We study how the structure and variability of magnetohydrodynamic ( MHD ) turbulence in accretion discs converge with domain size .
Our results are based on a series of vertically stratified local simulations , computed using the Athena MHD code , that have fixed spatial resolution , but varying radial and azimuthal extent ( from \Delta R = 0.5 H to 16 H , where H is the vertical scale height ) .
We show that elementary local diagnostics of the turbulence , including the Shakura-Sunyaev \alpha parameter , the ratio of Maxwell stress to magnetic energy , and the ratio of magnetic to fluid stresses , converge to within the precision of our measurements for spatial domains of radial size L _ { x } \geq 2 H .
We obtain \alpha \simeq 0.02 - 0.03 , consistent with other recent determinations .
Very small domains ( L _ { x } = 0.5 H ) return anomalous results , independent of spatial resolution .
This convergence with domain size , however , is only valid for a limited set of diagnostics : larger spatial domains admit the emergence of dynamically important mesoscale structures .
In our largest simulations , the Maxwell stress shows a significant large scale non-local component , while the density develops long-lived axisymmetric perturbations ( “ zonal flows ” ) at the 20 % level .
Most strikingly , the variability of the disc in fixed-sized patches decreases strongly as the simulation volume increases , while variability in the magnetically dominated corona remains constant .
Comparing our largest local simulations to global simulations with comparable spatial resolution , we find generally good agreement .
There is no direct evidence that the presence of curvature terms or radial gradients in global calculations materially affect the turbulence , except to perhaps introduce an outer radial scale for mesoscale structures .
The demonstrated importance of mean magnetic fields – seen in both large local and global simulations – implies , however , that the growth and saturation of these fields is likely of critical importance for the evolution of accretion discs .