Statistical Theory of Anisotropic Magnetohydrodynamic Turbulence: An Approach to Strong Shear Alfven Turbulence by Direct‐Interaction Approximation

Abstract

We develop a statistical theory of strong, homogeneous, and anisotropic MHD turbulence within a framework of the Eulerian direct-interaction approximation (DIA). Analysis is concentrated on stationary shear Alfven turbulence of which the mean magnetic field is uniform. We assume that timescales of the Alfven wave and the energy cascade in the energy-containing range (ECR) are much larger than the cascade timescale in the inertial range (InR). Thereby, we can obtain DIA equations governing the time evolution of the propagator and the correlation functions in analytically solvable form. The solutions of the DIA equations include an Alfven oscillation factor depending on kz and a relaxation factor depending on k⊥. Here kz and k⊥ are the wavenumbers parallel to and perpendicular to the mean magnetic field, respectively. Applying the result to the DIA equation of spectral energy transfer, we can show high anisotropy of energy cascades; that is, energy cascades to higher kz modes are inhibited, hence only k⊥ cascades occur. Thus, InR extends to much larger k⊥ than ECR while the kz-band broadening is suppressed. Motivated by this, we assume the functional form of the energy spectrum of InR to be E(k⊥,kz) ∝ k-μ⊥δ(kz) and find that μ = 5/2; hence we obtain the corresponding one-dimensional spectrum, k⊥ ∫∞-∞ dkzE(k⊥,kz) ∝ k-3/2⊥. Furthermore, we show the importance of the three-wave resonances for energy cascades in MHD turbulence. Our theory probably suffers spurious convection effects, and therefore so does the resultant energy spectrum. Nevertheless, it can be a good starting point toward refined theories applicable to real astrophysical MHD turbulence.