Angular momentum transport and large eddy simulations in magnetorotational turbulence: the small Pm limit

Context. Angular momentum transport in accretion discs is often believed to be due to magnetohydrodynamic turbulence mediated by the magnetorotational instability (MRI). Despite an abundant literature on the MRI, the parameters governing the saturation amplitude of the turbulence are poorly understood and the existence of an asymptotic behaviour in the Ohmic diffusion regime has not been clearly established. Aims. We investigate the properties of the turbulent state in the small magnetic Prandtl number limit. Since this is extremely computationally expensive, we also study the relevance and range of applicability of the most common subgrid scale models for this problem. Methods. Unstratified shearing box simulations are performed both in the compressible and incompressible limits, with a resolution up to 800 cells per disc scale height. This is the highest resolution ever attained for a simulation of MRI turbulence. Different magnetic field geometry and a wide range of dimensionless dissipative coefficients are considered. We also systematically investigate the relevance of using large eddy simulations (LES) in place of direct numerical simulations. Results. In the presence of a mean magnetic field threading the domain, angular momentum transport converges to a finite value in the small Pm limit. When the mean vertical field amplitude is such that β (the ratio between the thermal and magnetic pressure) equals 103, we find α ~ 3.2 × 10-2 when Pm approaches zero. In the case of a mean toroidal field for which β = 100, we find α ~ 1.8 × 10-2 in the same limit. Implicit LES and the Chollet-Lesieur closure model both reproduce these results for the α parameter and the power spectra. A reduction in computational cost by a factor of at least 16 (and up to 256) is achieved when using such methods. Conclusions. MRI turbulence operates efficiently in the small Pm limit provided there is a mean magnetic field. Implicit LES offers a practical and efficient means of investigation of this regime but should be used with care, particularly in the case of a vertical field. The Chollet-Lesieur closure model is perfectly suited for simulations done with a spectral code