Hostname: page-component-745bb68f8f-lrblm Total loading time: 0 Render date: 2025-01-10T06:06:19.171Z Has data issue: false hasContentIssue false
  • English
  • Français

Vacillatory convection in a rotating spherical fluid shell at infinite Prandtl number

Published online by Cambridge University Press:  26 April 2006

Keke Zhang
Keke Zhang
Affiliation:
Department of Earth Sciences, University of Leeds, Leeds, LS2 9JT, UK
Get access Rights & Permissions [Opens in a new window]

Abstract

The temporal and spatial behaviour of three-dimensional convection at infinite Prandtl number, in a rotating spherical fluid shell of radius ratio η = ri/ro = 0.4, has been investigated numerically for a range of Taylor number for which mc = 2, 3 and 4 are the critical values of azimuthal wavenumber at the onset of convection. When the Rayleigh number, R, exceeds a critical value, R1c, primary nonlinear solutions with the predominant wavenumber m0 = 4 in the form of azimuthally travelling waves give way to secondary solutions in the form of steadily drifting mixed-mode convection with two predominant wavenumbers, m0 = 2 and 4. The secondary bifurcation solution becomes unstable at another critical value, R2c, that leads to the tertiary solution in which the dominant wavelength of convection vacillates periodically between the two competing scales characterized by the azimuthal wavenumbers m0 = 2 and 4. Instabilities and bifurcations associated with the evolution from a static state to wavenumber vacillation are discussed for a representative Taylor number of T = 104. It is also shown that the interaction between the two spatially resonant wavenumbers m = 2 and 4 is much stronger than the interaction between the non-resonant wavenumbers m = 3 and 4 even though Rc(m = 3) is much closer to Rc(m = 4) than Rc(m = 2). For the convection of the dominant wavenumber m0 = 2, the analysis is focused on the range of Taylor number T > T4, where T4 is the Taylor number at which the critical wavenumber mc changes from 2 to 4 at the onset of convection. The m0 = 2 steadily drifting nonlinear solutions, which are unstable at small amplitudes owing to the Eckhaus-type instability, gain their stability at large amplitudes at R1c through nonlinear effects, and lose their stability at a higher Rayleigh number, R2c, to the amplitudevacillating instability which leads to a periodic change in the amplitude of convection with little fluctuation in the pattern of flow.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 228 , July 1991 , pp. 607 - 628
Copyright
© 1991 Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Braginsky, S. I. 1963 Constructure of the F layer and reasons for convection in the Earth's core. Dokl. Akad. Nauk SSSR (Engl. transl.) 149, 1311–1314.Google Scholar
Busse, F. H. 1970 Thermal instabilities in rapidly rotating systems. J. Fluid Mech. 44, 441–460.Google Scholar
Busse, F. H. 1978 Nonlinear properties of thermal convection. Rep. Prog. Phys. 41, 1929–1967.Google Scholar
Carrigan, C. R. & Busse, F. H. 1983 An experimental and theoretical investigation of the onset of convection in rotating spherical shells. J. Fluid Mech. 126, 287–305.Google Scholar
Chamberlain, J. A. & Carrigan, C. R. 1986 An experimental investigation of convection in a rotating sphere subject to time varying thermal boundary conditions. Geophys. Astrophys. Fluid Dyn. 41, 17–41.Google Scholar
Chandrasekhar, S. 1961 Hydrodynamic and Hydromagnetic Stability. Clarendon.
Dangelmayr, G. & Knobloch, E. 1987 The Takens–Bogdanov bifurcation with. O(2)-symmetry. Phil. Trans. R. Soc. Lond. A 322, 243–279.Google Scholar
Gilman, P. A. 1975 Linear simulations of Boussinesq convection in a deep rotating spherical shell. J. Atmos. Sci. 32, 1331–1352.Google Scholar
Gilman, P. A. 1977 Nonlinear dynamics of Boussinesq convection in a deep rotating spherical shell. I. Geophys. Astrophys. Fluid Dyn. 8, 93–135.Google Scholar
Gubbins, D. 1977 Energetics of the earth's core. J. Geophys. 43, 453–464.Google Scholar
Hart, J. E., Glatzmaier, G. A. & Toomre, J. 1986 Space-laboratory and numerical simulations of thermal convection in a rotating hemispherical shell with radial gravity. J. Fluid Mech. 173, 519–544.Google Scholar
Hide, R. 1958 An experimental study of thermal convection in a rotating fluids. Phil. Trans. R. Soc. Lond. A250, 442–478.Google Scholar
Hide, R. & Mason, P. J. 1975 Sloping convection in a rotating fluid. Adv. Phys. 24, 47–100.Google Scholar
Loper, D. E. 1978 The gravitationally powered dynamo. Geophys. J. R. Astron. Soc. 54, 389–404.Google Scholar
Mckenzie, D. 1987 The symmetry of convective transitions in space and time. J. Fluid Mech. 191, 287–339.Google Scholar
Proctor, M. R. E. & Jones, C. A. 1988 The interaction of two spatially resonant patterns in thermal convection. Part 1. Exact 1:2 resonance. J. Fluid Mech. 188, 301–335.Google Scholar
Roberts, P. H. 1968 On the thermal instability of a self-gravitating fluid sphere containing heat sources. Phil. Trans. R. Soc. Lond. A263, 93–117.Google Scholar
Young, R. E. 1974 Finite amplitude thermal convection in a spherical shell. J. Fluid Mech. 63, 695–721.Google Scholar
Zhang, K. 1991 Spiralling columnar convection in rapidly rotating spherical fluid shells. J. Fluid Mech. (submitted).Google Scholar
Zhang, K. & Busse, F. 1988 Finite amplitude convection and magnetic field generation in a rotating spherical shell. Geophys. Astrophys. Fluid Dyn. 44, 33–53.Google Scholar
Zhang, K. & Busse, F. 1989 Convection driven magnetohydrodynamic dynamos in rotating spherical shells. Geophys. Astrophys. Fluid Dyn. 49, 97–116.Google Scholar
Zhang, K. & Busse, F. 1990 Generation of magnetic fields by convection in a rotating spherical fluid shell of infinite Prandtl numbers. Phys. Earth Planet. Inter. 59, 208–222.Google Scholar
Zhang, K., Busse, F. & Hircshing, W. 1989 Numerical models in the theory of geomagnetisim. In Geomagnetism and Paleomagnetism (ed. F. J. Lowes et al.), pp. 347–358. Kluwer.