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Linear theory of compressible convection in rapidly rotating spherical shells, using the anelastic approximation

Published online by Cambridge University Press:  26 August 2009

C. A. JONES ,
K. M. KUZANYAN  and
R. H. MITCHELL
C. A. JONES*
Affiliation:
School of Mathematics, University of Leeds, Leeds LS2 9JT, UK
K. M. KUZANYAN
Affiliation:
School of Mathematics, University of Leeds, Leeds LS2 9JT, UK
R. H. MITCHELL
Affiliation:
School of Engineering, Computing and Mathematics, University of Exeter, North Park Road, Exeter EX4 4QF, UK
*
†Email address for correspondence: c.a.jones@maths.leeds.ac.uk
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Abstract

The onset of compressible convection in rapidly rotating spherical shells is studied in the anelastic approximation. An asymptotic theory valid at low Ekman number is developed and compared with numerical solutions of the full equations. Compressibility is measured by the number of density scale heights in the shell. In the Boussinesq problem, the location of the onset of convection is close to the tangent cylinder when there is no internal heating only a heat flux emerging from below. Compressibility strongly affects this result. With only a few scale heights or more of density present, there is onset of convection near the outer shell. Compressibility also strongly affects the frequencies and preferred azimuthal wavenumbers at onset. Compressible convection, like Boussinesq convection, shows strong spiralling in the equatorial plane at low Prandtl number. We also explore how higher-order linear modes penetrate inside the tangent cylinder at higher Rayleigh numbers and compare modes both symmetric and antisymmetric about the equator.

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Papers
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Journal of Fluid Mechanics , Volume 634 , 10 September 2009 , pp. 291 - 319
Copyright
Copyright © Cambridge University Press 2009

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References

REFERENCES

Al-Shamali, F. M., Heimpel, M. H. & Aurnou, J. M. 2004 Varying the spherical shell geometry in rotating thermal convection. Geophys. Astrophys. Fluid Dyn. 98, 153–169.Google Scholar
Anufriev, A. P., Jones, C. A. & Soward, A. M. 2005 The Boussinesq and anelastic liquid approximations for convection in the Earth's core. Phys. Earth Planet. Inter. 152, 163–190.Google Scholar
Batchelor, G. K. 1967 An Introduction to Fluid Dynamics. Cambridge University Press.Google Scholar
Braginsky, S. I. & Roberts, P. H. 1995 Equations governing convection in earth's core and the geodynamo. Geophys. Astrophys. Fluid Dyn. 79, 1–97.Google Scholar
Busse, F. H. 1970 Thermal instabilities in rapidly rotating systems. J. Fluid Mech. 44, 441–460.Google Scholar
Busse, F. H. & Hood, L. L. 1982 Differential rotation driven by convection in a rapidly rotating annulus. Geophys. Astrophys. Fluid Dyn. 21, 59–74.Google Scholar
Chandrasekhar, S. 1961 Hydrodynamic and Hydromagnetic Stability. Oxford University Press.Google Scholar
Clune, T. C., Elliot, J. R., Miesch, M. S., Toomre, J. & Glatzmeier, G. A. 1999 Computational aspects of a code to study rotating turbulent convection in spherical shells. Parallel Comput. 25, 361–380.Google Scholar
Davidson, P. A. 2004 Turbulence: An Introduction for Scientists and Engineers. Oxford University Press.Google Scholar
Dormy, E., 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
Drew, S. J., Jones, C. A. & Zhang, K. 1995 Onset of convection in a rapidly rotating compressible fluid spherical shell. Geophys. Astrophys. Fluid Dyn. 80, 241–254.Google Scholar
Evonuk, M. 2008 The role of density stratification in generating zonal flow structures in a rotating fluid. Astrophys. J. 673, 1154–1159.Google Scholar
Evonuk, M. & Glatzmaier, G. A. 2004 2D Studies of various approximations used for modeling convection in giant planets. Geophys. Astrophys. Fluid Dyn. 98, 241–255.Google Scholar
Geiger, G. & Busse, F. H. 1981 On the onset of convection in slowly rotating fluid shells. Geophys. Astrophys. Fluid Dyn. 18, 147–156.Google Scholar
Gillet, N. & Jones, C. A. 2006 The quasi-geostrophic model for rapidly rotating spherical convection outside the tangent cylinder. J. Fluid Mech. 554, 343–369.Google Scholar
Gilman, P. A. & Glatzmaier, G. A. 1981 Compressible convection in a rotating spherical shell. Part 1. Anelastic equations. Astrophys. J. Suppl. Ser. 45, 335–349.Google Scholar
Glatzmaier, G. A. & Gilman, P. A. 1981 a Compressible convection in a rotating spherical shell. Part 2. A Linear Anelastic Model. Astrophys. J. Suppl. Ser. 45, 351–380.Google Scholar
Glatzmaier, G. A. & Gilman, P. A. 1981 b Compressible convection in a rotating spherical shell. Part 4. Effects of viscosity, conductivity, boundary conditions, and zone depth. Astrophys. J. Suppl. Ser. 47, 103–115.Google Scholar
Gough, D. O. 1969 The anelastic approximation for thermal convection. J. Atmos. Sci. 26, 448–456.Google Scholar
Guillot, T. 1999 a A comparison of the interiors of Jupiter and Saturn. Planet. Space Sci. 47, 1183–1200.Google Scholar
Guillot, T. 1999 b Interior of giant planets inside and outside the solar system. Science 286, 72–77.Google Scholar
Heimpel, M., Aurnou, J. & Wicht, J. 2005 Simulation of equatorial and high-latitude jets on Jupiter in a deep convection model. Nature 438, 193–196.Google Scholar
Jones, C. A., Rotvig, J. & Abdulrahman, A. 2003 Multiple jets and zonal flow on Jupiter. Geophys. Res. Lett. 30, 1–1.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.Google Scholar
Landau, L. D. & Lifshitz, E. M. 1959 Fluid Mechanics. Pergamon.Google Scholar
Lantz, S. R. 1992 Dynamical Behavior of Magnetic Fields in a Stratified, Convecting Fluid Layer, PhD Thesis, Cornell University.Google Scholar
Miesch, M. S., Elliott, J. R., Toomre, J., Clune, T. L., Glatzmaier, G. A. & Gilman, P. A. 2000 Three-dimensional spherical simulations of solar convection. Part 1. Differential rotation and pattern evolution achieved with laminar and turbulent states. Astrophys. J. 532, 593–615.Google 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
Rotvig, J. & Jones, C. A. 2006 Multiple jets and bursting in the rapidly rotating convecting two-dimensional annulus model with nearly plane-parallel boundaries. J. Fluid Mech. 567, 117–140.Google Scholar
Tilgner, A. & Busse, F. H. 1997 Finite-amplitude convection in rotating fluid shells. J. Fluid Mech. 332, 359–376.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., Liao, X. & Busse, F. H. 2007 Asymptotic solutions of convection in rapidly rotating non-slip spheres. J. Fluid Mech. 578, 371–380.Google Scholar