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On Krein space related perturbation theory for MHD α 2 ‐dynamos
Author(s) -
Kirillov Oleg N.,
Günther Uwe
Publication year - 2006
Publication title -
pamm
Language(s) - English
Resource type - Journals
ISSN - 1617-7061
DOI - 10.1002/pamm.200610299
Subject(s) - dynamo , magnetohydrodynamics , eigenvalues and eigenvectors , physics , perturbation (astronomy) , perturbation theory (quantum mechanics) , galerkin method , mathematical analysis , boundary value problem , space (punctuation) , mathematical physics , classical mechanics , dynamo theory , mathematics , magnetic field , quantum mechanics , nonlinear system , linguistics , philosophy
The spectrum of the spherically symmetric α 2 –dynamo is studied in the case of idealized boundary conditions. Starting from the exact analytical solutions of models with constant α –profiles a perturbation theory and a Galerkin technique are developed in a Krein‐space approach. With the help of these tools a very pronounced α –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. An estimation technique is developed for obtaining the critical α –profiles at which the eigenvalues enter the right spectral half‐plane with non‐vanishing imaginary components (at which overcritical oscillatory dynamo regimes form). (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)