Publications Repository - Helmholtz-Zentrum Dresden-Rossendorf
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Asymptotic methods for spherically symmetric MHD α2-dynamosKirillov, O.; Günther, U.
We consider two models of spherically-symmetric MHD α2-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 α₀ are linear functions of α₀ and form a mesh in the (α₀,Re 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 α₀. For non-degenerate (simple) eigenvalues perturbation gradients are found at any particular α₀. We present a detailed study of the spectral behavior of the α2-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 the spectral deformations due to a hidden underlying invariance.
Keywords: MHD dynamo, homotopic deformation, homotopic invariance, non-Hermitian operator, spectral analysis
6th International Congress on Industrial and Applied Mathematics (iciam 07), 16.-20.07.2007, Zürich, Switzerland