Unified theory of resistive and inertial ballooning modes in three-dimensional configurations

A linear stability theory of non-ideal MHD ballooning modes is investigated using a two fluid model for arbitrary three-dimensional electron-ion plasmas. Resistive-inertia ballooning mode (RIBM) eigenvalues and eigenfunctions are calculated for a variety of equilibria including axisymmetric shifted circular geometry (ŝ−α model) as well as for three dimensional configurations of interest relevant to the Helically Symmetric Stellarator (HSX). For typical HSX parameters, characteristic growth rates exceed the electron collision frequency. In this regime, electron inertia effects dominate plasma resistivity and produce an instability whose growth rate scales with the electromagnetic skin depth. However, the resistive and inertia effects become unimportant as the plasma β is increased close to the transition to an ideal mode. Analytic calculations of RIBM stability using a two scale analysis are presented. This work generalizes previous calculations used for axisymmetric ŝ − α geometry [R. H. Hastie, J. J. Ramos and F. Porcelli Phys.Plasmas 10, 4405 (2003)] to general three-dimensional geometry. Both analytic and numerical results show that in the absence of drift effects RIBM modes are purely growing and persist in regimes where ideal MHD ballooning modes are stable. It is found that the magnitudes of the linear growth rates are not sensitive to the magnetic configuration that exists in HSX plasmas. ∗Department of Physics, Lehigh University, 16 Memorial Drive East, Bethlehem Pennsylvania 18015