The equilibrium range cascades of wind-generated waves (original) (raw)
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Physica D: Nonlinear Phenomena, 2012
We consider a general model of Hamiltonian wave systems with triple resonances, in the standard kinetic limit of a continuum of weakly interacting dispersive waves with random phases. We show in this limit that the leadingorder, asymptotically valid dynamical equation for multimode amplitude distributions is not the well-known equation of Peierls (also, Brout & Prigogine and Zaslavskii & Sagdeev), but is instead a reduced equation containing only a subset of the terms in that equation. Our equations are consistent with the Peierls equation in that the additional terms in the latter vanish as inverse powers of volume in the large-box limit. The equations that we derive are the direct analogue of the Boltzmann hierarchy obtained from the BBGKY hierarchy in the low-density limit for gases. We show that the asymptotic multimode equations possess factorized solutions for factorized initial data, which correspond to preservation in time of the property of "random phases & amplitudes". The factors satisfy the equations for the 1-mode probability density functions (PDF's) previously derived by Choi et al. and Jakobsen & Newell. Analogous to the Klimontovich density in the kinetic theory of gases, we introduce the concepts of the "empirical spectrum" and the "empirical 1-mode PDF". We show that the factorization of the hierarchy equations implies that these quantities are self-averaging: they satisfy the wave-kinetic closure equations of the spectrum and 1-mode PDF for almost any selection of phases and amplitudes from the initial ensemble. We show that both of these closure equations satisfy an H-theorem for an entropy defined by Boltzmann's prescription S = k B log W. We also characterize the general solutions of our multimode distribution equations, for initial conditions with random phases but with no statistical assumptions on the amplitudes. Analogous to a result of Spohn for the Boltzmann hierarchy, these are "super-statistical solutions" that correspond to ensembles of solutions of the wave-kinetic closure equations with random initial conditions or random forces. On the basis of our results, we discuss possible kinetic explanations of intermittency and non-Gaussian statistics in wave turbulence. In particular, we advance the explanation of a "superturbulence" produced by stochastic or turbulent solutions of the wave kinetic equations themselves.
Role of non-resonant interactions in the evolution of nonlinear random water wave fields
Journal of Fluid Mechanics, 2006
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 longterm 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.
2016
Wave-kinetic theory has been developed to describe the statistical dynamics of weakly nonlinear, dispersive waves. In the first part of this dissertation, we derive the wave-kinetic equations formally from a general model of Hamiltonian wave systems, in the standard limit of a continuum of weakly interacting dispersive waves with random phases. In this asymptotic limit we show that the correct dynamical equation for multi-mode amplitude distributions is not the well-known Peierls equation but is instead a reduced equation with only a subset of the terms in that equation. The equations that we derive are the direct analogue of the Boltzmann hierarchy obtained from the BBGKY hierarchy in the low-density limit for gases. We show that the asymptotic multi-mode equations possess factorized solutions for factorized initial data, which correspond to preservation in time of the property of "random phases & amplitudes". The factors satisfy the equations for the 1-mode probability density functions (PDF's) previously derived by Jakobsen & Newell and Choi et al. Analogous to the Klimontovich density in the kinetic theory of gases, we introduce the concepts of the "empirical spectrum" and the "empirical 1-mode PDF". We show ii ABSTRACT that the factorization of the hierarchy equations implies that these quantities are self-averaging: they satisfy the wave-kinetic equations of the spectrum and 1-mode PDF for almost any selection of phases and amplitudes from the initial ensemble. We show that both of these equations satisfy an H-theorem for an entropy defined by Boltzmann's prescription S = k B log W. We also characterize the general solutions of our multi-mode distribution equations, for initial conditions with random phases but with no statistical assumptions on the amplitudes. Analogous to a result of Spohn for the Boltzmann hierarchy, these are "super-statistical solutions" that correspond to ensembles of solutions of the wave-kinetic equations with random initial conditions or random forces. On the basis of our results, we discuss possible kinetic explanations of intermittency and non-Gaussian statistics in wave turbulence. In particular, we advance the explanation of a "super-turbulence" produced by stochastic or turbulent solutions of the wave-kinetic equations themselves. In the second part of the dissertation, we investigate a key assumption of wavekinetic theory-dispersivity. We show that systems which are generally dispersive can have resonant sets of wave modes with identical group velocities, leading to a local breakdown of dispersivity. This shows up as a geometric singularity of the resonant manifold and possibly as an infinite phase measure in the collision integral. Such singularities occur widely for classical wave systems, including acoustical waves, Rossby waves, helical waves in rotating fluids, light waves in nonlinear optics and also in quantum transport, e.g. kinetics of electron-hole excitations (matter waves) iii ABSTRACT in graphene. These singularities are the exact analogue of the critical points found by Van Hove in 1953 for phonon dispersion relations in crystals. The importance of these singularities in wave kinetics depends on the dimension of phase space D = (N − 2)d (d physical space dimension, N the number of waves in resonance) and the degree of degeneracy δ of the critical points. Following Van Hove, we show that non-degenerate singularities lead to finite phase measures for D > 2 but produce divergences when D ≤ 2 and possible breakdown of wave kinetics if the collision integral itself becomes too large (or even infinite). Similar divergences and possible breakdown can occur for degenerate singularities, when D − δ ≤ 2, as we find for several physical examples, including electron-hole kinetics in graphene. When the standard kinetic equation breaks down, one must develop a new singular wave kinetics. We discuss approaches from pioneering 1971 work of Newell & Aucoin on multi-scale perturbation theory for acoustic waves and field-theoretic methods based on exact Schwinger-Dyson integral equations for the wave dynamics.
On the kinetic equation in Zakharov's wave turbulence theory for capillary waves
2017
The wave turbulence equation is an effective kinetic equation that describes the dynamics of wave spectrum in weakly nonlinear and dispersive media. Such a kinetic model has been derived by physicists in the sixties, though the well-posedness theory remains open, due to the complexity of resonant interaction kernels. In this paper, we provide a global unique radial strong solution, the first such a result, to the wave turbulence equation for capillary waves.
Modelling Transient Sea States with the Generalised Kinetic Equation
Rogue and Shock Waves in Nonlinear Dispersive Media, 2016
For historical and technical reasons evolution of random weakly nonlinear wave fields so far has been studied primarily in a quasi-stationary environment, where the main modelling tool is the kinetic equation. In the context of oceanic waves sharp changes of wind do occur quite often and can generate transient sea states with characteristic timescales of up to hundreds of wave periods. It is of great fundamental and practical interest to understand wave field behaviour during shortlived and transient events. At present nothing is known about such ephemeral sea states. One, but not the only, reason was that there were no adequate modelling tools. The generalised kinetic equation (gKE) derived without assumptions of quasistationarity seems to fill this gap. Here we study transient events with the gKE aiming to understand what is going during such events and capabilities of the gKE in capturing them. We find how wave spectra evolve being subjected to sharp changes of wind, while tracing in parallel the concomitant evolution of higher moments characterizing the field departure from gaussianity. We demonstrated the capability of the gKE to capture short-lived events, in particular, we found sharp brief increase of kurtosis during squalls, which suggests significant increase of the likelihood of freak waves during such events. Although the study was focussed upon wind wave context the approach is generic and is transferrable to random weakly nonlinear wave fields of other nature.