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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: thibaut.clarte@ens-lyon.fr
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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
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Copyright
© The Author(s), 2021. Published by Cambridge University Press

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