Nonlinear Development of Phase-Mixed Alfven Waves

We derive an equation governing the nonlinear propagation of a linearly polarized Alfven wave in a two-dimensional, anisotropic, slightly compressible, highly magnetized, viscous plasma, where nonlinearities arise from the interaction of the Alfven wave with fast and slow magnetoacoustic waves. The phase mixing of such a wave has been suggested as a mechanism for heating the outer solar atmosphere (Heyvaerts and Priest, 1983). We find that cubic wave damping dominates shear linear dissipation whenever the Alfven wave velocity amplitude delta v(y) exceeds a few times ten metres per second. In the nonlinear regime, phase-mixed waves are marginally stable, while non-phase-mixed waves of wavenumber k(n) are damped over a timescale k(n) Re(0)vertical bar delta v(y)/v(A)vertical bar(-2), Re(0) being the Reynolds number corresponding to the Braginskij viscosity coefficient eta(0) and v(A) the Alfven speed. Dissipation is most effective where beta = (v(s)/v(A))(2) similar or equal to 1, v(s) being the speed of sound.