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Asymptotic solutions of convection in rapidly rotating non-slip spheres

Published online by Cambridge University Press:  26 April 2007

KEKE ZHANG ,
XINHAO LIAO  and
F. H. BUSSE
KEKE ZHANG
Affiliation:
Department of Mathematical Sciences, University of Exeter, EX4 4QE, UK
XINHAO LIAO
Affiliation:
Shanghai Astronomical Observatory, Chinese Academy of Sciences, Shanghai 200030, China
F. H. BUSSE
Affiliation:
Institute of Physics, University of Bayreuth, D-95440 Bayreuth, Germany
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Abstract

Asymptotic solutions describing the onset of convection in rotating, self-gravitating Boussinesq fluid spheres with no-slip boundary conditions, valid for asymptotically small Ekman numbers and for all values of the Prandtl number, are derived. Central to the asymptotic analysis is the assumption that the leading-order convection can be represented, dependent on the size of the Prandtl number, by either a single quasi-geostrophic-inertial-wave mode or by a combination of several quasi-geostrophic-inertial-wave modes, and is controlled or influenced by the effect of the oscillatory Ekman boundary layer. Comparisons between the asymptotic solutions and the corresponding fully numerical simulations show a satisfactory quantitative agreement.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 578 , 10 May 2007 , pp. 371 - 380
Copyright
Copyright © Cambridge University Press 2007

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References

REFRENCES

Busse, F. H. 1970 Thermal instabilities in rapidly rotating systems. J. Fluid Mech. 44, 441–460.CrossRefGoogle Scholar
Busse, F. H. 2002 Convective flows in rapidly rotating spheres and their dynamo action. Phys. Fluids 14, 1301–1314.CrossRefGoogle Scholar
Busse, F. H., Zhang, K. & Liao, X. 2005 On slow inertial waves in the solar convection zone. Astrophys. J. 631, L171–L174.CrossRefGoogle Scholar
Chan, K. H. Li, L. & Liao, X. 2006. Modelling the core convection using finite element and finite difference methods. Phys. Earth Planet. Inter. 157, 124–138.CrossRefGoogle Scholar
Chandrasekhar, S. 1961 Hydrodynamic and Hydromagnetic Stability. Clarendon.Google Scholar
Christensen, U. R. 2002 Zonal flow driven by strongly supercritical convection in rotating spherical shells J. Fluid Mech. 470, 115–133.CrossRefGoogle 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.CrossRefGoogle Scholar
Greenspan, H. P. 1968 The Theory of Rotating Fluids. Cambridge University Press.Google Scholar
Hollerbach, R. & Kerswell, R. R. 1995 Oscillatory internal shear layers in rotating and precessing flows. J. Fluid Mech. 298, 327–339.CrossRefGoogle 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
Kudlick, M. D. 1966 On transient motions in a contained, rotating fluid. PhD Thesis, Department of Mathematics, MIT.Google Scholar
Roberts, P. H. 1968 On the thermal instability of a self-gravitating fluid sphere containing heat sources. Phil. Trans. R. Soc. Lond. A 263, 93–117.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
Tilgner, A. & Busse, F. H. 1997 Finite amplitude convection in rotating spherical fluid shells. J. Fluid Mech. 332, 359–376CrossRefGoogle 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
Zhang, K. & Liao, X. 2004 A new asymptotic method for the analysis of convection in a rotating sphere. J. Fluid Mech. 518, 319–346.CrossRefGoogle Scholar