Effect of shoaling length on rogue wave occurrence

The impact of shoaling on linear water waves is well-known, but it has only been recently found to significantly amplify both the intensity and frequency of rogue waves in nonlinear irregular wave trains atop coastal shoals. At least qualitatively, this effect has been partially attributed to the ``rapid'' nature of the shoaling process, i.e., shoaling occurs over a distance far shorter than that required for waves to modulate themselves and adapt to the reduced water depth. Through the development of a theoretical model and highly accurate nonlinear simulations, we disentangle the respective effects of the slope length and the slope gradient of a shoal and focus on the slope length to investigate the rapidness of the shoaling process on the evolution of key statistical and spectral sea-state parameters. Provided the shoal slope is 1/10 or steeper, our results indicate that the non-equilibrium dynamics is involved even for rather short shoaling lengths and becomes dominant in the regime of large lengths. When the non-equilibrium dynamics governs the wave evolution, further extending the slope length no longer influences the statistical and spectral measures. Thus, the shoaling effect on rogue waves is mainly driven by the slope magnitude rather than the slope length. Moreover, the simulations show that a higher cut-off frequency of the wave spectrum has a smaller impact on wave statistics than expected for a flat bottom in deep water and that insufficient attenuation of low-frequency waves at the downstream domain boundary has notable influence on wave statistics atop the shoal.