On the effects of viscosity and anisotropic resistivity on the damping of Alfvén waves

Abstract

The equations of magnetohydrodynamics (MHD) are written for non-uniform viscosity and resistivity – the latter in the cases of Ohmic and anisotropic resistivity. In the case of Ohmic (anisotropic) diffusivity, there is (are) one (two) transverse components of the velocity and magnetic field perturbation(s), leading to a second-order (fourth-order) dissipative Alfvén- wave equation. In the more general case of dissipative Alfvén waves with isotropic viscosity and anisotropic resistivity, the fourth-order wave equation may be replaced by two decoupled second-order equations for right- and left-polarized waves, whose dispersion relations show that the first resistive diffusivity causes dissipation like the viscosity, whereas the second resistive diffusivity causes a change in propagation speed. The second resistive diffusivity invalidates the equipartition of kinetic and magnetic energy, modifies the energy flux through the propagation speed, and also changes the ratio of viscous to resistive dissipation. If the directions of propagation and polarization are equal (i.e. for right-polarized upward-propagating or left-polarized downward-propagating waves), the magnetic energy increases relative to the kinetic energy, the resistive dissipation increases relative to the viscous dissipation, and the total energy density and flux increase relative to the case of isotropic resistivity; the reverse is the case for opposite directions of propagation, i.e. upward-propagating left-polarized waves and downward-propagating right-polarized waves, which can lead to the existence of a critical layer. The role of the viscosity and first and second resistive diffusiveness on the dissipation of Alfvén waves is discussed with reference to the solar atmosphere.