  [Abridged] 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 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.
