Lagrangian versus Eulerian Methods for Toroidally-Magnetized Isothermal Disks

TL;DR

The paper tests whether the sustained toroidal magnetic fields seen in multi-physics disk simulations are a numerical artifact of resolution by re-running the G25 isothermal, ideal-MHD collapse problem with Lagrangian methods (MFM/MFV). It demonstrates that, at high resolution, Lagrangian results reproduce the Eulerian high-resolution collapse and midplane flux loss; at low resolution, Lagrangian methods still show collapse toward a thin midplane, whereas Eulerian methods do not, due to differing vertical-resolving capabilities. The authors argue this behavior is a natural consequence of the moving-mesh nature of Lagrangian codes, analogous to known differences in Jeans fragmentation, and that the sustained toroidal fields in more complex simulations are likely not solely a resolution effect. The findings imply that the persistent toroidal magnetization in recent multi-scale, multi-physics disks is robust to the numerical method and that the key physics (e.g., boundary/initial conditions and poloidal fields) should be the focus of further investigation.

Abstract

A number of simulations have seen the emergence of strongly-toroidally-magnetized accretion disks from interstellar medium inflows. Recently, Guo et al. 2025 (G25) studied an idealized test problem of toroidally-magnetized disks in isothermal ideal MHD with an Eulerian static-mesh method, and argued the midplane behavior changes qualitatively (with a significant loss of toroidal magnetic flux) when the the thermal scale-length is resolved ($Δx < H_{\rm thermal}$). We rerun the G25 test problem with two Lagrangian methods: meshless finite-mass, and meshless finite-volume. We show that Lagrangian methods reproduce the high-resolution ($Δx \ll H_{\rm thermal}$) Eulerian G25 results. At low resolution ($Δx \gg H_{\rm thermal}$), behaviors differ: Lagrangian methods still lose flux and evolve 'as close as possible' to the converged solution, while Eulerian methods show no evolution. We argue this difference in convergence behavior is related to the ability of Lagrangian codes to follow flows to an arbitrarily thin midplane layer, analogous to the well-studied difference in Jeans fragmentation problems. This and results from other higher-resolution simulations and different codes suggest that the sustained midplane toroidal fields seen in recent Lagrangian multi-scale, multi-physics simulations cannot be a numerical resolution effect, and some physical difference between those simulations and the G25 test problem explains their different behaviors.

Figures (1)

• Figure 1: G25 collapsing disk test problem (§ \ref{['sec:methods']}). Top: Time evolution in an example with unresolved thermal scale-height owing to extremely low-$\beta$ ($h=10^{-4}$), with Lagrangian MFM. Around the circularization radius, we plot the Alfvén scale-height $v_{A,\,\phi}/v_{\rm K}$ and midplane density scale-height $H/R$ in cylinders, with the Cartesian-equivalent resolution $\Delta x^{\rm C} = ({\rm Volume}/N_{\rm cells})^{1/3}$ and median Lagrangian resolution near the midplane $\langle \Delta x\rangle^{\rm mid}$ (both in units of $R$), at times after the initial disk forms until "collapse" as defined by G25 occurs. Middle: Vertical thermal ($P_{\rm th} = \rho\,c_{s}^{2} \propto \rho\,h^{2}$) and magnetic ($P_{\rm B} = B_{\phi}^{2}/2$) pressure profiles at the scale-height minimum of Lagrangian simulations with $h=0.05$. We show MFM at three resolution levels, plus an MFV run and a run without constrained-gradient MHD. We compare an exponential profile $\rho \propto \exp{(-|z|/\sqrt{2}\,h)}$. Converged solutions collapse to midplane $\beta\sim1$ and scale-height $\sim \sqrt{2}\,h$. Bottom: Same, for runs with $h=10^{-4}$. The behavior seen is similar to the "high-resolution" ($\Delta x^{\rm C} < h$) tests in G25, where $H/R$ decreases after tens of orbits, $\rho(z)$ forms a steep peak, and $P_{B}$ features a decrement around $|z|\sim 0$. With Lagrangian methods, collapse begins even with $\Delta x^{\rm C} \gg h$, until the disk is $\sim 1$-cell thick.