Turbulent cascades in anisotropic magnetohydrodynamics

The cascade behavior of turbulent magnetohydrodynamics with a strong background magnetic field is examined and compared with direct numerical solutions at high Reynolds number. Resonant interactions give rise to qualitatively different behavior for modes below a characteristic wave number $k_L$ defined in terms of the background field. Modes with parallel wave number above $k_L$ are passively driven by the longer wavelength modes, even when the majority of the energy is contained in the passive wave numbers. The passive modes do not cascade to higher parallel wave numbers, so the parallel wave number spectrum is not a power law and does not extend to dissipation scales. Energy is cascaded normally to small perpendicular scales, but more rapidly in the case of the passive modes, so an anisotropic spectrum develops from isotropic initial conditions. For a finite system with minimum wave number $>k_L,$ the only dynamically controlling mode is the vertical average, or mean mode. The mean mode evolves with two-dimensional dynamics, forming coherent current structures which are mirrored by the passive modes. Because of the differential decay rates, the mean mode dominates at long times. Quantitative comparisons are made to numerical solutions of reduced magnetohydrodynamics.