SINCE the seminal papers by Iroshnikov (1964) and Kraichnan (1965), MHD turbulence is thought to have its origin in the stochastic collisions of counter propagating Alfvén waves [105]. The phenomenological model based on this idea predicts an isotropic energy spectrum different from that of hydrodynamics based on the interaction of vortices. However, in the 1980s it was realized that in the presence of a large-scale magnetic field B0 – a necessary condition for the generation of Alfvén waves – the energy redistribution mechanism is non-isotropic with a weakening of the turbulent cascade along the B0 direction (Montgomery & Turner, 1981; Shebalin et al., 1983).
This result called into question the Iroshnikov–Kraichnan isotropic spectrum, a prediction based on three-wave interactions.
A first unsuccessful attempt to develop a theory for Alfvén wave turbulence (Sridhar & Goldreich, 1994) led to some confusion (and a strong debate) about the elementary bricks of MHD turbulence because this theory involves four-wave interactions (Montgomery and Matthaeus, ApJ, 1995).
We published the rigorous theory of Alfvén wave turbulence in 2000 [13,20], which is based on an asymptotic development of statistical quantities (two-point correlations) in Fourier space [139]. We explained why three-wave resonant interactions are dominant at main order in the nonlinear transfer of energy from large to small scales, and how these transfers become non-isotropic with a cascade frozen along B0. The subtle point is that three-wave interactions involve the slow mode (k//=0, k// being the component of k along B0). We found the exact stationary solution (Kolmogorov-Zakharov spectrum) which scales in the simplest case as kperp-2. We showed that Alfvén wave turbulence becomes strong at small scales. The numerical simulation of the wave turbulence equations revealed the existence, during the non-stationary phase, of an energy spectrum not compatible with the stationary solution. This is the first time that this spectral anomalous was detected in turbulence. It is now found in many systems, in weak and strong turbulence, and is understood as a self-similar solution of the second kind [102]. By solving this major problem of plasma physics, the Alfvén wave turbulence theory has become a reference (third most cited paper of J. Plasma Physics created in 1968) as it clarifies the foundation of MHD turbulence. Since then, Alfvén wave turbulence has been mentioned to explain measurements in the Jovian magnetosphere (Saur et al., 2002) and to describe the solar coronal loops (Rappazzo et al., 2007; [64]). Over the last ten years, I have returned to this fundamental problem to demonstrate the feasibility of such a regime using 3D direct numerical simulations. This is a non-trivial task as it requires the use of massive numerical resources and the development of specific tools dedicated to wave turbulence. With young researchers, we have succeeded in reproducing this regime. The very detailed study also allowed us to reveal new properties, including the transition from weak to strong wave turbulence described by the critical balance phenomenology [66,79,101,103].
