Phase mixing of Alfvén waves propagating in non-reflective magnetic plasma configurations

Abstract

The ability of phase mixing to provide efficient damping of Alfven waves even in weakly dissipative plasmas made it a popular mechanism for explaining the solar coronal heating. Initially it was studied in the equilibrium configurations with the straight magnetic field lines and the Alfven speed only varying in the direction perpendicular to the magnetic field. Later the analysis of the Alfven wave phase mixing was extended in various directions. In particular it was studied in two-dimensional planar magnetic plasma equilibria. Analytical investigation was carried out under the assumption that the wavelength is much smaller than the characteristic scale of the background quantity variation. This assumption enabled using the Wentzel, Kramers, and Brillouin (WKB) method. When it is not satisfied the study was only carried out numerically. In general, even the wave propagation in a one-dimensional inhomogeneous equilibrium can be only studied numerically. However there is one important exception, so-called non-reflective equilibria. In these equilibria the wave equation with the variable phase speed reduces to the Klein-Gordon equation with constant coefficients. In this paper we apply the theory of non-reflective wave propagation to studying the Alfven wave phase mixing in two-dimensional planar magnetic plasma equilibria. Using curvilinear coordinates we reduce the equation describing the Alfven wave phase mixing to the equation that becomes a one-dimensional wave equation in the absence of dissipation. This equation is further reduced to the equation which is the one-dimensional Klein-Gordon equation in the absence of dissipation. Then we show that this equation has constant coefficients when a particular relation between the plasma density and magnetic field magnitude is satisfied. Using the derived Klein-Gordon-type equation we study the phase mixing in various non-reflective equilibria. We emphasise that our analysis is valid even when the wavelength is comparable with the characteristic scale of the background quantity variation. In particular, we study the Alfven wave damping due to phase mixing in an equilibrium with constant plasma density and exponentially divergent magnetic field lines. We confirm the result previously obtained in the WKB approximation that there is enhanced Alfven wave damping in this equilibrium with the damping length proportional to ln(Re), where Re is the Reynolds number. Our theoretical results are applied to heating of coronal plumes. We show that, in spite of enhanced damping, Alfven waves with periods of the order of one minute can be efficiently damped in the lower corona, at the height about 200 Mm, only if the shear viscosity is increased by about 6 orders of magnitude in comparison with its value given by the classical plasma theory. We believe that such increase of the shear viscosity can be provided by the turbulence.