Determining role of Krein signature for three-dimensional Arnold tongues of oscillatory dynamos

Oleg N. Kirillov, Uwe Günther, and Frank Stefani
Phys. Rev. E 79, 016205 – Published 20 January 2009

Abstract

Using a homotopic family of boundary eigenvalue problems for the mean-field α2 dynamo with helical turbulence parameter α(r)=α0+γΔα(r) and homotopy parameter β[0,1], we show that the underlying network of diabolical points for Dirichlet (idealized, β=0) boundary conditions substantially determines the choreography of eigenvalues and thus the character of the dynamo instability for Robin (physically realistic, β=1) boundary conditions. In the (α0,β,γ) space the Arnold tongues of oscillatory solutions at β=1 end up at the diabolical points for β=0. In the vicinity of the diabolical points the space orientation of the three-dimensional 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 α profiles leading to spectral exceptional points, which are important ingredients in recent theories of polarity reversals of the geomagnetic field.

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  • Received 6 June 2008

DOI:https://doi.org/10.1103/PhysRevE.79.016205

©2009 American Physical Society

Authors & Affiliations

Oleg N. Kirillov1,*, Uwe Günther2,†, and Frank Stefani2,‡

  • 1Technische Universität Darmstadt, D-64289 Darmstadt, Germany
  • 2Forschungszentrum Dresden-Rossendorf, P.O. Box 510119, D-01314 Dresden, Germany

  • *kirillov@dyn.tu-darmstadt.de
  • u.guenther@fzd.de
  • f.stefani@fzd.de

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Issue

Vol. 79, Iss. 1 — January 2009

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