The probabilistic structure of the geodynamo

TL;DR

This paper introduces a probabilistic framework for modeling Earth's magnetic field reversals, deriving equations to predict reversal rates and field properties from stochastic sources in the magnetohydrodynamic equations.

Contribution

It develops a novel analytical approach using stochastic processes and Fokker-Planck equations to understand geomagnetic reversals, overcoming limitations of previous simulations and calculations.

Findings

Derivation of a differential equation for the joint probability distribution of magnetic modes.

Reduction to a Fokker-Planck equation and further simplification for dipole amplitude.

Quantitative predictions of reversal rates and field characteristics based on physical parameters.

Abstract

One of the most intriguing features of Earth's axial magnetic dipole field, well-known from the geological record, is its occasional and unpredictable reversal of polarity. Understanding the phenomenon is rendered very difficult by the highly non-linear nature of the underlying magnetohydrodynamic problem. Numerical simulations of the liquid outer core, where regeneration occurs, are only able to model conditions that are far from Earth-like. On the analytical front, the situation is not much better; basic calculations, such as relating the average rate of reversals to various core parameters, have apparently been intractable. Here, we present a framework for solving such problems. Starting with the magnetic induction equation, we show that by considering its sources to be stochastic processes with fairly general properties, we can derive a differential equation for the joint probability distribution of the dominant toroidal and poloidal modes. This can be simplified to a Fokker-Planck equation and, with the help of an adiabatic approximation, reduced even further to an equation for the dipole amplitude alone. From these equations various quantities related to the magnetic field, including the average reversal rate, field strength, and time to complete a reversal, can be computed as functions of a small number of numerical parameters. These parameters in turn can be computed from physical considerations or constrained by paleomagnetic, numerical, and experimental data.