Universality of the intermittency parameter and unresolved cascade mechanisms in two-dimensional bacterial turbulence

Dense suspensions of swimming bacteria form chaotic, vortex-filled flows at effectively zero Reynolds number. A 2024 experimental analysis of two-dimensional bacterial turbulence (Huang, Phys. Fluids 36, 065149) found that the coarse-grained energy-dissipation and enstrophy fields obey Kolmogorov-1962 lognormal multifractal statistics, with intermittency parameters ($\mu_\varepsilon = 0.26 \pm 0.01$, $\mu_\Omega = 0.36 \pm 0.02$) comparable to those of high-Reynolds-number hydrodynamic turbulence, and raised four open issues for future work: identify the mechanism injecting enstrophy at nearly all scales; determine systematically how scaling exponents and intermittency parameters depend on bacterial type, concentration, fluid viscosity, and model parameters; determine whether models of active flows reproduce not only the large-scale coherent structure but all the detailed scaling relations reported; and establish whether the intermittency parameter $\mu$ is genuinely universal across classical and active turbulence, together with a rigorous theoretical explanation of any observed universality.

Background

Bacterial (active) turbulence. Suspensions of swimming bacteria such as Bacillus subtilis spontaneously organize into swirling, chaotic mesoscale flows even though the Reynolds number (the ratio of inertial to viscous forces) is of order $10^{-5}$ in the source experiment, so ordinary inertial turbulence is impossible. The driving comes from the bacteria acting as immersed force dipoles (active stresses) rather than from a hydrodynamic instability; such systems are studied under the name active matter, and their turbulent-like state is called active or bacterial turbulence. The benchmark experiment is the quasi-two-dimensional suspension of Wensink et al. (PNAS 109, 14308 (2012)), whose velocity fields are also the dataset analyzed by the source paper.

Two-dimensional cascades. In classical two-dimensional turbulence, Kraichnan's picture splits the flow into two cascades: kinetic energy is transferred upscale (an inverse energy cascade, generating large coherent vortices) while enstrophy — one half the squared vorticity, $\Omega = |\nabla \times \mathbf{u}|^2/2$, a measure of rotational motion — is transferred downscale to small scales. In a stationary inertial range the flux of the cascaded quantity is constant across scales.

Kolmogorov 1941 and 1962 intermittency. Kolmogorov's 1941 theory (K41) predicts self-similar velocity statistics, with structure functions (moments of velocity increments over a separation $r$) scaling as power laws determined by dimensional analysis. Landau's objection — that the local energy dissipation rate $\varepsilon$ fluctuates — led to Kolmogorov's 1962 refined theory (K62), in which the coarse-grained dissipation $\varepsilon_\ell$ over a region of size $\ell$ is assumed lognormally distributed. Its moment scaling is $M_{\varepsilon,q}(\ell) = \langle \varepsilon_\ell^q \rangle \sim \ell^{-K(q)}$ with the parabolic spectrum $K(q) = (\mu/2)(q^2 - q)$; the single number $\mu > 0$ is the intermittency parameter, controlling how strongly dissipation concentrates on small scales, and it also fixes the anomalous velocity-structure-function exponents through the refined-similarity relation $\zeta(q) = qh - K(qh)$, where $h$ is the Hurst (roughness) exponent of the velocity field ($h = 1/3$ for the Eulerian velocity, $h = 1/2$ for the Lagrangian one). A multifractal description generalizes this; the lognormal form is the special case of a parabolic $K(q)$. A known shortcoming of the lognormal model is non-realizability: it predicts unbounded velocities (moments that grow too fast at high order), which is why alternative prescriptions such as the She-Leveque log-Poisson model exist.

What the source paper established. Huang (Phys. Fluids 36, 065149 (2024)) applied filter-space (coarse-graining) analysis to the experimental two-dimensional bacterial velocity fields and found: (i) an inverse energy cascade generates the large-scale coherent vortices, but the kinetic energy is dissipated on all scales because the motion lies below the fluid viscosity scale, so no constant energy flux exists; (ii) the mean enstrophy flux is positive (enstrophy is injected) at nearly all scales, which cannot be accounted for by the advection term $\mathbf{u} \cdot \nabla \mathbf{u}$ alone; (iii) the coarse-grained energy-dissipation and enstrophy fields both display lognormal multifractal statistics with $\mu_\varepsilon = 0.26 \pm 0.01$ and $\mu_\Omega = 0.36 \pm 0.02$, comparable to three-dimensional hydrodynamic turbulence ($\mu_\varepsilon \simeq 0.21$-$0.23$, $\mu_\Omega \simeq 0.33$), the forward cascade of two-dimensional turbulence ($\mu \simeq 0.31$), and even lithosphere deformation at Reynolds number $\sim 10^{-24}$ ($\mu \simeq 0.30$); and (iv) the joint scaling exponents of the dissipation and enstrophy multifractal measures follow the lognormal 'ellipse model'. Earlier analyses of the same bacterial data by other methods had given $\mu \simeq 0.26$ (Qiu et al., Hilbert-based) and $\mu \simeq 0.20$ (Wang and Huang, streamline-based). Unlike the classical two-dimensional inverse cascade, which is non-intermittent, in this active flow the intermittency correction and the inverse energy cascade coexist.

The open issues. The paper's discussion states that these observations leave unresolved: the physical mechanism injecting enstrophy across scales; the systematic dependence of the measured exponents and $\mu$ on controllable parameters (bacterial type, concentration, fluid viscosity, and the parameters of active-flow models), noting that the scaling ranges are short precisely because the flow sits below the viscosity scale; whether models of active flows can reproduce not only the large-scale coherent structure but all the detailed scaling relations presented in the work; and, centrally, whether $\mu$ is genuinely universal across classical and active turbulence — a question requiring both broader datasets and a rigorous theoretical argument, since the classical side itself has evidence of Reynolds-number dependence of intermittency parameters within the lognormal framework (Geneste et al., Entropy 21, 326 (2019)). Separately, the paper remarks in passing that other multifractal models could fit the same scaling exponents as well but chooses not to discuss them, so that comparison is not one of the paper's own future-work issues.

Problem Statement

The problem, as posed by the source paper, is to turn the empirical observation that two-dimensional bacterial turbulence displays Kolmogorov-1962 lognormal intermittency with an intermittency parameter comparable to classical turbulence into established science. The paper's discussion and conclusions raise four open questions, followed here by one added comparative test:

1. Enstrophy-injection mechanism. Identify the physical mechanism responsible for the observed positive mean enstrophy flux at nearly all scales in two-dimensional bacterial turbulence. The scale-by-scale enstrophy budget computed from the experimental velocity fields is not closed by the advection term $\mathbf{u} \cdot \nabla \mathbf{u}$ alone; candidate explanations discussed in the source — additional nonlinear interactions from the omitted active (force-dipole) or higher-order terms, or alignment between vorticity gradients and vorticity transport — are not captured by the experimental data, and the paper suggests that this imbalance deserves more careful study in the future using proper model equations.

2. Systematic parameter dependence. Determine, through targeted experimental and numerical parameter studies, how the scaling exponents (of the coarse-grained dissipation and enstrophy moments and of the velocity structure functions) and the intermittency parameters depend on bacterial type, bacterial concentration, fluid viscosity, and the parameters of active-flow models — acknowledging that the scaling ranges are short because the flow lies below the fluid viscosity scale and because of measurement limitations.

3. Model reproduction. Determine whether models of active flows reproduce not only the large-scale coherent structure but all the detailed scaling relations presented in the source — the scale-by-scale energy and enstrophy fluxes, the lognormal $K(q)$ and its scaling relations, and the joint dissipation-enstrophy ellipse model. The source states that this is beyond the topic of its own work.

4. Universality of $\mu$. Establish whether the intermittency parameter $\mu$ is genuinely universal across classical and active turbulence: extract $\mu$ for a broader set of turbulent and turbulence-like systems under consistent conventions, and either provide a rigorous theoretical explanation of the observed common value (approximately 0.2 to 0.4 across systems so different as high-Reynolds-number three-dimensional turbulence, the two-dimensional forward cascade, bacterial turbulence, and lithosphere deformation at Reynolds number $\sim 10^{-24}$, while accommodating known Reynolds-number trends on the classical side), or demonstrate that $\mu$ varies systematically with physical or control parameters beyond experimental uncertainty.

5. Added comparative test: multifractal model comparison. The source remarks in passing that other multifractal models could also fit the same scaling exponents but does not discuss them. Test whether realizable alternative prescriptions (e.g., log-Poisson/She-Leveque-type or other convex spectra) describe the measured $K(q)$ and the joint dissipation-enstrophy ellipse model as well as, or better than, the lognormal form over the accessible ranges.

The four open questions raised by the source are items 1-4, and each is independently attackable and independently meaningful: a physical mechanism for the positive mean enstrophy flux (1); measured dependences of the scaling exponents and $\mu$ on the listed controls (2); an active-flow model that reproduces the detailed scaling relations (3); and a broader $\mu$ comparison backed either by a rigorous theoretical argument or by evidence that $\mu$ varies systematically (4). Resolving any one of them on its own is a substantive result, while a complete resolution of the task addresses all four. Item 5 is an added comparison that the source mentions but does not pursue; it is an optional extension and is not part of the four.

References

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