SPECTRUM OF THE BALLOONING SCHRODINGER-EQUATION

The ballooning Schrodinger equation (BSE) is a model equation for inve stigating global modes that can, when approximated by a Wentzel-Kramer s-Brillouin (WKB) ansatz, be described by a ballooning formalism local ly to a field line. This second-order differential equation with coeff icients periodic in the independent variable theta(k) is assumed to ap ply 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. th e apparent discontinuity between quantization formulae for 'trapped' a nd 'passing' modes, whose ray paths have different topologies, is remo ved 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 t hat trapped modes go over to passing modes, reducing the maximum growt h rate by averaging over theta(k).