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Bénard convection in a slowly rotating penny-shaped cylinder subject to constant heat flux boundary conditions

Published online by Cambridge University Press:  02 November 2022

A.M. Soward Open the ORCID record for A.M. Soward [Opens in a new window] ,
L. Oruba Open the ORCID record for L. Oruba [Opens in a new window]  and
E. Dormy Open the ORCID record for E. Dormy [Opens in a new window]
A.M. Soward*
Affiliation:
School of Mathematics, Statistics and Physics, Newcastle University, Newcastle upon Tyne NE1 7RU, UK
L. Oruba*
Affiliation:
Laboratoire Atmosphères Milieux Observations Spatiales (LATMOS/IPSL), Sorbonne Université, UVSQ, CNRS, Paris, France
E. Dormy*
Affiliation:
Département de Mathématiques et Applications, UMR-8553, École Normale Supérieure, CNRS, PSL University, 75005 Paris, France
*
†Email addresses for correspondence: andrew.soward@ncl.ac.uk, ludivine.oruba@latmos.ipsl.fr, Emmanuel.Dormy@ens.fr
†Email addresses for correspondence: andrew.soward@ncl.ac.uk, ludivine.oruba@latmos.ipsl.fr, Emmanuel.Dormy@ens.fr
†Email addresses for correspondence: andrew.soward@ncl.ac.uk, ludivine.oruba@latmos.ipsl.fr, Emmanuel.Dormy@ens.fr
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Abstract

We consider axisymmetric Boussinesq convection in a shallow cylinder of radius $L$ and depth $H (\ll L)$, which rotates with angular velocity $\varOmega$ about its axis of symmetry aligned to the vertical. Constant heat flux boundary conditions, top and bottom, are adopted, for which the onset of instability occurs on a long horizontal length scale provided that $\varOmega$ is sufficiently small. We investigate the nonlinear development by well-established two-scale asymptotic expansion methods. Comparisons of the results with the direct numerical simulations (DNS) of the primitive governing equations are good at sufficiently large Prandtl number $\sigma$. As $\sigma$ is reduced, the finite amplitude range of applicability of the asymptotics reduces in concert. Though the large meridional convective cell, predicted by the DNS, is approximated adequately by the asymptotics, the azimuthal flow fails almost catastrophically, because of significant angular momentum transport at small $\sigma$, exacerbated by the cylindrical geometry. To appraise the situation, we propose hybrid methods that build on the meridional streamfunction $\psi$ derived from the asymptotics. With $\psi$ given, we solve the now linear azimuthal equation of motion for the azimuthal velocity $v$ by DNS. Our ‘hybrid’ methods enable us to explain features of the flow at large Rayleigh number, found previously by Oruba et al. (J. Fluid Mech., vol. 812, 2017, pp. 890–904).

JFM classification

Geophysical and Geological Flows: Atmospheric flows Geophysical and Geological Flows: Rotating flows
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 951 , 25 November 2022 , A5
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Copyright
© The Author(s), 2022. Published by Cambridge University Press

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References

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