Homotopic Arnold tongues deformation of the MHD α 2 ‐dynamo

We consider a mean–field α 2 –dynamo with helical turbulence parameter α( r )=α 0 +γΔα( r ) and a boundary homotopy with parameter β∈[0,1] interpolating between Dirichlet (idealized, β=0) and Robin (physically realistic, β=1) boundary conditions. It is shown that the zones of oscillatory solutions at β=1 end up at the diabolical points for β=0 under the homotopic deformation. The underlying network of the diabolical points for β=0 substantially determines the choreography of eigenvalues and thus the character of the dynamo instability for β=1. Using perturbation theory we derive the first–order approximations to the resonance (Arnold's) tongues in the α 0 βγ‐space, which turn out to be cones in the vicinity of the diabolical points, selected by the Fourier coefficients of Δα( r ). The space orientation of the 3D tongues 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 oscillatory dynamos, and it explicitly predicts the locations of the spectral exceptional points, which are important ingredients in the recent theories of polarity reversals of the geomagnetic field. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)