Quasi-modes as dissipative magnetohydrodynamic eigenmodes: Results for one-dimensional equilibrium states

Quasi-modes, which are important for understanding the MI-ID wave behavior of solar and astrophysical magnetic plasmas, are computed as eigenmodes of the linear dissipative MHD equations. This eigenmode computation is carried out with a simple numerical scheme, which is based on analytical solutions to the dissipative MHD equations in the quasi-singular resonance layer. Nonuniformity in magnetic field and plasma density gives rise to a continuous spectrum of resonant frequencies. Global discrete eigenmodes with characteristic frequencies lying within the range of the continuous spectrum may couple to localized resonant Alfven waves. In ideal MHD, these modes are not eigenmodes of the Hermitian ideal MHD operator, but are found as a temporal dominant, global, exponentially decaying response to an initial perturbation. In dissipative MHD, they are really eigenmodes with damping becoming independent of the dissipation mechanism in the limit of vanishing dissipation. An analytical solution of these global modes is found in the dissipative layer around the resonant Alfvenic position. Using the analytical solution to cross the quasi-singular resonance layer, the required numerical effort of the eigenvalue scheme is limited to the integration of the ideal MI-ID equations in regions away from any singularity. The presented scheme allows for a straightforward parametric study. The method is checked with known ideal quasi-mode frequencies found for a one-dimensional box model for the Earth's magnetosphere (Zhu & Kivelson). The agreement is excellent. The dependence of the oscillation frequency on the wavenumbers for a one-dimensional slab model for coronal loops found by Ofman, Davila, & Steinolfson is also easily recovered.