Freak waves in weakly nonlinear unidirectional wave trains over a sloping bottom in shallow water

Abstract

Using a Boussinesq model with improved linear dispersion, we show numerical evidence that bottom non-uniformity can provoke significantly increased probability of freak waves as a wave field propagates into shallower water, in agreement with recent experimental results [K. Trulsen, H. Zeng, and O. Gramstad, "Laboratory evidence of freak waves provoked by non-uniform bathymetry," Phys. Fluids 24, 097101 (2012)]. Increased values of skewness, kurtosis, and probability of freak waves can be found on the shallower side of a bottom slope, with a maximum close to the end of the slope. The increased probability of freak waves is typically seen to endure some distance into the shallower domain, before it decreases and reaches a stable value depending on the depth. The maxima of the statistical parameters are observed both in the case where there is a region of constant depth after the slope, and in the case where the uphill slope is immediately followed by a downhill slope. In the case that waves propagate over a slope from shallower to deeper water, however, we do not find any increase in freak wave occurrence.

Statistics of Extreme Waves in Coastal Waters: Large Scale Experiments and Advanced Numerical Simulations

“…In line with previous studies [23,[25][26][27], second-and third-order effects appear in the frequency spectrum, and local maximum values of the statistical parameters skewness and kurtosis were observed in the shallower region. It was experimentally observed that the second-order effects rapidly weaken after the end of the slope, as shown Figure 4a.…”

“…The evidence of the link between a water depth transition and a higher probability of the occurrence of extreme waves has been shown both experimentally [23,24], and numerically [25][26][27]. There are also studies dealing with extreme wave statistics in coastal areas with different shapes of variable bathymetry.…”

Extreme waves induced by strong depth transitions: Fully nonlinear results

“…The prediction obtained in such theoretical framework showed a transition between two statistically steady states on each side of the depth transition, but it did not allow to observe the enhanced probability of extreme wave events that experimental observations revealed. 3 Gramstad et al 5 and Sergeeva et al 6 obtained trends qualitatively similar to the experimental data using one-directional Boussinesq 5 and Korteweg-de Vries (KdV) 6 models. In particular, they document the emergence of extreme-wave activity near the shallow side of bottom slopes.…”

The physics of anomalous (‘rogue’) ocean waves

“…It was observed experimentally that waves travelling from deeper water to shallower water seem to have an excess of rogue waves at the top of the slope, shown schematically in figure 12 [98]. Numerical experiments have shown qualitative agreement [99].…”