On the numerical convergence of MRI simulations

Abstract

The magnetorotational instability (MRI) plays a crucial role in the evolution of many types of accretion disks. It is often studied using ideal-MHD numerical simulations. In principle, such simulations should be numerically converged, i.e. their properties should not change with resolution. Convergence is often assessed via the MRI quality factor, $Q$, the ratio of the Alfvén length to the grid-cell size. If it is above a certain threshold, the simulation is deemed numerically converged. In this paper we argue that the quality factor is not a good indicator of numerical convergence. First, we test the performance of the quality factor on simulations known to be unconverged, i.e. local ideal-MHD simulations with zero net-flux, and show that their $Q$s are well over the typical convergence threshold. The quality-factor test thus fails in these cases. Second, we take issue with the linear theory underpinning the use of $Q$, which posits a constant vertical field. This is a poor approximation in real nonlinear simulations, where the vertical field can vary rapidly in space and generically exhibits zeros. We calculate the linear MRI modes in such cases and show that the MRI can reach near-maximal growth rates at arbitrarily small scales. Yet, the quality factor assumes a single and well-defined scale, near the Alfvén length, below which the MRI cannot grow. We discuss other criticisms and suggest a modified quality factor that addresses some, though not all, of these issues.