Astron. Astrophys. 332, 339-352 (1998)

Grand minima and equatorial symmetry breaking in axisymmetric dynamo models

John M. Brooke ^{1, 2}, Jaan Pelt ^{3, 4}, Reza Tavakol ⁵ and Andrew Tworkowski <sup>5

1</sup> Manchester Computing, The University, Manchester M13 9PL, UK
² Mathematics Department, The University, Manchester M13 9PL, UK
³ Department of Physical Sciences, Astronomy Division, University of Oulu, P.O. Box 333, FIN-90571 Oulu, Finland
⁴ Tartu Observatory, EE2444, Toravere, Estonia
⁵ Astronomy Unit, School of Mathematical Sciences, Queen Mary & Westfield College, Mile End Road, London E1 4NS, UK

Received 8 April 1997 / Accepted 10 November 1997

Abstract

We consider the manner in which time-periodic solutions of an axisymmetric dynamo model can undergo breaking of equatorial symmetry, i.e. loss of pure dipolar or quadrupolar symmetry. By considering the symmetry group underlying the solutions, we show that the fluctuations responsible for the symmetry-breaking can be constrained such that they are in resonance with the former solution. They can then be amplified until they are comparable in magnitude to the former solution. If the bifurcation is supercritical, the amplitude of the fluctuation is stabilised and a stable mixed parity limit cycle is formed. If it is subcritical it gives rise to a recently identified form of intermittency, called icicle intermittency. This produces episodes in which the original solution and the fluctuation are almost exactly synchronised and the fluctuation grows exponentially in amplitude, interrupted by brief episodes where synchronicity is lost and the amplitude of the fluctuation declines rapidly by several orders of magnitude. During these latter episodes there is a significant dip in the amplitude of the total magnetic field. This model-independent analysis can produce quantitative predictions for the behaviour of this bifurcation and we provide evidence for this behaviour by analysing timeseries from four different mean-field dynamo models, where intermittency is observed without the need for stochastic, or chaotically driven, forcing terms in the dynamo equations. We compare these results with recent work on intermittency in dynamo models and consider their relevance to the intermittency present in solar and stellar cycles.

Key words: MHD chaos Sun: magnetic fields stars: magnetic fields

Send offprint requests to: John Brooke

Contents

• 1. Grand minima and intermittency
  • 1.1. Modelling intermittent behaviour
  • 1.2. Mechanisms for intermittency
  • 1.3. Structure of this paper
• 2. The model
  • 2.1. Dynamo models in shells and tori
  • 2.2. Spherical shell models
  • 2.3. Torus models
  • 2.4. Intermittent behaviour in the models
• 3. Theory of the icicle intermittency
  • 3.1. Symmetry-breaking, period-doubling and intermittency
  • 3.2. The symmetry breaking bifurcation
• 4. Testing the predictions in the dynamo models
  • 4.1. Phase Dispersion Analysis (PDA)
  • 4.2. Subcritical bifurcation of the torus dynamo
  • 4.3. Supercritical bifurcation in the shell dynamo
  • 4.4. Coherent behaviour in the other series
  • 4.5. Multiple attractors and intermittency
• 5. Summary and comparison with other work
• Acknowledgements
• References

© European Southern Observatory (ESO) 1998

Online publication: March 10, 1998
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