An Expansion Method for Computing Axisymmetric Sunspot Oscillations

Abstract

A method is proposed to solve for the linear axisymmetric oscillations of a general, axisymmetric, potential, magnetostatic sunspot equilibrium. The basic approach is to express the solution as a series of terms that are products of a prescribed radial planform times an unknown function of the vertical spatial coordinate and time. If the potential sunspot magnetic field is strictly uniform and aligned with the prevailing gravitational stratification, then a single term in the proposed series solution suffices, and the familiar problem first considered by Ferraro & Plumpton is readily recovered. For less trivial magnetic fields, which possess both vertical and radial gradients, the proposed series solution does not truncate after a finite number of terms, but the equations that determine the unknown functions of the vertical coordinate and time enjoy the advantage of being separable partial differential equations, which can be attacked through the solution of subordinate ordinary differential equations by the method of separation of variables. It is also demonstrated that the proposed series solution encompasses the thin flux tube expansion. Consequently, a rigorous mathematical basis is provided for this popular method employed to describe the dynamics of slender magnetic flux tubes, which proves useful in understanding the intrinsic astrophysical limitations of the approach. Whether this proposed method of solution is also a practical and efficient means to calculate the oscillation modes of axisymmetric sunspot equilibria is not answered here but will be addressed in a forthcoming companion paper.