Oscillating \alpha^2-dynamos and the reversal phenomenon of the global geodynamo

A geodynamo-model based on an \alpha-effect which has been computed under conditions suitable for the geodynamo is constructed. For a highly restricted class of radial \alpha-profiles the linear \alpha2-model exhibits oscillating solutions on a timescale given by the turbulent diffusion time. The basic properties of the periodic solutions are presented and the influence of the inner core size on the characteristics of the critical range that allows for oscillating solutions is shown. Reversals are interpreted as half of such an oscillation. They are rather seldom events because they can only occur if the \alpha-profile exists long enough within the small critical range that allows for periodic solutions. Due to strong fluctuations on the convective timescale the probability of such a reversal is very small. Finally, a simple non-linear mean-field model with reasonable input parameters based on simulations of Giesecke et al. (2005) demonstrates the plausibility of the presented theory with a long-time series of a (geo-)dynamo reversal sequence.