Abstract. 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 B 0ê . Numerically and analytically, we find energy spectra E ± ∼ k n± ⊥ , 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 is the statistical mechanics of random waves with a broadband spectrum interacting via non-linearity. To understand its difference from non-random well-tuned coherent waves, one could compare the sound of thunder to a piece of classical music. Wave turbulence is surprisingly common and important in a great variety of physical settings, starting with the most familiar ocean waves to waves at quantum scales or to much longer waves in astrophysics. We will provide a basic overview of the wave turbulence ideas, approaches and main results emphasising the physics of the phenomena and using qualitative descriptions avoiding, whenever possible, involved mathematical derivations. In particular, dimensional analysis will be used for obtaining the key scaling solutions in wave turbulence -Kolmogorov-Zakharov (KZ) spectra.

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.

Weak turbulence of shear-Alfvén 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-Alfvén waves are important.

A phenomenological turbulence model in which the energy spectrum obeys a nonlinear diffusion equation is analyzed. The general steady state contains a nonlinear mixture of the constant-flux Kolmogorov and fluxless thermodynamic components. Such "warm cascade" solutions describe a bottleneck phenomenon of spectrum stagnation near the dissipative scale. Transient self-similar solutions describing a finite-time formation of steady cascades are analyzed and found to exhibit nontrivial scaling behavior.

We present experimental results for water wave turbulence excited by piston-like programmed wavemakers in a water flume with dimensions 6 × 12 × 1.5 meters. Our main finding is that for a wide range of excitation amplitudes the energy spectrum has a power-law scaling, E ω ∼ ω −ν . These scalings were achieved in up to one-decade wide frequency range, which is significantly wider than the range available in field observations and in numerical simulations. However, exponent ν appears to be non-universal. It depends on the wavefield intensity and ranges from about 6.5 for weak forcing to about 3.5 for large levels of wave excitations. We discuss our results in the context of the key theoretical predictions, such as Zakharov-Filonenko spectrum ν = −4, Phillips spectrum ν = −5, Kuznetsov's revision of Phillips spectrum (leading to ν = −4) and Nazarenko's prediction ν = −6 for weak turbulence in finite basins. We measured Probability Density Function of the surface elevation and good agreement with the Tayfun shape except values near the maximum which we attribute to an anisotropy and inhomogeneity caused by the finite flume size. We argue that the wavenumber discreteness, due to the finite-size of the flume, prevents four-wave resonant interactions. Therefore, statistical evolution of the water surface in the laboratory is significantly different than in the open ocean conditions.

We consider superfluid turbulence near absolute zero of temperature generated by classical means, e.g. towed grid or rotation but not by counterflow. We argue that such turbulence consists of a polarized tangle of mutually interacting vortex filaments with quantized vorticity. For this system we predict and describe a bottleneck accumulation of the energy spectrum at the classical-quantum crossover scale ℓ. Demanding the same energy flux through scales, the value of the energy at the crossover scale should exceed the Kolmogorov-41 spectrum by a large factor ln 10/3 (ℓ/a0) (ℓ is the mean intervortex distance and a0 is the vortex core radius) for the classical and quantum spectra to be matched in value. One of the important consequences of the bottleneck is that it causes the mean vortex line density to be considerably higher that based on K41 alone, and this should be taken into account in (re)interpretation of new (and old) experiments as well as in further theoretical studies.

Time evolution equation for the Probability Distribution Function (PDF) is derived for system of weakly interacting waves. It is shown that a steady state for such system may correspond to strong intermittency.