A new eigenvalue problem associated with the two-dimensional Newcomb equation without continuous spectra

Abstract

A new eigenvalue problem associated with the two-dimensional Newcomb equation in an axisymmetric toroidal plasma, such as a tokamak, has been posed and solved numerically by using a finite element method. In the formulation of the eigenvalue problem, the weight functions (the kinetic energy integral) and the boundary conditions at rational surfaces are chosen such that the spectra of the eigenvalue problem are comprised of only the real and denumerable eigenvalues (point spectra) without continuous spectra. Applications to the ideal m=1 mode have verified that this formulation is able to identify stable states as well as unstable states, and that the numerically obtained eigenfunctions show the singular behavior predicted by the theory at rational surfaces.