Determining role of Krein signature for three-dimensional Arnold tongues of oscillatory dynamos.
Using a homotopic family of boundary eigenvalue problems for the mean-field alpha;{2} dynamo with helical turbulence parameter alpha(r)=alpha_{0}+gammaDeltaalpha(r) and homotopy parameter beta[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 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 alpha profiles leading to spectral exceptional points, which are important ingredients in recent theories of polarity reversals of the geomagnetic field.
Kirillov
Oleg N
ON
Technische Universität Darmstadt, D-64289 Darmstadt, Germany. kirillov@dyn.tu-darmstadt.de
Günther
Uwe
U
Stefani
Frank
F