Non-stationary resonant Alfvén surface waves in one-dimensional magnetic plasmas

Abstract

This paper uses incompressible visco-resistive MHD to study the propagation of linear resonant waves in an inhomogeneous plasma. The background density and magnetic field are assumed to depend only on one spartial Cartesian coordinate, and the magnetic field is taken to be unidirectional and perpendicular to the direction of inhomogeneity. The equation that governs the component of the velocity normal to the plane formed by the direction of the inhomogeneity and the magnetic field is derived under the assumption that the coefficients of viscosity and resistivity are sufficiently small that dissipation of energy is confined to a narrow dissipative layer. The solutions to this equation are obtained in the form of decaying normal surface modes with wavelengths much larger than the characteristic scale of the inhomogeneity. The effect of non-stationarity inside the dissipative layer is taken into account, and valid solutions are found even when the ratio of the thickness of the dissipative layer to the inhomogeneity length scale is of the order of or smaller than the ratio of the inhoinogeneity length scale to the wavelength. These solutions are the generalization of the solutions obtained by Mok and Einaudi, which are only valid when the first ratio is much larger than the second. The rate of wave damping is shown to be independent of the values of the viscosity and the resistivity. However, the behaviour of the solutions in the dissipative layer depends strongly on the viscosity and the resistivity. In the case that the effect of dissipation dominates the effect of non-stationarity, the solutions behave in the dissipative layer as found by Mok and Einaudi. When the effect of dissipation is steadily decreased in comparison with the effect of nonstationarity, the solutions become more and more oscillatory, and their amplitudes grow very rapidly in the dissipative layer. Eventually, when nonstationarity dominates dissipation, the amplitudes of the solutions become so large in the dissipative layer in comparison with those outside the dissipative layer that practically all the energy of the perturbations is concentrated in the dissipative layer.

Resonant Alfvén waves in partially ionized plasmas of the solar atmosphere

“…We find the remarkable result that these jumps are the same as obtained for ideal and dissipative MHD (e.g., Sakurai et al 1991;Goossens et al 1995;Ruderman et al 1995). This means that ion-neutral collisions do not modify the jumps of the plasma perturbations across the resonant layer.…”

“…Functions F (τ) and G(τ) play here the role of the universal functions found in a number of previous investigations of resonant waves in both stationary and non-stationary states and for different dissipative processes (see, e.g., Mok & Einaudi 1985;Goossens et al 1995;Ruderman et al 1995;Tirry & Goossens 1996;Erdélyi et al 1995;Wright & Allan 1996;Ruderman & Wright 1999;Vanlommel et al 2002). A comprehensive review on the importance and properties of the universal F (τ) and G(τ) functions can be found in Goossens et al (2011).…”

On the nature of kink MHD waves in magnetic flux tubes

“…This is clearly illustrated in a recent analytical seismological study by Goossens et al (2008) which complemented a fully numerical seismology investigation by Arregui et al (2007). The eigenfunctions in the thin dissipative layer can be described by the functions F(τ) and G(τ) defined by for the driven problem and the functionsF(τ) and G(τ) defined by Ruderman et al (1995) for the incompressible eigenvalue problem and by Tirry & Goossens (1996) for the compressible eigenvalue problem. In the dissipative layer the MHD kink waves are highly Alfvénic.…”

“…In view of that conclusion it is difficult to understand why a kink mode can be called fast as fast waves are absent from incompressible plasmas. The eigenfunctions in the thin dissipative layer can be described by the functionsF(τ) andG(τ) which were first introduced by Ruderman et al (1995) for non-stationary incompressible resonant Alfvén waves in planar plasmas. The conclusion is the same as in the previous section.…”

Damping of coronal loop kink oscillations due to mode conversion

“…Ruderman et al 1995) Ruderman et al (1995) reconsidered the problem studied by Mok & Einaudi (1985) when this condition is not satisfied. They showed that, while the solution describing the plasma motion in the dissipative layer is very much different from that obtained for the driven problem, the damping rate of the surface mode remains the same.…”