A field theory approach to the statistical kinematic dynamo

Variations in the geomagnetic field occur on a vast range of time scales, from milliseconds to millions of years. The advent of satellite measurements has allowed for detailed studies of short timescale geomagnetic field behaviour, but understanding its long timescale evolution remains challenging due to the sparsity of the paleomagnetic record. This paper introduces a field theory framework for studying magnetic field generation as a result of stochastic fluid motions. Starting from a stochastic kinematic dynamo model (the Kazantsev kinematic model), we derive statistical properties of the magnetic field that may be compared to observations from the paleomagnetic record. The fluid velocity is taken to be a Kraichnan field with general covariance, which acts as a random forcing obeying Gaussian statistics. Using the Martin-Siggia-Rose-Janssen-de Dominicis formalism, we compute the average magnetic field response function for fluid velocities with short correlation time. From this we obtain an estimate for the turbulent contribution to the magnetic diffusivity, and find that it is consistent with results from mean-field dynamo theory. This framework presents much promise for studying the geomagnetic field in a stochastic context.