Mean-field view on geodynamo models

Abstract

Mean-field theory provides a useful description of magnetohydrodynamic processes leading to large-scale magnetic fields in various cosmic objects. In mean-field theory, the coefficients occurring in the expansion of the mean electromotive force in terms of the mean field and its derivatives are used to analyse and to simulate dynamo action. In this study, we consider dynamo processes in a rotating spherical shell similar to the Earth`s outer core; mean fields are then defined by azimuthal averaging. We present two different techniques for determining mean-field coefficients and apply both to a simulation of rotating magnetoconvection and a simple quasi-stationary dynamo (the benchmark example). In both examples, the tensorial mean-field coefficients are highly anisotropic and demonstrate the existence of an α2 -mechanism along with a strong γ-effect operating outside the inner core tangent cylinder. The turbulent diffusivity exceeds the molecular one by at least one order of magnitude in the benchmark example. The first technique is also applied to two highly time-dependent dynamos, one exhibiting a strongly columnar and the other a fully developed convection pattern. The resulting time-averaged mean-field coefficients resemble those obtained in the magnetoconvection and benchmark example which indicates that similar dynamo processes take place. The temporal fluctuations of mean-field coefficients occur on the convective turnover timescale. These fluctuations are particularly large in the fully developed regime, because the velocity field lacks any equatorial symmetry.With the aim of comparing mean-field simulations with corresponding direct numerical simulations, a two-dimensional mean-field model involving all the previously determined mean-field coefficients has been constructed. In the magnetoconvection and benchmark example, the match between direct numerical simulations and mean-field simulations is best if at least 17 mean-field coefficients are kept. In the magnetoconvection example, the azimuthally averaged magnetic field resulting from a direct numerical simulation is in good agreement with the result given by the mean-field model. However, this match is not satisfactory in the benchmark example. Here, the traditional representation of the mean electromotive force including no higher than first-order derivatives is not justified. The lack of a clear scale separation renders the traditional mean-field approach inappropriate in this example.