Oscillations and Evolution of Curved Current-carrying Loops in the Solar Corona

The dynamics of curved, slender, current-carrying solar coronal loops whose footpoints are rooted in the photosphere are examined. By integrating the MHD equations over the cross section of a loop, a set of ordinary differential equations are obtained which describe its temporal evolution. The normal modes of oscillation are calculated. When the height of the loop apex above the photosphere exceeds a critical fraction of its footpoint separation, the loop is globally stable and oscillates with a frequency that depends strongly on the loop geometry and curvature, magnetic field, density, and temperature. Low-lying loops tend to be unstable and will rise to become more nearly semicircular. It is shown that loop curvature introduces a new mode of oscillation that differs from the familiar Alfvén wave. Future observations of such oscillations (either as Doppler shifts or through changes in the emission measure) could serve as a useful diagnostic of coronal magnetic field strengths. The evolution of a loop when magnetic flux is injected rapidly at one footpoint is also studied. The loop rises and oscillates about a new equilibrium, with the major radius oscillating at a much lower frequency than the minor radius. The ratio of these frequencies can serve as another diagnostic of the properties of coronal magnetic fields. These results are applied to a number of currently observed solar phenomena such as winking filaments and prominence oscillations, flare oscillations, and oscillations seen at the limb in coronal lines.