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Weakly nonlinear mode-interactions in spherical Rayleigh-Benard convection

Authors
  • Mestel, A
  • Mannix, P
Publication Date
Mar 30, 2019
Source
Spiral - Imperial College Digital Repository
Keywords
License
Unknown
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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 = Rac. Solving the linear stability problem, it is shown that degenerate points (d = dc; Rac) exist where two modes simultaneously become unstable. Considering the weakly nonlinear evolution of these modes, it is found that spatial resonances play a crucial role in determining the preferred convection pattern for neighbouring modes (` : ` 1) and non-neighbouring even modes (` : ` 2). Deriving coupled amplitude equations relevant at all degeneracies we outline the in uence of changes in d; Ra and Prandtl number Pr. A particular conclusion is that only even modes have pure mode solutions, and that odd modes exist only as a component of mixed mode solutions. The mode-dependent in uence of Pr on the saturation of mixed mode solutions is shown to be markedly di erent in the limits Pr ! 0 and Pr ! 1. 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 of oscillations in a spherical annulus, is explained using the structure of the equations derived.

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