Systematic study of rogue wave probability distributions in a fourth‐order nonlinear Schrödinger equation

Abstract

[1] Nonlinear instability and refraction by ocean currents are both important mechanisms that go beyond the Rayleigh approximation and may be responsible for the formation of freak waves. In this paper, we quantitatively study nonlinear effects on the evolution of surface gravity waves on the ocean, to explore systematically the effects of various input parameters on the probability of freak wave formation. The fourth-order current-modified nonlinear Schrödinger equation (CNLS 4 ) is employed to describe the wave evolution. By solving CNLS 4 numerically, we are able to obtain quantitative predictions for the wave height distribution as a function of key environmental conditions such as average steepness, angular spread, and frequency spread of the local sea state. Additionally, we explore the spatial dependence of the wave height distribution, associated with the buildup of nonlinear development.

Linear and nonlinear rogue wave statistics in the presence of random currents

“…In the following examples, equation ( 21) is integrated numerically with the current set to zero, in order to investigate systematically and quantitatively the effect of nonlinear focusing. In nature, the interplay between linear and nonlinear mechanisms is also of considerable interest, and may give rise to even stronger enhancement in the probability of rogue wave occurrence than either effect individually, as demonstrated in section 4 (see also [42,43]).…”

“…An example of this spatial dependence appears in figure 11. Thus, a more accurate model for the total wave height distribution consists of a sum of several K-distributions, or equivalently the tail of the full distribution may be modelled by a K-distribution multiplied by a prefactor C < 1, as discussed in [43]. Nevertheless, as seen in figure 8, equation ( 13) correctly describes wave height probabilities at the ±20% level of accuracy, allows for an extremely simple one-parameter characterization of the wave height distribution, and facilitates easy comparison between the effects of linear and nonlinear focusing.…”

Changes in significant and maximum wave heights in the Norwegian Sea

“…The nonlinearity of wave--wave interaction has also been found to affect the crest height and trough depth distributions, but not the peak--to--trough wave height distributions in observations [Tayfun, 1983;Casas--Prat and Holthuijsen, 2010]. More recent laboratory and theoretical work has suggested that nonlinearity may also have some effect on wave height distribution, depending upon the state of wave development (Sluryaev and Sergeeva, 2012;Ying and Kaplan, 2012). Forristall (1978) and Gemmrich and Garrett (2011) have shown that the Weibull distribution provides a better estimate of the observed largest wave heights, i.e.…”

Branched flow and caustics in nonlinear waves

“…Here, one rather expects a superposition of plane waves where the propagation direction and the wavenumber vary in a certain range. This will affect the probability distribution of the wave intensities, and in general reduces the chance to observe rogue waves, as described in [21,22,24] for geometrical and nonlinear focusing mechanisms individually. However, in order to focus our study on the principles of the interplay of caustics and branched flows with nonlinear focusing mechanisms, we limit the initial conditions to a single plane wave.…”

“…Especially the formation of branched flows in the presence of modulation instability needs to be understood. First numerical observations have indicated that the combination of the two effects can indeed lead to higher rogue wave probabilities than the individual mechanisms [22,24].…”