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# Spontaneous mirror-symmetry breaking induces inverse energy cascade in 3D active fluids

Edited by David A. Weitz, Harvard University, Cambridge, MA, and approved January 4, 2017 (received for review September 2, 2016)

### This article has a correction. Please see:

## Significance

Turbulence provides an important mechanism for energy redistribution and mixing in interstellar gases, planetary atmospheres, and the oceans. Classical turbulence theory suggests for ordinary 3D fluids or gases, such as water or air, that larger vortices can transform into smaller ones but not vice versa, thus limiting energy transfer from smaller to larger scales. Our calculations predict that bacterial suspensions and other pattern-forming active fluids can deviate from this paradigm by creating turbulent flow structures that spontaneously break mirror symmetry. These results imply that the collective dynamics of swimming microorganisms can enhance fluid mixing more strongly than previously thought.

## Abstract

Classical turbulence theory assumes that energy transport in a 3D turbulent flow proceeds through a Richardson cascade whereby larger vortices successively decay into smaller ones. By contrast, an additional inverse cascade characterized by vortex growth exists in 2D fluids and gases, with profound implications for meteorological flows and fluid mixing. The possibility of a helicity-driven inverse cascade in 3D fluids had been rejected in the 1970s based on equilibrium-thermodynamic arguments. Recently, however, it was proposed that certain symmetry-breaking processes could potentially trigger a 3D inverse cascade, but no physical system exhibiting this phenomenon has been identified to date. Here, we present analytical and numerical evidence for the existence of an inverse energy cascade in an experimentally validated 3D active fluid model, describing microbial suspension flows that spontaneously break mirror symmetry. We show analytically that self-organized scale selection, a generic feature of many biological and engineered nonequilibrium fluids, can generate parity-violating Beltrami flows. Our simulations further demonstrate how active scale selection controls mirror-symmetry breaking and the emergence of a 3D inverse cascade.

Turbulence, the chaotic motion of liquids and gases, remains one of the most widely studied phenomena in classical physics (1, 2). Turbulent flows determine energy transfer and material mixing over a vast range of scales, from the interstellar medium (3, 4) and solar winds (5) to the Earth’s atmosphere (6, 7), ocean currents (8), and our morning cup of coffee. Of particular recent interest is the interplay of turbulence and active biological matter (9), owing to its relevance for carbon fixation and nutrient transport in marine ecosystems (10). Although much has been learned about the statistical and spectral properties of turbulent flows both experimentally (11⇓–13) and theoretically (14⇓⇓⇓⇓⇓⇓–21) over the last 75 years, several fundamental physical and mathematical (22) questions still await their answer. One of the most important among them, with profound implications for the limits of hydrodynamic mixing, concerns whether 3D turbulent flows can develop an inverse cascade that transports energy from smaller to larger scales (19, 23, 24).

Kolmogorov’s 1941 theory of turbulence (14) assumes that turbulent energy transport in 3D proceeds primarily from larger to smaller scales through the decay of vortices. This forward (Richardson) cascade is a consequence of the fact that the 3D inviscid Euler equations conserve energy (1). In 1967, Kraichnan (17) realized that the presence of a second conserved quantity, enstrophy, in 2D turbulent flows implies the existence of two dual cascades (25): a vorticity-induced cascade to smaller scales and an inverse energy cascade to larger scales (20, 26). Two years later, Moffatt (27) discovered a new invariant of the 3D Euler equations, which he termed helicity. Could helicity conservation generate an inverse turbulent cascade in 3D? Building on thermodynamic considerations, Kraichnan (23) argued in 1973 that this should not be possible, but he also conceded that turbulent flows do not necessarily follow equilibrium statistics. Since then, insightful theoretical studies (19, 24) have elucidated other important conditions for the emergence of helicity-driven inverse cascades in 3D fluids, in particular identifying mirror-symmetry breaking as a key mechanism (24). However, no natural or artificially engineered fluid system exhibiting this phenomenon has been identified to date.

Here, we predict that fluid flows in active nonequilibrium liquids, such as bacterial suspensions, can spontaneously break mirror symmetry, resulting in a 3D inverse cascade. Broken mirror symmetry plays an important role in nature, exemplified by the parity-violating weak interactions (28) in the standard model of particle physics, by the helical structure of DNA (29) or, at the macroscale, by chiral seed pods (30). Another, fluid-based realization (31) of a spontaneously broken chiral symmetry was recently observed in confined bacterial suspensions (32, 33), which form stable vortices of well-defined circulation when the container dimensions match the correlation scale

To demonstrate the existence of a helicity-driven inverse cascade in 3D active bulk fluids, we first verify analytically the existence of exact parity-violating Beltrami-flow (43⇓–45) solutions. We then confirm numerically that active bulk flows starting from random initial conditions approach attractors that spontaneously break mirror symmetry and are statistically close to Beltrami-vector fields. Finally, we demonstrate that the broken mirror symmetry leads to an inverse cascade with triad interactions as predicted by Waleffe (19) about 25 years ago.

## Results

### Theory.

We consider pattern-forming nonequilibrium fluids consisting of a passive solvent component, such as water, and a stress-generating active component, which could be bacteria (34), ATP-driven microtubules (38), or Janus particles (46, 47). In contrast to earlier studies, which analyzed the velocity field of the active matter component (35, 48, 49), we focus here on the incompressible solvent velocity field *Model Justification*). Such higher-order stresses arise naturally from diagrammatic expansions (57). Similar 1D and 2D models have been studied in the context of soft-mode turbulence and seismic waves (58⇓–60).

The parameters **1** reduce to the standard Navier–Stokes equations with kinematic viscosity **1c** defines the simplest ansatz for an active stress tensor that is isotropic, selects flow patterns of a characteristic scale (Fig. 1), and yields a stable theory at small and large wavenumbers (36, 37). The active-to-passive phase transition corresponds to a sign change from *A*). The resulting nonequilibrium flow structures can be characterized in terms of the typical vortex size *A*)

Specifically, we find ^{−1} for flows measured in *Bacillus subtilis* suspensions (34, 35) and ^{−1} for ATP-driven microtubule–network suspensions (38) (*Comparison with Experiments*). We emphasize, however, that truncated polynomial stress tensors of the form **1c** can provide useful long-wavelength approximations for a broad class of pattern-forming liquids, including magnetically (61), electrically (62), thermally (46, 63, 64), or chemically (47, 65) driven flows.

### Exact Beltrami-Flow Solutions and Broken-Mirror Symmetry.

The higher-order Navier–Stokes equations defined by Eq. **1** are invariant under the parity transformation **1b** onto helicity eigenstates (19) yields the evolution equation for the mode amplitudes ** k** modes and

**4**, arbitrary superpositions of modes with identical wavenumber

*A*, we obtain exact stationary solutions

**5**) correspond to Beltrami flows (43⇓–45), obeying

*B*).

Although the exact solutions **1**. As we demonstrate next, simulations with random initial conditions do indeed converge to statistically stationary flow states that spontaneously break mirror symmetry and are close to Beltrami flows.

### Spontaneous Mirror-Symmetry Breaking in Time-Dependent Solutions.

We simulate the full nonlinear Eq. **1** on a periodic cubic domain (size *Numerical Methods*). Simulations are performed for typical bacterial parameters ^{−1}, ^{−1}, and ^{−1}, corresponding to active fluids with a small (S), an intermediate (I), and a wide (W) range of energy injection scales. A small bandwidth means that the active stresses inject energy into a narrow shell in Fourier space, whereas a wide bandwidth means energy is pumped into a wide range of Fourier modes (Fig. 1*A*). All simulations are initiated with weak incompressible random flow fields. For all three values of *C* shows results from 150 runs for *A* and *B*.

### Beltrami-Flow Attractors.

Having confirmed spontaneous parity violation for the time-dependent solutions of Eq. **1**, we next characterize the chaotic flow attractors. To this end, we measure and compare the histograms of the angles between the local velocity field *C*). Recalling that perfect alignment, described by **1**, we indeed find that the numerically computed flow fields exhibit *D*). Keeping

## Discussion

### Spontaneous Parity Breaking vs. Surgical Mode Removal.

Important previous studies identified bifurcation mechanisms (66⇓–68) leading to parity violation in 1D and 2D (69) continuum models of pattern-forming nonequilibrium systems (70, 71). The above analytical and numerical results generalize these ideas to 3D fluid flows, by showing that an active scale selection mechanism can induce spontaneous helical mirror-symmetry breaking. Such self-organized parity violation can profoundly affect energy transport and mixing in 3D active fluids, which do not satisfy the premises of Kraichnan’s thermodynamic no-go argument (23). An insightful recent study (24), based on the classical Navier–Stokes equation, found that an ad hoc projection of solutions to positive or negative helicity subspaces can result in an inverse cascade but it has remained an open question whether such a surgical mode removal can be realized experimentally in passive fluids. By contrast, active fluids spontaneously achieve helical parity breaking (Fig. 1*C*) by approaching Beltrami-flow states (Fig. 2 *C* and *D*), suggesting the possibility of a self-organized inverse energy cascade even in 3D. Before testing this hypothesis we recall that the model defined by Eq. **1** merely assumes the existence of linear active stresses to account for pattern scale selection as observed in a wide range of microbial suspensions (35, 38, 54, 72), but does not introduce nonlinearities beyond those already present in the classical Navier–Stokes equations. That is, energy redistribution in the solvent fluid is governed by the advective nonlinearities as in conventional passive liquids.

### Inverse Cascade in 3D Active Fluids.

To quantify how pattern scale selection controls parity breaking and energy transport in active fluids, we analyzed large-scale simulations (*A* and *B*) for different values of the activity bandwidth *A*) while keeping the pattern scale *A*) corresponds to the energy injection range in Fourier space and provides a natural separation between large flow scales (blue domain I) and small flow scales (blue domain III). Consequently, the forward cascade corresponds to a net energy flux from domain II to domain III, whereas an inverse cascade transports energy from domain II to I. We calculate energy spectra **2**, which yields a natural splitting into cumulative energy and flux contributions *Numerical Methods*). Time-averaged spectra and fluxes are computed for each simulation run after the system has relaxed to a statistically stationary state (Fig. S2). For a small injection bandwidth *A*), depending on the initial conditions. Moreover, in addition to the expected 3D forward transfer, the simulation data for *B*) in domain I. As evident from the blue curves in Fig. 3 *A* and *B*, this inverse cascade is facilitated by the helical modes that carry most of the energy. For a large injection bandwidth *E*), but the energy transported to larger scales becomes negligible relative to the forward cascade, as contributions from opposite-helicity modes approximately cancel in the long-wavelength domain I (Fig. 3*F*). Results for the intermediate case *E*), demonstrating how the activity bandwidth—or, equivalently, the pattern selection range—controls both parity violation and inverse cascade formation in an active fluid. The upward transfer is noninertial at intermediate scales, as indicated by the wavenumber dependence of the energy flux (Fig. 3*B*). At very large scales *Cascade Characteristics*). In contrast to the energy-mediated 2D inverse cascade in passive fluids, the helicity-driven 3D inverse cascade in active fluids is linked to the formation of extended vortex chain complexes that move collectively through the fluid (Movie S1 and *Cascade Characteristics*).

### Triad Interactions.

Our numerical flux measurements confirm directly the existence of a self-sustained 3D inverse cascade induced by spontaneous parity violation, consistent with earlier projection-based arguments for the classical Navier–Stokes equations (24). An inverse energy cascade can exist in 3D active fluids because mirror-symmetry breaking favors only a subclass of all possible triad interactions, which describe advective energy transfer in Fourier space between velocity modes **4**). To analyze in detail which triads are spontaneously activated in a pattern-forming nonequilibrium fluid, we consider combinations *A* and distinguish modes by their helicity index *Numerical Methods*)

For active fluids, Fourier space is naturally partitioned into three regions (Fig. 1*A*) and there are *Numerical Methods*) for small (*C* and *G*. For reflection-symmetric turbulent flows, these two tables would remain unchanged under an upside-down flip (*C*), which persists in weakened form for *G*). Specifically, we observe for *D*). These cumulative triads visualize dominant entries of the tables in Fig. 3 *C* and *G* and represent the total contributions from all triadic interactions between modes with given helicity indexes and with “legs” lying in the specified spectral domain. The observed energy transfer directions, with energy flowing out of the intermediate domain II when the small-scale modes carry the same helicity index, are in agreement with a turbulent instability mechanism proposed by Waleffe (19). Interestingly, however, our numerical results show that both “R”-interaction channels *D*); when one surgically projects the full dynamics onto states with fixed parity, only the +++ channel remains (24). By contrast, for a wide bandwidth *H*) favor the forward cascade. Hence, the inverse energy cascade in 3D active fluids is possible because only a subset of triadic interactions is active in the presence of strong mirror-symmetry breaking. This phenomenon is controlled by the spectral bandwidth of the scale selection mechanism.

### Enhanced Mixing.

Eq. **1** describe a 3D isotropic fluid capable of transporting energy from smaller to larger scales. Previously, self-organized inverse cascades were demonstrated only in effectively 2D flows (6, 18, 20, 25, 73⇓⇓⇓⇓⇓–79). The 2D inverse cascade has been intensely studied in meteorology (6, 7), a prominent example being Jovian atmospheric dynamics (80), because of its importance for the mixing of thin fluid layers (81⇓–83). Analogously, the 3D inverse cascade and the underlying Beltrami-flow structure are expected to enhance mixing and transport in active fluids. Arnold (43) showed that steady solutions of the incompressible Euler equations include Beltrami-type ABC flows (44) characterized by chaotic streamlines. Similarly, the Beltrami structure of the active-flow attractors of Eq. **1** implies enhanced local mixing. Combined with the presence of an inverse cascade, which facilitates additional large-scale mixing through the excitation of long-wavelength modes, these results suggest that active biological fluids, such as microbial suspensions (35, 54, 72), can be more efficient at stirring fluids and transporting nutrients than previously thought.

## Conclusions

To detect Beltrami flows in biological or engineered active fluids, one has to construct histograms and spectra as shown in Figs. 2 *C* and *D* and 3 *A* and *E* from experimental fluid velocity and helicity data, which is possible with current fluorescence imaging techniques (13, 35). Moreover, helical tracer particles (84) can help distinguish left-handed and right-handed flows. The above analysis predicts that Beltrami-flow structures, mirror-symmetry breaking, and the inverse 3D cascade appear more pronounced when the pattern selection is focused in a narrow spectral range. Our simulations further suggest that the relaxation time required for completion of the mirror-symmetry breaking process depends on the domain size (Fig. S5). For small systems, the relaxation is exponentially fast, whereas for large domains relaxation proceeds in two stages, first exponentially and then linearly. In practice, it may therefore be advisable to accelerate relaxation by starting experiments from rotating initial conditions.

## Methods

Eq. **1** was solved numerically in the vorticity-vector potential form with periodic boundary conditions using a spectral code with 3/2 anti-aliasing (*Numerical Methods*). Tables in Fig. 3 were calculated using the Littlewood–Paley decomposition and collocation.

## Comparison with Experiments

The generalized Navier–Stokes equations defined in Eq. **1** of the main text aim to provide an effective three-parameter description of solvent flows driven by an active component. Although the flow structures seen in the simulations look visually similar to those observed in experiments on bacterial and other active suspensions, a quantitative comparison with experimental data is needed to evaluate the practical applicability of the theory.^{*} To contribute toward closing the gap between theory and experiments, we performed systematic parameter scans, comparing fluid flow statistics measured in our simulations with recently reported experimental data for two different classes of active fluids: (*i*) concentrated quasi-3D suspensions of swimming *B. subtilis* bacteria (35) and (*ii*) ATP-driven microtubule networks (38). This analysis identified specific parameter values

### Bacterial Suspensions.

The experiments reported in ref. 35 studied dense suspensions of rod-like *B. subtilis* bacteria swimming in a quasi-3D microfluidic channel (height *A–C*).

Fig. S1*A* compares the experimentally measured velocity distribution for bacteria (open circles) and solvent tracer particles (solid circles) with the statistics of the five-parameter model for the bacterial velocity field considered in ref. 35 (black line labeled “theory”) and our generalized Navier–Stokes model (blue line). As discussed by the authors of ref. 35, their model for the bacterial dynamics fails to capture the tails of the velocity distributions as it includes an effective fourth-order velocity potential (representing steric alignment interactions) that dominates the tails of velocity distributions in their simulations. By contrast, our generalized Navier–Stokes model accurately captures the experimentally measured Gaussian velocity probability distribution functions (PDFs) over the whole range of the available experimental data (see Fig. S1*C* legend for a summary of fit parameters).

Fig. S1 *B* compares the equal-time (in-plane) velocity correlation functions (VCFs) for the bacteria, tracer particles, and the theories. As mentioned in ref. 35, the VCFs for tracer particles become unreliable at large distances

Fig. S1 *C* compares the simulation results for the velocity autocorrelation functions (VACFs) with the corresponding experimental results at different bacterial activities (35) due to oxygen depletion. PTV-based VACFs were not given in ref. 35 as a specific tracer particle typically spends only a short time in the 2D field of view of the microscope before diffusing out of view. As evident from Fig. S1 *C*, our generalized Navier–Stokes model can correctly reproduce the functional form of the PIV-based VACFs at high (green), intermediate (blue), and low (magenta) activities. With regard to a future quantitative characterization and classification of active fluids, we find it encouraging that a three-parameter model can account for the key velocity statistics reported in ref. 35.

Another interesting experimental observation reported but not rationalized in ref. 35 is the linear scaling of kinetic energy and enstrophy (figure 2d in ref. 35). We note that such a linear scaling is consistent with the Beltrami-like flows found in our simulations, which satisfy *C* and *D* of the main text). Generally, we hope that the good agreement between the generalized Navier–Stokes model defined in Eqs. **1** of the main text and the experimental data for *B. subtilis* will stimulate additional 3D measurements on bacterial suspension in the near future, to test the Beltrami flow prediction directly and to explore the possibility of spontaneous mirror-symmetry breaking in detail.

### ATP-Driven Microtubule Networks.

The generalized Navier–Stokes equations defined in Eq. **1** of the main text merely assume that active stresses in an otherwise passive fluid lead to scale selection. They should therefore also apply to other types of active fluids, including ATP-driven microtubule suspensions. To test this hypothesis, we performed additional simulations to compare our model with experimental data published recently in ref. 38. The authors of this study report VCF data for tracer particles diffusing in fluid flows driven by predominantly extensile microtubule–kinesin bundles that form complex, approximately isotropic networks. The flows created by these active networks exhibit turbulent vortices on scales larger than the typical bundle-bending radii, suggesting that these flows are generated by the collective extensile dynamics of the bundles. Fig. S1 *D* shows the experimental VCF data reported in ref. 38 (colored circles and lines) and a fit (black solid line) obtained from simulations of our generalized Navier–Stokes model, using the parameters specified in the legend.^{†} Different ATP-controlled activity levels can be reproduced in our model through a trivial adjustment of the velocity scale *B* and *D*), corroborating the idea that active suspensions can be robustly described by the leading-order terms of stress tensor expansions. More generally, the good agreement between the generalized Navier–Stokes model and two microscopically distinct active fluids supports the view that the main results and predictions of our study apply to a broad range of pattern-forming nonequilibrium fluids.

## Model Justification

The generalized Navier–Stokes model defined in Eq. **1** of the main text describes the solvent flows in active suspensions through effective higher-order stresses that account phenomenologically for the experimentally observed flow-pattern scale selection (35, 38). By contrast, derivations of effective higher-order continuum models (88) often focus on the complementary problem of obtaining a higher-order equation for the orientational order-parameter fields of the active component by ignoring nonlinear inertial effects in the fluid. The generalized Navier–Stokes Eq. **1** in the main text avoid the latter oversimplification and assume a linear response between orientational order-parameter fields and ambient fluid (72).

### Inertial Effects.

The standard argument for neglecting inertial terms in the Navier–Stokes equations for dilute microbial suspensions is based on the typical length scale and swimming speed of a single bacterium and the viscosity of water (89). This argument is certainly correct for very low bacterial volume fractions when collective dynamical effects are negligible. The argument becomes invalid, however, at sufficiently high concentrations when collective effects dominate the suspension dynamics. There are three reasons for this: First, collective locomotion speeds of bacteria at moderate-to-high concentrations (

### Active Stresses.

At the fundamental continuum level, the fluid dynamics of a passive solvent are described by the Navier–Stokes equations. The effect of the active components on the fluid can be written as a collection of point forces entering on the right-hand side (rhs) of the Navier–Stokes equations. An important feature of intrinsically driven active suspensions (in contrast to externally forced colloidal suspensions) is given by the experimentally confirmed fact that bacteria and other microbes achieve locomotion through shape changes that require zero net force (89, 92⇓–94). Considering, for instance, the simplest force dipole model, this means that forces can be paired, so that monopole contributions cancel and the leading-order contributions entering the Navier–Stokes equations take the form of divergences of stress tensors (92).

### Slaving and Linear Response.

In sufficiently dense suspensions, the net effective stress tensor depends on the collective dynamics, which are typically characterized through orientational order-parameter fields. If the fluid flows generated by the collective action of the active constituents dominate over their individual swimming dynamics, then one can assume that tensorial order parameters become “slaved” to the solvent dynamics. In this case, assuming a linear and isotropic response, one arrives at closure conditions for polar and nematic order parameters * p* and

*of the form*

**Q***and*

**p***, and by truncating at order*

**Q****1c**of the main text. This reasoning can be formalized systematically through diagrammatic expansion techniques (57).

^{‡}The successful comparison with the experiments above suggests that such truncated stress tensors can capture essential aspects of the collective dynamics.

## Numerical Methods

Numerical simulations were performed using a Fourier spectral method with a 3/2 rule to avoid aliasing when calculating the advection term through collocation (85). We typically used grids of size **1** provides strong damping **1** by using the Hodge decomposition (86) and solving the corresponding vorticity-vector potential problem **S2** are evolved in time, using a third-order semiimplicit backward differentiation time-stepping scheme (86), calculating the nonlinear advection term explicitly while inverting the linear part implicitly. The discretized Eq. **S2** maintain *C* and *G* efficiently, we decompose the velocity field into Littlewood–Paley components and use collocation.

### Vorticity-Vector Potential Formulation.

We find the vorticity-vector potential formulation of the system 1 of the main text on the three-torus, * v*, the decomposition takes the form

*is an element of the 3D space of harmonic vector fields, which implies on a torus that*

**H***as the fluid center of mass motion. By working in the center of mass frame, we are left with*

**H****1**of the main text gives

We can simplify the advection term by using the following standard identity **S4** gives

### Characteristic Scales.

The linear growth rate associated with the operator

### Nondimensionalization in Numerical Simulations.

For simulation purposes, we rescale time and space as **S8** then become (after dropping the tildes)

Setting

### Time Discretization.

For the time stepping, we use the third-order semiimplicit backward differentiation scheme introduced by Ascher et al. (87),

### Space Discretization.

We work with a Fourier spectral method. If we denote the rhs of Eq. **S14a** by * v*, and we can set

If we initiate the simulations with divergence-free fields, then the update rule S16 preserves this property in exact arithmetic. Nevertheless, numerical errors will always build up after several iterations in double-precision arithmetic. We project back onto the divergence-free manifold every several steps by mimicking gauge transformation. Suppose

### Calculation of Shell Interactions.

We next explain how the energy spectra, fluxes, and energy flow tables are calculated numerically (Fig. 3 of the main text and Fig. S3). To establish notation, we first recall the derivation of the energy balance equation as given in Waleffe (19). Expanding the velocity and pressure fields in Fourier series, Eq. **1** of the main text give **3** of the main text. To find the equation for the energy in mode * k* we relabel

The energy in shell

The corresponding evolution equation is

We used the fact that the sum over all modes can be split into radial and shell parts

Symmetrizing as

The quantity

Projecting the velocity field onto the helical modes reveals additional substructure (19). The energy spectrum splits into two helical components

The energy flow and energy flux split into eight components, one for each possible assignment of the helicity index over the triads

We now consider time averages. For a quantity **S22a** to

We numerically estimate the discrete stationary spectra **S32** and the total flux from

For plotting purposes, we connect the discrete energy spectra to their continuous definitions. The mean kinetic energy in the system of size

The spectral domains I, II, and III in Fig. 1 *A* of the main text have finite thickness. To calculate the energy flow between the regions, we have to sum over shells contained in a given region. For example, * v* by keeping only the Fourier amplitudes supported on the region

**S21**. We symmetrize in the last to indexes, by defining

To split *C* and *G* in the main text and Fig. S3 *C*), we adopt a procedure analogous to that used to estimate the energy spectra.

## Cascade Characteristics

The phenomenology of the inverse cascade in passive 2D turbulent flows is often characterized in terms of vortex mergers. By contrast, in active fluids with a well-defined vortex scale *E*), manifests itself in the flow-field structure of a 3D active fluid. Our simulations demonstrate that pattern-forming nonequilibrium fluids can achieve energy transport to larger scales by forming chain-like vortex complexes that propagate through the fluid (Movie S1). To illustrate this phenomenon in more detail, Fig. S4 *A–D* shows two horizontal 2D *A* and *B*, the flow field is visualized through the perpendicular *C* and *D* through the local helicity *A* and *B* indicate in-plane portions of filaments consisting of alternating vortices that correspond to 3D filamentous clusters of high helicity in Fig. S4 *C* and *D*. The kinetic energy transported to large scales manifests itself as the formation and motion of such vortex chains (Movie S1). These results illustrate that the helicity-driven 3D inverse cascade in active fluids is distinctly different from the energy-driven 2D inverse cascade in passive fluids.

A detailed spectral characterization of this helicity-driven 3D active turbulence can be obtained by analyzing the upward energy transfer into region I in Fourier space (defined in Fig. 1 *A* of the main text). Fig. S4 *E* shows the absolute value of the energy flux for an active fluid with small bandwidth **1** of the main text effectively reduces to the classical Navier–Stokes equations.

## Acknowledgments

We thank Aden Forrow, Francis Woodhouse, Luca Biferale, Moritz Linkmann, and Michael Tribelsky for helpful discussions.

## Footnotes

- ↵
^{1}To whom correspondence should be addressed. Email: dunkel{at}math.mit.edu.

Author contributions: J.S. and J.D. designed research; J.S. performed research; and J.S. and J.D. wrote the paper.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission.

↵*We thank an anonymous reviewer for insisting on a detailed comparison with experiments. The comparison presented here benefited from the fact that one of us (J.D.) was involved in the original analysis of the experimental data in ref. 35.

↵

^{†}These parameters agree well with the typical velocity, length, and time scales expected from microbial suspensions.↵

^{‡}Structurally similar sixth-order hydrodynamic equations are obtained by systematically reducing magneto-hydrodynamic models in the vicinity of flow bifurcations (Geoffrey Vasil, personal communication).This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1614721114/-/DCSupplemental.

Freely available online through the PNAS open access option.

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