Implementation of a high-order combined compact difference scheme in problems of thermally driven convection and dynamo in rotating spherical shells

Geophysical and Astrophysical Fluid Dynamics

Volumn 106, Issue 3, 2012, Pages 231-249

  Takahashi, Futoshi ^a  

^a TOKYO INSTITUTE OF TECHNOLOGY   (Japan)

Author keywords

Combined compact difference; Core convection; Numerical simulation; Planetary dynamos

Indexed keywords

COMBINED COMPACT DIFFERENCE; COMPACT DIFFERENCE SCHEME; CORE CONVECTION; DYNAMO ACTION; FINITE DIFFERENCE SCHEME; GOVERNING EQUATIONS; GRID POINTS; HIGH-ORDER; NON-UNIFORM GRIDS; NON-UNIFORM MESH; NUMERICAL COMPARISON; RAYLEIGH NUMBER; SECOND DERIVATIVES; SOLUTION METHODS; SPHERICAL HARMONICS; SPHERICAL SHELL; STABILITY OF THE SCHEME; THERMALLY DRIVEN;

COMPUTER SIMULATION; HARMONIC ANALYSIS; INTERACTIVE DEVICES;

DC GENERATORS;

ACCURACY ASSESSMENT; CORE (PLANETARY); GEODYNAMO; MANTLE CONVECTION; MODELING; RAYLEIGH NUMBER;

EID: 84860892421     PISSN: 03091929     EISSN: 10290419     Source Type: Journal    
DOI: 10.1080/03091929.2011.565337     Document Type: Article

References (32)

1. Bullard, EC and Gellman, H. 1954. Homogeneous dynamos and terrestrial magnetism. Philos. Trans. R. Soc. Lond. A, 247: 213-278.
2. Chan, KH, Li, L and Liao, X. 2006. Modelling the core convection using finite element and finite difference methods. Phys. Earth Planet. Inter., 157: 124-138.
3. Chu, PC and Fan, C. 1998. A three-point combined compact difference scheme. J. Comput. Phys., 140: 370-399.
4. Chu, PC and Fan, C. 1999. A three-point six-order nonuniform combined compact difference scheme. J. Comput. Phys., 148: 663-674.
5. Christensen, UR, Aubert, J, Cardin, P, Dormy, E, Gibbons, S, Glatzmaier, GA, Grote, E, Honkura, Y, Jones, C, Kono, M, Matsushima, M, Sakuraba, A, Takahashi, F, Tilgner, A, Wicht, J and Zhang, K. 2001. A numerical dynamo benchmark. Phys. Earth Planet. Inter., 128: 25-34.
6. Christensen, UR and Wicht, J. 2007. "Numerical dynamo simulations". In Treatise on Geophysics, Edited by: Olson, P. Vol. 8, 245-282. Amsterdam: Elsevier B.V.
7. Fournier, A, Bunge, H-P, Hollerbach, H and Vilotte, J-P. 2005. A Fourier-spectral element algorithm for thermal convection in rotating axisymmetric containers. J. Comput. Phys., 204: 462-489.
8. Glatzmaier, GA. 1984. Numerical simulations of steller convective dynamos: I. The model and method. J. Comput. Phys., 55: 461-484.
9. Glatzmaier, GA and Roberts, PH. 1995. A three-dimensional self-consistent computer simulation of a geomagnetic field reversal. Nature, 337: 203-209.
10. Harder, H and Hansen, U. 2005. A finite-volume solution method for thermal convection and dynamo problems in spherical shells. Goephys. J. Int., 161: 522-532.
11. Hejda, P and Reshetnyak, M. 2003. Control volume method for the dynamo problem in the sphere with the free rotating inner core. Stud. Geophys. Geod., 47: 147-159.
12. Hollerbach, R. 2000. A spectral solution of the magneto-convection equations in spherical geometry. Int. J. Numer. Meth. Fluids., 32: 773-797.
13. Jackson, A, Jonkers, ART and Walker, MR. 2000. Four centuries of geomagnetic secular variation from historical records. Phil. Trans. R. Soc. Lond. A, 358: 957-990.
14. Kageyama, A, Miyagoshi, T and Sato, T. 2008. Formation of current coils in geodynamo simulations. Nature, 454: 1106-1109.
15. Kageyama, A, Sato, T and the Complexity Simulation Group. 1995. Computer simulation of a magnetohydrodynamic dynamo: II. Phys. Plasmas, 2: 1421-1431.
16. Kono, M and Roberts, PH. 2002. Recent geodynamo simulations and observations of the geomagnetic field. Rev. Geophys., 40: 1013
17. Kuang, W and Bloxham, J. 1999. Numerical modeling of magnetohydrodynamic convection in a rapidly rotating spherical shell: weak and strong field dynamo action. J. Comput. Phys., 153: 51-81.
18. Lele, SK. 1992. Compact finite difference schemes with spectral-like resolution. J. Comput. Phys., 103: 16-42.
19. Mahesh, K. 1998. A family of high order finite difference schemes with good spectral resolution. J. Comput. Phys., 145: 332-358.
20. Matsui, H and Okuda, H. 2004. Development of a simulation code for MHD dynamo processes using the GeoFEM platform. Int. J. Comput. Fluid Dyn., 18: 323-332.
21. Matsui, H and Okuda, H. 2005. MHD dynamo simulation using the GeoFEM platform -- verification by the dynamo benchmark test. Int. J. Comput. Fluid Dyn., 19: 15-22.
22. Nishikawa, N and Kusano, K. 2002. Effect of density stratification on the thermal convection in a rotating spherical shell. Astrophys. J., 581: 745-759.
23. Oishi, Y, Sakuraba, A and Hamano, Y. 2007. Numerical method for geodynamo simulations based on Fourier expansion in longitude and finite difference in meridional plane. Phys. Earth Planet. Inter., 164: 208-220.
24. Olson, P, Christensen, UR and Glatzmaier, GA. 1999. Numerical modeling of the geodynamo: mechanisms of field generation and equilibration. J. Geophys. Res., 104: 10383-10404.
25. Sakuraba, A and Roberts, PH. 2009. Generation of a strong magnetic field using uniform heat flux at the surface of the core. Nature Geosci., 2: 802-805.
26. Sengupta, TK, Ganeriwal, G and De, S. 2003. Analysis of central and upwind compact schemes. J. Comput. Phys., 192: 677-694.
27. Sengupta, TK, Lakshmanan, V and Vijay, VVSN. 2009. A new combined stable and dispersion relation preserving compact scheme for non-periodic problems. J. Comput. Phys., 228: 3048-3071.
28. Takahashi, F, Matsushima, M and Honkura, Y. 2005. Simulations of a quasi-Taylor state geomagnetic field including polarity reversals on the Earth Simulator. Science, 309: 459-461.
29. Takahashi, F, Matsushima, M and Honkura, Y. 2008a. Scale variability in convection-driven MHD dynamos at low Ekman number. Phys. Earth Planet. Inter., 167: 168-178.
30. Takahashi, F, Tsunakawa, H, Matsushima, M, Mochizuki, N and Honkura, Y. 2008b. Effects of thermally heterogeneous structure in the lowermost mantle on the geomagnetic field strength. Earth Planet. Sci. Lett., 272: 738-746.
31. Tilgner, A. 1999. Spectral methods for the simulations of incompressible flows in spherical shells. Int. J. Numer. Meth. Fluids., 30: 713-724.
32. Wicht, J, Stellmach, S and Harder, H. 2009. "Numerical models of the geodynamo: from fundamental Cartesian models to 3D simulations of field reversals". In Geomagnetic Field Variations, Edited by: Glassmeier, K., Soffel, H. and Negendank, J. 107-158. Berlin: Springer.