The two-dimensional structure of thin accretion disks

The two-dimensional structure of a thin accretion disk in the vicinity of a Schwarzschild black hole after passing a marginally stable orbit (r< 3rg is discussed in terms of the Grad-Shafranov hydrodynamic equation. The accretion disk is shown to be sharply compressed as the sonic surface is approached, so the mass flow here is no longer radial. As a result, the dynamic forces ρ[(v ∇)v]θ, which are equal in magnitude to the pressure gradient ∇θP on the sonic surface, become significant in vertical balance. Therefore, the disk thickness in the supersonic region (and, in particular, near the black-hole horizon) may be assumed to be determined not by the pressure gradient but by the shape of ballistic trajectories.