- Physical Review Journals
All JournalsPhysics Magazine

Physical Review E

Reuse & Permissions

It is not necessary to obtain permission to reuse this article or its components as it is available under the terms of the Creative Commons Attribution 4.0 International license. This license permits unrestricted use, distribution, and reproduction in any medium, provided attribution to the author(s) and the published article's title, journal citation, and DOI are maintained. Please note that some figures may have been included with permission from other third parties. It is your responsibility to obtain the proper permission from the rights holder directly for these figures.

Export citation

Export citation

Choose format for download:

Download Citation
  • Editors' Suggestion
  • Open Access

Multiscale data assimilation in turbulent models

Francesco Fossella1,2,3,*, Luca Biferale2,3, Alberto Carrassi4, Massimo Cencini3,5, and Vikrant Gupta6

  • *Contact author: francesco.fossella@telecom-paris.fr
PDF

Phys. Rev. E 113, 024208 – Published 17 February, 2026

DOI: https://doi.org/10.1103/c1wn-n1j6

Abstract

We explore the potential of data assimilation (DA) within the multiscale framework of a shell model of turbulence, with a focus on the ensemble Kalman filter (EnKF). The central objective is to understand how measuring mesoscales (i.e., inertial-range scales) enhances the prediction of both large-scale and small-scale intermittent variables, by systematically varying observation frequency and the set of measured scales. We demonstrate that measurements conducted at frequencies that exceed those of the observed scales enable full synchronization of larger scales, provided that at least two adjacent mesoscales are measured. In addition, we benchmark the EnKF against two other DA methods, namely, nudging and ensemble four‐dimensional variational method. EnKF is clearly superior to the former and comparable with the latter but achieving the result with a lower computational complexity. Moreover, our results underscore the need for a tailored, scale-aware inflation technique to stabilize the assimilation process, preventing filter divergence and ensuring robust convergence.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (56)

  1. E. N. Lorenz, Tellus 21, 289 (1969).
  2. G. Boffetta and S. Musacchio, Phys. Rev. Lett. 119, 054102 (2017).
  3. E. Aurell, G. Boffetta, A. Crisanti, G. Paladin, and A. Vulpiani, Phys. Rev. E 53, 2337 (1996).
  4. D. Bandak, A. A. Mailybaev, G. L. Eyink, and N. Goldenfeld, Phys. Rev. Lett. 132, 104002 (2024).
  5. A. Pikovsky, M. Rosenblum, and J. Kurths, Synchronization—A Universal Concept in Nonlinear Sciences (Cambridge University Press, Cambridge, UK, 2001).
  6. O. Talagrand, J. Meteorol. Soc. Jpn. 75, 191 (1997).
  7. A. Carrassi, M. Bocquet, L. Bertino, and G. Evensen, WIREs Clim. Change 9, e535 (2018).
  8. G. Evensen, F. C. Vossepoel, and P. Jan van Leeuwen, Data Assimilation Fundamentals: A Unified Formulation of the State and Parameter Estimation Problem (Springer, Berlin, 2022).
  9. A. Carrassi, M. Bocquet, J. Demaeyer, C. Grudzien, P. Raanes, and S. Vannitsem, in Data Assimilation for Atmospheric, Oceanic and Hydrologic Applications (Vol. IV), edited by S. K. Park and L. Xu (Springer, Cham, 2022).
  10. G. Evensen, J. Geophys. Res. 99, 10143 (1994).
  11. G. Evensen, Ocean Dyn. 53, 343 (2003).
  12. U. Frisch, Turbulence: The Legacy of A.N. Kolmogorov (Cambridge University Press, Cambridge, UK, 1995).
  13. T. Bohr, M. H. Jensen, G. Paladin, and A. Vulpiani, Dynamical Systems Approach to Turbulence (Cambridge University Press, Cambridge, UK, 1998).
  14. P. D. Ditlevsen, Turbulence and Shell Models (Cambridge University Press, Cambridge, UK, 2010).
  15. V. S. L'vov, E. Podivilov, A. Pomyalov, I. Procaccia, and D. Vandembroucq, Phys. Rev. E 58, 1811 (1998).
  16. M. Asch, M. Bocquet, and M. Nodet, Data Assimilation: Methods, Algorithms, and Applications (Society for Industrial and Applied Mathematics, Philadelphia, PA, 2016).
  17. G. A. Gottwald and A. J. Majda, Nonlinear Process. Geophys. 20, 705 (2013).
  18. W. Sacher and P. Bartello, Mon. Weather Rev. 136, 3035 (2008).
  19. J. L. Anderson and S. L. Anderson, Mon. Weather Rev. 127, 2741 (1999).
  20. P. N. Raanes, M. Bocquet, and A. Carrassi, Q. J. R. Meteorol. Soc. 145, 53 (2019).
  21. R. A. Anthes, J. Atmos. Sci. 31, 702 (1974).
  22. P. Courtier, J.-N. Thépaut, and A. Hollingsworth, Q. J. R. Meteorol. Soc. 120, 1367 (1994).
  23. R. N. Bannister, Q. J. R. Meteorol. Soc. 143, 607 (2017).
  24. M. Bocquet and A. Carrassi, Tellus A 69, 1304504 (2017).
  25. T. M. Hamill, in Predictability of Weather and Climate, edited by T. Palmer and R. Hagedorn (Cambridge University Press, Cambridge, UK, 2006), pp. 124–156.
  26. M. Tanguay, P. Bartello, and P. Gauthier, Tellus A 47, 974 (1995).
  27. N. Chen, Y. Li, and E. Lunasin, Chaos 31, 103123 (2021).
  28. M. Tondeur, A. Carrassi, S. Vannitsem, and M. Bocquet, J. Stat. Phys. 179, 1161 (2020).
  29. L. Garcia-Oliva, A. Carrassi, and F. Counillon, Nonlinear Process. Geophys. 32, 439 (2024).
  30. E. N. Lorenz, in Proceedings on Seminar on Predictability (ECMWF, 1996), Vol. 1, pp. 1–18.
  31. E. B. Gledzer, Sov. Phys. Dokl. 18, 216 (1973) [english translation of Dokl. Akad. Nauk SSSR 209, 1046 (1973)].
  32. L. Biferale, Annu. Rev. Fluid Mech. 35, 441 (2003).
  33. G. Burgers, P. J. van Leeuwen, and G. Evensen, Mon. Weather Rev. 126, 1719 (1998).
  34. J. Lei, P. Bickel, and C. Snyder, Mon. Weather Rev. 138, 1293 (2010).
  35. C. K. Wikle and L. M. Berliner, Physica D 230, 1 (2007).
  36. D. H. Dini and D. P. Mandic, IEEE Trans. Neural Netw. Learn. Syst. 23, 775 (2012).
  37. P. J. van Leeuwen, Q. J. R. Meteorol. Soc. 146, 2815 (2020).
  38. R. E. Kalman, J. Basic Eng. 82, 35 (1960).
  39. C. Grudzien, A. Carrassi, and M. Bocquet, Nonlinear Process. Geophys. 25, 633 (2018).
  40. I. Pasmans, Y. Chen, A. Carrassi, and C. K. R. T. Jones, Q. J. R. Meteorol. Soc. 150, 2820 (2024).
  41. B. Ménétrier, T. Montmerle, Y. Michel, and L. Berre, Mon. Weather Rev. 143, 1622 (2015).
  42. F. Zhang, C. Snyder, and S. Juanzhen, Mon. Weather Rev. 132, 1238 (2004).
  43. E. Olson and E. S. Titi, J. Stat. Phys. 113, 799 (2003).
  44. A. Azouani, E. Olson, and E. S. Titi, J. Nonlinear Sci. 24, 277 (2014).
  45. K. Yoshida, J. Yamaguchi, and Y. Kaneda, Phys. Rev. Lett. 94, 014501 (2005).
  46. Y. Li, J. Zhang, G. Dong, and N. S. Abdullah, J. Fluid Mech. 885, A9 (2020).
  47. P. Clark Di Leoni, A. Mazzino, and L. Biferale, Phys. Rev. X 10, 011023 (2020).
  48. X. Zou, I. M. Navon, and F. X. Ledimet, Q. J. R. Meteorol. Soc. 118, 1163 (1992).
  49. D. R. Stauffer and J.-W. Bao, Tellus A 45, 358 (1993).
  50. D. Pazó, A. Carrassi, and J. M. López, Q. J. R. Meteorol. Soc. 142, 1290 (2016).
  51. L. Agasthya, P. C. Di Leoni, and L. Biferale, Phys. Fluids 34, 015128 (2022).
  52. P. Clark Di Leoni, L. Agasthya, M. Buzzicotti, and L. Biferale, Eur. Phys. J. E 46, 16 (2023).
  53. K. Um, R. Brand, Y. Fei, P. Holl, and N. Thuerey, Adv. Neural Inf. Process. Syst. 33, 6111 (2020).
  54. A. Freitas, K. Um, M. Desbrun, M. Buzzicotti, and L. Biferale, Phys. Rev. Fluids 10, 044602 (2025).
  55. S. Cheng, C. Quilodrán-Casas, S. Ouala, A. Farchi, C. Liu, P. Tandeo, R. Fablet, D. Lucor, B. Iooss, J. Brajard, D. H. Xiao, T. Janjic, W. P. Ding, Y. K. Guo, A. Carrassi, M. Bocquet, and R. Arcucci, IEEE/CAA J. Autom. Sinica 10, 1361 (2023).
  56. https://doi.org/10.5281/zenodo.16312540.

Outline

Information

Phys. Rev. E 113, 024208– Published 17 February, 2026
  • Received 22 July 2025
  • Accepted 13 January 2026
Crossmark - Check for updates button
Creative Commons Logo

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

Published by the American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation