Nonstationary wave turbulence

In the early 1960s, it was established that the stochastic initial value problem for weakly coupled wave systems has a natural asymptotic closure induced by the dispersive properties of the waves and the large separation of linear and nonlinear time scales. One is thereby led to kinetic equations for the redistribution of spectral densities via three-and four-wave resonances together with a nonlinear renormalization of the frequency. The kinetic equations have equilibrium solutions which are much richer than the familiar thermodynamic, Fermi-Dirac or Bose-Einstein spectra and admit in addition finite flux (Kolmogorov-Zakharov) solutions which describe the transfer of conserved densities (e.g. energy) between sources and sinks. There is much one can learn from the kinetic equations about the behavior of particular systems of interest including insights in connection with the phenomenon of intermittency. What we would like to convince you is that what we call weak or wave turbulence is every bit as rich as the macho turbulence of 3D hydrodynamics at high Reynolds numbers and, moreover, is analytically more tractable. It is an excellent paradigm for the study of many-body Hamiltonian systems which are driven far from equilibrium by the presence of external forcing and damping. In almost all cases, it contains within its solutions behavior which invalidates the premises on which the theory is based in some spectral range. We give some new results concerning the dynamic breakdown of the weak turbulence description and discuss the fully nonlinear and intermittent behavior which follows. These results may also be important for proving or disproving the global existence of solutions for the underlying partial differential equations. Wave turbulence is a subject to which many have made important contributions. But no contributions have been more fundamental than those of Volodja Zakharov whose 60th birthday we celebrate at this meeting. He was the first to appreciate that the kinetic equations admit a far richer class of solutions than the fluxless thermodynamic solutions of equilibrium systems and to realize the central roles that finite flux solutions play in non-equilibrium systems. It is appropriate, therefore, that we call these Kolmogorov-Zakharov (KZ) spectra. : S 0 1 6 7 -2 7 8 9 ( 0 1 ) 0 0 1 9 2 -0 1. The set up: the equation for the Fourier amplitudes. 2. Moments, cumulants.

Physical Review Letters, 1994

The Kolmogorov relation for the third order structure function is used to derive the energy spectrum in the far dissipation range (k~oo). With no unspecified constants and methods from matched asymptotic expansions, a uniformly valid form for the inertial through the dissipative ranges is obtained. An analogous energy spectrum is presented. This is compared with the results of physical and numerical experiments on the energy spectra E(k) The. theoretical predictions are found to deviate by not more than a few percent from data in the entire range of wave numbers where the energy spectrum E(k) varies by more than 30 orders of magnitude.

Arxiv preprint arXiv:1104.5413, 2011

A novel D-model of wave turbulence is presented which allows to reproduce in a single frame various nonlinear wave phenomena such as intermittency, formation and direction of energy cascades, possible growth of nonlinearity due to direct energy cascades, etc. depending on the initial state. No statistical assumptions are used, all effects are due to the behavior of distinct modes. Classical energy spectra Eø ∼ ø −ν , ν = const > 0, for dispersion function of the form ø ∼ k α , α > 0 are obtained as a particular case of a more general form of energy spectra: Eø ∼ ø −ν , 2 + α −1 ≤ ν ≤ 2(2 + α −1 ) where magnitude of ν is defined by the parameters of the initial excitation. D-model is a generic model which can be expanded into a hierarchy of more refined models including dissipation, forcing, etc. D-model can be applied to the experimental and theoretical study of numerous wave turbulent systems appearing in hydrodynamics, nonlinear optics, electrodynamics, convection theory, etc.

Nonlinearity, 2015

In the framework of the focusing Nonlinear Schrodinger (NLS) equation we study numerically the nonlinear stage of the modulation instability (MI) of the condensate. As expected, the development of the MI leads to formation of "integrable turbulence" . We study the time evolution of it's major characteristics averaged across realizations of initial data -the condensate solution seeded by small random noise with fixed statistical properties. The measured quantities are: (1) waveaction spectrum and spatial correlation function, (2) the probability density function (PDF) of wave amplitudes and their momenta, and (3) kinetic and potential energies.

Physical Review E, 2010

We present a systematic study of the dynamical scaling process leading to the establishment of the Kolmogorov-Zakharov (KZ) spectrum in weak 3-wave turbulence. In the finite capacity case, in which the transient spectrum reaches infinite frequency in finite time, the dynamical scaling exponent is anomalous in the sense that it cannot be determined from dimensional considerations. As a consequence, the transient spectrum preceding the establishment of the steady state is steeper than the KZ spectrum. Constant energy flux is actually established from right to left in frequency space after the singularity of the transient solution. From arguments based on entropy production, a steeper transient spectrum is heuristically plausible. 94.05.Lk Wave turbulence [1] concerns the statistical mechanics of systems containing large numbers of dispersive waves which interact conservatively. Such wave systems occur in a variety of contexts in nature and engineering. Commonly cited examples include gravity waves on the ocean surface, waves in Bose-Einstein condensates and magnetohydrodynamic waves in strongly magnetized plasmas. For a short review of applications see [2]. Often the waves are driven by external forcing which supplies energy (or possibly another conserved quantity such as wave-action) at a scale which is widely separated from the characteristic scale of dissipation. Since interactions among waves are conservative, turbulence results. That is to say, the physics is dominated by the flux of energy between the forcing scale and dissipation scale. This flux is mediated by the wave interactions. Theoretically, wave turbulence is more tractable than its hydrodynamic cousin. In the limit of weak nonlinearity, the theory is asymptotically closed under relatively mild assumptions on the initial wave statistics (see for a review). Asymptotic closure, resulting from the interplay of weak nonlinearity and dispersion, allows a consistent derivation of a kinetic equation describing the long time asymptotics of the frequency-space wave action density, N ω (t) , which in turn, determines the leading order behaviour of all the higher order cumulants in both Fourier and physical spaces. In this article, we limit our discussion to the so-called 3-wave kinetic equation (3WKE) describing cases where the dominant nonlinearity is quadratic and the dispersion relation admits three-wave resonances.

Theoretical and Mathematical Physics, 1996

We construct an approximate solution to the equation of energy spectrum balance for developed homogeneous isotropic turbulence in the dissipative range. This equation is closed in the framework of the statistical model we use. The model is based on the principle of maximum randomness, renormalization group technique, and e-expansion. We calculate the energy spectra and the transfer functions in the cases where the wave numbers relate to the energy-containing and inertial ranges. The Kolmogorov constant is found to be CK = 1.55. It is shown that the one-dimensional longitudinal energy spectrum calculated without any free fitting parameters and normalized in yon Karman-Hovart energy units, is in good agreement with experimental data on decaying turbulence.

This Letter demonstrates that the kinetic equations for wave turbulence, the long time statistical behavior of a sea of weakly coupled, dispersive waves, will almost always develop solutions for which the theory fails due to strongly nonlinear and intermittent events either at small or large scales.  (A.C. Newell). 0375-9601/01/$ -see front matter  2001 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 5 -9 6 0 1 ( 0 1 ) 0 0 0 1 6 -0

arXiv (Cornell University), 2021

The present work considers systems whose dynamics are governed by the nonlinear interactions among groups of 6 nonlinear waves, such as those described by the unforced quintic nonlinear Schrödinger equation. Specific parameter regimes in which ensemble-averaged dynamics of such systems with finite size are accurately described by a wave kinetic equation, as used in wave turbulence theory, are theoretically predicted. In addition, the underlying reasons that the wave kinetic equation may be a poor predictor of wave dynamics outside these regimes are also discussed. These theoretical predictions are directly verified by comparing ensemble averages of solutions to the dynamical equation with corresponding solutions of the wave kinetic equation.

Journal of Turbulence, 2004

Homogeneous isotropic turbulence subject to linearly increasing forcing is investigated as a unit problem for statistically unsteady turbulence. The transient spectral dynamics is analysed using a closure theory. A long time asymptotic state is found with k −7/3 corrections to the energy spectrum as proposed by Yoshizawa. Although the cancellation of O(Re 1/2 ) terms underlying the standard dissipation rate equation is confirmed in this asymptotic state, it is found that this equation cannot predict the transient dynamics accurately. The discrepancies are explained in terms of the basic mechanisms of vortex stretching and enstrophy destruction responsible for the evolution of the dissipation rate.

Journal of Experimental and Theoretical Physics, 1990

On the basis of an analysis of weak-turbulence Kolmogorov spectra, it is suggested that drift-wave turbulence is nonlocal: The dominant interaction is an interaction with a zonal flow, rather than a neighboring-scale interaction. The equation for nonlocal drift turbulence is derived. Its physical consequences are analyzed. In particular, two new effects are found. First, as the medium evolves, the k-space spectrum of the drift-wave turbulence separates into two unconnected components: an intense zonal flow and a jet of small-scale turbulence concentrated on an unclosed curve. Second, the very existence of a small-scale weak turbulence fixes the level of the zonal-flow turbulence.