SINCE the seminal papers of Iroshnikov and Kraichnan published in the 1960s, it is believed that incompressible MHD turbulence finds its origin in the
stochastic collisions of counter-propagating MHD waves called Alfvén waves [105]. The phenomenological model based on this idea leads to the prediction of a total
energy spectrum in k-3/2 (with k the wavenumber). This scaling law differs from that of hydrodynamics (energy spectrum in k-5/3) for which
the dynamics is driven by the interactions of eddies. Later in the 1980s, it was realized that in presence of a large-scale magnetic field B0
– a condition to generate small-scale Alfvén waves – the previous mechanism of redistribution of energy leads certainly to anisotropy with a weaker cascade
along B0 (see e.g. Montgomery & Turner, PoF, 1981; Shebalin et al., JPP, 1983). This result called into question the Iroshnikov–Kraichnan’s
spectrum prediction based on three-wave interactions. In 1994 a theory of weak MHD turbulence was proposed for four-wave interactions (Sridhar & Goldreich,
ApJ, 1994) by claiming that the three-wave interactions are actually absent and a new scaling for the energy spectrum was proposed. The confusion was maximum
at that time because several studies explained, with phenomenological arguments or numerical simulations, why three-wave interactions have probably an active
role in MHD turbulence (see e.g. Ng & Bhattacharjee, ApJ, 1996).
To close the debate, it was necessary to rigorously derive a theory of weak MHD turbulence. Such a theory requires the use of high level mathematical tools [139] based on asymptotic developments of statistical quantities (like two-point correlations) in the Fourier space. We published
the theory in 2000 [13,20]. We explained why the three-wave interactions are indeed dominant in the nonlinear transfer of energy from large to small scales. These
transfers are highly anisotropic with a cascade along the uniform magnetic field B0 completely frozen. We analytically found (exact results)
the constant-flux stationary solutions of the problem which is, in the simplest case called balanced turbulence, a total energy spectrum of power law index -2.
Finally, we explained why weak MHD turbulence becomes necessarily strong at small scale. The regime of weak MHD turbulence was mentioned to explain e.g. the
measures made in the Jovian magnetosphere where a strong large-scale magnetic field is found (Saur et al., A&A, 2002). It is also considered as the regime
encounters in the solar corona (coronal loops and young solar wind) (see e.g. [64]). More recently, the weak MHD turbulence regime was verified numerically in
great details with massive three-dimensional direct numerical simulations and the transition to the strong regime was also detected at small scale [66,79,101,103].
