The modern theory of accretion disks is dominated by the discovery of the magnetorotational instability (MRI). While hydrodynamic disks satisfy Rayleigh's criterion and there exists no known unambiguous route to turbulence in such disks, a weakly magnetized disk of plasma is subject to the MRI and will become turbulent. This MRI-driven magnetohydrodnamic turbulence generates a strong anisotropic correlation between the radial and azimuthal magnetic fields which drives angular momentum outwards. Accretion disks perform two vital functions in various astrophysical systems: an intermediate step in the gravitational collapse of a rotating gas, where the disk transfers angular momentum outwards and allows material to fall inwards; and as a power source, where the gravitational potential energy of infalling matter can be converted to luminosity. Accretion disks are important in astrophysical processes at all scales in the universe. Studying accretion from first principles is difficult, as analytic treatments of turbulent systems have proven quite limited. As such, computer simulations are at the forefront of studying systems this far into the non-linear regime.;While computational work is necessary to study accretion disks, it is no panacea. Fully three-dimensional simulations of turbulent astrophysical systems require an enormous amount of computational power that is inaccessible even to sophisticated modern supercomputers. These limitations have necessitated the use of local models, in which a small spatial region of the full disk is simulated, and constrain numerical resolution to what is feasible. These compromises, while necessary, have the potential to introduce numerical artifacts in the resulting simulations. Understanding how to disentangle these artifacts from genuine physical phenomena and to minimize their effect is vital to constructing simulations that can make reliable astrophysical predictions and is the primary concern of the work presented here.;The use of local models is predicated on the assumption that these models accurately capture the dynamics of a small patch of a global astrophysical disk. This assumption is tested in detail through the study of local regions of global simulations. To reach resolutions comparable to those used in local simulations an orbital advection algorithm, a semi-Lagrangian reformulation of the fluid equations, is used which allows an order of magnitude increase in computational efficiency. It is found that the turbulence in global simulations agrees at intermediate- and small-scales with local models and that the presence of magnetic flux stimulates angular momentum transport in global simulations in a similar manner to that observed for local ones. However, the importance of this flux-stress connection is shown to cast doubt on the validity of local models due to their inability to accurately capture the temporal evolution of the magnetic flux seen in global simulations.;The use of orbital advection allows the ability to probe previously-inaccessible resolutions in global simulations and is the basis for a rigorous resolution study presented here. Included are the results of a study utilizing a series of global simulations of varying resolutions and initial magnetic field topologies where a collection of proposed metrics of numerical convergence are explored. The resolution constraints necessary to establish numerical convergence of astrophysically-important measurements are presented along with evidence suggesting that the use of proper azimuthal resolution, while computationally-demanding, is vital to achieving convergence. The majority of the proposed metrics are found to be useful diagnostics of MRI-driven turbulence, however they suffer as metrics of convergence due to their dependence on the initial magnetic field topology. In contrast to this, the magnetic tilt angle, a measure of the planar anisotropy of the magnetic field, is found to be a powerful tool for diagnosing convergence independent of initial magnetic field topology.
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