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

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.