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Uncertainty growth in stably stratified turbulence

Mrinal Jyoti Powdel* and Samriddhi Sankar Ray†

  • *Contact author: mrinal.jyoti@icts.res.in
  • †Contact author: samriddhisankarray@gmail.com
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Phys. Rev. Fluids 11, 064615 – Published 23 June, 2026

DOI: https://doi.org/10.1103/xblr-vpl3

Abstract

We investigate uncertainty growth and chaotic dynamics in statistically steady, stably stratified three-dimensional turbulence. Using direct numerical simulations of the Boussinesq equations, we quantify the divergence of initially infinitesimal perturbations via twin simulations and decorrelator diagnostics. At short times, perturbations exhibit exponential growth, allowing us to define a (largest) Lyapunov exponent. We systematically examine how this exponent depends on stratification strength, quantified by the Brunt-Väisälä frequency and the Froude number, in a parameter regime relevant to oceanic flows. We find that increasing stratification leads to a monotonic reduction of the Lyapunov exponent, indicating suppressed chaoticity. Despite this reduction, uncertainty growth retains the universal temporal sequence observed in homogeneous isotropic turbulence—initial decay, exponential growth, and saturation. The growth phase is characterized by self-similar decorrelator spectra, but exhibits strong anisotropy: uncertainty spreads much more slowly along the stratification direction than horizontally, with the disparity increasing with stratification strength. An analysis of the decorrelator evolution equation reveals that the suppression of chaos arises primarily from strain-mediated alignment dynamics rather than direct buoyancy coupling. Our results provide a quantitative characterization of predictability and uncertainty growth in stratified turbulence and highlight the utility of decorrelator-based methods for anisotropic geophysical flows.

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References (63)

  1. D. Ruelle, Microscopic fluctuations and turbulence, Phys. Lett. A 72, 81 (1979).
  2. U. Frisch and G. Parisi, Turbulence and predictability of geophysical fluid dynamics and climate dynamics, in Proceedings of the International School of Physics Enrico Fermi, Course LXXXVIII, Varenna (North-Holland, Amsterdam, 1985).
  3. U. Frisch, Turbulence: The legacy of A. N. Kolmogorov (Cambridge University Press, Cambridge, 1995).
  4. A. Crisanti, M. H. Jensen, A. Vulpiani, and G. Paladin, Intermittency and predictability in turbulence, Phys. Rev. Lett. 70, 166 (1993).
  5. J. Bec, L. Biferale, G. Boffetta, M. Cencini, S. Musacchio, and F. Toschi, Lyapunov exponents of heavy particles in turbulence, Phys. Fluids 18, 091702 (2006).
  6. M. Cencini, F. Cecconi, and A. Vulpiani, Chaos (World Scientific, Singapore, 2009).
  7. S. S. Ray, Non-intermittent turbulence: Lagrangian chaos and irreversibility, Phys. Rev. Fluids 3, 072601(R) (2018).
  8. R. D. J. G. Ho, D. Clark, and A. Berera, Chaotic measures as an alternative to spectral measures for analysing turbulent flow, Atmosphere 15, 1053 (2024).
  9. A. Das, S. Chakrabarty, A. Dhar, A. Kundu, D. A. Huse, R. Moessner, S. S. Ray, and S. Bhattacharjee, Light-cone spreading of perturbations and the butterfly effect in a classical spin chain, Phys. Rev. Lett. 121, 024101 (2018).
  10. E. N. Lorenz, The predictability of a flow which possesses many scales of motion, Tellus 21, 289 (1969).
  11. C. E. Leith and R. H. Kraichnan, Predictability of turbulent flows, J. Atmos. Sci. 29, 1041 (1972).
  12. P. Mohan, N. Fitzsimmons, and R. D. Moser, Scaling of Lyapunov exponents in homogeneous isotropic turbulence, Phys. Rev. Fluids 2, 114606 (2017).
  13. S. Mukherjee, J. Schalkwijk, and H. J. J. Jonker, Predictability of dry convective boundary layers: An LES study, J. Atmos. Sci. 73, 2715 (2016).
  14. G. Boffetta and S. Musacchio, Chaos and predictability of homogeneous-isotropic turbulence, Phys. Rev. Lett. 119, 054102 (2017).
  15. A. Berera and R. D. J. G. Ho, Chaotic properties of a turbulent isotropic fluid, Phys. Rev. Lett. 120, 024101 (2018).
  16. J. Ge, J. Rolland, and J. C. Vassilicos, The production of uncertainty in three-dimensional Navier–Stokes turbulence, J. Fluid Mech. 977, A17 (2023).
  17. J. Ge, J. Rolland, and J. C. Vassilicos, The interscale behaviour of uncertainty in three-dimensional Navier–Stokes turbulence, J. Fluid Mech. 1017, A29 (2025).
  18. J. R. C. King, R. J. Poole, C. P. Fonte, and S. J. Lind, Uncertainty in elastic turbulence, J. Fluid Mech. 1020, A2 (2025).
  19. V. de Jesus Valadão, F. De Lillo, S. Musacchio, and G. Boffetta, Scaling and predictability in surface quasi-geostrophic turbulence, J. Fluid Mech. 1017, A38 (2025).
  20. A. Banerjee, R. Mukherjee, S. D. Murugan, S. Bhattacharjee, and S. S. Ray, Intermittent fluctuations determine the nature of chaos in turbulence, Phys. Rev. Lett. (2026), doi:10.1103/1dj2-zw28.
  21. S. D. Murugan, D. Kumar, S. Bhattacharjee, and S. S. Ray, Many-body chaos in thermalized fluids, Phys. Rev. Lett. 127, 124501 (2021).
  22. S. Mukherjee, R. K. Singh, M. James, and S. S. Ray, Intermittency, fluctuations and maximal chaos in an emergent universal state of active turbulence, Nat. Phys. 19, 891 (2023).
  23. P. A. Davidson, Turbulence in Rotating, Stratified and Electrically Conducting Fluids (Cambridge University Press, Cambridge, 2013).
  24. J. J. Riley and M.-P. Lelong, Fluid motions in the presence of strong stable stratification, Annu. Rev. Fluid Mech. 32, 613 (2000).
  25. C. Staquet and J. Sommeria, Internal gravity waves: From instabilities to turbulence, Annu. Rev. Fluid Mech. 34, 559 (2002).
  26. C. P. Caulfield, Layering, instabilities, and mixing in turbulent stratified flows, Annu. Rev. Fluid Mech. 53, 113 (2021).
  27. C.-C. P. Caulfield, Open questions in turbulent stratified mixing: Do we even know what we do not know? Phys. Rev. Fluids 5, 110518 (2020).
  28. P. F. Linden, Mixing in stratified fluids, Geophys. Astrophys. Fluid Dyn. 13, 3 (1979).
  29. J. J. Riley, R. W. Metcalfe, and M. A. Weissman, Direct numerical simulations of homogeneous turbulence in density‐stratified fluids, AIP Conf. Proc. 76, 79 (1981).
  30. H. J. S. Fernando, The growth of a turbulent patch in a stratified fluid, J. Fluid Mech. 190, 55 (1988).
  31. E. C. Itsweire and K. N. Helland, Spectra and energy transfer in stably stratified turbulence, J. Fluid Mech. 207, 419 (1989).
  32. B. R. Ruddick, T. J. McDougall, and J. S. Turner, The formation of layers in a uniformly stirred density gradient, Deep Sea Res. A. Oceanogr. Res. Papers 36, 597 (1989).
  33. Y.-G. Park, J. A. Whitehead, and A. Gnanadeskian, Turbulent mixing in stratified fluids: Layer formation and energetics, J. Fluid Mech. 279, 279 (1994).
  34. F. Nicolleau and J. C. Vassilicos, Turbulent diffusion in stably stratified non-decaying turbulence, J. Fluid Mech. 410, 123 (2000).
  35. P. Billant and J.-M. Chomaz, Self-similarity of strongly stratified inviscid flows, Phys. Fluids 13, 1645 (2001).
  36. J.-P. Laval, J. C. McWilliams, and B. Dubrulle, Forced stratified turbulence: Successive transitions with Reynolds number, Phys. Rev. E 68, 036308 (2003).
  37. E. Lindborg, The energy cascade in a strongly stratified fluid, J. Fluid Mech. 550, 207 (2006).
  38. E. Lindborg and G. Brethouwer, Vertical dispersion by stratified turbulence, J. Fluid Mech. 614, 303 (2008).
  39. P. Augier, S. Galtier, and P. Billant, Kolmogorov laws for stratified turbulence, J. Fluid Mech. 709, 659 (2012).
  40. Y. Kimura and J. R. Herring, Energy spectra of stably stratified turbulence, J. Fluid Mech. 698, 19 (2012).
  41. C. Rorai, P. D. Mininni, and A. Pouquet, Stably stratified turbulence in the presence of large-scale forcing, Phys. Rev. E 92, 013003 (2015).
  42. C. Herbert, R. Marino, D. Rosenberg, and A. Pouquet, Waves and vortices in the inverse cascade regime of stratified turbulence with or without rotation, J. Fluid Mech. 806, 165 (2016).
  43. A. V. Glazunov, E. V. Mortikov, K. V. Barskov, E. V. Kadantsev, and S. S. Zilitinkevich, Layered structure of stably stratified turbulent shear flows, Izv. Atmos. Oceanic Phys. 55, 312 (2019).
  44. A. Maffioli, A. Delache, and F. S. Godeferd, Signature and energetics of internal gravity waves in stratified turbulence, Phys. Rev. Fluids 5, 114802 (2020).
  45. V. Labarre, P. Augier, G. Krstulovic, and S. Nazarenko, Internal gravity waves in stratified flows with and without vortical modes, Phys. Rev. Fluids 9, 024604 (2024).
  46. A. K. Varanasi, Lagrangian intermittency and vertical confinement in stably stratified turbulence, arXiv:2503.22445.
  47. M. F. Diaz and M. L. Waite, Predictability of decaying stratified turbulence, Phys. Fluids 36, 065138 (2024).
  48. Y. Peng, X. Xu, Q. Shao, H. Weng, H. Niu, Z. Li, C. Zhang, P. Li, X. Zhong, and J. Yang, Applications of finite-time Lyapunov exponent in detecting Lagrangian coherent structures for coastal ocean processes: A review, Front. Mar. Sci. 11, 1345260 (2024).
  49. S. Hariri, Analysis of mixing structures in the Adriatic Sea using finite-size Lyapunov exponents, Geophys. Astrophys. Fluid Dyn. 116, 20 (2022).
  50. J. H. Bettencourt, C. López, and E. Hernández-García, Characterization of coherent structures in three-dimensional turbulent flows using the finite-size Lyapunov exponent, J. Phys. A: Math. Theor. 46, 254022 (2013).
  51. G. K. Vallis, Atmospheric and Oceanic Fluid Dynamics: Fundamentals and Large-Scale Circulation, 2nd ed. (Cambridge University Press, Cambridge, 2017).
  52. M. L. Waite and P. Bartello, Stratified turbulence dominated by vortical motion, J. Fluid Mech. 517, 281 (2004).
  53. A. Maffioli, G. Brethouwer, and E. Lindborg, Mixing efficiency in stratified turbulence, J. Fluid Mech. 794, R3 (2016).
  54. G. Brethouwer, P. Billant, E. Lindborg, and J.-M. Chomaz, Scaling analysis and simulation of strongly stratified turbulent flows, J. Fluid Mech. 585, 343 (2007).
  55. W. J. Emery, W. G. Lee, and L. Magaard, Geographic and seasonal distributions of Brunt–Väisälä frequency and Rossby Radii in the North Pacific and North Atlantic, J. Phys. Oceanogr. 14, 294 (1984).
  56. R. C. Millard, W. B. Owens, and N. P. Fofonoff, On the calculation of the Brunt-Väisäla frequency, Deep Sea Res. A. Oceanogr. Res. Papers 37, 167 (1990).
  57. K. Shahzadi and N. Pinardi, SeaDataCloud Brunt–Väisälä Frequency profiles for the Atlantic and Pacific Oceans, Ref. Product Information Document (PIDoc), SeaDataCloud 2021, https://doi.org/10.13155/79296.
  58. E. M. Stanley and R. C. Batten, Viscosity of sea water at moderate temperatures and pressures, J. Geophys. Res. 74, 3415 (1969).
  59. R.-C. Lien and M. C. Gregg, Observations of turbulence in a tidal beam and across a coastal ridge, J. Geophys. Res.: Oceans 106, 4575 (2001).
  60. J. Jiménez, Oceanic turbulence at millimeter scales, Oceanogr. Lit. Rev 3, 598 (1998).
  61. S. D. Murugan and S. S. Ray, Genesis of thermalization in the three-dimensional, incompressible, Galerkin-truncated Euler equation, Phys. Rev. Fluids 8, 084605 (2023).
  62. H. Hanazaki and J. C. R. Hunt, Linear processes in unsteady stably stratified turbulence, J. Fluid Mech. 318, 303 (1996).
  63. R. D. J. G. Ho, A. Berera, and D. Clark, Chaotic behavior of Eulerian magnetohydrodynamic turbulence, Phys. Plasmas 26, 042303 (2019).

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Phys. Rev. Fluids 11, 064615– Published 23 June, 2026
  • Received 2 January 2026
  • Accepted 8 May 2026
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