A Reynolds-like Criterion for the Onset of Universal Active Turbulence

Beyond a critical Toner–Tu drive, bacterial turbulence exhibits universal statistics, including a parameter-independent energy spectrum $E(k)\sim k^{-3\mathrm{/}2}$. We show that the same universal state can be reached not only by increasing activity, but also by strengthening nonlinear advection, weakening nonlinear saturation, or tuning the Swift–Hohenberg instability. Motivated by the competition between these processes, we identify the independent dimensionless groups governing the dynamics and construct a Reynolds-like control parameter from them. A single critical value, calibrated from the known activity-controlled transition, predicts the onset of universality along the other parameter routes and collapses them onto a common threshold. These results identify a dynamical organizing parameter for the onset of universal active turbulence and provide a Reynolds-like measure for the Toner–Tu–Swift–Hohenberg universality class.