Magnetically Arrested Accretion (MAD)

• MAD is a state of black hole accretion characterized by saturated poloidal magnetic flux that chokes inflow and triggers flux eruptions.
• GRMHD simulations and horizon-scale polarimetric observations confirm that magnetic stresses in MADs regulate disk structure and enable efficient jet launching.
• MAD models unify disk-jet coupling across mass scales, influencing spectral variability, angular momentum transport, and energetic feedback.

A magnetically arrested accretion flow (commonly: Magnetically Arrested Disk, MAD) forms when the poloidal magnetic flux accumulated in the innermost region of a black hole accretion disk becomes dynamically dominant, ultimately impeding further advection of field and gas. In this regime, magnetic stresses choke the inflow, regulate accretion through flux eruptions, and fundamentally reconfigure both disk and jet structure. The MAD state is robustly realized in high-resolution general relativistic magnetohydrodynamics (GRMHD) simulations and is strongly favored by horizon-scale polarimetric observations of M87* and Sgr A*.

1. Fundamental MAD Criteria, Thermodynamics, and Field Structure

The core defining feature of a MAD is the saturation of the poloidal magnetic flux on the horizon, typically quantified by the dimensionless parameter

$\phi_{\rm BH} \equiv \Phi_{\rm BH}/\sqrt{\dot M r_g^2 c} \simeq 50\mbox{--}60,$

where $\Phi_{\rm BH}$ is the horizon-threading poloidal flux, $\dot M$ the mass accretion rate, $r_g=GM/c^2$ the gravitational radius. This saturation is robust across a variety of black hole spins, accretion thicknesses, and boundary conditions (Begelman et al., 2021, Jacquemin-Ide et al., 29 Oct 2025, Salas et al., 1 May 2024).

In the saturated MAD state:

• The magnetic pressure in the inner flow balances or exceeds the ram (inflow) pressure, i.e., $\beta \equiv P_{\rm gas}/P_{\rm mag} \lesssim 1$ throughout $r \lesssim 10\,r_g$. Sub-Keplerian rotation emerges with $\Omega/\Omega_K \sim 0.4\mbox{--}0.6$ (Li et al., 27 Nov 2024, Begelman et al., 2021), the inner flow becomes vertically compressed, and the accretion velocity drops to $v_R \sim 0.01$–$0.1\,v_K$.
• The magnetic field is dominated by toroidal ($B_\phi$) in volume average, with $B_\phi^2 \gg B_r^2 + B_\theta^2$; the poloidal flux, though essential to sustain large-scale dynamo action, is dynamically subdominant except near the axis (Begelman et al., 2021, Zhang et al., 2023).
• Non-axisymmetric instabilities, including magnetic Rayleigh–Taylor (RT) and interchange/interfacial modes, facilitate intermittent accretion by breaking up the magnetic barrier and regulating flux escape (Chatterjee et al., 2022, Marshall et al., 2017, Salas et al., 1 May 2024).
• MADs can be sustained in both prograde and retrograde flows; disk spin and tilt modulate, but do not prevent, flux saturation (Chatterjee et al., 2023, Aktar et al., 15 Jun 2024).
• The transition to MAD occurs across a variety of accretion regimes—radiatively inefficient (ADAF), radiatively efficient (thin), and super-Eddington (slim) flows—though most simulations to date focus on RIAFs (Mocz et al., 2014, Li et al., 27 Nov 2024).

2. Magnetic Flux Transport, Diffusivity, and the Dynamics of Flux Eruptions

The global evolution of the large-scale poloidal flux $\Phi(r,t)$ in a MAD is governed by a slow, nearly balanced equilibrium between mean advective inflow and turbulent diffusivity. The net flux transport velocity,

$v_\Phi = v_{\rm adv} + v_{\rm diff},$

has $v_\Phi/V_K \ll 1$ (typically $\sim 10^{-2}$), explaining the persistent, quasi-steady saturation of $\phi_{\rm BH}$ (Jacquemin-Ide et al., 29 Oct 2025). This balance produces the defining long recurrence timescale of MAD flux eruption events,

$$t_{\rm rec} \sim \frac{r_g}{v_\Phi} \sim 10^3\,r_g/c,$$

in excellent agreement with both simulation and observed flare timescales in Sgr A* (Jacquemin-Ide et al., 29 Oct 2025, Chatterjee et al., 2022).

Turbulent resistivity $\eta$ can be measured from the turbulent EMF as $\mathcal{E}_\phi = -\eta J_\phi$. The turbulent magnetic Prandtl number,

$\mathcal{P}_m \equiv \nu_T/\eta \sim 3,$

matches predictions from MRI turbulence (Jacquemin-Ide et al., 29 Oct 2025). During a flux eruption—the transition from accumulated surplus flux to outflow, typically accompanied by reconnection—the instantaneous value of $v_\Phi$ spikes, and the MAD parameter $\phi_{\rm BH}$ decreases by $\Delta \phi \sim 5$–$10$, after which accretion refills the magnetosphere (Salas et al., 1 May 2024, Chatterjee et al., 2022).

3. Instabilities, Accretion Mechanisms, and Angular Momentum Transport

The highly magnetized inner flow of a MAD is subject to a range of MHD instabilities:

• Magnetic Rayleigh–Taylor (RT) instability produces low-density, highly magnetized "bubbles" that intermittently rise through the disk, disrupting coherent field geometry and sustaining turbulence even when the axisymmetric MRI is suppressed (Marshall et al., 2017, Salas et al., 1 May 2024).
• Non-axisymmetric interchange (or "toroidal convection") enables both poloidal and toroidal flux to be episodically expelled. MRI remains active at lower levels, contributing to the angular momentum transport (Begelman et al., 2021, Jacquemin-Ide et al., 29 Oct 2025).
• The dominant angular momentum transport in MADs is due to large-scale Maxwell stress ($-B_r B_\phi/4\pi$), supplemented by turbulent (RT-driven) Maxwell stress and enhanced by strong vertical ($T^{\theta}_{\phi}$) flux during eruptions (Chatterjee et al., 2022, Scepi et al., 2023, Marshall et al., 2017).
• In thin MADs ($h/r\sim0.03$–$0.1$), these instabilities produce powerful, laminar winds—where the bulk of the accretion torque is carried by the wind, and the inflow velocity $v_r$ can become transonic, in contrast to standard $\alpha$-disk expectations (Scepi et al., 2023).

4. Jet Formation, Efficiency, and Energetic Feedback

MADs are the most efficient known channel for launching relativistic jets:

• Jet efficiency in the Blandford–Znajek paradigm scales as $\eta_{\rm jet} \propto \phi_{\rm BH}^2 \Omega_H^2$, with $\phi_{\rm BH}\sim 50$ driving $\eta_{\rm jet}\gtrsim 1$ ($P_{\rm jet} > \dot M c^2$) for high-spin ($a_* \gtrsim 0.8$) Kerr holes (Narayan et al., 2021, Janiuk et al., 2022).
• The angular momentum and energy loss due to these jets causes secular black hole spin-down in prograde MADs, with the rate $da_*/dt$ determined by the net Maxwell stress and dependent nonlinearly on $a_*$ (Narayan et al., 2021, Zhang et al., 2023).
• Jets from MADs display power-law width profiles, typically parabolic, and their geometry (width, opening angle) correlates with the disk magnetic flux and spin (Narayan et al., 2021).
• Ultra-fast outflows (UFOs) with velocities $v \sim 0.2$–$0.5\,c$ are launched by the reconnection-driven flux eruptions in MADs, exhibiting broad, stochastic velocity distributions—distinct from periodic outflows induced by orbiting secondaries (Suková et al., 2023, Scepi et al., 2023).

5. Spectral, Variability, and Observational Diagnostics

The spectral energy distributions (SED) and time variability of MADs provide indirect but robust observational diagnostics:

• At fixed accretion rates, the SED of MADs is systematically brighter (by factors $\sim$2–5 in X-rays) than for SANE disks, but differences in spectral shape are subtle, making it difficult to distinguish MAD/SANE solely by broadband spectra (Xie et al., 2019, Aktar et al., 15 Jun 2024).
• High Compton-thick columns, mildly relativistic outflows, synchrotron/far-infrared peaks, and coronal heating in extended vertical regions are robust MAD features, reflecting the laminar wind torque and vertical magnetic support (Scepi et al., 2023, Xie et al., 2019).
• Recurrence intervals for large flares (X-ray, IR) are set by the flux transport-equilibrium cycle ($t_{\rm rec}\sim1000$–$2000\,r_g/c$), matching observed flare periodicities in Sgr A* and M87* (Jacquemin-Ide et al., 29 Oct 2025, Scepi et al., 2021).
• High-resolution and polarimetric imaging (EHT) directly reveals the strong horizon-scale flux, rapid field reversals, and structures such as spiral magnetic arms and fast, unresolved variability in polarization fraction and angle—consistent with simulated MAD states (Salas et al., 1 May 2024, Nalewajko et al., 10 Oct 2024).

6. MADs Across Mass Scales, Accretion Regimes, and Cosmic Context

The MAD paradigm unifies accretion physics from stellar-mass X-ray binaries (e.g., MAXI J1820+070) to AGN and even GRB engines:

• Observed delays between X-ray, radio, and optical flares in XRB outbursts (8–17 days in MAXI J1820+070) reflect the formation and growth of a MAD zone and the onset of wind- or disk-instability-driven propagation fronts (You et al., 2023).
• In low-Eddington systems (FR I radio galaxies, M87*, Sgr A*), only flows with sufficient external fields ($\beta_{\rm out} \lesssim 100$ at $R \sim 10^3 R_s$) can become MADs; the presence of an inner MAD dramatically increases jet power, accounting for radio-loud/radio-quiet dichotomies and powerful jets from ADAF+MAD interiors (Li et al., 27 Nov 2024, Mocz et al., 2014).
• MADs in GRB and AGN simulations produce structured, two-component jets and variability with $P(f)\propto f^{-1.7}$, scaling naturally with black hole mass $M$ (Janiuk et al., 2022). The MAD state provides a basis for unified time-domain models of AGN, microquasar, and TDE variability.
• State transitions in X-ray binaries can be mapped to global magnetic field inversions in MADs, where jet quenching and transient ejections are driven by reconnection-induced loss and regrowth of horizon-scale flux (Dexter et al., 2013).

7. Current Challenges, Numerical Considerations, and Future Directions

Critical outstanding issues include:

• Verification of MAD structure in radiatively efficient, thin and slim disks remains incomplete; all current flux–$\dot M$ scaling in such regimes rests on extrapolation from RIAF results (Mocz et al., 2014, Xie et al., 2019).
• MAD properties, such as flux saturation, cycle times, and heating partition, exhibit convergence at moderate ($512^3$) resolution; however, the fine-scale RT fingers, plasmoid-mediated reconnection, and jet-sheath mixing require much higher resolution ($N_\theta>1000$), which modifies limb-brightening, polarization, and flare spectra at the percent level (Salas et al., 1 May 2024).
• The suppression of Lense–Thirring precession in thick, misaligned MADs, alongside the replacement of precession-driven QPOs with flux-eruption-driven flares, suggests that observed quasi-periodic signals in AGN/XRBs are best understood in the context of magnetic flux eruption cycles rather than coherent disk-body precession (Chatterjee et al., 2023).
• The precise role of turbulent magnetic Prandtl number, resistivity, and the nature of cross-field transport (diffusive vs. advective) in setting both flux recurrence and the secular growth of $\phi_{\rm BH}$ is now directly accessible to simulation-based measurement (Jacquemin-Ide et al., 29 Oct 2025).

Overall, MADs represent a dynamical attractor solution for black hole accretion flows supplied with sufficient net vertical magnetic flux. Their essential physics—horizon-scale flux saturation, slow quasi-steady advection–diffusion equilibrium, flux eruptions via RT and reconnection, powerful, structured jets, and regulated angular momentum transport—underpins the emerging paradigm unifying disk–jet coupling, multiwavelength time variability, and polarimetric structure from microquasars to radio galaxies and GRBs. As observational techniques (EHT, GRAVITY) probe the horizon and theoretical efforts reach ever higher fidelity, MADs provide a quantitative, predictive framework for black hole accretion and feedback across the full cosmic mass spectrum (Jacquemin-Ide et al., 29 Oct 2025, Salas et al., 1 May 2024, Chatterjee et al., 2022, Suková et al., 2023, Mocz et al., 2014, Xie et al., 2019).