Intermittency measurement in two-dimensional bacterial turbulence.

Abstract

In this paper, an experimental velocity database of a bacterial collective motion, e.g., Bacillus subtilis, in turbulent phase with volume filling fraction 84% provided by Professor Goldstein at Cambridge University (UK), was analyzed to emphasize the scaling behavior of this active turbulence system. This was accomplished by performing a Hilbert-based methodology analysis to retrieve the scaling property without the β-limitation. A dual-power-law behavior separated by the viscosity scale ℓ_{ν} was observed for the qth-order Hilbert moment L_{q}(k). This dual-power-law belongs to an inverse-cascade since the scaling range is above the injection scale R, e.g., the bacterial body length. The measured scaling exponents ζ(q) of both the small-scale (k>k_{ν}) and large-scale (k<k_{ν}) motions are convex, showing the multifractality. A log-normal formula was put forward to characterize the multifractal intensity. The measured intermittency parameters are μ_{S}=0.26 and μ_{L}=0.17, respectively, for the small- and large-scale motions. It implies that the former cascade is more intermittent than the latter one, which is also confirmed by the corresponding singularity spectrum f(α) versus α. Comparison with the conventional two-dimensional Ekman-Navier-Stokes equation, a continuum model indicates that the origin of the multifractality could be a result of some additional nonlinear interaction terms, which deservers a more careful investigation.