Demystifying Flux Eruptions: Magnetic Flux Transport in Magnetically Arrested Disks

Abstract Magnetically arrested disks (MADs) are a compelling model for explaining variability in low-luminosity active galactic nuclei (AGN), including horizon-scale outbursts. 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 ≪ 1, where V k is the local Keplerian velocity. Using this framework, we analytically derive a recurrence timescale for flux eruptions, t rec ≃ 1500 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, P m , defined as the ratio of turbulent viscosity to turbulent resistivity. We find P m ≃ 3 , consistent with shearing-box simulations of magnetorotational instability-driven turbulence. While flux eruptions excite large-scale nonaxisymmetric 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.