Active Turbulence: Dynamics & Models

• Active turbulence is the spatiotemporal chaotic flow in active matter, driven by internal energy injection from entities like bacteria and microswimmers.
• Continuum models augmenting classical hydrodynamics with active stress tensors explain the emergence, defect dynamics, and vortex formation observed in these systems.
• Experimental and numerical studies reveal non-classical scaling laws and anomalous tracer dispersions that deepen our understanding of energy transfer in active fluids.

Active turbulence is the regime of spatiotemporal chaos and self-organized, multiscale flow phenomena that emerge in fluids energized internally by active constituents, such as bacteria, synthetic microswimmers, cytoskeletal filaments with molecular motors, or active colloidal suspensions. This state arises at vanishingly small Reynolds number due to intrinsic driving and instability mechanisms absent from classical inertial turbulence. The hallmark signatures of active turbulence are persistent mesoscale vortices, jets, and dynamic topological defects, underpinned by the self-organized injection and dissipation of energy at characteristic intrinsic length scales rather than via a classical inertial energy cascade. Major theoretical, numerical, and experimental efforts have converged to establish active turbulence as a universal feature of a wide variety of wet and dry active materials (Alert et al., 2021, Thampi et al., 2016, Gascó et al., 2023, Sahoo et al., 25 Feb 2026).

1. Continuum Models and Governing Hydrodynamics

The generic description of active turbulence derives from hydrodynamic equations augmenting classical (Navier–Stokes or Stokes) momentum balance with active stress tensors and order-parameter evolution for orientational (polar or nematic) fields.

• Nematic (Beris–Edwards) active fluids are governed by:
  • $$\partial_t Q_{ij} + u_k \partial_k Q_{ij} - S_{ij} = \Gamma H_{ij}$$
  • $$\rho (\partial_t u_i + u_k \partial_k u_i) = \partial_j \Pi_{ij}$$
  • where $$\Pi_{ij} = -P \delta_{ij} + 2\eta E_{ij} + \Pi_{ij}^{\rm passive} - \zeta Q_{ij}$$.
  • The active stress $-\zeta Q_{ij}$ arises from force-dipole activity (e.g., extensile rods for $\zeta>0$, contractile for $\zeta<0$) (Thampi et al., 2016).
• Polar (Toner–Tu–Swift–Hohenberg, TTSH) and generalized Navier–Stokes models for suspensions of self-propelled entities:
  • $$\partial_t v + \lambda (v \cdot \nabla) v = -\nabla P - [\alpha + \beta |v|^2] v + \Gamma_0 \nabla^2 v - \Gamma_2 \nabla^4 v$$
  • For fields of spatially varying activity, $\alpha(x,t)$ may be dynamically advected and diffused by the flow (Sahoo et al., 25 Feb 2026).
• Microswimmer suspensions often use explicit particle-resolved or continuum stresslet models, with activity entering as a force-dipole (stresslet) tensor:
  • $\mathbf{\sigma}^{(a)} = \zeta \mathbf{P}\mathbf{P}$ (with polarization $\mathbf{P}$), or as collective force-densities arising from pairwise forces between swimmer elements (Gascó et al., 2023, Bardfalvy et al., 2019).

Boundary conditions, substrate friction terms, and coupling to a passive solvent or substrate (e.g., via a frictional drag $$\rho (\partial_t u_i + u_k \partial_k u_i) = \partial_j \Pi_{ij}$$0, or frictional exchange $$\rho (\partial_t u_i + u_k \partial_k u_i) = \partial_j \Pi_{ij}$$1) control the momentum decay and screening length (Thampi et al., 2014, Spera et al., 27 Nov 2025).

2. Instability, Transition, and Pattern-Forming Mechanisms

Active turbulence invariably requires the crossing of a critical threshold in activity (e.g., the activity coefficient $$\rho (\partial_t u_i + u_k \partial_k u_i) = \partial_j \Pi_{ij}$$2 or dimensionless activity number $$\rho (\partial_t u_i + u_k \partial_k u_i) = \partial_j \Pi_{ij}$$3) at which the uniform ordered or quiescent phase becomes absolutely unstable to long-wavelength fluctuations. The instability is generically governed by a competition between active stress and elastic, viscous, and frictional restoring forces.

• Thresholds and Bifurcations:
  • For nematics, instability arises when $$\rho (\partial_t u_i + u_k \partial_k u_i) = \partial_j \Pi_{ij}$$4 with the fastest growing wavelength $$\rho (\partial_t u_i + u_k \partial_k u_i) = \partial_j \Pi_{ij}$$5 (Thampi et al., 2016).
  • In polar TTSH models, negative "effective viscosity" ($$\rho (\partial_t u_i + u_k \partial_k u_i) = \partial_j \Pi_{ij}$$6) and positive Swift–Hohenberg ($$\rho (\partial_t u_i + u_k \partial_k u_i) = \partial_j \Pi_{ij}$$7) select finite bands of unstable modes. The critical activity is set by the competition of driving and dissipation (Sahoo et al., 25 Feb 2026).
  • The transition from laminar to active turbulence can be discontinuous, marked by a sharp jump in the mean-squared velocity and bistability, as shown by a jump at a critical activity number $$\rho (\partial_t u_i + u_k \partial_k u_i) = \partial_j \Pi_{ij}$$8 in 2D active nematics (Hillebrand et al., 10 Jan 2025).
• Pattern Formation and Defect Proliferation:
  • Post-instability, the nonlinear evolution produces bending walls (lines of director distortion), which fragment into $$\rho (\partial_t u_i + u_k \partial_k u_i) = \partial_j \Pi_{ij}$$9 disclination defects. The continual cycle of wall formation, defect creation, defect motion and annihilation constitutes the backbone of spatiotemporal chaos (Thampi et al., 2016, Čopar et al., 2019).
• Role of Substrate, Friction, and Dimensionality:
  • Increased substrate friction screens momentum over a length $$\Pi_{ij} = -P \delta_{ij} + 2\eta E_{ij} + \Pi_{ij}^{\rm passive} - \zeta Q_{ij}$$0, reducing wall/defect spacing and leading to jammed, banded states at large friction (Thampi et al., 2014).
  • In 3D confined geometries (e.g., droplets), turbulence is regulated by the nature of surface anchoring and dimensional control parameters (activity number, Ericksen number). Defect morphologies include closed loops or bulk-spanning segments, whose dynamics comprise breakups, reconnections, coalescence, and annihilation (Čopar et al., 2019).

3. Statistical Properties, Scaling Laws, and Spectra

Active turbulence lacks an inertial energy cascade in the Kolmogorov sense; instead, it features distinct, system-specific power-law spectra, often with universal integer exponents fixed by symmetry, dimensionality, and the nature of activity.

System/Class	Energy Spectrum $$\Pi_{ij} = -P \delta_{ij} + 2\eta E_{ij} + \Pi_{ij}^{\rm passive} - \zeta Q_{ij}$$1	Range/Regime
2D active nematics	$$\Pi_{ij} = -P \delta_{ij} + 2\eta E_{ij} + \Pi_{ij}^{\rm passive} - \zeta Q_{ij}$$2	$$\Pi_{ij} = -P \delta_{ij} + 2\eta E_{ij} + \Pi_{ij}^{\rm passive} - \zeta Q_{ij}$$3: sub-vortex
2D active nematics	$$\Pi_{ij} = -P \delta_{ij} + 2\eta E_{ij} + \Pi_{ij}^{\rm passive} - \zeta Q_{ij}$$4	$$\Pi_{ij} = -P \delta_{ij} + 2\eta E_{ij} + \Pi_{ij}^{\rm passive} - \zeta Q_{ij}$$5: vortex-gas
3D active nematics	$$\Pi_{ij} = -P \delta_{ij} + 2\eta E_{ij} + \Pi_{ij}^{\rm passive} - \zeta Q_{ij}$$6	$$\Pi_{ij} = -P \delta_{ij} + 2\eta E_{ij} + \Pi_{ij}^{\rm passive} - \zeta Q_{ij}$$7 (Gascó et al., 2023)
Sparse swimmers (wet polar)	$$\Pi_{ij} = -P \delta_{ij} + 2\eta E_{ij} + \Pi_{ij}^{\rm passive} - \zeta Q_{ij}$$8	Inertial-range, under certain conditions
TTSH (homogeneous, high-$$\Pi_{ij} = -P \delta_{ij} + 2\eta E_{ij} + \Pi_{ij}^{\rm passive} - \zeta Q_{ij}$$9)	$-\zeta Q_{ij}$0	Intermediate range (Sahoo et al., 25 Feb 2026)
Binary/Passive coupling	$-\zeta Q_{ij}$1	Passive phase, drag transfer (Ahmed et al., 1 Nov 2025)

• In 3D particle-resolved simulations of microswimmers, the spectrum is $-\zeta Q_{ij}$2 in the developed regime, with velocity PDFs exhibiting a Lévy (power-law) tail, in contrast to the stretched/tempered distributions in inertial turbulence (Gascó et al., 2023, Bardfalvy et al., 2019).
• For swarming bacterial systems at moderate Re, a clear Kolmogorov $-\zeta Q_{ij}$3 window is observed, demonstrating that direct energy cascade statistics can arise even in an internally driven system when collective interactions dominate (Bourgoin et al., 2019).
• In 2D active nematics with sufficiently strong activity and flow-alignment, broad elastic spectra promote the growth of kinetic energy that triggers an inverse energy cascade, leading to coexistence of active and inertial turbulence (Rorai et al., 2022).
• Binary active–passive mixtures display momentum transfer via interfacial drag, yielding a steeper $-\zeta Q_{ij}$4 tail in the passive component (Ahmed et al., 1 Nov 2025), and coupling to a Newtonian substrate acts as a low-pass filter that further steepens the substrate spectrum to $-\zeta Q_{ij}$5 at large $-\zeta Q_{ij}$6 (Spera et al., 27 Nov 2025).

4. Lagrangian and Persistence Statistics

Active turbulence produces anomalous particle and tracer dynamics distinct from inertial turbulence.

• Tracer persistence time inside coherent vortices follows a Weibull distribution, with parameters determined by activity magnitude, while in the turbulent background, exit times are exponentially distributed, reflecting Poissonian statistics (Manoharan et al., 2023).
• Single-particle mean-squared displacement exhibits superdiffusive scaling, $-\zeta Q_{ij}$7, consistent with Lévy-walk dynamics, due to emergent streaky flow structures (Singh et al., 2021, Sahoo et al., 25 Feb 2026).
• Pair-dispersion of tracers deviates from the classical Richardson $-\zeta Q_{ij}$8 law, featuring instead an initial exponential separation (Lyapunov regime) crossing over to diffusive statistics at long times, without a robust power-law regime (Singh et al., 2021).
• In defect-free active nematics, dynamical arrest occurs in extensile, flow-aligning regimes: labyrinthine networks of stabilized domain walls form, freezing chaotic motion, in contrast to the perpetuated active turbulence mediated by defect proliferation in defect-laden systems (Lavi et al., 2024).

5. Physical Mechanisms and Energetic Pathways

Active turbulence arises from fundamentally different mechanisms than inertial turbulence:

• Energy Injection is localized and self-organized, driven by the continuous conversion of chemical to mechanical energy at the scale of the constituent particles.
• Instability Mechanisms:
  • In nematic systems, bend/splay instabilities break uniform order and produce walls; defects then drive jets and vorticity.
  • In polar/TTSH systems, advective nonlinearity and flow-alignment produce inverse transfer of orientational (and thus active stress) fluctuations, which in turn source mesoscale flows (Gascó et al., 2023, Sahoo et al., 25 Feb 2026).
• Spectral Energy Transfer:
  • Unlike the inertial (Kolmogorov) cascade, energy is injected and rapidly dissipated at intrinsic scales; classical scale-local transfer is suppressed, and the energy balance is local in $-\zeta Q_{ij}$9-space (Alert et al., 2021).
  • In special geometries (e.g., active turbulence on curved surfaces), coherent vortex-chain networks can mediate upscale energy transfer (distinct from the 2D inverse cascade), as observed in spherical geometries (Mickelin et al., 2017).

6. Connections, Extensions, and Advanced Scenarios

• Active–Elastic Analogy: The hydrodynamics of elastic turbulence in polymer solutions closely map onto contractile active nematic equations; both systems exhibit transverse instabilities, defect dynamics, and flow jammed states at high effective activity, providing a unifying continuum description (Dzanica et al., 13 Jan 2026).
• Phase Separation Coupling: Cahn–Hilliard phase separation in active-passive fluids leads to microphase domain formation. The resulting patterns, vortex statistics, and energy spectra are controlled by the competition between active stress injection, viscous dissipation, elasticity, and interfacial tension (Ahmed et al., 1 Nov 2025).
• Coexisting Regimes: In regimes where both effective Re and activity are large, simultaneous active and inertial turbulence can coexist, as documented in extensile, strongly flow-aligning 2D active nematics (Rorai et al., 2022).
• Impact of Curvature: On curved substrates, particularly spheres, the topology and size of coherent structures are dictated by geometric constraints, leading to anomalous chaining of vortices and alternative energy transfer mechanisms (Mickelin et al., 2017).

7. Experimental, Numerical, and Theoretical Advances

Significant progress has been achieved via large-scale simulations (e.g., particle-resolved LB, multiparticle collision dynamics, hybrid lattice Boltzmann, and spectral methods), advanced experimental diagnostics (tracer tracking, velocimetry, structure function analysis), and the development of minimal and shell models capturing the essential ingredients.

• Universal Features:
  • The active length scale $\zeta>0$0 or related expressions governs vortex size, wall spacing, and spectral crossovers across diverse systems.
  • Velocity-velocity correlation functions, vortex area/length distribution, and energy spectra provide system- and regime-diagnostic fingerprints (Thampi et al., 2016, Alert et al., 2021, Gascó et al., 2023).
• Open Questions:
  • The universality of scaling exponents in various classes and the emergence of cascade-like features in highly interacting or high-Re systems remain under investigation (Sahoo et al., 25 Feb 2026, Rorai et al., 2022, Bourgoin et al., 2019).
  • The interplay of topology, confinement, and substrate coupling in dictating dynamics and pattern statistics is a vibrant research direction (Čopar et al., 2019, Spera et al., 27 Nov 2025, Lavi et al., 2024).
  • Connections between active and elastic turbulence, the effect of heterogeneous or time-dependent activity fields, and the emergence of glassy/arrested states represent frontier challenges (Sahoo et al., 25 Feb 2026, Dzanica et al., 13 Jan 2026, Lavi et al., 2024).

Active turbulence thus constitutes a unifying, fundamentally non-equilibrium paradigm, bridging soft active matter, hydrodynamics, and statistical physics. Its regime diagrams, scaling laws, and defect kinetics provide insight into both synthetic and biological systems, including bacterial swarms, cytoskeletal networks, and active emulsions, and will continue to inform theoretical modeling, materials design, and experimental exploration across the discipline.