  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 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.
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.
