Poloidal Field Amplification through Compression-Shear Dynamics in Schwarzschild Accretion: Pathways to MAD States

TL;DR

We address how large-scale poloidal magnetic fields are amplified in black hole accretion flows under strong gravity and rotational shear. The authors develop a semi-analytical general-relativistic framework in Schwarzschild spacetime, parameterizing rotational support with $ξ$ to track Lagrangian magnetic-field evolution under ideal MHD and flux freezing. They show a dichotomy: sub-Keplerian flows maximize radial flux amplification via compression, while Keplerian flows maximize poloidal-winding via shear, with GR dramatically shortening growth timescales near the horizon and pushing the field toward a non-radial, MAD-like configuration. The work provides analytic insight that complements GRMHD simulations and informs interpretation of horizon-scale observations and jet-launching physics.

Abstract

The amplification of magnetic fields in black hole accretion flows governs key high-energy phenomena such as magnetically arrested disks and relativistic jets. We develop a semi-analytical general relativistic framework that extends classical compressional amplification models by incorporating rotational shear, and apply it to large-scale poloidal magnetic field evolution in accretion flows around a Schwarzschild black hole. By parameterizing the azimuthal velocity as a fraction of the Keplerian value ($ξ\in [0,1]$), from purely radial infall ($ξ=0$) to Keplerian rotation ($ξ=1$), we examine the combined effects of radial compression and shear. Purely radial flows maximize amplification of both $B_r$ and $B_θ$ due to strong compression. In rotating flows, a distinct dichotomy emerges: sub-Keplerian regimes ($ξ<1$) preferentially enhance $B_r$, whereas Keplerian rotation strengthens $B_θ$ via shear. The transition from subsonic outer regions to supersonic relativistic inner regions further accelerates magnetic growth, revealing effects absent in earlier analytical treatments. These results show that rotational support controls both amplification efficiency and magnetic geometry, with sub-Keplerian phases particularly favorable for advecting the radial flux required for MAD formation. This work provides an analytical bridge between classical accretion theory and modern GRMHD simulations, with implications for X-ray binaries, AGNs, and EHT-scale systems.

Figures (7)

• Figure 1: Comparison of the time evolution of the physical components of the magnetic field for Keplerian flow ($\xi=1.0$) at different polar angles, measured at the fixed radial location $x=0.3$.
• Figure 2: Comparison of the time evolution of the physical components of the magnetic field for sub-Keplerian flow ($\xi=0.8$) at different polar angles, measured at the fixed radial location $x=0.3$.
• Figure 3: Physical poloidal magnetic field components at the fixed strong-gravity location $x=0.3$ as a function of polar angle $\theta$, measured in the local orthonormal (ZAMO) frame. (a) Radial component $_r B$ for Keplerian flow ($\xi=1.0$): slightly weaker than the sub-Keplerian case due to centrifugal barrier effects. (b) Polar component $_\theta B$ for Keplerian flow ($\xi=1.0$): strongest near the equatorial plane ($\theta \approx 7\pi/16$), reflecting efficient amplification by relativistic rotational shear. (c) Radial component $_r B$ for sub-Keplerian flow ($\xi=0.8$): significantly stronger across all angles, confirming radial compression as the dominant mechanism. (d) Polar component $_\theta B$ for sub-Keplerian flow ($\xi=0.8$): substantially reduced compared to the Keplerian case owing to weaker differential rotation. Panels (a,c) and (b,d) together illustrate the kinematic dichotomy: sub-Keplerian flows favor radial flux accumulation, whereas Keplerian flows maximize polar field winding.
• Figure 4: Comparison of the variations in the radial ($_r B$) and polar ($_\theta B$) components of the Keplerian flow ($\xi=1.0$) magnetic field as a function of radius for different polar angles, measured at the fixed observer time $t=25\,r_g/c$.
• Figure 5: Comparison of the variations in the radial ($_r B$) and polar ($_\theta B$) components of the sub-Keplerian flow ($\xi=0.8$) magnetic field as a function of radius for different polar angles, measured at the fixed observer time $t=25\,r_g/c$.