Krein space related perturbation theory for MHD α²-dynamos

The mean field α²-dynamo of magnetohydrodynamics (MHD) plays a similarly paradigmatic role in MHD dynamo theory like the harmonic oscillator in quantum mechanics. In its kinematic regime this dynamo is described by a linear induction equation for the magnetic field. For spherically symmetric α²-profiles α(r) the vector of the magnetic field can be decomposed into poloidal and toroidal components and expanded in spherical harmonics. In the present contribution the spectrum of the spherically symmetric α²-dynamo is studied in the case of idealized boundary conditions. Starting from the exact analytical solutions of models with constant α²-profiles a perturbation theory and a Galerkin technique are developed in a Krein-space approach. With the help of these tools a very pronounced α²-resonance pattern is found in the deformations of the spectral mesh as well as in the unfolding of the diabolical points located at the nodes of this mesh. Non-oscillatory as well as oscillatory dynamo regimes are obtained. Finally, Fréchet derivative (gradient) based methods are developed, suitable for further numerical investigations of Krein-space related setups like MHD α²-dynamo or models of PT−symmetric quantum mechanics.