Turbulence is ubiquitous, from oceanic currents to small-scale biological and quantum systems. Self-sustained turbulent motion in microbial suspensions presents an intriguing example of collective dynamical behavior among the simplest forms of life and is important for fluid mixing and molecular transport on the microscale. The mathematical characterization of turbulence phenomena in active nonequilibrium fluids proves even more difficult than for conventional liquids or gases. It is not known which features of turbulent phases in living matter are universal or system-specific or which generalizations of the Navier-Stokes equations are able to describe them adequately. Here, we combine experiments, particle simulations, and continuum theory to identify the statistical properties of self-sustained meso-scale turbulence in active systems. To study how dimensionality and boundary conditions affect collective bacterial dynamics, we measured energy spectra and structure functions in dense Bacillus subtilis suspensions in quasi-2D and 3D geometries. Our experimental results for the bacterial flow statistics agree well with predictions from a minimal model for self-propelled rods, suggesting that at high concentrations the collective motion of the bacteria is dominated by short-range interactions. To provide a basis for future theoretical studies, we propose a minimal continuum model for incompressible bacterial flow. A detailed numerical analysis of the 2D case shows that this theory can reproduce many of the experimentally observed features of self-sustained active turbulence.low Reynolds number swimming | velocity increment distributions | scaling S imple forms of life, like amoebae or bacteria, self-organize into remarkable macroscopic patterns (1, 2), ranging from extended networks (3, 4) to complex vortices (5-10) and swarms (11). These structures often bear a striking resemblance to assemblies of higher organisms [e.g., flocks of birds (12) or schools of fish (13,14)] and present important biological model systems to study nonequilibrium phases and their transitions (15)(16)(17). A particularly interesting manifestation of collective behavior in microbial suspensions is the emergence of meso-scale turbulent motion (7,8,18,19). Driven by the microorganisms' self-propulsion and their mutual interactions, such self-sustained "active turbulence" can have profound effects on nutrient mixing and molecular transport in microbiological systems (2,(20)(21)(22). However, in spite of recent progress (19,(23)(24)(25), the phenomenology of turbulent bacterial dynamics is scarcely understood and a commonly accepted theoretical description is lacking (2,16,26). The latter fact may not be surprising given that a comprehensive mathematical characterization of turbulence in conventional fluids has remained elusive after more than a century of intense research (27).In view of the various physical and chemical pathways through which bacteria may communicate (1, 11, 28), a basic yet unsolved problem is to identify those interactions...

Self-sustained turbulent structures have been observed in a wide range of living fluids, yet no quantitative theory exists to explain their properties. We report experiments on active turbulence in highly concentrated 3D suspensions of Bacillus subtilis and compare them with a minimal fourth-order vector-field theory for incompressible bacterial dynamics. Velocimetry of bacteria and surrounding fluid, determined by imaging cells and tracking colloidal tracers, yields consistent results for velocity statistics and correlations over 2 orders of magnitude in kinetic energy, revealing a decrease of fluid memory with increasing swimming activity and linear scaling between kinetic energy and enstrophy. The best-fit model allows for quantitative agreement with experimental data.