Cascade properties of shear Alfvén wave turbulence

Abstract

Nonlinear three-wave interactions of linear normal modes are investigated for two-dimensional incompressible magnetohydrodynamics and the weakly three-dimensional Strauss equations in the case where a strong uniform background field B0 is present. In both systems the only resonant interaction affecting Alfvén waves is caused by the shear of the background field plus the zero frequency (ω=k ⋅ B0=0) components of the perturbation. It is shown that the Alfvén waves are cascaded in wavenumber space by a mechanism equivalent to the resonant absorption at the Alfvén resonance. For large wavenumbers perpendicular to B0, the cascade is described by Hamilton’s ray equations, dk/dt=−∂ω/∂r, where ω includes the effects of the zero frequency perturbations.