Equilibria as boundary value problems under Lie transformations

Abstract

The use of transformations recently gained attention in obtaining invariant solutions to the equilibrium problem of plasma physics. In all of the cases considered, the new solutions were related to a (Generalized) Grad–Shafranov equation. In the same context, the present study focuses on the issue of an axisymmetric, toroidal plasma equilibrium as a boundary value problem associated with new solutions obtained by means of Lie group transformations. It appears that in all the cases examined, only a single infinitesimal generator of the symmetry group permits closed boundary that remains invariant under the transformation. The respective equilibrium, in addition to a peculiar axisymmetric magnetically confined plasma with current hole reaching the axis of symmetry, describes a planet's magnetosphere for low heights.