Sergey Nazarenko

A weak turbulence theory for incompressible mhd

We derive a weak turbulence formalism for incompressible MHD. Three-wave interactions lead to a system of kinetic equations for the spectral densities of energy and helicity. We find energy spectra solution of the kinetic equations. The constants of the spectra are computed exactly and found to depend on the amount of correlation between the velocity and the magnetic field. Comparison with several numerical simulations and models is also made.

A weak turbulence theory for incompressible magnetohydrodynamics

We derive a weak turbulence formalism for incompressible magnetohydrodynamics. Three-wave interactions lead to a system of kinetic equations for the spectral densities of energy and helicity. The kinetic equations conserve energy in all wavevector planes normal to the applied magnetic field B0 ê‖. Numerically and analytically, we find energy spectra E± ∼ k± ⊥ , such that n+ +n− = −4, where E± are the spectra of the Elsässer variables z± = v ± b in the two-dimensional case (k‖ = 0). The constants of the spectra are computed exactly and found to depend on the amount of correlation between the velocity and the magnetic field. Comparison with several numerical simulations and models is also made.

Wave turbulence and intermittency

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.

Anisotropic turbulence of shear-Alfvén waves

Weak turbulence of shear-Alfven waves is considered in the limit of strongly anisotropic pulsations that are elongated along the external magnetic field. The kinetic equation thus derived agrees with the Galtier et al. formulation of the full three-dimensional helical case when taking the proper limit. This new approach allows for significant simplification, and, as a result, the applicability conditions for the weak turbulence theory are now more transparent. It thus provides an attractive theoretical framework for describing anisotropic MHD turbulence in astrophysical contexts where a strong magnetic field is present and for which shear-Alfven waves are important.

Nonlocality and intermittency in three-dimensional turbulence

Numerical simulations are used to determine the influence of the nonlocal and local interactions on the intermittency corrections in the scaling properties of three-dimensional turbulence. We show that neglect of local interactions leads to an enhanced small-scale energy spectrum and to a significantly larger number of very intense vortices (“tornadoes”) and stronger intermittency (e.g., wider tails in the probability distribution functions of velocity increments and greater anomalous corrections). On the other hand, neglect of the nonlocal interactions results in even stronger small-scale spectrum but significantly weaker intermittency. Thus, the amount of intermittency is not determined just by the mean intensity of the small scales, but it is nontrivially shaped by the nature of the scale interactions. Namely, the role of the nonlocal interactions is to generate intense vortices responsible for intermittency and the role of the local interactions is to dissipate them. Based on these observations, a new...

Optical wave turbulence and the condensation of light

In an optical experiment, we report a wave turbulence regime that, starting with weakly nonlinear waves with randomized phases, shows an inverse cascade of photons toward the lowest wavenumbers. We show that the cascade is induced by a six-wave resonant interaction process and is characterized by increasing nonlinearity. At low wavenumbers the nonlinearity becomes strong and leads to modulational instability developing into solitons, whose number is decreasing farther along the beam.

On the nonlocal turbulence of drift type waves

Abstract Two new effects the drift turbulence can display are disclosed: (1) the turbulence spectrum in k -space separates into unconnected components of large and small scales, (2) the very presence of weak small-scale turbulence imposes rigid restrictions on powerful large-scale components.

Statistics of surface gravity wave turbulence in the space and time domains

We present experimental results on simultaneous space-time measurements for the gravity wave turbulence in a large laboratory flume. We compare these results with predictions of the weak turbulence theory (WTT) based on random waves, as well as with predictions based on the coherent singular wave crests. We see that both wavenumber and the frequency spectra are not universal and dependent on the wave strength, with some evidence in favor of WTT at larger wave intensities when the finite flume effects are minimal. We present further theoretical analysis of the role of the random and coherent waves in the wave probability density function (PDF) and the structure functions (SFs). Analyzing our experimental data we found that the random waves and the coherent structures/breaks coexist: the former show themselves in a quasi-gaussian PDF core and in the low-order SFs, and the latter - in the PDF tails and the high-order SFs. It appears that the x-space signal is more intermittent than the t-space signal, and the x-space SFs capture more singular coherent structures than do the t-space SFs. We outline an approach treating the interactions of these random and coherent components as a turbulence cycle characterized by the turbulence fluxes in both the wavenumber and the amplitude spaces.

Modulational instability of Rossby and drift waves and generation of zonal jets

We study the modulational instability of geophysical Rossby and plasma drill waves within the Charney-Hasegawa-Mima (CH M) model both theoretically, using truncated (four-mode and three-mode) models, and numerically, using direct simulations of CHM equation in the Fourier space. We review the linear theory of Gill (Geophys. Fluid Dyn., vol. 6, 1974, p. 29) and extend it to show that for strong primary waves the most unstable modes are perpendicular to the primary wave, which correspond to generation of a zonal flow if the primary wave is purely meridional. For weak waves, the maximum growth occurs for off-zonal inclined modulations that are close to being in three-wave resonance with the primary wave. Our numerical simulations confirm the theoretical predictions of the linear theory as well as the nonlinear jet pinching predicted by Manin & Nazarenko (Pit vs. Fluids, vol. 6, 1994, p. 1158). We find that, for strong primary waves, these narrow zonal jets further roll up into Karman-like vortex streets, and at this moment the truncated models fail. For weak primary waves, the growth of the unstable mode reverses and the system oscillates between a dominant jet and a dominate primary wave, so that the truncated description holds for longer. The two-dimensional vortex streets appear to be more stable than purely one-dimensional zonal jets, and their zonal-averaged speed can reach amplitudes much stronger than is allowed by the Rayleigh-Kuo instability criterion for the one-dimensional case. In the long term, the system transitions to turbulence helped by the vortex-pairing instability (for strong waves) and the resonant wave wave interactions (for weak waves).

Noisy spectra, long correlations, and intermittency in wave turbulence.

We study the k-space fluctuations of the wave action about its mean spectrum in the turbulence of dispersive waves. We use a minimal model based on the random phase approximation (RPA) and derive evolution equations for the arbitrary-order one-point moments of the wave intensity in the wave-number space. The first equation in this series is the familiar kinetic equation for the mean wave-action spectrum, whereas the second and higher equations describe the fluctuations about this mean spectrum. The fluctuations exhibit a nontrivial dynamics if some long coordinate-space correlations are present in the system, as it is the case in typical numerical and laboratory experiments. Without such long-range correlations, the fluctuations are trivially fixed at their Gaussian values and cannot evolve even if the wave field itself is non-Gaussian in the coordinate space. Unlike the previous approaches based on smooth initial k-space cumulants, the RPA model works even for extreme cases where the k-space fluctuations are absent or very large and intermittent. We show that any initial non-Gaussianity at small amplitudes propagates without change toward the high amplitudes at each fixed wave number. At each fixed amplitude, however, the probability distribution function becomes Gaussian at large time.