The geodynamo as a bistable oscillator

Our intent is to provide a simple and quantitative understanding of the var iability of the axial dipole component of the geomagnetic field on both sho rt and long time scales. To this end we study the statistical properties of a prototype nonlinear mean field model. An azimuthal average is employed, so that (1) we address only the axisymmetric component of the field, and (2 ) the dynamo parameters have a random component that fluctuates on the (fas t) eddy turnover time scale, Numerical solutions with a rapidly fluctuating a reproduce several features of the geomagnetic field: (1) a variable, dom inantly dipolar field with additional fine structure due to excited overton es, and sudden reversals during which the field becomes almost quadrupolar, (2) aborted reversals and excursions, (3) intervals between reversals havi ng a Poisson distribution. These properties are robust, and appear regardle ss of the type of nonlinearity and the model parameters. A technique is pre sented for analysing the statistical properties of dynamo models of this ty pe. The Fokker-Planck equation for the amplitude a of the fundamental dipol e mode shows that a behaves as the position of a heavily damped particle in a bistable potential proportional to (1-a(2))(2), subject to random forcin g. The dipole amplitude oscillates near the bottom of one well and makes oc casional jumps to the other. These reversals are induced solely by the over tones. Theoretical expressions are derived for the statistical distribution of the dipole amplitude, the variance of the dipole amplitude between reve rsals, and the mean reversal rate. The model explains why the reversal rate increases with increasing secular variation, as observed. Moreover, the pr esent reversal rate of the geodynamo, once per (2-3) x 10(5) year, is shown to imply a secular variation of the axial dipole moment of similar to 15% (about the current value). The theoretical dipole amplitude distribution ag rees well with the Sint-800 data.