The Spherically Symmetric a2–dynamo and Some of its Spectral Peculiarities

Abstract

A brief overview is given of recent results on the spectral properties of spherically symmetric MHD α2-dynamos. In particular, the spectra of sphere-confined fluid or plasma configurations with physically realistic boundary conditions (BCs) (surrounding vacuum) and with idealized BCs (super-conducting surrounding) are discussed. The subjects comprise third-order branch points of the spectrum, self-adjointness of the dynamo operator in a Krein space as well as the resonant unfolding of diabolical points. It is sketched how certain classes of dynamos with a strongly localized α-profile embedded in a conducting surrounding can be mode decoupled by a diagonalization ofthe dynamo operator matrix. A mapping of the dynamo eigenvalue problem to that of a quantum mechanical Hamiltonian with energy dependent potential is used to obtain qualitative information about the spectral behavior. Links to supersymmetric Quantum Mechanics and to the Dirac equation are indicated.

Keywords

• MHD dynamo
• operator spectrum
• Krein space
• boundary conditions
• supersymmetric Quantum Mechanics
• diabolical points
• resonance
• KdV soliton potential