- 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

Cascade of mesostrophy in turbulence with reduced vortex stretching

Wouter J. T. Bos

PDF

Phys. Rev. Fluids 11, 074605 – Published 17 July, 2026

DOI: https://doi.org/10.1103/qhck-4dky

Abstract

In three-dimensional turbulence, vortex stretching is the central mechanism enabling the transfer of kinetic energy toward smaller scales. If vortex stretching is absent in three-dimensional turbulence, energy is no longer conserved and enstrophy cascades to smaller scales. In this paper we propose a system that interpolates between these two cases by reducing the strength of vortex stretching in the Navier-Stokes equations. The resulting dynamics yield a cascade that is neither a pure energy cascade nor a pure enstrophy cascade. We refer to this process as a mesostrophy cascade. We formulate a consistent picture describing this cascade and the characteristic scales involved in the mesostrophy balance. We illustrate this picture via numerical integrations of the eddy-damped quasinormal Markovian model with reduced vortex stretching.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (63)

  1. G. I. Taylor, Statistical theory of turbulence, Proc. R. Soc. London, Ser. A 151, 421 (1935).
  2. L. Onsager, Statistical hydrodynamics, Il Nuovo Cimento 6, 279 (1949).
  3. A. N. Kolmogorov, The local structure of turbulence in incompressible viscous fluid for very large Reynolds numbers, Dokl. Akad. Nauk. SSSR 30, 301 (1941).
  4. L. F. Richardson, Atmospheric diffusion shown on a distance-neighbour graph, Proc. R. Soc. London, Ser. A 110, 709 (1926).
  5. U. Frisch, Turbulence, the Legacy of A. N. Kolmogorov (Cambridge University Press, Cambridge, UK, 1995).
  6. P. A. Davidson, K. Morishita, and Y. Kaneda, On the generation and flux of enstrophy in isotropic turbulence, J. Turbul. 9, N42 (2008).
  7. P. Baj, F. A. Portela, and D. W. Carter, On the simultaneous cascades of energy, helicity, and enstrophy in incompressible homogeneous turbulence, J. Fluid Mech. 952, A20 (2022).
  8. W. J. T. Bos, Three-dimensional turbulence without vortex stretching, J. Fluid Mech. 915, A121 (2021).
  9. T. Wu and W. J. T. Bos, Cascades of enstrophy and helicity in turbulence without vortex stretching, Phys. Rev. Fluids 7, 094601 (2022).
  10. G. K. Batchelor, Computation of the energy spectrum in homogeneous two-dimensional turbulence, Phys. Fluids 12, II-233 (1969).
  11. C. E. Leith, Diffusion approximation for two-dimensional turbulence, Phys. Fluids 11, 671 (1968).
  12. R. H. Kraichnan, Inertial ranges in two-dimensional turbulence, Phys. Fluids 10, 1417 (1967).
  13. T. Wu and W. J. T. Bos, Statistical mechanics of the Euler equations without vortex stretching, J. Fluid Mech. 929, A11, (2021).
  14. G. Joyce and D. Montgomery, Negative temperature states for the two-dimensional guiding center plasma, J. Plasma Phys. 10, 107 (1973).
  15. T. Wu, T. David, and W. J. T. Bos, Point-vortex statistical mechanics applied to turbulence without vortex stretching, J. Stat. Mech. (2023) 113203.
  16. A. Brissaud, U. Frisch, J. Léorat, M. Lesieur, and A. Mazure, Helicity cascades in fully developed isotropic turbulence, Phys. Fluids 16, 1366 (1973).
  17. J. C. André and M. Lesieur, Influence of helicity on the evolution of isotropic turbulence at high Reynolds number, J. Fluid Mech. 81, 187 (1977).
  18. Q. Chen, S. Chen, and G. L. Eyink, The joint cascade of energy and helicity in three-dimensional turbulence, Phys. Fluids 15, 361 (2003).
  19. F. Plunian, A. Teimurazov, R. Stepanov, and M. K. Verma, Inverse cascade of energy in helical turbulence, J. Fluid Mech. 895, A13 (2020).
  20. W. J. T. Bos, X. Shao, T. Wu, and L. Fang, The role of vortex stretching in drag reduction of polymer-laden turbulent flow, J. Fluid Mech. 1019, A32 (2025).
  21. T. von Kármán, Mechanische aenlichkeit und turbulenz, Nachr. Ges. Wiss. Goettingen, Math.-Phys. Kl. 1930, 58 (1930).
  22. B. E Owolabi, D. J. C. Dennis, and R. J. Poole, Turbulent drag reduction by polymer additives in parallel-shear flows, J. Fluid Mech. 827, R4 (2017).
  23. P. S. Virk, Drag reduction fundamentals, AIChE J. 21, 625 (1975).
  24. M. E. McCormick and R. Bhattacharyya, Drag reduction of a submersible hull by electrolysis, Nav. Eng. J. 85, 11 (1973).
  25. J.-L. Marié, A simple analytical formulation for microbubble drag reduction, Physicochem. Hydrodyn. 8, 213 (1987).
  26. X. Shen, S. L Ceccio, and M. Perlin, Influence of bubble size on micro-bubble drag reduction, Exp. Fluids 41, 415 (2006).
  27. J. Mathieu and J. F. Scott, An Introduction to Turbulent flow (Cambridge University Press, Cambridge, UK, 2000).
  28. P. Sagaut and C. Cambon, Homogeneous Turbulence Dynamics (Springer, New York, 2008), Vol. 10.
  29. L. S. G. Kovaznay, Spectrum of locally isotropic turbulence, J. Aeronaut. Sci. 15, 745 (1948).
  30. R. Rubinstein and T. Clark, Reassessment of the Classical Turbulence Closures (Cambridge Scholars Publishing, Newcastle upon Tyne, England, 2022).
  31. G. K. Batchelor, The theory of Homogeneous Turbulence (Cambridge University Press, Cambridge, UK, 1953).
  32. F. Liu, L. Fang, and L. Shao, The role of velocity derivative skewness in understanding non-equilibrium turbulence, Chin. Phys. B 29, 114702 (2020).
  33. C. E. Leith, Diffusion approximation to inertial energy transfer in isotropic turbulence, Phys. Fluids 10, 1409 (1967).
  34. V. S. L'vov and S. Nazarenko, Differential model for 2D turbulence, JETP Lett. 83, 541 (2006).
  35. W. Heisenberg, Zur statistischen Theorie der Turbulenz, Z. Phys. 124, 628 (1948).
  36. R. Rubinstein and T. T. Clark, A generalized Heisenberg model for turbulent spectral dynamics, Theor. Comput. Fluid Dyn. 17, 249 (2004).
  37. R. H. Kraichnan, The structure of isotropic turbulence at very high Reynolds numbers, J. Fluid Mech. 5, 497 (1959).
  38. R. H. Kraichnan, Lagrangian-history closure approximation for turbulence, Phys. Fluids 8, 575 (1965).
  39. S. A. Orszag, Analytical theories of turbulence, J. Fluid Mech. 41, 363 (1970).
  40. M. Lesieur, Turbulence in Fluids (Kluwer, Dordrecht, 1990).
  41. U. Frisch, M. Lesieur, and P. L. Sulem, Crossover dimensions for fully developed turbulence, Phys. Rev. Lett. 37, 1312 (1976).
  42. W. J. T. Bos and L. Fang, Dependence of turbulent advection on the Lagrangian correlation time, Phys. Rev. E 91, 043020 (2015).
  43. A. Briard, L. Biferale, and T. Gomez, Closure theory for the split energy-helicity cascades in homogeneous isotropic homochiral turbulence, Phys. Rev. Fluids 2, 102602(R) (2017).
  44. T. Gotoh, Y. Watanabe, Y. Shiga, T. Nakano, and E. Suzuki, Statistical properties of four-dimensional turbulence, Phys. Rev. E 75, 016310 (2007).
  45. D. Clark, R. D. J. G. Ho, and A. Berera, Effect of spatial dimension on a model of fluid turbulence, J. Fluid Mech. 912, A40 (2021).
  46. G. Sahoo, A. Alexakis, and L. Biferale, Discontinuous transition from direct to inverse cascade in three-dimensional turbulence, Phys. Rev. Lett. 118, 164501 (2017).
  47. U. Frisch, A. Pomyalov, I. Procaccia, and S. S. Ray, Turbulence in noninteger dimensions by fractal Fourier decimation, Phys. Rev. Lett. 108, 074501 (2012).
  48. A. Alexakis and L. Biferale, Cascades and transitions in turbulent flows, Phys. Rep. 767, 1 (2018).
  49. R. H. Kraichnan, Inertial-range transfer in two-and three-dimensional turbulence, J. Fluid Mech. 47, 525 (1971).
  50. G. Boffetta and S. Musacchio, Evidence for the double cascade scenario in two-dimensional turbulence, Phys. Rev. E 82, 016307 (2010).
  51. L. Mydlarski and Z. Warhaft, On the onset of high-Reynolds-number grid-generated wind tunnel turbulence, J. Fluid Mech. 320, 331 (1996).
  52. W. J. T. Bos, L. Chevillard, J. F. Scott, and R. Rubinstein, Reynolds number effect on the velocity increment skewness in isotropic turbulence, Phys. Fluids 24, 015108 (2012).
  53. S. Tang, L. Danaila, and R. A. Antonia, Finite Reynolds number effect on small-scale statistics in decaying grid turbulence, Atmosphere 15, 540 (2024).
  54. T. D. Lee, On some statistical properties of hydrodynamical and magnetohydrodynamical fields, Quart. Appl. Math. 10, 69 (1952).
  55. H. K. Moffatt, The degree of knottedness of tangled vortex lines, J. Fluid Mech. 35, 117 (1969).
  56. R. H. Kraichnan, Helical turbulence and absolute equilibrium, J. Fluid Mech. 59, 745 (1973).
  57. A. Briard and T. Gomez, Dynamics of helicity in homogeneous skew-isotropic turbulence, J. Fluid Mech. 821, 539 (2017).
  58. A. Venaille, T. Dauxois, and S. Ruffo, Violent relaxation in two-dimensional flows with varying interaction range, Phys. Rev. E 92, 011001(R) (2015).
  59. W. Agoua, X.-Y. Yin, T. Wu, and Wouter J. T. Bos, Coexistence of two equilibrium configurations in two-dimensional turbulence, Phys. Rev. Fluids 10, 034604 (2025).
  60. E. J. Hinch, Mechanical models of dilute polymer solutions in strong flows, Phys. Fluids 20, S22 (1977).
  61. S. Douady, Y. Couder, and M. E. Brachet, Direct observation of the intermittency of intense vorticity filaments in turbulence, Phys. Rev. Lett. 67, 983 (1991).
  62. P. C. Valente, C. D. Silva, and F. T. Pinho, The effect of viscoelasticity on the turbulent kinetic energy cascade, J. Fluid Mech. 760, 39 (2014).
  63. P. C. Valente, C. B. da Silva, and F. T. Pinho, Energy spectra in elasto-inertial turbulence, Phys. Fluids 28, 075108 (2016).

Outline

Information

Phys. Rev. Fluids 11, 074605– Published 17 July, 2026
  • Received 24 September 2025
  • Accepted 3 June 2026
Crossmark - Check for updates button

©2026 American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation