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Bacterial swarms display intriguing dynamical states like active turbulence.Using a hydrodynamic model we now show that such dense active suspensionsmanifest super-diffusion, via L\'evy walks, which masquerades as a crossoverfrom ballistic to diffusive scaling in measurements ofmean-squared-displacements, and is tied to the emergence of hitherto undetectedoscillatory streaks in the flow. Thus, while laying the theoretical frameworkof an emergent advantageous strategy in the collective behaviour ofmicroorganisms, our study underlines the essential differences between activeand inertial turbulence.

Anomalous diffusion and L\'evy walks distinguish active from inertial turbulence