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

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. © 2009 The American Physical Society.