Nonlinear statistics evolution of extreme wave fields over strongly reflective plane beaches

The description of complex wave processes, in addition to the shoaling problem, is often cumbersome even for the evolution of regular waves. For reflection under the regime of wave breaking, the surf similarity is generally accepted as the leading parameter controlling the reflection rates and types of breakers. While little is known about the effect of reflection rates on the formation of extreme nonlinear waves, some debate has arisen regarding whether high reflection rates enhance the nonlinearity at the tail of the wave height distribution through its Gram-Charlier approximation proxies (excess kurtosis and skewness). In this work, we provide theoretical evidence that at very steep beaches of smooth composition, the reflection rate nearing unity will tend to stabilize the excess kurtosis otherwise generated by shoaling and controlled in magnitude by the bottom slope magnitude. We further verified this result through fully nonlinear numerical simulations, reaching a good agreement.