On the numerical convergence of MRI simulations

TL;DR

This work demonstrates that the conventional MRI quality factor $Q$ can falsely indicate convergence in ideal MHD simulations, particularly for zero-net-flux configurations, because turbulence sets scales beyond the grid and is not captured by a fixed MRI lengthscale. By combining a critical evaluation of the linear MRI with a magnetic-null analysis, the authors show that significant growth can occur at arbitrarily small scales, undermining the foundation of $Q$ as a convergence metric. They reproduce large-$Q$ behavior in unconverged ZNF simulations and develop a linear-theory framework in the presence of magnetic zeros, arguing that a global diagnostic based on the large-scale net vertical flux $B_{ ext{net}}(R)$ and a corresponding quality factor $\\mathcal{Q}(R)$ offers a more robust, though not perfect, path toward assessing convergence in global MRI turbulence. The results imply that current $Q$-based convergence criteria may overestimate resolvability, and they advocate for diagnostics tied to large-scale magnetic structure and nonlinear energy injection scales to better gauge numerically converged MRI turbulence in accretion disks.

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.

Figures (14)

• Figure 1: Distribution of $B_y$ (first row) and $Q_z$ (second row) in the $(x,z)$ plane $y=0$ for the last outputs of simulations STD64, STD128 and STD256. In the bottom row all cells with $Q_z < 8$ are shown in black. From left to right, the increase in resolution leads to the appearance of smaller structures.
• Figure 2: Evolution of the space-averaged $\left< Q_z \right>_{x,y,z}$ over time.
• Figure 3: Points: evolution of $\left< Q_z \right>_{t,x,y,z}$ with resolution. Red line: best fit of those points ($l_{A_z} \propto \delta z^{0.25}$). Blue line: hypothetical resolved case ($l_{A_z}$=constant). Orange line: hypothetical non-resolved case ($l_{A_z} \propto \delta z$).
• Figure 4: Points: evolution of $\left< Q_z \right>_{t,x,y,z}$ with vertical aspect ratio. Line: best fit of those points ($l_{A_z} \propto (L_z / L_x)^{0.58}$). All simulations have the same resolution as STD64: $(N_x,N_y,N_z) = \left( 64,100,64(L_z/L_x)\right)$.
• Figure 5: Time and space-averaged stress $\alpha$ as a function of resolution for ZNF stratified simulations. Dashed lines are fits of $\alpha \propto (H/ \delta z)^{-a}$ to each dataset, where $a$ is 0.32 (this work), 0.34 Ryan2017, 0.33 Bodo2014 and 0.41 Davis2010. Error bars are standard deviations with time of the space-averaged stresses, assumed to be stationary processes.