Krein space related perturbation theory for MHD alpha-2-dynamos and resonant unfolding of diabolical points

TL;DR

This paper develops a Krein-space perturbation theory for alpha-2 dynamo spectra, revealing resonance patterns and unfolding diabolical points, with methods applicable to MHD and PT-symmetric quantum models.

Contribution

It introduces a novel Krein-space approach combining analytical solutions, perturbation theory, and numerical techniques for studying dynamo spectra and related systems.

Findings

Discovery of pronounced alpha-resonance patterns in spectral deformations

Identification of conditions for eigenvalues to become overcritical and oscillatory

Development of gradient-based methods for further numerical analysis

Abstract

The spectrum of the spherically symmetric alpha-2 dynamo is studied in the case of idealized boundary conditions. Starting from the exact analytical solutions of models with constant alpha-profiles a perturbation theory and a Galerkin technique are developed in a Krein-space approach. With the help of these tools a very pronounced alpha-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. A Fourier component based estimation technique is developed for obtaining the critical alpha-profiles at which the eigenvalues enter the right spectral half-plane with non-vanishing imaginary components (at which overcritical oscillatory dynamo regimes form). Finally, Frechet derivative (gradient) based methods are developed, suitable for further numerical investigations of Krein-space related setups like MHD alpha-2-dynamos or models of PT-symmetric quantum mechanics.