Incompressible Homogeneous Anisotropic Turbulence: Pure Rotation

  • Pierre Sagaut
  • Claude Cambon


This chapter is devoted to purely rotating turbulence. An exhaustive review of physical and numerical experiments is provided, and the limits of single-point models are investigated. The Coriolis force induces dispersive and anisotropic inertial waves: It is shown that, in the absence of energy production, their impact on dynamics is on the interscale energy transfer, with a particular relevance of triadic closures ranging from inertial wave turbulence to anisotropic EDQNM. The trend toward two-dimensionalization is quantified and discussed at length. A recent study using Quasi-Normal Scale Elimination is presented and discussed, new experimental results in terms of both frequency and wave vector are reported, for a support to the wave turbulence concept. Extension of wave turbulence to bounded flows is presented, together with a new discussion of the description and the active role of helicity. At last, a recent investigation of the threshold scale to recover isotropy at smallest scales is incorporated.


  1. Aupoix, B., Cousteix, J., Liandrat, J.: Effects of rotation on isotropic turbulence. In: Bradbury, L.J.S., Durst, F., Launder, B.E., Schmidt, F.W., Whitelaw, J.H. (eds.) Turbulent Shear Flows. Springer, New York (1983)Google Scholar
  2. Bardina, J., Ferziger, J.M., Rogallo, R.S.: Effect of rotation on isotropic turbulence: computation and modelling. J. Fluid Mech. 154, 321–326 (1985)ADSCrossRefGoogle Scholar
  3. Baroud, C.N., Plapp, B.B., Swinney, H.L.: Scaling in three-dimensional and quasi-two-dimensional rotating turbulent flows. Phys. Fluids 15(8), 2091–2104 (2003)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  4. Bartello, P., Métais, O., Lesieur, M.: Coherent structures in rotating three-dimensional turbulence. J. Fluid Mech. 273, 1–29 (1994)ADSMathSciNetCrossRefGoogle Scholar
  5. Bellet, F., Godeferd, F.S., Scott, J.F., Cambon, C.: Wave-turbulence in rapidly rotating flows. J. Fluid Mech. 562, 83–121 (2006)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  6. Benney, D.J., Newell, A.C.: Random wave closure. Stud. Appl. Math. 48, 29–53 (1969)CrossRefzbMATHGoogle Scholar
  7. Biferale, L., Bonaccorso, F., Mazzitelli, I.M., et al.: Coherent structures and extreme events in rotating multiphase turbulent flows. Phys. Rev. X 6, 041036 (2016)Google Scholar
  8. Bourouiba, L., Bartello, P.: The intermediate Rossby number range and 2D–3D transfers in rotating decaying homogeneous turbulence. J. Fluid Mech. 587, 131–161 (2007)ADSCrossRefzbMATHGoogle Scholar
  9. Callies, J., Ferrari, R., Buehler, O.: Transition from geostrophic turbulence to inertia-gravity waves in the atmospheric energy spectrum. PNAS 111(48), 17033–17038 (2014)ADSCrossRefGoogle Scholar
  10. Cambon, C.: Etude spectrale d’un champ turbulent incompressible soumis à des effets couplés de déformation et rotation imposés extérieurement. Thèse de Doctorat d’Etat, Université Lyon I, France (1982)Google Scholar
  11. Cambon, C., Jacquin, L.: Spectral approach to non-isotropic turbulence subjected to rotation. J. Fluid Mech. 202, 295–317 (1989)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  12. Cambon, C., Scott, J.F.: Linear and nonlinear models of anisotropic turbulence. Annu. Rev. Fluid Mech. 31, 1–53 (1999)ADSMathSciNetCrossRefGoogle Scholar
  13. Cambon, C., Jacquin, L., Lubrano, J.-L.: Towards a new Reynolds stress model for rotating turbulent flows. Phys. Fluids A 4, 812–824 (1992)ADSCrossRefzbMATHGoogle Scholar
  14. Cambon, C., Mansour, N.N., Godeferd, F.S.: Energy transfer in rotating turbulence. J. Fluid Mech. 337, 303–332 (1997)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  15. Cambon, C., Rubinstein, R., Godeferd, F.S.: Advances in wave-turbulence: rapidly rotating flows. New J. Phys. 6, 73 (2004)ADSMathSciNetCrossRefGoogle Scholar
  16. Cambon, C., Godeferd, F.S., Nicolleau, F., Vassilicos, J.C.: Turbulent diffusion in rapidly rotating flows with and without stable stratification. J. Fluid Mech. 499, 231–255 (2004)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  17. Chen, Q., Chen, S., Eyink, G.S., Holm, D.D.: Resonant interactions in rotating homogeneous three-dimensional turbulence. J. Fluid Mech. 542, 139–164 (2005)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  18. Davidson, P.A., Stapelhurst, P.J., Dalziel, S.B.: On the evolution of eddies in a rapidly rotating system. J. Fluid Mech. 557, 135–144 (2006)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  19. Delache, A., Cambon, C., Godeferd, F.S.: Scale by scale anisotropy in freely decaying rotating turbulence. Phys. Fluids 26, 025104 (2014)ADSCrossRefGoogle Scholar
  20. Dickinson, S.C., Long, R.R.: Oscillating grid-turbulence including effects of rotation. J. Fluid Mech. 126, 315–333 (1984)ADSCrossRefGoogle Scholar
  21. Galperin, B., Sukoriansky, S.: Turbulence in rotating fluids and the Nastrom and Gage spectrum. Invited talk Sixth International Conference Turbulence Mixing and Beyond, ICTP, Trieste (Italy), 14–18 Aug 2017Google Scholar
  22. Galtier, S.: A weak inertial wave-turbulence theory. Phys. Rev. E 68, 1–4 (2003)CrossRefGoogle Scholar
  23. Gence, J.N., Frick, C.: Naissance des corrélations triples de vorticité dans une turbulence statistiquement homogène soumise à une rotation. C. R. Acad. Sci. Paris Série II b 329(5), 351–356 (2001)ADSzbMATHGoogle Scholar
  24. Godeferd, F.S., Lollini, L.: Direct numerical simulations of turbulence with confinment and rotation. J. Fluid Mech. 393, 257–308 (1999)ADSCrossRefzbMATHGoogle Scholar
  25. Greenspan, H.P.: The Theory of Rotating Fluids. Cambridge University Press, Cambridge (1968)zbMATHGoogle Scholar
  26. Hopfinger, E.J., Browand, F.K., Gagne, Y.: Turbulence and waves in a rotating tank. J. Fluid Mech. 125, 505 (1982)ADSCrossRefGoogle Scholar
  27. Ibbetson, A., Tritton, D.: Experiments of rotation in a rotating fluid. J. Fluid Mech. 68, 639–672 (1975)ADSCrossRefGoogle Scholar
  28. Jacquin, L., Leuchter, O., Cambon, C., Mathieu, J.: Homogeneous turbulence in the presence of rotation. J. Fluid Mech. 220, 1–52 (1990)ADSCrossRefzbMATHGoogle Scholar
  29. Kaneda, Y., Ishida, T.: Suppression of vertical diffusion in strongly stratified turbulence. J. Fluid Mech. 402, 311–327 (2000)Google Scholar
  30. Kassinos, S.C., Reynolds, W.C., Rogers, M.M.: One-point turbulence structure tensors. J. Fluid Mech. 428, 213–248 (2001)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  31. Liechtenstein, L., Godeferd, F.S., Cambon, C.: Nonlinear formation of structures in rotating stratified turbulence. J. Turbul. 6, 1–18 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  32. Mansour, N.N., Cambon, C., Speziale, C.G.: Single point modelling of initially isotropic turbulence under uniform rotation. Center for Turbulence Research, Stanford University, Annual Research Briefs (1991)Google Scholar
  33. Mc Ewan, A.D.: Inertial oscillations in a rotating fluid cylinder. J. Fluid Mech. 40, 603–639 (1970)ADSCrossRefGoogle Scholar
  34. Mc Ewan, A.D.: Angular momentum diffusion and the initialization of cyclones. Nat. Ser. 260, 126 (1976)CrossRefGoogle Scholar
  35. Mininni, P., Rosenberg, D., Pouquet, A.: Isotropisation at small scales of rotating helically driven turbulence. J. Fluid Mech. 699, 263–279 (2012)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  36. Morinishi, Y., Nakabayashi, K., Ren, S.K.: Dynamics of anisotropy on decaying homogeneous turbulence subjected to system rotation. Phys. Fluids 13(10), 2912–2922 (2001)ADSCrossRefzbMATHGoogle Scholar
  37. Morize, C., Moisy, F., Rabaud, M.: Decaying grid-generated turbulence in a rotating tank. Phys. Fluids 17(9), 095105 (2005)ADSCrossRefzbMATHGoogle Scholar
  38. Mowbray, D.E., Rarity, B.S.H.: A theoretical and experimental investigation of the phase configuration of internal waves of small amplitude in a density stratified liquid. J. Fluid Mech. 28, 1–16 (1967)ADSCrossRefGoogle Scholar
  39. Nastrom, G.D., Gage, K.S.: A climatology of atmospheric wavenumber spectra of wind and temperature observed by commercial aircraft. J. Atmos. Sci. 42, 950–960 (1985)ADSCrossRefGoogle Scholar
  40. Orszag, S.A.: Analytical theories of turbulence. J. Fluid Mech. 41, 363–386 (1970)ADSCrossRefzbMATHGoogle Scholar
  41. Praud, O., Sommeria, J., Fincham, A.: Decaying grid turbulence in a rotating stratified fluid. J. Fluid Mech. 547, 389–412 (2006)ADSCrossRefzbMATHGoogle Scholar
  42. Proudman, J.: On the motion of solids in a liquid possessing vorticity. Proc. R. Soc. Lond. A 92, 408 (1916)ADSCrossRefzbMATHGoogle Scholar
  43. Salhi, A., Cambon, C.: Anisotropic phase-mixing in homogeneous turbulence in a rapidly rotating or in a strongly stratified fluid: an analytical study. Phys. Fluid 19, 055102 (2007)ADSCrossRefzbMATHGoogle Scholar
  44. Scott, J.F.: Wave turbulence in a rotating channel. J. Fluid Mech. 741, 316–349 (2014)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  45. Simand, C.: Etude de la turbulence au voisinage d’ un vortex, Ph.D. Thesis, Ecole Normale Supérieure de Lyon, defended on 28 Aug 2002 (2002)Google Scholar
  46. Skamarock, W.C., Park, S.-H., Klemp, J.B., Snyder, C.: Atmospheric kinetic energy spectra from global high-resolution nonhydrostatic simulations. J. Atmos. Sci. 71, 4369–4381 (2014)ADSCrossRefGoogle Scholar
  47. Smith, L.M., Lee, Y. : On near resonances and symmetry breaking in forced rotating flows at moderate Rossby number. J. Fluid Mech. 535, 111–142 (2005)Google Scholar
  48. Squires, K.D., Chasnov, J.R., Mansour, N.N., Cambon, C.: The asymptotic state of rotating homogeneous turbulence at high Reynolds number. In: 74 th Fluid Dynamics Symposium on Application of Direct and Large Eddy Simulation to Transition and Turbulence, Chania, Greece, AGARD CP 551, 4-1– 4-9 (2000)Google Scholar
  49. Staplehurst, P.J., Davidson, P.A., Dalziel, S.B.: Structure formation in homogeneous, freely-decaying, rotating turbulence. J. Fluid Mech. 598, 81–103 (2008)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  50. Sukoriansky, S., Galperin, B.: QNSE theory of turbulence anisotropization and onset of the inverse energy cascade by solid body rotation. J. Fluid Mech. 805, 384–421 (2016)ADSMathSciNetCrossRefGoogle Scholar
  51. Taylor, G.I.: Experiments on the motion of solid bodies in rotating fluids. Proc. R. Soc. London A 104, 213 (1921)ADSCrossRefGoogle Scholar
  52. Traugott, S.S.: Influence of solid body rotation on screen produced turbulence. NACA Technical Report 4135 (1958)Google Scholar
  53. van Bokhoven, L.J.A., Cambon, C., Liechtenstein, L., Godeferd, F.S., Clercx, H.J.H.: Refined vorticity statistics of decaying rotating three-dimensional turbulence. J. Turbul. 9, 1–24 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  54. Waleffe, F.: Inertial transfers in the helical decomposition. Phys. Fluids A 5, 677–685 (1993)ADSCrossRefzbMATHGoogle Scholar
  55. Wigeland, R.A., Nagib, H.M.: Grid generated turbulence with and without rotation about the stream wise direction. IIT Fluids and Heat Transfer Report R78-1 (1978)Google Scholar
  56. Woods, J.D.: Do waves limit turbulent diffusion in the ocean? Nature 288, 219–224 (1980)ADSCrossRefGoogle Scholar
  57. Yang, X., Domaradzki, J.A.: LES of decaying rotating turbulence. Phys. Fluids 16(11), 4088–4104 (2004)ADSCrossRefzbMATHGoogle Scholar
  58. Yarom, E., Sharon, E.: Experimental observation of steady inertial wave turbulence in deep rotating flows. Nat. Phys. 10(7), 510–514 (2014)CrossRefGoogle Scholar
  59. Zakharov, V.E., L’vov, V.S., Falkowich, G.: Kolmogoroff Spectra of Turbulence I. Wave Turbulence. Springer Series in Nonlinear Dynamics. Springer, Berlin (1992)CrossRefGoogle Scholar
  60. Zeman, O.: A note on the spectra and decay of rotating homogeneous turbulence. Phys. Fluids 6, 3221 (1994)ADSCrossRefzbMATHGoogle Scholar
  61. Zhou, Y.: A phenomenological treatment of rotating turbulence. Phys. Fluids 7(8), 2092–2094 (1995)ADSMathSciNetCrossRefzbMATHGoogle Scholar

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Authors and Affiliations

  1. 1.Laboratoire de Mécanique, Modélisation et Procédés Propres, UMR CNRS 7340, Ecole Centrale de MarseilleAix-Marseille UniversitéMarseilleFrance
  2. 2.Laboratoire de Mécanique des Fluides et d’Acoustique, UMR CNRS 5509Ecole Centrale de LyonÉcullyFrance

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