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

Kirillov, O. N. and Guenther, U. and Stefani, F. (2009):
Determining role of Krein signature for three dimensional Arnold tongues of oscillatory dynamos.
79(1), In: Physical Review E, American Institute of Physics, pp. 016205, [Online-Edition: http://link.aps.org/doi/10.1103/PhysRevE.79.016205],
[Article]

Abstract

Using a homotopic family of boundary eigenvalue problems for the mean-field <span class='mathrm'>α<sup>2</sup></span> dynamo with helical turbulence parameter <span class='mathrm'>α(r)=α<sub>0</sub>+γvarphi(r)</span> and homotopy parameter <span class='mathrm'>β ∈ [0,1]</span>, we show that the underlying network of diabolical points for Dirichlet (idealized, <span class='mathrm'>β=0</span>) boundary conditions substantially determines the choreography of eigenvalues and thus the character of the dynamo instability for Robin (physically realistic, <span class='mathrm'>β=1</span>) boundary conditions. In the (α<sub>0</sub>,β,γ) space the Arnold tongues of oscillatory solutions at <span class='mathrm'>β=1</span> end up at the diabolical points for <span class='mathrm'>β=0</span>. 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 profiles leading to spectral exceptional points, which are important ingredients in recent theories of polarity reversals of the geomagnetic field.

Item Type: Article
Erschienen: 2009
Creators: Kirillov, O. N. and Guenther, U. and Stefani, F.
Title: Determining role of Krein signature for three dimensional Arnold tongues of oscillatory dynamos
Language: English
Abstract:

Using a homotopic family of boundary eigenvalue problems for the mean-field <span class='mathrm'>α<sup>2</sup></span> dynamo with helical turbulence parameter <span class='mathrm'>α(r)=α<sub>0</sub>+γvarphi(r)</span> and homotopy parameter <span class='mathrm'>β ∈ [0,1]</span>, we show that the underlying network of diabolical points for Dirichlet (idealized, <span class='mathrm'>β=0</span>) boundary conditions substantially determines the choreography of eigenvalues and thus the character of the dynamo instability for Robin (physically realistic, <span class='mathrm'>β=1</span>) boundary conditions. In the (α<sub>0</sub>,β,γ) space the Arnold tongues of oscillatory solutions at <span class='mathrm'>β=1</span> end up at the diabolical points for <span class='mathrm'>β=0</span>. 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 profiles leading to spectral exceptional points, which are important ingredients in recent theories of polarity reversals of the geomagnetic field.

Journal or Publication Title: Physical Review E
Volume: 79(1)
Publisher: American Institute of Physics
Divisions: 16 Department of Mechanical Engineering
16 Department of Mechanical Engineering > Dynamics and Vibrations
Date Deposited: 12 Mar 2009 12:54
Official URL: http://link.aps.org/doi/10.1103/PhysRevE.79.016205
Additional Information:

Department of Mechanical Engineering, Dynamics group

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