Slope effect on rogue wave occurrence: saturation at steep shoals and unifying picture

Shoaling surface gravity waves is a process that still calls for a thorough understanding of how it enhances rogue wave formation. Though commonly reduced to water waves passing over a step, the influence of the slope steepness on rogue wave enhancement over a shoal has been demonstrated in numerical simulations. We analytically tackle this with non-equilibrium physics of a spatially varying energy density. While the shoal causes an energy density redistribution and enhances rogue wave occurrence due to a decrease in water depth, the slope effect on the exceedance probability can be interpreted as a second redistribution of the wave statistics. In the presence of a strong departure from a zero-mean water level due to a set-down/set-up the potential energy density is affected by a slope-induced correction. In the case of a shoal, such energy disturbance decreases the total potential energy due to a set-down as compared to linear homogeneous waves, thereby increasing the effect of the energy redistribution. Conversely, a set-up induced by wave-breaking would cause the potential energy density to increase, and so we would observe a decrease in the exceedance probability.

 

Increasing the slope increases the amplification of rogue wave probability until this amplification saturates at steep slopes. The response of the set-down to the steep slope transition past the saturation point is slower than the depth transition itself, because the effect of lowering the mean water level over the slope balances the pace of the depth transition itself. In contrast, a larger down slope of a subsequent de-shoal zone leads to a stronger decrease in the rogue wave probability. This is because the faster increase of the set-up due to steeper slopes is not balanced by the depth transition, as the mean depth will increase rather than decrease. Thus, a strong asymmetry between shoaling and de-shoaling zones develops. We show that models based on a step can effectively describe the physics of steep finite slopes owing to the saturation of the rogue wave amplification at steep slopes.

 

Our framework poses a clear unifying picture for wave statistics and energetics transitioning from deep to shallow waters. Waves propagating in deep water will not have their energy affected by the slope and tend to keep a constant rogue wave probability, while in intermediate water the wave energy density will be redistributed due to depth effects on the steepness, vertical asymmetry, and mean water level, ultimately increasing rogue wave likelihoods. Finally, in shallow water the effects on steepness and vertical asymmetry still exist, but the quick divergence of the super-harmonics halts the energy redistribution while the set-up inverts the effect of the latter. Therefore, in the absence of any ocean process besides shoaling, we unify within a single physical mechanism the seemingly contradictory observations of Gaussian statistics in deep water, super-Gaussian (i.e. above) in intermediate water and sub-Gaussian (i.e. below) in shallow waters.