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

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.

22 pages, 7 figures, presented at the 4th International Workshop on Pseudo-Hermitian Hamiltonians in Quantum Physics, November 23-25, 2005, Stellenbosch, South Africa; minor typos corrected, refs added

References in corpus (7)

• Coupling of eigenvalues of complex matrices at diabolic and exceptional points
• MHD alpha^2-dynamo, Squire equation and PT-symmetric interpolation between square well and harmonic oscillator
• Unfolding of eigenvalue surfaces near a diabolic point due to a complex perturbation
• Why dynamos are prone to reversals
• Isospectrality of spherical MHD dynamo operators: pseudo-Hermiticity and a no-go theorem
• Third order spectral branch points in Krein space related setups: PT-symmetric matrix toy model, MHD alpha^2-dynamo, and extended Squire equation
• The MHD alpha^2-dynamo, Z_2-graded pseudo-Hermiticity, level crossings and exceptional points of branching type

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