Spectrum of the ballooning Schrödinger equation

Abstract

The ballooning Schrödinger equation (BSE) is a model equation for investigating global modes that can, when approximated by a Wentzel - Kramers - Brillouin (WKB) ansatz, be described by a ballooning formalism locally to a field line. This second-order differential equation with coefficients periodic in the independent variable is assumed to apply even in cases where simple WKB quantization conditions break down, thus providing an alternative to semiclassical quantization. Also, it provides a test bed for developing more advanced WKB methods, e.g. the apparent discontinuity between quantization formulae for `trapped' and `passing' modes, whose ray paths have different topologies, is removed by extending the WKB method to include the phenomena of tunnelling and reflection. The BSE is applied to instabilities with shear in the real part of the local frequency, so that the dispersion relation is inherently complex. As the frequency shear is increased, it is found that trapped modes go over to passing modes, reducing the maximum growth rate by averaging over .