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Scaling behaviour of small-scale dynamos driven by Rayleigh–Bénard convection

Published online by Cambridge University Press:  09 March 2021

M. Yan Open the ORCID record for M. Yan [Opens in a new window] ,
S.M. Tobias Open the ORCID record for S.M. Tobias [Opens in a new window]  and
M.A. Calkins
M. Yan*
Affiliation:
Department of Physics, University of Colorado, Boulder, CO80309, USA
S.M. Tobias
Affiliation:
Department of Applied Mathematics, University of Leeds, Leeds LS2 9JT, UK
M.A. Calkins
Affiliation:
Department of Physics, University of Colorado, Boulder, CO80309, USA
*
†Email address for correspondence: [email protected]
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Abstract

A numerical investigation of convection-driven dynamos is carried out in the plane layer geometry. Dynamos with different magnetic Prandtl numbers $Pm$ are simulated over a broad range of the Rayleigh number $Ra$. The heat transport, as characterized by the Nusselt number $Nu$, shows an initial departure from the heat transport scaling of non-magnetic Rayleigh–Bénard convection (RBC) as the magnetic field grows in magnitude; as $Ra$ is increased further, the data suggest that $Nu$ grows approximately as $Ra^{2/7}$, but with a smaller prefactor in comparison with RBC. Viscous ($\epsilon _u$) and ohmic ($\epsilon _B$) dissipation contribute approximately equally to $Nu$ at the highest $Ra$ investigated; both ohmic and viscous dissipation approach a Reynolds-number-dependent scaling of the form $Re^a$, where $a \approx 2.8$. The ratio of magnetic to kinetic energy approaches a $Pm$-dependent constant as $Ra$ is increased, with the constant value increasing with $Pm$. The ohmic dissipation length scale depends on $Ra$ in such a way that it is always smaller, and decreases more rapidly with increasing $Ra$, than the viscous dissipation length scale for all investigated values of $Pm$.

JFM classification

MHD and Electrohydrodynamics: Dynamo theory MHD and Electrohydrodynamics: Magneto convection
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 915 , 25 May 2021 , A15
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Copyright
© The Author(s), 2021. Published by Cambridge University Press

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References

REFERENCES

Ahlers, G., Grossmann, S. & Lohse, D. 2009 Heat transfer and large scale dynamics in turbulent Rayleigh–Bénard convection. Rev. Mod. Phys. 81 (2), 503–537.CrossRefGoogle Scholar
Aurnou, J.M. & Olson, P. 2001 Experiments on Rayleigh–Bénard convection, magnetoconvection, and rotating magnetoconvection in liquid gallium. J. Fluid Mech. 430, 283–307.CrossRefGoogle Scholar
Brummell, N.H., Cattaneo, F. & Tobias, S.M. 2001 Linear and nonlinear dynamo properties of time-dependent ABC flows. Fluid Dyn. Res. 28 (4), 237–265.CrossRefGoogle Scholar
Bushby, P.J. & Favier, B. 2014 Mesogranulation and small-scale dynamo action in the quiet Sun. Astron. Astrophys. 562, A72.CrossRefGoogle Scholar
Bushby, P.J., Favier, B., Proctor, M.R.E. & Weiss, N.O. 2012 Convectively driven dynamo action in the quiet sun. Geophys. Astrophys. Fluid Dyn. 106 (4–5), 508–523.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
Castaing, B., Gunaratne, G., Heslot, F., Kadanoff, L., Libchaber, A., Thomae, S., Wu, X., Zaleski, S. & Zanetti, G. 1989 Scaling of hard thermal turbulence in Rayleigh–Bénard convection. J. Fluid Mech. 204, 1–30.CrossRefGoogle Scholar
Cattaneo, F. 1999 On the origin of magnetic fields in the quiet photosphere. Astrophys. J. Lett. 515 (1), L39.CrossRefGoogle Scholar
Cattaneo, F., Emonet, T. & Weiss, N. 2003 On the interaction between convection and magnetic fields. Astrophys. J. 588 (2), 1183–1198.CrossRefGoogle Scholar
Chandrasekhar, S. 1961 Hydrodynamic and Hydromagnetic Stability. Oxford University Press.Google Scholar
Cheng, J.S., Stellmach, S., Ribeiro, A., Grannan, A., King, E.M. & Aurnou, J.M. 2015 Laboratory-numerical models of rapidly rotating convection in planetary cores. Geophys. J. Intl 201, 1–17.CrossRefGoogle Scholar
Childress, S. & Soward, A.M. 1972 Convection-driven hydromagnetic dynamo. Phys. Rev. Lett. 29 (13), 837–839.CrossRefGoogle Scholar
Cioni, S., Chaumat, S. & Sommeria, J. 2000 Effect of a vertical magnetic field on turbulent Rayleigh–Bénard convection. Phys. Rev. E 62 (4), R4520–R4523.CrossRefGoogle ScholarPubMed
Cioni, S., Ciliberto, S. & Sommeria, J. 1997 Strongly turbulent Rayleigh–Bénard convection in mercury: comparison with results at moderate Prandtl number. J. Fluid Mech. 335, 111–140.CrossRefGoogle Scholar
Fauve, S. & Pétrélis, F. 2007 Scaling laws of turbulent dynamos. C. R. Phys. 8 (1), 87–92.CrossRefGoogle Scholar
French, M., Becker, A., Lorenzen, W., Nettelmann, N., Bethkenhagen, M, Wicht, J. & Redmer, R. 2012 Ab initio simulations for material properties along the Jupiter adiabat. Astrophys. J. Suppl. 202 (1), 5.CrossRefGoogle Scholar
Goluskin, D. & Spiegel, E.A. 2012 Convection driven by internal heating. Phys. Lett. A 377 (1–2), 83–92.CrossRefGoogle Scholar
Grossmann, S. & Lohse, D. 2000 Scaling in thermal convection: a unifying theory. J. Fluid Mech. 407, 27–56.CrossRefGoogle Scholar
Haugen, N.E.L., Brandenburg, A. & Dobler, W. 2004 Simulations of nonhelical hydromagnetic turbulence. Phys. Rev. E 70 (1), 016308.CrossRefGoogle ScholarPubMed
Jones, C.A. 2011 Planetary magnetic fields and fluid dynamos. Annu. Rev. Fluid Mech. 43, 583–614.CrossRefGoogle Scholar
Jones, C.A. & Roberts, P.H. 2000 Convection-driven dynamos in a rotating plane layer. J. Fluid Mech. 404, 311–343.CrossRefGoogle Scholar
Käpylä, P.J., Käpylä, M.J. & Brandenburg, A. 2018 Small-scale dynamos in simulations of stratified turbulent convection. Astron. Nachr. 339 (2–3), 127–133.CrossRefGoogle Scholar
Kraichnan, R.H. 1962 Turbulent thermal convection at arbitrary Prandtl number. Phys. Fluids 5 (11), 1374–1389.CrossRefGoogle Scholar
Lohse, D. & Toschi, F. 2003 Ultimate state of thermal convection. Phys. Rev. Lett. 90 (3), 034502.CrossRefGoogle ScholarPubMed
Malkus, W.V.R. 1954 The heat transport and spectrum of thermal turbulence. Proc. R. Soc. Lond. A 225 (1161), 196–212.Google Scholar
Marti, P., Calkins, M.A. & Julien, K. 2016 A computationally efficient spectral method for modeling core dynamics. Geochem. Geophys. Geosys. 17 (8), 3031–3053.CrossRefGoogle Scholar
Meneguzzi, M., Frisch, U. & Pouquet, A. 1981 Helical and nonhelical turbulent dynamos. Phys. Rev. Lett. 47 (15), 1060–1064.CrossRefGoogle Scholar
Meneguzzi, M. & Pouquet, A. 1989 Turbulent dynamos driven by convection. J. Fluid Mech. 205, 297–318.CrossRefGoogle Scholar
Moffatt, H.K. 1970 Dynamo action associated with random inertial waves in a rotating conducting fluid. J. Fluid Mech. 44, 705–719.CrossRefGoogle Scholar
Ossendrijver, M. 2003 The solar dynamo. Astron. Astrophys. Rev. 11 (4), 287–367.CrossRefGoogle Scholar
Pandey, A., Scheel, J.D. & Schumacher, J. 2018 Turbulent superstructures in Rayleigh–Bénard convection. Nat. Commun. 9 (1), 1–11.CrossRefGoogle ScholarPubMed
Parker, E.N. 1955 Hydromagnetic dynamo models. Astrophys. J. 122, 293–314.CrossRefGoogle Scholar
Pope, S.B. 2000 Turbulent Flows. Cambridge University Press.CrossRefGoogle Scholar
Pozzo, M., Davies, C.J., Gubbins, D. & Alfé, D. 2013 Transport properties for liquid silicon-oxygen-iron mixtures at Earth's core conditions. Phys. Rev. B 87, 014110.CrossRefGoogle Scholar
Qiu, X.L. & Tong, P. 2001 Large-scale velocity structures in turbulent thermal convection. Phys. Rev. E 64 (3), 036304.CrossRefGoogle ScholarPubMed
Rincon, F. 2019 Dynamo theories. J. Plasma Phys. 85 (4), 205850401.CrossRefGoogle Scholar
Schaeffer, N., Jault, D., Nataf, H.-C. & Fournier, A. 2017 Turbulent geodynamo simulations: a leap towards earth's core. Geophys. J. Intl 211 (1), 1–29.CrossRefGoogle Scholar
Scheel, J.D. & Schumacher, J. 2017 Predicting transition ranges to fully turbulent viscous boundary layers in low Prandtl number convection flows. Phys. Rev. Fluids 2, 123501.CrossRefGoogle Scholar
Schekochihin, A.A., Boldyrev, S.A. & Kulsrud, R.M. 2002 a Spectra and growth rates of fluctuating magnetic fields in the kinematic dynamo theory with large magnetic Prandtl numbers. Astrophys. J. 567 (2), 828–852.CrossRefGoogle Scholar
Schekochihin, A.A., Cowley, S.C., Hammett, G.W., Maron, J.L. & McWilliams, J.C. 2002 b A model of nonlinear evolution and saturation of the turbulent MHD dynamo. New J. Phys. 4 (1), 84.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
Schekochihin, A.A., Maron, J.L., Cowley, S.C. & McWilliams, J.C. 2002 c The small-scale structure of magnetohydrodynamic turbulence with large magnetic Prandtl numbers. Astrophys. J. 576 (2), 806–813.CrossRefGoogle Scholar
Sheyko, A., Finlay, C.C. & Jackson, A. 2016 Magnetic reversals from planetary dynamo waves. Nature 551, 551–554.CrossRefGoogle Scholar
Shraiman, B.I. & Siggia, E.D. 1990 Heat transport in high-Rayleigh-number convection. Phys. Rev. A 42 (6), 3650–3653.CrossRefGoogle ScholarPubMed
Simitev, R. & Busse, F.H. 2005 Prandtl-number dependence of convection-driven dynamos in rotating spherical fluid shells. J. Fluid Mech. 532, 365–388.CrossRefGoogle Scholar
Spalart, P.R., Moser, R.D. & Rogers, M.M. 1991 Spectral methods for the Navier–Stokes equations with one infinite and two periodic directions. J. Comput. Phys. 96, 297–324.CrossRefGoogle Scholar
Spiegel, E.A. 1965 Convective instability in a compressible atmosphere. I. Astrophys. J. 141, 1068–1090.CrossRefGoogle Scholar
Steenbeck, M., Krause, F. & Rädler, K.-H. 1966 A calculation of the mean electromotive force in an electrically conducting fluid in turbulent motion, under the influence of coriolis forces. Z. Naturforsch. 21a, 369–376.CrossRefGoogle Scholar
Stevens, R.J., Blass, A., Zhu, X., Verzicco, R. & Lohse, D. 2018 Turbulent thermal superstructures in Rayleigh–Bénard convection. Phys. Rev. Fluids 3, 041501.CrossRefGoogle Scholar
Stevens, R.J., Lohse, D. & Verzicco, R. 2011 Prandtl and Rayleigh number dependence of heat transport in high Rayleigh number thermal convection. J. Fluid Mech. 688, 31–43.CrossRefGoogle Scholar
Tobias, S.M. 2019 The turbulent dynamo. arXiv:1907.03685.Google Scholar
Tobias, S.M., Cattaneo, F. & Boldyrev, S.B. 2012 MHD dynamos and turbulence. In Ten Chapters in Turbulence (ed. P.A. Davidson, Y. Kaneda & K.R. Sreenivasan), pp. 351–404. Cambridge University Press.CrossRefGoogle Scholar
Tobias, S.M., Cattaneo, F., Boldyrev, S., Davidson, P.A., Kaneda, Y. & Sreenivasan, K.R. 2013 Ten Chapters in Turbulence, vol. 1 (1.4). Cambridge University Press, p. 1.Google Scholar
Vogt, T., Horn, S., Grannan, A.M. & Aurnou, J.M. 2018 Jump rope vortex in liquid metal convection. Proc. Natl Acad. Sci. USA 115 (50), 12674–12679.CrossRefGoogle ScholarPubMed
Yan, M., Calkins, M.A., Maffei, S., Julien, K., Tobias, S.M. & Marti, P. 2019 Heat transfer and flow regimes in quasi-static magnetoconvection with a vertical magnetic field. J. Fluid Mech. 877, 1186–1206.CrossRefGoogle Scholar
Zhu, X., Mathai, V., Stevens, R.J., Verzicco, R. & Lohse, D. 2018 Transition to the ultimate regime in two-dimensional Rayleigh–Bénard convection. Phys. Rev. Lett. 120 (14), 144502.CrossRefGoogle ScholarPubMed
Zürner, T., Liu, W., Krasnov, D. & Schumacher, J. 2016 Heat and momentum transfer for magnetoconvection in a vertical external magnetic field. Phys. Rev. E 94 (4), 043108.CrossRefGoogle Scholar