Hostname: page-component-cd9895bd7-gvvz8 Total loading time: 0 Render date: 2024-12-19T06:54:35.408Z Has data issue: false hasContentIssue false
  • English
  • Français

The effects of a Robin boundary condition on thermal convection in a rotating spherical shell

Published online by Cambridge University Press:  17 May 2021

Thibaut T. Clarté Open the ORCID record for Thibaut T. Clarté [Opens in a new window] ,
Nathanaël Schaeffer Open the ORCID record for Nathanaël Schaeffer [Opens in a new window] ,
Stéphane Labrosse Open the ORCID record for Stéphane Labrosse [Opens in a new window]  and
Jérémie Vidal Open the ORCID record for Jérémie Vidal [Opens in a new window]
Thibaut T. Clarté*
Affiliation:
Univ. Grenoble Alpes, Univ. Savoie Mont Blanc, CNRS, IRD, IFSTTAR, ISTerre, 38000Grenoble, France Université de Lyon, ENSL, UCBL, Laboratoire LGLTPE, 15 parvis René Descartes, BP7000, 69342Lyon, CEDEX 07, France
Nathanaël Schaeffer
Affiliation:
Univ. Grenoble Alpes, Univ. Savoie Mont Blanc, CNRS, IRD, IFSTTAR, ISTerre, 38000Grenoble, France
Stéphane Labrosse
Affiliation:
Université de Lyon, ENSL, UCBL, Laboratoire LGLTPE, 15 parvis René Descartes, BP7000, 69342Lyon, CEDEX 07, France
Jérémie Vidal
Affiliation:
Univ. Grenoble Alpes, Univ. Savoie Mont Blanc, CNRS, IRD, IFSTTAR, ISTerre, 38000Grenoble, France
*
†Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

Convection in a spherical shell is widely used to model fluid layers of planets and stars. The choice of thermal boundary conditions in such models is not always straightforward. To understand the implications of this choice, we report on the effects of the thermal boundary condition on thermal convection, in terms of instability onset, fully developed transport properties and flow structure. We impose a Robin boundary condition, enforcing linear coupling between the temperature anomaly and its radial derivative, with the Biot number ${{\textit {Bi}}}$ as a proportionality factor in non-dimensional form. Varying ${{\textit {Bi}}}$ allows us to transition from fixed temperature for ${{\textit {Bi}}}=+\infty$, to imposed heat flux for ${{\textit {Bi}}}=0$. We find that the onset of convection is only affected by ${{\textit {Bi}}}$ in the non-rotating case. Far from onset, considering an effective Rayleigh number and a generalized Nusselt number, we show that the Nusselt and Péclet numbers follow standard universal scaling laws, independent of ${{\textit {Bi}}}$ in all cases considered. However, for the non-rotating limit, the large-scale flow structure keeps the signature of the boundary condition with more vigorous large scales for smaller ${{\textit {Bi}}}$, even though the global heat transfer and kinetic energy are the same. For all practical purposes, the Robin condition can be safely replaced by a fixed flux when ${{\textit {Bi}}} \lesssim 0.03$ and by a fixed temperature for ${{\textit {Bi}}}\gtrsim 30$. For turbulent rapidly rotating convection, the thermal boundary condition does not seem to have any impact, once the effective numbers are considered and a reference temperature profile has been chosen.

JFM classification

Geophysical and Geological Flows: Rotating flows Turbulent Flows: Turbulent convection
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 918 , 10 July 2021 , A36
Check for updates
Copyright
© The Author(s), 2021. Published by Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

Bénard, H. 1900 Les tourbillons cellulaires dans une nappe liquide propageant de la chaleur par convection en régime permanent. Rev. Gen. Sci. Pures Appl. Bull. Assoc. 11, 1309–1328.Google Scholar
Boltzmann, L. 1884 Ableitung des stefan'schen gesetzes, betreffend die abhängigkeit der wärmestrahlung von der temperatur aus der electromagnetischen lichttheorie. Ann. Phys. 258 (6), 291–294.CrossRefGoogle Scholar
Busse, F.H. 1970 Thermal instabilities in rapidly rotating systems. J. Fluid Mech. 44 (3), 441–460.CrossRefGoogle Scholar
Busse, F.H. 1989 Fundamentals of thermal convection. In Mantle Convection. Plate Tectonics and Global Dynamics (ed. W.R. Peltier), The fluid mechanics of astrophysics and geophysics, vol. 4, chap. 2, pp. 23–95. Gordon and Breach.Google Scholar
Busse, F.H. & Carrigan, C.R. 1976 Laboratory simulation of thermal convection in rotating planets and stars. Science 191 (4222), 81–83.CrossRefGoogle ScholarPubMed
Busse, F.H. & Or, A.C. 1986 Convection in a rotating cylindrical annulus: thermal Rossby waves. J. Fluid Mech. 166, 173–187.CrossRefGoogle Scholar
Busse, F.H. & Riahi, N. 1980 Nonlinear convection in a layer with nearly insulating boundaries. J. Fluid Mech. 96 (2), 243–256.CrossRefGoogle Scholar
Calkins, M.A., Hale, K., Julien, K., Nieves, D., Driggs, D. & Marti, P. 2015 The asymptotic equivalence of fixed heat flux and fixed temperature thermal boundary conditions for rapidly rotating convection. J. Fluid Mech. 784, R2.CrossRefGoogle Scholar
Calzavarini, E., Lohse, D., Toschi, F. & Tripiccione, R. 2005 Rayleigh and Prandtl number scaling in the bulk of Rayleigh–Bénard turbulence. Phys. Fluids 17 (5), 055107.CrossRefGoogle Scholar
Chandrasekhar, S. 1953 The instability of a layer of fluid heated below and subject to Coriolis forces. P. R. Soc. A 217 (1130), 306–327.Google Scholar
Chandrasekhar, S. 1961 Hydrodynamic and Hydrodynamic Stability. Oxford University Press.Google Scholar
Chapman, C.J. & Proctor, M.R.E. 1980 Nonlinear Rayleigh–Bénard convection between poorly conducting boundaries. J. Fluid Mech. 101 (4), 759–782.CrossRefGoogle Scholar
Christensen, U.R. 2002 Zonal flow driven by strongly supercritical convection in rotating spherical shells. J. Fluid Mech. 470, 115–133.CrossRefGoogle Scholar
Collins, G.C. & Goodman, J.C. 2007 Enceladus’ south polar sea. Icarus 189 (1), 72–82.CrossRefGoogle Scholar
Cross, M.C. & Hohenberg, P.C. 1993 Pattern formation outside of equilibrium. Rev. Mod. Phys. 65 (3), 851–1112.CrossRefGoogle Scholar
Dziewonski, A.M. & Anderson, D.L. 1981 Preliminary reference earth model. Phys. Earth Planet. Inter. 25 (4), 297–356.CrossRefGoogle Scholar
Falsaperla, P. & Mulone, G. 2010 Stability in the rotating Bénard problem with Newton–Robin and fixed heat flux boundary conditions. Mech. Res. Commun. 37 (1), 122–128.CrossRefGoogle Scholar
Garcia, F., Chambers, F.R.N. & Watts, A.L. 2018 Onset of low Prandtl number thermal convection in thin spherical shells. Phys. Rev. Fluids 3 (2), 024801.CrossRefGoogle Scholar
Garcia, F., Sánchez, J. & Net, M. 2008 Antisymmetric polar modes of thermal convection in rotating spherical fluid shells at high Taylor numbers. Phys. Rev. Lett. 101 (19), 194501.CrossRefGoogle ScholarPubMed
Gastine, T., Wicht, J. & Aubert, J. 2016 Scaling regimes in spherical shell rotating convection. J. Fluid Mech. 808, 690–732.CrossRefGoogle Scholar
Gastine, T., Wicht, J. & Aurnou, J.M. 2015 Turbulent Rayleigh–Bénard convection in spherical shells. J. Fluid Mech. 778, 721–764.CrossRefGoogle Scholar
Gillet, N. & Jones, C.A. 2006 The quasi-geostrophic model for rapidly rotating spherical convection outside the tangent cylinder. J. Fluid Mech. 554, 343–369.CrossRefGoogle Scholar
Gilman, P.A. 1977 Nonlinear dynamics of Boussinesq convection in a deep rotating spherical shell-I. Geophys. Astrophys. Fluid Dyn. 8 (1), 93–135.CrossRefGoogle Scholar
Grigné, C., Labrosse, S. & Tackley, P.J. 2007 Convection under a lid of finite conductivity: heat flux scaling and application to continents. J. Geophys. Res. Solid Earth 112 (8), B08402.Google Scholar
Guervilly, C., Cardin, P. & Schaeffer, N. 2019 Turbulent convective length scale in planetary cores. Nature 570 (7761), 368–371.CrossRefGoogle ScholarPubMed
Guillou, L. & Jaupart, C. 1995 On the effect of continents on mantle convection. J. Geophys. Res. 100 (B12), 24217–24238.CrossRefGoogle Scholar
Ishiwatari, M., Takehiro, S. & Hayashi, Y. 1994 The effects of thermal conditions on the cell sizes of two-dimensional convection. J. Fluid Mech. 281, 33–50.CrossRefGoogle Scholar
Iyer, K.P., Scheel, J.D., Schumacher, J. & Sreenivasan, K.R. 2020 Classical 1/3 scaling of convection holds up to $Ra = 1015$. Proc. Natl Acad. Sci. USA 117 (14), 7594–7598.CrossRefGoogle ScholarPubMed
Johnston, H. & Doering, C.R. 2009 Comparison of turbulent thermal convection between conditions of constant temperature and constant flux. Phys. Rev. Lett. 102, 064501.CrossRefGoogle ScholarPubMed
Kaplan, E.J., Schaeffer, N., Vidal, J. & Cardin, P. 2017 Subcritical thermal convection of liquid metals in a rapidly rotating sphere. Phys. Rev. Lett. 119, 094501.CrossRefGoogle Scholar
King, E.M. & Aurnou, J.M. 2013 Turbulent convection in liquid metal with and without rotation. Proc. Natl Acad. Sci. USA 110 (17), 6688–6693.CrossRefGoogle ScholarPubMed
King, E.M., Stellmach, S. & Aurnou, J.M. 2012 Heat transfer by rapidly rotating Rayleigh–Bénard convection. J. Fluid Mech. 691, 568–582.CrossRefGoogle Scholar
Kreith, F. 2000 The CRC Handbook of Thermal Engineering. Springer.Google Scholar
Labrosse, S., Hernlund, J.W. & Hirose, K. 2015 Fractional melting and freezing in the deep mantle and implications for the formation of a basal magma ocean. Early Earth: Accret. Differ. Geophys. Monogr. 212, 123–142.Google Scholar
Long, R.S., Mound, J.E., Davies, C.J. & Tobias, S.M. 2020 Scaling behaviour in spherical shell rotating convection with fixed-flux thermal boundary conditions. J. Fluid Mech. 889, A7.CrossRefGoogle Scholar
Maas, C. & Hansen, U. 2015 Effects of earth's rotation on the early differentiation of a terrestrial magma ocean. J. Geophys. Res. 120 (11), 7508–7525.CrossRefGoogle Scholar
Monville, R., Vidal, J., Cébron, D. & Schaeffer, N. 2019 Rotating double-diffusive convection in stably stratified planetary cores. Geophys. J. Intl 219 (Supplement 1), S195–S218.CrossRefGoogle Scholar
O'Sullivan, C.T. 1990 Newton's law of cooling – a critical assessment. Am. J. Phys. 58 (10), 956–960.CrossRefGoogle Scholar
Phillips, A.C. 2013 The Physics of Stars. John Wiley & Sons.Google Scholar
Plumley, M. & Julien, K. 2019 Scaling laws in Rayleigh–Benard convection. Earth Space Sci. 6 (9), 1580–1592.CrossRefGoogle Scholar
Rayleigh, Lord 1916 On convection currents in a horizontal layer of fluid, when the higher temperature is on the under side. Lond. Edinb. Dubl. Phil. Mag. 32 (192), 529–546.CrossRefGoogle Scholar
Savijärvi, H. 1994 Meteorology of the terrestrial planets. Surv. Geophys. 15 (6), 755–773.CrossRefGoogle Scholar
Schaeffer, N. 2013 Efficient spherical harmonic transforms aimed at pseudospectral numerical simulations. Geochem. Geophys. Geosyst. 14 (3), 751–758.CrossRefGoogle Scholar
Snyder, D., Gier, E. & Carmichael, I. 1994 Experimental determination of the thermal conductivity of molten CaMgSi$_2$O$_6$ and the transport of heat through magmas. J. Geophys. Res. 99 (B8), 15503–15516.CrossRefGoogle Scholar
Solomatov, V. 2007 9.04 – magma oceans and primordial mantle differentiation. In Treatise on Geophysics (ed. G. Schubert), pp. 91–119. Elsevier.CrossRefGoogle Scholar
Sotin, C. & Tobie, G. 2004 Internal structure and dynamics of the large icy satellites. C. R. Phys. 5 (7), 769–780.CrossRefGoogle Scholar
Sparrow, E.M., Goldstein, R.J. & Jonsson, V.K. 1964 Thermal instability in a horizontal fluid layer: effect of boundary conditions and non-linear temperature profile. J. Fluid Mech. 18 (4), 513–528.CrossRefGoogle Scholar
Takehiro, S., Ishiwatari, M., Nakajima, K. & Hayashi, Y. 2002 Linear stability of thermal convection in rotating systems with fixed heat flux boundaries. Geophys. Astrophys. Fluid Dyn. 96 (6), 439–459.CrossRefGoogle Scholar
Verzicco, R. & Sreenivasan, K.R. 2008 A comparison of turbulent thermal convection between conditions of constant temperature and constant heat flux. J. Fluid Mech. 595, 203–219.CrossRefGoogle Scholar
Vidal, J. & Schaeffer, N. 2015 Quasi-geostrophic modes in the Earth's fluid core with an outer stably stratified layer. Geophys. J. Intl 202 (3), 2182–2193.CrossRefGoogle Scholar
Yadav, R.K., Gastine, T., Christensen, U.R., Duarte, L.D.V. & Reiners, A. 2015 Effect of shear and magnetic field on the heat-transfer efficiency of convection in rotating spherical shells. Geophys. J. Intl 204 (2), 1120–1133.CrossRefGoogle Scholar