On the numerical convergence of MRI simulations

TL;DR

This paper critically examines the common use of the MRI quality factor as a convergence indicator in simulations, demonstrating its limitations and proposing a modified metric for better assessment.

Contribution

The authors identify shortcomings of the MRI quality factor in convergence tests and introduce a revised metric to improve the reliability of MRI simulation analysis.

Findings

The MRI quality factor often fails to indicate convergence in zero-net-flux simulations.

Linear MRI theory assumptions do not hold in nonlinear, variable magnetic field conditions.

A modified quality factor is proposed to better assess numerical convergence in 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 Alfv\'en 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 Alfv\'en 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.