Evolution of skewness and kurtosis of weakly nonlinear unidirectional waves over a sloping bottom

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.