Magnetically Arrested Disk (MAD)

• MAD is an accretion regime where dynamically significant magnetic flux accumulates near black holes, regulating inflow and prompting episodic magnetic eruptions.
• Analytic models and GRMHD simulations show that when magnetic pressure rivals gas pressure, the disk structure alters, slowing accretion and facilitating jet-launching systems.
• Observational diagnostics, including polarimetry and potential Zeeman splitting of Fe Kα lines, are key to identifying MAD states despite ongoing debates over MRI suppression.

Searching arXiv for recent and foundational papers on magnetically arrested disks to ground the article in current literature. Magnetically arrested disk (MAD) denotes an accretion regime in which large-scale magnetic flux accumulates near a black hole until the magnetic field becomes dynamically significant and regulates, impedes, or partially arrests inflow. In this state, the inner flow is strongly magnetized, the magnetic pressure is comparable to or exceeds the gas pressure, and accretion proceeds through a combination of slowed inflow, turbulent transport, interchange-like behavior, and episodic magnetic flux eruptions rather than through an unmodified weak-field accretion pattern (Li et al., 2024). MADs are commonly discussed in contrast to standard and normal evolution (SANE) flows, and they have been studied in analytic models, GRMHD simulations, radiation-MHD calculations, and multiwavelength observational interpretations spanning radiatively inefficient flows, thin disks, radio galaxies, X-ray binaries, and jet-launching systems (Xie et al., 2019).

1. Concept and defining criteria

The classical formulation of a MAD emphasizes the accumulation of dynamically significant net poloidal magnetic flux near the black hole horizon and the associated reduction of the inflow speed once magnetic forces become comparable to gravity (Li et al., 2024). In analytic and semi-analytic descriptions, the flow becomes “arrested” when the radial magnetic force rises sufficiently to impede accretion, while in GRMHD usage the state is often identified by saturation of the horizon-threading magnetic flux and by plasma beta values indicating strong magnetization (Aktar et al., 12 Mar 2026).

Several operational diagnostics recur across the literature. One is the normalized magnetic flux at the horizon, written in one simulation study as $\dot{\phi}_{\rm acc}$, with the MAD state identified when $\dot{\phi}_{\rm acc} \gtrsim 50$ in cgs Gaussian units; another is the spatially averaged plasma beta, with $\beta_{\rm ave} \lesssim 1$ indicating magnetic pressure comparable to or exceeding gas pressure (Aktar et al., 12 Mar 2026). A related horizon-flux diagnostic is the dimensionless magnetic flux $\phi_{\rm BH}$, for which values of order $20$–$50$ are quoted from GRMHD simulations in the context of observational diagnostics of inner-disk magnetic field strength (Inoue, 2023). In an ADAF-based formation model, the transition to an inner MAD region is judged by the near-freezing of the radial-velocity gradient, expressed as

$$\frac{d}{dR}v_R \lesssim 0.01 \frac{d}{dR}v_{\rm k},$$

with the resulting arrested region appearing at several ten Schwarzschild radii outside the horizon (Li et al., 2024).

A point of terminology is important. Some papers use “MAD” primarily for a high horizon-flux state, while others argue that no single scalar diagnostic suffices. “The SANE, the MAD, and the Chimera” argues that MAD-like behavior is not captured by any single diagnostic, but by a dynamical coupling among horizon flux, jet power, magnetic support, Maxwell transport, surface-layer flow, disk morphology, and eruption activity (Wong et al., 8 Jul 2026). This suggests that the meaning of “arrested” is best understood dynamically rather than as a threshold in one number alone.

2. Magnetic-flux accumulation and formation pathways

A central mechanism in MAD formation is inward transport of magnetic flux by accretion. In advection-dominated accretion flows, the radial velocity is high enough that inward transport of poloidal magnetic flux is much more efficient than in thin disks, allowing dynamically significant flux to accumulate near the black hole (Li et al., 2024). This picture appears in both analytic and numerical work: externally supplied large-scale magnetic field is advected inward by the accreting gas, and once an upper limit to the sustainable magnetic flux is reached, the flow becomes magnetically arrested (Mondal et al., 2018).

The ADAF-based treatment of flux transport uses a mass-accretion profile with outflows,

$$\dot{M}_{\rm acc}(R)=\dot{M}_{\rm out}\left(\frac{R}{R_{\rm out}}\right)^s,$$

together with radial momentum, angular momentum, energy, and induction equations, including a magnetic force term

$$g_{\mathrm{m}} = \frac{B_R^{\mathrm{s}} B_z}{2\pi \Sigma},$$

and a magnetic torque

$T_m = \frac{B_z B_\varphi^{\rm s}}{2\pi} R .$

The flux-transport equation is written as

$\dot{\phi}_{\rm acc} \gtrsim 50$0

with outer conditions set at $\dot{\phi}_{\rm acc} \gtrsim 50$1 by self-similar ADAF solutions and inner conditions given by zero torque at the horizon and a transonic solution near $\dot{\phi}_{\rm acc} \gtrsim 50$2 (Li et al., 2024).

That formalism yields a threshold statement: MAD is unlikely to form through inward flux advection when the external magnetic field is too weak, specifically for $\dot{\phi}_{\rm acc} \gtrsim 50$3 around $\dot{\phi}_{\rm acc} \gtrsim 50$4 (Li et al., 2024). The same work reports that the radial extent of the inner MAD region increases as $\dot{\phi}_{\rm acc} \gtrsim 50$5 decreases, and that establishing a MAD throughout the entire disk would require $\dot{\phi}_{\rm acc} \gtrsim 50$6 to approach $\dot{\phi}_{\rm acc} \gtrsim 50$7–$\dot{\phi}_{\rm acc} \gtrsim 50$8 (Li et al., 2024). A plausible implication is that flux supply at large radii is not a minor boundary detail but a controlling parameter for whether the MAD state exists at all.

More recent GRMHD analysis of flux transport emphasizes that fully developed MADs maintain a statistical quasi-steady state in which inward advection and outward diffusion nearly balance. In that formulation, the axisymmetric poloidal flux obeys

$\dot{\phi}_{\rm acc} \gtrsim 50$9

with

$\beta_{\rm ave} \lesssim 1$0

and the empirical result that $\beta_{\rm ave} \lesssim 1$1, while $\beta_{\rm ave} \lesssim 1$2 (Jacquemin-Ide et al., 29 Oct 2025). Flux eruptions then appear as small departures from this near-cancellation rather than as evidence that the disk has ceased transporting flux.

3. Disk structure, force balance, and transport physics

The inner MAD region differs structurally from both weak-field hot accretion flows and standard thin-disk expectations. In the ADAF-to-MAD model, the radial velocity in the MAD region drops to roughly $\beta_{\rm ave} \lesssim 1$3 of the value in a normal ADAF, becoming subsonic and nearly constant with radius outside the innermost transonic zone, while the angular velocity becomes highly sub-Keplerian, $\beta_{\rm ave} \lesssim 1$4, and the gas-to-magnetic pressure ratio falls to $\beta_{\rm ave} \lesssim 1$5 (Li et al., 2024). The same solution gives an inner MAD that is thinner, denser by about an order of magnitude, and characterized by a steeper density profile than a standard ADAF (Li et al., 2024).

A long-standing interpretive issue concerns which magnetic component actually regulates the MAD state. “What really makes an accretion disc MAD” argues that MADs are characterized and regulated mainly by a dynamically important toroidal field, not by the net vertical flux alone (Begelman et al., 2021). In that picture, the vertical field is comparatively weak throughout most of the accretion flow except in magnetized bubbles near the hole, while the toroidal field dominates the pressure and dynamics outside the innermost $\beta_{\rm ave} \lesssim 1$6 (Begelman et al., 2021). The paper generalizes the Høiland stability criteria by introducing

$\beta_{\rm ave} \lesssim 1$7

and argues that the disc self-organizes toward marginal radial convective instability rather than toward a simple vertical-field “magnetospheric radius” balance (Begelman et al., 2021). It also states that MRI is not suppressed in MADs and is probably responsible for the existence of the strong toroidal field (Begelman et al., 2021).

That claim directly intersects with an older thin-disk MAD interpretation in which large-scale ordered fields marginally suppress the axisymmetric MRI and angular momentum transport is instead driven by the magnetic Rayleigh-Taylor instability and the turbulence it generates (Marshall et al., 2017). In the thin $\beta_{\rm ave} \lesssim 1$8 simulation analyzed there, low-density highly magnetized bubbles decrease the stress in the disk during bubble episodes, but the dominant stress component remains the turbulent magnetic field rather than the mean large-scale field (Marshall et al., 2017). The resulting picture is not purely laminar even when ordered fields are strong.

A separate line of thin-MAD work emphasizes wind-driven transport. In simulations with controlled thermal thicknesses $\beta_{\rm ave} \lesssim 1$9, $\phi_{\rm BH}$0, and $\phi_{\rm BH}$1, thin MADs are reported to remain near Keplerian in radial force balance, not mostly supported by magnetic field radially, while turbulent magnetic pressure becomes crucial for vertical support in the thinnest case (Scepi et al., 2023). For $\phi_{\rm BH}$2, the density scale height is $\phi_{\rm BH}$3 larger than the thermal scale height, the midplane density is up to two orders of magnitude lower than in standard theory, and wind-driven magnetic torque dominates net angular-momentum extraction even though MRI turbulence remains present (Scepi et al., 2023). This does not negate strong-field regulation; it specifies that the way magnetic fields reshape the disk depends on whether one is examining radial support, vertical support, or torque balance.

4. Variability, flux eruptions, and outflows

MADs are associated with episodic magnetic-flux eruptions, non-axisymmetric structures, and intermittent accretion through localized channels. High-resolution convergence studies show that time-averaged disk properties are broadly robust, but also that higher resolution reveals inward angular-momentum transport attributed to turbulent convection during flux eruptions, together with wave-like features in the jet sheath and increased sheath variability (Salas et al., 2024). The same study argues that plasmoid-mediated reconnection becomes explicitly resolved at the highest resolutions, enabling physical reconnection rates when the Lundquist number exceeds $\phi_{\rm BH}$4, with

$\phi_{\rm BH}$5

and suggesting direct consequences for heating and particle acceleration (Salas et al., 2024).

A new transport-based interpretation of flux eruptions derives an analytic recurrence timescale

$\phi_{\rm BH}$6

with the long recurrence attributed to the smallness of the net transport velocity $\phi_{\rm BH}$7 relative to the larger advective and diffusive contributions (Jacquemin-Ide et al., 29 Oct 2025). The same work reports the first measurement of turbulent resistivity in MADs and finds a turbulent magnetic Prandtl number $\phi_{\rm BH}$8, consistent with shearing-box MRI turbulence (Jacquemin-Ide et al., 29 Oct 2025). This suggests that even strongly magnetized MADs retain a turbulent transport phenomenology that can be compared quantitatively with more standard accretion-flow turbulence models.

Outflow phenomenology is correspondingly diverse. In “MAD UFOs,” 3D GRMHD simulations show that an outflow launched in the MAD state can be accelerated to relativistic velocities and persist throughout the simulation, with stochastic behavior and an approximately flat velocity distribution spanning $\phi_{\rm BH}$9 and $20$0 (Suková et al., 2023). Thin-MAD simulations report terminal wind velocities of $20$1–$20$2 for $20$3, with mass-loss and energy efficiencies consistent with ultra-fast outflows observed in AGN (Scepi et al., 2023). Earlier analytic work on magnetized advective flows describes “magnetic barriers” that can decelerate or repel inflow, and proposes single-barrier, double-barrier, and episodic ejection behaviors as possible ingredients of jet and outflow production (Mondal et al., 2018).

Large-scale magnetic reconnection can also change the accretion state itself. In simulations of global field inversion, reconnection of the MAD field in the inner radii destroys the steady jet, increases the accretion rate, moves the effective inner disk edge inward, and launches a transient hot outflow whose power can be comparable to the steady Blandford-Znajek jet (Dexter et al., 2013). This has been used to interpret high-luminosity hard-to-soft state transitions in black hole X-ray binaries (Dexter et al., 2013).

5. Radiation, spectra, and high-energy emission

MAD radiative properties are not reducible to a single universal signature. A two-temperature radiative calculation based on MHD-informed magnetic geometry finds that, at the same accretion rate, MADs and SANEs have similar spectral energy distributions, although MADs are systematically brighter and have somewhat higher radiative efficiency for a given $20$4 (Xie et al., 2019). In that treatment, the critical accretion rate above which the hot solution ceases to exist is lower in MAD than in SANE by a factor of $20$5–$20$6, and the maximum luminosity reachable in MAD is comparable but slightly lower than in SANE (Xie et al., 2019). The practical consequence is that spectral modeling alone is difficult to use as a MAD diagnostic.

Radiative cooling can itself modify the dynamics. A GRMHD study including synchrotron cooling introduces a critical accretion rate

$20$7

with normalization depending on saturated magnetic flux and the electron-to-proton temperature ratio, but not on black-hole mass (Singh et al., 2024). Above this threshold, the MAD parameter and jet efficiency vary by a factor of $20$8 and the force balance shifts so that magnetic contributions increase as thermal pressure support decreases (Singh et al., 2024). This suggests that “radiatively inefficient MAD” and “cooling-modified MAD” should not be conflated.

Radiation-RMHD simulations aimed at AGN accretion similarly find that the MAD state persists across black-hole spins and that the total radiative luminosity is significantly higher than the luminosities from synchrotron and bremsstrahlung alone (Aktar et al., 12 Mar 2026). In those calculations the flow is considered MAD when both the horizon-flux and plasma-beta criteria are satisfied, and the overall dynamics and SED are reported to be qualitatively similar across spins, with electron temperatures in the jet region reaching $20$9K (Aktar et al., 12 Mar 2026).

MADs have also been proposed as high-energy particle accelerators. In radio galaxies, hadronic models posit that magnetic reconnection or turbulence in a magnetically dominated MAD efficiently accelerates non-thermal protons and electrons, with proton synchrotron producing GeV gamma rays, secondary processes contributing at higher energies, and pair production in the black-hole magnetosphere screening any vacuum gap (Kimura et al., 2020). The same model is used to explain the gamma-ray emission of M87 and NGC 315 and to predict contributions to the extragalactic gamma-ray background and a multi-PeV neutrino background (Kimura et al., 2020). In quiescent black-hole binaries, MADs formed inside a RIAF have likewise been proposed as sites of non-thermal particle acceleration, broadband synchrotron emission, and proton acceleration up to PeV energies (Kimura et al., 2021).

6. Observational interpretations and outstanding debates

Observational claims for MADs are necessarily indirect, because the state is defined by magnetic flux and dynamical force balance rather than by a single spectral line or luminosity ratio. Event Horizon Telescope polarimetry of M87$50$0 and Sgr A$50$1 has been interpreted as suggesting a dynamically strong, ordered magnetic field typical of a MAD (Salas et al., 2024). For radio-loud AGN with thin disks, analysis of the 3CRR sample finds that the average X-ray luminosity is about $50$2 times higher than in radio-quiet AGN with matching optical-UV luminosity, broadly consistent with a comparison MAD sample, and that the radio–X-ray relation and inferred jet efficiencies are consistent with MAD expectations (Li et al., 2024). That work therefore suggests that probably all the quasars and at least a fraction of HERGs in 3CRR, and perhaps all RLAGNs with strong radio emission, contain a MAD (Li et al., 2024).

For stellar-mass systems, archival observations of the 2018 outburst of MAXI J1820+070 have been interpreted as evidence for MAD formation, based on a radio lag of $50$3 days and an optical lag of $50$4 days relative to the X-ray flare (You et al., 2023). In that scenario, the expanding ADAF amplifies and advects magnetic flux until the field at the ISCO reaches the MAD value, with the radio flare marking maximized jet power and the optical delay attributed to a revived thermal-viscous instability in the outer disk (You et al., 2023).

A more direct proposed signature is Zeeman splitting of relativistically broadened Fe K$50$5 lines. In a two-phase inner-disk geometry with a magnetically dominated corona and cold reflector, the splitting is estimated as

$50$6

and is argued to be detectable in X-ray binaries with XRISM or Athena if the system is in the MAD configuration (Inoue, 2023). This proposal does not claim that existing observations have already confirmed MAD, but it specifies an in-principle magnetic-field diagnostic.

Several debates remain open. One concerns whether MRI is suppressed in MADs: some studies describe suppression of the axisymmetric MRI and replacement by magnetic Rayleigh-Taylor or interchange-driven transport (Marshall et al., 2017), whereas others argue that non-axisymmetric MRI remains active and is central to sustaining the dominant toroidal field (Begelman et al., 2021). Another concerns definition: some authors use horizon magnetic flux saturation as the hallmark of MAD, whereas others emphasize that high flux and strong jets can occur without the full standard MAD phenomenology, as illustrated by the “Chimera” flow (Wong et al., 8 Jul 2026). A further unresolved issue is regime generality. “Interpreting MAD within multiple accretion regimes” argues that the characteristic MAD flux–accretion relationship is established by simulations in radiatively inefficient flows, and that extending it to thin and slim disks remains an assumption requiring dedicated GRMHD validation (Mocz et al., 2014).

Taken together, the literature presents MAD not as a single phenomenological template but as a strong-flux accretion state whose observable consequences depend on magnetic-flux supply, disk thickness, radiative efficiency, spin, transport physics, and accretion history. This suggests that the most reliable identification of a MAD combines magnetic-flux diagnostics, dynamical evidence for magnetic regulation of the inner flow, and jet or variability properties, rather than relying on any isolated observable.