Expressions for perturbed vacuum potential energy for 3D linear MHD stability

Abstract

The calculation of the perturbed vacuum potential energy in an ideal Magnetohydrodynamical (MHD) system consisting of plasma surrounded by vacuum is a crucial ingredient for the treatment of peeling modes. In this work, detailed theoretical expressions are derived to describe this, making use of the potential representation of the magnetic field in the vacuum, which leads to a Laplace equation that can be solved through the boundary element method. The complications that arise from the singular integrals that appear are studied thoroughly. Special attention is also given to the case of axisymmetric equilibria, which leads to more accurate but also more complicated analytical expressions than the general 3D expressions. In the near future, the resulting expressions will be used directly in 3D ideal MHD stability codes such as PB3D so that it can treat all 3D ideal high-n MHD stability, including peeling modes.