Showing papers by "Vladimir E. Zakharov published in 2006"

Freak waves as nonlinear stage of Stokes wave modulation instability

Abstract: Numerical simulation of evolution of nonlinear gravity waves is presented. Simulation is done using two-dimensional code, based on conformal mapping of the fluid to the lower half-plane. We have considered two problems: (i) modulation instability of wave train and (ii) evolution of NLSE solitons with different steepness of carrier wave. In both cases we have observed formation of freak waves.

Flux balance and self-similar laws of wind wave growth

Abstract: is a core of all the modern wind-wave prediction models It describes evolution of wave action spectral density N(k, t) due to four-wave nonlinear resonant interactions (the so-called collision integral Snl) and external forcing Sf The fundamental solutions of the conservative Hasselmann equation have been found in the sixties (Zakharov & Filonenko 1966, Zakharov 1966, Zakharov & Zaslavskii 1982) These solutions represent remarkable examples of the so-called flux solutions that provide constant fluxes of the first integrals: energy, action and momentum from infinitely small or large scales to opposite infinity Energy and momentum are conserved formally only within the Hasselmann equation: they can leak to high frequency Thus, the equation is not “complete”, describe an “open system” and, in this sense, is not physically correct The external input Sf appears to be of key importance for the Hasselmann equation and makes the Hasselmann equation both physically and mathematically correct This is in contrast with the classic Boltzmann equation for ideal gas dynamics which solutions are unique at any initial data and conserve action, energy and momentum A correct account of external forcing is not a trivial problem for the case of wind-driven waves: the wave growth and dissipation are governed by multiple physical mechanisms which mathematical and physical description is not well elaborated yet Experimental parameterizations of Sf by different authors give dispersion of magnitudes that exceeds the magnitude of Sf itself (see Hsiao & Shemdin 1983, Plant 1982, Snyder et al 1981, Stewart 1974, Donelan & Pierson-jr 1987) The basic feature of wind-wave dynamics can help to resolve the difficulty of description of external forcing: in a wide range of physical conditions the nonlinear transfer term Snl dominates over external forcing term Sf Thus, an asymptotic method can be developed for the Hasselmann equation (1) The procedure “split” the wind-wave balance into two parts: spectra are described by conservative Hasselmann equation

Freak waves. Theory and experiments

Abstract: We present numerical simulation of several problem related to free surface hydrodynamics: a) when nonlinear Shredinger approximation breaks, b) nonlinear stage of modulation instability freak wave formation, c) stability analysis of the free surface hydrodynamics. Simulation is done using two-dimensional code, based on conformal mapping of the fluid to the lower half-plane.