The influence of line-tying on coronal perturbations in a gravitationally stratified equilibrium

We study the influence of gravitational stratification of the solar atmosphere on the stability of coronal magnetic structures. In particular we question whether the (presumably stabilizing) influence of the anchoring of the magnetic field lines in the solar photosphere ('line-tying') can be adequately modeled by either 'rigid wall' or 'flow through' boundary conditions on the coronal perturbations, as is commonly done. Using the ideal magnetohydrodynamic (MHD) model without gravitational effects, inertial line-tying alone cannot lead to a full stabilization, as marginal stability cannot be crossed by including only the rapid density increase at the photospheric interface. We demonstrate, using the (localized) ballooning ordering, that when gravity and the corresponding intrinsically stable stratification of the photosphere is included, the points of marginal stability are no longer independent of the density. The sharp increase in density and associated decrease in pressure scale height at the solar surface leads to a stabilizing effect, which may result in a full transition from unstable to stable modes. Gravitational effects imply that rigid wall conditions represent photospheric field line anchoring better than flow-through conditions for determining the stability or modes of oscillation of a coronal equilibrium. Applying rigid wall conditions gives good approximations for frequencies that are much larger than photospheric time scales when the plasma is stable, and growth rates when the plasma is unstable. At the same time we show however that near marginal stability, even when gravity is included, rigid wall conditions are still violated.