Geometry and Topology of Turbulence in Active Nematics

Geometry and topology of turbulence in active nematics

Luca Giomi1 1Instituut-Lorentz, Universiteit Leiden, P.O. Box 9506, 2300 RA Leiden, The Netherlands The problem of low Reynolds number turbulence in active nematic fluids is theoretically addressed. Using numerical simulations I demonstrate that an incompressible turbulent flow, in two-dimensional active nematics, consists of an ensemble of vortices whose areas are exponentially distributed within a range of scales. Building on this evidence, I construct a mean-field theory of active turbulence by which several measurable quantities, including the spectral densities and the correlation functions, can be analytically calculated. Because of the profound connection between the flow geometry and the topological properties of the nematic director, the theory sheds light on the mechanisms leading to the proliferation of topological defects in active nematics and provides a number of testable predictions. A hypothesis, inspired by Onsager’s statistical hydrodynamics, is finally introduced to account for the equilibrium probability distribution of the vortex sizes.

The paradigm of “active matter” [1–3] has had notable successes over the past decade in describing self- organization in a surprisingly broad class of biological and bio-inspired systems: from flocks of starlings [4,5] to robots [6], down to bacterial colonies [7–11], motile colloids [12, 13] and the cell cytoskeleton [14–16]. Ac- tive systems are generic non-equilibrium assemblies of anisotropic components that are able to convert stored or ambient energy into motion. Because of the interplay between internal activity and the interactions between the constituents, these systems exhibit a spectacular va- riety of collective behaviors that are entirely self-driven and do not require a central control mechanism. A particularly interesting manifestation of collective behavior in active systems is the emergence of spatio- temporal chaos. In active bio-fluids, such as bacterial suspensions or cytoskeletal mixtures, the chaotic dynamics takes place through the formation of structures, such as jets or swirls, reminiscent of turbulence in Newtonian fluids, in spite of the undisputed predominance of dissi- pation over inertia at the microscopic scale. Examples of low Reynolds number turbulence in active fluids were first reported for the case of bacterial suspensions [7–11], where this is believed to have an important impact on FIG. 1: (A) A two-dimensional active nematic suspensions of nutrient mixing and molecular transport at the microbi- microtubules bundles and kinesin at the water-oil interface. ological scale. The white scale bar corresponds to 100 µm. [Courtesy of the Dogic Lab]. (B-D) Numerical simulations of an extensile ac- Recently, a series of remarkable experiments on active nematic obtained from and integration of Eqs. (1). (B) tomyosin motility assays [17] and suspensions of micro- Flow velocity (black streamlines) and vorticity (background tubles bundles and kinesin [18, 19] (see Fig.1A), have color). (C) Schlieren texture constructed from the director unveiled a profound link between the topological struc- field n. The red and blue dots mark respectively the +1/2 arXiv:1409.1555v2 [cond-mat.soft] 3 Aug 2015 ture of the orientationally ordered constituents and the and −1/2 disclinations. (D) Clockwise rotating (blue) and flow dynamics, suggesting that active turbulence could counterclockwise rotating (red) vortices, detected by measur- ing the Okubo-Weiss field as described in the text. be mediated by unbound pairs of topological defects. Ac- cording to this picture, the strong distortion associated with a defect, fueled by the active stresses, determines a local shear flow which in turn drives the unbinding of a profound and yet unsolved problem. more defect pairs. Similar patterns have been observed The current efforts toward understanding active tur- in “living liquid crystals” obtained from the combination bulence rely on the use of continuum models, such as of swimming bacteria and lyotropic liquid crystals [20]. the Toner-Tu or Swift-Hohenberg model [10, 11] or the Whether these examples of active turbulence are different equations of active nematodynamics [21–25]. Both of realizations of the same universal mechanism or substan- these approaches have been shown to be able to account tially different forms of spatio-temporal chaos, represents for the occurrence of self-sustained low Reynolds number 2 turbulence such as that observed in the experiments on microscopic models [30, 31] in the form: bacteria and cytoskeletal fluids, although a systematic Dvi 2 comparison between theory and experiments is still in its ρ = η∇ vi − ∂ip + ∂jσij , (1a) infancy. The recent numerical work by Thampi et al. Dt DQij −1 [21–23], in particular, has provided a convincing demon- = λSuij + Qikωkj − ωikQkj + γ Hij . (1b) stration of the correlation between defects dynamics and Dt turbulence in active nematics. The interplay between Here D/Dt = ∂t+v·∇ indicates the material derivative, p defects and turbulence has been further investigated in is the pressure, η the shear viscosity, λ the flow alignment Ref. [24, 25] and, following a different approach, in Ref. parameter and γ the rotational viscosity [32]. In Eq. [26]. Agent-based simulations have also been recently (1b) uij = (∂ivj + ∂jvi)/2 and ωij = (∂ivj − ∂jvi)/2, are employed to highlight the interplay between defects and the strain rate and vorticity tensors corresponding to the dynamics in granular active nematics [27]. The overlap symmetric and antisymmetric parts of the velocity gra- between these “dry” systems and active liquid crystals dient, while Hij = −δFLdG/δQij is the so-called molec- remains, however, unclear. ular tensor, governing the relaxational dynamics of the In this article I report an exhaustive numerical study nematic phase and obtained from the two-dimensional of turbulence in active nematics. As a starting point, Landau-de Gennes free energy [32]: I demonstrate that, as for inertial turbulence, low 1 Z F = d2r K|∇Q|2 + C tr Q2(tr Q2 − 1) , (2) Reynolds number turbulence in active nematics is, in LdG 2 fact, a multiscale phenomenon characterized by the formation of vortices spanning a range of length scales. with K and C material constants. Finally, the stress e a Within this active range, the areas of the vortices are tensor σij = σij + σij is the sum of the elastic stress e exponentially distributed, while their vorticity is approx- σij = −λHij + QikHkj − HikQkj, due to the entropic imately constant. Building on these observations, I then elasticity of the nematic phase, and an active contribu- a formulate a mean-field theory of turbulence in active ne- tion σij = αQij describing the contractile (α > 0) and matics that allows the analytical calculation of several extensile (α < 0) stresses exerted by the active parti- measurable quantities, including the mean kinetic energy cles in the direction of the director field. The Ericksen E and enstrophy, their corresponding spectral densities and stress σij = −∂iQkl δFLdG/δ(∂jQkl) has been neglected the velocity and vorticity correlation functions. The con- because of higher order in the derivatives of Qij com- e nection between the topological structure of the nematic pared to σij. This simplification is known not to have phase and the geometry of the flow is then elucidated appreciable consequences in the fluid mechanics of two- through a quantitative description of the defect statis- dimensional active nematics [28, 29]. tics. Eqs. (1) have been numerically integrated in a square domain of size L with periodic boundary conditions. To render the equations dimensionless, all the variables have been normalized by the typical scales associated with the viscous flow. Distances are then scaled by the system size L, time by the time scale of viscous dissipa- I. RESULTS tion τ = ρL2/η and stress by the viscous stress scale Σ = η/τ. Finally, low Reynolds number is imposed by setting Dv /Dt = ∂ v in Eq. (1a). The integra- A. Active nematdynamics i t i tion is performed by finite differences on a square grid of 256×256 points via a fourth-order Runge-Kutta method. Let us consider an incompressible uniaxial active ne- To make contact with the recent and ongoing experi- matic liquid crystal in two spatial dimensions. The two- ments on microtubule suspensions [18, 19], I restrict the dimensional setting is appropriate to describe experi- discussion to the case of extensile systems (α < 0). The ments such as that by Sanchez et al.[18], where the contractile case was found to be nearly identical and is microtubule bundles are confined to a water-oil interface briefly described in AppendixD. Unless stated otherwise forming a dense active nematic monolayer, but also of the parameter values used in the numerical simulations 4 considerable interest in its own right. Let then ρ and are λ = 0.1, K = 1, γ = 10 and C = 4 × 10 , in the v be the density and velocity of an incompressible ne- previously described units. matic fluid. Incompressibility requires ∇ · v = 0. Ne- matic order is described by the alignment tensor Qij = B. Active range S(ninj − δij/2), with n the director and 0 ≤ S ≤ 1 the nematic order parameter. The tensor Qij is by construc- tion traceless and symmetric and has only two indepen- Eqs. (1) contain two important length scales, in ad- dent components in two dimensions. The hydrodynamic dition to the system size L. These are the coherence p equations of an active nematic can be constructed from length of the nematic phase `n = K/C and the ac- p phenomenological arguments [14, 28, 29], or derived from tive length scale à = K/|α|. The former determines 3 how quickly the nematic order parameter drops in the neighborhood of a topological defect and can be taken as a measure of the defect core radius. The quantity à, on the other hand, is the length scale over which active and passive stresses balance, leading to spontaneous elastic distortion and hydrodynamic flow [22, 23, 25]. As a consequence, a quiescent uniformly oriented configuration becomes unstable to a laminar flowing state once à ∼ L [15, 28, 29, 33]. As à becomes lower than the system size L, the laminar flow is eventually replaced by a turbulent flow. Depending on the values of the various parameters in Eqs. (1), the onset of turbulence can be characterized by the formation of “walls”, narrow regions where the director is highly distorted, and their breakup into pairs of ±1/2 disclinations [21, 25]. The number FIG. 2: Number of vortices n(a)∆a (with ∆a/L2 = 1.5 × of unbound defects increases with activity until satura- 10−5) with area in between a and ∆a as a function of a, tion when à ≈ `n, as the reduction of the nematic order obtained, in extensile systems, from a numerical integration of parameter due to the defects compensates the activity Eqs. (1) for various α values. The shaded regions surrounding increase [25]. Here we overlook the problem of the onset the curves correspond to the statistical error obtained from and focus on the regime where turbulence is fully devel- five simulations with different (disordered) initial conditions. The data show a prominent exponential distribution in the oped, but still far from saturation: thus `n à L. range amin < a < L, with amin the area of the smallest active Experimentally, à depends on the microscopic details vortex. Inset: Average vorticity of an individual vortex as a of the system as well as the abundance of the biochem- function of its area. ical fuel powering the active stresses. For instance, in the microtuble-kinesin suspension shown in Fig.1A, à ≈ 100 µm (i.e. the typical length scale associated tity to characterize two-dimensional flows [36, 37]. To with bending deformations), while the microtubles them- identify the vortex cores, on the other hand, one can selves (hence `n) are approximatively 1.5 µm in length calculate the angle the velocity field rotates in one loop [18]. Similar à values have been probed in experiments around each cell of the computational grid [38]. If the cell with actomyosin motility assays, using filaments of ap- contains the core of a vortex, this angle is equal to 2π, proximatively 5 µm in length [17]. The latter is also the regardless of whether the vortex is left- or right-handed. typical length of the Bacillus subtilis cells used in Ref. The combination of these two criteria allows to formulate [10] to investigate bacterial turbulence, while in this case the following vortex detection algorithm: 1) from the ve- à ≈ 10 µm, which is, thus, much closer to the lower locity field v the cores of the vortices are initially located; bound of the range of length scales analyzed here. 2) the area of a vortex is then defined as the area of the Fig.1B shows the typical structure of the turbulent region surrounding a vortex core where Q < 0. Fig.1D flow arising from Eqs. (1) for large (negative) values shows the vortices detected by this method. of the active stress α. The velocity field appears to be Fig.2 shows the density function n(a) versus the vor- decomposed in vortices of various sizes and shapes, while tex area a obtained from a numerical integration of Eqs. the director field is highly distorted by the presence of (1). The data show a prominent exponential distribution several ±1/2 disclination pairs (Fig.1C). of the form: In order to demonstrate the multiscale structure of the N n(a) = exp(−a/a∗) , a < a < a . (3) flow, I measure the distribution of the vortex area. Call- Z min max ing a the area of a vortex, this can be described by a 2 density function n(a), such that dN = da n(a) is the to- where amin and amax ∼ L are, respectively, the min- ∗ tal number of vortices of area in between a and a + da. imal and maximal area of an active vortex and a a The area of a vortex can be measured from the numeri- suitable scale parameter. By analogy with the inertial cal data by introducing the so-called Okubo-Weiss field range in classic turbulence, hereafter, we will refer to [34, 35] Q = (∂2 ψ)2 − (∂2ψ)(∂2ψ), with ψ the stream this range as the active range. Here by active vortex we xy x y mean a vortex resulting directly from mechanical work function, such that vx = ∂yψ and vy = −∂xψ. The quantity Q is related to the Lyapunov exponent of a tracer performed by the active stresses. Beside active vortices, particle advected by the flow: where Q > 0, the distance other vortices might form due to the strong shear in between two initially close particles will diverge expo- the space between active vortices. These secondary vor- nentially in time, while for Q < 0 the trajectories will tices are expected to lie outside the active range, thus R amax where a < amin. The quantities N = da n(a) remain close. Thus coherent regions in the flow are de- amin a fined as regions in which Q < 0 [34]. The Okubo-Weiss and Z = R max da exp(−a/a∗) in Eq. (3) represent re- amin field has been widely used for analyzing atmospheric and spectively the total number of active vortices and a nor- oceanic circulations and realized as an important quan- malization constant. In the Conclusions I will speculate 4

∗ FIG. 3: (A) The areas amin (red tones) and a (blue tones) appearing in the vortex probability distribution Eq. (3) for various activity and Frank constant K values. The collapse ∗ 2 of the data demonstrates that amin ≈ a ∼ à. (B) Vortices mean vorticity ωv versus activity for various K values. As expected, ωv grows linearly with activity, with a prefactor weakly dependent on the Frank constant. FIG. 4: Probability distribution function of the velocity components (A) and vorticity (B). All the data are normalized about the physical origin of this exponential distribution. by their corresponding standard deviation. The black solid The vortices mean vorticity ω = (1/a) R d2r ω(r) is v vortex line√ represents a unit-variance Gaussian function: f(x) = shown in the inset of Fig.2 as a function of a. Un- 1/ 2π exp(−x2/2). Velocity (C) and vorticity (D) correla- like n(a), ωv remains roughly constant across the scales tion functions for various√ activity values. The distance is and shows some dependence only for large activity val- normalized by rmax = 2/2 L, corresponding to the maximal ues, where the mean vortex size is substantially smaller distance between two points on a periodic square of size L. than the size of the system (see the blue line in the inset of Fig.2). As activity is increased, the vortices become smaller ity probability density function (PDF) exhibits a visible ∗ and faster as indicated by the dependence of amin, a deviation from Gaussianity along the tails (see Fig.4C). and ωv on α. Being à the only length scale asso- Fig.4C-D show the normalized velocity and vorticity 2 ciated with activity, intuitively one could expect that correlation functions: Cvv(r) = hv(0) · v(r)i/h|v (0)|i ∗ 2 2 amin ≈ a ∼ à. Analogously, the balance of active and and Cωω(r) = hω(0)ω(r)i/hω (0)i, where the angular viscous stresses over the scale of a vortex, suggests that brackets h · i indicate an average over space and time. ωv ∼ α/η. These expectations are confirmed from the These quantities have played a central role in the study numerical data shown in Fig.3. of active turbulence starting from the experimental work It is useful to recall that the microtubules-kinesin sus- by Sanchez et al.[18]. In this work it was argued that, pensions studied in [18, 19] consist of an active nematic after rescaling by the mean-squared value, the correla- monolayer at the interface of a three-dimensional bulk tion functions no longer depend on activity, suggesting fluid. As was investigated in a classic paper by Stone that the underlying geometrical structure of the flow is and Ajdari [39], the frictional damping exerted by the due to a passive mechanism, while activity controls only surrounding fluid dissipates momentum through a force the flow average speed. This scenario found support in of the form ffri = −ξv in Eq. (1a). Such a frictional in- the numerical work of Thampi et al.[21], who also pro- teraction removes kinetic energy from the flow at scales p vided an elegant interpretation based on the creation and `fri = η/ξ and is expected to have no effect on the annihilation dynamics of topological defects. The numer- global properties of the flow as long as `fri à. ical data shown in Fig.4, combined with that reported in the previous section, demonstrate that the geometrical structure of the flow does in fact depend on activity C. Statistical geometry of the flow through the active length scale à. Such a dependence, which is clearly marked by the intersection of the curves The multiscale organization and the exponential distri- in Fig.4C-D, is, however, subtle and could have been bution of the vortex areas have striking consequences on missed before due to the limited activity range explored. the overall statistical properties of the flow. From a gross An analytical approximation of the correlation functions application of the central limit theorem we could expect Cvv(r) and Cωω(r) is given in AppendixC. the velocity components to be Gaussianly distributed. To gain further insight about how activity affects the The numerical data shown in Fig.4A support this expec- flow, I measured the average kinetic energy hv2i/2 and tation. As in classic high Reynolds number turbulence, enstrophy hω2i/2 per unit area for varying α values (Fig. on the other hand, vorticity and, in general, any func- 5A-B). These exhibit, respectively, a clear linear and tion of the velocity gradients do not obey the Gaussian quadratic dependence on activity. These scaling prop- distribution due to the spatial correlation introduced by erties can be straightforwardly understood from the ge- the derivatives [40]. In this particular case, the vortic- ometrical picture previously described. As the vorticity 5

D. Mean-ﬁeld theory

The spectral structure of turbulence in two- dimensional active nematics as well as short-scale velocity and vorticity correlation can be satisfactorily described within a mean-field approximation. This approach was introduced by Benzi et al.[34, 41] to account for the emergence of self-similar coherent structures in two-dimensional decaying turbulence and can be extended to the non-self-similiar case discussed here. We consider a two-dimensional flow whose vorticity field can be decomposed in a discrete number of vortices of radius Ri and vorticity ωi(r) = ωv,if(r/Ri), with r the distance from the vortex center and ωv,i a constant. P Then: ω(r) = i ωv,if(|r − ri|/Ri), where ri is the position of the i−th vortex center. The power spectrum FIG. 5: Enstrophy (A) and energy (B) per unit area ver- of the function ω(r) can then be expressed as: sus activity. The data show, respectively, quadratic and linear scaling. (C) Enstrophy and (D) energy spectra for three 2 X −ik·(ri−rj ) 2 2 ˆ ˆ activity values. The wavelength is normalized to unity at |ωˆ(k)| = e ωv,iωv,j Ri Rj F (kRi)F (kRj) . kmin = 2π/L. ij (6) ˆ R ∞ where F (kR) = 1/(2π) 0 dξ ξf(ξ)J0(κR ξ), with J0 a is decomposed in a discrete number of vortices having Bessel function of the first kind, is a dimensionless vortex structure factor (see AppendixA). Now, if we neglect the ωv ≈ α/η, the total enstrophy can be expressed as: spatial correlation between the vortices only the diagonal Z Z terms in the sum survive upon averaging. Then: 1 2 2 1 2 1 2 Ωtot = d r ω (r) = da n(a) a ωv = N a ωv , 2 2 2 2 X 2 4 ˆ2 h|ω(k)| i = ωv,iRi F (κRi) i where N is the total number of vortices and ( · ) = Z R 2 4 ˆ2 da n(a)( · )/N indicates the vortex ensemble average. ≈ dR n(R) ωv(R)R F (κR) , (7) Thus: where we have replaced the summation with an integral 1 2 Ωtot 2 2 over the vortex population. Finally, the enstrophy spec- hω i = ≈ ωv ∼ α , (4) 2 L2 tral density, can be calculated from the vorticity power spectrum as: Ω(k) = 4π3k h|ω(k)|2i (see AppendixB). 2 where we use the fact that a ≈ L /N. Analogously, Now, the distribution function n(R) can be obtained the total kinetic energy of a single vortex is given by: from Eq. (3) upon setting a = πR2, so that n(R) = 2 2 Ev(a) ≈ 1/(16π) ωva , with the approximation becom- |da/dR|n(a). Note that assuming the vortices to have ing an equality in the case of a circular vortex. Av- circular shape is not generally correct as it is not guar- eraging over the vortex ensemble thus gives Etot = anteed that the ansatz used to parametrize the vorticity 2 2 1/(16π) ωvN a , from which: field would hold in general. Nonetheless, based on the existence of a single characteristic length scale à, one 1 a2 could hope that both approximations would affect the hv2i ≈ ω2 ∼ α . (5) 2 v a accuracy of the calculation only through irrelevant pref- actors. Next, using the fact that ωv does not depend While changing the resolution of the vortex ensem- on R and replacing n(R) into Eq. (7) yields, after some ble, activity does not affect the spectral structure of the algebraic manipulations: flow. Fig.5C-D, show the enstrophy Ω( k) and energy 5 4 2 ∗ Z 2 2 8π ωvN R 5 − ξ 2 E(k) = Ω(k)/k spectra [40] for various activity val- Ω(κ) = dξ ξ e ( κ ) Fˆ (ξ) , (8) ues. Analogously to what is observed in bacterial tur- Z κ bulence [10], the spectra are nonmonotonic with a peak ∗ p ∗ ∗ around ka = 2π/à dividing the growing regime at small where we have set R = a /π ∼ à and κ = kR . k−values from the decay regime at large k−values. In Now, consistent with the previous assumption about the the latter regime, the data show a clear power-law decay shape of a vortex, we can choose f(r/R) = 1 for r/R ≤ 1 with Ω(k) ∼ k−2 and E(k) ∼ k−4. In the next section, and f(r/R) = 0 otherwise, then the structure factor can we illustrate the origin of these exponents in a mean-field be easily calculated in the form Fˆ(ξ) = J1(ξ)/(2πξ) (see framework. AppendixA). Using this in Eq. (8) and extending for 6

FIG. 6: The flow field generated by a +1/2 (A) and −1/2 disclination (B). The white lines indicate the orientation of FIG. 7: (A) Mean-squared displacement of +1/2 (red tones) the director field. The black arrows correspond to the flow and −1/2 (blue tones) disclinations versus time for various velocity while the background color indicates the vorticity. activity values. Number (B), mean free path (C) and rates The flow is obtained from an analytical solution of the in- of creation and annihilation (D) of ±1/2 disclinations versus compressible Stokes equation in the presence of a body force activity. f ± = ∇ · (αQ±) and Q± the nematic tensor associated with a ±1/2 disclination [25].

defects serve then as a template for the turbulent flow, simplicity the integration to the whole positive real axis which in turn advects the defects themselves leading to yields: chaotic mixing. To provide a quantitative description of the defect chaotic dynamics I measure the mean-squared 2 2 2 − κ κ κ displacement (MSD) of ±1/2 disclinations as a function Ω(κ) = Cκe 2 I − I , (9) 0 2 1 2 of time (Fig.7A). For both positively and negatively charged defects this shows a substantially diffusive be- where I0 and I1 are modified Bessel functions of the first havior, with a slight superdiffusive trend in the short time 2 2 ∗ 5 kind [42] and C = π ωvNR /(2Z) is a quantity inde- dynamics of +1/2 disclinations, due to the self-propulsion pendent on κ. The spectral structure of the turbulent provided by the self-induced dipolar flow (Fig.6A and flow is encoded in the asymptotic behavior of the func- Ref. [25]). −2 tion in Eq. (9). For κ 1, Ω(κ) ∼ κ (see AppendixB) The total number of topological defects Nd is evi- in perfect agreement with the numerical data. The en- dently proportional to the number of vortices. Thus 2 ergy spectral density can be calculated straightforwardly Nd ∼ L / a ∼ α, consistently with what we find nu- from Ω(κ), this yields E(κ) ∼ κ−4. For κ ≈ 0, on the merically (see Fig.7B). As already mentioned, this lin- other hand, Eq. (9) yields Ω(κ) ∼ κ and E(κ) ∼ κ−1. ear growth in the defect population tends to saturate These predictions are very difficult to compare with the when the active length scale à approaches the defects numerical data as they refer to the narrow range of the core radius, proportional to `n (not shown here) [25]. spectrum (i.e. k < k < 10k , with k = 2π/L) Analogously, the defect mean-free path (Fig.7C) is min min min √ p 2 preceding the crossover region. Λ ∼ L /Nd ∼ 1/ α. From the spectra Ω(k) and E(k), the correlation func- Fig.7D, shows the defect creating and annihilation tions C (r) and C (r) can be easily determined (see vv ωω rates νc and νa versus activity (in units of the viscous AppendixC). Because of the mean-field approximation, time scale τ). For large activity values these exhibit a however, the accuracy of this calculation is limited to the quadratic dependence on α: ν ≈ ν ∼ α2. This behav- ∗ c a range 0 < r < R , where the spatial correlation between ior can be understood by noticing that topological defects vortices is negligible. moves predominantly along the edge of the vortices at approximatively constant angular velocity ωv ≈ α/η. Dur- ing this circulation, they might approach an oppositely E. Topological structure of active turbulence charged defect and annihilate. The elastic, Coulomb-like, attraction between oppositely charged defects takes over As I mentioned in the Introduction, the geometry of only when these have become very close to each other, the flow field is strictly connected with the topological thus the typical time scale of annihilation, ta, is predom- structure of the nematic phase [18, 21–24]. As it was inantly dictated by the active circulation: ta ∼ 1/ωv. 2 stressed in Ref. [25], the configuration of the nematic From this we might expect that νa ≈ Nd/ta ∼ α . Once director in the neighborhood of a ±1/2 disclination de- turbulence reaches a steady state, the defects creation termines the local vortex structure (Fig.6). Topological and annihilation balance, hence νc ≈ νa. 7

The interplay between defects and vortices illustrated While some questions have been answered in this work, here for two-dimensional active nematics is both remark- others remain open. How do energy and enstrophy flow able and unique, due to the asymmetric structure of semi- across the scales? In two-dimensional inertial turbulence integer disclinations and, simultaneously, the absence of it is well known that enstrophy flows toward the small vortex stretching in two-dimensional fluids [40]. In three- scale where it is eventually dissipated, while energy flows dimensional active nematics, for instance, disclination toward the large scale where it is either dissipated by lines will give rise to tubular vortices according to the frictional interactions with the wall or condensed in large same mechanism illustrated here for the two-dimensional coherent structures [40]. In complex fluids, on the other case. Because of vortex stretching, however, these vor- hand, kinetic energy can be converted into elastic en- tices will tend to lengthen with a consequent redistri- ergy and be dissipated or stored via mechanisms that do bution of energy toward the small scale. Whether the not require cascading. While the existence of an active effect of this redistribution will be only to bias the func- range does imply that of energy and enstrophy flux across tion n(a) toward small a values or more dramatic is, at scales, the organization of such a flux remains unknown. the moment, impossible to predict. In active polar liquid Another important question, which I deliberately crystals, on the other hand, the active stress associated saved for the end, concerns the origin of the exponen- with a +1 disclination does not give rise to a flow, due to tial distribution of the vortex areas. Perhaps the most the O(2) symmetry of this configuration. The prolifera- natural explanation appeals to the interpretation of the tion of +1 defects is then expected to hinder turbulence vortex population as an equilibrium ensemble, subject to rather than fueling it. While this property, evidently, the laws of statistical mechanics. This reasoning is not does not prevent low Reynolds number turbulence from new in two-dimensional turbulence, but goes back to the developing in active polar liquid crystals, we can expect pioneering works of Onsager [43] and of Joyce and Mont- it to affect the statistics of the vortices and therefore the gomery [44, 45] (see also Refs. [46, 47] for a review). To spectral properties of the turbulent flow. clarify this concept let us consider a system of N active vortices having the same absolute vorticity and let ni be P the number of vortices of area ai, so that i ni = N. II. DISCUSSION AND CONCLUSIONS A microscopic configuration is then characterized by a ∞ set of occupancy numbers {ni} and, as the vortices i=1 Q In this article I have report a thorough numerical and are indistinguishable, there are W = N!/ i ni! different analytical investigation of low Reynolds number turbu- ways to realize the same microscopic configuration. In lence in two-dimensional active nematics. Spectacular the limit of large N, corresponding to fully developed ac- S R experimental realizations of this system are found in cy- tive turbulence, W ∼ e , where S = − da n(a) log n(a) toskeletal fluids of microtubules and kinesin at the water- is an analog of the Shannon-Gibbs entropy for the vortex oil interface [18] or incapsulated in a lipid vesicle [19]. ensemble. Since the vortices all have the same vorticity, a For large enough activity values (corresponding to high macroscopic state can be arguably identified by the their R R concentrations of motors or adenosine triphosaphate in total number N = da n(a) and area A = da n(a) a, cytoskeletal fluids), these systems are known to develop with a = A/N. The most probable (N, A)−macrostate is a chaotic spontaneous flow reminiscent of turbulence in that maximizing the entropy S for fixed N and A; hence, 2 viscous fluids (see Fig.1A). n(a) = N/a exp(−a/a). Finally, setting a ∼ à yields an Here I demonstrate that, as for inertial turbulence, low expression equivalent to Eq. (3). Reynolds number turbulence in active fluids is in fact a This hypothesis is inevitably naive and yet incredibly multiscale phenomenon characterized by the appearance fascinating in suggesting an unexpected connection be- of vortices spanning a range of length scales. Within this tween the simplest and the most complex forms of mat- active range the vortex areas follow the exponential dis- ter. tribution, whose characteristic length scale à is set by the balance between active and elastic stresses. This pe- culiar geometrical structure of the flow leaves a strong signature on all the relevant physical observables. The mean kinetic energy, for instance, scales linearly with ac- Acknowledgments tivity (and not quadratically as one could have naively expected from a comparison with the laminar case, where v ∼ αL/η) because the vortices become smaller as activ- This work is supported by The Netherlands Organiza- ity is increased. Furthermore, the enstrophy and energy tion for Scientific Research (NWO/OCW). I am grate- spectra scale as k−2 and k−4, rispectively, thus in net con- ful to Luca Heltai, Cristina Marchetti, Sumesh Thampi, trast with two-dimensional inertial turbulence [40]. The Vincenzo Vitelli and Julia Yeomans for several useful con- statistics of the vortices, finally, completely determines versations and especially indebted with Stephen DeCamp that of the defects (and vice versa) making possible the and Zvonimir Dogic for the experimental image of Fig. formulation of various scaling relations amenable to ex- 1A as well as all the wonderful work that inspired this perimental scrutiny. research. 8

Appendix A: Vortex form factor

The dimensionless vortex structure factor Fˆ(kR), introduced in the mean-field calculation, is defined from the Fourier transform of the vorticity profile function f(r/R), where r is the distance from the vortex center. Thus: Z d2r r fˆ(k) = e−ik·rf (2π)2 R 1 Z ∞ r = dr rf J0(kr) , (A1) 2π 0 R where J0 is a zeroth-order Bessel function of the first kind. Now, the simplest approximation of the function f FIG. 8: Normalized vorticity (red dots) and velocity (gray is evidently: dots) correlation functions obtained from a numerical integration of Eqs. (1). The theoretical curves (MFT) correspond ( 1 r/R ≤ 1 ∗ r to Eqs. (C3) and (C6), with R = 0.034 rmax obtained from f = (A2) ∗ R a fit of the data. An extrapolation of R from the scale pa- 0 r/R > 1 , rameter a∗, as defined in Eq. (3), can be obtained by setting ∗ p ∗ ∗ representing a circular vortex of radius R. Placing this R = a /π, this gives R = 0.054 rmax slightly larger than into Eq. (A1) yields: the value obtained from a fit of the correlation function. This slight discrepancy is presumably due to the inevitable sys- Z R ˆ 1 R tematic error in the calculation of the vortex area from the f(k) = dr rJ0(kr) = J1(kR) . (A3) Okubo-Weiss field as well as the circular approximation of the 2π 0 2πk vortex shape. The dimensionless function Fˆ(kR) is then defined from fˆ(k) = R2Fˆ(kR); hence: Appendix C: Correlation functions 1 Fˆ(kR) = J1(kR) . (A4) 2πkR The spectral densities E(k) and Ω(k) and the correlation functions hv(0) · v(r)i and hω(0)ω(r)i are related by Appendix B: Asymptotic behavior of the spectral the Weiner-Kinchin theorem [40]. This implies that: densities 1 Ω(k) = ∆ kd−1F{hω(0)ω(r)i} , (C1) 2 d The expression of the enstrophy spectrum, as obtained where ∆ is the d−dimensional solid angle and F denotes within the mean-field approximation, is given by: d Fourier transformation. An equivalent expression holds 2 2 2 − κ κ κ for the vorticity spectrum and, in general, for the power Ω(κ) = Cκe 2 I − I . (B1) 0 2 1 2 spectrum of any random field given its two-point correla- Now, small κ limit can be easily determined by consid- tion function. For the special case of a two-dimensional 2 2 vorticity field with azimuthal symmetry, Eq. (C1) yields: ering that, for κ 1, I0(κ /2) ≈ exp(−κ /2) ≈ 1 and 2 I1(κ /2) ≈ 0. Therefore: 1 Z ∞ Ω(κ) ∼ κ , κ 1 . (B2) Ω(k) = dr krJ0(kr)hω(0)ω(r)i . (C2) 2 0 To calculate the large κ limit we can use the following asymptotic expansion of the modified Bessel function: If the spectrum is known, Eq. (C2) can be inverted to obtain the vorticity correlation function Cωω(r) = ex 1 − 4ν2 2 I (x) ≈ √ 1 + + ··· . (B3) hω(0)ω(r)i/h|ω(0)| i. This yields: ν 8x 2πx 2 Z ∞ From this we obtain: Cωω(r) = 2 dk J0(kr)Ω(k) , (C3) hω i 0 2 2 2 κ κ κ e 2 2 R ∞ I0 − I1 ≈ √ . (B4) where hω i/2 = 0 dk Ω(k) is the mean enstrophy per 2 2 πκ3 unit area. The same expression holds for the velocity The exponential term exactly cancels that in Eq. (B1) correlation function upon replacing Ω(k) with E(k) and resulting in a simple power-law behavior: the normalization factor with the mean energy per unit 2 R ∞ −2 area: hv i/2 = 0 dk E(k). Ω(κ) ∼ κ , κ 1 . (B5) Now, placing the expression for Ω(k) given into Eq. The asymptotic behavior of the energy spectrum fol- (B1) in (C3) yields: lows directly from this by virtue of the relation Ω(k) = r 2 C (r) = erfc , (C4) k E(k). ωω 2R∗ 9

R r 0 0 where h(r) = 0 dr r Cωω(r). Fig.8 shows a comparison between the normalized velocity and vorticity correlation functions obtained from a numerical integration of Eqs. (1) and their mean-field approximations given in Eqs. (C3) and (C6). For small distances, the agreement is remarkable. This, however, breaks down for r R∗, where the spatial correlation between neighboring vortices (which is neglected in the mean-field framework) becomes crucial. For instance, Cωω(r) becomes negative when r is larger than the average vortex diameter, due to the fact that a given central vortex is surrounded by vortices of opposite vorticity (see the red dots in Fig.8). This feature is clearly absent in the mean-field calculation and the resulting correlation function decays without sign changes (solid black line in Fig.8).

Appendix D: Extensile versus contractile

FIG. 9: (A) Number of defects and scale factor a∗ versus activity for contractile (gray dots) and extensile (red dots) systems. (B) The scale factor a∗ appearing in the vortex area The numerical results presented in the main text de- p distribution n(a) versus the active length scale à = K/|α| scribe the case of extensile active nematics (α < 0), such for for contractile (gray dots) and extensile (red dots) systems. as the suspensions of microtubule bundles and kinesin pi- oneered by Sanchez et al.[18] and recently employed by Keber et al. in the fabrication of active vesicles [19]. The where erfc(x) = 1 − erf(x) is the complementary error behavior of contractile active nematics (α > 0) is nearly ∗ function [42] while 2R ∼ à represent the mean diameter identical to the extensile case. For a given activity mag- of an active vortex. nitude |α| the average number of defects in contractile The simple algebraic relation between energy and en- and extensile systems (therefore the spatial organization strophy spectra, translates to real space into the follow- of the flow) is essentially the same (Fig.9A). ing differential relation between the velocity and vorticity correlation functions [47]: The only notable difference appears to be the offset in ∗ 2 2 the linear relation between a and à (Fig.9B). Such ∇ hv(0) · v(r)i = −hω(0)ω(r)i . (C5) an offset is presumably due to the asymmetry between contractile and extensile systems at the onset of turbu- For an azimuthally symmetric function, this implies: lence [25], which is in turn related to the asymmetry in 2 Z r 0 the linear instability of the quiescent state. I remand the hω i 0 h(r ) Cvv(r) = 1 − 2 dr 0 , (C6) reader to Refs. [25, 33] for a detailed explanation. hv i 0 r

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