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Fundamentals of laminar free convection in internally heated fluids at values of the Rayleigh–Roberts number up to $10^{9}$

Published online by Cambridge University Press:  11 May 2018

Kenny Vilella Open the ORCID record for Kenny Vilella [Opens in a new window] ,
Angela Limare ,
Claude Jaupart ,
Cinzia G. Farnetani ,
Loic Fourel  and
Edouard Kaminski
Kenny Vilella*
Affiliation:
Institut de Physique du Globe, Univ. Paris Diderot, Sorbonne Paris Cité, CNRS, F-75005 Paris, France Institute of Earth Sciences, Academia Sinica, Taipei 11529, Taiwan
Angela Limare
Affiliation:
Institut de Physique du Globe, Univ. Paris Diderot, Sorbonne Paris Cité, CNRS, F-75005 Paris, France
Claude Jaupart
Affiliation:
Institut de Physique du Globe, Univ. Paris Diderot, Sorbonne Paris Cité, CNRS, F-75005 Paris, France
Cinzia G. Farnetani
Affiliation:
Institut de Physique du Globe, Univ. Paris Diderot, Sorbonne Paris Cité, CNRS, F-75005 Paris, France
Loic Fourel
Affiliation:
Institut de Physique du Globe, Univ. Paris Diderot, Sorbonne Paris Cité, CNRS, F-75005 Paris, France Geological Survey of Norway, 7040 Trondheim, Norway
Edouard Kaminski
Affiliation:
Institut de Physique du Globe, Univ. Paris Diderot, Sorbonne Paris Cité, CNRS, F-75005 Paris, France
*
†Email address for correspondence: [email protected]
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Abstract

Motions in the solid mantle of silicate planets are predominantly driven by internal heat sources and occur in laminar regimes that have not been systematically investigated. Using high-resolution numerical simulations conducted in three dimensions for a large range of Rayleigh–Roberts numbers ( $5\times 10^{3}\leqslant Ra_{H}\leqslant 10^{9}$ ), we have determined the characteristics of flow in internally heated fluid layers with both rigid and free slip boundaries. Superficial planforms evolve with increasing $Ra_{H}$ from a steady-state tessellation of hexagonal cells with axial downwellings to time-dependent clusters of thin linear downwellings within large areas of nearly isothermal fluid. The transition between the two types of planforms occurs as a remarkable flow polarity reversal over a small $Ra_{H}$ range, such that downwellings go from isolated cylindrical structures encircled by upwellings to thin interconnected linear segments outlining polygonal cells. In time-dependent regimes at large values of $Ra_{H}$ , linear downwellings dominate the flow field at shallow depth but split and merge at intermediate depths into nearly cylindrical plume-like structures that go through the whole layer. With increasing $Ra_{H}$ , the number of plumes per unit area and their velocities increase whilst the amplitude of thermal anomalies decreases. Scaling laws for the main flow characteristics are derived for $Ra_{H}$ values in a $10^{6}$ – $10^{9}$ range. For given $Ra_{H}$ , plumes are significantly colder, narrower and wider apart beneath free boundaries than beneath rigid ones. From the perspective of planetary studies, these results alert to the dramatic changes of convective planform that can occur along secular cooling.

JFM classification

Convection: Convection Nonlinear Dynamical Systems: Pattern formation Convection: Plumes/thermals
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 846 , 10 July 2018 , pp. 966 - 998
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Copyright
© 2018 Cambridge University Press 

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