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Critical balance and scaling of strongly stratified turbulence at low Prandtl number

Published online by Cambridge University Press:  30 January 2023

Valentin A. Skoutnev*
Affiliation:
Department of Astrophysical Sciences, Princeton University, Princeton, NJ 08544, USA
*
Email address for correspondence: valentinskoutnev@gmail.com

Abstract

We extend the scaling relations of strongly (stably) stratified turbulence from the geophysical regime of unity Prandtl number to the astrophysical regime of extremely small Prandtl number applicable to stably stratified regions of stars and gas giants. A transition to a new turbulent regime is found to occur when the Prandtl number drops below the inverse of the buoyancy Reynolds number, i.e. $Pr\,Rb<1$, which signals a shift of the dominant balance in the buoyancy equation. Application of critical balance arguments then derives new predictions for the anisotropic energy spectrum and dominant balance of the Boussinesq equations in the $Pr\,Rb\ll 1$ regime. We find that all the standard scaling relations from the unity $Pr$ limit of strongly stratified turbulence simply carry over if the Froude number, $Fr$, is replaced by a modified Froude number, $Fr_M\equiv Fr/(Pr\,Rb)^{1/4}$. The geophysical and astrophysical regimes are thus smoothly connected across the $Pr\,Rb=1$ transition. Applications to vertical transport in stellar radiative zones and modification to the instability criterion for the small-scale dynamo are discussed.

Type
JFM Papers
Copyright
© The Author(s), 2023. Published by Cambridge University Press

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References

REFERENCES

Aerts, C., Mathis, S. & Rogers, T.M. 2019 Angular momentum transport in stellar interiors. Annu. Rev. Astron. Astrophys. 57, 3578.CrossRefGoogle Scholar
Augier, P., Billant, P. & Chomaz, J.-M. 2015 Stratified turbulence forced with columnar dipoles: numerical study. J. Fluid Mech. 769, 403443.CrossRefGoogle Scholar
Bartello, P. & Tobias, S.M. 2013 Sensitivity of stratified turbulence to the buoyancy Reynolds number. J. Fluid Mech. 725, 122.CrossRefGoogle Scholar
de Bruyn Kops, S.M. & Riley, J.J. 2019 The effects of stable stratification on the decay of initially isotropic homogeneous turbulence. J. Fluid Mech. 860, 787821.CrossRefGoogle Scholar
Billant, P. & Chomaz, J.-M. 2000 a Experimental evidence for a new instability of a vertical columnar vortex pair in a strongly stratified fluid. J. Fluid Mech. 418, 167188.CrossRefGoogle Scholar
Billant, P. & Chomaz, J.-M. 2000 b Theoretical analysis of the zigzag instability of a vertical columnar vortex pair in a strongly stratified fluid. J. Fluid Mech. 419, 2963.CrossRefGoogle Scholar
Billant, P. & Chomaz, J.-M. 2001 Self-similarity of strongly stratified inviscid flows. Phys. Fluids 13 (6), 16451651.CrossRefGoogle Scholar
Bois, P.A. 1991 Asymptotic aspects of the Boussinesq approximation for gases and liquids. Geophys. Astrophys. Fluid Dyn. 58 (1–4), 4555.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, 343368.CrossRefGoogle Scholar
Chini, G.P., Michel, G., Julien, K., Rocha, C.B. & Colm-cille, P.C. 2022 Exploiting self-organized criticality in strongly stratified turbulence. J. Fluid Mech. 933, A22.CrossRefGoogle Scholar
Cope, L., Garaud, P. & Caulfield, C.P. 2020 The dynamics of stratified horizontal shear flows at low Péclet number. J. Fluid Mech. 903, A1.CrossRefGoogle Scholar
Dewan, E. 1997 Saturated-cascade similitude theory of gravity wave spectra. J. Geophys. Res. 102 (D25), 2979929817.CrossRefGoogle Scholar
Falder, M., White, N.J. & Caulfield, C.P. 2016 Seismic imaging of rapid onset of stratified turbulence in the south Atlantic ocean. J. Phys. Oceanogr. 46 (4), 10231044.CrossRefGoogle Scholar
Fernando, H.J.S. 1991 Turbulent mixing in stratified fluids. Annu. Rev. Fluid Mech. 23 (1), 455493.CrossRefGoogle Scholar
Ferrari, R. & Wunsch, C. 2009 Ocean circulation kinetic energy: reservoirs, sources, and sinks. Annu. Rev. Fluid Mech. 41, 253282.CrossRefGoogle Scholar
Garaud, P. 2020 Horizontal shear instabilities at low Prandtl number. Astrophys. J. 901 (2), 146.CrossRefGoogle Scholar
Garaud, P. 2021 Journey to the center of stars: the realm of low Prandtl number fluid dynamics. Phys. Rev. Fluids 6 (3), 030501.CrossRefGoogle Scholar
Garaud, P., Gagnier, D. & Verhoeven, J. 2017 Turbulent transport by diffusive stratified shear flows: from local to global models. I. Numerical simulations of a stratified plane Couette flow. Astrophys. J. 837 (2), 133.CrossRefGoogle Scholar
Garaud, P., Gallet, B. & Bischoff, T. 2015 The stability of stratified spatially periodic shear flows at low Péclet number. Phys. Fluids 27 (8), 084104.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, 229238.CrossRefGoogle Scholar
Goldreich, P. & Sridhar, S. 1995 Toward a theory of interstellar turbulence. 2: strong Alfvenic turbulence. Astrophys. J. 438, 763775.CrossRefGoogle Scholar
Gregg, M.C. 2021 Ocean Mixing. Cambridge University Press.CrossRefGoogle Scholar
Gregg, M.C., D'Asaro, E.A., Riley, J.J. & Kunze, E. 2018 Mixing efficiency in the ocean. Ann. Rev. Mar. Sci. 10, 443473.CrossRefGoogle ScholarPubMed
Howard, L.N. & Maslowe, S.A. 1973 Stability of stratified shear flows. Boundary-Layer Meteorol. 4 (1), 511523.CrossRefGoogle Scholar
Iskakov, A.B., Schekochihin, A.A., Cowley, S.C., McWilliams, J.C. & Proctor, M.R.E. 2007 Numerical demonstration of fluctuation dynamo at low magnetic Prandtl numbers. Phys. Rev. Lett. 98 (20), 208501.CrossRefGoogle ScholarPubMed
Ivey, G.N., Winters, K.B. & Koseff, J.R. 2008 Density stratification, turbulence, but how much mixing? Annu. Rev. Fluid Mech. 40, 169184.CrossRefGoogle Scholar
Khani, S. 2018 Mixing efficiency in large-eddy simulations of stratified turbulence. J. Fluid Mech. 849, 373394.CrossRefGoogle Scholar
Lefauve, A. & Linden, P.F. 2022 Experimental properties of continuously forced, shear-driven, stratified turbulence. Part 1. Mean flows, self-organisation, turbulent fractions. J. Fluid Mech. 937, A34.CrossRefGoogle Scholar
Legaspi, J.D. & Waite, M.L. 2020 Prandtl number dependence of stratified turbulence. J. Fluid Mech. 903, A12.CrossRefGoogle Scholar
Legg, S. 2021 Mixing by oceanic lee waves. Annu. Rev. Fluid Mech. 53, 173201.CrossRefGoogle Scholar
Lignières, F. 1999 The small-Péclet-number approximation in stellar radiative zones. Astron. Astrophys. 348, 933939.Google Scholar
Lignières, F. 2019 Turbulence in stably stratified radiative zone. arXiv:1911.07813CrossRefGoogle Scholar
Lilly, D.K. 1983 Stratified turbulence and the mesoscale variability of the atmosphere. J. Atmos. Sci. 40 (3), 749761.2.0.CO;2>CrossRefGoogle Scholar
Lin, J.-T. & Pao, Y.-H. 1979 Wakes in stratified fluids. Annu. Rev. Fluid Mech. 11 (1), 317338.CrossRefGoogle Scholar
Lindborg, E. 2006 The energy cascade in a strongly stratified fluid. J. Fluid Mech. 550, 207242.CrossRefGoogle Scholar
Lindborg, E. & Brethouwer, G. 2007 Stratified turbulence forced in rotational and divergent modes. J. Fluid Mech. 586, 83108.CrossRefGoogle Scholar
Lucas, D., Caulfield, C.P. & Kerswell, R.R. 2017 Layer formation in horizontally forced stratified turbulence: connecting exact coherent structures to linear instabilities. J. Fluid Mech. 832, 409437.CrossRefGoogle Scholar
Maeder, A. & Meynet, G. 2000 The evolution of rotating stars. Annu. Rev. Astron. Astrophys. 38 (1), 143190.CrossRefGoogle Scholar
Maffioli, A. & Davidson, P.A. 2016 Dynamics of stratified turbulence decaying from a high buoyancy Reynolds number. J. Fluid Mech. 786, 210233.CrossRefGoogle Scholar
Monismith, S.G., Koseff, J.R. & White, B.L. 2018 Mixing efficiency in the presence of stratification: when is it constant? Geophys. Res. Lett. 45 (11), 56275634.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, 134153.CrossRefGoogle Scholar
Okino, S. & Hanazaki, H. 2020 Direct numerical simulation of turbulence in a salt-stratified fluid. J. Fluid Mech. 891, A19.CrossRefGoogle Scholar
Ozmidov, R.V. 1992 Variability of shelf and adjacent seas. J. Mar. Syst. 3 (4–5), 417421.CrossRefGoogle Scholar
Pinsonneault, M. 1997 Mixing in stars. Annu. Rev. Astron. Astrophys. 35 (1), 557605.CrossRefGoogle Scholar
Prat, V., Guilet, J., Viallet, M. & Müller, E. 2016 Shear mixing in stellar radiative zones—II. Robustness of numerical simulations. Astron. Astrophys. 592, A59.CrossRefGoogle Scholar
Prat, V. & Lignières, F. 2013 Turbulent transport in radiative zones of stars. Astron. Astrophys. 551, L3.CrossRefGoogle Scholar
Prat, V. & Lignières, F. 2014 Shear mixing in stellar radiative zones—I. Effect of thermal diffusion and chemical stratification. Astron. Astrophys. 566, A110.CrossRefGoogle Scholar
Richardson, L.F. 1920 The supply of energy from and to atmospheric eddies. Proc. R. Soc. Lond. A 97 (686), 354373.Google Scholar
Riley, J.J. & de BruynKops, S.M. 2003 Dynamics of turbulence strongly influenced by buoyancy. Phys. Fluids 15 (7), 20472059.CrossRefGoogle Scholar
Riley, J.J. & Lindborg, E. 2008 Stratified turbulence: a possible interpretation of some geophysical turbulence measurements. J. Atmos. Sci. 65 (7), 24162424.CrossRefGoogle Scholar
Riley, J.J. & Lindborg, E. 2010 Recent progress in stratified turbulence. In Ten Chapters in Turbulence (ed. P.A. Davidson, Y. Kaneda & K.R. Sreenivasan), pp. 269–317. Cambridge University Press.CrossRefGoogle Scholar
Salaris, M. & Cassisi, S. 2017 Chemical element transport in stellar evolution models. R. Soc. Open Sci. 4 (8), 170192.CrossRefGoogle ScholarPubMed
Salehipour, H., Peltier, W.R. & Caulfield, C.P. 2018 Self-organized criticality of turbulence in strongly stratified mixing layers. J. Fluid Mech. 856, 228256.CrossRefGoogle Scholar
Salehipour, H., Peltier, W.R. & Mashayek, A. 2015 Turbulent diapycnal mixing in stratified shear flows: the influence of Prandtl number on mixing efficiency and transition at high Reynolds number. J. Fluid Mech. 773, 178223.CrossRefGoogle Scholar
Schekochihin, A.A. 2022 MHD turbulence: a biased review. J. Plasma Phys. 88 (5), 155880501.CrossRefGoogle Scholar
Schekochihin, A.A., Iskakov, A.B., Cowley, S.C., McWilliams, J.C., Proctor, M.R.E. & Yousef, T.A. 2007 Fluctuation dynamo and turbulent induction at low magnetic Prandtl numbers. New J. Phys. 9 (8), 300.CrossRefGoogle Scholar
Skoutnev, V., Squire, J. & Bhattacharjee, A. 2021 Small-scale dynamo in stably stratified turbulence. Astrophys. J. 906 (1), 61.CrossRefGoogle Scholar
Smyth, W.D. & Moum, J.N. 2013 Marginal instability and deep cycle turbulence in the eastern equatorial pacific ocean. Geophys. Res. Lett. 40 (23), 61816185.CrossRefGoogle Scholar
Smyth, W.D., Nash, J.D. & Moum, J.N. 2019 Self-organized criticality in geophysical turbulence. Sci. Rep. 9 (1), 18.CrossRefGoogle ScholarPubMed
Spedding, G.R., Browand, F.K. & Fincham, A.M. 1996 The long-time evolution of the initially turbulent wake of a sphere in a stable stratification. Dyn. Atmos. Oceans 23 (1-4), 171182.CrossRefGoogle Scholar
Spiegel, E.A. & Veronis, G. 1960 On the Boussinesq approximation for a compressible fluid. Astrophys. J. 131, 442.CrossRefGoogle Scholar
Thorpe, S.A. 2005 The Turbulent Ocean. Cambridge University Press.CrossRefGoogle Scholar
Waite, M.L. 2011 Stratified turbulence at the buoyancy scale. Phys. Fluids 23 (6), 066602.CrossRefGoogle Scholar
Waite, M.L. & Bartello, P. 2004 Stratified turbulence dominated by vortical motion. J. Fluid Mech. 517, 281308.CrossRefGoogle Scholar
Waite, M.L., von Larcher, T. & Williams, P. 2014 Direct numerical simulations of laboratory-scale stratified turbulence. In Modelling Atmospheric and Oceanic Flows: Insights from Laboratory Experiments (ed. T. von Larcher & P. Williams), American Geophysical Union.CrossRefGoogle Scholar
Zahn, J.-P. 1974 Rotational instabilities and stellar evolution. Symposium-International Astronomical Union 59, 185195.CrossRefGoogle Scholar
Zahn, J.-P. 1992 Circulation and turbulence in rotating stars. Astron. Astrophys. 265, 115132.Google Scholar
Zeldovich, Y.B. & Ruzmaikin, A. 1980 Magnetic field of a conducting fluid in two-dimensional motion. Zh. Eksp. Teor. Fiz. 78, 980986.Google Scholar