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Turbulent Rayleigh–Bénard convection in spherical shells

Published online by Cambridge University Press:  10 August 2015

Thomas Gastine ,
Johannes Wicht  and
Jonathan M. Aurnou
Thomas Gastine*
Affiliation:
Max Planck Institut für Sonnensystemforschung, Justus-von-Liebig-Weg 3, 37077 Göttingen, Germany
Johannes Wicht
Affiliation:
Max Planck Institut für Sonnensystemforschung, Justus-von-Liebig-Weg 3, 37077 Göttingen, Germany
Jonathan M. Aurnou
Affiliation:
Department of Earth, Planetary and Space Sciences, University of California, Los Angeles, CA 90095-1567, USA
*
†Email address for correspondence: gastine@mps.mpg.de
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Abstract

We simulate numerically Boussinesq convection in non-rotating spherical shells for a fluid with a Prandtl number of unity and for Rayleigh numbers up to $10^{9}$. In this geometry, curvature and radial variations of the gravitational acceleration yield asymmetric boundary layers. A systematic parameter study for various radius ratios (from ${\it\eta}=r_{i}/r_{o}=0.2$ to ${\it\eta}=0.95$) and gravity profiles allows us to explore the dependence of the asymmetry on these parameters. We find that the average plume spacing is comparable between the spherical inner and outer bounding surfaces. An estimate of the average plume separation allows us to accurately predict the boundary layer asymmetry for the various spherical shell configurations explored here. The mean temperature and horizontal velocity profiles are in good agreement with classical Prandtl–Blasius laminar boundary layer profiles, provided the boundary layers are analysed in a dynamical frame that fluctuates with the local and instantaneous boundary layer thicknesses. The scaling properties of the Nusselt and Reynolds numbers are investigated by separating the bulk and boundary layer contributions to the thermal and viscous dissipation rates using numerical models with ${\it\eta}=0.6$ and with gravity proportional to $1/r^{2}$. We show that our spherical models are consistent with the predictions of Grossmann & Lohse’s (J. Fluid Mech., vol. 407, 2000, pp. 27–56) theory and that $\mathit{Nu}(\mathit{Ra})$ and $\mathit{Re}(\mathit{Ra})$ scalings are in good agreement with plane layer results.

JFM classification

Boundary Layers: Boundary Layers Geophysical and Geological Flows: Geophysical and Geological Flows
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Journal of Fluid Mechanics , Volume 778 , 10 September 2015 , pp. 721 - 764
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© 2015 Cambridge University Press 

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