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Transport and emergent stratification in the equilibrated Eady model: the vortex-gas scaling regime

Published online by Cambridge University Press:  12 September 2022

Basile Gallet Open the ORCID record for Basile Gallet [Opens in a new window] ,
Benjamin Miquel Open the ORCID record for Benjamin Miquel [Opens in a new window] ,
Gabriel Hadjerci ,
Keaton J. Burns Open the ORCID record for Keaton J. Burns [Opens in a new window] ,
Glenn R. Flierl  and
Raffaele Ferrari
Basile Gallet*
Affiliation:
Université Paris-Saclay, CNRS, SPEC, CEA, 91191 Gif-sur-Yvette, France
Benjamin Miquel
Affiliation:
Université Paris-Saclay, CNRS, SPEC, CEA, 91191 Gif-sur-Yvette, France Univ Lyon, CNRS, Ecole Centrale de Lyon, INSA Lyon, Université Claude Bernard Lyon 1, LMFA, UMR5509, 69130 Ecully, France
Gabriel Hadjerci
Affiliation:
Université Paris-Saclay, CNRS, SPEC, CEA, 91191 Gif-sur-Yvette, France
Keaton J. Burns
Affiliation:
Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
Glenn R. Flierl
Affiliation:
Department of Earth, Atmospheric, and Planetary Sciences, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
Raffaele Ferrari
Affiliation:
Department of Earth, Atmospheric, and Planetary Sciences, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
*
†Email address for correspondence: [email protected]
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Abstract

We numerically and theoretically investigate the Boussinesq Eady model, where a rapidly rotating density-stratified layer of fluid is subject to a meridional temperature gradient in thermal wind balance with a uniform vertically sheared zonal flow. Through a suite of numerical simulations, we show that the transport properties of the resulting turbulent flow are governed by quasigeostrophic (QG) dynamics in the rapidly rotating strongly stratified regime. The ‘vortex gas’ scaling predictions put forward in the context of the two-layer QG model carry over to this fully three-dimensional system: the functional dependence of the meridional flux on the control parameters is the same, the two adjustable parameters entering the theory taking slightly different values. In line with the QG prediction, the meridional heat flux is depth-independent. The vertical heat flux is such that turbulence transports buoyancy along isopycnals, except in narrow layers near the top and bottom boundaries, the thickness of which decreases as the diffusivities go to zero. The emergent (re)stratification is set by a simple balance between the vertical heat flux and diffusion along the vertical direction. Overall, this study demonstrates how the vortex-gas scaling theory can be adapted to quantitatively predict the magnitude and vertical structure of the meridional and vertical heat fluxes, and of the emergent stratification, without additional fitting parameters.

JFM classification

Geophysical and Geological Flows: Baroclinic flows Geophysical and Geological Flows: Geostrophic turbulence Geophysical and Geological Flows: Ocean processes
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 948 , 10 October 2022 , A31
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Copyright
© The Author(s), 2022. Published by Cambridge University Press

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References

REFERENCES

Abernathey, R., Marshall, J., Mazloff, M. & Shuckburgh, E. 2010 Enhancement of mesoscale eddy stirring at steering levels in the southern ocean. J. Phys. Oceanogr. 40 (1), 170–184.CrossRefGoogle Scholar
Arbic, B.K. & Flierl, G.R. 2004 Baroclinically unstable geostrophic turbulence in the limits of strong and weak bottom ekman friction: application to midocean eddies. J. Phys. Oceanogr. 34, 2257–2273.2.0.CO;2>CrossRefGoogle Scholar
Bachman, S. & Fox-Kemper, B. 2013 Eddy parameterization challenge suite I: eady spindown. Ocean Model. 64, 12–28.CrossRefGoogle Scholar
Blumen, W. 1978 Uniform potential vorticity flow: part I. Theory of wave interactions and two-dimensional turbulence. J. Atmos. Sci. 35 (5), 774–783.2.0.CO;2>CrossRefGoogle Scholar
Burns, K.J., Vasil, G.M., Oishi, J.S., Lecoanet, D. & Brown, B.P. 2020 Dedalus: a flexible framework for numerical simulations with spectral methods. Phys. Rev. Res. 2, 023068.Google Scholar
Callies, J., Flierl, G., Ferrari, R. & Fox-Kemper, B. 2016 The role of mixed-layer instabilities in submesoscale turbulence. J. Fluid Mech. 788, 5–41.CrossRefGoogle Scholar
Carnevale, G.F., McWilliams, J.C., Pomeau, Y., Weiss, J.B. & Young, W.R. 1991 Evolution of vortex statistics in two-dimensional turbulence. Phys. Rev. Lett. 66, 2735–2737.CrossRefGoogle ScholarPubMed
Charney, J.G. & Stern, M.E. 1962 On the stability of internal Baroclinic jets in a rotating atmosphere. J. Atmos. Sci. 19 (2), 159–172.2.0.CO;2>CrossRefGoogle Scholar
Eady, E.T. 1949 Long waves and cyclone waves. Tellus 1 (3), 33–52.CrossRefGoogle Scholar
Gallet, B., Campagne, A., Cortet, P.-P. & Moisy, F. 2014 Scale-dependent cyclone-anticyclone asymmetry in a forced rotating turbulence experiment. Phys. Fluids 26 (3), 035108.CrossRefGoogle Scholar
Gallet, B. & Ferrari, R. 2020 The vortex gas scaling regime of baroclinic turbulence. Proc. Natl Acad. Sci. USA 117, 4491–4497.CrossRefGoogle ScholarPubMed
Gallet, B. & Ferrari, R. 2021 A quantitative scaling theory for meridional heat transport in planetary atmospheres and oceans. AGU Adv. 2 (3), e2020AV000362.CrossRefGoogle Scholar
Garner, S.T., Nakamura, N. & Held, I.M. 1992 Nonlinear equilibration of two-dimensional eady waves: a new perspective. J. Atmos. Sci. 49 (21), 1984–1996.2.0.CO;2>CrossRefGoogle Scholar
Gent, P.R. & Mcwilliams, J.C. 1990 Isopycnal mixing in ocean circulation models. J. Phys. Oceanogr. 20 (1), 150–155.Google Scholar
Hakim, G.J., Snyder, C. & Muraki, D.J. 2002 A new surface model for cyclone-anticyclone asymmetry. J. Atmos. Sci. 59 (16), 2405–2420.2.0.CO;2>CrossRefGoogle Scholar
Held, I.M. 1999 The macroturbulence of the troposphere. Tellus 51, 59–70.CrossRefGoogle Scholar
Held, I.M., Pierrehumbert, R.T., Garner, S.T. & Swanson, K.L. 1995 Surface quasi-geostrophic dynamics. J. Fluid Mech. 282, 1–20.CrossRefGoogle Scholar
Held, I.M. & Suarez, M.J. 1994 A proposal for the intercomparison of the dynamical cores of atmospheric general circulation models. Bull. Am. Meteorol. Soc. 75 (10), 1825–1830.2.0.CO;2>CrossRefGoogle Scholar
Hoskins, B.J. & Bretherton, F.P. 1971 Atmospheric frontogenesis models: some solutions. Q. J. R. Meteorol. Soc. 97 (412), 139–153.CrossRefGoogle Scholar
Klein, P., Hua, B.-L., Lapeyre, G., Capet, X., Le Gentil, S. & Sasaki, H. 2008 Upper ocean turbulence from high-resolution 3d simulations. J. Phys. Oceanogr. 38 (8), 1748–1763.CrossRefGoogle Scholar
Lapeyre, G. 2017 Surface quasi-geostrophy. Fluids 2 (1), 7.CrossRefGoogle Scholar
Larichev, V.D. & Held, I.M. 1995 Eddy amplitudes and fluxes in a homogeneous model of fully developed Baroclinic instability. J. Phys. Oceanogr. 25, 2285–2297.2.0.CO;2>CrossRefGoogle Scholar
Liu, J. & Schneider, T. 2010 Mechanisms of jet formation on the giant planets. J. Atmos. Sci. 67 (11), 3652–3672.CrossRefGoogle Scholar
Marshall, J. & Radko, T. 2003 Residual-mean solutions for the antarctic circumpolar current and its associated overturning circulation. J. Phys. Oceanogr. 33 (11), 2341–2354.2.0.CO;2>CrossRefGoogle Scholar
Miquel, B. 2021 Coral: a parallel spectral solver for fluid dynamics and partial differential equations. J. Open Source Softw. 6 (65), 2978.CrossRefGoogle Scholar
Molemaker, M.J., McWilliams, J.C. & Capet, X. 2010 Balanced and unbalanced routes to dissipation in an equilibrated Eady flow. J. Fluid Mech. 654, 35–63.CrossRefGoogle Scholar
Muraki, D.J., Snyder, C. & Rotunno, R. 1999 The next-order corrections to quasigeostrophic theory. J. Atmos. Sci. 56 (11), 1547–1560.2.0.CO;2>CrossRefGoogle Scholar
Nakamura, N. & Held, I.M. 1989 Nonlinear equilibration of two-dimensional Eady waves. J. Atmos. Sci. 46 (19), 3055–3064.2.0.CO;2>CrossRefGoogle Scholar
Nikurashin, M. & Vallis, G.K. 2011 A theory of deep stratification and overturning circulation in the ocean. J. Phys. Oceanogr. 41 (3), 485–502.CrossRefGoogle Scholar
Nikurashin, M. & Vallis, G.K. 2012 A theory of the interhemispheric meridional overturning circulation and associated stratification. J. Phys. Oceanogr. 42 (10), 1652–1667.CrossRefGoogle Scholar
Nowlin, W.D. Jr. & Klinck, J.M. 1986 The physics of the antarctic circumpolar current. Rev. Geophys. 24 (3), 469–491.CrossRefGoogle Scholar
Pedlosky, J. 1979 Geophysical Fluid Dynamics. SpringerGoogle Scholar
Phillips, N.A. 1954 Energy transformations and meridional circulations associated with simple Baroclinic waves in a two-level, quasi-geostrophic model. Tellus 6 (3), 274–286.CrossRefGoogle Scholar
Ragone, F. & Badin, G. 2016 A study of surface semi-geostrophic turbulence: freely decaying dynamics. J. Fluid Mech. 792, 740–774.CrossRefGoogle Scholar
Read, P., Kennedy, D., Lewis, N., Scolan, H., Tabataba-Vakili, F., Wang, Y., Wright, S. & Young, R. 2020 Baroclinic and barotropic instabilities in planetary atmospheres: energetics, equilibration and adjustment. Nonlinear Process. Geophys. 27 (2), 147–173.CrossRefGoogle Scholar
Rivière, P., Treguier, A.-M. & Klein, P. 2004 Effects of bottom friction on nonlinear equilibration of an oceanic Baroclinic jet. J. Phys. Oceanogr. 34 (2), 416–432.2.0.CO;2>CrossRefGoogle Scholar
Salmon, R. 1980 Baroclinic instability and geostrophic turbulence. Geophys. Astrophys. Fluid Dyn. 15, 157–211.Google Scholar
Salmon, R. 1998 Lectures on Geophysical Fluid Dynamics. Oxford University Press.CrossRefGoogle Scholar
Siegelman, L., Klein, P., Rivière, P., Thompson, A.F., Torres, H.S., Flexas, M. & Menemenlis, D. 2020 Enhanced upward heat transport at deep submesoscale ocean fronts. Nat. Geosci. 13 (1), 50–55.CrossRefGoogle Scholar
Smith, K.S. & Marshall, J. 2009 Evidence for enhanced eddy mixing at middepth in the southern ocean. J. Phys. Oceanogr. 39 (1), 50–69.Google Scholar
Stone, P.H. 1971 Baroclinic stability under non-hydrostatic conditions. J. Fluid Mech. 45 (4), 659–671.CrossRefGoogle Scholar
Thompson, A.F. & Young, W.R. 2006 Scaling Baroclinic eddy fluxes: vortices and energy balance. J. Phys. Oceanogr. 36, 720–736.CrossRefGoogle Scholar
Vallis, G.K. 2006 Atmospheric and Oceanic Fluid Dynamics: Fundamentals and Large-Scale Circulation. Cambridge University Press.CrossRefGoogle Scholar
Volkov, D.L., Fu, L.-L. & Lee, T. 2010 Mechanisms of the meridional heat transport in the southern ocean. Ocean Dyn. 60 (4), 791–801.CrossRefGoogle Scholar
Wolfe, C.L. & Cessi, P. 2010 What sets the strength of the middepth stratification and overturning circulation in eddying ocean models? J. Phys. Oceanogr. 40 (7), 1520–1538.CrossRefGoogle Scholar
Wunsch, C. 1997 The vertical partition of oceanic horizontal kinetic energy. J. Phys. Oceanogr. 27 (8), 1770–1794.2.0.CO;2>CrossRefGoogle Scholar
Young, W.R. 2012 An exact thickness-weighted average formulation of the Boussinesq equations. J. Phys. Oceanogr. 42 (5), 692–707.CrossRefGoogle Scholar