Roman Pleshkov

A parametric study of two-disk dynamo is carried out in three-dimensional parameter spaces: the magnetic Prandtl number Pm (the ratio of viscous to Ohmic dissipation), the dimensionless dissipation μ, and the ratio α of torques applied to two disks. At Pm of the order of unity, chaotic solutions do not exist at all. Small inclusions of chaotic modes appear at Pm∼0.1. A further decrease in Pm leads to the formation of a wideband of chaos in the (μ,α) plane with superimposed oblique windows of stationary solutions. The rare-reversal chaos is a regime characterized by long-lived quasi-stationary states with weak oscillations of variables, ending with a sharp burst of oscillations with a possible transition to the other stationary solution and corresponding change of field polarity, i.e., a reversal. This regime was detected at Pm≤0.0001 only in a narrow domain. The longest time interval of constant polarity was observed in simulations for Pm=1.37×10−9, α=1.1, and μ=5.39 lasts 9000 dimensionless units of time. It has been found that, in a two-disk dynamo system, for any chaotic mode, there is an unambiguous dependence of the current oscillation amplitude on the time remaining until the next field reversal. Thus, despite the fact that the duration of each interval is a random value, at any stage of the field evolution, for a known set of governing parameters, it is possible to determine the time of the next reversal by recording the current amplitude of oscillations.

A systematic comparative study of two-disk dynamo models without friction (the Rikitake model) and with friction has been carried out. It is shown that the sets of chaotic and periodic modes realized in both models are qualitatively similar. We introduce a simple measure of the complexity of a solution, which allows finding the chaotic mode characterized by long-lived quasistationary states with very weak oscillations of variables, ending with a sharp burst of oscillations with a possible transition to the attraction area of the other stationary solution. This transition corresponds to the change in the magnetic field sign. Such a behavior of the field is somewhat similar to the dynamics of the Earth's large-scale field in terms of the chaotic change of zones from one polarity to another. This rare reversal chaos was found in both two-disk dynamo models (without friction and with friction), but in the viscous case, it appears only at weak friction.