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Onset of fully compressible convection in a rapidly rotating spherical shell

Published online by Cambridge University Press:  01 July 2019

Shuang Liu ,
Zhen-Hua Wan Open the ORCID record for Zhen-Hua Wan [Opens in a new window] ,
Rui Yan ,
Chao Sun Open the ORCID record for Chao Sun [Opens in a new window]  and
De-Jun Sun
Shuang Liu
Affiliation:
Department of Modern Mechanics, University of Science and Technology of China, Hefei, Anhui 230027, China
Zhen-Hua Wan*
Affiliation:
Department of Modern Mechanics, University of Science and Technology of China, Hefei, Anhui 230027, China
Rui Yan
Affiliation:
Department of Modern Mechanics, University of Science and Technology of China, Hefei, Anhui 230027, China
Chao Sun
Affiliation:
Center for Combustion Energy, Key Laboratory for Thermal Science and Power Engineering of Ministry of Education, Department of Energy and Power Engineering, Tsinghua University, Beijing 100084, China
De-Jun Sun*
Affiliation:
Department of Modern Mechanics, University of Science and Technology of China, Hefei, Anhui 230027, China
*
†Email addresses for correspondence: [email protected], [email protected]
†Email addresses for correspondence: [email protected], [email protected]
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Abstract

The onset of thermal convection in a rapidly rotating spherical shell is studied by linear stability analysis based on the fully compressible Navier–Stokes equations. Compressibility is quantified by the number of density scale heights $N_{\unicode[STIX]{x1D70C}}$, which measures the intensity of density stratification of the motionless, polytropic base state. The nearly adiabatic flow with polytropic index $n=1.499<n_{a}=1.5$ is considered, where $n_{a}$ is the adiabatic polytropic index. By investigating the stability of the base state with respect to the disturbance of specified wavenumber, the instability process is found to be sensitive to the Prandtl number $Pr$ and to $N_{\unicode[STIX]{x1D70C}}$. For large $Pr$ and small $N_{\unicode[STIX]{x1D70C}}$, the quasi-geostrophic columnar mode loses stability first; while for relatively small $Pr$ a new quasi-geostrophic compressible mode is identified, which becomes unstable first under strong density stratification. The inertial mode can also occur first for relatively small $Pr$ and a certain intensity of density stratification in the parameter range considered. Although the Rayleigh numbers $Ra$ for the onsets of the quasi-geostrophic compressible mode and columnar mode are different by several orders of magnitude, we find that they follow very similar scaling laws with the Taylor number. The critical $Ra$ for convection onset is found to be always positive, in contrast with previous results based on the widely used anelastic model that convection can occur at negative $Ra$. By evaluating the relative magnitude of the time derivative of density perturbation in the continuity equation, we show that the anelastic approximation in the present system cannot be applied in the small-$Ra$ and large-$N_{\unicode[STIX]{x1D70C}}$ regime.

JFM classification

Convection: Buoyancy-driven instability Geophysical and Geological Flows: Rotating flows
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 873 , 25 August 2019 , pp. 1090 - 1115
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Copyright
© 2019 Cambridge University Press 

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