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Asymptotic methods for spherically symmetric MHD α 2 ‐dynamos
Author(s) -
Günther U.,
Kirillov O.N.
Publication year - 2007
Publication title -
pamm
Language(s) - English
Resource type - Journals
ISSN - 1617-7061
DOI - 10.1002/pamm.200700964
Subject(s) - eigenvalues and eigenvectors , physics , degenerate energy levels , magnetohydrodynamics , gravitational singularity , constant (computer programming) , mathematical analysis , perturbation (astronomy) , mathematical physics , dynamo , spectrum (functional analysis) , classical mechanics , mathematics , magnetic field , quantum mechanics , programming language , computer science
We consider two models of spherically‐symmetric MHD α 2 –dynamos; one with idealized boundary conditions (BCs); and one with physically realistic BCs. As it has been shown in our previous work, the eigenvalues λ of a model with idealized BCs and constant α–profile α 0 are linear functions of α 0 and form a mesh in the ( α 0 , λ )–plane. The nodes of the spectral mesh correspond to double‐degenerate eigenvalues of algebraic and geometric multiplicity 2 (diabolical points). It was found that perturbations of the constant α –profile lead to a resonant unfolding of the diabolical points with selection rules of the resonant unfolding defined by the Fourier coefficients of the perturbations. In the present contribution we present new exact results on the spectrum of the model with physically realistic BCs and constant α . For non‐degenerate (simple) eigenvalues perturbation gradients are found at any particular α 0 . We briefly discuss the spectral behavior of the α 2 –dynamo operator over a family of homotopic deformations of the BCs between idealized ones and physically realistic ones. Furthermore, we demonstrate that although the spectral singularities are lifted, a memory about their locations remains deeply imprinted in the homotopic family of spectral deformations due to a hidden underlying invariance. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)