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Turbulent mixing in the vertical magnetic Rayleigh–Taylor instability

Published online by Cambridge University Press:  11 January 2024

A. Briard Open the ORCID record for A. Briard [Opens in a new window] ,
B.-J. Gréa Open the ORCID record for B.-J. Gréa [Opens in a new window]  and
F. Nguyen Open the ORCID record for F. Nguyen [Opens in a new window]
A. Briard*
Affiliation:
CEA, DAM, DIF, 91297 Arpajon, France
B.-J. Gréa
Affiliation:
CEA, DAM, DIF, 91297 Arpajon, France Laboratoire de la Matière en Conditions Extrêmes, Université Paris-Saclay, 91680 Bruyères-le-Châtel, France
F. Nguyen
Affiliation:
CEA, DAM, DIF, 91297 Arpajon, France
*
†Email address for correspondence: antoine.briard@cea.fr
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Abstract

The presence of a mean magnetic field aligned with the direction of the acceleration greatly modifies the development of the Rayleigh–Taylor instability (RTI). High resolution direct numerical simulations of the Boussinesq–Navier–Stokes equations under the magnetohydrodynamics approximation reveal that, after an initial damping of the perturbations at the interface between the two miscible fluids, a rapid increase of the mixing layer is observed. Structures are significantly stretched in the vertical direction because magnetic tension prevents small-scale shear instabilities. When the vertical turbulent velocity exceeds the Alfvén velocity, the flow transitions to turbulence, structures break and an enhanced mixing occurs with strong dissipation. Afterwards, the mixing zone slows down and its growth rate is decreased compared to the hydrodynamic case. For larger magnitudes of the mean magnetic field, a strong anisotropy persists, and an increased fraction of potential energy injected into the system is lost into turbulent magnetic energy: as a consequence, the mixing zone growth rate is decreased even more. This phenomenology is embedded in a general buoyancy-drag equation, derived from simplified equations that reflect the large-scale dynamics, in which the drag coefficient is increased by the presence of turbulent magnetic energy.

JFM classification

Turbulent Flows: Stratified turbulence Mixing: Turbulent mixing MHD and Electrohydrodynamics: MHD turbulence
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 979 , 25 January 2024 , A8
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Copyright
© The Author(s), 2024. Published by Cambridge University Press

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References

Alexakis, A. 2013 Large-scale magnetic fields in magnetohydrodynamics turbulence. Phys. Rev. Lett. 110, 084502.CrossRefGoogle Scholar
Andrews, M.J. & Spalding, D.B. 1990 A simple experiment to investigate two-dimensional mixing by Rayleigh–Taylor instability. Phys. Fluids A 2, 922.CrossRefGoogle Scholar
Aslangil, D., Banerjee, A. & Lawrie, A.G.W. 2016 Numerical investigation of initial condition effects on Rayleigh–Taylor instability with acceleration reversals. Phys. Rev. E 94, 053114.CrossRefGoogle ScholarPubMed
Baldwin, K.A., Scase, M.M. & Hill, R.J.A. 2015 The inhibition of the Rayleigh–Taylor instability by rotation. Nature Sci. Rep. 5, 11706.Google ScholarPubMed
Beresnyak, A. 2014 Spectra of strong magnetohydrodynamic turbulence from high-resolution simulations. Astrophys. J. Lett. 784, L20.CrossRefGoogle Scholar
Biskamp, D. 2003 Magnetohydrodynamic Turbulence. Cambridge University Press.CrossRefGoogle Scholar
Boffetta, G., Borgnino, M. & Musacchio, S. 2020 Scaling of Rayleigh–Taylor mixing in porous media. Phys. Rev. Fluids 99, 062501(R).CrossRefGoogle Scholar
Boffetta, G., De Lillo, F. & Musacchio, S. 2010 Nonlinear diffusion model for Rayleigh–Taylor mixing. Phys. Rev. Lett. 104, 034505.CrossRefGoogle ScholarPubMed
Boffetta, G., Magnani, M. & Musacchio, S. 2019 Suppression of Rayleigh–Taylor turbulence by time-periodic acceleration. Phys. Rev. E 99, 033110.CrossRefGoogle ScholarPubMed
Boffetta, G. & Mazzino, A. 2017 Incompressible Rayleigh–Taylor turbulence. Annu. Rev. Fluid Mech. 49, 119.CrossRefGoogle Scholar
Boffetta, G., Mazzino, A. & Musacchio, S. 2016 Rotating Rayleigh–Taylor turbulence. Phys. Rev. Fluids 1, 054405.CrossRefGoogle Scholar
Boffetta, G., Mazzino, A., Musacchio, S. & Vozella, L. 2009 Kolmogorov scaling and intermittency in Rayleigh–Taylor turbulence. Phys. Rev. E 79, 065301(R).CrossRefGoogle ScholarPubMed
Boldyrev, S., Perez, J.C., Borovsky, J.E. & Podesta, J.J. 2011 Spectral scaling laws in magnetohydrodynamic turbulence simulations and in the solar wind. Astrophys. J. Lett. 741, L19.CrossRefGoogle Scholar
Briard, A. & Gomez, T. 2018 The decay of isotropic magnetohydrodynamics turbulence and the effects of cross-helicity. J. Plasma Phys. 84, 905840110.CrossRefGoogle Scholar
Briard, A., Gomez, T. & Cambon, C. 2016 Spectral modelling for passive scalar dynamics in homogeneous anisotropic turbulence. J. Fluid Mech. 799, 159–199.CrossRefGoogle Scholar
Briard, A., Gostiaux, L. & Gréa, B.-J. 2020 The turbulent Faraday instability in miscible fluids. J. Fluid Mech. 883, A57.CrossRefGoogle Scholar
Briard, A., Gréa, B.-J. & Nguyen, F. 2022 Growth rate of the turbulent magnetic Rayleigh–Taylor instability. Phys. Rev. E 106, 065201.CrossRefGoogle ScholarPubMed
Briard, A., Iyer, M. & Gomez, T. 2017 Anisotropic spectral modeling for unstably stratified homogeneous turbulence. Phys. Rev. Fluids 2, 044604.CrossRefGoogle Scholar
Burlot, A., Gréa, B.-J., Godeferd, F.S., Cambon, C. & Griffond, J. 2015 Spectral modelling of high Reynolds number unstably stratified homogeneous turbulence. J. Fluid Mech. 765, 17–44.CrossRefGoogle Scholar
Cabot, W.H. & Cook, A.W. 2006 Reynolds number effects on Rayleigh–Taylor instability with possible implications for type-Ia supernovae. Nat. Phys. 2, 562–568.CrossRefGoogle Scholar
Carlyle, J. & Hillier, A. 2017 The non-linear growth of the magnetic Rayleigh–Taylor instability. Astron. Astrophys. 605, A101.CrossRefGoogle Scholar
Chae, J. 2010 Dynamics of vertical threads and descending knots in a hedgerow prominence. Astrophys. J. 714, 618–629.CrossRefGoogle Scholar
Chandrasekhar, S. 1961 X: the stability of superimposed fluids: the Rayleigh–Taylor instability. In Hydrodynamic and Hydromagnetic Stability, pp. 428–477. Oxford University Press.Google Scholar
Cook, A.W., Cabot, W. & Miller, P.L. 2004 The mixing transition in Rayleigh–Taylor instability. J. Fluid Mech. 511, 333–362.CrossRefGoogle Scholar
Cook, A.W. & Dimotakis, P.E. 2001 Transition stages of Rayleigh–Taylor instability between miscible fluids. J. Fluid Mech. 443, 69–99.CrossRefGoogle Scholar
Cox, C.I., Gull, S.F. & Green, D.A. 1991 Numerical simulations of the ’jet’ in the Crab Nebula. Mon. Not. R. Astron. Soc. 250, 750–759.CrossRefGoogle Scholar
Davies, R.M. & Taylor, G.I. 1950 The mechanics of large bubbles rising through extended liquids and through liquids in tubes. Proc. R. Soc. Lond. 200, 375.Google Scholar
Davies Wykes, M.S. & Dalziel, S.B. 2014 Efficient mixing in stratified flows: experimental study of Rayleigh–Taylor unstable interface within an otherwise stable stratification. J. Fluid Mech. 756, 1027–1057.CrossRefGoogle Scholar
Davies Wykes, M.S., Hughes, G.O. & Dalziel, S.B. 2015 On the meaning of mixing efficiency for buoyancy-driven mixing in stratified turbulent flows. J. Fluid Mech. 781, 261–275.CrossRefGoogle Scholar
Dimonte, G. 2000 Spanwise homogeneous buoyancy-drag model for Rayleigh–Taylor mixing and experimental evaluation. Phys. Plasma 7, 2255.CrossRefGoogle Scholar
Dimonte, G., et al. 2004 A comparative study of the turbulent Rayleigh–Taylor instability using high-resolution three-dimensional numerical simulations: the Alpha-Group collaboration. Phys. Fluids 16, 1668.CrossRefGoogle Scholar
Galtier, S., Politano, H. & Pouquet, A. 1997 Self-similar energy decay in magnetohydrodynamic turbulence. Phys. Rev. Lett. 79 (15), 2807–2810.CrossRefGoogle Scholar
Goldreich, P. & Sridhar, S. 1995 Toward a theory of interstellar turbulence. 2: strong alfvenic turbulence. Astrophys. J. 438, 763–775.CrossRefGoogle Scholar
Goncharov, V.N. 2002 Analytical model of nonlinear, single-mode, classical Rayleigh–Taylor instability at arbitrary Atwood numbers. Phys. Rev. Lett. 88, 134502.CrossRefGoogle ScholarPubMed
Gréa, B.-J. 2013 The rapid acceleration model and the growth rate of a turbulent mixing zone induced by Rayleigh–Taylor instability. Phys. Fluids 25, 015118.CrossRefGoogle Scholar
Gréa, B.-J. & Briard, A. 2023 Inferring the magnetic field from the Rayleigh–Taylor instability. Astrophys. J. 958 (2), 164.CrossRefGoogle Scholar
Gréa, B.-J., Burlot, A., Godeferd, F.S., Griffond, J., Soulard, O. & Cambon, C. 2016 Dynamics and structure of unstably stratified homogeneous turbulence. J. Turbul. 17 (7), 651–663.CrossRefGoogle Scholar
Griffond, J., Gréa, B.-J. & Soulard, O. 2014 Unstably stratified homogeneous turbulencen as a tool for turbulent mixing modeling. Trans. ASME J. Fluids Engng 136, 091201.CrossRefGoogle Scholar
Griffond, J., Gréa, B.-J. & Soulard, O. 2015 Numerical investigation of self-similar unstably stratified homogeneous turbulence. J. Turbul. 16, 167–183.CrossRefGoogle Scholar
Haerendel, G. & Berger, T. 2011 A droplet model of quiescent prominence downflows. Astrophys. J. 731 (2), 82.CrossRefGoogle Scholar
Hillier, A. 2018 The magnetic Rayleigh–Taylor instability in solar prominences. Rev. Mod. Plasma Phys. 2, 1–47.CrossRefGoogle Scholar
Hillier, A. 2020 Ideal MHD Instabilities, with a Focus on the Rayleigh–Taylor and Kelvin–Helmholtz Instabilities, pp. 1–36. Springer.CrossRefGoogle Scholar
Hughes, D.W. & Tobias, S.M. 2001 On the instability of magnetohydrodynamic shear flows. Proc. R. Soc. Lond. 457, 1365–1384.CrossRefGoogle Scholar
Hunt, J.C.R. & Carruthers, D.J. 1990 Rapid distortion theory and the ‘problems’ of turbulence. J. Fluid Mech. 212, 497–532.CrossRefGoogle Scholar
Inoue, T., Shimoda, J., Ohira, Y. & Yamazaki, R. 2013 The origin of radially aligned magnetic fields in young supernova remnants. Astrophys. J. Lett. 772 (2), L20.CrossRefGoogle Scholar
Iroshnikov, P.S. 1964 Turbulence of a conducting fluid in a strong magnetic field. Sov. Astron. 7 (4), 566–571.Google Scholar
Isobe, H., Miyagoshi, T., Shibata, K. & Yokoyama, T. 2005 Filamentary structure on the sun from the magnetic Rayleigh–Taylor instability. Nature 434, 478–481.CrossRefGoogle ScholarPubMed
Jenkins, J.M. & Keppens, R. 2022 Resolving the solar prominence/filament paradox using the magnetic Rayleigh–Taylor instability. Nature Astron. 6, 942–950.CrossRefGoogle Scholar
Jun, B.-I. & Norman, M.L. 1996 On the origin of radial magnetic fields in young supernova remnants. Astrophys. J. 472, 245–256.CrossRefGoogle Scholar
Jun, B.-I., Norman, M.L. & Stone, J.M. 1995 The non-linear growth of the magnetic Rayleigh–Taylor instability. Astrophys. J. 453, 332–349.CrossRefGoogle Scholar
Kadoch, B., Kolomenskiy, D., Angot, P. & Schneider, K. 2012 A volume penalization method for incompressible flows and scalar advection-diffusion with moving obstacles. J. Comput. Phys. 231, 4365–4383.CrossRefGoogle Scholar
Keppens, R., Xia, C. & Porth, O. 2015 Solar prominences: ‘double doublek boil and bubble’. Astrophys. J. Lett. 806, L13.CrossRefGoogle Scholar
Kord, A. & Capecelatro, J. 2019 Optimal perturbations for controlling the growth of a Rayleigh–Taylor instability. J. Fluid Mech. 876, 150–185.CrossRefGoogle Scholar
Kraichnan, R.H. 1965 Inertial-range spectrum of hydromagnetic turbulence. Phys. Fluids 8 (7), 1385–1387.CrossRefGoogle Scholar
Leroy, J.L. 1989 Observation of Prominence Magnetic Fields, pp. 77–113. Springer.Google Scholar
Livescu, D. 2013 Numerical simulations of two-fluid turbulent mixing at large density ratios and applications to the Rayleigh–Taylor instability. Phil. Trans. R. Soc. A 371, 20120185.CrossRefGoogle Scholar
Llor, A. & Bailly, P. 2003 A new turbulent two-field concept for modeling Rayleigh–Taylor, Richtmyer–Meshkov, and Kelvin–Helmholtz mixing layers. Laser Part. Beams 21, 311–315.CrossRefGoogle Scholar
Lumley, J.L. 1967 Similarity and the turbulent energy spectrum. Phys. Fluids 10 (4), 855–858.CrossRefGoogle Scholar
Maffioli, A., Brethouwer, G. & Lindborg, E. 2016 Mixing efficiency in stratified turbulence. J. Fluid Mech. 794, R3.CrossRefGoogle Scholar
Mason, J., Perez, J.C., Boldyrev, S. & Cattaneo, F. 2012 Numerical simulations of strong incompressible magnetohydrodynamic turbulence. Phys. Plasma 19, 055902.CrossRefGoogle Scholar
Matsakos, T., Uribe, A. & Königl, A. 2015 Classification of magnetized star-planet interactions: bow shocks, tails, and inspiraling flows. Astron. Astrophys. 578, A6.CrossRefGoogle Scholar
Miura, A. & Pritchett, P.L. 1982 Nonlocal stability analysis of the MHD Kelvin–Helmholtz instability in a compressible plasma. J. Geophys. Res. Space Phys. 87 (A9), 7431–7444.CrossRefGoogle Scholar
Morales, J.A., Leroy, M., Bos, W.J.T. & Schneider, K. 2014 Simulation of confined magnetohydrodynamic flows with Dirichlet boundary conditions using a pseudo spectral method with volume penalization. J. Comput. Phys. 274, 64–94.CrossRefGoogle Scholar
Morgan, B.E. & Black, W.J. 2020 Parametric investigation of the transition to turbulence in Rayleigh–Taylor mixing. Physica D 402, 132223.CrossRefGoogle Scholar
Muller, W.-C. & Grappin, R. 2005 Spectral energy dynamics in magnetohydrodynamic turbulence. Phys. Rev. Lett. 95, 114502.CrossRefGoogle ScholarPubMed
O'Gorman, P.A. & Pullin, D. 2005 Effect of Schmidt number on the velocity-scalar cospectrum in isotropic turbulence with a mean scalar gradient. J. Fluid Mech. 532, 111–140.CrossRefGoogle Scholar
Pekurovsky, D. 2012 P3DFFT: a framework for parallel computations of Fourier transforms in three dimensions. J. Sci. Comput. 34 (4), C192–C209.Google Scholar
Peltier, W.R. & Caufield, C.P. 2003 Mixing efficiency in stratified shear flows. Annu. Rev. Fluid Mech. 35, 135–167.CrossRefGoogle Scholar
Perez, J.C., Mason, J., Boldyrev, S. & Cattaneo, F. 2012 On the energy spectrum of strong magnetohydrodynamic turbulence. Phys. Rev. X 2, 041005.Google Scholar
Porth, O., Komissarov, S. & Keppens, R. 2014 Rayleigh–Taylor instability in magnetohydrodynamic simulations of the Crab nebula. Mon. Not. R. Astron. Soc. 443, 547–558.CrossRefGoogle Scholar
Poujade, O. & Peybernes, M. 2010 Growth rate of Rayleigh–Taylor turbulent mixing layers with the foliation approach. Phys. Rev. E 81, 016316.CrossRefGoogle ScholarPubMed
Ramshaw, J.D. 1998 Simple model for linear and nonlinear mixing at unstable fluid interfaces in spherical geometry. Phys. Rev. E 60, 1775.CrossRefGoogle Scholar
Rayleigh, L. 1882 Investigation of the character of the equilibrium of an incompressible heavy fluid of variable density. Proc. Lond. Math. Soc. 14, 170.CrossRefGoogle Scholar
Remington, B.A., et al. 2019 Rayleigh–Taylor instabilities in high-energy density settings on the national ignition facility. Proc. Natl Acad. Sci. 116 (37), 18233–18238.CrossRefGoogle ScholarPubMed
Ruderman, M.S., Terradas, J. & Ballester, J.L. 2014 Rayleigh–Taylor instabilities with sheared magnetic fields. Astrophys. J. 785, 110.CrossRefGoogle Scholar
Ryutova, M., Berger, T., Frank, Z., Tarbell, T. & Title, A. 2010 Observation of plasma instabilities in quiescent prominences. Solar Phys. 267, 75–94.CrossRefGoogle Scholar
Scase, M.M., Baldwin, K.A. & Hill, R.J.A. 2017 Rotating Rayleigh–Taylor instability. Phys. Rev. Fluids 2, 024801.CrossRefGoogle Scholar
Sharp, D.H. 1984 An overview of Rayleigh–Taylor instability. Physica 12D, 3–18.Google Scholar
Soulard, O. 2012 Implications of the Monin-Yaglom relation for Rayleigh–Taylor turbulence. Phys. Rev. Lett. 109, 254501.CrossRefGoogle ScholarPubMed
Soulard, O. & Gréa, B.-J. 2017 Influence of zero-modes on the inertial-range anisotropy of Rayleigh–Taylor and unstably stratified homogeneous turbulence. Phys. Rev. Fluids 2, 074603.CrossRefGoogle Scholar
Soulard, O. & Griffond, J. 2012 Inertial range anisotropy in Rayleigh–Taylor turbulence. Phys. Fluids 24, 025101.CrossRefGoogle Scholar
Soulard, O., Griffond, J. & Gréa, B.-J. 2014 Large-scale analysis of self-similar unstably stratified homogeneous turbulence. Phys. Fluids 26, 015110.CrossRefGoogle Scholar
Soulard, O., Griffond, J. & Gréa, B.-J. 2015 Large-scale analysis of unconfined self-similar Rayleigh–Taylor turbulence. Phys. Fluids 27, 095103.CrossRefGoogle Scholar
Soulard, O., Griffond, J. & Gréa, B.-J. 2016 Influence of the mixing parameter on the second-order moments of velocity and concentration in Rayleigh–Taylor turbulence. Phys. Fluids 28, 065107.CrossRefGoogle Scholar
Sreenivasan, K.R. 1996 The passive scalar spectrum and the Obukhov-Corrsin constant. Phys. Fluids 8 (1), 189–196.CrossRefGoogle Scholar
Stone, J.M. & Gardiner, T. 2007 a The magnetic Rayleigh–Taylor instability in three dimensions. Astrophys. J. 671, 1726–1735.CrossRefGoogle Scholar
Stone, J.M. & Gardiner, T. 2007 b Nonlinear evolution of the magnetohydrodynamic Rayleigh–Taylor instability. Phys. Fluids 19, 094104.CrossRefGoogle Scholar
Taylor, G.I. 1950 The instability of liquid surfaces when accelerated in a direction perpendicular to their planes. Proc. R. Soc. Lond. 192, 201.Google Scholar
Terradas, J., Oliver, R. & Ballester, J.L. 2012 The role of Rayleigh–Taylor instabilities in filament threads. Astron. Astrophys. 541, A102.CrossRefGoogle Scholar
Vickers, E., Ballai, I. & Erdélyi, R. 2020 Magnetic Rayleigh–Taylor instability at a contact discontinuity with an oblique magnetic field. Astron. Astrophys. 634, A96.CrossRefGoogle Scholar
Walsh, C.A. & Clark, D.S. 2021 Biermann battery magnetic fields in ICF capsules: total magnetic flux generation. Phys. Plasmas 28, 092705.CrossRefGoogle Scholar
Winters, K., Lombard, P.N., Riley, J.J. & D'Asaro, A. 1995 Available potential energy and mixing in density-stratified fluids. J. Fluid Mech. 289, 115–128.CrossRefGoogle Scholar
Youngs, D.L. 1994 Numerical simulation of mixing by Rayleigh–Taylor and Richtmyer–Meshkov instabilities. Laser Part. Beams 12, 725–750.CrossRefGoogle Scholar
Zhou, Y. 2017 Rayleigh–Taylor and Richtmyer–Meshkov instability induced flow, turbulence and mixing I. Phys. Rep. 720–722, 1–136.Google Scholar
Zhou, Y., Matthaeus, W.H. & Dmitruk, P. 2004 Colloquium: magnetohydrodynamic turbulence and time scales in astrophysical and space plasmas. Rev. Mod. Phys. 76, 1015–1035.CrossRefGoogle Scholar
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