1932

Abstract

Understanding inhomogeneous and anisotropic fluid flows requires mathematical and computational tools that are tailored to such flows and distinct from methods used to understand the canonical problem of homogeneous and isotropic turbulence. We review some recent developments in the theory of inhomogeneous and anisotropic turbulence, placing special emphasis on several kinds of quasi-linear approximations and their corresponding statistical formulations. Aspects of quasi-linear theory that have received insufficient attention in the literature are discussed, and open questions are framed.

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2023-01-19
2024-04-19
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Literature Cited

  1. Ait-Chaalal F, Schneider T, Meyer B, Marston J. 2016. Cumulant expansions for atmospheric flows. New J. Phys. 18:2025019
    [Google Scholar]
  2. Allawala A, Marston JB. 2016. Statistics of the stochastically-forced Lorenz attractor by the Fokker-Planck equation and cumulant expansions. Phys. Rev. E 94:052218
    [Google Scholar]
  3. Allawala A, Tobias SM, Marston JB. 2020. Dimensional reduction of direct statistical simulation. J. Fluid Mech. 898:533–18
    [Google Scholar]
  4. Arakawa A. 1966. Computational design for long-term numerical integration of the equations of fluid motion: two-dimensional incompressible flow. Part I. J. Comput. Phys. 1:119–43
    [Google Scholar]
  5. Bakas NA, Ioannou PJ. 2011. Structural stability theory of two-dimensional fluid flow under stochastic forcing. J. Fluid Mech. 682:332–61
    [Google Scholar]
  6. Bakas NA, Ioannou PJ. 2013. Emergence of large scale structure in barotropic β-plane turbulence. Phys. Rev. Lett. 110:22224501
    [Google Scholar]
  7. Bakas NA, Ioannou PJ. 2014. A theory for the emergence of coherent structures in beta-plane turbulence. J. Fluid Mech. 740:312–41
    [Google Scholar]
  8. Barkley D. 2006. Linear analysis of the cylinder wake mean flow. Europhys. Lett. 75:5750–56
    [Google Scholar]
  9. Barkley D. 2016. Theoretical perspective on the route to turbulence in a pipe. J. Fluid Mech. 803:15–80
    [Google Scholar]
  10. Batchelor GK. 1953. The Theory of Homogeneous Turbulence Cambridge, UK: Cambridge Univ. Press
  11. Batchelor GK, Proudman I. 1954. The effect of rapid distortion of a fluid in turbulent motion. Q. J. Mech. Appl. Math. 7:83–103
    [Google Scholar]
  12. Bengana Y, Tuckerman LS. 2021. Frequency prediction from exact or self-consistent mean flows. Phys. Rev. Fluids 6:6063901
    [Google Scholar]
  13. Bouchet F, Laurie J, Zaboronski O 2014. Langevin dynamics, large deviations and instantons for the quasi-geostrophic model and two-dimensional Euler equations. J. Stat. Phys. 156:61066–92
    [Google Scholar]
  14. Bouchet F, Marston JB, Tangarife T. 2018. Fluctuations and large deviations of Reynolds stresses in zonal jet dynamics. Phys. Fluids 30:015110–20
    [Google Scholar]
  15. Bouchet F, Nardini C, Tangarife T. 2013. Kinetic theory of jet dynamics in the stochastic barotropic and 2D Navier-Stokes equations. J. Stat. Phys. 153:4572–625
    [Google Scholar]
  16. Bretheim JU, Meneveau C, Gayme DF. 2015. Standard logarithmic mean velocity distribution in a band-limited restricted nonlinear model of turbulent flow in a half-channel. Phys. Fluids 27:011702
    [Google Scholar]
  17. Canet L, Delamotte B, Wschebor N. 2016. Fully developed isotropic turbulence: nonperturbative renormalization group formalism and fixed-point solution. Phys. Rev. E 93:6299–26
    [Google Scholar]
  18. Chen N, Majda AJ. 2017. Beating the curse of dimension with accurate statistics for the Fokker–Planck equation in complex turbulent systems. PNAS 114:4912864–69
    [Google Scholar]
  19. Child A, Hollerbach R, Marston B, Tobias S 2016. Generalised quasilinear approximation of the helical magnetorotational instability. J. Plasma Phys. 82:3905820302
    [Google Scholar]
  20. Chini GP, Malecha Z, Dreeben TD. 2014. Large-amplitude acoustic streaming. J. Fluid Mech. 744:329–51
    [Google Scholar]
  21. Chini GP, Michel G, Julien K, Rocha CB, Caulfield CcP 2022. Exploiting self-organized criticality in strongly stratified turbulence. J. Fluid Mech. 933:A22
    [Google Scholar]
  22. Connaughton C, Nazarenko S, Quinn B. 2015. Rossby and drift wave turbulence and zonal flows: the Charney–Hasegawa–Mima model and its extensions. Phys. Rep. 604:1–71
    [Google Scholar]
  23. Constantinou NC, Farrell BF, Ioannou PJ. 2014. Emergence and equilibration of jets in beta-plane turbulence: applications of stochastic structural stability theory. J. Atmos. Sci. 71:1818–42
    [Google Scholar]
  24. Constantinou NC, Farrell BF, Ioannou PJ. 2016. Statistical state dynamics of jet–wave coexistence in barotropic beta-plane turbulence. J. Atmos. Sci. 73:52229–53
    [Google Scholar]
  25. Davidson PA. 2015. Turbulence: An Introduction for Scientists and Engineers Oxford: Oxford Univ. Press
  26. Delplace P, Marston JB, Venaille A. 2017. Topological origin of equatorial waves. Science 358:63661075–77
    [Google Scholar]
  27. Delsole T, Farrell BF. 2017. The quasi-linear equilibration of a thermally maintained, stochastically excited jet in a quasigeosrophic model. J. Atmos. Sci. 53:131781–97
    [Google Scholar]
  28. Domaradzki JA, Orszag SA. 1987. Numerical solutions of the direct interaction approximation equations for anisotropic turbulence. J. Sci. Comput. 2:227–48
    [Google Scholar]
  29. Dupuis N, Canet L, Eichhorn A, Metzner W, Pawlowski J et al. 2021. The nonperturbative functional renormalization group and its applications. Phys. Rep. 910:1–114
    [Google Scholar]
  30. Eyink GL, Sreenivasan KR. 2006. Onsager and the theory of hydrodynamic turbulence. Rev. Mod. Phys. 78:87–135
    [Google Scholar]
  31. Falkovich G, Gawdzki K, Vergassola M. 2001. Particles and fields in fluid turbulence. Rev. Mod. Phys. 73:4913–75
    [Google Scholar]
  32. Farrell BF, Ioannou PJ. 1993. Stochastic forcing of the linearized Navier–Stokes equations. Phys. Fluids 5:2600–9
    [Google Scholar]
  33. Farrell BF, Ioannou PJ. 2003. Structural stability of turbulent jets. J. Atmos. Sci. 60:2101–18
    [Google Scholar]
  34. Farrell BF, Ioannou PJ. 2007. Structure and spacing of jets in barotropic turbulence. J. Atmos. Sci. 64:3652–65
    [Google Scholar]
  35. Farrell BF, Ioannou PJ, Jiménez J, Constantinou NC, Lozano-Durán A, Nikolaidis MA. 2016. A statistical state dynamics-based study of the structure and mechanism of large-scale motions in plane Poiseuille flow. J. Fluid Mech. 809:290–315
    [Google Scholar]
  36. Frederiksen JS, Okane TJ. 2018. Markovian inhomogeneous closures for Rossby waves and turbulence over topography. J. Fluid Mech. 858:45–70
    [Google Scholar]
  37. Fried BD, Gell-Mann M, Jackson JD, Wyld HW 1960. Longitudinal plasma oscillations in an electric field. J. Nuclear Energy C 1:4190
    [Google Scholar]
  38. Frisch U. 1995. Turbulence: The Legacy of A.N. Kolmogorov Cambridge, UK: Cambridge Univ. Press
  39. Frishman A. 2017. The culmination of an inverse cascade: mean flow and fluctuations. Phys. Fluids 29:12125102
    [Google Scholar]
  40. Frishman A, Herbert C. 2018. Turbulence statistics in a two-dimensional vortex condensate. Phys. Rev. Lett. 120:20204505
    [Google Scholar]
  41. Hasselmann K. 1966. Feynman diagrams and interaction rules of wave–wave scattering processes. Rev. Geophys. 4:1–32
    [Google Scholar]
  42. Held IM, Phillips PJ. 1987. Linear and nonlinear barotropic decay on the sphere. J. Atmos. Sci. 44:200–7
    [Google Scholar]
  43. Hernández CG, Yang Q, Hwang Y. 2022a. Generalised quasilinear approximations of turbulent channel flow. Part 1. Streamwise nonlinear energy transfer. J. Fluid Mech. 936:A33
    [Google Scholar]
  44. Hernández CG, Yang Q, Hwang Y. 2022b. Generalised quasilinear approximations of turbulent channel flow. Part 2. Spanwise scale interactions. J. Fluid Mech. 944:A34
    [Google Scholar]
  45. Herring JR. 1963. Investigation of problems in thermal convection. J. Atmos. Sci. 20:4325–38
    [Google Scholar]
  46. Hunt JCR, Carruthers DJ. 1990. Rapid distortion theory and the ‘problems’ of turbulence. J. Fluid Mech. 212:497–532
    [Google Scholar]
  47. Hwang Y, Cossu C. 2010a. Amplification of coherent streaks in the turbulent Couette flow: an input–output analysis at low Reynolds number. J. Fluid Mech. 643:333–48
    [Google Scholar]
  48. Hwang Y, Cossu C. 2010b. Linear non-normal energy amplification of harmonic and stochastic forcing in the turbulent channel flow. J. Fluid Mech. 664:51–73
    [Google Scholar]
  49. Kaspi Y, Galanti E, Showman AP, Stevenson DJ, Guillot T et al. 2020. Comparison of the deep atmospheric dynamics of Jupiter and Saturn in light of the Juno and Cassini gravity measurements. Space Sci. Rev. 216:584
    [Google Scholar]
  50. Kellam C. 2019. Generalized quasilinear simulation of turbulent channel flow PhD Thesis, Univ. N.H. Durham, N.H:.
  51. Kraichnan RH. 1959. The structure of isotropic turbulence at very high Reynolds numbers. J. Fluid Mech. 5:497–543
    [Google Scholar]
  52. Kraichnan RH. 1961. Dynamics of nonlinear stochastic systems. J. Math. Phys. 2:124–48. Erratum. 1962. J. Math. Phys. 3:205
    [Google Scholar]
  53. Kraichnan RH. 1964. Decay of isotropic turbulence in the direct-interaction approximation. Phys. Fluids 7:1030–48
    [Google Scholar]
  54. Kraichnan RH. 1980. Realizability inequalities and closed moment equations. Ann. N.Y. Acad. Sci. 357:37–46
    [Google Scholar]
  55. Kraichnan RH 1985. Decimated amplitude equations in turbulence dynamics. Theoretical Approaches to Turbulence DL Dwoyer, MY Hussaini, RG Voight 91–135 New York: Springer
    [Google Scholar]
  56. Krause F, Raedler K. 1980. Mean-Field Magnetohydrodynamics and Dynamo Theory Oxford: Pergamon
  57. Laurie J, Boffetta G, Falkovich G, Kolokolov I, Lebedev V. 2014. Universal profile of the vortex condensate in two-dimensional turbulence. Phys. Rev. Lett. 113:25254503
    [Google Scholar]
  58. Laurie J, Bouchet F 2015. Computation of rare transitions in the barotropic quasi-geostrophic equations. New J. Phys. 17:015009
    [Google Scholar]
  59. Ledoux P, Schwarzchild M, Spiegel EA. 1961. On the spectrum of turbulent convection. Astrophys. J. 133:184–97
    [Google Scholar]
  60. Legras B. 1980. Turbulent phase shift of Rossby waves. Geophys. Astrophys. Fluid Dyn. 15:253–81
    [Google Scholar]
  61. Li K, Marston J, Saxena S, Tobias SM. 2021a. Direct statistical simulation of the Lorenz63 system. Chaos 32:043111
    [Google Scholar]
  62. Li K, Marston JB, Tobias SM. 2021b. Direct statistical simulation of low-order dynamo systems. Proc. R. Soc. A 477:20210427
    [Google Scholar]
  63. Malkus WVR. 1954. The heat transport and spectrum of thermal turbulence. Proc. R. Soc. A 225:1161196–212
    [Google Scholar]
  64. Mantic-Lugo V, Arratia C, Gallaire F. 2015. A self-consistent model for the saturation dynamics of the vortex shedding around the mean flow in the unstable cylinder wake. Phys. Fluids 27:7074103
    [Google Scholar]
  65. Markeviciute VK, Kerswell RR. 2022. Improved assessment of the statistical stability of turbulent flows using extended Orr-Sommerfeld stability analysis. arXiv:2201.01540 [physics.flu-dyn]
  66. Marston JB. 2010. Statistics of the general circulation from cumulant expansions. Chaos 20:4041107
    [Google Scholar]
  67. Marston JB. 2012. Planetary atmospheres as nonequilibrium condensed matter. Annu. Rev. Condensed Matter Phys. 3:285–310
    [Google Scholar]
  68. Marston JB, Chini GP, Tobias SM. 2016. Generalized quasilinear approximation: application to zonal jets. Phys. Rev. Lett. 116:21214501
    [Google Scholar]
  69. Marston JB, Conover E, Schneider T. 2008. Statistics of an unstable barotropic jet from a cumulant expansion. J. Atmos. Sci. 65:61955–66
    [Google Scholar]
  70. Marston JB, Qi W, Tobias SM. 2019a. Direct statistical simulation of a jet. arXiv:1412.0381v2 [physics.flu-dyn]
  71. Marston JB, Qi W, Tobias SM 2019b. Direct statistical simulation of a jet. Zonal Jets: Phenomenology, Genesis and Physics B Galerpin, PL Read 332–46 Cambridge, UK: Cambridge Univ. Press
    [Google Scholar]
  72. McKeon BJ, Sharma AS. 2010. A critical-layer framework for turbulent pipe flow. J. Fluid Mech. 658:336–82
    [Google Scholar]
  73. Meliga P. 2017. Harmonics generation and the mechanics of saturation in flow over an open cavity: a second-order self-consistent description. J. Fluid Mech. 826:503–21
    [Google Scholar]
  74. Michel G, Chini GP. 2019. Multiple scales analysis of slow–fast quasi-linear systems. Proc. R. Soc. A 475:20180630
    [Google Scholar]
  75. Moffatt H, Dormy E. 2019. Self-Exciting Fluid Dynamos Cambridge, UK: Cambridge Univ. Press
  76. Nikolaidis MA, Ioannou PJ, Farrell BF, Lozano-Durán A. 2021. POD-based study of structure and dynamics in turbulent plane Poiseuille flow: comparing quasi-linear simulations to DNS. arXiv:2109.02465 [physics.flu-dyn]
  77. Nivarti GV, Kerswell RR, Marston JB, Tobias SM. 2022. Non-equivalence of quasilinear dynamical systems and their statistical closures. arXiv:2202.04127 [physics.flu-dyn]
  78. Noerdlinger PD. 1963. Quasi-linear theory of plasma oscillations in an electric field. Phys. Fluids 6:81196–98
    [Google Scholar]
  79. O'Gorman PA, Schneider T. 2007. Recovery of atmospheric flow statistics in a general circulation model without nonlinear eddy-eddy interactions. Geophys. Res. Lett. 34:L22801
    [Google Scholar]
  80. Okane TJ, Frederiksen JS. 2004. The QDIA and regularized QDIA closures for inhomogeneous turbulence over topography. J. Fluid Mech. 504:133–65
    [Google Scholar]
  81. Orszag SA. 1970. Analytical theories of turbulence. J. Fluid Mech. 41:363–86
    [Google Scholar]
  82. Orszag SA 1977. Lectures on the statistical theory of turbulence. Fluid Dynamics: Les Houches 1973 R Balian, J-L Peube 235–74 London: Gordon Breach Sci. Pub.
    [Google Scholar]
  83. Parker JB. 2021. Topological phase in plasma physics. J. Plasma Phys. 87:2835870202
    [Google Scholar]
  84. Parker JB, Krommes JA. 2013. Zonal flow as pattern formation. Phys. Plasmas 20:10100703
    [Google Scholar]
  85. Parker JB, Krommes JA. 2014. Generation of zonal flows through symmetry breaking of statistical homogeneity. New J. Phys. 16:035006
    [Google Scholar]
  86. Parker JB, Marston JB, Tobias SM, Zhu Z. 2020. Topological gaseous plasmon polariton in realistic plasma. Phys. Rev. Lett. 124:19195001
    [Google Scholar]
  87. Pausch M, Yang Q, Hwang Y, Eckhardt B. 2019. Quasilinear approximation for exact coherent states in parallel shear flows. Fluid Dyn. Res. 51:011402
    [Google Scholar]
  88. Plumb RA. 1977. The interaction of two internal waves with the mean flow: implications for the theory of the quasi-biennial oscillation. J. Atmos. Sci. 34:121847–58
    [Google Scholar]
  89. Plumley M, Calkins MA, Julien K, Tobias SM. 2018. Self-consistent single mode investigations of the quasi-geostrophic convection-driven dynamo model. J. Plasma Phys. 84:4735840406
    [Google Scholar]
  90. Plummer A, Marston JB, Tobias SM. 2019. Joint instability and abrupt nonlinear transitions in a differentially rotating plasma. J. Plasma Phys. 85:905850113
    [Google Scholar]
  91. Pope SB. 2000. Turbulent Flows Cambridge, UK: Cambridge Univ. Press
  92. Saad Y. 2003. Iterative Methods for Sparse Linear Systems New York: SIAM. , 2nd ed..
  93. Scott RK, Dritschel DG. 2012. The structure of zonal jets in geostrophic turbulence. J. Fluid Mech. 711:576–98
    [Google Scholar]
  94. Shepherd TG. 1990. Symmetries, conservation laws, and Hamiltonian structure in geophysical fluid dynamics. Adv. Geophys. 32:287–38
    [Google Scholar]
  95. Shih HY, Hsieh TL, Goldenfeld N. 2015. Ecological collapse and the emergence of travelling waves at the onset of shear turbulence. Nat. Phys. 12:3245–48
    [Google Scholar]
  96. Skitka J, Marston JB, Fox-Kemper B. 2020. Reduced-order quasilinear model of ocean boundary-layer turbulence. J. Phys. Oceanogr. 50:3537–58
    [Google Scholar]
  97. Spears BK, Brase J, Bremer PT, Chen B, Field J et al. 2018. Deep learning: a guide for practitioners in the physical sciences. Phys. Plasmas 25:8080901
    [Google Scholar]
  98. Spiegel EA. 1962. Thermal turbulence at very small Prandtl number. J. Geophys. Res. 67:83063–70
    [Google Scholar]
  99. Squire J, Bhattacharjee A. 2015. Statistical simulation of the magnetorotational dynamo. Phys. Rev. Lett. 114:085002
    [Google Scholar]
  100. Srinivasan K, Young WR. 2012. Zonostrophic instability. J. Atmos. Sci. 69:1633–56
    [Google Scholar]
  101. Thomas VL, Farrell BF, Ioannou PJ, Gayme DF. 2015. A minimal model of self-sustaining turbulence. Phys. Fluids 27:10105104
    [Google Scholar]
  102. Thomas VL, Lieu BK, Jovanović MR, Farrell BF, Ioannou PJ, Gayme DF. 2014. Self-sustaining turbulence in a restricted nonlinear model of plane Couette flow. Phys. Fluids 26:10105112
    [Google Scholar]
  103. Thomson W. 1880. On gravitational oscillations of rotating water. Proc. R. Soc. Edinb. 10:92–100
    [Google Scholar]
  104. Tobias S. 2021. The turbulent dynamo. J. Fluid Mech. 912:P1
    [Google Scholar]
  105. Tobias S, Dagon K, Marston J. 2011. Astrophysical fluid dynamics via direct statistical simulation. Astrophys. J. 727:2127
    [Google Scholar]
  106. Tobias S, Marston JB. 2013. Direct statistical simulation of out-of-equilibrium jets. Phys. Rev. Lett. 110:10104502
    [Google Scholar]
  107. Tobias SM, Marston JB. 2017. Three-dimensional rotating Couette flow via the generalised quasilinear approximation. J. Fluid Mech. 810:412–28
    [Google Scholar]
  108. Tobias SM, Oishi JS, Marston JB. 2018. Generalized quasilinear approximation of the interaction of convection and mean flows in a thermal annulus. Proc. R. Soc. Lond. A 474:221920180422
    [Google Scholar]
  109. Touchette H. 2009. The large deviation approach to statistical mechanics. Phys. Rep. 478:1–31–69
    [Google Scholar]
  110. Tretiak K, Plumley M, Calkins M, Tobias S 2022. Efficiency gains of a multi-scale integration method applied to a scale-separated model for rapidly rotating dynamos. Comput. Phys. Commun. 273:108253
    [Google Scholar]
  111. Turton SE, Tuckerman LS, Barkley D. 2015. Prediction of frequencies in thermosolutal convection from mean flows. Phys. Rev. E 91:4043009
    [Google Scholar]
  112. Vedenov AA. 1963. Quasi-linear theory of a plasma. Sov. At. Energy 13:591–612
    [Google Scholar]
  113. Vedenov AA, Velikhov EP, Sagdeev RZ. 1961. Quasilinear theory of plasma oscillations. Proceedings of the IAEA Conference on Plasma Physics and Controlled Nuclear Fusion Research465–75 Vienna: IAEA
    [Google Scholar]
  114. Venaille A, Delplace P. 2021. Wave topology brought to the coast. Phys. Rev. Res. 3:4043002
    [Google Scholar]
  115. Venturi D. 2018. The numerical approximation of nonlinear functionals and functional differential equations. Phys. Rep. 732:1–102
    [Google Scholar]
  116. Willis AP, Kerswell RR. 2007. Turbulent dynamics of pipe flow captured in a reduced model: puff relaminarization and localized ‘edge’ states. J. Fluid Mech. 619:213–33
    [Google Scholar]
  117. Woillez E, Bouchet F. 2017. Theoretical prediction of Reynolds stresses and velocity profiles for barotropic turbulent jets. EPL 118:554002–7
    [Google Scholar]
  118. Yaglom AM. 1994. A.N. Kolmogorov as a fluid mechanician and founder of a school in turbulence research. Annu. Rev. Fluid Mech. 26:1–23
    [Google Scholar]
  119. Yokoi N. 2018. Turbulence, transport and reconnection Lectures given at CISM Courses and Lectures: Advanced Topics in MHD June 11–15, Udine, Italy
  120. Zare A, Jovanović MR, Georgiou TT. 2017. Colour of turbulence. J. Fluid Mech. 812:636–80
    [Google Scholar]
  121. Zhang C, Lawrence A, Marston B, Kushner PJ. 2019. Infinite U(1) symmetry of the quasi-linear approximation Paper presented at 22nd Conference on Atmospheric and Oceanic Fluid Dynamics June 24–28 Portland, ME:
  122. Zhou Y. 2021. Turbulence theories and statistical closure approaches. Phys. Rep. 935:1–117
    [Google Scholar]
  123. Zhu Z, Li C, Marston JB. 2021. Topology of rotating stratified fluids with and without background shear flow. arXiv:2112.04691 [physics.flu-dyn]
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