Closure tests for mean field magnetohydrodynamics using a self consistent reduced model

TL;DR

This paper tests the validity of two mean-field magnetohydrodynamics approximations, SOCA and tau, using a nonlinear model of turbulent flow to compare their predictions against numerical simulations across various parameters.

Contribution

It provides a systematic comparison of SOCA and tau approximations against direct simulations, highlighting their applicability and limitations in different dynamical regimes.

Findings

Tau approximation is suitable when a small-scale dynamo is present.

Catastrophic quenching of alpha occurs at high magnetic Prandtl numbers.

SOCA overestimates alpha at high Pm compared to exact solutions.

Abstract

The mean electromotive force and alpha effect are computed for a forced turbulent flow using a simple nonlinear dynamical model. The results are used to check the applicability of two basic analytic ansatze of mean-field magnetohydrodynamics - the second order correlation approximation (SOCA) and the tau approximation. In the numerical simulations the effective Reynolds number Re is 2-20, while the magnetic Prandtl number varies from 0.1 to $10^{7}$. We present evidence that the $\tau$ approximation may be appropriate in dynamical regimes where there is a small-scale dynamo. Catastrophic quenching of the $\alpha$ effect is found for high $P_{m}$. Our results indicate that for high $P_{m}$ SOCA gives a very large value of the $\alpha$ coefficient compared with the ``exact'' solution. The discrepancy depends on the properties of the random force that drives the flow, with a larger difference occuring for $\delta$-correlated force compared with that for a steady random force.