On the essential spectrum of ideal magnetohydrodynamics. Dedicated to harold grad on the occasion of his sixtieth birthday

Abstract

The essential spectrum of magnetohydrodynamics (MHD) is shown to arise from waves propagating one-dimensionally along magnetic field lines. Different polarizations of these waves give rise to the "Alfven" and "ballooning" spectra. The essential spectrum of an axisymmetric equilibrium when a single azimuthal mode number is considered consists of the Alfven spectrum only, while the ballooning modes appear as intervals of accumulation of discrete eigenvalues with different mode numbers. We derive some necessary and one sufficient conditions for stability and show some examples where the criteria coincide to yield a simple condition for stability.

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

“…In addition to a point (discrete) spectrum of unstable modes (ω 2 < 0) there are the Alfvén and slow-magnetosonic continuous spectra on the stable side of the origin (ω 2 > 0) and the possibility of dense sets of accumulation points on the unstable side. In mathematical spectral theory the stable continua and unstable accumulation "continua" [4] are characterized [5] as belonging to the essential spectrum. (For a self-adjoint operator L, the essential spectrum is the set of λ-values for which the range of L − λ * Electronic address: robert.dewar@anu.edu.au is not a closed set and/or the dimensionality of the null space of L − λ is infinite.…”

Shear Alfvén mode resonances in nonaxisymmetric toroidal low-pressure plasmas. II. Singular modes in the shear Alfvén continuum

“…It has been established that the spectra of modes, both in the shear Alfvén continuum and in the ballooning continuum, are again determined by eigenvalue problems for 2 which contain only derivatives along magnetic field lines. [8][9][10][11] However, unlike in the spatially symmetric configurations, individual field lines in configurations without symmetry are not equivalent. The continuum physics therefore is primarily connected with individual field lines.…”

The accumulation continua in ideal magnetohydrodynamics

“…It is a curious fact that modes with large mode numbers were first discovered in two-dimensional equilibria, [7][8][9] and that their theory was then generalized to three-dimensional ones, [10][11][12][13][14][15] but that little attention has been paid to onedimensional ones ͑plane slab or circular cylinder, turned into a topological torus by imposing appropriate periodicity conditions͒. The various continua, on the other hand, are most easily visualized in one-dimensional equilibria.…”

“…In addition, unstable continua may occur 5,6 that consist of accumulation points of eigenfrequencies of modes with large variation in the pressure surfaces ͑large mode numbers͒. [7][8][9][10][11][12][13][14][15] Modes with large mode numbers are localized at magnetic field lines where the perturbations have finite variation. They were often called ''ballooning modes'' because the eigenfunctions are believed to concentrate at those sections of the field line where its curvature is unfavorable for stability.…”