Global simulations of magnetorotational turbulence - I. Convergence and the quasi-steady state

Magnetorotational turbulence provides a viable mechanism for angular momentum transport in accretion discs. We present global, three-dimensional (3D), magnetohydrodynamic accretion disc simulations that investigate the dependence of the turbulent stresses on resolution. Convergence in the time-and volume-averaged stress-to-gas-pressure ratio, 〈αp̄ 〉 at a value of ~0.04, is found for a model with radial, vertical and azimuthal resolution of 12-51, 27 and 12.5 cells per scaleheight (the simulation mesh is such that cells per scaleheight varies in the radial direction). The gas pressure dependence of the quasi-steady state stress level is also examined using models with different scaleheight-to-radius aspect ratio (H/R), revealing a weak dependence of 〈αp̄〉 on pressure. A control volume analysis is performed on the main body of the disc (|z| < 2H) to examine the production and removal of magnetic energy. Maxwell stresses in combination with the mean disc rotation are mainly responsible for magnetic energy production, whereas turbulent dissipation (facilitated by numerical resistivity) predominantly removes magnetic energy from the disc. Re-casting the magnetic energy equation in terms of the power injected by Maxwell stresses on the boundaries of, and by Lorentz forces within, the control volume highlights the importance of the boundary conditions (of the control volume). The different convergence properties of shearing-box and global accretion discsimulations can be readily understood on the basis of choice of boundary conditions and the magnetic field configuration. Periodic boundary conditions restrict the establishment of large-scale gradients in the magnetic field, limiting the power that can be delivered to the disc by Lorentz forces and by stresses at the surfaces. The factor of 3 lower resolution required for convergence in 〈αp〉 for our global disc models compared to stratified shearing-boxes is explained by this finding.