In this study, we have developed a new eigenvalue code to solve the three-variables ideal magnetohydrodynamic (MHD) equation in flux coordinate system with the help of a symbolic vector analysis module and an automatic numerical discretization module, which were developed with the use of symbolic computation technique and were greatly facilitated the development of toroidal full MHD eigenvalue code SCELT. It has been used to solve for unstable modes and the present work is dedicated to the more difficult challenge of MHD continua, which ordinarily come with spatial singularities. With this new code, we were able to perform numerical calculations to determine the continuum spectrum of geodesic acoustic modes (GAMs), as well as obtain eigenfunctions for both the perturbed plasma displacement and perturbed magnetic field, incorporating a sufficient number of components. Through our global calculations, we have provided numerical evidence of singularities in the eigenfunctions, characterized by the type and . Moreover, we have examined the spatial variations of GAM fluctuations in regions both near and far from the singular surface. Here represents a flux function that characterizes the magnetic flux surfaces and defines the singular surface. Additionally, we have investigated the effects of finite aspect ratio, non-circular cross sections as well as the asymmetry on the frequency and eigenfunction of continuum GAMs through numerical analysis.
Low-frequency magnetohydrodynamics and geodesic acoustic modes in toroidally rotating tokamak plasmas