Demystifying flux eruptions: Magnetic flux transport in magnetically arrested disks

TL;DR

This work develops a GRMHD-based framework for magnetic flux transport in magnetically arrested disks, introducing a net flux transport velocity $v_\Phi$ that captures the balance between inward advection and outward diffusion. By Reynolds-averaging the GRMHD equations, the authors derive a vertically local and radially integrated flux transport equation and validate it with two 3D MAD simulations, showing a statistical quasi-steady state where $|v_{\rm adv}| \approx |v_{\rm diff}| \gg |v_\Phi|$. They analytically derive a recurrence timescale $t_{\rm rec} \sim 1500\, r_g/c$ for flux eruptions and verify it against simulations, while revealing that diffusion is driven by MRI-like turbulence with turbulent resistivity yielding a turbulent Prandtl number $\mathcal{P}_m \sim 1$–$5$. The study also measures the turbulent resistivity, examines the azimuthal structure of diffusion, and evaluates flux-transport criteria, finding that a Jacobian-based criterion $\mathcal{D}_{\rm Jac}$ best tracks the simulated behavior. Together these results illuminate the physics of magnetic flux transport in MADs, connect disk turbulence to horizon-scale variability, and offer a pathway to semi-analytic models of AGN variability and jet launching.

Abstract

Magnetically arrested disks (MADs) are a compelling model for explaining variability in low-luminosity active galactic nuclei (AGN), including horizon-scale outbursts like those observed in Sagittarius A*. MADs experience powerful flux eruptions-episodic ejections of magnetic flux from the black hole horizon-that may drive the observed luminosity variations. In this work, we develop and validate a new formalism describing large-scale magnetic field transport in general relativistic magnetohydrodynamic simulations of MADs with geometrical thicknesses of $h/R=0.1$ and $h/R=0.3$. We introduce a net flux transport velocity, $v_Φ$, which accounts for both advective and diffusive processes. We show that MADs maintain a statistical quasi-steady state where advection and diffusion nearly balance. Flux eruptions appear as small deviations from this equilibrium, with $v_Φ/V_k\ll1$, where $V_k$ is the local Keplerian velocity. Using this framework, we analytically derive a recurrence timescale for flux eruptions, $t_{\rm rec}\sim1500\, r_g/c$. This timescale closely matches simulation results. The smallness of $v_Φ$ explains the long recurrence times of flux eruptions compared to other system timescales. We also take a closer look at the diffusion of the magnetic field by performing the first measurement of turbulent resistivity in MADs. We then estimate the turbulent magnetic Prandtl number, defined as the ratio of turbulent viscosity to turbulent resistivity. We find $\mathcal{P}_m\sim3$, consistent with shearing-box simulations of magneto rotational instability-driven turbulence. While flux eruptions excite large-scale non-axisymmetric modes and locally enhance turbulent resistivity, magnetic field diffusion is dominated by smaller-scale turbulent motions. These results provide new insight into the nature of AGN variability and the fundamental physics of magnetic field transport.

Figures (13)

• Figure 1: (i)–(iii) Evolution of $_{\rm BH}$ for the thin and thick disk simulations, respectively. Dashed purple vertical lines mark local minima, and orange dotted lines mark local maxima; note the different temporal scales. (ii)–(iv) Comparison of $-\frac{ _t \Phi}{ _r \Phi}$ (dotted blue) and $v_\Phi$ (solid red) at $r_H$ for the thin and thick disk models, respectively. The same vertical lines as in (i)–(iii) indicate minima and maxima of $_{\rm BH}$; minima are followed by inward flux advection ($v_\Phi<0$), while maxima are followed by outward flux diffusion ($v_\Phi>0$). The close agreement between the solid red and dashed blue curves supports the validity of our flux transport framework.
• Figure 2: (i)–(iv) $r$–$t$ spacetime diagrams of the magnetic flux $\Phi(r,t)$ for the thin and thick disk models, respectively. Contours trace magnetic field lines, revealing cyclical inward and outward transport related to, but distinct from, flux eruptions (see text). (ii)–(v) $r$–$t$ spacetime diagrams of the flux transport velocity $v_\Phi(r,t)$ for the thin and thick disks. Red and blue regions correspond to inward advection and outward diffusion, respectively, and their patterns closely match those seen in $\Phi$ panels (i)–(iv), indicating consistency between flux evolution and transport velocity. (iii)–(vi) $r$–$t$ spacetime diagrams of the Alfvén velocity associated with the vertical magnetic field, $V_A^ /V_K$. Flux eruptions appear as regions with elevated $V_A^ /V_K$, with a clear temporal correlation between these high-$V_A^ /V_K$ events and the diffusion phases identified in panels (iii)–(iv). This agreement further supports the validity of our flux transport framework.
• Figure 3: Time evolution of $v_{\rm adv}$, $v_{\rm diff}$, and their sum $v_\Phi$ at $r = r_H$, normalized to the local Keplerian velocity $V_k$, for the thin (i) and thick (ii) disk models. In both cases, $v_{\rm adv} < 0$ and $v_{\rm diff} > 0$, showing inward advection and outward diffusion as expected, supporting the validity of our framework. We also observe $v_{\rm diff} \simeq |v_{\rm adv}| \gg |v_\Phi|$, indicating that net flux transport arises from small deviations between advection and diffusion. On average, $v_\Phi \sim 0$, consistent with a statistical steady state of the magnetic structure, though strong fluctuations remain. Dashed purple vertical lines mark local minima, and orange dotted lines mark local maxima; obtained from Fig. \ref{['fig:model_valid']}.
• Figure 4: (a) Time-averaged radial profiles of $v_\Phi$, $v_{\rm adv}$, and $v_{\rm diff}$, normalized to the local Keplerian velocity, for both disk models. Across all radii, $v_{\rm adv} < 0$ and $v_{\rm diff} > 0$, with magnitudes much larger than $v_\Phi$, showing that net flux transport arises from a small difference between advection and diffusion. The profiles of $v_{\rm adv}/V_k$ and $v_{\rm diff}/V_k$ are relatively flat, suggesting that in a fully steady-state simulation, these profiles remain flat for $r \gtrsim 10\,r_g$. (b) Since $v_\Phi$ is stochastic, its time average tends to $v_\Phi \sim 0$. To better illustrate its structure, the positive and negative contributions, $v_\Phi^+$ and $v_\Phi^-$, are separately time-averaged and normalized to $V_k$, these are also called the net outward and inward velocities respectively. They are smaller than $v_{\rm adv}$ and $v_{\rm diff}$ by a factor of $\sim 3$–$5$, indicating that the residual net flux transport is much smaller than the total advection and diffusion contributions. The profiles of $v_\Phi^{+,-}/V_k$ are roughly constant over $r_H \lesssim r \lesssim 40\,\,r_g$.
• Figure 5: Vertical profiles of flux transport velocities at $r=6\,r_g$. Panel (a): $\mathcal{A}^ /\langle B^ \rangle_$ (advection velocity $v_{\rm adv}$), $\mathcal{E}^ /\langle B^ \rangle_$ (diffusion velocity $v_{\rm diff}$), and $E^ /\langle B^ \rangle_$ (total flux transport velocity $v_\Phi$), all normalized to the local Keplerian velocity. Both thin and thick disk models show Gaussian-like profiles for $\mathcal{A}^ /\langle B^ \rangle_$ and $\mathcal{E}^ /\langle B^ \rangle_$, while the time-averaged $E^ /\langle B^ \rangle_$ remains close to zero. Thick disks display a slightly shallower vertical decrease compared to thin disks, as expected. Panel (b): latitudinal profiles of the positive ($E^{ ,+}/\langle B^ \rangle_$) and negative ($E^{ ,-}/\langle B^ \rangle_$) contributions to the magnetic flux velocity, alongside the net $E^{ }/\langle B^ \rangle_$. The positive and negative contributions are each an order of magnitude smaller than the advection and diffusion terms, consistent with the near cancellation in the total transport velocity.