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Phase dynamics and their role determining energy flux in hydrodynamic shell models

Santiago J. Benavides1,2,* and Miguel D. Bustamante3,†

  • 1School of Aeronautics and Space Engineering, Universidad Politécnica de Madrid, Madrid, Spain
  • 2School of Mathematics and Maxwell Institute for Mathematical Sciences, University of Edinburgh, Edinburgh, United Kingdom
  • 3School of Mathematics and Statistics, University College Dublin, Belfield, Dublin 4, Ireland

  • *Contact author: Santiago.Benavides@ed.ac.uk
  • †Contact author: miguel.bustamante@ucd.ie
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Phys. Rev. Fluids 11, 044601 – Published 2 April, 2026

DOI: https://doi.org/10.1103/y9cr-4698

Abstract

The transfer of energy and other conserved quantities across scales, also known as flux or spectral flux, is a central aspect of out-of-equilibrium systems such as turbulent hydrodynamic flows. Despite its role in the few predictive theories that exist, a dynamical understanding of what determines said flux (and its direction in scale) has yet to be established. In this study we work towards this understanding by investigating how the dynamics of complex Fourier velocity phases influence the flux of conserved quantities in hydrodynamic shell models. The phase dynamics, like energy evolution, are influenced by contributions from all neighboring triads, making the full problem intractable. Instead, we assume that the dynamics of the triad phases are determined solely by the so-called self-interaction term and treat the other neighboring triad terms as noise. This transforms the phase dynamics into that of a noisy phase oscillator, which we solve analytically to predict phase statistics. We validate this assumption with a suite of shell model simulations. Our results give us analytical predictions for the energy flux, when the energy spectrum is given. We prove that all shell models that conserve energy along with a sign-indefinite quadratic quantity (this includes three-dimensional turbulence analogs) undergo a forward energy cascade and further show that the phase dynamics prevent the shell model analog of two-dimensional turbulence from forming an inverse cascade of energy.

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References (89)

  1. U. Frisch, Turbulence: The Legacy of A. N. Kolmogorov (Cambridge University Press, Cambridge, 1995).
  2. J. Jiménez, Cascades in wall-bounded turbulence, Annu. Rev. Fluid Mech. 44, 27 (2012).
  3. P. A. Davidson, Turbulence in Rotating, Stratified and Electrically Conducting Fluids (Cambridge University Press, Cambridge, 2013).
  4. A. Alexakis and L. Biferale, Cascades and transitions in turbulent flows, Phys. Rep. 767–769, 1 (2018).
  5. M. K. Verma, Energy Transfers in Fluid Flows: Multiscale and Spectral Perspectives (Cambridge University Press, Cambridge, 2019).
  6. G. Boffetta and R. E. Ecke, Two-dimensional turbulence, Annu. Rev. Fluid Mech. 44, 427 (2012).
  7. J. Marston and S. Tobias, Recent developments in theories of inhomogeneous and anisotropic turbulence, Annu. Rev. Fluid Mech. 55, 351 (2023).
  8. G. L. Mellor and T. Yamada, Development of a turbulence closure model for geophysical fluid problems, Rev. Geophys. 20, 851 (1982).
  9. E. Kalnay, Atmospheric Modeling, Data Assimilation and Predictability (Cambridge University Press, Cambridge, 2003).
  10. C. Wunsch and R. Ferrari, Vertical mixing, energy, and the general circulation of the oceans, Annu. Rev. Fluid Mech. 36, 281 (2004).
  11. T. Dauxois, T. Peacock, P. Bauer, C. P. Caulfield, C. Cenedese, C. Gorlé, G. Haller, G. N. Ivey, P. F. Linden, E. Meiburg, N. Pinardi, N. M. Vriend, and A. W. Woods, Confronting grand challenges in environmental fluid mechanics, Phys. Rev. Fluids 6, 020501 (2021).
  12. B. A. Storer, M. Buzzicotti, H. Khatri, S. M. Griffies, and H. Aluie, Global energy spectrum of the general oceanic circulation, Nat. Commun. 13, 5314 (2022).
  13. G. L. Eyink, Multi-scale gradient expansion of the turbulent stress tensor, J. Fluid Mech. 549, 159 (2006).
  14. N. A. K. Doan, N. Swaminathan, P. A. Davidson, and M. Tanahashi, Scale locality of the energy cascade using real space quantities, Phys. Rev. Fluids 3, 084601 (2018).
  15. M. Carbone and A. D. Bragg, Is vortex stretching the main cause of the turbulent energy cascade? J. Fluid Mech. 883, R2 (2020).
  16. P. L. Johnson, Energy transfer from large to small scales in turbulence by multiscale nonlinear strain and vorticity interactions, Phys. Rev. Lett. 124, 104501 (2020).
  17. P. L. Johnson, On the role of vorticity stretching and strain self-amplification in the turbulence energy cascade, J. Fluid Mech. 922, A3 (2021).
  18. P. L. Johnson and M. Wilczek, Multiscale velocity gradients in turbulence, Annu. Rev. Fluid Mech. 56, 463 (2024).
  19. D. Park and A. Lozano-Durán, The coherent structure of the energy cascade in isotropic turbulence, Sci. Rep. 15, 14 (2025).
  20. R. H. Kraichnan, Inertial ranges in two‐dimensional turbulence, Phys. Fluids 10, 1417 (1967).
  21. R. H. Kraichnan, Inertial-range transfer in two- and three-dimensional turbulence, J. Fluid Mech. 47, 525 (1971).
  22. J. C. Mcwilliams, The vortices of two-dimensional turbulence, J. Fluid Mech. 219, 361 (1990).
  23. S. Chen, R. E. Ecke, G. L. Eyink, M. Rivera, M. Wan, and Z. Xiao, Physical mechanism of the two-dimensional inverse energy cascade, Phys. Rev. Lett. 96, 084502 (2006).
  24. G. L. Eyink, A turbulent constitutive law for the two-dimensional inverse energy cascade, J. Fluid Mech. 549, 191 (2006).
  25. Z. Xiao, M. Wan, S. Chen, and G. L. Eyink, Physical mechanism of the inverse energy cascade of two-dimensional turbulence: A numerical investigation, J. Fluid Mech. 619, 1 (2009).
  26. J. Jiménez, Collective organization and screening in two-dimensional turbulence, Phys. Rev. Fluids 6, 084601 (2021).
  27. J. G. Ballouz and N. T. Ouellette, Tensor geometry in the turbulent cascade, J. Fluid Mech. 835, 1048 (2018).
  28. J. G. Ballouz and N. T. Ouellette, Geometric constraints on energy transfer in the turbulent cascade, Phys. Rev. Fluids 5, 034603 (2020).
  29. P. Vieillefosse, Local interaction between vorticity and shear in a perfect incompressible fluid, J. Phys. (Paris) 43, 837 (1982).
  30. P. Vieillefosse, Internal motion of a small element of fluid in an inviscid flow, Physica A 125, 150 (1984).
  31. A. van Kan, Phase transitions in anisotropic turbulence, Chaos 34, 122103 (2024).
  32. R. Fjørtoft, On the changes in the spectral distribution of kinetic energy for twodimensional, nondivergent flow, Tellus A 5, 225 (1953).
  33. E. Hopf, Statistical hydromechanics and functional calculus, J. Ration. Mech. Anal. 1, 87 (1952).
  34. T. D. Lee, On some statistical properties of hydrodynamical and magneto-hydrodynamical fields, Q. Appl. Math. 10, 69 (1952).
  35. R. H. Kraichnan, Helical turbulence and absolute equilibrium, J. Fluid Mech. 59, 745 (1973).
  36. R. H. Kraichnan and D. Montgomery, Two-dimensional turbulence, Rep. Prog. Phys. 43, 547 (1980).
  37. R. H. Kraichnan, Statistical dynamics of two-dimensional flow, J. Fluid Mech. 67, 155 (1975).
  38. L. Bourouiba, Model of a truncated fast rotating flow at infinite Reynolds number, Phys. Fluids 20, 075112 (2008).
  39. C. Herbert, A. Pouquet, and R. Marino, Restricted equilibrium and the energy cascade in rotating and stratified flows, J. Fluid Mech. 758, 374 (2014).
  40. L. M. Smith and F. Waleffe, Transfer of energy to two-dimensional large scales in forced, rotating three-dimensional turbulence, Phys. Fluids 11, 1608 (1999).
  41. L. J. A. van Bokhoven, H. J. H. Clercx, G. J. F. van Heijst, and R. R. Trieling, Experiments on rapidly rotating turbulent flows, Phys. Fluids 21, 096601 (2009).
  42. E. Yarom, Y. Vardi, and E. Sharon, Experimental quantification of inverse energy cascade in deep rotating turbulence, Phys. Fluids 25, 085105 (2013).
  43. M. Buzzicotti, H. Aluie, L. Biferale, and M. Linkmann, Energy transfer in turbulence under rotation, Phys. Rev. Fluids 3, 034802 (2018).
  44. F. Waleffe, The nature of triad interactions in homogeneous turbulence, Phys. Fluids 4, 350 (1992).
  45. F. Waleffe, Inertial transfers in the helical decomposition, Phys. Fluids 5, 677 (1993).
  46. L. Biferale, S. Musacchio, and F. Toschi, Inverse energy cascade in three-dimensional isotropic turbulence, Phys. Rev. Lett. 108, 164501 (2012).
  47. L. Biferale, S. Musacchio, and F. Toschi, Split energy–helicity cascades in three-dimensional homogeneous and isotropic turbulence, J. Fluid Mech. 730, 309 (2013).
  48. A. Alexakis, Helically decomposed turbulence, J. Fluid Mech. 812, 752 (2017).
  49. G. Sahoo, A. Alexakis, and L. Biferale, Discontinuous transition from direct to inverse cascade in three-dimensional turbulence, Phys. Rev. Lett. 118, 164501 (2017).
  50. A. Alexakis and L. Biferale, λ-Navier–Stokes turbulence, Philos. Trans. R. Soc. A 380, 20210243 (2022).
  51. H. K. Moffatt, Note on the triad interactions of homogeneous turbulence, J. Fluid Mech. 741, R3 (2014).
  52. L. Biferale, M. Buzzicotti, and M. Linkmann, From two-dimensional to three-dimensional turbulence through two-dimensional three-component flows, Phys. Fluids 29, 111101 (2017).
  53. K. Seshasayanan, S. J. Benavides, and A. Alexakis, On the edge of an inverse cascade, Phys. Rev. E 90, 051003(R) (2014).
  54. K. Seshasayanan and A. Alexakis, Critical behavior in the inverse to forward energy transition in two-dimensional magnetohydrodynamic flow, Phys. Rev. E 93, 013104 (2016).
  55. S. J. Benavides and A. Alexakis, Critical transitions in thin layer turbulence, J. Fluid Mech. 822, 364 (2017).
  56. T. Pestana and S. Hickel, Regime transition in the energy cascade of rotating turbulence, Phys. Rev. E 99, 053103 (2019).
  57. A. van Kan and A. Alexakis, Critical transition in fast-rotating turbulence within highly elongated domains, J. Fluid Mech. 899, A33 (2020).
  58. M. Buzzicotti, B. P. Murray, L. Biferale, and M. D. Bustamante, Phase and precession evolution in the Burgers equation, Eur. Phys. J. E 39, 34 (2016).
  59. S. Moradi, B. Teaca, and J. Anderson, Role of phase synchronisation in turbulence, AIP Adv. 7, 115213 (2017).
  60. B. P. Murray and M. D. Bustamante, Energy flux enhancement, intermittency and turbulence via Fourier triad phase dynamics in the 1-D Burgers equation, J. Fluid Mech. 850, 624 (2018).
  61. J.-A. Arguedas-Leiva, E. Carroll, L. Biferale, M. Wilczek, and M. D. Bustamante, Minimal phase-coupling model for intermittency in turbulent systems, Phys. Rev. Res. 4, L032035 (2022).
  62. B. Protas, D. Kang, and M. D. Bustamante, Alignments of triad phases in extreme one-dimensional Burgers flows, Phys. Rev. E 109, 055104 (2024).
  63. C. Wang, L. Fang, Z. Wang, and C. Xu, Role of Fourier phase dynamics in decaying turbulence, Phys. Rev. Fluids 9, 114603 (2024).
  64. D. Kang, B. Protas, and M. D. Bustamante, Alignments of triad phases in 1D Burgers and 3D Navier-Stokes flows, arXiv:2105.09425.
  65. J. M. Reynolds-Barredo, D. E. Newman, P. W. Terry, and R. Sanchez, Fourier signature of filamentary vorticity structures in two-dimensional turbulence, Europhys. Lett. 115, 34002 (2016).
  66. L. Biferale, Shell models of energy cascade in turbulence, Annu. Rev. Fluid Mech. 35, 441 (2003).
  67. P. D. Ditlevsen, Turbulence and Shell Models (Cambridge University Press, Cambridge, 2010).
  68. E. Aurell, G. Boffetta, A. Crisanti, P. Frick, G. Paladin, and A. Vulpiani, Statistical mechanics of shell models for two-dimensional turbulence, Phys. Rev. E 50, 4705 (1994).
  69. P. D. Ditlevsen and I. A. Mogensen, Cascades and statistical equilibrium in shell models of turbulence, Phys. Rev. E 53, 4785 (1996).
  70. T. Gilbert, V. S. L'vov, A. Pomyalov, and I. Procaccia, Inverse cascade regime in shell models of two-dimensional turbulence, Phys. Rev. Lett. 89, 074501 (2002).
  71. O. D. Gürcan, P. Morel, S. Kobayashi, R. Singh, S. Xu, and P. H. Diamond, Logarithmic discretization and systematic derivation of shell models in two-dimensional turbulence, Phys. Rev. E 94, 033106 (2016).
  72. A. A. Mailybaev, Hidden scale invariance of intermittent turbulence in a shell model, Phys. Rev. Fluids 6, L012601 (2021).
  73. A. A. Mailybaev, Shell model intermittency is the hidden self-similarity, Phys. Rev. Fluids 7, 034604 (2022).
  74. X. M. de Wit, G. Ortali, A. Corbetta, A. A. Mailybaev, L. Biferale, and F. Toschi, Extreme statistics and extreme events in dynamical models of turbulence, Phys. Rev. E 109, 055106 (2024).
  75. E. Carroll, Fourier phase symchronization in minimal models for turbulence, Ph.D. thesis, University College Dublin, 2023.
  76. L. Manfredini and Ö. D. Gürcan, Nonlinear phase synchronization and the role of spacing in shell models Phys. Rev. E 113, 015101 (2026).
  77. R. Benzi, L. Biferale, and G. Parisi, On intermittency in a cascade model for turbulence, Physica D 65, 163 (1993).
  78. V. S. L'vov, E. Podivilov, A. Pomyalov, I. Procaccia, and D. Vandembroucq, Improved shell model of turbulence, Phys. Rev. E 58, 1811 (1998).
  79. G. L. Eyink, S. Chen, and Q. Chen, Gibbsian hypothesis in turbulence, J. Stat. Phys. 113, 719 (2003).
  80. L. Schimansky-Geier and C. Zülicke, Harmonic noise: Effect on bistable systems, Z. Phys. B 79, 451 (1990).
  81. N. G. Van Kampen, Stochastic Processes in Physics and Chemistry (Elsevier, Amsterdam, 1992), Vol. 1.
  82. G. B. Arfken, H. J. Weber, and F. E. Harris, Chapter 14 - Bessel Functions, Mathematical Methods for Physicists (Seventh Edition) (Academic Press, Boston, 2013), pp. 643–713.
  83. S. J. Benavides, s-benavides/MHD_shell_model: MHD_shell_model v1.0Z (Z for zenodo) Zenodo, Geneva (2025), https://zenodo.org/records/15800547.
  84. J. Y. N. Cho and E. Lindborg, Horizontal velocity structure functions in the upper troposphere and lower stratosphere: 1. Observations, J. Geophys. Res.: Atmos. 106, 10223 (2001).
  85. L. Sorriso-Valvo, R. Marino, V. Carbone, A. Noullez, F. Lepreti, P. Veltri, R. Bruno, B. Bavassano, and E. Pietropaolo, Observation of inertial energy cascade in interplanetary space plasma, Phys. Rev. Lett. 99, 115001 (2007).
  86. B. Qiu, T. Nakano, S. Chen, and P. Klein, Bi-directional energy cascades in the Pacific Ocean from equator to Subarctic Gyre, Geophys. Res. Lett. 49, e2022GL097713 (2022).
  87. D. Balwada, J.-H. Xie, R. Marino, and F. Feraco, Direct observational evidence of an oceanic dual kinetic energy cascade and its seasonality, Sci. Adv. 8, eabq2566 (2022).
  88. R. Marino and J.-H. Xie, Flux equipartition in astrophysical plasma turbulence, Sci. Adv. 11, eadv8988 (2025).
  89. S. J. Benavides and M. D. Bustamante, Data and scripts for figures in Benavides & Bustamante “Phase dynamics and their role determining energy flux in hydrodynamic shell models”, Figshare, London (2026), https://doi.org/10.6084/m9.figshare.29469659.

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Phys. Rev. Fluids 11, 044601– Published 2 April, 2026
  • Received 11 August 2025
  • Revised 21 November 2025
  • Accepted 24 February 2026
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