Saturation of rogue wave amplification over steep shoals

Shoaling surface gravity waves induce rogue wave formation. Though commonly reduced to water waves passing over a step, nonequilibrium physics allows finite slopes to be considered in this problem. Using nonhomogeneous spectral analysis of a spatially varying energy density ratio, we describe the dependence of the amplification as a function of the slope steepness. Increasing the slope increases the amplification of rogue wave probability until this amplification saturates at steep slopes. In contrast, the increase of the down slope of a subsequent de-shoal zone leads to a monotonic decrease in the rogue wave probability, thus featuring a strong asymmetry between shoaling and de-shoaling zones. Due to the saturation of the rogue wave amplification at steep slopes, our model is applicable beyond its range of validity up to a step, thus elucidating why previous models based on a step could describe the physics of steep finite slopes. We also explain why the rogue wave probability increases over a shoal while it is lower in shallower water.