Force-Free Magnetic Field in a Torus of Arbitrary Aspect Ratio

Abstract

The helical force-free equation, ∇ × B = αB, has been solved analytically in a toroidal coordinates system for a torus of arbitrary aspect ratio without the approximation of a large aspect ratio. The three-dimensional force-free equation is reduced to a scalar Helmholtz equation. A set of analytical solutions for the Helmholtz equation in the torus is presented. With these solutions, the eigenvalues have been obtained for an aspect ratio R/a ≥ 7.5 and toroidal mode number −5 ≤ n ≤ 14. The difference in the eigenvalue between a torus and a cylinder becomes large in the case of a small aspect ratio and a large toroidal mode number. However, the smallest eigenvalues and the corresponding toroidal wave numbers are found to be in close agreement with those of a cylinder for R/a ≥ 1.5.