Skip to main content

Thank you for visiting nature.com. You are using a browser version with limited support for CSS. To obtain the best experience, we recommend you use a more up to date browser (or turn off compatibility mode in Internet Explorer). In the meantime, to ensure continued support, we are displaying the site without styles and JavaScript.

  • Article
  • Published:

Intermittency, fluctuations and maximal chaos in an emergent universal state of active turbulence

Nature Physics volume 19, pages 891–897 (2023)Cite this article

Subjects

Abstract

The phenomenon of active turbulence, a complex organization of matter driven at the scale of its constituent agents, is puzzling. Specifically, the lack of scale-separation in low-Reynolds-number active flows breaks away from the familiar notions of the energy cascade and approximate scale-invariance of inertial turbulence. Here, using a generalized hydrodynamic model developed for bacterial turbulence, we provide compelling analytical and numerical evidence that, beyond a critical drive, active turbulence indeed attains universality akin to inertial turbulence. In this asymptotic state, the energy spectrum scales as k−3/2, reminiscent of some classes of inertial turbulence. The flow also exhibits spatio-temporal intermittency beyond the transition, as seen from non-Gaussian fluctuations in velocity differences. With these tell-tale fingerprints, active turbulence is placed closer in phenomenology to inertial turbulence than previously held. We show, however, that as a consequence of a finite range of scales, the degree of chaoticity and hence mixing efficiency saturates to a maximum in the asymptotic regime, unlike unbounded chaos in inertial turbulence. We conclude that active turbulence, depending on the level of drive, can switch between fundamentally distinct non-universal and universal states.

This is a preview of subscription content, access via your institution

Access options

Buy this article

  • Purchase on Springer Link
  • Instant access to full article PDF
Buy now

Prices may be subject to local taxes which are calculated during checkout

Fig. 1: Intermittency arises beyond a critical level of activity.
Fig. 2: Effective energy transfer timescale develops scaling.
Fig. 3: Universal energy spectrum scaling beyond a critical activity.
Fig. 4: Nature of Eulerian chaos in mildly and highly active turbulence.

Similar content being viewed by others

Data availability

Energy spectra and their scaling exponents, as well as twin-simulation decorrelators, have been shared in a public repository on Open Science Framework, which may be accessed at https://osf.io/tcsyw/ with https://doi.org/10.17605/OSF.IO/TCSYW. Field data, which are large-sized, are available from the corresponding author upon request.

Code availability

Simulations were performed using our in-house codes, which are available from the corresponding author upon request.

References

  1. Wensink, H. H. et al. Meso-scale turbulence in living fluids. Proc. Natl. Acad. Sci. USA 109, 14308–14313 (2012).

    Article  ADS  MATH  Google Scholar 

  2. Marchetti, M. C. et al. Hydrodynamics of soft active matter. Rev. Mod. Phys. 85, 1143–1189 (2013).

    Article  ADS  Google Scholar 

  3. Ramaswamy, S. Active matter. J. Stat. Mech. Theory Exp. 2017, 054002 (2017).

    Article  MathSciNet  MATH  Google Scholar 

  4. Alert, R., Casademunt, J. & Joanny, J.-F. Active turbulence. Annu. Rev. Condens. Matter Phys. 13, 143–170 (2022).

    Article  ADS  Google Scholar 

  5. James, M., Suchla, D. A., Dunkel, J. & Wilczek, M. Emergence and melting of active vortex crystals. Nat. Commun. 12, 5630 (2021).

  6. Mukherjee, S., Singh, R. K., James, M. & Ray, S. S. Anomalous diffusion and Lévy walks distinguish active from inertial turbulence. Phys. Rev. Lett. 127, 118001 (2021).

    Article  ADS  Google Scholar 

  7. Puggioni, L., Boffetta, G. & Musacchio, S. Giant vortex dynamics in confined active turbulence. Phys. Rev. E 106, 055103 (2022).

  8. Frisch, U. Turbulence: The Legacy of A. N. Kolmogorov (Cambridge Univ. Press, 1996).

  9. Bhattacharjee, J. K. & Kirkpatrick, T. R. Activity induced turbulence in driven active matter. Phys. Rev. Fluids 7, 034602 (2022).

    Article  ADS  Google Scholar 

  10. Humphries, N. E. et al. Environmental context explains Lévy and Brownian movement patterns of marine predators. Nature 465, 1066–1069 (2010).

    Article  ADS  Google Scholar 

  11. Volpe, G. & Volpe, G. The topography of the environment alters the optimal search strategy for active particles. Proc. Natl. Acad. Sci. USA 114, 11350–11355 (2017).

    Article  ADS  Google Scholar 

  12. Ariel, G. et al. Swarming bacteria migrate by Lévy walk. Nat. Commun. 6, 8396 (2015).

    Article  ADS  Google Scholar 

  13. Singh, R. K., Mukherjee, S. & Ray, S. S. Lagrangian manifestation of anomalies in active turbulence. Phys. Rev. Fluids 7, 033101 (2022).

    Article  ADS  Google Scholar 

  14. Linkmann, M., Boffetta, G., Marchetti, M. C. & Eckhardt, B. Phase transition to large scale coherent structures in two-dimensional active matter turbulence. Phys. Rev. Lett. 122, 214503 (2019).

    Article  ADS  Google Scholar 

  15. Rorai, C., Toschi, F. & Pagonabarraga, I. Coexistence of active and hydrodynamic turbulence in two-dimensional active nematics. Phys. Rev. Lett. 129, 218001 (2022).

    Article  ADS  Google Scholar 

  16. Sokolov, A. & Aranson, I. S. Physical properties of collective motion in suspensions of bacteria. Phys. Rev. Lett. 109, 248109 (2012).

    Article  ADS  Google Scholar 

  17. Toner, J. & Tu, Y. Long-range order in a two-dimensional dynamical XY model: how birds fly together. Phys. Rev. Lett. 75, 4326–4329 (1995).

    Article  ADS  Google Scholar 

  18. Toner, J. & Tu, Y. Flocks, herds and schools: a quantitative theory of flocking. Phys. Rev. E 58, 4828–4858 (1998).

    Article  ADS  MathSciNet  Google Scholar 

  19. CP, S. & Joy, A. Friction scaling laws for transport in active turbulence. Phys. Rev. Fluids 5, 024302 (2020).

    Article  ADS  Google Scholar 

  20. Bratanov, V., Jenko, F. & Frey, E. New class of turbulence in active fluids. Proc. Natl. Acad. Sci. USA 112, 15048–15053 (2015).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  21. Ilkanaiv, B., Kearns, D. B., Ariel, G. & Be’er, A. Effect of cell aspect ratio on swarming bacteria. Phys. Rev. Lett. 118, 158002 (2017).

    Article  ADS  Google Scholar 

  22. Kiran, K. V., Gupta, A., Verma, A. K. & Pandit, R. Irreversibility in bacterial turbulence: insights from the mean-bacterial-velocity model. Phys. Rev. Fluids 8, 023102 (2023).

  23. Verma, M. K. Energy Transfers in Fluid Flows: Multiscale and Spectral Perspectives (Cambridge Univ. Press, 2019); https://doi.org/10.1017/9781316810019

  24. Verma, M. K. Variable energy flux in turbulence. J. Phys. A Math. Theor. 55, 013002 (2021).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  25. Mukherjee, S., Schalkwijk, J. & Jonker, H. J. J. Predictability of dry convective boundary layers: an LES study. J. Atmos. Sci. 73, 2715–2727 (2016).

    Article  ADS  Google Scholar 

  26. Boffetta, G. & Musacchio, S. Chaos and predictability of homogeneous-isotropic turbulence. Phys. Rev. Lett. 119, 054102 (2017).

    Article  ADS  Google Scholar 

  27. Berera, A. & Ho, R. D. J. G. Chaotic properties of a turbulent isotropic fluid. Phys. Rev. Lett. 120, 024101 (2018).

    Article  ADS  Google Scholar 

  28. Das, A. et al. Light-cone spreading of perturbations and the butterfly effect in a classical spin chain. Phys. Rev. Lett. 121, 024101 (2018).

    Article  ADS  Google Scholar 

  29. Murugan, S. D., Kumar, D., Bhattacharjee, S. & Ray, S. S. Many-body chaos in thermalized fluids. Phys. Rev. Lett. 127, 124501 (2021).

    Article  ADS  Google Scholar 

  30. Lorenz, E. N. The predictability of a flow which possesses many scales of motion. Tellus 21, 289–307 (1969).

    Article  ADS  Google Scholar 

  31. Iroshnikov, P. S. Turbulence of a conducting fluid in a strong magnetic field. Sov. Astron. 7, 566 (1964).

    ADS  MathSciNet  Google Scholar 

  32. Kraichnan, R. H. Inertial range spectrum of hydromagnetic turbulence. Phys. Fluids 8, 1385–1387 (1965).

    Article  ADS  Google Scholar 

  33. Nazarenko, S. Wave turbulence. Contemp. Phys. 56, 359–373 (2015).

    Article  ADS  Google Scholar 

  34. Zakharov, V. E. & Sagdeev, R. Z. Spectrum of acoustic turbulence. Sov. Phys. Doklady 15, 439 (1970).

    ADS  MATH  Google Scholar 

  35. Pan, N. & Banerjee, S. Energy transfer in simple and active binary fluid turbulence—a false friend of incompressible MHD turbulence. Preprint at arXiv https://arxiv.org/abs/2206.12782 (2022).

  36. Buzzicotti, M., Biferale, L., Frisch, U. & Ray, S. S. Intermittency in fractal fourier hydrodynamics: lessons from the Burgers equation. Phys. Rev. E 93, 033109 (2016).

    Article  ADS  MathSciNet  Google Scholar 

  37. Alert, R., Joanny, J.-F. & Casademunt, J. Universal scaling of active nematic turbulence. Nat. Phys. 16, 682–688 (2020).

    Article  Google Scholar 

  38. Bourgoin, M. et al. Kolmogorovian active turbulence of a sparse assembly of interacting Marangoni surfers. Phys. Rev. X 10, 021065 (2020).

    MathSciNet  Google Scholar 

  39. Kokot, G. et al. Active turbulence in a gas of self-assembled spinners. Proc. Natl. Acad. Sci. USA 114, 12870–12875 (2017).

    Article  ADS  Google Scholar 

  40. Saghatchi, R., Yildiz, M. & Doostmohammadi, A. Nematic order condensation and topological defects in inertial active nematics. Phys. Rev. E 106, 014705 (2022).

    Article  ADS  Google Scholar 

  41. Petroff, A. P., Wu, X.-L. & Libchaber, A. Fast-moving bacteria self-organize into active two-dimensional crystals of rotating cells. Phys. Rev. Lett. 114, 158102 (2015).

    Article  ADS  Google Scholar 

  42. Reas, C. & Fry, B. Processing: a Programming Handbook for Visual Designers and Artists (MIT Press, 2007).

  43. Pearson, M. Generative Art: a Practical Guide using Processing (Simon & Schuster, 2011).

  44. James, M. & Wilczek, M. Vortex dynamics and Lagrangian statistics in a model for active turbulence. Eur. Phys. J. E 41, 21 (2018).

    Article  Google Scholar 

  45. James, M., Bos, WouterJ. T. & Wilczek, M. Turbulence and turbulent pattern formation in a minimal model for active fluids. Phys. Rev. Fluids 3, 061101 (2018).

    Article  ADS  Google Scholar 

Download references

Acknowledgements

The simulations were performed on the ICTS clusters Tetris and Contra. S.M. and S.S.R. thank J. Bec, S. D. Murugan and J. Picardo for insightful discussions and suggestions. S.S.R. acknowledges SERB-DST (India) projects MTR/2019/001553, STR/2021/000023 and CRG/2021/002766 for financial support. M.J. gratefully acknowledges support from grant no. STR/2021/000023 and the hospitality of ICTS-TIFR. S.S.R. thanks the Isaac Newton Institute for Mathematical Sciences for support and hospitality during the programme ‘Mathematical aspects of turbulence: where do we stand?’ (EPSRC grant no. EP/R014604/1), when a part of this work was done. We acknowledge the support of the DAE, Government of India, under projects nos. 12-R&D-TFR-5.10-1100 and RTI4001.

Author information

Authors and Affiliations

  1. International Centre for Theoretical Sciences, Tata Institute of Fundamental Research, Bangalore, India

    Siddhartha Mukherjee, Rahul K. Singh & Samriddhi Sankar Ray

  2. Complex Fluids and Flows Unit, Okinawa Institute of Science and Technology Graduate University, Okinawa, Japan

    Rahul K. Singh

  3. Machine Learning Genoa Center (MaLGa), Department of Civil, Chemical and Environmental Engineering, University of Genoa, Genoa, Italy

    Martin James

Authors
  1. Siddhartha Mukherjee
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Rahul K. Singh
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Martin James
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Samriddhi Sankar Ray
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

S.S.R. and S.M. designed the research. S.M. performed the simulations and analysis. S.M., R.K.S., M.J. and S.S.R. contributed to interpretation of the results and writing the manuscript.

Corresponding authors

Correspondence to Siddhartha Mukherjee, Rahul K. Singh, Martin James or Samriddhi Sankar Ray.

Ethics declarations

Competing interests

The authors declare no competing interests.

Peer review

Peer review information

Nature Physics thanks Alexey Snezhko and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.

Additional information

Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Mukherjee, S., Singh, R.K., James, M. et al. Intermittency, fluctuations and maximal chaos in an emergent universal state of active turbulence. Nat. Phys. 19, 891–897 (2023). https://doi.org/10.1038/s41567-023-01990-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1038/s41567-023-01990-z

This article is cited by

Search

Advanced search

Quick links

Nature Briefing

Sign up for the Nature Briefing newsletter — what matters in science, free to your inbox daily.

Get the most important science stories of the day, free in your inbox. Sign up for Nature Briefing