A comparison between Markovian and non-Markovian closures in simulations of nonlinear dynamos with application to the protogalactic dynamo

Nonhelical magnetohydrodynamic (MHD) turbulence and the nonlinear dynamo are investigated numerically using two statistical closures: the direct interaction approximation, or DIA (Kraichnan), and the realizable Markovian closure, or RMC (Bowman et al.). The RMC is an approximate version of the DIA that can be used to simulate wider ranges of wavenumbers for longer stretches of time. In a controlled numerical experiment, it is found that the DIA and RMC lead to comparable results in simulations of the nonlinear dynamo. This helps justify the use of the RMC and other Markovian closures for the dynamo problem, thereby supporting the result of Pouquet, Frisch, & Leorat that an initially weak magnetic field grows up to rough equipartition with the kinetic energy on all scales. The result obtained by Pouquet, Frisch, & Leorat is also reproduced directly with a numerical DIA calculation. A separate DIA calculation produces k-(3/2) spectra for steady state isotropic MHD turbulence, in accord with the physical arguments of Kraichnan. A rough preliminary calculation based upon the RMC suggests that the magnetic energy grows to approximately 6% of the kinetic energy during the collapse of the protogalaxy in the theory of Kulsrud et al. In a technical aside, a geometric algorithm is presented for calculating wavenumber-bin volume factors in three-dimensional closure calculations.