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Convection in a rapidly rotating spherical shell with an imposed laterally varying thermal boundary condition

Published online by Cambridge University Press:  10 December 2009

CHRISTOPHER J. DAVIES ,
DAVID GUBBINS  and
PETER K. JIMACK
CHRISTOPHER J. DAVIES*
Affiliation:
School of Earth and Environment, University of Leeds, Leeds LS2 9JT, UK
DAVID GUBBINS
Affiliation:
School of Earth and Environment, University of Leeds, Leeds LS2 9JT, UK
PETER K. JIMACK
Affiliation:
School of Computing, University of Leeds, Leeds LS2 9JT, UK
*
†Email address for correspondence: c.davies@see.leeds.ac.uk
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Abstract

We investigate thermally driven convection in a rotating spherical shell subject to inhomogeneous heating on the outer boundary, extending previous results to more rapid rotation rates and larger amplitudes of the boundary heating. The analysis explores the conditions under which steady flows can be obtained, and the stability of these solutions, for two boundary heating modes: first, when the scale of the boundary heating corresponds to the most unstable mode of the homogeneous problem; second, when the scale is larger. In the former case stable steady solutions exhibit a two-layer flow pattern at moderate rotation rates, but at very rapid rotation rates no steady solutions exist. In the latter case, stable steady solutions are always possible, and unstable solutions show convection rolls that cluster into nests that are out of phase with the boundary anomalies and remain trapped for many thermal diffusion times.

JFM classification

Convection: Convection
Type
Papers
Information
Journal of Fluid Mechanics , Volume 641 , 25 December 2009 , pp. 335 - 358
Copyright
Copyright © Cambridge University Press 2009

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References

REFERENCES

Busse, F. H. 1970 Thermal instabilities in rapidly rotating systems. J. Fluid Mech. 44, 441–460.CrossRefGoogle Scholar
Chandrasekhar, S. 1961 Hydrodynamic and Hydromagnetic Stability. Dover.Google Scholar
Dormy, E., Soward, S., 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
Gibbons, S. J. & Gubbins, D. 2000 Convection in the Earth's core driven by lateral variations in core-mantle boundary heat flux. Geophys. J. Intl 142, 631–642.CrossRefGoogle Scholar
Gibbons, S. J., Gubbins, D. & Zhang, K 2007 Convection in a rotating spherical shell with inhomogeneous heat flux at the outer boundary. Geophys. Astrophys. Fluid Dyn. 101, 347–370.CrossRefGoogle Scholar
Glatzmaier, G. A. & Gilman, P. A. 1981 Compressible convection in a rotating spherical shell. IV. Effects of viscosity, conductivity, boundary conditions, and zone depth. Astrophys. J. Suppl. Ser. 47, 103–115.CrossRefGoogle Scholar
Grote, E. & Busse, F. H. 2001 Dynamics of convection and dynamos in rotating spherical fluid shells. Fluid Dyn. Res. 28, 349–368.CrossRefGoogle Scholar
Jackson, A., Jonker, A. R. T. & Walker, M. R. 2000 Four centuries of geomagnetic secular variation from historical records. Phil. Trans. R. Soc. Lond. A 358, 957–990.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
Lehoucq, R. B., Sorensen, D. C. & Yang, C. 1998 ARPACK Users Guide: Solution of Large Scale Eigenvalue Problems by the Implicitly Restarted Arnoldi Method. Philedelphia, PA. SIAM, http://www.caam.rice.edu/software/ARPACK/UG/ug.htm.CrossRefGoogle Scholar
Masters, G., Johnson, S., Laske, G. & Bolton, H. F. 1996 A shear-velocity model of the mantle. Phil. Trans. R. Soc. Lond. A 354, 1385–1411.Google Scholar
Nakagawa, T. & Tackley, P. J. 2007 Lateral variations in CMB heat flux and deep mantle seismic velocity caused by a thermal-chemical-phase boundary layer in 3D spherical convection. Earth Planet. Sci. Lett. 271, 348–358.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
Sun, Z.-P., Schubert, G. & Glatzmaier, G. A. 1993 Transitions to chaotic thermal convection in a rapidly rotating spherical fluid shell. Geophys. Astrophys. Fluid Dyn. 69, 95–131.CrossRefGoogle Scholar
Takehiro, S., Ishiwatari, M., Nakajima, K. & Hayashi, Y. 1999 The effects of thermal boundary conditions on convection in rapidly rotating spherical shells. Theor. Appl. Mech. 48, 415–421.Google Scholar
Takehiro, S., Ishiwatari, M. Nakajima, K. & Hayashi, Y. 2002 Linear stability of thermal convection in rotating systems with fixed heat flux boundaries. Geophys. Astrophys. Fluid Dyn. 96, 439–459.CrossRefGoogle Scholar
Tilgner, A. & Busse, F. H. 1997 Finite-amplitude convection in rotating spherical fluid shells. J. Fluid Mech. 332, 359–376.CrossRefGoogle Scholar
Verhulst, F. 1993 Nonlinear differential equations and dynamical systems, 3rd edn. Springer.Google Scholar
Willis, A. P., Sreenivasan, B. & Gubbins, D. 2007 Thermal core-mantle interaction: exploring regimes for ‘locked’ dynamo action. Phys. Earth Planet. Intl 165, 83–92.CrossRefGoogle Scholar
Zhang, K. 1991 Vascillatory convection in a rotating spherical shell at infinite Prandtl number. J. Fluid Mech. 228, 607–628.Google Scholar
Zhang, K. 1992 a Convection in a rapidly rotating spherical shell at infinite Prandtl number: transition to vacillating flows. Phys. Earth Planet. Intl 72, 236–248.CrossRefGoogle Scholar
Zhang, K. 1992 b Spiralling columnar convection in rapidly rotating spherical fluid shells. J. Fluid Mech. 236, 535–556.CrossRefGoogle Scholar
Zhang, K. & Gubbins, D. 1992 On convection in the Earth's core driven by lateral temperature variations in the lower mantle. Geophys. J. Intl 108, 247–255.CrossRefGoogle Scholar
Zhang, K. & Gubbins, D. 1993 Convection in a rotating spherical fluid shell with an inhomogeneous temperature boundary condition at infinite Prandtl number. J. Fluid Mech. 250, 209–232.CrossRefGoogle Scholar
Zhang, K. & Gubbins, D. 1996 Convection in a rotating spherical fluid shell with an inhomogeneous temperature boundary condition at finite Prandtl number. Phys. Fluids 8, 1141–1148.CrossRefGoogle Scholar
Zhang, K. & Liao, X. 2004 A new asymptotic method for the analysis of convection in a rapidly rotating sphere. J. Fluid Mech. 518, 319–346.CrossRefGoogle Scholar
Zhang, K., Liao, X. & Busse, F. H. 2007 Asymptotic solutions of convection in rapidly rotating non-slip spheres. J. Fluid Mech. 578, 371–380.CrossRefGoogle Scholar