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Weakly nonlinear mode interactions in spherical Rayleigh–Bénard convection

Published online by Cambridge University Press:  09 July 2019

P. M. Mannix Open the ORCID record for P. M. Mannix [Opens in a new window]  and
A. J. Mestel Open the ORCID record for A. J. Mestel [Opens in a new window]
P. M. Mannix*
Affiliation:
Department of Mathematics, Imperial College London, London SW7 2AZ, UK
A. J. Mestel
Affiliation:
Department of Mathematics, Imperial College London, London SW7 2AZ, UK
*
†Email address for correspondence: [email protected]
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Abstract

In an annular spherical domain with separation $d$, the onset of convective motion occurs at a critical Rayleigh number $Ra=Ra_{c}$. Solving the axisymmetric linear stability problem shows that degenerate points $(d=d_{c},Ra_{c})$ exist where two modes simultaneously become unstable. Considering the weakly nonlinear evolution of these two modes, it is found that spatial resonances play a crucial role in determining the preferred convection pattern for neighbouring modes $(\ell ,\ell \pm 1)$ and non-neighbouring even modes $(\ell ,\ell \pm 2)$. Deriving coupled amplitude equations relevant to all degeneracies, we outline the possible solutions and the influence of changes in $d,Ra$ and Prandtl number $Pr$. Using direct numerical simulation (DNS) to verify all results, time periodic solutions are also outlined for small $Pr$. The $2:1$ periodic signature observed to be general for oscillations in a spherical annulus is explained using the structure of the equations. The relevance of all solutions presented is determined by computing their stability with respect to non-axisymmetric perturbations at large $Pr$.

JFM classification

Instability: Absolute/convective instability
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 874 , 10 September 2019 , pp. 359 - 390
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Copyright
© 2019 Cambridge University Press 

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Mannix and Mestel supplementary movie 1

DNS of the odd-odd $\ell=1, m=3$ periodic solution for $\mathrm{Ra}_c = 440.852, \epsilon =5.85$ and $\Pran = 0.05$. Even and odd mode components of the solution oscillate with their frequencies in a $2:1$ ratio.

Download Mannix and Mestel supplementary movie 1(Video)
Video 1.4 MB

Mannix and Mestel supplementary movie 2

DNS of the even-even $\ell=2, m=4$ periodic solution for $\mathrm{Ra}_c = 1515.41, \epsilon = 0.847$ and $\Pran = 0.05$. Even and odd mode components of the solution oscillate with their frequencies in a $2:1$ ratio.

Download Mannix and Mestel supplementary movie 2(Video)
Video 1.3 MB

Mannix and Mestel supplementary movie 3

DNS of the even-even $\ell=4, m=6$ periodic solution for $\mathrm{Ra}_c = 3425.94, \epsilon =0.4594$ and $\Pran = 0.05$. Even and odd mode components of the solution oscillate with their frequencies in a $2:1$ ratio.

Download Mannix and Mestel supplementary movie 3(Video)
Video 1.6 MB