Nonlocal phenomenology for anisotropic magnetohydrodynamic turbulence

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(parallel to) and k(perpendicular to) interact with eddies with wavenumbers q(parallel to), q(perpendicular to) such that a resonance condition between the wavenumbers q(parallel to), q(perpendicular to) and k(parallel to), k(perpendicular to) holds. As a consequence, energy from the eddy with wavenumbers k(parallel to) and k(perpendicular to) cascades due to interactions with eddies located in the resonant manifold whose wavenumbers are determined by q(parallel to) similar or equal to epsilon(1/3)k(perpendicular to)(2/3)/B and q(perpendicular to) similar or equal to k(perpendicular to), and energy will cascade along the lines k(parallel to) similar or equal to k(0) + k(perpendicular to)(2/3) epsilon(1/3)/B. For a uniform energy injection rate in the parallel direction, the resulting energy spectrum is E(k(parallel to),k(perpendicular to)) similar or equal to epsilon(2/3) k(parallel to)(-1)k(perpendicular to)(5/3). For a general forcing, however, the model suggests a nonuniversal behavior. The connections with previous models, numerical simulations and weak turbulence theory are discussed.