Rikitake two‐disk dynamo system: Statistical properties and growth of instability

Abstract

The apparent randomness in the observed behaviors of the geomagnetic field (polarity reversals, intensity fluctuations, etc.) can be simulated either by simplified physical models (e.g., disk dynamo models) or by stochastic models (e.g., the Cox model). The Rikitake two‐disk dynamo is particularly interesting as a simulator, since it displays aperiodic changes of current direction resembling the geomagnetic reversals. The behaviors of the Rikitake system are investigated in some detail, with various values for parameters (μ, k). The current intensity fluctuates around one of the two stationary states, and the amplitude of oscillation grows monotonically with time, following a growth curve common to models with the same k. When the amplitude becomes so big that the peak of oscillation exceeds some threshold value, the system flips to the opposite polarity and the oscillation resumes around the new stationary state. The growth of the instability and the eventual flip of the polarity occur as deterministic process as dictated by the differential equation governing the system, and yet the system behavior appears almost random for a wide range of parameters (μ, k). The randomness enters as the growth rate of the amplitude is exceptionally large just before the polarity reversal, and the system behavior in the next polarity interval depends critically on the largest amplitude gained in the previous interval. The two statistical distributions, those of polarity intervals and field intensity, can be classified into three types which are distinctly different from each other. The transition between these types occur almost in parallel in the two distributions in the (μ, k) space.