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Characteristics of a precessing flow under the influence of a convecting temperature field in a spheroidal shell

  • Jan Vormann (a1) and Ulrich Hansen (a1)


We present results from direct numerical simulations of flows in spherical and oblate spheroidal shells, driven both by precession and thermal convection, with Ekman number $Ek=10^{-4}$ , non-diffusive Rayleigh numbers from $Ra=0.1$ to $Ra=10$ and unity Prandtl number. The applied precessional forcing spans seven orders of magnitude. Our experiments show a clear transition between a convective state and a precessing flow that can be approximated by a reduced dynamical model. The change in the flow is apparent in visualizations and a decomposition of the velocity into symmetric and antisymmetric components. For the flow dominated by precession, some parameter combinations show two stable solutions that can be realized by a hysteresis or a strong thermal forcing. An increase of the Rayleigh number at a constant precession rate exhibits established scaling properties for the heat transfer, with exponents $2/7$ and $6/5$ .

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Abramowitz, M. & Stegun, I. A. 1972 Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 10th edn. Dover.
Agnor, C. B., Canup, R. M. & Levison, H. F. 1999 On the character and consequences of large impacts in the late stage of terrestrial planet formation. Icarus 142 (1), 219237.
Ahlers, G., Grossmann, S. & Lohse, D. 2009 Heat transfer and large scale dynamics in turbulent Rayleigh–Bénard convection. Rev. Mod. Phys. 81, 503537.
Aitta, A. 2012 Venus’ internal structure, temperature and core composition. Icarus 218 (2), 967974.
Andrault, D., Monteux, J., Le Bars, M. & Samuel, H. 2016 The deep Earth may not be cooling down. Earth Planet. Sci. Lett. 443, 195203.
Aurnou, J. M. 2007 Planetary core dynamics and convective heat transfer scaling. Geophys. Astrophys. Fluid Dyn. 101 (5), 327345.
Busse, F. H. 1968 Steady fluid flow in a precessing spheroidal shell. J. Fluid Mech. 33 (4), 739751.
Cébron, D. 2015 Bistable flows in precessing spheroids. Fluid Dyn. Res. 47, 025504.
Cébron, D., Laguerre, R., Noir, J. & Schaeffer, N. 2019 Precessing spherical shells: flows, dissipation, dynamo and the lunar core. Geophys. J. Intl 219 (suppl. 1), S34S57.
Cébron, D., Maubert, P. & Le Bars, M. 2010 Tidal instability in a rotating and differentially heated ellipsoidal shell. Geophys. J. Intl 182 (3), 13111318.
Cheng, J. S., Aurnou, J. M., Julien, K. & Kunnen, R. P. J. 2018 A heuristic framework for next-generation models of geostrophic convective turbulence. Geophys. Astrophys. Fluid Dyn. 112 (4), 277300.
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), 117.
Christensen, U. R. & Wicht, J. 2015 Numerical dynamo simulations. In Core Dynamics, 2nd edn (ed. Gerald, S.), Treatise on Geophysics, vol. 8, pp. 245277. Elsevier.
Ćuk, M. & Stewart, S. T. 2012 Making the Moon from a fast-spinning Earth: a giant impact followed by resonant despinning. Science 338 (6110), 10471052.
Deville, M. O., Fischer, P. F. & Mund, E. H. 2002 High-Order Methods for Incompressible Fluid Flow, 1st edn. Cambridge Monographs on Applied and Computational Mathematics, vol. 9. Cambridge University Press.
Dormy, E., Soward, A. M., Jones, C. A., Jault, D. & Cardin, P. 2004 The onset of thermal convection in rotating spherical shells. J. Fluid Mech. 501, 4370.
Dwyer, C. A., Stevenson, D. J. & Nimmo, F. 2011 A long-lived lunar dynamo driven by continuous mechanical stirring. Nature 479 (7372), 212214.
Dziewonski, A. M. & Anderson, D. L. 1981 Preliminary reference Earth model. Phys. Earth Planet. Inter. 25 (4), 297356.
Ernst-Hullermann, J., Harder, H. & Hansen, U. 2013 Finite volume simulations of dynamos in ellipsoidal planets. Geophys. J. Intl 195 (3), 13951405.
Evonuk, M. 2015 Convection in deformed bodies: the effect of equatorial ellipticity on convective behavior. Earth Planet. Sci. Lett. 430, 249259.
Favier, B., Grannan, A. M., Bars, M. L. & Aurnou, J. M. 2015 Generation and maintenance of bulk turbulence by libration-driven elliptical instability. Phys. Fluids 27, 066601.
Fischer, P. F. 1997 An overlapping schwarz method for spectral element solution of the incompressible Navier–Stokes equations. J. Comput. Phys. 133 (1), 84101.
Fischer, P. F. & Lottes, J. W. 2005 Hybrid Schwarz-multigrid methods for the spectral element method: extensions to Navier–Stokes. In Domain Decomposition Methods in Science and Engineering (ed. Barth, T. J. et al. ), Lecture Notes in Computational Science and Engineering, vol. 40. Springer.
Fuller, M. D. 2017 Evidence for a geodynamo driven by thermal energy in the outermost core and by precession deeper in the outer core. Intl J. Earth Sci. Geophys. 3 (1), 012.
Goepfert, O. & Tilgner, A. 2016 Dynamos in precessing cubes. New J. Phys. 18, 103019.
Greenspan, H. P. & Howard, L. N. 1963 On a time-dependent motion of a rotating fluid. J. Fluid Mech. 17 (3), 385404.
Hollerbach, R. & Kerswell, R. R. 1995 Oscillatory internal shear layers in rotating and precessing flows. J. Fluid Mech. 298, 327339.
Hvoždara, M. & Kohút, I. 2012 Gravity field due to a homogeneous oblate spheroid: simple solution form and numerical calculations. Contrib. Geophys. Geod. 41 (4), 307327.
Jones, C. A. 2015 Thermal and compositional convection in the outer core. In Core Dynamics, 2nd edn (ed. Schubert, G.), Treatise on Geophysics, vol. 8, pp. 115159. Elsevier.
Julien, K., Aurnou, J. M., Calkins, M. A., Knobloch, E., Marti, P., Stellmach, S. & Vasil, G. M. 2016 A nonlinear model for rotationally constrained convection with Ekman pumping. J. Fluid Mech. 798, 5087.
Julien, K., Legg, S., McWilliams, J. & Werne, J. 1996 Hard turbulence in rotating Rayleigh–Bénard convection. Phys. Rev. E 53 (6), R5557R5560.
Karniadakis, G. & Sherwin, S. 2013 Spectral/hp Element Methods for Computational Fluid Dynamics: Second Edition. Oxford University Press.
King, E. M., Soderlund, K. M., Christensen, U. R., Wicht, J. & Aurnou, J. M. 2010 Convective heat transfer in planetary dynamo models. Geochem. Geophys. Geosyst. 11, Q06016.
King, E. M., Stellmach, S. & Aurnou, J. M. 2012 Heat transfer by rapidly rotating Rayleigh–Bénard convection. J. Fluid Mech. 691, 568582.
King, E. M., Stellmach, S., Noir, J., Hansen, U. & Aurnou, J. M. 2009 Boundary layer control of rotating convection systems. Nature 457 (7227), 301304.
Le Bars, M. 2016 Flows driven by libration, precession, and tides in planetary cores. Phys. Rev. Fluids 1 (6), 060505.
Lemasquerier, D., Grannan, A. M., Vidal, J., Cébron, D., Favier, B., Le Bars, M. & Aurnou, J. M. 2017 Libration-driven flows in ellipsoidal shells. J. Geophys. Res. 122 (9), 19261950.
Lin, Y., Marti, P., Noir, J. & Jackson, A. 2016 Precession-driven dynamos in a full sphere and the role of large scale cyclonic vortices. Phys. Fluids 28 (6), 066601.
Lorenzani, S. & Tilgner, A. 2001 Fluid instabilities in precessing spheroidal cavities. J. Fluid Mech. 447, 111128.
Lorenzani, S. & Tilgner, A. 2003 Inertial instabilities of fluid flow in precessing spheroidal shells. J. Fluid Mech. 492, 363379.
Malkus, W. V. 1954 The heat transport and spectrum of thermal turbulence. Proc. R. Soc. Lond. A 225 (1161), 196212.
Malkus, W. V. 1968 Precession of the Earth as the cause of geomagnetism: experiments lend support to the proposal that precessional torques drive the earth’s dynamo. Science 160 (3825), 259264.
Noir, J., Cardin, P., Jault, D. & Masson, J.-P. 2003 Experimental evidence of non-linear resonance effects between retrograde precession and the tilt-over mode within a spheroid. Geophys. J. Intl 154, 407416.
Noir, J. & Cébron, D. 2013 Precession-driven flows in non-axisymmetric ellipsoids. J. Fluid Mech. 737, 412439.
Olson, P. 2013 The new core paradox. Science 342 (6157), 431432.
Plumley, M. & Julien, K. 2019 Scaling laws in Rayleigh–Bénard convection. Earth Space Sci. 6, 15801592.
Plumley, M., Julien, K., Marti, P. & Stellmach, S. 2016 The effects of Ekman pumping on quasi-geostrophic Rayleigh–Bénard convection. J. Fluid Mech. 803, 5171.
Reddy, K. S., Favier, B. & Bars, M. L. 2018 Turbulent kinematic dynamos in ellipsoids driven by mechanical forcing. Geophys. Res. Lett. 45 (4), 17411750.
Rivoldini, A., Van Hoolst, T., Verhoeven, O., Mocquet, A. & Dehant, V. 2011 Geodesy constraints on the interior structure and composition of Mars. Icarus 213 (2), 451472.
Scheel, J. D. & Schumacher, J. 2014 Local boundary layer scales in turbulent Rayleigh–Bénard convection. J. Fluid Mech. 758, 344373.
Scheel, J. D. & Schumacher, J. 2016 Global and local statistics in turbulent convection at low Prandtl numbers. J. Fluid Mech. 802, 147173.
Shraiman, B. I. & Siggia, E. D. 1990 Heat transport in high-Rayleigh-number convection. Phys. Rev. A 42 (6), 36503653.
Smith, D. E., Zuber, M. T., Phillips, R. J., Solomon, S. C., Hauck, S. A., Lemoine, F. G., Mazarico, E., Neumann, G. A., Peale, S. J., Margot, J.-L. et al. 2012 Gravity field and internal structure of Mercury from MESSENGER. Science 336 (6078), 214217.
Stefan, C., Dobrica, V. & Demetrescu, C. 2017 Core surface sub-centennial magnetic flux patches: characteristics and evolution. Earth Planets Space 69 (1), 146.
Stevenson, D. J. 2003 Planetary magnetic fields. Earth Planet. Sci. Lett. 208 (1), 111.
Tilgner, A. 2005 Precession driven dynamos. Phys. Fluids 17 (3), 034104.
Tilgner, A. & Busse, F. H. 2001 Fluid flows in precessing spherical shells. J. Fluid Mech. 426, 387396.
Tricarico, P. 2014 Multi-layer hydrostatic equilibrium of planets and synchronous moon: theory and application to Ceres and to solar system moons. Astrophys. J. 782 (2), 99.
Van Hoolst, T. 2015 Rotation of the terrestrial planets. In Physics of Terrestrial Planets and Moons, 2nd edn (ed. Gerald, S.), Treatise on Geophysics, vol. 10, pp. 121151. Elsevier.
Vanyo, J. P. & Dunn, J. R. 2000 Core precession: flow structures and energy. Geophys. J. Intl 142 (2), 409425.
Vormann, J. & Hansen, U. 2018 Numerical simulations of bistable flows in precessing spheroidal shells. Geophys. J. Intl 213 (2), 786797.
Weber, R. C., Lin, P.-Y., Garnero, E. J., Williams, Q. & Lognonné, P. 2011 Seismic detection of the lunar core. Science 331 (6015), 309312.
Wei, X. 2016 The combined effect of precession and convection on the dynamo action. Astrophys. J. 827 (2), 123.
Wei, X. & Tilgner, A. 2013 Stratified precessional flow in spherical geometry. J. Fluid Mech. 718, R2.
Zhang, K., Liao, X. & Earnshaw, P. 2004 On inertial waves and oscillations in a rapidly rotating spheroid. J. Fluid Mech. 504, 140.
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Characteristics of a precessing flow under the influence of a convecting temperature field in a spheroidal shell

  • Jan Vormann (a1) and Ulrich Hansen (a1)


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