Three-dimensional thermal convection in a spherical shell

Independent pseudo-spectral and Galerkin numerical codes are used to investigate three-dimensional infinite Prandtl number thermal convection of a Boussinesq fluid in a spherical shell with constant gravity and an inner to outer radius ratio equal to 0.55. The shell is heated entirely from below and has isothermal, stress-free boundaries. Nonlinear solutions are validated by comparing results from the two codes for an axisymmetric solution at Rayleigh number Ra = 14250 and three fully three-dimensional solutions at Ra = 2000, 3500 and 7000 (the onset of convection occurs at Ra = 712). In addition, the solutions are compared with the predictions of a slightly nonlinear analytic theory. The axisymmetric solution is equatorially symmetric and has two convection cells with upwelling at the poles. Two dominant planforms of convection exist for the three-dimensional solutions: a cubic pattern with six upwelling cylindrical plumes, and a tetrahedral pattern with four upwelling plumes. The cubic and tetrahedral patterns persist for Ra at least up to 70000. Time dependence does not occur for these solutions for Ra [les ] 70000, although for Ra > 35000 the solutions have a slow asymptotic approach to steady state. The horizontal and vertical structure of the velocity and temperature fields, and the global and three-dimensional heat flow characteristics of the various solutions are investigated for the two patterns up to Ra = 70000. For both patterns at all Ra, the maximum velocity and temperature anomalies are greater in the upwelling regions than in the downwelling ones and heat flow through the upwelling regions is almost an order of magnitude greater than the mean heat flow. The preferred mode of upwelling is cylindrical plumes which change their basic shape with depth. Downwelling occurs in the form of connected two-dimensional sheets that break up into a network of broad plumes in the lower part of the spherical shell. Finally, the stability of the two patterns to reversal of flow direction is tested and it is found that reversed solutions exist only for the tetrahedral pattern at low Ra.

[1]  G. Glatzmaier.  Numerical simulations of mantle convection: time-dependent, three-dimensional, compressible, spherical shell , 1988 .

[2]  G. Schubert,et al.  Geoid and topography for infinite Prandtl number convection in a spherical shell , 1988 .

[3]  F. Busse,et al.  Mixed-mode patterns of bifurcations from spherically symmetric basic states , 1988 .

[4]  D. Yuen,et al.  Chaotic axisymmetrical spherical convection and large-scale mantle circulation , 1987 .

[5]  G. Schubert,et al.  Plume formation in the D″-layer and the roughness of the core–mantle boundary , 1987, Nature.

[6]  R. Jeanloz,et al.  The Melting Curve of Iron to 250 Gigapascals: A Constraint on the Temperature at Earth's Center , 1987, Science.

[7]  A. Dziewoński,et al.  Global Images of the Earth's Interior , 1987, Science.

[8]  Andrea Morelli,et al.  Topography of the core–mantle boundary and lateral homogeneity of the liquid core , 1987, Nature.

[9]  Juri Toomre,et al.  Space-laboratory and numerical simulations of thermal convection in a rotating hemispherical shell with radial gravity , 1986, Journal of Fluid Mechanics.

[10]  D. Yuen,et al.  The onset of time‐dependent convection in spherical shells as a clue to chaotic convection in the Earth's mantle , 1986 .

[11]  C. Bowin.  Topography at the core‐mantle boundary , 1986 .

[12]  M. Richards,et al.  Coupling of the core dynamo and mantle: Thermal or topographic? , 1986 .

[13]  T. Jordan,et al.  Aspherical structure of the core‐mantle boundary from PKP travel times , 1986 .

[14]  M. Rabinowicz,et al.  Three-dimensional convection in spherical shells , 1986 .

[15]  Jeremy Bloxham,et al.  The secular variation of Earth's magnetic field , 1985, Nature.

[16]  John R. Baumgardner,et al.  Three-dimensional treatment of convective flow in the earth's mantle , 1985 .

[17]  M. Rabinowicz,et al.  Transitions to a two mode axisymmetrical spherical convection : Application to the Earth's mantle , 1985 .

[18]  G. Schubert,et al.  Convective motions in a spherical shell , 1985, Journal of Fluid Mechanics.

[19]  Gary A. Glatzmaier,et al.  Numerical Simulations of Stellar Convective Dynamos. I. The Model and Method , 1984 .

[20]  Adam M. Dziewonski,et al.  Mapping the lower mantle: Determination of lateral heterogeneity in P velocity up to degree and order 6 , 1984 .

[21]  John H. Woodhouse,et al.  Mapping the upper mantle: Three‐dimensional modeling of earth structure by inversion of seismic waveforms , 1984 .

[22]  T. Jordan,et al.  Slab penetration into the lower mantle , 1984 .

[23]  F. Busse.  Quadrupole convection in the lower mantle , 1983 .

[24]  F. Busse,et al.  Patterns of convection in spherical shells. Part 2 , 1982, Journal of Fluid Mechanics.

[25]  F. Busse,et al.  Finite prandtl number convection in spherical shells , 1982 .

[26]  P. Olson.  Mantle convection with spherical effects , 1981 .

[27]  G. Schubert,et al.  Whole planet cooling and the radiogenic heat source contents of the Earth and Moon , 1980 .

[28]  G. Schubert,et al.  Infinite Prandtl number thermal convection in a spherical shell , 1980, Journal of Fluid Mechanics.

[29]  F. Busse,et al.  Non-linear properties of thermal convection , 1978 .

[30]  E. R. Oxburgh,et al.  Mechanisms of continental drift , 1978 .

[31]  F. D. Stacey.  Physics of the earth , 1977 .

[32]  F. Busse.  Patterns of convection in spherical shells , 1975, Journal of Fluid Mechanics.

[33]  R. Young.  Finite-amplitude thermal convection in a spherical shell , 1974, Journal of Fluid Mechanics.

[34]  P. Silver,et al.  Deep Slabs, Geochemical Heterogeneity, and the Large-Scale Structure of Mantle Convection: Investigation of an Enduring Paradox , 1988 .

[35]  P. Roberts.  Convection in Spherical Systems , 1987 .

[36]  G. Schubert,et al.  Character and stability of axisymmetric thermal convection in spheres and spherical shells. [model for heat transfer in planetary interiors , 1983 .

[37]  F. Busse.  Transition to turbulence in Rayleigh-Beénard convection , 1981 .

[38]  G. Schubert,et al.  Thermal convection of an internally heated infinite Prandtl number fluid in a spherical shell. [earth core-mantle-surface model , 1980 .

[39]  D. Turcotte,et al.  Finite-amplitude thermal convection within a self-gravitating fluid sphere. , 1972 .