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Effects of geometry on the convection with core-cooling


We study the dynamical (three-dimensional box, axisymmetric and spherical shell geometry) and parameterized models of the mantle convection with the core-cooling. The viscosity is constant in space and dependent on the volume averaged mantle temperature. Core is treated as a hot bath. To understand the process of cooling, we use the ‘local’ Rayleigh (Ral ) and Nusselt (Nul) numbers, which are defined in each thermal boundary layer. In the dynamical calculations, we check the various combinations of Ral and Nul, and find that the local Rayleigh number either at the top or bottom surface may control both the top and bottom local Nusselt numbers. This result suggests that the core-cooling in this case may be controlled by the flow either at top or bottom boundary layer. The least-square-fitting of Nul-Ral relationship shows that its power-law index is around 0.3, despite of the different geometry. Comparing the thermal history calculated by the dynamical and parameterized models, we find that the parameterized convection theory based on the local Ra-Nu relationship obtained by the dynamical calculation is useful for investigating the thermal history of the mantle and core. Applying the parameterized theory to the Earth, we find that the plausible Urey ratio is smaller than that obtained by the previous works which ignored the bottom thermal boundary layer.


  1. Arkani-Hamed, J., Effects of the core cooling on the internal dynamics and thermal evolution of terrestrial planets, J. Geophys. Res., 99, 12109–12119, 1994.

  2. Christensen, U. R., Heat transport by variable viscosity convection and implications for the Earth’s thermal evolution, Phys. Earth Planet. Inter., 35, 264–282, 1984.

  3. Christensen, U. R., Thermal evolution models for the earth, J. Geophys. Res., 90, 2995–3007, 1985.

  4. Davies, G. F., Thermal histories of convective Earth models and constraints on radiogenic heat production in the Earth, J. Geophys. Res., 85, 2517–2530, 1980.

  5. Davies, G. F., Cooling the core and mantle by plume and plate flows, Geophys. J. Int., 115, 132–146, 1993.

  6. Honda, S., A simple parameterized model of Earth’s thermal history with the transition from layered to whole mantle convection, Earth Planet. Sci. Lett., 131, 357–369, 1995.

  7. Honda, S., Local Rayleigh and Nusselt numbers for cartesian convection with temperature-dependent viscosity, Geophys. Res. Lett., 23, 2445–2448, 1996.

  8. Honda, S. and Y. Iwase, Comparison of the dynamical and parameterized models of mantle convection including core-cooling, Earth Planet. Sci. Lett., 139, 133–146, 1996.

  9. Honda, S. and D. A. Yuen, Cooling model of mantle convection with phase changes: effects of aspect ratio and initial conditions, J. Phys. Earth, 42, 165–186, 1994.

  10. Howard, L. N., Convection at high Rayleigh number, in Proc. 11th Int. Cong. Appl. Math., edited by H. Görtler, pp. 1109–1115, Springer-Verlag, New York, 1966.

  11. Iwase, Y., Three-dimensional infinite Prandtl number convection in a spherical shell with temperature-dependent viscosity, J. Geomag. Geoelectr., 48, 1499–1514, 1996.

  12. Iwase, Y. and S. Honda, An interpretation of the Nusselt-Rayleigh number relationship for convection in a spherical shell, Geophys. J. Int., 130, 801–804, 1997.

  13. McKenzie, D. P. and N. O. Weiss, Speculations on the thermal and tectonic history of the earth, Geophys. J. R. Astron Soc., 42, 131–174, 1975.

  14. Nakakuki, T., Studies of convection in the mantle with the phase and the chemical boundaries by numerical simulations, Ph.D. Thesis, Ocean Research Institute, Univ. Tokyo, 1993.

  15. Patankar, S. V., Numerical Heat Transfer and Fluid Flow, 197pp., Hemisphere Pub. Corp., New York, 1980.

  16. Ratcliff, J. T., G. Schubert, and A. Zebib, Steady tetrahedral and cubic patterns of spherical-shell convection with temperature-dependent viscosity, J. Geophys. Res., 101, 25473–25484, 1996.

  17. Schubert, G., Numerical models of mantle convection, Annu. Rev. Fluid Mech., 24, 359–394, 1992.

  18. Schubert, G., P. Cassen, and R. E. Young, Subsolidus convective cooling histories of terrestrial planets, Icarus, 38, 192–211, 1979.

  19. Stacey, F. D., Cooling of the earth—A constraint on paleotectonic hypotheses, in Evolution of the Earth, edited by R. J. O’Connell and W. S. Fyfe, Geodyn. Ser., vol. 5, pp. 272–276, AGU, Washington, 1981.

  20. Steinbach, V., D. A. Yuen, and W. Zhao, Instabilities from phase transitions and the timescales of mantle convection, Geophys. Res. Lett., 20, 1119–1122, 1993.

  21. Stevenson, D. J., T. Spohn, and G. Schubert, Magnetism and thermal evolution of the terrestrial planets, Icarus, 54, 466–489, 1983.

  22. Tackley, P. J., Effects of strongly temperature-dependent viscosity on time-dependent, three-dimensional models of mantle convection, Geophys. Res. Lett., 20, 2187–2190, 1993.

  23. Turcotte, D. L. and G. Schubert, Geodynamics, 450pp., John Wiley, New York, 1982.

  24. van den Berg, A. P. and D. A. Yuen, Convectively induced transition in mantle rheological behavior, Geophys. Res. Lett., 22, 1549–1552, 1995.

  25. Yuen, D. A., S. Balachandar, V. C. Steinbach, S. Honda, D. M. Reuteler, J. J. Smedsmo, and G. S. Lauer, Non-equilibrium effects of core-cooling and time-dependent internal heating on mantle flush events, Nonlinear Process. Geophys., 2, 206–221, 1995.

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Correspondence to Yasuyuki Iwase.

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Iwase, Y., Honda, S. Effects of geometry on the convection with core-cooling. Earth Planet Sp 50, 387–395 (1998).

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  • Nusselt Number
  • Rayleigh Number
  • Thermal History
  • Spherical Shell
  • Thermal Boundary Layer