On the viscous and resistive dissipation of magnetohydrodynamic waves

Abstract

The dissipative magnetohydrodynamic (MHD) equations, in the presence of Ohmic electrical resistance and bulk and shear viscosities, are linearized relative to a mean state of rest, and eliminated for the velocity perturbation. The result is a vector dissipative MHD equation consisting of (i) nondissipative terms, involving second-order space and time derivatives; (ii) dissipative terms, linear or quadratic in the diffusivities, involving fourth- and fifth-order space–time derivatives. Considering plane wave solutions, the dispersion matrix shows that Alfvén waves decouple, and are damped by Ohmic electrical resistance and shear viscosity. The latter two, and the bulk viscosity, damp the coupled slow and fast modes, which satisfy a dispersion relation of degree five in the frequency; it reduces to degree three in the case of magnetosonic waves, i.e., wave vector transverse to the direction of propagation. The roots of these dispersion relations of degree three to five, are obtained by a novel method, which applies first to the case of weak damping, and then can be extended to strong damping, e.g., taking into account products of diffusivities. For most applications, e.g., MHD waves in the solar atmosphere, the second-order effects of products of diffusivities are small.