Mean-field concept and direct numerical simulations of rotating magnetoconvection and the geodynamo

Abstract

Mean-field theory describes magnetohydrodynamic processes leading to large-scale magnetic fields in various cosmic objects. In this study magnetoconvection and dynamo processes in a rotating spherical shell are considered. Mean fields are defined by azimuthal averaging. In the framework of mean-field theory, the coefficients which determine the traditional representation of the mean electromotive force, including derivatives of the mean magnetic field up to the first order, are crucial for analyzing and simulating dynamo action. Two methods are developed to extract mean-field coefficients from direct numerical simulations of the mentioned processes. While the first method does not use intrinsic approximations, the second one is based on the second-order correlation approximation. There is satisfying agreement of the results of both methods for sufficiently slow fluid motions. Both methods are applied to simulations of rotating magnetoconvection and a quasi-stationary geodynamo. The mean-field induction effects described by these coefficients, e.g. the α-effect, are highly anisotropic in both examples. An α 2 -mechanism is suggested 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 geodynamo example. With the aim to compare mean-field simulations with corresponding direct numerical simulations, a two-dimensional mean-field model involving all previously determined mean-field coefficients was constructed. Various tests with different sets of mean-field coefficients reveal their action and significance. In the magnetoconvection and geodynamo examples considered here, the match between direct numerical simulations and mean-field simulations is only satisfying if a large number of mean-field coefficients are involved. In the magnetoconvection example, the azimuthally averaged magnetic field resulting from the numerical simulation is in good agreement with its counterpart in the mean-field model. However, this match is not completely satisfactory in the geodynamo case anymore. Here the traditional representation of the mean electromotive force ignoring higher than first-order spatial derivatives of the mean magnetic field is no longer a good approximation.

Simulations of astrophysical dynamos

“…However, in the case of Euler potentials it is not clear what happens and whether this method can be used to simulate even the ideal MHD equations, given that each numerical scheme will introduce some type of diffusion. In this short review, we have also attempted to clarify why numerical calculations of α effect and turbulent diffusion using the standard test-field method (Schrinner et al, 2007) would yield values that can only reproduce a correct growth rate in the case of vanishing growth. In all other case, a representation in terms of integral kernels has to be used.…”

“…This provides an important resource for ideas of what one might have been missing in various contexts. Here we just mention the case of the Roberts flow, for which α ij and η ij have been determined using the test-field methods (Schrinner et al, 2007). One might then expect that the growth rate obtained from the underlying dissipation rate, should agree with the value obtained from the direct calculation.…”

Mean‐field dynamos: The old concept and some recent developments

“…The method was developed by Schrinner et al (2005Schrinner et al ( , 2007 in the context of this task: Consider a simple geodynamo model, with the magnetic field maintained by convection. Define mean fields by averaging over the azimuthal coordinate; they are then axisymmetric.…”

Advances in Theory and Simulations of Large-Scale Dynamos

“…This method is known as the test-field method and was developed by Schrinner et al (2005Schrinner et al ( , 2007 to calculate all tensor components from snapshots of simulations of the geodynamo in a spherical shell. This method was later applied to time-dependent turbulence in triply-periodic Cartesian domains, both with shear and no helicity (Brandenburg 2005 The test-field method has recently been criticized by Cattaneo & Hughes (2008) on the grounds that the test fields are arbitrary pre-determined mean fields.…”