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Spatial evolution of the kurtosis of steep unidirectional random waves. (English) Zbl 1461.76068

Summary: We study the evolution of unidirectional water waves from a randomly forced input condition with uncorrelated Fourier components. We examine the kurtosis of the linearised free surface as a convenient proxy for the probability of a rogue wave. We repeat the laboratory experiments of M. Onorato et al. [“Observation of strongly non-Gaussian statistics for random sea surface gravity waves in wave flume experiments”, Phys. Rev. E (3) 70, No. 6, Article ID 067302, 4 p. (2004; doi:10.1103/PhysRevE.70.067302)], both experimentally and numerically, and extend the parameter space in our numerical simulations. We consider numerical simulations based on the modified nonlinear Schrödinger equation and the fully nonlinear water wave equations, which are in good agreement. For low steepness, existing analytical models based on the nonlinear Schrödinger equation (NLS) are found to be accurate. For cases which are steep or have very narrow bandwidths, these analytical models over-predict the rate at which excess kurtosis develops. In these steep cases, the kurtosis in both our experiments and numerical simulations peaks before returning to an equilibrium level. Such transient maxima are not predicted by NLS-based analytical models. Above a certain threshold of steepness, the steady-state value of kurtosis is primarily dependent on the spectral bandwidth. We also examine how the average shape of extreme events is modified by nonlinearity over the evolution distance, showing significant asymmetry during the initial evolution, which is greatly reduced once the spectrum has reached equilibrium. The locations of the maxima in asymmetry coincide approximately with the locations of the maxima in kurtosis.

MSC:

76B15 Water waves, gravity waves; dispersion and scattering, nonlinear interaction
76E30 Nonlinear effects in hydrodynamic stability
86A05 Hydrology, hydrography, oceanography
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