On the probability of wave breaking in deep waters

An analytical expression for the spectral probability density pB(Σ) of breaking waves in deep water, with or without swell, is derived on the basis of a joint distribution of wave frequencies and amplitudes, and a modified Banner and Phillips [1974. Journal of Fluid Mechanics 66, 625-640] breaking criterion. The joint probability density and the derived marginal amplitude, frequency and conditional densities, when normalized properly, all show (within the proper spectral bandwidth range) similarities with other pertinent distributions. The derived breaking probability is found to depend on the spectrum bandwidth, θ, and consequently on the significant slope, §, and/or the wave age, cp/u*, and is not restricted to narrow-band seas. The variation of breaking probability, with §, at the normalized spectral peak frequency Σp has no direct dependence on cp/u*, whereas its counterpart variations at frequencies 2Σp, and 2.5Σp both show weak dependence on the latter parameter for large values of wave age (representing mature wave fields), in accord with field observations. An overall (average) breaking probability, B, characterizing the wave field (locally) also is derived and compared with counterpart-published expressions, and their similarities and differences are properly discussed.