The probabilistic structure of the geodynamo

Many recent analyses of paleomagnetic data have modeled the dipole field at Earth's surface as a Langevin stochastic process governed by the Fokker-Planck equation. This has proved to be a useful way to characterize both observational and numerical data sets. However, the justification for the method has only been that the Fokker-Planck/Langevin system is well-understood and that it produces data qualitatively similar to the real system. In this work, we start with the underlying equation governing the geomagnetic field, the dynamo equation, and we derive the Fokker-Planck equation for the dipole field, showing exactly what assumptions need to be made about the source terms and time scales for this type of approach to hold. We find that these assumptions are generally quite mild, the most important being that the sources are short-lived compared with time scales of interest, have a short correlation time, and that no single event has a large impact on the field. Our derivation also gives us forms for the diffusion and drift coefficients that relate data sets to processes occurring inside the core, such as magnetic quenching and decay. It is therefore possible that our approach may eventually be used to constrain core properties with paleomagnetic data, and we give an example of how it could be used to estimate the conductivity.