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The collective motion of microswimmers in suspensions induce patterns ofvortices on scales that are much larger than the characteristic size of amicroswimmer, attaining a state called bacterial turbulence. Hydrodynamicturbulence acts on even larger scales and is dominated by inertial transport ofenergy. Using an established modification of the Navier-Stokes equation thataccounts for the small scale forcing of hydrodynamic flow by microswimmers, westudy the properties of a dense supensions of microswimmers in two dimensions,where the conservation of enstrophy can drive an inverse cascade through whichenergy is accumulated on the largest scales. We find that the dynamical andstatistical properties of the flow show a sharp transition to the formation ofvortices at the largest length scale. The results show that 2d bacterial andhydrodynamic turbulence are separated by a subcritical phase transition.

Phase transition to large scale coherent structures in 2d active matter turbulence