Numerical Methods for Mantle Convection

Treatise on Geophysics

Volumn 7, Issue , 2007, Pages 227-252

  Zhong, S J ^a   Yuen, D A ^b   Moresi, L N ^c  

^a UNIVERSITY OF COLORADO   (United States)

^b UNIVERSITY OF MINNESOTA   (United States)

^c MONASH UNIVERSITY   (Australia)

Author keywords

Finite difference; Finite elements; Finite volume; Mantle convection; Numerical modeling; Spectral methods

Indexed keywords

EID: 80054860965     PISSN: None     EISSN: None     Source Type: Book    
DOI: 10.1016/B978-044452748-6.00118-8     Document Type: Chapter

References (145)

1. Albers M. A local mesh refinement multigrid method for 3-D convection problems with strongly variable viscosity. Journal of Computational Physics 2000, 160:126-150.
2. Atanga J., Silvester D. Iterative methods for stabilized mixed velocity-pressure finite elements. International Journal of Numerical Methods in Fluids 1992, 14:71-81.
3. Auth C., Harder H. Multigrid solution of convection problems with strongly variable viscosity. Geophysical Journal International 1999, 137:793-804.
4. Balachandar S., Yuen D.A. Three-dimensional fully spectral numerical method for mantle convection with depth-dependent properties. Journal of Computational Physics 1994, 132:62-74.
5. Balachandar S., Yuen D.A., Reuteler D.M., Lauer G. Viscous dissipation in three dimensional convection with temperature-dependent viscosity. Science 1995, 267:1150-1153.
6. Bastian P., Wieners C. Multigrid methods on adaptively refined grids. Computing in Science and Engineering 2006, 8(6):44-55.
7. Bathe K.-J. Finite Element Procedures 1996, Prentice Hall, Englewood Cliffs, NY, 1037 pp.
8. Baumgardner J.R. Three dimensional treatment of convectionflow in the Earth's mantle. Journal of Statistical Physics 1985, 39(5/6):501-511.
9. Bercovici D. The generation of plate tectonics from mantle convection. Earth and Planetary Science Letters 2003, 205:107-121.
10. Bercovici D. A source-sink model of the generation of plate tectonics from non-Newtonian mantle flow. Journal of Geophysical Research 1995, 100:2013-2030.
11. Bergen B., Gradl T., Huelsemann F., Ruede U. A massively parallel multigrid method for finite elements. Computing in Science and Engineering 2006, 8(6):56-62.
12. Berger M., Oliger J. Adaptive mesh refinement for hyperbolic partial differential equations. Journal of Computational Physics 1984, 53:484-512.
13. Blankenbach B., Busse F., Christensen U., et al. A benchmark comparison of mantle convection codes. Geophysical Journal International 1989, 98:23-38.
14. Bollig EF, Jensen PA, Lyness MD, et al. (in press) VLAB: Web Services, Portlets, and Workflows for enabling cyber-infrastructure in computational mineral physics. Physics of the Earth and Planetary Interiors.
15. Brandt A. Multi-level adaptive solutions to boundary-value problems. Mathematics of Computation 1999, 31:333-390.
16. Briggs W.L., Henson V.E., McCormick S.F. A Multigrid Tutorial 2000, SIAM Press, 2000. 2nd edn.
17. Brooks AN and Petrov-Galerkin A (1981) Finite Element Formulation for Convection Dominated Flows. PhD Thesis, California Institute of Technology, Pasadena, CA.
18. Bruegmann B., Tichy W. Numerical solutions of orbiting black holes. Physical Review Letters 2004, 92:211101.
19. Bunge H.P., Richards M.A., Lithgow-Bertelloni C., Baumgardner J.R., Grand S.P., Romanowicz B.A. Time scales and heterogeneous structure in geodynamic Earth models. Science 1998, 280:91-95.
20. Bunge H.-P., Richards M.A., Baumgardner J.R. A sensitive study of 3-dimensional spherical mantle convection at 108 Rayleigh number: Effects of depth-dependent viscosity, heating mode, and an endothermic phase change. Journal of Geophysical Research 1997, 102:11991-12007.
21. Busse F.H. Fundamentals of thermal convection. Mantle Convection, Plate Tectonics and Global Dynamics 1989, 23-109. Gordon and Breach, New York. W.R. Peltier (Ed.).
22. Christensen U. Mixing by time-dependent convection. Earth and Planetary Science Letters 1989, 95:382-394.
23. Christensen U.R. Convection with pressure- and temperature-dependent non-Newtonian rheology. Geophysical Journal of the Royal Astronomical Society 1984, 77:343-384.
24. Christensen U.R., Yuen D.A. Layered convection induced by phase changes. Journal of Geophysical Research 1985, 90:1029110300.
25. Christensen U.R., Yuen D.A. The interaction of a subducting lithosphere with a chemical or phase boundary. Journal of Geophysical Research 1984, 89(B6):4389-4402.
26. High-Performance Computing Requirements for the Computational Solid-Earth Sciences 2005, 101. http://www.geoprose.com/computational_SES.html, R.E. Cohen (Ed.).
27. Cserepes L., Rabinowicz M., Rosemberg-Borot C. Three-dimensional infinite Prandtl number convection in one and two layers with implication for the Earth's gravity field. Journal of Geophysical Research 1988, 93:12009-12025.
28. Cserepes L., Rabinowicz M. Gravity and convection in a two-layer mantle. Earth and Planetary Science Letters 1985, 76:193-207.
29. Daubeschies I., Grossmann A., Meyer Y. Painless nonorthogonal expansions. Journal of Mathematical Physics 1985, 27:1271-1283.
30. Davaille A. Simultaneous generation of hotspots and superswells by convection in a heterogeneous planetary mantle. Nature 1999, 402:756-760.
31. Davies G.F. Penetration of plates and plumes through the mantle transition zone. Earth and Planetary Science Letters 1995, 133:507-516.
32. Desai C.S., Abel J.F. Introduction to the Finite Element Method. A Numerical Method for Engineering Analysis 1972, Academic Press, New York.
33. Erlebacher C., Yuen D.A., Dubuffet F. Current trends and demands in visualization in the geosciences. Visual Geosciences 2001, 6:59. 10.1007/s10069-001-1019-y.
34. Erlebacher G., Yuen D.A., Lu Z., Bollig E.F., Pierce M., Pallickara S. A grid framework for visualization services in the Earth Sciences. Pure and Applied Geophysics 2006, 163:2467-2483.
35. Fornberg B. High-order finite differences and the pseudospectral method on staggered grids. SIAM Journal on Numerical Analysis 1990, 27:904-918.
36. Fornberg B.A. A Practical Guide to Pseudospectral Methods 1995, Cambridge University Press, Cambridge.
37. Gable C.W., O'Connell R.J., Travis B.J. Convection in three dimensions with surface plates: Generation of toroidal flow. Journal of Geophysical Research 1991, 96:8391-8405.
38. Gerya TV and Yuen DA (in press) Robust characteristics method for modeling multiphase visco-elastoplastic thermo-mechanical problems. Physics of the Earth and Planetary Interiors.
39. Gerya T.V., Connolly J.A.D., Yuen D.A., Gorczyk W., Capel A.M. Seismic implications of mantle wedge plumes. Physics of the Earth and Planetary Interiors 2006, 156:59-74.
40. Gerya T.V., Yuen D.A. Characteristics-based marker method with conservative finite-differences schemes for modeling geological flows with strongly variable transport properties. Physics of the Earth and Planetary Interiors 2003, 140:295-320.
41. Glatzmaier G.A. Numerical simulations of mantle convection: Time-dependent, three-dimensional, compressible, spherical shell. Geophysics and Astrophysics Fluid Dynamics 1988, 43:223-264.
42. Glatzmaier G.A., Roberts P.H. A three-dimensional self-consistent computer simulation of a geomagnetic field reversal. Nature 1995, 377:203-209.
43. Golub G.H., van Loan C.F. Matrix Computations 1989, The Johns Hopkins University Press, Baltimore, MD, 642 pp.
44. Griffiths D.V., Smith I.M. Numerical Methods for Engineers 2006, Chapman and Hall, New York. 2nd edn.
45. Gurnis M., Zhong S., Toth J. On the competing roles of fault reactivation and brittle failure in generating plate tectonics from mantle convection. The History and Dynamics of Global Plate Motions 2000, 73-94. American Geophysical Union, Washington, DC. M.A. Richards, R. Gordon, R. van der Hilst (Eds.).
46. Gurnis M., Davies G.F. Mixing in numerical models of mantle convection incorporating plate kinematics. Journal of Geophysical Research 1986, 91:6375-6395.
47. Hackbush W. Multigrid Methods and Applications 1985, Springer Verlag, Berlin.
48. Hager B.H., Richards M.A. Long-wavelength variations in Earth's geoid: Physical models and dynamical implications. Philosophical Transactions of the Royal Society of London A 1989, 328:309-327.
49. Hager B.H., O'Connell R.J. A simple global model of plate dynamics and mantle convection. Journal of Geophysical Research 1981, 86:4843-4878.
50. Hansen C.D., Johnson C.R. The Visualization Handbook 2005, Elsevier, Amsterdam, 962 pp.
51. Hansen U., Yuen D.A. Numerical simulations of thermal chemical instabilities and lateral heterogeneities at the core-mantle boundary. Nature 1988, 334:237-240.
52. Harder H., Hansen U. A finite-volume solution method for thermal convection and dynamo problems in spherical shells. Geophysical Journal International 2005, 161:522-537.
53. Hughes T.J.R. The Finite Element Method 2000, Dover Publications, New York, 682 pp.
54. Ita J.J., King S.D. The influence of thermodynamic formulation on simulations of subduction zone geometry and history. Geophysical Research Letters 1998, 25:1463-1466.
55. Jarvis G.T., McKenzie D.P. Convection in a compressible fluid with infinite Prandtl number. Journal of Fluid Mechanics 1980, 96:515-583.
56. Kameyama M.C., Yuen D.A. 3-D convection studies on the thermal state of the lower mantle with post perovskite transition. Geophysical Research Letters 2006, 33. 10.1029/2006GL025744.
57. Kageyama A., Sato T. The 'Yin and Yang Grid': An overset grid in spherical geometry. Geochemistry Geophysics Geosystem 2004, 5(9):Q09005.
58. Karato S., Wu P. Rheology of the upper mantle: A synthesis. Science 1993, 260:771-778.
59. Kellogg L.H., Hager B.H., van der Hilst R.D. Compositional stratification in the deep mantle. Science 1999, 283:1881-1884.
60. Kellogg L.H. Mixing in the mantle. Annual Review of Earth and Space Sciences 1992, 20:365-388.
61. King S.D., Gable C.W., Weinstein S.A. Models of convection-driven tectonic plates: A comparison of methods and results. Geophysical Journal International 1992, 109:481-487.
62. King S.D., Raefsky A., Hager B.H. ConMan: Vectorizing a finite element code for incompressible two-dimensional convection in the Earth's mantle. Physics of the Earth and Planetary Interiors 1990, 59:195-207.
63. King S.D., Hager B.H. The relationship between plate velocity and trench viscosity in Newtonian and power-law subduction calculations. Geophysical Research Letters 1990, 17:2409-2412.
64. King S.D., Lowman J.P., Gable C.W. Episodic tectonic plate reorganizations driven by mantle convection. Earth and Planetary Science Letters 2002, 203:83-91.
65. Komatitsch D., Tromp J. Introduction to the spectral element method for three-dimensional seismic wave propagation. Geophysical Journal International 1999, 139:806-822.
66. Kowalik Z., Murty T.S. Numerical Modeling of Ocean Dynamics 1993, World Scientific Publishing, Toh Tuck Link, Singapore, 481 pp.
67. Kuang W., Bloxham J. Numerical modeling of magnetohydrodynamic convection in a rapidly rotating spherical shell: Weak and strong field dynamo action. Journal of Computational Physics 1999, 153:51-81.
68. Larsen T.B., Yuen D.A., Moser J., Fornberg B. A higher-order finite-difference method applied to large Rayleigh number mantle convection. Geophysical and Astrophysical Fluid Dynamics 1997, 84:53-83.
69. Lenardic A., Kaula W.M. A numerical treatment of geodynamic viscous flow problems involving the advection of material interfaces. Journal of Geophysical Research 1993, 98:8243-8269.
70. Lynch D.R. Numerical Partial Differential Equations for Environmental Scientists and Engineers: A First Practical Course 2005, Springer Verlag, Berlin, 388 pp.
71. Machetel Ph., Rabinowicz M., Bernadet P. Three-dimensional convection in spherical shells. Geophysical and Astrophysical Fluid Dynamics 1986, 37:57-84.
72. Maday Y., Patera A. Spectral -element methods for the incompressible Navier-Stokes equations. State of the Art Survey in Computational Mechanics 1989, 71-143. ASME, New York. A. Noor, J. Oden (Eds.).
73. Malevsky A.V. Spline-characteristic method for simulation of convective turbulence. Journal of Computational Physics 1996, 123:466-475.
74. Malevsky A.V., Yuen D.A. Characteristics-based methods applied to infinite Prandtl number thermal convection in the hard turbulent regime. Physics Fluids A 1991, 3(9):2105-2115.
75. Malevsky A.V., Yuen D.A. Plume structures in the hard-turbulent regime of three-dimensional infinite Prandtl number convection. Geophysical Research Letters 1993, 20:383-386.
76. McKenzie D.P., Roberts J.M., Weiss N.O. Convection in the Earth's mantle: Towards a numerical solution. Journal of Fluid Mechanics 1974, 62:465-538.
77. McNamara A.K., Zhong S. Thermochemical structures within a spherical mantle: Superplumes or piles?. Journal of Geophysical Research 2004, 109:B07402. 10.1029/2003JB002847.
78. Monnereau M., Yuen D.A. How flat is the lower-mantle temperature gradient?. Earth and Planetary Science Letters 2002, 202:171-183.
79. Moresi L.N., Dufour F., Muhlhaus H.-B. A Lagrangian integration point finite element method for large deformation modeling of viscoelastic geomaterials. Journal of Computational. Physics 2003, 184:476-497.
80. Moresi L.N., Solomatov V.S. Mantle convection with a brittle lithosphere: Thoughts on the global tectonic styles of the Earth and Venus. Geophysical Journal International 1998, 133:669-682.
81. Moresi L.N., Solomatov V.S. Numerical investigation of 2D convection with extremely large viscosity variation. Physics of Fluids 1995, 9:2154-2164.
82. Moresi L., Gurnis M. Constraints on the lateral strength of slabs from three-dimensional dynamic flow models. Earth and Planetary Science Letters 1996, 138:15-28.
83. Moresi L.N., Zhong S., Gurnis M. The accuracy of finite element solutions of Stokes' flow with strongly varying viscosity. Physics of the Earth and Planetary Interiors 1996, 97:83-94.
84. Murakami M., Hirose K., Kawamura K., Sata N., Ohishi Y. Post-perovskite phase transition in MgSiO3. Science 2004, 304:855-858.
85. Ogawa M., Schubert G., Zebib A. Numerical simulations of 3-dimensional thermal convection in a fluid with strongly temperature-dependent viscosity. Journal of Fluid Mechanics 1991, 233:299-328.
86. Ohno N, Kageyama A, and Kusano K (in press) Virtual reality visualization by CAVE with VFIVE and VTK. Journal of Plasma Physics.
87. Oosterlee C.W., Gaspar-Lorenz F.J. Multigrid methods for the Stokes system. Computing in Science and Engineering 2006, 8(6):34-43.
88. Osher S., Fedkiw R.P. Level Set Methods and Dynamic Implicit Surfaces 2003, Springer Verlag, Berlin, 275 pp.
89. Pantakar S.V. Numerical Heat Transfer and Fluid Flow 1980, Hemisphere Publishing Corporation, New York.
90. Pantakar S.V., Spalding D.B. A calculation procedure for heat, mass and momentum transfer in three-dimensional parabolic flows. International Journal of Heat and Mass Transfer 1972, 15:1787-2000.
91. Parmentier E.M., Sotin C. Three-dimensional numerical experiments on thermal convection in a very viscous fluid: Implications for the dynamics of a thermal boundary layer at high Rayleigh number. Physics of Fluids 2000, 12:609-617.
92. Parmentier E.M., Turcotte D.L., Torrance K.E. Studies of finite amplitude non-Newtonian thermal convection with application to convection in the Earth's mantle. Journal of Geophysical Research 1976, 81:1839-1846.
93. Parmentier E.M., Turcotte D.L., Torrance K.E. Numerical experiments on the structure of mantle plumes. Journal of Geophysical Research 1975, 80:4417-4425.
94. Peyret R., Taylor T.D. Computational Methods for Fluid Flow 1983, Springer Verlag, Berlin, 350 pp.
95. Press W.H., Flannery B.P., Teukolsky S.A., Vetterling W.A. Numerical Recipes in FORTRAN 1992, Cambridge University Press, New York.
96. Ramage A., Wathen A.J. Iterative solution techniques for the Stokes and Navier-Stokes equations. International Journal of Numerical Methods in Fluids 1994, 19:67-83.
97. Ratcliff J.T., Schubert G., Zebib A. Steady tetrahedral and cubic patterns of spherical-shell convection with temperature-dependent viscosity. Journal of Geophysical Research 1996, 101:25473-25484.
98. Richter F.M. Finite amplitude convection through a phase boundary. Geophysical Journal of the Royal Astronomical Society 1973, 35:265-276.
99. Richtmeyer R.D., Morton K.W. Difference Methods for Initial Value Problems 1967, Interscience Publishers, New York, 405 pp.
100. Rudolf M., Gerya T.V., Yuen D.A., De Rosier S. Visualization of multiscale dynamics of hydrous cold plumes at subduction zones. Visual Geosciences 2004, 9:59-71.
101. Saad Y., Schultz M.H. GMRES: A generalized minimal residual algorithm for solving nonsymmetric linear systems. SIAM Journal on Scientific Statistical Computing 1986, 7:856-869.
102. Schmalholz S.M., Podladchikov Y.Y., Schmid D.W. A spectral/finite-difference for simulating large deformations of heterogeneous, viscoelastic materials. Geophysical Journal International 2001, 145:199-219.
103. Schott B., Schmeling H. Delamination and detachment of a lithospheric root. Tectonophysics 1998, 296:225-247.
104. Schubert G., Turcotte D.L., Olson P. Mantle Convection in the Earth and Planets 2001, Cambridge University Press, Cambrideg, UK.
105. Schubert G., Yuen D.A., Turcotte D.L. Role of phase transitions in a dynamic mantle. Geophysical Journal of the Royal Astronomical Society 1975, 42:705-735.
106. Smolarkiewicz P.K. A simple positive definite advection transport algorithm with small implicit diffusion. Monthly Weather Review 1983, 111:479-489.
107. Smolarkiewicz P.K. A fully multidimensional positive definite advection scheme with small implicit diffusion. Journal of Computational Physics 1984, 54:325-339.
108. Spalding D.B. A novel finite difference formulation for differential expressions involving both first and second derivatives. International Journal of Numerical Methods in Engineering 1972, 4:551-559.
109. Spiegelman M., Katz R.F. A semi-Lagrangian Cranck-Nicolson algorithm for the numerical solution of advection-diffusion problems. Geochemistry, Geophysics, Geosystems 2006, 7. 10.1029/2005GC001073.
110. Stemmer K., Harder H., Hansen U. A new method to simulate convection with strongly temperature-dependent and pressure-dependent viscosity in a spherical shell: Applications to the Earth's mantle. Physics of the Earth and Planetary Interiors 2006, 157:223-249.
111. Sulsky D., Chen Z., Schreyer H.L. A particle method for history-dependent materials. Computational Methods in Applied Mechanics and Engineering 1994, 118:179-196.
112. Tackley P.J. Self-consistent generation of tectonic plates in time-dependent, three-dimensional mantle convection simulations. Part 1: Pseudoplastic yielding. Geochemistry Geophysics Geosystems 2000, 1(8). 10.1029/2000GC000043.
113. Tackley P.J. Three-dimensional simulations of mantle convection with a thermo-chemical basal boundary layer: D″. The Core-Mantle Region 1998, 231-253. AGU, Washington, DC. M. Gurnis (Ed.).
114. Tackley P.J. Self-consistent generation of tectonic plates in three-dimensional mantle convection. Earth and Planetary Science Letters 1998, 157:9-22.
115. Tackley PJ (1994) Three-Dimensional Models of Mantle Convection: Influence of Phase Transitions and Temperature-Dependent Viscosity. PhD Thesis, California Institute of Technology, Pasadena, CA.
116. Tackley P.J. Effects of strongly temperature-dependent viscosity on time-dependent, 3-dimensional models of mantle convection. Geophysical Research Letters 1993, 20:2187-2190.
117. Tackley P.J., King S.D. Testing the tracer ratio method for modeling active compositional fields in mantle convection simulations. Geochemistry Geophysics Geosystems 2003, 4. 10.1029/2001GC000214.
118. Tackley P.J., Stevenson D.J., Glatzmeir G.A., Schubert G. Effects of an endothermic phase transition at 670km depth on spherical mantle convection. Nature 1993, 361:137-160.
119. Torrance K.E., Turcotte D.L. Thermal convection with large viscosity variations. Journal of Fluid Mechanics 1971, 47:113-125.
120. Travis B.J., Olson P., Schubert G. The transition from two-dimensional to three-dimensional planforms in infinite-Prandtl-number thermal convection. Journal of Fluid Mechanics 1990, 216:71-92.
121. Trompert R.A., Hansen U. Mantle convection simulations with rheologies that generate plate-like behavior. Nature 1998, 395:686-688.
122. Trompert R.A., Hansen U. The application of a finite-volume multigrid method to 3-dimensional flow problems in a highly viscous fluid with a variable viscosity. Geophysical and Astrophysical Fluid Dynamics 1996, 83:261-291.
123. Trottenberg U., Oosterlee C., Schueller A. Multigrid 2001, Academic Press, Orlando, FL, 631 pp.
124. Turcotte D.L., Torrance K.E., Hsui A.T. Convection in the Earth's mantle in Methods. Computational Physics 1973, vol. 13:431-454. Academic Press, New York. B.A. Bolt (Ed.).
125. van den Berg A.P., van Keken P.E., Yuen D.A. The effects of a composite non-Newtonian and Newtonian rheology on mantle convection. Geophysical Journal International 1993, 115:62-78.
126. van Keken P.E., Zhong S. Mixing in a 3D spherical model of present-day mantle convection. Earth and Planetary Science Letters 1999, 171:533-547.
127. van Keken P.E., King S.D., Schmeling H., Christensen U.R., Neumeister D., Doin M.-P. A comparison of methods for the modeling of thermochemical convection. Journal of Geophysical Research 1997, 102:22477-22496.
128. Verfurth R. A combined conjugate gradient-multigrid algorithm for the numerical solution of the Stokes problem. IMA Journal of Numerical Analysis 1984, 4:441-455.
129. Weinberg R.F., Schmeling H. Polydiapirs: Multi-wavelength gravity structures. Journal of Structural Geology 1992, 14:425-436.
130. Yavneh I. Why multigrid methods are so efficient?. Computing in Science and Engineering 2006, 8:12-23.
131. Yoshida M., Kageyama A. Application of the Yin-Yang grid to a thermal convection of a Boussinesq fluid with infinite Prandtl number in a three-dimensional spherical shell. Geophysical Research Letters 2004, 31. 10.1029/2004GL019970.
132. Yuen D.A., Balachandar S., Hansen U. Modelling mantle convection: A significant challenge in geophysical fluid dynamics. Geophysical and Astrophysical Convection 2000, 257-294. P.A. Fox, R.M. Kerr (Eds.).
133. Zhang S., Christensen U. Some effects of lateral viscosity variations on geoid and surface velocities induced by density anomalies in the mantle. Geophysical Jornal International 1993, 114:531-547.
134. Zhang S., Yuen D.A. The influences of lower mantle viscosity stratification on 3-D spherical-shell mantle convection. Earth and Planetary Science Letters 1995, 132:157-166.
135. Zhang S., Yuen D.A. Various influences on plumes and dynamics in time-dependent, compressible, mantle convection in 3-D spherical shell. Physics of the Earth and Planetary Interiors 1996, 94:241-267.
136. Zhang S., Yuen D.A. Intense local toroidal motion generated by variable viscosity compressible convection in 3-D spherical shell. Geophysical Research Letters 1996, 23:3135-3138.
137. Zhong S. Constraints on thermochemical convection of the mantle from plume heat flux, plume excess temperature and upper mantle temperature. Journal of Geophysical Research 2006, 111:B04409. 10.1029/2005JB003972.
138. Zhong S. Dynamics of thermal plumes in 3D isoviscous thermal convection. Geophysical Journal International 2005, 154(162):289-300.
139. Zhong S., Hager B.H. Entrainment of a dense layer by thermal plumes. Geophysical Journal International 2003, 154:666-676.
140. Zhong S., Zuber M.T., Moresi L.N., Gurnis M. Role of temperature dependent viscosity and surface plates in spherical shell models of mantle convection. Journal of Geophysical Research 2000, 105:11063-11082.
141. Zhong S., Gurnis M., Moresi L.N. The role of faults, nonlinear rheology, and viscosity structure in generating plates from instantaneous mantle flow models. Journal of Geophysical Research 1998, 103:15255-15268.
142. Zhong S., Gurnis M. The role of plates and temperature-dependent viscosity in phase change dynamics. Journal of Geophysical Research 1994, 99:15903-15917.
143. Zienkiewicz O.C., Taylor R.L., Zhu J.Z. The Finite Element Method: Its Basis and Fundamentals 2005, Butterworth and Heinemann, Woburn, MA. 6th edn.
144. Computational Infrastructure for Geodynamics. http://www.geodynamics.org.
145. Laboratory for Computational Science and Engineering, University of Minnesota. http://www.lcse.umn.edu.