Nonlocal Phenomenology for Anisotropic Magnetohydrodynamic Turbulence

Abstract

A nonlocal cascade model for anisotropic magnetohydrodynamic (MHD) turbulence in the presence of a uniform magnetic field B is proposed. The model takes into account that (1) energy cascades in an anisotropic manner, and as a result a different estimate for the cascade rate in the direction parallel and perpendicular to the B field is made, and (2) the interactions that result in the cascade are between different scales. Eddies with wavenumbers k∥ and k⊥ interact with eddies with wavenumbers q∥, q⊥ such that a resonance condition between the wavenumbers q∥, q⊥ and k∥, k⊥ holds. As a consequence, energy from the eddy with wavenumbers k∥ and k⊥ cascades due to interactions with eddies located in the resonant manifold whose wavenumbers are determined by q∥ 1/3k/B and q⊥ k⊥, and energy will cascade along the lines k∥ k0 + k1/3/B. For a uniform energy injection rate in the parallel direction, the resulting energy spectrum is E(k∥,k⊥) 2/3kk. For a general forcing, however, the model suggests a nonuniversal behavior. The connections with previous models, numerical simulations and weak turbulence theory are discussed.