Phase ordering, topological defects, and turbulence in the three-dimensional incompressible Toner-Tu equation

We investigate the phase-ordering dynamics of the incompressible Toner-Tu equation in three dimensions. We show that the phase ordering proceeds via defect merger events and the dynamics is controlled by the Reynolds number Re. At low Re, the dynamics is similar to that of the Ginzburg-Landau equation. At high Re, turbulence controls phase ordering. In particular, we observe a forward energy cascade from the coarsening length scale to the dissipation scale, clustering of defects, and multiscaling in velocity correlations.