Ballooning mode spectrum in general toroidal systems

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Abstract
A WKB formalism for constructing normal modes of short‐wavelength ideal hydromagnetic, pressure‐driven instabilities (ballooning modes) in general toroidal magnetic containment devices with sheared magnetic fields is developed. No incompressibility approximation is made. A dispersion relation is obtained from the eigenvalues of a fourth‐order system of ordinary differential equations to be solved by integrating along a line of force. Higher‐order calculations are performed to find the amplitude equation and the phase change at a caustic. These conform to typical WKB results. In axisymmetric systems, the ray equations are integrable, and semiclassical quantization leads to a growth rate spectrum consisting of an infinity of discrete eigenvalues, bounded above by an accumulation point. However, each eigenvalue is infinitely degenerate. In the nonaxisymmetric case, the rays are unbounded in a four‐dimensional phase space, and semiclassical quantization breaks down, leading to broadening of the discrete eigenvalues and the accumulation point of the axisymmetric unstable spectrum into continuum bands. Analysis of a model problem indicates that the broadening of the discrete eigenvalues is numerically very small, the dominant effect being broadening of the accumulation point.

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