Probability distributions of surface gravity waves during spectral changes

Abstract

Simulations have been performed with a fairly narrow band numerical gravity wave model (higher-order NLS type) and a computational domain of dimensions 128 × 128 typical wavelengths. The simulations are initiated with s 6 × 10 4 Fourier modes corresponding to truncated JONSWAP spectra and different angular distributions giving both short-and long-crested waves. A development of the spectra on the so-called Benjamin-Feir timescale is seen, similar to the one reported by Dysthe et al. (J. Fluid Mech. vol. 478, 2003, P. 1). The probability distributions of surface elevation and crest height are found to fit theoretical distributions found by Tayfun (J. Geophys. Res. vol. 85, 1980Res. vol. 85, , p. 1548) very well for elevations up to four standard deviations (for realistic angular spectral distributions). Moreover, in this range of the distributions, the influence of the spectral evolution seems insignificant. For the extreme parts of the distributions a significant correlation with the spectral change can be seen for very long-crested waves. For this case we find that the density of large waves increases during spectral change, in agreement with a recent experimental study by Onorato et al. (J. Fluid Mech. 2004 submitted).

Gravity Wave Turbulence in a Laboratory Flume

“…The PDF for stronger nonlinearities was obtained by Tayfun [15] using a model where the wave field is made of independent weakly nonlinear Stokes waves whose first harmonics are gaussian. Tayfun distributions where found to be in good agreement with numerical simulations with wide-angle quasi-isotropic wavefields [16] and to much lesser extent in narrow-angle distributions [17,18]. Numerically obtained PDF of spectral intensity in a small band around ω equal to the eight minimal wave frequencies [8,10].…”

Emergence of coherent wave groups in deep-water random sea

“…Also in this case results are available from both laboratory experiments [25,29] and numerical simulations [15,33,12,28,32,36]. A central finding of such studies consists of the intensification of extreme wave activity for high values of the Benjamin-Feir index (see § 2.1), i.e., when the power spectrum of the surface elevation is sharply peaked around the fundamental wavenumber.…”

“…Other recent studies [34] suggest that mechanisms may exist in the ocean capable of leading to narrow-banded spectra, therefore making this kind of regime of potential interest for oceanic waves. Narrow-banded spectra correspond to slowly modulated wave fields in physical space, in this condition modulational (or Benjamin-Feir, or side-band) instability arises and develops until a nonlinear saturation is attained [33,32]. As a result of such an instability, the spectral band-width broadens, and the associated value of the BenjaminFeir index decreases below its critical value.…”

“…This fact, together with observations of high wave amplifications occurring during the disruption of uniform wave trains undergoing Benjamin-Feir instability [6], underpins the idea that a connection exists between rogue waves and Benjamin-Feir instability. Different authors [33,37,32] have suggested that rogue waves are a result of such transient evolution.…”

Weak Statistical Constraints for Variational Stereo Imaging of Oceanic Waves

“…The solution is obtained in the context of active surfaces by deforming an initial surface via a gradient descent PDE derived from the necessary optimality conditions of the energy functional, the so-called Euler-Lagrange (EL) equations. In parallel to the advances in vision tools, the oceanographic community has developed statistical and spectral models for the characterization of oceanic sea states [9,19,15,16] that clearly indicate that oceanic waves are quasi-Gaussian in nature.…”