Single-spectrum prediction of kurtosis of water waves in a nonconservative model

Abstract

We study statistical properties after a sudden episode of wind for water waves propagating in one direction. A wave with random initial conditions is propagated using a forced-damped higher order Nonlinear Schrödinger equation (NLS). During the wind episode, the wave action increases, the spectrum broadens, the spectral mean shifts up and the Benjamin-Feir index (BFI) and the kurtosis increase. Conversely, after the wind episode, the opposite occurs for each quantity. The kurtosis of the wave height distribution is considered the main parameter that can indicate whether rogue waves are likely to occur in a sea state, and the BFI is often mentioned as a means to predict the kurtosis. However, we find that while there is indeed a quadratic relation between these two, this relationship is dependent on the details of the forcing and damping. Instead, a simple and robust quadratic relation does exist between the kurtosis and the bandwidth. This could allow for a single-spectrum assessment of the likelihood of rogue waves in a given sea state. In addition, as the kurtosis depends strongly on the damping and forcing coefficients, by combining the bandwidth measurement with the damping coefficient, the evolution of the kurtosis after the wind episode can be predicted. * maura.brunetti@unige.ch

Evolution of Spectral Distributions in Deep-Water Constant Vorticity Flows

“…Finally, we study the behavior of the kurtosis as a function of the BFI parameter. The quadratic relationship found in [5] and [16] is shown to hold, though the nature of the fit is clearly dependent on the vorticity, conforming to the dependence of the fit on physical parameters found in [16].…”

“…was necessary to remove artificial instabilities. Similar issues with Dysthe-like equations have been noted in [16] among other sources; see Appendix for a discussion of this issue. Moreover, the coefficientsα j are readily derived, but their forms are cumbersome and thus we omit explicitly writing them for the sake of readability.…”

“…Our results likewise support this. However, also shown in [5] and followed up on in several later works, see [3,15,16], it is shown that the kurtosis of the corresponding quasi-steady distribution of the surface waves, sayk(τ f ), where we take τ f to be the timescale over which MI acts, is a quadratic function of the the BFI. We likewise aim to establish a similar relationship, thereby maintaining the utility of the BFI insofar as it then helps predict a key feature of nonlinear-random wave fields.…”

“…Later work has simulated the impact of wind and damping on the interplay between the BFI and the kurtosis of a wave field; see [15,16]. However, the impact of vorticity was not taken into account in these works, thereby ignoring a central mechanism for the transfer of energy across length scales in oceanic flows.…”

“…Moving beyond just examining the properties of the NLSE with vorticity, we look at the statistical properties of solutions to a higher order model, the vor-Dysthe equation (VDE) derived in [17]. As shown in [3,5] and [16], the higher order terms associated with the Dysthe equation can have significant impacts on the statistics of the waves, so it is a nontrivial question to examine how the vorticity interacts with these higher order nonlinearities. Moreover, as shown in [9], the NLSE can rapidly lose any quantitative predictive power in experimental settings while the Dysthe equation provides far more accurate results over a relatively wide range of experimental conditions.…”