Statistical characterization of the interchange-instability spectrum of a separable ideal-magnetohydrodynamic model system

A Suydam-unstable circular cylinder of plasma with periodic boundary conditions in the axial direction is studied within the approximation of linearized ideal magnetohydrodynamics (MHD). The normal mode equations are completely separable, so both the toroidal Fourier harmonic index n and the poloidal index m are good quantum numbers. The full spectrum of eigenvalues in the range 1≤m≤mmax is analyzed quantitatively, using asymptotics for large m, numerics for all m, and graphics for qualitative understanding. The density of eigenvalues scales like mmax2 as mmax→∞. Because finite-m corrections scale as 1/mmax2, their inclusion is essential in order to obtain the correct statistics for the distribution of eigenvalues. Near the largest growth rate, only a single radial eigenmode contributes to the spectrum, so the eigenvalues there depend only on m and n as in a two-dimensional system. However, unlike the generic separable two-dimensional system, the statistics of the ideal-MHD spectrum departs somewhat from the Poisson distribution, even for arbitrarily large mmax. This departure from Poissonian statistics may be understood qualitatively from the nature of the distribution of rational numbers in the rotational transform profile