Coupling between nonlinear Alfvén waves and reduced magnetohydrodynamics for compressible fluids

A reduced description of magnetohydrodynamic (MHD) turbulence in a plasma permeated by a strong uniform magnetic field is derived using a reductive perturbative expansion method. This leads to a coupled system of equations for the velocity and magnetic fields in the planes perpendicular to the ambient field, described by the usual reduced MHD equations [Zank and W. H. Matthaeus, J. Plasma Phys. 48, 85–100 (1992)] and for parallel propagating Alfvén waves, which on the characteristic time and length scale of the transverse flow, undergo cumulative nonlinear effects described by a generalized Derivative Nonlinear Schrödinger equation (DNLS). The cases β far from unity and β≈1 are addressed separately. In the latter case, the nonlinear Alfvén waves drive a mean longitudinal velocity whose magnitude is comparable to that of the transverse components. A modulational analysis for a quasi-monochromatic plane Alfvén wave at β far from unity illustrates the effect of the transverse MHD flow on the Alfvén wave filamentation.