Phase mixing of Alfvén waves propagating in non-reﬂective magnetic plasma conﬁgurations
The ability of phase mixing to provide eﬃcient damping of Alfvén waves even in weakly dissipative plasmas made it a popular mechanismforexplainingthesolarcoronalheating.Initiallyitwasstudiedintheequilibriumconﬁgurationswiththestraightmagnetic ﬁeldlinesandtheAlfvénspeedonlyvaryinginthedirectionperpendiculartothemagneticﬁeld.LatertheanalysisoftheAlfvénwave 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 Alfvén wave phase mixing in two-dimensional planar magnetic plasma equilibria. Using curvilinear coordinates we reduce the equation describing the Alfvén wave phase mixing to the equationthatbecomesaone-dimensionalwaveequationintheabsenceofdissipation.Thisequationisfurtherreducedtotheequation 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 Alfvén 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 Alfvén 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, Alfvén 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.