Wave-breaking probability - Breaking and Dissipation of Ocean Surface Waves - page 175

density of wave steepness (see Babanin & Young , 2005 ; Babanin et al. , 2007c ). This will
be discussed in Section 5.3.2 in the context of breaking probabilities for spectral waves and
its value is given by (5.36) .
Therefore, a characteristic breaking threshold always exists and bears a principal signif-
icance in wave-breaking dynamics and statistics. Breaking probability will depend on the
characteristic steepness or its equivalent, but only if it exceeds the threshold value, and the
probability appears to depend on the excess above the threshold steepness rather than on
the steepness itself (see Sections 5.1.4 and 5.3.1 ).
From the point of view of nonlinear hydrodynamics of wave trains/fields, however,
this threshold does not signify any fundamental transition from one type of behaviour
to another. As mentioned at the beginning of this section with respect to two-dimensional
trains, the non-breaking trains, with their characteristic steepness being below the threshold
(5.11) , will exhibit all the same nonlinear dynamic features as the breaking trains (except
for the breaking itself of course). The breaking/non-breaking segregation is not because
of a transition to a different underlying physics of wave evolution, but due to the abil-
ity/inability of the nonlinear wave system to produce individual waves that exceed the
limiting steepness (2.47) .
Differences between the breaking threshold (5.11) and (5.19) and breaking criterion
(2.47) should be reiterated, emphasised and stressed at this stage. The threshold is a mean
steepness of the wave train/field, whereas the criterion is a limiting steepness for individual
waves in this train/field. That is if the mean steepness is above (5.11) and (5.19) , there
will appear individual waves reaching up to the steepness of (2.47) . Therefore, individual
waves can be steeper than (5.11) and (5.19) , but that does not warrant an eventual breaking
unless the mean steepness exceeds this level. Once these individual waves break, the mean
steepness may fall below the threshold and the breaking will stop, unless the steepness is
pumped up, for example, by the wind, by an adverse current, by the bottom proximity etc.
High individual waves will still be produced, but they will not be reaching the limit and
will not be breaking unless there is an input of energy into the wave system (e.g. wind input
in the field or mechanical generation of waves in the laboratory).
5.3 Spectral waves
Now that the meaning of the breaking threshold is understood ( Section 5.2 above), the
breaking probability of spectral waves, i.e. waves with continuous distribution of energy
along different wave periods/lengths and different directions can be analysed. While it
is tempting to extend an analogy of the distance to breaking (which, with some caution,
can be interpreted as breaking probability) of the monochromatic trains, parameterised
in Section 5.1.4 , to the spectral waves, this is not a straightforward exercise because the
amplitudes and therefore steepness of waves with a specific wavenumber are not defined in
the continuous-spectrum environment. An even less straightforward notion is at the spectral
tail where there is not even a characteristic bandwidth that can be employed to produce a
definition of a characteristic steepness similar to that for the spectral peak (5.17) - (5.18) .