Abstract
<title>Abstract</title> <p> Magnetic fields are dynamically important in many black-hole accretion flows. They contribute to vertical support, transport angular momentum through Maxwell stress, heat the corona, and help determine whether an accretion flow is weakly magnetized, magnetically elevated, or magnetically arrested. This paper develops a self-consistent analytic treatment of low-beta plasma in a disk around a non-rotating Schwarzschild black hole, with plasma beta defined as the ratio of gas pressure to magnetic pressure. The Schwarzschild geometry is used as a control case because it has an event horizon and an innermost stable circular orbit but no frame dragging or Blandford-Znajek spin-energy extraction. Within a local, isothermal, constant-beta model, we derive the magnetic-pressure fraction, the effective hydrostatic scale height, the Alfvén-to-sound-speed ratio, and the exact fastest-growing wavelength of the ideal, axisymmetric, vertical-field magnetorotational instability (MRI) in a Newtonian Keplerian patch. The derivation distinguishes this exact wavelength from the frequently used estimate 2 <italic>πv</italic> <sub>A</sub> / <italic>Ω</italic> . As beta decreases, the disk becomes more strongly supported by magnetic pressure and the fastest-growing vertical MRI mode becomes too long to fit comfortably within one scale height. This does not imply universal stabilization: strong fields suppress parts of the local MRI spectrum while permitting magnetic buoyancy, reconnection, interchange motions, and global flux accumulation. Recent horizon-scale polarimetry of M87* and Sagittarius A* constrains ordered and time-dependent magnetic-field structure, but beta remains an inferred, model-dependent parameter. The results are analytic scalings, not a replacement for radiative general-relativistic magnetohydrodynamic simulations. </p>