A Simple Stochastic Model for Dipole Moment Fluctuations in Numerical Dynamo Simulations

Earth's axial dipole field changes in a complex fashion on many different time scales ranging from less than a year to tens of million years. Documenting, analysing, and replicating this intricate signal is a challenge for data acquisition, theoretical interpretation, and dynamo modeling alike. Here we explore whether axial dipole variations can be described by the superposition of a slow deterministic drift and fast stochastic fluctuations, i.e. by a Langevin-type system. The drift term describes the time averaged behavior of the axial dipole variations, whereas the stochastic part mimics complex flow interactions. The statistical behavior of the system is described by a Fokker-Planck equation which allows useful predictions, including the average rates of dipole reversals and excursions. We analyze several numerical dynamo simulations, most of which have been integrated particularly long in time, and also the palaeomagnetic model PADM2M which covers the past 2Myr. The results show that the Langevin description provides a viable statistical model of the axial dipole variations on time scales longer than about 1kyr. For example, the axial dipole probability distribution and the average reversal rate are successfully predicted. The exception is PADM2M where the stochastic model reversal rate seems too low. The dependence of the drift on the axial dipole moment reveals the nonlinear interactions that establish the dynamo balance. A separate analysis of inductive and diffusive magnetic effects in three dynamo simulations suggests that the classical quadratic quenching of induction predicted by mean-field theory seems at work.