Quantum Chaos? Genericity and Nongenericity in the MHD Spectrum of Nonaxisymmetric Toroidal Plasmas

Abstract

The eigenmode spectrum is a fundamental starting point for the analysis of plasma stability and the onset of turbulence, but the characterization of the spectrum even for the simplest plasma model, ideal magnetohydrodynamics (MHD), is not fully understood. This is especially true in configurations with no continuous geometric symmetry, such as in a real tokamak when the discrete nature of the external magnetic field coils is taken into account, or the alternative fusion concept, the stellarator, where axisymmetry is deliberately broken to provide a nonzero winding number (rotational transform) on each invariant torus of the magnetic field line dynamics (assumed for present purposes to be an integrable Hamiltonian system). Quantum (wave) chaos theory provides tools for characterizing the spectrum statistically, from the regular spectrum of the separable case (integrable semiclassical dynamics) to that where the semiclassical ray dynamics is so chaotic that no simple classification of the individual eigenvalues is possible (quantum chaos). The MHD spectrum exhibits certain nongeneric properties, which we show, using a toy model, to be understable from the number-theoretic properties of the asymptotic spectrum in the limit of large toroidal and poloidal mode (quantum) numbers when only a single radial mode number is retained. Much more realistically, using the ideal MHD code CAS3D, we have constructed a data set of several hundred growth-rate eigenvalues for an interchange-unstable three-dimensional stellarator equilibrium with a rather flat, nonmonotonic rotational transform profile. A statistical analysis of eigenvalue spacings shows evidence of generic quantum chaos, which we attribute to the mixing effect of having a large number of radial mode numbers.