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Asymptotic theory for torsional convection in rotating fluid spheres

Published online by Cambridge University Press:  19 January 2017

Keke Zhang ,
Kameng Lam  and
Dali Kong
Keke Zhang*
Affiliation:
College of Engineering, Mathematics and Physical Sciences, University of Exeter, Exeter EX4 4QE, UK Lunar and Planetary Science Laboratory, Macau University of Science and Technology, Macau
Kameng Lam
Affiliation:
Lunar and Planetary Science Laboratory, Macau University of Science and Technology, Macau
Dali Kong
Affiliation:
College of Engineering, Mathematics and Physical Sciences, University of Exeter, Exeter EX4 4QE, UK
*
†Email address for correspondence: [email protected]
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Abstract

This paper is concerned with the classical, well-studied problem of convective instabilities in rapidly rotating, self-gravitating, internally heated Boussinesq fluid spheres. Sanchez et al. (J. Fluid Mech., vol. 791, 2016, R1) recently found, unexpectedly via careful numerical simulation, that non-magnetic convection in the form of axially symmetric, equatorially antisymmetric torsional oscillation is physically preferred in a special range of small Prandtl number for rapidly rotating fluid spheres with the stress-free boundary condition. We derive an asymptotic solution describing convection-driven torsional oscillation – whose flow velocity and pressure are fully analytical and in closed form – that confirms the result of the numerical analysis and is in quantitative agreement with the numerical solution. We also demonstrate, through the derivation of a different asymptotic solution, that convection-driven torsional oscillation cannot occur in rapidly rotating fluid spheres with the no-slip boundary condition.

JFM classification

Convection: Buoyancy-driven instability Convection: Convection Geophysical and Geological Flows: Geophysical and Geological Flows
Type
Rapids
Information
Journal of Fluid Mechanics , Volume 813 , 25 February 2017 , R2
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Copyright
© 2017 Cambridge University Press 

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References

Busse, F. H. 1970 Thermal instabilities in rapidly rotating systems. J. Fluid Mech. 44, 441–460.Google Scholar
Busse, F. H. & Simitev, R. 2004 Inertial convection in rotating fluid spheres. J. Fluid Mech. 498, 23–30.Google Scholar
Chandrasekhar, S. 1961 Hydrodynamic and Hydromagnetic Stability. Clarendon.Google Scholar
Dormy, A. M., Soward, A. M., Jones, C. A., Jault, D. & Cardin, P. 2004 The onset of thermal convection in rotating spherical shells. J. Fluid Mech. 501, 43–70.Google Scholar
Greenspan, H. P. 1968 The Theory of Rotating Fluids. Cambridge University Press.Google Scholar
Jones, C. A., Soward, A. M. & Mussa, A. I. 2000 The onset of thermal convection in a rapidly rotating sphere. J. Fluid Mech. 405, 157–179.CrossRefGoogle Scholar
Landeau, R. & Aubert, A. 2000 Equatorially asymmetric convection inducing a hemispherical magnetic field in rotating spheres and implications for the past Martian dynamo. Phys. Earth Planet. Inter. 185, 61–73.CrossRefGoogle Scholar
Roberts, P. H. 1968 On the thermal instability of a rotating–fluid sphere containing heat sources. Phil. Trans. R. Soc. Lond. A 263, 93–117.Google Scholar
Sanchez, J., Garcia, F. & Net, M. 2016 Critical torsional modes of convection in rotating fluid spheres at high Taylor numbers. J. Fluid Mech. 791, R1.Google Scholar
Soward, A. M. 1977 On the finite amplitude thermal instability of a rapidly rotating fluid sphere. Geophys. Astrophys. Fluid Dyn. 9, 19–74.CrossRefGoogle Scholar
Stewartson, K. & Roberts, P. H. 1963 On the motion of liquid in a spheroidal cavity of a precessing rigid body. J. Fluid Mech. 17, 1–20.Google Scholar
Zhang, K. 1992 Spiralling columnar convection in rapidly rotating spherical fluid shells. J. Fluid Mech. 236, 535–556.Google Scholar
Zhang, K. 1994 On coupling between the Poincaré equation and the heat equation. J. Fluid Mech. 268, 211–229.Google Scholar
Zhang, K. 1995 On coupling between the Poincaré equation and the heat equation: non-slip boundary condition. J. Fluid Mech. 284, 239–256.CrossRefGoogle Scholar
Zhang, K., Earnshaw, P., Liao, X. & Busse, F. H. 2001 On inertial waves in a rotating fluid sphere. J. Fluid Mech. 437, 103–119.CrossRefGoogle Scholar