Energetics and stochastics of extreme waves breaking over a symmetrical breakwater

Rogue wave formation and enhancement over coastal areas have been documented over the last decade. This is in apparent contradiction with the observed low rogue wave probability near the surf zone. Existing theories and experiments describe rogue wave amplification in this regime; however, they do so without considering wave breaking processes. To address this gap, we consider fully nonlinear effects of wave breaking through the proxy of height-to-depth ratio, spatial changes on the wavenumber through the Wentzel–Kramers–Brillouin approximation, and slope-corrected refraction mild-slope equations on the energetics of irregular wave fields traveling over a breakwater. We show that by increasing the significant wave height toward the breaking limit, the kinetic energy grows faster than the variance of the surface elevation due to nonlinearity. Thus, it decreases the kurtosis, albeit not to the point of getting sub-Gaussian statistics. We thereby resolve the apparent paradox of the occurrence probability of rogue waves increasing at the beginning of shoaling but subsequently decreasing when wave breaking becomes dominant. Motivated by these theoretical developments, we experimentally probe inhomogeneous wave fields nearing the wave-breaking regime. We conduct unidirectional irregular wave experiments in a 30-m-long wave flume, generating broad-banded waves over a symmetric submerged breakwater featuring a bottom slope of 1/5, allowing detailed characterization of spectral evolution and the persistence of elevated excess kurtosis even near the breaking limit for this steep slope. Through this analysis, we confirm that the excess kurtosis can still be large if the bottom slope is steep, and its maximum value is at least four times larger than in other ocean processes, occurring atop the breakwater and about half of its deep water peak wavelength distance after the shoal. As these conditions are typical near shorelines, this understanding is key to coastal areas.