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Global Magnetorotational Instability with Inflow. I. Linear Theory and the Role of Boundary Conditions

We formulate a model system suitable for the systematic numerical investigation of global aspects of the magnetorotational instability and nonlinear dynamo action in accretion disks. The model consists of a cylindrical annulus occupied by an incompressible fluid with explicit viscosity and resistivity. Boundary conditions are imposed that permit an accretion flow appropriate to the stresses acting within the fluid to develop freely through the annulus. A steady basic state is identified in which a slow, steady accretion flow is driven by the explicit viscosity. We investigate the linear theory of this state subject to different choices of boundary conditions. The choice of boundary conditions is a crucial factor in determining the nature and growth rate of the instabilities. It is found that very rapidly growing wall modes occur generically, drawing energy artificially from outside the computational domain. However, by carefully selecting boundary conditions for which the total pressure is constrained at the radial boundaries, the wall modes are found to have growth rates bounded by the local properties of the magnetorotational instability. The resulting model provides the basis for a systematic exploration of nonlinear behavior in our future work.