Klein-Gordon equations for horizontal transverse oscillations in two-dimensional coronal loops

Abstract

We present a theory for hydromagnetic waves in an axi-symmetric background magnetic field in which the wave equations for the horizontal transverse magnetic field and velocity perturbations can be transformed into Klein-Gordon (KG) equations. For harmonic time variations, the KG equations become a set of ordinary differential equations that can be solved along any individual field line, subject to boundary conditions at the two ends. The solutions provide the spatial (latitudinal) profiles of the transverse magnetic field and velocity oscillations, especially in the horizontal direction, along the field line. In particular, we examine the KG solutions for two background field geometries: a local dipole field line, and a stretched global dipole field line which may approximate coronal loop geometries in the solar corona. The results yield the oscillation frequencies in agreement with observations (periods on the order of minutes), and the spatial profiles which are characteristic of a propagating type near the center of the loop and a possible evanescent type towards the footpoints of the loop. The latter solution arises when the oscillation frequency is less than a critical cut-off frequency which varies spatially along the loop. The oscillation amplitude is also affected by an adiabatic growth/decay factor along the loop. We discuss the implications of our results and future applications to coronal loop oscillations.