On the dynamics of magnetorotational turbulent stresses

The turbulent stresses that lead to angular momentum transport in accretion discs have often been treated as resulting from an isotropic effective viscosity, related to the pressure through the alpha parametrization of Shakura & Sunyaev. This simple approach may be adequate for the simplest aspects of accretion disc theory, and was necessitated historically by an incomplete understanding of the origin of the turbulence. More recently, Balbus & Hawley have shown that the magnetorotational instability provides a robust mechanism of generating turbulent Reynolds and Maxwell stresses in sufficiently ionized discs. The alpha viscosity model fails to describe numerous aspects of this process. This paper introduces a new analytical model that aims to represent more faithfully the dynamics of magnetorotational turbulent stresses and bridge the gap between analytical studies and numerical simulations. Covariant evolutionary equations for the mean Reynolds and Maxwell tensors are presented, which correctly include the linear interaction with the mean flow. Non-linear and dissipative effects, in the absence of an imposed magnetic flux and in the limit of large Reynolds number and magnetic Reynolds number, are modelled through five non-linear terms that represent known physical processes and are strongly constrained by symmetry properties and dimensional considerations. The resulting model explains the development of statistically steady, anisotropic turbulent stresses in the shearing sheet, a local representation of a differentially rotating disc, in agreement with numerical simulations. It also predicts that purely hydrodynamic turbulence is not sustained in a flow that adequately satisfies Rayleigh’s stability criterion. The model is usually formally hyperbolic and therefore ‘causal’, and guarantees the realizability of the stress tensors. It should be particularly useful in understanding the dynamics of warped, eccentric and tidally distorted discs, non-Keplerian accretion flows close to black holes, and a variety of time-dependent accretion phenomena.