Determining role of Krein signature for 3D Arnold tongues of oscillatory dynamos

Abstract

Using a homotopic family of boundary eigenvalue problems for the mean-field ${\alpha }^2$-dynamo with helical turbulence parameter $\alpha (r)={\alpha }_0+\gamma \Delta \alpha (r)$ and homotopy parameter $\beta \in [0,1]$, we show that the underlying network of diabolical points for Dirichlet (idealized, $\beta =0$) boundary conditions substantially determines the choreography of eigenvalues and thus the character of the dynamo instability for Robin (physically realistic, $\beta =1$) boundary conditions. In the $({\alpha }_0,\beta ,\gamma )-$space the Arnold tongues of oscillatory solutions at $\beta =1$ end up at the diabolical points for $\beta =0$. In the vicinity of the diabolical points the space orientation of the 3D tongues, which are cones in first-order approximation, is determined by the Krein signature of the modes involved in the diabolical crossings at the apexes of the cones. The Krein space induced geometry of the resonance zones explains the subtleties in finding $\alpha$-profiles leading to spectral exceptional points, which are important ingredients in recent theories of polarity reversals of the geomagnetic field.

arXiv comment4 pages, 3 figures, presented at the GAMM 2008, Bremen, Germany Introduction extended, refs added

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• Standard and helical magnetorotational instability: How singularities create paradoxal phenomena in MHD
• On the spectrum of the magnetohydrodynamic mean-field alpha^2-dynamo operator
• Perturbation of multiparameter non-self-adjoint boundary eigenvalue problems for operator matrices
• Unfolding the conical zones of the dissipation-induced subcritical flutter for the rotationally symmetrical gyroscopic systems
• Membrane flutter induced by radiation of surface gravity waves on a uniform flow
• Graphical Krein Signature Theory and Evans-Krein Functions