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Rayleigh–Bénard convection in a creeping solid with melting and freezing at either or both its horizontal boundaries

Published online by Cambridge University Press:  03 May 2018

Stéphane Labrosse Open the ORCID record for Stéphane Labrosse [Opens in a new window] ,
Adrien Morison ,
Renaud Deguen  and
Thierry Alboussière Open the ORCID record for Thierry Alboussière [Opens in a new window]
Stéphane Labrosse*
Affiliation:
Université de Lyon, ENSL, UCBL, CNRS, LGL-TPE, 46 allée d’Italie, F-69364 Lyon, France
Adrien Morison
Affiliation:
Université de Lyon, ENSL, UCBL, CNRS, LGL-TPE, 46 allée d’Italie, F-69364 Lyon, France
Renaud Deguen
Affiliation:
Université de Lyon, ENSL, UCBL, CNRS, LGL-TPE, 46 allée d’Italie, F-69364 Lyon, France
Thierry Alboussière
Affiliation:
Université de Lyon, ENSL, UCBL, CNRS, LGL-TPE, 46 allée d’Italie, F-69364 Lyon, France
*
†Email address for correspondence: stephane.labrosse@ens-lyon.fr
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Abstract

Solid-state convection can take place in the rocky or icy mantles of planetary objects, and these mantles can be surrounded above or below or both by molten layers of similar composition. A flow towards the interface can proceed through it by changing phase. This behaviour is modelled by a boundary condition taking into account the competition between viscous stress in the solid, which builds topography of the interface with a time scale $\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D702}}$ , and convective transfer of the latent heat in the liquid from places of the boundary where freezing occurs to places of melting, which acts to erase topography, with a time scale $\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D719}}$ . The ratio $\unicode[STIX]{x1D6F7}=\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D719}}/\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D702}}$ controls whether the boundary condition is the classical non-penetrative one ($\unicode[STIX]{x1D6F7}\rightarrow \infty$ ) or allows for a finite flow through the boundary (small $\unicode[STIX]{x1D6F7}$ ). We study Rayleigh–Bénard convection in a plane layer subject to this boundary condition at either or both its boundaries using linear and weakly nonlinear analyses. When both boundaries are phase-change interfaces with equal values of $\unicode[STIX]{x1D6F7}$ , a non-deforming translation mode is possible with a critical Rayleigh number equal to $24\unicode[STIX]{x1D6F7}$ . At small values of $\unicode[STIX]{x1D6F7}$ , this mode competes with a weakly deforming mode having a slightly lower critical Rayleigh number and a very long wavelength, $\unicode[STIX]{x1D706}_{c}\sim 8\sqrt{2}\unicode[STIX]{x03C0}/3\sqrt{\unicode[STIX]{x1D6F7}}$ . Both modes lead to very efficient heat transfer, as expressed by the relationship between the Nusselt and Rayleigh numbers. When only one boundary is subject to a phase-change condition, the critical Rayleigh number is $\mathit{Ra}_{c}=153$ and the critical wavelength is $\unicode[STIX]{x1D706}_{c}=5$ . The Nusselt number increases approximately two times faster with the Rayleigh number than in the classical case with non-penetrative conditions, and the average temperature diverges from $1/2$ when the Rayleigh number is increased, towards larger values when the bottom boundary is a phase-change interface.

JFM classification

Convection: Buoyancy-driven instability Geophysical and Geological Flows: Mantle convection Phase change: Solidification/melting

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Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 846 , 10 July 2018 , pp. 5 - 36
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© 2018 Cambridge University Press 

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