Hostname: page-component-cd9895bd7-q99xh Total loading time: 0 Render date: 2024-12-28T06:09:55.310Z Has data issue: false hasContentIssue false
  • English
  • Français

Restricted equilibrium and the energy cascade in rotating and stratified flows

Published online by Cambridge University Press:  09 October 2014

Corentin Herbert ,
Annick Pouquet  and
Raffaele Marino
Corentin Herbert*
Affiliation:
National Center for Atmospheric Research, PO Box 3000, Boulder, CO 80307, USA
Annick Pouquet
Affiliation:
National Center for Atmospheric Research, PO Box 3000, Boulder, CO 80307, USA Laboratory for Atmospheric and Space Physics, University of Colorado, Boulder, CO 80309, USA
Raffaele Marino
Affiliation:
National Center for Atmospheric Research, PO Box 3000, Boulder, CO 80307, USA Institute for Chemical-Physical Processes - IPCF/CNR, Rende (CS), 87036, Italy
*
†Email address for correspondence: cherbert@ucar.edu
Get access Rights & Permissions [Opens in a new window]

Abstract

Most turbulent flows appearing in nature (e.g. geophysical and astrophysical flows) are subjected to strong rotation and stratification. These effects break the symmetries of classical, homogenous isotropic turbulence. In doing so, they introduce a natural decomposition of phase space in terms of wave modes and potential vorticity modes. The appearance of a new time scale, associated with the propagation of waves, hinders the understanding of energy transfers across scales. For instance, it is difficult to predict a priori whether the energy cascades downscale as in homogeneous isotropic turbulence or upscale as expected from balanced dynamics. In this paper, we suggest a theoretical approach based on equilibrium statistical mechanics for the ideal system, inspired by the restricted partition function formalism introduced in metastability studies. We focus on the qualitative features of the inviscid system, taking into account either all the modes or just the slow modes. Specifically, we show that at absolute equilibrium, i.e. when all the modes are considered, no negative temperature states exist, and the isotropic energy spectrum is close to equipartition. By contrast, when the statistics is restricted to the contributions of the slow modes, we find that in the presence of rotation, there exists a regime of negative temperature featuring an infrared divergence in both the isotropic and the axisymmetric average energy spectrum, characteristic of an inverse cascade regime. Such regimes are not allowed for purely stratified flows, even in the restricted ensemble, because the slow manifold then partitions into modes that carry potential vorticity on the one hand, and hydrostatically balanced but vorticity-free modes, the so-called vertical shear horizontal flows, on the other hand, which forbid the appearance of negative temperatures.

JFM classification

Turbulent Flows: Rotating turbulence Turbulent Flows: Stratified turbulence Turbulent Flows: Turbulence theory
Type
Papers
Information
Journal of Fluid Mechanics , Volume 758 , 10 November 2014 , pp. 374 - 406
Copyright
© 2014 Cambridge University Press 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Aluie, H. & Kurien, S. 2011 Joint downscale fluxes of energy and potential enstrophy in rotating stratified Boussinesq flows. Europhys. Lett. 96 (4), 44006.CrossRefGoogle Scholar
Baer, F. & Tribbia, J. J. 1977 On complete filtering of gravity modes through nonlinear initialization. Mon. Weath. Rev. 105, 1536–1539.2.0.CO;2>CrossRefGoogle Scholar
Bartello, P. 1995 Geostrophic adjustment and inverse cascades in rotating stratified turbulence. J. Atmos. Sci. 52, 4410–4428.2.0.CO;2>CrossRefGoogle Scholar
Bartello, P. & Tobias, S. M. 2013 Sensitivity of stratified turbulence to the buoyancy Reynolds number. J. Fluid Mech. 725, 1–22.CrossRefGoogle Scholar
Billant, P. & Chomaz, J.-M. 2001 Self-similarity of strongly stratified inviscid flows. Phys. Fluids 13 (6), 1645–1651.CrossRefGoogle Scholar
Boffetta, G. & Ecke, R. E. 2012 Two-dimensional turbulence. Annu. Rev. Fluid Mech. 44, 427–451.CrossRefGoogle Scholar
Bouchet, F. 2008 Simpler variational problems for statistical equilibria of the 2D Euler equation and other systems with long range interactions. Physica D 237, 1976–1981.CrossRefGoogle Scholar
Bouchet, F. & Venaille, A. 2012 Statistical mechanics of two-dimensional and geophysical flows. Phys. Rep. 515, 227–295.CrossRefGoogle Scholar
Bourouiba, L. 2008 Model of a truncated fast rotating flow at infinite Reynolds number. Phys. Fluids 20, 075112.CrossRefGoogle Scholar
Brethouwer, G., Billant, P., Lindborg, E. & Chomaz, J. M. 2007 Scaling analysis and simulation of strongly stratified turbulent flows. J. Fluid Mech. 585, 343–368.CrossRefGoogle Scholar
Campa, A., Dauxois, T. & Ruffo, S. 2009 Statistical mechanics and dynamics of solvable models with long-range interactions. Phys. Rep. 480, 57–159.CrossRefGoogle Scholar
Capocaccia, D., Cassandro, M. & Olivieri, E. 1974 A study of metastability in the Ising model. Commun. Math. Phys. 39 (3), 185–205.CrossRefGoogle Scholar
Charney, J. G. 1971 Geostrophic turbulence. J. Atmos. Sci. 28, 1087–1094.2.0.CO;2>CrossRefGoogle Scholar
Chen, Q., Chen, S. & Eyink, G. 2003 The joint cascade of energy and helicity in three-dimensional turbulence. Phys. Fluids 15, 361–374.CrossRefGoogle Scholar
Chen, Q., Chen, S., Eyink, G. L. & Holm, D. D. 2005 Resonant interactions in rotating homogeneous three-dimensional turbulence. J. Fluid Mech. 542, 139–164.CrossRefGoogle Scholar
Craya, A.1958 Contribution à l’analyse de la turbulence associée à des vitesses moyennes. Tech. Rep. Publ. Sci. Tech. Ministère de l’Air 345. Ministère de l’Air, Paris, France.Google Scholar
Dauxois, T., Ruffo, S., Arimondo, E. & Wilkens, M.(Eds) 2002 Dynamics and Thermodynamics of Systems with Long Range Interactions, Lecture Notes in Physics, vol. 602. Springer.CrossRefGoogle Scholar
Delache, A., Cambon, C. & Godeferd, F. 2014 Scale by scale anisotropy in freely decaying rotating turbulence. Phys. Fluids 26 (2), 025104.CrossRefGoogle Scholar
Dhar, D. & Lebowitz, J. L. 2010 Restricted equilibrium ensembles: exact equation of state of a model glass. Europhys. Lett. 92 (2), 20008.CrossRefGoogle Scholar
Dubrulle, B. & Valdettaro, L. 1992 Consequences of rotation in energetics of accretion disks. Astron. Astrophys. 263, 387–400.Google Scholar
Ellis, R. S., Haven, K. & Turkington, B. 2000 Large deviation principles and complete equivalence and nonequivalence results for pure and mixed ensembles. J. Stat. Phys. 101, 999–1064.CrossRefGoogle Scholar
Embid, P. F. & Majda, A. J. 1998 Low Froude number limiting dynamics for stably stratified flow with small or finite Rossby numbers. Geophys. Astrophys. Fluid Dyn. 87 (1–2), 1–50.CrossRefGoogle Scholar
Errico, R. M. 1984 The statistical equilibrium solution of a primitive-equation model. Tellus A 36 (1), 42–51.CrossRefGoogle Scholar
Fox, D. G. & Orszag, S. A. 1973 Inviscid dynamics of two-dimensional turbulence. Phys. Fluids 16, 169–171.CrossRefGoogle Scholar
Godeferd, F. S. & Cambon, C. 1994 Detailed investigation of energy transfers in homogeneous stratified turbulence. Phys. Fluids 6 (6), 2084–2100.CrossRefGoogle Scholar
Godoy-Diana, R., Chomaz, J.-M. & Billant, P. 2004 Vertical length scale selection for pancake vortices in strongly stratified viscous fluids. J. Fluid Mech. 504, 229–238.CrossRefGoogle Scholar
Herbert, C. 2013 Additional invariants and statistical equilibria for the 2D Euler equations on a spherical domain. J. Stat. Phys. 152, 1084–1114.CrossRefGoogle Scholar
Herbert, C. 2014 Restricted partition functions and inverse energy cascades in parity symmetry breaking flows. Phys. Rev. E 89, 013010.CrossRefGoogle ScholarPubMed
Herbert, C., Dubrulle, B., Chavanis, P.-H. & Paillard, D. 2012a Phase transitions and marginal ensemble equivalence for freely evolving flows on a rotating sphere. Phys. Rev. E 85, 056304.CrossRefGoogle ScholarPubMed
Herbert, C., Dubrulle, B., Chavanis, P.-H. & Paillard, D. 2012b Statistical mechanics of quasi-geostrophic flows on a rotating sphere. J. Stat. Mech. 2012, P05023.CrossRefGoogle Scholar
Herring, J. R. 1974 Approach of axisymmetric turbulence to isotropy. Phys. Fluids 17, 859–872.CrossRefGoogle Scholar
Herring, J. R. 1977 On the statistical theory of two-dimensional topographic turbulence. J. Atmos. Sci. 34, 1731–1750.2.0.CO;2>CrossRefGoogle Scholar
Herring, J. R., Kerr, R. M. & Rotunno, R. 1994 Ertel’s potential vorticity in unstratified turbulence. J. Atmos. Sci. 51, 35–47.2.0.CO;2>CrossRefGoogle Scholar
Herring, J. R. & Métais, O. 1989 Numerical experiments in forced stably stratified turbulence. J. Fluid Mech. 202 (1), 97–115.CrossRefGoogle Scholar
Holloway, G. 1986 Eddies, waves, circulation, and mixing: statistical geofluid mechanics. Annu. Rev. Fluid Mech. 18, 91–147.CrossRefGoogle Scholar
Julien, K., Knobloch, E., Milliff, R. & Werne, J. 2006 Generalized quasi-geostrophy for spatially anisotropic rotationally constrained flows. J. Fluid Mech. 555, 233–274.CrossRefGoogle Scholar
Khinchin, A. 1949 The Mathematical Foundations of Statistical Mechanics. Dover.Google Scholar
Kiessling, M. K. H. & Lebowitz, J. L. 1997 The micro-canonical point vortex ensemble: beyond equivalence. Lett. Math. Phys. 42, 43–56.CrossRefGoogle Scholar
Kolmogorov, A. N. 1941 The local structure of turbulence in incompressible viscous fluid for very large Reynolds’ numbers. Dokl. Akad. Nauk SSSR 30, 301–305.Google Scholar
Kraichnan, R. H. 1967 Inertial ranges in two-dimensional turbulence. Phys. Fluids 10, 1417–1423.CrossRefGoogle Scholar
Kraichnan, R. H. 1973 Helical turbulence and absolute equilibrium. J. Fluid Mech. 59, 745–752.CrossRefGoogle Scholar
Kraichnan, R. H. 1975 Statistical dynamics of two-dimensional flow. J. Fluid Mech. 67, 155–175.CrossRefGoogle Scholar
Kraichnan, R. H. & Montgomery, D. C. 1980 Two-dimensional turbulence. Rep. Prog. Phys. 43, 547–619.CrossRefGoogle Scholar
Lamriben, C., Cortet, P.-P. & Moisy, F. 2011 Direct measurements of anisotropic energy transfers in a rotating turbulence experiment. Phys. Rev. Lett. 107 (2), 024503.CrossRefGoogle Scholar
Lee, T. D. 1952 On some statistical properties of hydrodynamical and magneto-hydrodynamical fields. Q. Appl. Maths 10, 69–74.CrossRefGoogle Scholar
Leith, C. E. 1980 Nonlinear normal mode initialization and quasi-geostrophic theory. J. Atmos. Sci. 37, 958–968.2.0.CO;2>CrossRefGoogle Scholar
Lindborg, E. 2006 The energy cascade in a strongly stratified fluid. J. Fluid Mech. 550, 207–242.CrossRefGoogle Scholar
Lindborg, E. & Brethouwer, G. 2007 Stratified turbulence forced in rotational and divergent modes. J. Fluid Mech. 586, 83–108.CrossRefGoogle Scholar
Lorenz, E. N. 1980 Attractor sets and quasi-geostrophic equilibrium. J. Atmos. Sci. 37, 1685–1699.2.0.CO;2>CrossRefGoogle Scholar
Lorenz, E. N. 1986 On the existence of a slow manifold. J. Atmos. Sci. 43, 1547–1558.2.0.CO;2>CrossRefGoogle Scholar
Lorenz, E. N. 1992 The slow manifold—what is it? J. Atmos. Sci. 49, 2449–2451.2.0.CO;2>CrossRefGoogle Scholar
Lorenz, E. N. & Krishnamurthy, V. 1987 On the nonexistence of a slow manifold. J. Atmos. Sci. 44, 2940–2950.2.0.CO;2>CrossRefGoogle Scholar
Lucarini, V., Blender, R., Herbert, C., Pascale, S., Ragone, F. & Wouters, J.2014 Mathematical and physical ideas for climate science. Rev. Geophys. in press arXiv:1311.1190.CrossRefGoogle Scholar
Machenhauer, B. 1977 On the dynamics of gravity oscillations in a shallow water model, with application to normal mode initialization. Beitr. Phys. Atmos. 50, 253–271.Google Scholar
Mahalov, A. & Zhou, Y. 1996 Analytical and phenomenological studies of rotating turbulence. Phys. Fluids 8 (8), 2138–2152.CrossRefGoogle Scholar
Marino, R., Mininni, P. D., Rosenberg, D. & Pouquet, A. 2013 Inverse cascades in rotating stratified turbulence: fast growth of large scales. Europhys. Lett. 102, 44006.CrossRefGoogle Scholar
McWilliams, J. C. 2006 Fundamentals of Geophysical Fluid Dynamics. Cambridge University Press.Google Scholar
Merryfield, W. J. 1998 Effects of stratification on quasi-geostrophic inviscid equilibria. J. Fluid Mech. 354, 345–356.CrossRefGoogle Scholar
Merryfield, W. J. & Holloway, G. 1996 Inviscid quasi-geostrophic flow over topography: testing statistical mechanical theory. J. Fluid Mech. 309, 85–91.CrossRefGoogle Scholar
Métais, O., Bartello, P., Garnier, E., Riley, J. J. & Lesieur, M. 1996 Inverse cascade in stably stratified rotating turbulence. Dyn. Atmos. Oceans 23 (1), 193–203.CrossRefGoogle Scholar
Miller, J. 1990 Statistical mechanics of Euler equations in two dimensions. Phys. Rev. Lett. 65, 2137–2140.CrossRefGoogle ScholarPubMed
Mininni, P. D., Dmitruk, P., Matthaeus, W. H. & Pouquet, A. 2011 Large-scale behavior and statistical equilibria in rotating flows. Phys. Rev. E 83, 016309.CrossRefGoogle ScholarPubMed
Mininni, P. D. & Pouquet, A. 2010a Rotating helical turbulence. I. Global evolution and spectral behavior. Phys. Fluids 22, 035105.CrossRefGoogle Scholar
Mininni, P. D. & Pouquet, A. 2010b Rotating helical turbulence. II. Intermittency, scale invariance, and structures. Phys. Fluids 22, 035106.CrossRefGoogle Scholar
Mininni, P. D., Rosenberg, D. & Pouquet, A. 2012 Isotropization at small scales of rotating helically driven turbulence. J. Fluid Mech. 699, 263–279.CrossRefGoogle Scholar
Montgomery, D. & Turner, L. 1982 Two-and-a-half-dimensional magnetohydrodynamic turbulence. Phys. Fluids 25 (2), 345–349.CrossRefGoogle Scholar
Nazarenko, S. V. 2010 Wave Turbulence, Lecture Notes in Physics, vol. 825. Springer.Google Scholar
Newell, A. C. & Rumpf, B. 2011 Wave turbulence. Annu. Rev. Fluid Mech. 43 (1), 59–78.CrossRefGoogle Scholar
Pedlosky, J. 1987 Geophysical Fluid Dynamics. Springer.CrossRefGoogle Scholar
Penrose, O. & Lebowitz, J. L. 1971 Rigorous treatment of metastable states in the van der Waals–Maxwell theory. J. Stat. Phys. 3 (2), 211–236.CrossRefGoogle Scholar
Penrose, O. & Lebowitz, J. L. 1979 Towards a rigorous molecular theory of metastability. In Fluctuation Phenomena (ed. Montroll, E. W. & Lebowitz, J. L.), chap. 5, pp. 293–340. North-Holland.CrossRefGoogle Scholar
Polzin, K. L. & Lvov, Y. V. 2011 Toward regional characterizations of the oceanic internal wavefield. Rev. Geophys. 49 (4), RG4003.CrossRefGoogle Scholar
Pouquet, A. & Marino, R. 2013 Geophysical turbulence and the duality of the energy flow across scales. Phys. Rev. Lett. 111, 234501.CrossRefGoogle Scholar
Pouquet, A. & Mininni, P. D. 2010 The interplay between helicity and rotation in turbulence: implications for scaling laws and small-scale dynamics. Phil. Trans. R. Soc. A 368, 1635–1662.CrossRefGoogle ScholarPubMed
Rhines, P. B. 1979 Geostrophic turbulence. Annu. Rev. Fluid Mech. 11, 401–441.CrossRefGoogle Scholar
Richardson, L. F. 1922 Weather Prediction by Numerical Process. Cambridge University Press.Google Scholar
Riley, J. J. & de Bruyn Kops, S. M. 2003 Dynamics of turbulence strongly influenced by buoyancy. Phys. Fluids 15, 2047–2059.CrossRefGoogle Scholar
Robert, R. & Sommeria, J. 1991 Statistical equilibrium states for two-dimensional flows. J. Fluid Mech. 229, 291–310.CrossRefGoogle Scholar
Salmon, R. 1998 Lectures on Geophysical Fluid Dynamics. Oxford University Press.CrossRefGoogle Scholar
Salmon, R. 2010 The shape of the main thermocline, revisited. J. Mar. Res. 68 (3–4), 541–568.CrossRefGoogle Scholar
Salmon, R., Holloway, G. & Hendershott, M. C. 1976 The equilibrium statistical mechanics of simple quasi-geostrophic models. J. Fluid Mech. 75, 691–703.CrossRefGoogle Scholar
Seyler, C. E. Jr, Salu, Y., Montgomery, D. & Knorr, G. 1975 Two-dimensional turbulence in inviscid fluids or guiding center plasmas. Phys. Fluids 18, 803–813.CrossRefGoogle Scholar
Smith, L. M., Chasnov, J. R. & Waleffe, F. 1996 Crossover from two- to three-dimensional turbulence. Phys. Rev. Lett. 77 (12), 2467–2470.CrossRefGoogle ScholarPubMed
Smith, L. M. & Waleffe, F. 1999 Transfer of energy to two-dimensional large scales in forced, rotating three-dimensional turbulence. Phys. Fluids 11 (6), 1608–1622.CrossRefGoogle Scholar
Smith, L. M. & Waleffe, F. 2002 Generation of slow large scales in forced rotating stratified turbulence. J. Fluid Mech. 451, 145–168.CrossRefGoogle Scholar
Staquet, C. & Riley, J. J. 1989 On the velocity field associated with potential vorticity. Dyn. Atmos. Oceans 14, 93–123.CrossRefGoogle Scholar
Teitelbaum, T. & Mininni, P. D. 2009 Effect of helicity and rotation on the free decay of turbulent flows. Phys. Rev. Lett. 103, 14501.CrossRefGoogle ScholarPubMed
Touchette, H., Ellis, R. S. & Turkington, B. 2004 An introduction to the thermodynamic and macrostate levels of nonequivalent ensembles. Physica A 340, 138–146.CrossRefGoogle Scholar
Turkington, B. 1999 Statistical equilibrium measures and coherent states in two-dimensional turbulence. Commun. Pure Appl. Maths 52, 781–809.3.0.CO;2-C>CrossRefGoogle Scholar
Vallis, G. K. 2006 Atmospheric and Oceanic Fluid Dynamics: Fundamentals and Large-Scale Circulation. Cambridge University Press.CrossRefGoogle Scholar
Vanneste, J. 2013 Balance and spontaneous wave generation in geophysical flows. Annu. Rev. Fluid Mech. 45, 147–172.CrossRefGoogle Scholar
Vautard, R. & Legras, B. 1986 Invariant manifolds, quasi-geostrophy, and initialization. J. Atmos. Sci 43, 565–584.2.0.CO;2>CrossRefGoogle Scholar
Venaille, A. & Bouchet, F. 2011 Solvable phase diagrams and ensemble inequivalence for two-dimensional and geophysical turbulent flows. J. Stat. Phys. 143, 346–380.CrossRefGoogle Scholar
Waite, M. L. 2011 Stratified turbulence at the buoyancy scale. Phys. Fluids 23 (6), 066602.CrossRefGoogle Scholar
Waite, M. L. 2013 Potential enstrophy in stratified turbulence. J. Fluid Mech. 722, R4.CrossRefGoogle Scholar
Waite, M. L. & Bartello, P. 2004 Stratified turbulence dominated by vortical motion. J. Fluid Mech. 517, 281–308.CrossRefGoogle Scholar
Waite, M. L. & Bartello, P. 2006 The transition from geostrophic to stratified turbulence. J. Fluid Mech. 568, 89–108.CrossRefGoogle Scholar
Waleffe, F. 1992 The nature of triad interactions in homogeneous turbulence. Phys. Fluids A 4, 350–363.CrossRefGoogle Scholar
Warn, T. 1986 Statistical mechanical equilibria of the shallow water equations. Tellus A 38 (1), 1–11.CrossRefGoogle Scholar
Warn, T. 1997 Nonlinear balance and quasi-geostrophic sets. Atmos.-Ocean 35, 135–145.CrossRefGoogle Scholar
Wingate, B., Embid, P. F., Holmes-Cerfon, M. & Taylor, M. 2011 Low Rossby limiting dynamics for stably stratified flow with finite Froude number. J. Fluid Mech. 676, 546–571.CrossRefGoogle Scholar
Yamazaki, Y., Kaneda, Y. & Rubinstein, R. 2002 Dynamics of inviscid truncated model of rotating turbulence. J. Phys. Soc. Japan 71 (1), 81–92.CrossRefGoogle Scholar
Zeman, O. 1994 A note on the spectra and decay of rotating homogeneous turbulence. Phys. Fluids 6, 3221–3223.CrossRefGoogle Scholar
Zhu, J.-Z., Yang, W. & Zhu, G.-Y. 2014 On one-chiral-sector-dominated turbulence states. J. Fluid Mech. 739, 479–501.CrossRefGoogle Scholar