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Energy cascades in rapidly rotating and stratified turbulence within elongated domains

Published online by Cambridge University Press:  20 December 2021

Adrian van Kan Open the ORCID record for Adrian van Kan [Opens in a new window]  and
Alexandros Alexakis Open the ORCID record for Alexandros Alexakis [Opens in a new window]
Adrian van Kan*
Affiliation:
Laboratoire de Physique de l'Ecole normale supérieure, ENS, Université PSL, CNRS, Sorbonne Université, Université de Paris, F-75005 Paris, France Department of Physics, University of California, Berkeley, CA 94720, USA
Alexandros Alexakis
Affiliation:
Laboratoire de Physique de l'Ecole normale supérieure, ENS, Université PSL, CNRS, Sorbonne Université, Université de Paris, F-75005 Paris, France
*
†Email address for correspondence: [email protected]
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Abstract

We study forced, rapidly rotating and stably stratified turbulence in an elongated domain using an asymptotic expansion at simultaneously low Rossby number $\mathit {Ro}\ll 1$ and large domain height compared with the energy injection scale, $h=H/\ell _{in}\gg 1$. The resulting equations depend on the parameter $\lambda =(h \mathit {Ro} )^{-1}$ and the Froude number $\mathit {Fr}$. An extensive set of direct numerical simulations (DNS) is performed to explore the parameter space $(\lambda,\mathit {Fr})$. We show that a forward energy cascade occurs in one region of this space, and a split energy cascade outside it. At weak stratification (large $\mathit {Fr}$), an inverse cascade is observed for sufficiently large $\lambda$. At strong stratification (small $\mathit {Fr}$) the flow becomes approximately hydrostatic and an inverse cascade is always observed. For both weak and strong stratification, we present theoretical arguments supporting the observed energy cascade phenomenology. Our results shed light on an asymptotic region in the phase diagram of rotating and stratified turbulence, which is difficult to attain by brute-force DNS.

JFM classification

Turbulent Flows: Turbulent transition Turbulent Flows: Stratified turbulence Turbulent Flows: Rotating turbulence
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 933 , 25 February 2022 , A11
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Copyright
© The Author(s), 2021. Published by Cambridge University Press

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References

REFERENCES

Alexakis, A. 2011 Two-dimensional behavior of three-dimensional magnetohydrodynamic flow with a strong guiding field. Phys. Rev. E 84 (5), 056330.CrossRefGoogle ScholarPubMed
Alexakis, A. 2015 Rotating Taylor–Green flow. J. Fluid Mech. 769, 46–78.CrossRefGoogle Scholar
Alexakis, A. & Biferale, L. 2018 Cascades and transitions in turbulent flows. Phys. Rep. 767, 1–101.CrossRefGoogle Scholar
Bartello, P. 1995 Geostrophic adjustment and inverse cascades in rotating stratified turbulence. J. Atmos. Sci. 52 (24), 4410–4428.2.0.CO;2>CrossRefGoogle Scholar
Benavides, S.J. & Alexakis, A. 2017 Critical transitions in thin layer turbulence. J. Fluid Mech. 822, 364–385.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 (1), 427–451.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
Calkins, M.A., Julien, K., Tobias, S.M. & Aurnou, J.M. 2015 A multiscale dynamo model driven by quasi-geostrophic convection. J. Fluid Mech. 780, 143–166.CrossRefGoogle Scholar
Campagne, A., Gallet, B., Moisy, F. & Cortet, P.-P. 2014 Direct and inverse energy cascades in a forced rotating turbulence experiment. Phys. Fluids 26 (12), 125112.CrossRefGoogle Scholar
Celani, A., Cencini, M., Mazzino, A. & Vergassola, M. 2004 Active and passive fields face to face. New J. Phys. 6 (1), 72.CrossRefGoogle Scholar
Celani, A., Musacchio, S. & Vincenzi, D. 2010 Turbulence in more than two and less than three dimensions. Phys. Rev. Lett. 104, 184506.CrossRefGoogle ScholarPubMed
Charney, J.G. 1971 Geostrophic turbulence. J. Atmos. Sci. 28 (6), 1087–1095.2.0.CO;2>CrossRefGoogle Scholar
Cushman-Roisin, B. & Beckers, J.-M. 2011 Introduction to Geophysical Fluid Dynamics: Physical and Numerical Aspects. Academic Press.Google Scholar
Davidson, P.A. 2013 Turbulence in Rotating, Stratified and Electrically Conducting Fluids. Cambridge University Press.CrossRefGoogle Scholar
Deusebio, E., Boffetta, G., Lindborg, E. & Musacchio, S. 2014 Dimensional transition in rotating turbulence. Phys. Rev. E 90 (2), 023005.CrossRefGoogle ScholarPubMed
Di Leoni, P.C., Alexakis, A., Biferale, L. & Buzzicotti, M. 2020 Phase transitions and flux-loop metastable states in rotating turbulence. Phys. Rev. Fluids 5 (10), 104603.CrossRefGoogle Scholar
Ertel, H. 1942 Ein neuer hydrodynamischer erhaltungssatz. Naturwissenschaften 30 (36), 543–544.CrossRefGoogle Scholar
Falkovich, G., Gawedzki, K. & Vergassola, M. 2001 Particles and fields in fluid turbulence. Rev. Mod. Phys. 73 (4), 913.CrossRefGoogle Scholar
Favier, B., Silvers, L.J. & Proctor, M.R.E. 2014 Inverse cascade and symmetry breaking in rapidly rotating Boussinesq convection. Phys. Fluids 26 (9), 096605.CrossRefGoogle Scholar
Frisch, U. 1995 Turbulence: The Legacy of AN Kolmogorov. Cambridge University Press.CrossRefGoogle Scholar
Galtier, S. 2003 Weak inertial-wave turbulence theory. Phys. Rev. E 68 (1), 015301.CrossRefGoogle ScholarPubMed
Gibson, C.H. 1991 Laboratory, numerical, and oceanic fossil turbulence in rotating and stratified flows. J. Geophys. Res. 96 (C7), 12549–12566.CrossRefGoogle Scholar
Greenspan, H.P.G., et al. 1968 The Theory of Rotating Fluids. CUP Archive.Google Scholar
Grooms, I., Julien, K., Weiss, J.B. & Knobloch, E. 2010 Model of convective Taylor columns in rotating Rayleigh–Bénard convection. Phys. Rev. Lett. 104 (22), 224501.CrossRefGoogle ScholarPubMed
Guervilly, C. & Hughes, D.W. 2017 Jets and large-scale vortices in rotating Rayleigh–Benard convection. Phys. Rev. Fluids 2 (11), 113503.CrossRefGoogle Scholar
Guervilly, C., Hughes, D.W. & Jones, C.A. 2014 Large-scale vortices in rapidly rotating Rayleigh–Bénard convection. arXiv:1403.7442.CrossRefGoogle Scholar
Herbert, C., Pouquet, A. & Marino, R. 2014 Restricted equilibrium and the energy cascade in rotating and stratified flows. arXiv:1401.2103.CrossRefGoogle Scholar
Herring, J.R. & Métais, O. 1989 Numerical experiments in forced stably stratified turbulence. J. Fluid Mech. 202, 97–115.CrossRefGoogle Scholar
Hoskins, B., Pedder, M. & Jones, D.W. 2003 The omega equation and potential vorticity. Q. J. R. Meteorol. Soc. 129 (595), 3277–3303.CrossRefGoogle Scholar
Hoskins, B.J., Draghici, I. & Davies, H.C. 1978 A new look at the $\omega$-equation. Q. J. R. Meteorol. Soc. 104 (439), 31–38.CrossRefGoogle Scholar
Hough, S.S. 1897 IX. On the application of harmonic analysis to the dynamical theory of the tides. Part I. On Laplace's ‘oscillations of the first species’ and the dynamics of ocean currents. Phil. Trans. R. Soc. Lond. A 189, 201–257.Google Scholar
Hua, B.L. & Haidvogel, D.B. 1986 Numerical simulations of the vertical structure of quasi-geostrophic turbulence. J. Atmos. Sci. 43 (23), 2923–2936.2.0.CO;2>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
Julien, K., Knobloch, E. & Plumley, M. 2018 Impact of domain anisotropy on the inverse cascade in geostrophic turbulent convection. J. Fluid Mech. 837, R4.CrossRefGoogle Scholar
Julien, K., Knobloch, E., Rubio, A.M. & Vasil, G.M. 2012 a Heat transport in low-Rossby-number Rayleigh–Bénard convection. Phys. Rev. Lett. 109 (25), 254503.CrossRefGoogle ScholarPubMed
Julien, K., Knobloch, E. & Werne, J. 1998 A new class of equations for rotationally constrained flows. Theor. Comput. Fluid Dyn. 11 (3–4), 251–261.CrossRefGoogle Scholar
Julien, K., Rubio, A.M., Grooms, I. & Knobloch, E. 2012 b Statistical and physical balances in low Rossby number Rayleigh–Bénard convection. Geophys. Astrophys. Fluid Dyn. 106 (4–5), 392–428.CrossRefGoogle Scholar
van Kan, A. & Alexakis, A. 2019 Condensates in thin-layer turbulence. J. Fluid Mech. 864, 490–518.CrossRefGoogle Scholar
van Kan, A. & Alexakis, A. 2020 Critical transition in fast-rotating turbulence within highly elongated domains. J. Fluid Mech. 899, A33.CrossRefGoogle Scholar
van Kan, A., Nemoto, T. & Alexakis, A. 2019 Rare transitions to thin-layer turbulent condensates. J. Fluid Mech. 878, 356–369.CrossRefGoogle Scholar
Kurien, S., Wingate, B. & Taylor, M.A. 2008 Anisotropic constraints on energy distribution in rotating and stratified turbulence. Europhys. Lett. 84 (2), 24003.CrossRefGoogle Scholar
Leith, C.E. 1980 Nonlinear normal mode initialization and quasi-geostrophic theory. J. Atmos. Sci. 37 (5), 958–968.2.0.CO;2>CrossRefGoogle Scholar
Linkmann, M., Hohmann, M. & Eckhardt, B. 2020 Non-universal transitions to two-dimensional turbulence. J. Fluid Mech. 892, A18.CrossRefGoogle Scholar
Maffei, S., Krouss, M.J., Julien, K. & Calkins, M.A. 2021 On the inverse cascade and flow speed scaling behaviour in rapidly rotating Rayleigh–Benard convection. J. Fluid Mech. 913, A18.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 (4), 44006.CrossRefGoogle Scholar
Marino, R., Mininni, P.D., Rosenberg, D.L. & Pouquet, A. 2014 Large-scale anisotropy in stably stratified rotating flows. Phys. Rev. E 90 (2), 023018.CrossRefGoogle ScholarPubMed
Marino, R., Pouquet, A. & Rosenberg, D. 2015 Resolving the paradox of oceanic large-scale balance and small-scale mixing. Phys. Rev. Lett. 114 (11), 114504.CrossRefGoogle ScholarPubMed
Maxworthy, T. & Browand, F.K. 1975 Experiments in rotating and stratified flows: oceanographic application. Annu. Rev. Fluid Mech. 7 (1), 273–305.CrossRefGoogle Scholar
McWilliams, J.C. 1989 Statistical properties of decaying geostrophic turbulence. J. Fluid Mech. 198, 199–230.CrossRefGoogle Scholar
Mininni, P.D., Rosenberg, D., Reddy, R. & Pouquet, A. 2011 A hybrid MPI–OpenMP scheme for scalable parallel pseudospectral computations for fluid turbulence. Parallel Comput. 37 (6–7), 316–326.CrossRefGoogle Scholar
Musacchio, S. & Boffetta, G. 2017 Split energy cascade in turbulent thin fluid layers. Phys. Fluids 29 (11), 111106.CrossRefGoogle Scholar
Musacchio, S. & Boffetta, G. 2019 Condensate in quasi-two-dimensional turbulence. Phys. Rev. Fluids 4 (2), 022602.CrossRefGoogle Scholar
Nazarenko, S.V. & Schekochihin, A.A. 2011 Critical balance in magnetohydrodynamic, rotating and stratified turbulence: towards a universal scaling conjecture. J. Fluid Mech. 677, 134–153.CrossRefGoogle Scholar
Oks, D., Mininni, P.D., Marino, R. & Pouquet, A. 2017 Inverse cascades and resonant triads in rotating and stratified turbulence. Phys. Fluids 29 (11), 111109.CrossRefGoogle Scholar
Pedlosky, J. 2013 Geophysical Fluid Dynamics. Springer Science & Business Media.Google Scholar
Pestana, T. & Hickel, S. 2019 Regime transition in the energy cascade of rotating turbulence. Phys. Rev. E 99 (5), 053103.CrossRefGoogle ScholarPubMed
Poujol, B., van Kan, A. & Alexakis, A. 2020 Role of the forcing dimensionality in thin-layer turbulent energy cascades. Phys. Rev. Fluids 5 (6), 064610.CrossRefGoogle Scholar
Pouquet, A., Marino, R., Mininni, P.D. & Rosenberg, D. 2017 Dual constant-flux energy cascades to both large scales and small scales. Phys. Fluids 29 (11), 111108.CrossRefGoogle Scholar
Pouquet, A., Rosenberg, D., Stawarz, J.E. & Marino, R. 2019 Helicity dynamics, inverse, and bidirectional cascades in fluid and magnetohydrodynamic turbulence: a brief review. Earth Space Sci. 6 (3), 351–369.CrossRefGoogle Scholar
Proudman, J. 1916 On the motion of solids in a liquid possessing vorticity. Proc. R. Soc. Lond. A 92 (642), 408–424.Google Scholar
Rhines, P.B. 1979 Geostrophic turbulence. Annu. Rev. Fluid Mech. 11 (1), 401–441.CrossRefGoogle Scholar
Rosenberg, D., Pouquet, A., Marino, R. & Mininni, P.D. 2015 Evidence for Bolgiano-Obukhov scaling in rotating stratified turbulence using high-resolution direct numerical simulations. Phys. Fluids 27 (5), 055105.CrossRefGoogle Scholar
Rubio, A.M., Julien, K., Knobloch, E. & Weiss, J.B. 2014 Upscale energy transfer in three-dimensional rapidly rotating turbulent convection. Phys. Rev. Lett. 112 (14), 144501.CrossRefGoogle ScholarPubMed
Sahoo, G., Alexakis, A. & Biferale, L. 2017 Discontinuous transition from direct to inverse cascade in three-dimensional turbulence. Phys. Rev. Lett. 118 (16), 164501.CrossRefGoogle ScholarPubMed
Sahoo, G. & Biferale, L. 2015 Disentangling the triadic interactions in Navier–Stokes equations. Eur. Phys. J. E 38 (10), 114.CrossRefGoogle ScholarPubMed
Salmon, R. 1980 Baroclinic instability and geostrophic turbulence. Geophys. Astrophys. Fluid Dyn. 15 (1), 167–211.CrossRefGoogle Scholar
Salmon, R. 1998 Lectures on Geophysical Fluid Dynamics. Oxford University Press.CrossRefGoogle Scholar
Seshasayanan, K. & Alexakis, A. 2016 Critical behavior in the inverse to forward energy transition in two-dimensional magnetohydrodynamic flow. Phys. Rev. E 93 (1), 013104.CrossRefGoogle ScholarPubMed
Seshasayanan, K. & Alexakis, A. 2018 Condensates in rotating turbulent flows. J. Fluid Mech. 841, 434–462.CrossRefGoogle Scholar
Seshasayanan, K., Benavides, S.J. & Alexakis, A. 2014 On the edge of an inverse cascade. Phys. Rev. E 90 (5), 051003.CrossRefGoogle ScholarPubMed
Smith, L.M., Chasnov, J.R. & Waleffe, F. 1996 Crossover from two-to three-dimensional turbulence. Phys. Rev. Lett. 77 (12), 2467.CrossRefGoogle ScholarPubMed
Smith, L.M. & Waleffe, F. 2002 Generation of slow large scales in forced rotating stratified turbulence. J. Fluid Mech. 451 (1), 145–168.CrossRefGoogle Scholar
Sozza, A., Boffetta, G., Muratore-Ginanneschi, P. & Musacchio, S. 2015 Dimensional transition of energy cascades in stably stratified forced thin fluid layers. Phys. Fluids 27 (3), 035112.CrossRefGoogle Scholar
Sprague, M., Julien, K., Knobloch, E. & Werne, J. 2006 Numerical simulation of an asymptotically reduced system for rotationally constrained convection. J. Fluid Mech. 551, 141–174.CrossRefGoogle Scholar
Sukhatme, J. & Smith, L.M. 2008 Vortical and wave modes in 3D rotating stratified flows: random large-scale forcing. Geophys. Astrophys. Fluid Dyn. 102 (5), 437–455.CrossRefGoogle Scholar
Taylor, G.I. 1917 Motion of solids in fluids when the flow is not irrotational. Proc. R. Soc. Lond. A 93 (648), 99–113.Google Scholar
Thomas, J. & Daniel, D. 2021 Forward flux and enhanced dissipation of geostrophic balanced energy. J. Fluid Mech. 911, A60.CrossRefGoogle Scholar
Trustrum, K. 1964 Rotating and stratified fluid flow. J. Fluid Mech. 19 (3), 415–432.CrossRefGoogle Scholar
Vallgren, A. & Lindborg, E. 2010 Charney isotropy and equipartition in quasi-geostrophic turbulence. J. Fluid Mech. 656, 448–457.CrossRefGoogle Scholar
Vallis, G.K. 2017 Atmospheric and Oceanic Fluid Dynamics. Cambridge University Press.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
Warhaft, Z. 2000 Passive scalars in turbulent flows. Annu. Rev. Fluid Mech. 32 (1), 203–240.CrossRefGoogle Scholar
Xia, H., Byrne, D., Falkovich, G. & Shats, M.G. 2011 Upscale energy transfer in thick turbulent fluid layers. Nat. Phys. 7 (4), 321.CrossRefGoogle Scholar
Xia, H., Shats, M. & Falkovich, G. 2009 Spectrally condensed turbulence in thin layers. Phys. Fluids 21 (12), 125101.CrossRefGoogle Scholar
Yokoyama, N. & Takaoka, M. 2017 Hysteretic transitions between quasi-two-dimensional flow and three-dimensional flow in forced rotating turbulence. Phys. Rev. Fluids 2 (9), 092602.CrossRefGoogle Scholar