Role of non-resonant interactions in the evolution of nonlinear random water wave fields

Abstract

We present the results of direct numerical simulations (DNS) of the evolution of nonlinear random water wave fields. The aim of the work is to validate the hypotheses underlying the statistical theory of nonlinear dispersive waves and to clarify the role of exactly resonant, nearly resonant and non-resonant wave interactions. These basic questions are addressed by examining relatively simple wave systems consisting of a finite number of wave packets localized in Fourier space. For simulation of the long-term evolution of random water wave fields we employ an efficient DNS approach based on the integrodifferential Zakharov equation. The non-resonant cubic terms in the Hamiltonian are excluded by the canonical transformation. The proposed approach does not use a regular grid of harmonics in Fourier space. Instead, wave packets are represented by clusters of discrete Fourier harmonics.The simulations demonstrate the key importance of near-resonant interactions for the nonlinear evolution of statistical characteristics of wave fields, and show that simulations taking account of only exactly resonant interactions lead to physically meaningless results. Moreover, exact resonances can be excluded without a noticeable effect on the field evolution, provided that near-resonant interactions are retained. The field evolution is shown to be robust with respect to the details of the account taken of near-resonant interactions. For a wave system initially far from equilibrium, or driven out of equilibrium by an abrupt change of external forcing, the evolution occurs on the ‘dynamical’ time scale, that is with quadratic dependence on nonlinearity $\varepsilon$, not on the $O(\varepsilon^{-4})$ time scale predicted by the standard statistical theory. However, if a wave system is initially close to equilibrium and evolves slowly in the presence of an appropriate forcing, this evolution is in quantitative accordance with the predictions of the kinetic equation. We suggest a modified version of the kinetic equation able to describe all stages of evolution.Although the dynamic time scale of quintet interactions $\varepsilon^{-3}$ is smaller than the kinetic time scale $\varepsilon^{-4}$, they are not included in the existing statistical theory, and their effect on the evolution of wave spectra is unknown. We show that these interactions can significantly affect the spectrum evolution, although on a time scale much larger than $O(\varepsilon^{-4})$. However, for waves of high but still realistic steepness $\varepsilon\,{\sim}\,0.25$, the scales of evolution are no longer separated. By tracing the evolution of high statistical moments of the wave field, we directly verify one of the main assumptions used in the derivation of the kinetic equation: the quasi-Gaussianity of the wave holds throughout the evolution, both with and without accounting for quintet interactions.The conclusions are not confined to water waves and are applicable to a generic weakly nonlinear dispersive wave field with prohibited triad interactions.

Phase-averaged equation for water waves

“…By comparison with Monte-Carlo simulation of the nonlinear Schrödinger equation and the Zakharov equation he found O. Gramstad and M. Stiassnie good agreement with his modified kinetic equation for evolution of one-dimensional spectra. Later, Annenkov & Shrira (2006b) proposed a more general phase-averaged equation, which describes the spectral evolution on the O( −2 ) time scale. To our knowledge no real applications of such a modified kinetic equation have been presented.…”

“…The equation is derived starting from the Zakharov equation and follows roughly the same procedure as in the derivation given by Annenkov & Shrira (2006b). However, compared to Annenkov & Shrira (2006b) we include some additional higher-order contributions in the statistical closure. The derivation shares many of the same concepts as the derivation of the kinetic equation, but uses more relaxed stochastic properties.…”

Incoherent Fermi-Pasta-Ulam Recurrences and Unconstrained Thermalization Mediated by Strong Phase Correlations

“…We note the interesting aspect that fast oscillatory energy transfers among modes can be described through quasiresonant (or nonresonant) interactions in the framework of a generalized version of the wave turbulence kinetic equation (see Refs. [71,100,101] and Chap. 7 in Ref.…”

Anisotropic Wave Turbulence for Reduced Hydrodynamics with Rotationally Constrained Slow Inertial Waves

“…Note that in the case of non-resonant interactions, due to the absence of the frequency delta function, a stationary Kolmogorov (constant flux) solution cannot be obtained. The reader is also referred to the works of Janssen [57] and Annenkov and Shrira [58] for a detailed theory of quasi-resonant interactions in four wave dispersive systems.…”