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.
By means of direct numerical simulations (DNS) based on the integrodifferential Zakharov equation, we study the long-term evolution of nonlinear random water wave fields. For the first time, formation of powerlike Kolmogorov-type spectra corresponding to weak-turbulent inverse cascade is demonstrated by DNS, and the evolution in time of the resulting spectra is quantitatively investigated. The predictions of the statistical theory for water waves, both qualitative (formation of the direct and inverse cascades, self-similar behavior) and quantitative (the spectra exponents, specific shape of self-similar functions, the rate of time evolution) are found to be in good agreement with the DNS results, except for the initial part of the evolution, where the established statistical theory is not applicable yet and the evolution has a much faster time scale.
Long‐term evolution of random wind waves is studied by direct numerical simulation within the framework of the Zakharov equation. The emphasis is on kurtosis as a single characteristics of field departure from Gaussianity. For generic wave fields generated by a steady or changing wind, kurtosis is found to be almost entirely due to bound harmonics. This observation enables one to predict the departure of evolving wave fields from Gaussianity, capitalizing on the already existing capability of wave spectra forecasting. Kurtosis rapidly adjusts to a sharp increase of wind and slowly decreases after a drop of wind. Typically kurtosis is in the range 0.1–0.3, which implies a tangible increase of freak wave probability compared to the Rayleigh distribution. Evolution of narrow‐banded fields is qualitatively different from the generic case of wind waves: statistics is essentially non‐Gaussian, which confirms that in this special case the standard kinetic equation paradigm is inapplicable.
Kinetic equations are widely used in many branches of science to describe the evolution of random wave spectra. To examine the validity of these equations, we study numerically the long-term evolution of water wave spectra without wind input using three different models. The first model is the classical kinetic (Hasselmann) equation (KE). The second model is the generalised kinetic equation (gKE), derived employing the same statistical closure as the KE but without the assumption of quasistationarity. The third model, which we refer to as the DNS-ZE, is a direct numerical simulation algorithm based on the Zakharov integrodifferential equation, which plays the role of the primitive equation for a weakly nonlinear wave field. It does not employ any statistical assumptions. We perform a comparison of the spectral evolution of the same initial distributions without forcing, with/without a statistical closure and with/without the quasistationarity assumption. For the initial conditions, we choose two narrow-banded spectra with the same frequency distribution and different degrees of directionality. The short-term evolution ($O(10^{2})$ wave periods) of both spectra has been previously thoroughly studied experimentally and numerically using a variety of approaches. Our DNS-ZE results are validated both with existing short-term DNS by other methods and with available laboratory observations of higher-order moment (kurtosis) evolution. All three models demonstrate very close evolution of integral characteristics of the spectra, approaching with time the theoretical asymptotes of the self-similar stage of evolution. Both kinetic equations give almost identical spectral evolution, unless the spectrum is initially too narrow in angle. However, there are major differences between the DNS-ZE and gKE/KE predictions. First, the rate of angular broadening of initially narrow angular distributions is much larger for the gKE and KE than for the DNS-ZE, although the angular width does appear to tend to the same universal value at large times. Second, the shapes of the frequency spectra differ substantially (even when the nonlinearity is decreased), the DNS-ZE spectra being wider than the KE/gKE ones and having much lower spectral peaks. Third, the maximal rates of change of the spectra obtained with the DNS-ZE scale as the fourth power of nonlinearity, which corresponds to the dynamical time scale of evolution, rather than the sixth power of nonlinearity typical of the kinetic time scale exhibited by the KE. The gKE predictions fall in between. While the long-term DNS show excellent agreement with the KE predictions for integral characteristics of evolving wave spectra, the striking systematic discrepancies for a number of specific spectral characteristics call for revision of the fundamentals of the wave kinetic description.
Modeling of nonlinear random wave fields in nature (and, in particular, their most common example-wind waves in the ocean) is one of the fundamental open problems of natural sciences. The existing theoretical approaches based on the kinetic equation paradigm assume a proximity to stationarity and homogeneity. In reality this assumption is often violated and how a wave field evolves is not known. We show by direct numerical simulation that after a strong perturbation the wave field evolves on the much faster O(epsilon;{-2}) "dynamic" [rather then O(epsilon;{-4}) "kinetic"] time scale; here epsilon is the characteristic wave steepness (epsilon<<1). The phenomenon of fast evolution is universal, and it must occur whenever there is a strong external perturbation.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.
hi@scite.ai
10624 S. Eastern Ave., Ste. A-614
Henderson, NV 89052, USA
Copyright © 2024 scite LLC. All rights reserved.
Made with 💙 for researchers
Part of the Research Solutions Family.