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

Abstract

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.

Numerical modeling of rogue waves in coastal waters

“…In some situations a significant increase in the rogue wave occurrence was observed indeed, though in other cases no significant changes were found. Zeng and Trulsen (2012) considered a "deeper" regime (kh > 1.2) of wave evolution compared to numerical simulations by Sergeeva et al (2011) (kh < 0.4) and exposed the reduced kurtosis and skewness for the shallower region of transformation. Laboratory experiments of Trulsen et al (2012) embrace the "transitional" zone when the waves travel from an intermediate depth to the shallow water with kh down to 0.54.…”

“…A suitable model is the nonlinear Schrödinger equation (NLS) for the case of wave evolution in space over uneven depth. The impact of bathymetry is taken into account through depth-dependent coefficients and a shoaling term (Djordjevic and Redekopp, 1978;Zeng and Trulsen, 2012): …”

“…An important feature of the measured time series, which are in the focus of the present paper, is variable bathymetry: the sea is considered as a finite-depth basin, and the depth variation is essential. The complicated picture of changes in nonlinear waves when they travel from deep to shallow water was revealed by Sergeeva et al (2011), Zeng and Trulsen (2012), and Trulsen et al (2012). Though nonlinear wave interactions in constant-depth water are believed to trigger more rogue waves than predicted by the linear quasiGaussian statistics, these publications proved that variabledepth conditions were able to further increase the probability of rogue waves.…”

Estimation of Freak Wave Occurrence From Deep to Shallow Water Regions

“…Although the characteristics of freak wave occurrence and its prediction in deep water regions become getting clear, there are a few studies about the characteristics of freak wave propagating from deep to shallow water regions (e.g. Janssen and Onorato, 2007;Zeng and Trulsen, 2012;Trulsen et al, 2012). The most of previous studies assumed a flat bottom and quasi stationary conditions for given water depth.…”

On quantitative errors of two simplified unsteady models for simulating unidirectional nonlinear random waves on large scale in deep sea

“…It is concluded that the reason of rogue wave occurrence is because of the interaction of quasisoliton coherent structures. Meanwhile, Zeng & Trulsen 59 developed a NLSE for uneven bottom to simulate random waves propagating over 143L0 for 159 T0, and indicated that a change of water depth can provoke a spatially non-uniform distribution of kurtosis on the lee side of the slope. More recently, Sergeeva,et al 60 simulated random waves in coastal regions on a scale of 20~40 L0 for up to 80 T0 based on a variable CNLSE, and found that rogue waves are likely to occur at deeper locations.…”