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Kink Oscillations of a Curved, Gravitationally Stratified Coronal Loop

Loops of magnetic field in the corona are observed to oscillate and these oscillations have been posited to be the superposition of resonant kink waves. To date, most analyses of these oscillations have concentrated on calculating the frequency shifts that result from spatial variation in the kink wave speed. Further, most have ignored gravity and treated the loop as a straight tube. Here we ignore spatial variation in the wave speed, but self-consistently include the effects of gravity and loop curvature in both the equilibrium loop model and in the wave equation. We model a coronal loop as an isolated, thin, magnetic fibril that is anchored at two points in the photosphere. The equilibrium shape of the loop is determined by a balance between magnetic buoyancy and magnetic tension, which is characterized by a Magnetic Bond Number ε, that is typically small |ε| << 1. This balance produces a loop that has a variable radius of curvature. The resonant kink waves of such a loop come in two polarizations that are decoupled from each other: waves with motion completely within the plane of the loop (normal oscillations) and waves with motions that are completely horizontal, perpendicular to the plane of the loop (binormal oscillations). We solve for the eigensolutions of both polarizations using perturbation theory for small Magnetic Bond Number. For modes of the same order, normal oscillations have smaller eigenfrequencies than binormal oscillations. The additional forces of buoyancy and magnetic tension from the curvature of the loop increase and decrease the mode frequencies, respectively. The ratio of the frequencies of the first overtone to the fundamental mode is modified by the inclusion of buoyancy and curvature. We find that the normal polarization possesses a frequency ratio that exceeds the canonical value of 2, whereas the binormal polarization has a ratio less than 2.