Comparison of flow variables for a non-self-gravitating disk and a self-gravitating disk with the spiral coordinates. The panels a–d present the Mach number (M), rotational velocity (q₂), sound speed or equivalent disk height (), and density of flow (q_ρ), respectively. We fixed the flow parameters (θ, q_2c, q_ρc) = (50° ,0.10, 10⁻⁴). For the calculation of a self-gravitating disk, we fixed the radial distance at r = 25. See the text for details.