Turbulence and Steady Flows in 3D Global Stratified MHD Simulations of   Accretion Disks
  We present full 2 Pi global 3-D stratified MHD simulations of accretion disks. We interpret our results in the context of proto-planetary disks. We investigate the turbulence driven by the magneto-rotational instability (MRI) using the PLUTO Godunov code in spherical coordinates with the accurate and robust HLLD Riemann solver. We follow the turbulence for more than 1500 orbits at the innermost radius of the domain to measure the overall strength of turbulent motions and the detailed accretion flow pattern. We find that regions within two scale heights of the midplane have a turbulent Mach number of about 0.1 and a magnetic pressure two to three orders of magnitude less than the gas pressure, while outside three scale heights the magnetic pressure equals or exceeds the gas pressure and the turbulence is transonic, leading to large density fluctuations. The strongest large-scale density disturbances are spiral density waves, and the strongest of these waves has m=5. No clear meridional circulation appears in the calculations because fluctuating radial pressure gradients lead to changes in the orbital frequency, comparable in importance to the stress gradients that drive the meridional flows in viscous models. The net mass flow rate is well-reproduced by a viscous model using the mean stress distribution taken from the MHD calculation. The strength of the mean turbulent magnetic field is inversely proportional to the radius, so the fields are approximately force-free on the largest scales. Consequently the accretion stress falls off as the inverse square of the radius. 
M. Flock, N. Dzyurkevich, H. Klahr, N. J. Turner, Th. Henning
astro-ph.EP