A singular finite element technique for calculating continuum damping of Alfv\'en eigenmodes

TL;DR

This paper introduces a singular finite element method to accurately compute continuum damping of Alfvén eigenmodes in magnetohydrodynamics, addressing the challenge of resonance poles.

Contribution

A novel finite element approach using Frobenius expansion for precise calculation of continuum damping in ideal MHD.

Findings

Method accurately computes complex mode frequencies.

Results closely match complex contour technique.

Applicable to large aspect ratio tokamaks.

Abstract

Damping due to continuum resonances can be calculated using dissipation-less ideal magnetohydrodynamics (MHD) provided that the poles due to these resonances are properly treated. We describe a singular finite element technique for calculating the continuum damping of Alfv\'{e}n waves. A Frobenius expansion is used to determine appropriate finite element basis functions on an inner region surrounding a pole due to the continuum resonance. The location of the pole due to the continuum resonance and mode frequency are calculated iteratively using a Galerkin method. This method is used to find the complex frequency and mode structure of a toroidicity-induced Alfv\'{e}n eigenmode (TAE) in a large aspect ratio circular tokamak and are shown to agree closely with a complex contour technique.