Exact shering box solutions of MHD flows with resistivity, viscosity and cooling

Axisymmetric incompressible modes of the magneto-rotational instability (MRI) with a vertical wavenumber are exact solutions of the non-linear local equations of motion for a disk (shearing box). They are referred to as "channel solutions". Here, we generalize a class of these solutions to include energy losses, viscous, and resistive effects. In the limit of zero shear, we recover the result that torsional Alfv\'en waves are exact solutions of the non-linear equations. Our method allows the extension of these solutions into the dissipative regime. These new solutions serve as benchmarks for simulations including dissipation and energy loss, and to calibrate numerical viscosity and resistivity in the Zeus3D code. We quantify the anisotropy of numerical dissipation and compute its scaling with time and space resolution. We find a strong dependence of the dissipation on the mean magnetic field that may affect the saturation state of the MRI as computed with Zeus3D. It is also shown that elongated grid cells generally preclude isotropic dissipation and that a Courant time step smaller than that which is commonly used should be taken to avoid spurious anti-diffusion of magnetic field.