Adiabatic accretion with subsonic turbulence and e_rot = 0.3 (brown: Ma ~ 0.35, i.e., Ta_t ~ 0.75; orange: 1/4 lower Mach number, i.e., Ta_t ~ 3): evolution of the accretion rate. The accretion rate is suppressed by a factor ~3 compared with the nonrotating model (dot-dashed), as in the purely rotating run (solid black; Sect. 3), since turbulence still induces local (but not global) vorticity. The chaotic eddies generate however higher variability, as long as Ta_t< 1. In the opposite regime, rotation drives the dynamics.