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

Oleg Kirillov·Uwe Guenther
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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.

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Oleg Kirillov, Uwe Guenther. 2006-02-24. Krein space related perturbation theory for MHD alpha-2-dynamos and resonant unfolding of diabolical points. https://doi.org/10.1088/0305-4470%2F39%2F32%2Fs08

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