On scaling laws for incompressible anisotropic MHD turbulence

A heuristic model is given for anisotropic magnetohydrodynamics (MHD) turbulence in the presence of a uniform external magnetic field $B_0 {\bf {\hat e}_{\pa}}$. The model is valid for both moderate and strong $B_0$ and is able to describe both the strong and weak wave turbulence regimes as well as the transition between them. The main ingredient of our model is the assumption of constant ratio at all scales between linear wave periods and nonlinear turnover timescales. Contrary to the model of critical balance introduced by Goldreich and Sridhar (1995), we do not assume in addition that this ratio be equal to unity at all scales which allows us to use the Iroshnikov-Kraichnan phenomenology. We are then able to recover the widely observed anisotropic scaling law $\kpa \propto \kpe^{2/3}$ between parallel and perpendicular wavenumbers (with reference to $B_0 {\bf {\hat e}_{\pa}}$) and we obtain the universal prediction, $3\alpha + 2\beta = 7$, for the total energy spectrum $E(\kpe,\kpa) \sim \kpe^{-\alpha} \kpa^{-\beta}$. In particular, with such a prediction we recover the weak Alfv\'en wave turbulence constant-flux solution and give, for the first time, a possible explanation to its precursor found numerically by Galtier et al (2000)