Accretion with heating, cooling, e_rot = 0.3, and varying levels of turbulence: evolution of the accretion rate (the Ṁ_• average in the top panel has 0.1 Myr step). The Mach number varies from the reference Ma ~ 0.35 (red) to 1/2 (magenta) and 1/4 (orange) of this value, i.e., Ta_t ≃ 0.7,1.5,3, respectively. The dashed line is the average net cooling rate (1 Myr step). As before, the solid black line is the adiabatic rotating model, and the runtime Bondi rate for the normalized plot is computed at r ≈ 1–2 kpc (Sect. 3). In the atmosphere with Ta_t< 1, CCA drives the dynamics, boosting the accretion rate up to 100 × the Bondi rate, which is consistent with the nonrotating CCA evolution shown in G13. As Ta_t > 1, the accretion flow shifts from turbulence-driven (linked to extended filaments and boosted accretion) to rotationally-driven (tied to a coherent disk and suppressed accretion).