Noise induced relaxation oscillations and Earth's magnetic field reversals

The magnetic field of the Earth is known to undergo polarity reversals with a mean reversal rate that varies between zero during the superchrons and around 5 per Myr in the present. Typically, these reversals have an asymmetric, saw-toothed shape with a slow decay and a fast recreation of the dipole. This asymmetry points to a possible connection with relaxation oscillations as they are well known from the van der Pol oscillator. A simple mean-field dynamo model with a spherically symmetric helical turbulence parameter which is quenched by the magnetic energy and disturbed by additional noise is analysed with view on this similarity. The basic features of geomagnetic polarity reversals are shown to be generic consequences of the dynamo action in the vicinity of branching points of the spectrum of the dynamo operator where two real eigenvalues coalesce and continue as a complex conjugated pair of eigenvalues. A comparison of the time series and the phase space trajectories with those of paleomagnetic measurements is carried out. For the case of highly supercritical dynamos a very good agreement with the paleomagnetic data is achieved. We explain also why such highly supercritical dynamos have a general tendency to self-tune into reversal prone states. The similar clustering properties of numerical and paleomagnetic reversal sequences are discussed. Spectral theory of the non-selfadjoint dynamo operator is also invoked to speculate that the growth of the inner core might be responsible for the long term changes of the reversal rate and for the occurrence of superchrons.