A. R. Osborne

Freak Waves in Random Oceanic Sea States

Freak waves are very large, rare events in a random ocean wave train. Here we study their generation in a random sea state characterized by the Joint North Sea Wave Project spectrum. We assume, to cubic order in nonlinearity, that the wave dynamics are governed by the nonlinear Schrödinger (NLS) equation. We show from extensive numerical simulations of the NLS equation how freak waves in a random sea state are more likely to occur for large values of the Phillips parameter alpha and the enhancement coefficient gamma. Comparison with linear simulations is also reported.

The nonlinear dynamics of rogue waves and holes in deep-water gravity wave trains

Abstract Rogue waves are rare “giant”, “freak”, “monster” or “steep wave” events in nonlinear deep water gravity waves which occasionally rise up to surprising heights above the background wave field. Holes are deep troughs which occur before and/or after the largest rogue crests. The dynamical behavior of these giant waves is here addressed as solutions of the nonlinear Schrodinger equation in both 1+1 and 2+1 dimensions. We discuss analytical results for 1+1 dimensions and demonstrate numerically, for certain sets of initial conditions, the ubiquitous occurrence of rogue waves and holes in 2+1 spatial dimensions. A typical wave field evidently consists of a background of stable wave modes punctuated by the intermittent upthrusting of unstable rogue waves.

Extreme wave events in directional, random oceanic sea states

We discuss the effect of the directional spreading on the occurrence of extreme wave events. We numerically integrate the envelope equation recently proposed by Trulsen et al. [Phys. Fluids 12, 2432 (2000)] as a weakly nonlinear model for realistic oceanic gravity waves. Initial conditions for numerical simulations are characterized by the spatial JONSWAP power spectrum for several values of the significant wave height, steepness, and directional spreading. We show that by increasing the directionality of the initial spectrum the appearance of extreme events is reduced.

Statistical properties of mechanically generated surface gravity waves: a laboratory experiment in a three-dimensional wave basin

A wave basin experiment has been performed in the MARINTEK laboratories, in one of the largest existing three-dimensional wave tanks in the world. The aim of the experiment is to investigate the effects of directional energy distribution on the statistical properties of surface gravity waves. Different degrees of directionality have been considered, starting from long-crested waves up to directional distributions with a spread of ±30° at the spectral peak. Particular attention is given to the tails of the distribution function of the surface elevation, wave heights and wave crests. Comparison with a simplified model based on second-order theory is reported. The results show that for long-crested, steep and narrow-banded waves, the second-order theory underestimates the probability of occurrence of large waves. As directional effects are included, the departure from second-order theory becomes less accentuated and the surface elevation is characterized by weak deviations from Gaussian statistics.

Modulational instability and non-Gaussian statistics in experimental random water-wave trains

We study random, long-crested surface gravity waves in the laboratory environment. Starting with wave spectra characterized by random phases we consider the development of the modulational instability and the consequent formation of large amplitude waves. We address both dynamical and statistical interpretations of the experimental data. While it is well known that the Stokes wave nonlinearity leads to non-Gaussian statistics, we also find that the presence of the modulational instability is responsible for the departure from a Gaussian behavior, indicating that, for particular parameters in the wave spectrum, coherent unstable modes are quite prevalent, leading to the occurrence of what might be called a “rogue sea.” Statistical results are also compared with ensemble numerical simulations of the Dysthe equation.

Solitons, cnoidal waves and nonlinear interactions in shallow-water ocean surface waves

Abstract We analyze shallow-water surface wave data from the Adriatic Sea using a nonlinear generalization of Fourier analysis based upon the periodic inverse scattering transform in the θ-function representation for the Korteweg-de Vries (KdV) equation. While linear Fourier analysis superposes sine waves, the nonlinear Fourier approach superposes cnoidal waves (the travelling wave solution to KdV) plus their mutual, nonlinear interactions . A new procedure is presented for the nonlinear low-pass and band-pass filtering of measured wave trains. We apply the approach to a measured time series and discuss the dynamics of solitons and the physics of the nonlinear interactions in terms of global, spatio-temporal phase shifts amongst the cnoidal waves.

Second-Order Theory and Setup in Surface Gravity Waves: A Comparison with Experimental Data

Abstract The second-order, three-dimensional, finite-depth wave theory is here used to investigate the statistical properties of the surface elevation and wave crests of field data from Lake George, Australia. A direct comparison of experimental and numerical data shows that, as long as the nonlinearity is small, the second-order model describes the statistical properties of field data very accurately. By low-pass filtering the Lake George time series, there is evidence that some energetic wave groups are accompanied by a setup instead of a setdown. A numerical study of the coupling coefficient of the second-order model reveals that such an experimental result is consistent with the second-order theory, provided directional spreading is included in the wave spectrum. In particular, the coupling coefficient of the second-order difference contribution predicts a setup as a result of the interaction of two waves with the same frequency but with different directions. This result is also confirmed by numerical...

Four-wave resonant interactions in the classical quadratic Boussinesq equations

We investigate theoretically the irreversibile energy transfer in flat bottom shallow water waves. Starting from the oldest weakly nonlinear dispersive wave model in shallow water, i.e. the original quadratic Boussinesq equations, and by developing a statistical theory (kinetic equation) of the aforementioned equations, we show that the four-wave resonant interactions are naturally part of the shallow water wave dynamics. These interactions are responsible for a constant flux of energy in the wave spectrum, i.e. an energy cascade towards high wavenumbers, leading to a power law in the wave spectrum of the form of k −3/4 . The nonlinear time scale of the interaction is found to be of the order of ( h / a ) 4 wave periods, with a the wave amplitude and h the water depth. We also compare the kinetic equation arising from the Boussinesq equations with the arbitrary-depth Hasselmann equation and show that, in the limit of shallow water, the two equations coincide. It is found that in the narrow band case, both in one-dimensional propagation and in the weakly two-dimensional case, there is no irreversible energy transfer because the coupling coefficient in the kinetic equation turns out to be identically zero on the resonant manifold.

Interaction of two quasi--monochromatic waves in shallow water

We study the nonlinear interaction of waves propagating in the same direction in shallow water characterized by a double-peaked power spectrum. The starting point is the prototypical equation for weakly nonlinear unidirectional waves in shallow water, i.e., the Korteweg–de Vries equation. In the framework of envelope equations, using a multiple-scale technique and under the hypothesis of narrow-banded spectra, a system of two coupled nonlinear Schrodinger equations is derived. The validity of the resulting model and the stability of their plane wave solutions is discussed. We show that when retaining higher order dispersive terms in the system, plane wave solutions become modulationally unstable.

The numerical inverse scattering transform for the periodic Korteweg-de Vries equation

Abstract We introduce inverse scattering transform (IST) algorithms for obtaining the spectrum of complex wave trains governed by the periodic Korteweg-de Vries (KdV) equation and for constructing solutions to this equation. A scattering matrix formulation is implemented to determine the Floquet spectrum and an iterated similarity transformation is exploited to compute the hyperelliptic osscilation modes. A linear superposition law of the hyperelliptic modes, a generalization of ordinary Fourier series, is used to construct general wave train solutions to the KdV equation. The algorithms should be useful for nonlinear Fourier analysis of computer generated wave trains and experimentally measured space or time series.