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

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 ($\xi \in [0,1]$), from purely radial infall ($\xi=0$) to Keplerian rotation ($\xi=1$), we examine the combined effects of radial compression and shear. Purely radial flows maximize amplification of both $B_r$ and $B_\theta$ due to strong compression. In rotating flows, a distinct dichotomy emerges: sub-Keplerian regimes ($\xi<1$) preferentially enhance $B_r$, whereas Keplerian rotation strengthens $B_\theta$ 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.