Kristian B. Dysthe

A modified nonlinear Schrödinger equation for broader bandwidth gravity waves on deep water

Abstract The modified nonlinear Schrodinger equation of Dysthe [ Proc. Roy. Soc. Lond. Ser. A, 369 , 105–114 (1979)] is extended by relaxing the narrow bandwidth constraint to make it more suitable for application to a realistic ocean wave spectrum. The new equation limits the bandwidth of unstable wave number perturbations of a Stokes wave in good agreement with the exact results of McLean et al. [ Phys. Rev. Lett. 46 , 817–820 (1981)]. Results are presented for the parametric bifurcation boundary between collinear and oblique most unstable side band perturbations of a Stokes wave.

Probability distributions of surface gravity waves during spectral changes

Simulations have been performed with a fairly narrow band numerical gravity wave model (higher-order nls type) and a computational domain of dimensions

On weakly nonlinear modulation of waves on deep water

128\times 128

Evolution of a narrow-band spectrum of random surface gravity waves

typical wavelengths. The simulations are initiated with

A thermal oscillating two‐stream instability

\thicksim6\times10^{4}

Frequency downshift in three-dimensional wave trains in a deep basin

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, 1980, 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).

Spatial Extreme Value Analysis of Nonlinear Simulations of Random Surface Waves

We propose a new approach for modeling weakly nonlinear waves, based on enhancing truncated amplitude equations with exact linear dispersion. Our example is based on the nonlinear Schrodinger (NLS) equation for deep-water waves. The enhanced NLS equation reproduces exactly the conditions for nonlinear four-wave resonance (the “figure 8” of Phillips) even for bandwidths greater than unity. Sideband instability for uniform Stokes waves is limited to finite bandwidths only, and agrees well with exact results of McLean; therefore, sideband instability cannot produce energy leakage to high-wave-number modes for the enhanced equation, as reported previously for the NLS equation. The new equation is extractable from the Zakharov integral equation, and can be regarded as an intermediate between the latter and the NLS equation. Being solvable numerically at no additional cost in comparison with the NLS equation, the new model is physically and numerically attractive for investigation of wave evolution.

Frequency Down-Shift Through Self Modulation and Breaking

Numerical simulations of the evolution of gravity wave spectra of fairly narrowbandwidth have been performed both for two and three dimensions. Simulationsusing the nonlinear Schr¨odinger (NLS) equation approximately verify the stabilitycriteria of Alber (1978) in the two-dimensional but not in the three-dimensional case.Using a m odified NLS equation (Trulsen et al. 2000) the spectra ‘relax’ towards aquasi-stationary state on a timescale (

Influence of variable Froude number on waves generated by ships in shallow water

A theory for the oscillating two‐stream instability, in which the Ohmic heating of the electrons constitutes the nonlinearity, is developed for an inhomogeneous and magnetized plasma. Its possible role in explaining short‐scale, field‐aligned irregularities observed in ionospheric heating experiments is emphasized. The theory predicts that the initial growth of such irregularities is centered around the level of upper hybrid resonance. Furthermore, plane disturbances nearly parallel to the magnetic meridian plane have the largest growth rates. Expressions for threshold, growth rate, and transverse scale of maximum growth are obtained. Special attention is paid to the transport theory, since the physical picture depends heavily on the kind of electron collisions which dominate. This is due to the velocity dependence of collison frequencies, which gives rise to the thermal forces.

Refraction of gravity waves by weak current gradients

The conservative evolution of weakly nonlinear narrow-banded gravity waves in deep water is investigated numerically with a modified nonlinear Schrodinger equation, for application to wide wave tanks. When the evolution is constrained to two dimensions, no permanent shift of the peak of the spectrum is observed. In three dimensions, allowing for oblique sideband perturbations, the peak of the spectrum is permanently downshifted. Dissipation or wave breaking may therefore not be necessary to produce a permanent downshift. The emergence of a standing wave across the tank is also predicted.