Non-homogeneous approximation for the kurtosis evolution of shoaling rogue waves

TL;DR

This paper extends spectral analysis to describe how bathymetric changes influence the evolution of excess kurtosis in rogue waves, providing bounds and insights into their probability in different water depths.

Contribution

It introduces a non-homogeneous correction for kurtosis evolution of rogue waves over shoals, enhancing understanding of rare event probabilities in varying bathymetric conditions.

Findings

In deep water, kurtosis remains stable despite shoaling.

Shallower water causes increased wave steepness and kurtosis.

Upper bounds for kurtosis growth are established.

Abstract

Bathymetric changes have been experimentally shown to affect the occurrence of rogue waves. We recently derived a non-homogeneous correction to the spectral analysis, allowing to describe the evolution of the rogue wave probability over a shoal. Here, we extend this work to the evolution of the excess kurtosis of the surface elevation, that plays a central role in estimating rare event probabilities. Furthermore, we provide an upper bound to the excess kurtosis. In intermediate and deep water regimes, a shoal does not affect wave steepness nor bandwidth significantly, so that the vertical asymmetry between crests and troughs, the excess kurtosis, and the exceedance probability of wave height stay rather constant. In contrast, in shallower water, a sharp increase in wave steepness increases the vertical asymmetry, resulting in a growth of both the tail of the exceedance probability and the excess kurtosis.