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On the onset of low-Prandtl-number convection in rotating spherical shells: non-slip boundary conditions

Published online by Cambridge University Press:  25 April 2008

MARTA NET ,
FERRAN GARCIA  and
JUAN SÁNCHEZ
MARTA NET
Affiliation:
Departament de Física Aplicada, Universitat Politècnica de CatalunyaJordi Girona Salgado s/n. Campus Nord. Mòdul B4, 08034 Barcelona, [email protected]; [email protected]
FERRAN GARCIA
Affiliation:
Departament de Física Aplicada, Universitat Politècnica de CatalunyaJordi Girona Salgado s/n. Campus Nord. Mòdul B4, 08034 Barcelona, [email protected]; [email protected]
JUAN SÁNCHEZ
Affiliation:
Departament de Física Aplicada, Universitat Politècnica de CatalunyaJordi Girona Salgado s/n. Campus Nord. Mòdul B4, 08034 Barcelona, [email protected]; [email protected]
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Abstract

Accurate numerical computations of the onset of thermal convection in wide rotating spherical shells are presented. Low-Prandtl-number (σ) fluids, and non-slip boundary conditions are considered. It is shown that at small Ekman numbers (E), and very low σ values, the well-known equatorially trapped patterns of convection are superseded by multicellular outer-equatorially-attached modes. As a result, the convection spreads to higher latitudes affecting the body of the fluid, and increasing the internal viscous dissipation. Then, from E < 10−5, the critical Rayleigh number (Rc) fulfils a power-law dependence Rc ~ E−4/3, as happens for moderate and high Prandtl numbers. However, the critical precession frequency (|ωc|) and the critical azimuthal wavenumber (mc) increase discontinuously, jumping when there is a change of the radial and latitudinal structure of the preferred eigenfunction. In addition, the transition between spiralling columnar (SC), and outer-equatorially-attached (OEA) modes in the (σ, E)-space is studied. The evolution of the instability mechanisms with the parameters prevents multicellular modes being selected from σ≳0.023. As a result, and in agreement with other authors, the spiralling columnar patterns of convection are already preferred at the Prandtl number of the liquid metals. It is also found that, out of the rapidly rotating limit, the prograde antisymmetric (with respect to the equator) modes of small mc can be preferred at the onset of the primary instability.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 601 , 25 April 2008 , pp. 317 - 337
Copyright
Copyright © Cambridge University Press 2008

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References

REFERENCES

Al-Shamali, F., Heimpel, M. & Arnou, J. 2004 Varying the spherical shell geometry in rotating thermal convection. Geophys. Astrophys. Fluid Dyn. 98, 153–169.Google Scholar
Ardes, M., Busse, F. H. & Wicht, J. 1997 Thermal convection in rotating spherical shells. PEPI 99, 55–67.Google Scholar
Aubert, J., Brito, D., Nataf, H.-C., Cardin, P. & Masson, J. P. 2001 A systematic experimental study of rapidly rotating spherical convection in water and liquid gallium. Phys. Earth Planet. Inter. 128, 51–74.CrossRefGoogle Scholar
Batiste, O., Mercader, I., Net, M. & Knobloch, E. 1999 Onset of oscillatory binary fluid convection in finite containers. PRE 59, 6730–6741.CrossRefGoogle ScholarPubMed
Busse, F. H. 1970 Thermal instabilities in rapidly rotating systems. J. Fluid Mech. 44, 441–460.CrossRefGoogle Scholar
Busse, F. H. & Simitev, R. 2004 Inertial convection in rotating fluid spheres. J. Fluid Mech. 498, 23–30.CrossRefGoogle 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.CrossRefGoogle Scholar
Ecke, R. E., Zhong, F. & Knobloch, E. 1992 Hopf bifurcation with broken reflection symmetry in rotating Rayleigh–Bénard convection. Europhys. Lett. 19, 177–182.CrossRefGoogle Scholar
Finlay, C. C. & Jackson, A. 2003 Equatorially dominated magnetic field change at the surface of the Earth's core. Science 300 (5628), 2084–2086.CrossRefGoogle ScholarPubMed
Gillet, N., Brito, D., Jault, D. & Nataf, H.-C. 2007 Experimental and numerical studies of convection in a rapidly rotating spherical shell. J. Fluid Mech. 580, 83–121.CrossRefGoogle Scholar
Herrmann, J. & Busse, F. H. 1997 Convection in a rotating cylindrical annulus. Part 4. Modulations and transitions to chaos at low Prandtl numbers. J. Fluid Mech. 350, 209–229.CrossRefGoogle Scholar
Jaletzky, M. 1999 Experimental study of rotating cylindrical annulus convection. PhD thesis, University of Bayreuth.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
Lehoucq, R. B., Sorensen, D. C. & Yang, C. 1998 ARPACK User's Guide: Solution of Large-Scale Eigenvalue Problems with Implicitly Restarted Arnoldi Methods. SIAM.CrossRefGoogle Scholar
Li, L., Zhang, P., Liao, X. & Zhang, K. 2005 Multiplicity of nonlinear thermal convection in a spherical shell. Phys. Rev. E 71, 016301(1)–016301(9).Google Scholar
Pino, D., Mercader, I. & Net, M. 2000 Thermal and inertial modes of convection in a rapidly rotating annulus. Phys. Rev. E 61, 1507–1517.Google Scholar
Pino, D., Net, M., Sánchez, J. & Mercader, I. 2001 Thermal Rossby waves in a rotating annulus: their stability. Phys. Rev. E 63, 056312(1)–056312(14).Google Scholar
Plaut, E. & Busse, F. H. 2002 Low-Prandtl-number convection in a rotating annulus. J. Fluid Mech. 464, 345–363.CrossRefGoogle Scholar
Plaut, E. & Busse, F. H. 2005 Multicellular convection in rotating annuli. J. Fluid Mech. 528, 119–133.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
Schnaubelt, M. & Busse, F. H. 1992 Convection in a rotating cylindrical annulus. Part 3. Vacillating and spatially modulated flow. J. Fluid Mech. 245, 155–173.Google Scholar
Secco, R. A. & Schloessin, H. H. 1989 The electrical resistivity of solid and liquid Fe at pressures up to 7 GPa. J. Geophy. Res. B 94, 5887–5894.CrossRefGoogle Scholar
Simitev, R. & Busse, F. H. 2003 Patterns of convection in rotating spherical shells. New J. Phys. 5, 97.1–97.20.CrossRefGoogle Scholar
Soward, A. M. 1977 On the finite amplitude thermal instability in a rapidly rotating fluid sphere. Geophys. Astrophys. Fluid Dyn. 9, 19–74.CrossRefGoogle Scholar
Wijs, G. A. d., Kresse, G., Vočadlo, L., Dobson, D., Alfè,, D., Gillan, M. J. & Price, G. D. 1998 The viscosity of liquid iron at the physical conditions of the Earth's core. Nature 392, 805–807.Google Scholar
Yano, J. I. 1992 Asymptotic theory of thermal convection in a rapidly rotating system. J. Fluid Mech. 243, 103–131.CrossRefGoogle Scholar
Zhang, K. 1992 Spiralling columnar convection in rapidly rotating spherical fluid shells. J. Fluid Mech. 236, 535–556.Google Scholar
Zhang, K. 1993 On equatorially trapped boundary inertial waves. J. Fluid Mech. 248, 203–217.Google Scholar
Zhang, K. 1994 On coupling between the Poincaré equation and the heat equation. J. Fluid Mech. 268, 211–229.CrossRefGoogle Scholar
Zhang, K. 1995 On coupling between the Poincaré equation and the heat equation: non-slip boundary condition. J. Fluid Mech. 284, 239–256.Google Scholar
Zhang, K. & Busse, F. H. 1987 On the onset of convection in rotating spherical shells. Geophys. Astrophys. Fluid Dyn. 39, 119–147.CrossRefGoogle Scholar
Zhang, K. & Jones, C. A. 1993 The influence of Ekman boundary layers on rotating convection. Geophys. Astrophys. Fluid Dyn. 71, 145–162.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.Google Scholar
Zhang, K., Liao, X. & Busse, F. 2007 Asymptotic solutions of convection in rapidly rotating non-slip spheres. J. Fluid Mech. 578, 371–380.CrossRefGoogle Scholar