Staircase model for magnetohydrodynamic spectrum and global instabilities in the Elmo bumpy torus

Abstract

In the staircase model a continuous equilibrium is approximated by a discontinuous one with jump surfaces separating regions of constant pressure, density, and magnetic rotation number. The dispersion relation, giving the growth rates or frequencies of eigenmodes of the linearized motion as solutions of a finite matrix eigenvalue problem, is derived for arbitrary axisymmetric straight staircase equilibria with no azimuthal magnetic field and with weak, long, and thin periodic mirrors. This relation is shown to describe global modes accurately, thus being well suited to obtaining their dependence upon the pressure and density profile with minimal numerical effort. Application to hollow pressure profiles typical of the Elmo bumpy torus yields global instabilities with growth rates too large to be compatible with observations, indicating that kinetic effects must be invoked to explain even the large-scale behavior of this device.