lead to
the limiting steepness (2.47) due to directional convergence of crests, is low. At the other

> 0 .