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Abstract
The spectral properties of the normal modes of linearized resistive magnetohydrodynamics are investigated in cylindrical geometry by solving the full system of MHD equations with a numerical shooting method. It is found that, in an inhomogeneous configuration, resistive shear Alfven waves have quite different properties from those of ideal shear Alfven waves. While ideal shear Alfven waves are stationary, the resistive shear Alfven waves are strongly damped. The spectrum of resistive shear Alfven waves is discrete, and the eigenvalues lie on specific lines originating from the accumulation points of the ideal continuum. The spectral lines of resistive shear Alfven waves have a 30 degree characteristic regardless of the equilibrium model employed. As the resistivity is reduced, the eigenvalues move toward the accumulation points, and the density of the eigenvalues on the spectral lines increases. The eigenmode structures of the resistive shear Alfven waves show many radial nodes, and the number of these nodes increases as the magnitude of the attenuation rate increases. It is also found that the resistive slow magnetosonic waves have similar spectral properties to the shear Alfven waves. Their eigenvalues lie on lines with a 30 degree characteristic and the eigenmodes are localized and highly oscillatory. In contrast, the fast magnetosonic waves have a regular behavior indicating that the effect of resistivity is small. The behavior of unstable tearing mode is studied in the presence of a stabilizing positive pressure gradient. It is found that the tearing mode's growth rate is reduced until it coalesces with a second resistive mode whose growth rate increases with the pressure gradient. Further increase of the pressure gradient results in overstable modes which essentially damped when the pressure gradient is greater than some critical value. A similar behavior of the unstable modes is found by varying the value of resistivity with a fixed positive pressure gradient. The locus of the eigenvalues is shown to be a heart-shaped curve. In the study of unstable modes, a numerical asymptotic matching method is utilized in addition to the shooting method.