Hostname: page-component-857557d7f7-v2cwp Total loading time: 0 Render date: 2025-12-11T05:42:07.511Z Has data issue: false hasContentIssue false
  • English
  • Français

Mean flow generation via non-resonant interactions in two-dimensional forced stratified turbulence

Published online by Cambridge University Press:  09 December 2025

Antonin Zoppi ,
Paul Billant Open the ORCID record for Paul Billant [Opens in a new window]  and
Leo Demaine
Antonin Zoppi
Affiliation:
LadHyX, CNRS, Ecole Polytechnique, Institut Polytechnique de Paris, 91120 Palaiseau, France
Paul Billant*
Affiliation:
LadHyX, CNRS, Ecole Polytechnique, Institut Polytechnique de Paris, 91120 Palaiseau, France
Leo Demaine
Affiliation:
LadHyX, CNRS, Ecole Polytechnique, Institut Polytechnique de Paris, 91120 Palaiseau, France
*
Corresponding author: Paul Billant, paul.billant@ladhyx.polytechnique.fr
Get access Rights & Permissions [Opens in a new window]

Abstract

We perform numerical simulations of two-dimensional strongly stratified flows in a square periodic domain $(y,z)$ forced by a steady mode with vorticity of the form $\sin (k_{\textit{y f}}y)\sin (k_{\textit{z f}}z)$, where $(k_{\textit{y f}},k_{\textit{z f}})$ are fixed wavenumbers. It is shown that such deterministic forcing can lead to a transition to turbulence and the emergence of horizontal layers (so-called vertically sheared horizontal flows, VSHFs) similarly as for random stochastic forcing. The flow characteristics are studied depending on the Froude and Reynolds numbers. Furthermore, the mechanisms of layers formation are disentangled. Triadic instabilities first lead to the growth of pairs of wavevectors that resonate with each of the four forced wavevectors. Quadratic interactions between these resonant modes and the forcing also drive the growth of several non-resonant modes at the same growth rate. Since the forcing comprises the wavevectors $\pm (k_{\textit{y f}},k_{\textit{z f}})$ and their mirror symmetric with respect to the horizontal $\pm (k_{\textit{y f}},-k_{\textit{z f}})$, there exist enslaved/bound modes with the same horizontal wavenumber and different vertical wavenumbers. Hence, the quadratic interactions between the latter modes force a second generation of modes among which some are VSHFs. Their growth rate is twice the growth rate of the primary resonant modes. Such a mechanism is similar to resonant quartets (Newell, J. Fluid Mech., 1969, vol. 35, no 2, pp. 255–271; Smith & Waleffe, Phys. Fluids, 1999, vol. 11, no 6, pp. 1608–1622). When the forcing is restricted to only the two wavevectors $\pm (k_{\textit{y f}},k_{\textit{z f}})$, the second generation of enslaved/bound modes all have a non-zero horizontal wavenumber. However, further quadratic interactions can force VSHF. Thus, horizontal layers also emerge, but with a growth rate equal to the number of quadratic interactions times the growth rate of the primary instability.

JFM classification

Geophysical and Geological Flows: Stratified flows Turbulent Flows: Stratified turbulence Instability: Transition to turbulence

Information

Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 1024 , 10 December 2025 , A59
Check for updates
Copyright
© The Author(s), 2025. Published by 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.)

Article purchase

Temporarily unavailable

References

Augier, P., Billant, P. & Chomaz, J.-M. 2015 Stratified turbulence forced with columnar dipoles: numerical study. J. Fluid Mech. 769, 403–443.10.1017/jfm.2015.76CrossRefGoogle Scholar
Benielli, D. & Sommeria, J. 1998 Excitation and breaking of internal gravity waves by parametric instability. J. Fluid Mech. 374, 117–144.10.1017/S0022112098002602CrossRefGoogle Scholar
Bouruet-Aubertot, P., Sommeria, J. & Staquet, C. 1995 Breaking of standing internal gravity waves through two-dimensional instabilities. J. Fluid Mech. 285, 265–301.10.1017/S0022112095000541CrossRefGoogle Scholar
Bouruet-Aubertot, P., Sommeria, J. & Staquet, C. 1996 Stratified turbulence produced by internal wave breaking: two-dimensional numerical experiments. Dyn. Atmos. Oceans 23 (1-4), 357–369.10.1016/0377-0265(95)00430-0CrossRefGoogle Scholar
Brethouwer, G., Billant, P., Lindborg, E. & Chomaz, J.-M. 2007 Scaling analysis and simulation of strongly stratified turbulent flows. J. Fluid Mech. 585, 343–368.10.1017/S0022112007006854CrossRefGoogle Scholar
Brunet, M., Gallet, B. & Cortet, P.-P. 2020 Shortcut to geostrophy in wave-driven rotating turbulence: the quartetic instability. Phys. Rev. Lett. 124 (12), 124501.10.1103/PhysRevLett.124.124501CrossRefGoogle ScholarPubMed
de Bruyn Kops, S.M. & Riley, J.J. 1998 Direct numerical simulation of laboratory experiments in isotropic turbulence. Phys. Fluids 10 (9), 2125–2127.10.1063/1.869733CrossRefGoogle Scholar
Caulfield, C.P. 2021 Layering, instabilities, and mixing in turbulent stratified flows. Annu. Rev. Fluid Mech. 53, 113–145.10.1146/annurev-fluid-042320-100458CrossRefGoogle Scholar
Dauxois, T., Joubaud, S., Odier, P. & Venaille, A. 2018 Instabilities of internal gravity wave beams. Annu. Rev. Fluid Mech. 50, 131–156.10.1146/annurev-fluid-122316-044539CrossRefGoogle Scholar
Deloncle, A. 2007 Three dimensional instabilities in stratified fluids. PhD thesis, Ecole Polytechnique.Google Scholar
Deloncle, A., Billant, P. & Chomaz, J.-M. 2008 Nonlinear evolution of the zigzag instability in stratified fluids: a shortcut on the route to dissipation. J. Fluid Mech. 599, 229–239.10.1017/S0022112007000109CrossRefGoogle Scholar
Farrell, B.F. & Ioannou, P.J. 2019 Statistical state dynamics: a new perspective on turbulence in shear flow. In Zonal Jets Phenomenology, Genesis, and Physics.(ed. B. Galpirin and P. L. Read), pp. 380–400. Cambridge University Press.10.1017/9781107358225.025CrossRefGoogle Scholar
Fitzgerald, J.G. & Farrell, B.F. 2018 a Statistical state dynamics of vertically sheared horizontal flows in two-dimensional stratified turbulence. J. Fluid Mech. 854, 544–590.10.1017/jfm.2018.560CrossRefGoogle Scholar
Fitzgerald, J.G. & Farrell, B.F. 2018 b Vertically sheared horizontal flow-forming instability in stratified turbulence: analytical linear stability analysis of statistical state dynamics equilibria. J. Atmos. Sci. 75 (12), 4201–4227.10.1175/JAS-D-18-0075.1CrossRefGoogle Scholar
Galmiche, M., Thual, O. & Bonneton, P. 2000 Wave/wave interaction producing horizontal mean flows in stably stratified fluids. Dyn. Atmos. Oceans 31 (1–4), 193–207.10.1016/S0377-0265(99)00033-0CrossRefGoogle Scholar
Galperin, B. & Read, P.L. 2019 Zonal Jets: Phenomenology, Genesis, and Physics. Cambridge University Press.10.1017/9781107358225CrossRefGoogle Scholar
Herbert, C., Marino, R., Rosenberg, D. & Pouquet, A. 2016 Waves and vortices in the inverse cascade regime of stratified turbulence with or without rotation. J. Fluid Mech. 806, 165–204.10.1017/jfm.2016.581CrossRefGoogle Scholar
Herring, J.R. & Métais, O. 1989 Numerical experiments in forced stably stratified turbulence. J. Fluid Mech. 202, 97–115.10.1017/S0022112089001114CrossRefGoogle Scholar
Koudella, C.R. & Staquet, C. 2006 Instability mechanisms of a two-dimensional progressive internal gravity wave. J. Fluid Mech. 548, 165–196.10.1017/S0022112005007524CrossRefGoogle Scholar
Kumar, A., Verma, K.M. & Sukhatme, J. 2017 Phenomenology of two-dimensional stably stratified turbulence under large-scale forcing. J. Turbul. 18 (3), 219–239.10.1080/14685248.2016.1271123CrossRefGoogle Scholar
Laval, J.-P., McWilliams, J.C. & Dubrulle, B. 2003 Forced stratified turbulence: successive transitions with Reynolds number. Phys. Rev. E 68 (3), 036308.10.1103/PhysRevE.68.036308CrossRefGoogle ScholarPubMed
Le Reun, T., Gallet, B., Favier, B. & Le Bars, M. 2020 Near-resonant instability of geostrophic modes: beyond Greenspan’s theorem. J. Fluid Mech. 900, R2.10.1017/jfm.2020.454CrossRefGoogle Scholar
LeBlond, P.H. & Mysak, L.A. 1981 Waves in the Ocean. Elsevier.Google Scholar
Lelong, M.-P. & Riley, J.J. 1991 Internal wave – vortical mode interactions in strongly stratified flows. J. Fluid Mech. 232, 1–19.10.1017/S0022112091003609CrossRefGoogle Scholar
Lemasquerier, D., Favier, B. & Le Bars, M. 2021 Zonal jets at the laboratory scale: hysteresis and Rossby waves resonance. J. Fluid Mech. 910, A18.10.1017/jfm.2020.1000CrossRefGoogle Scholar
Linares, M.C. 2020 Numerical study of 2d stratified turbulence forced by internal gravity waves. PhD thesis, Université Grenoble Alpes.Google Scholar
Lindborg, E. 2006 The energy cascade in a strongly stratified fluid. J. Fluid Mech. 550, 207–242.10.1017/S0022112005008128CrossRefGoogle Scholar
Maffioli, A. 2019 Asymmetry of vertical buoyancy gradient in stratified turbulence. J. Fluid Mech. 870, 266–289.10.1017/jfm.2019.240CrossRefGoogle Scholar
Maffioli, A., Brethouwer, G. & Lindborg, E. 2016 Mixing efficiency in stratified turbulence. J. Fluid Mech. 794, R3.10.1017/jfm.2016.206CrossRefGoogle Scholar
Manneville, P. 2014 Dissipative Structures and Weak Turbulence. Academic Press.Google Scholar
McEwan, A.D. 1971 Degeneration of resonantly-excited standing internal gravity waves. J. Fluid Mech. 50 (3), 431–448.10.1017/S0022112071002684CrossRefGoogle Scholar
McEwan, A.D., Mander, D.W. & Smith, R.K. 1972 Forced resonant second-order interaction between damped internal waves. J. Fluid Mech. 55 (4), 589–608.10.1017/S0022112072002034CrossRefGoogle Scholar
Newell, A.C. 1969 Rossby wave packet interactions. J. Fluid Mech. 35 (2), 255–271.10.1017/S0022112069001108CrossRefGoogle Scholar
Newell, A.C., Passot, T. & Lega, J. 1993 Order parameter equations for patterns. Annu. Rev. Fluid Mech. 25, 399–453.10.1146/annurev.fl.25.010193.002151CrossRefGoogle Scholar
Phillips, O.M. 1968 The interaction trapping of internal gravity waves. J. Fluid Mech. 34 (2), 407–416.10.1017/S0022112068001977CrossRefGoogle Scholar
Phillips, O.M. 1972 Turbulence in a strongly stratified fluid – is it unstable? Deep Sea Res. 19 (1), 79–81.Google Scholar
Plumb, R.A. & McEwan, A.D. 1978 The instability of a forced standing wave in a viscous stratified fluid: a laboratory analogue of the quasi-biennial oscillation. J. Atmos. Sci. 35 (10), 1827–1839.10.1175/1520-0469(1978)035<1827:TIOAFS>2.0.CO;22.0.CO;2>CrossRefGoogle Scholar
Posmentier, E.S. 1977 The generation of salinity finestructure by vertical diffusion. J. Phys. Oceanogr. 7 (2), 298–300.10.1175/1520-0485(1977)007<0298:TGOSFB>2.0.CO;22.0.CO;2>CrossRefGoogle Scholar
Remmel, M., Sukhatme, J. & Smith, L.M. 2014 Nonlinear gravity-wave interactions in stratified turbulence. Theor. Comput. Fluid Dyn. 28, 131–145.10.1007/s00162-013-0305-2CrossRefGoogle Scholar
Rhines, P.B. 1975 Waves and turbulence on a beta-plane. J. Fluid Mech. 69 (3), 417–443.10.1017/S0022112075001504CrossRefGoogle Scholar
Riley, J.J. & Lelong, M.-P. 2000 Fluid motions in the presence of strong stable stratification. Annu. Rev. Fluid Mech. 32 (1), 613–657.10.1146/annurev.fluid.32.1.613CrossRefGoogle Scholar
Semin, B., Facchini, G., Pétrélis, F. & Fauve, S. 2016 Generation of a mean flow by an internal wave. Phys. Fluids 28 (9), 096601.10.1063/1.4962937CrossRefGoogle Scholar
Semin, B., Garroum, N., Pétrélis, F. & Fauve, S. 2018 Nonlinear saturation of the large scale flow in a laboratory model of the quasibiennial oscillation. Phys. Rev. Lett. 121 (13), 134502.10.1103/PhysRevLett.121.134502CrossRefGoogle Scholar
Smith, L.M. 2001 Numerical study of two-dimensional stratified turbulence. Contemp. Math. 283, 91–106.10.1090/conm/283/04716CrossRefGoogle Scholar
Smith, L.M. & Waleffe, F. 1999 Transfer of energy to two-dimensional large scales in forced, rotating three-dimensional turbulence. Phys. Fluids 11 (6), 1608–1622.10.1063/1.870022CrossRefGoogle Scholar
Smith, L.M. & Waleffe, F. 2002 Generation of slow large scales in forced rotating stratified turbulence. J. Fluid Mech. 451, 145–168.10.1017/S0022112001006309CrossRefGoogle Scholar
Srinivasan, K. & Young, W.R. 2012 Zonostrophic instability. J. Atmos. Sci. 69 (5), 1633–1656.10.1175/JAS-D-11-0200.1CrossRefGoogle Scholar
Staquet, C. & Sommeria, J. 2002 Internal gravity waves: from instabilities to turbulence. Annu. Rev. Fluid Mech. 34 (1), 559–593.10.1146/annurev.fluid.34.090601.130953CrossRefGoogle Scholar
Sutherland, B.R. 2010 Internal Gravity Waves. Cambridge University Press.10.1017/CBO9780511780318CrossRefGoogle Scholar
Thorpe, S.A. 1968 On standing internal gravity waves of finite amplitude. J. Fluid Mech. 32 (3), 489–528.10.1017/S002211206800087XCrossRefGoogle Scholar
Waite, M.L. & Bartello, P. 2004 Stratified turbulence dominated by vortical motion. J. Fluid Mech. 517, 281–308.10.1017/S0022112004000977CrossRefGoogle Scholar
Wedi, N.P. & Smolarkiewicz, P.K. 2006 Direct numerical simulation of the Plumb–McEwan laboratory analog of the QBO. J. Atmos. Sci. 63 (12), 3226–3252.10.1175/JAS3815.1CrossRefGoogle Scholar