We show experimental evidence that as relatively long unidirectional waves propagate over a sloping bottom, from a deeper to a shallower domain, there can be a local maximum of kurtosis and skewness close to the shallower side of the slope. We also show evidence that the probability of large wave envelope has a local maximum near the shallower side of the slope. We therefore anticipate that the probability of freak waves can have a local maximum near the shallower side of a slope for relatively long unidirectional waves.

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.

Abstract.We consider the effect of slowly varying depth on the values of skewness and kurtosis of weakly nonlinear irregular waves propagating from deeper to shallower water. It is known that the equilibrium value of kurtosis decreases with decreasing depth for waves propagating on constant depth. Waves propagating over a sloping bottom must continually adjust toward a new equilibrium state. We demonstrate that weakly nonlinear waves may need a considerable horizontal propagation distance in order to adjust to a new shallower environment, therefore the kurtosis can be notably different from the equilibrium value for each corresponding depth both on top of and beyond a bottom slope. A change of depth can provoke a wake-like spatially non-uniform distribution of kurtosis on the lee side of the slope. As an application, we anticipate that the probability of freak waves on or near the edge of the continental shelf may exhibit a rather complicated spatial structure for wave fields entering from deep sea.