Journal of Earth Science

, Volume 29, Issue 1, pp 1–20 | Cite as

How Properties that Distinguish Solids from Fluids and Constraints of Spherical Geometry Suppress Lower Mantle Convection

Invited Article


The large magnitude of the dimensionless Rayleigh number (Ra ∼108) for Earth’s ∼3 000 km thick mantle is considered evidence of whole mantle convection. However, the current formulation assumes behavior characteristic of gases and liquids and also assumes Cartesian geometry. Issues arising from neglecting physical properties unique to solids and ignoring the spherical shapes for planets include: (1) Planet radius must be incorporated into Ra, in addition to layer thickness, to conserve mass during radial displacements. (2) The vastly different rates for heat and mass diffusion in solids, which result from their decoupled transport mechanisms, promote stability. (3) Unlike liquids, substantial stress is needed to deform solids, which independently promotes stability. (4) High interior compression stabilizes the mantle in additional minor ways. Therefore, representing conditions for convection in solid, self-gravitating spheroids, requires modifying formulae developed for bottomheated fluids near ambient conditions under an invariant gravitational field. To derive stability criteria appropriate to solid spheres, we use dimensional analysis, and consider the effects of geometry, force competition, and microscopic behavior. We show that internal heating has been improperly accounted for in the Ra. We conclude that the lower mantle is stable for two independent reasons: heat diffusion far outpaces mass diffusion (creep) and yield strength of solids at high pressure exceeds the effective deviatoric stress. We discuss the role of partial melt in lubricating plate motion, and explain why the Ra is not applicable to the multi-component upper mantle. When conduction is insufficient to transport heat in the Earth, melt production and ascent are expected, not convection of solid rock.

Key words

plasticity diffusion mantle convection stability criteria dimensional analysis geometry 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.



Support for AMH was provided by NSF (No. EAR- 1524495). The authors declare that no conflict of interest exists. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation. E. M. Criss is an employee of Panasonic Avionics Corporation, but prepared this article independent of his employment and without use of information, resources, or other support from Panasonic Avionics Corporation. The final publication is available at Springer via

References Cited

  1. Agee, C. B., 1998. Phase Transformations and Seismic Structure in the Upper Mantle and Transition Zone. Reviews in Mineralogy, 37: 165–204Google Scholar
  2. Anderson, D. L., 1989. Theory of the Earth. Blackwell Scientific, BostonGoogle Scholar
  3. Armienti, P., Gasperini, D., 2010. Isotopic Evidence for Chaotic Imprint in Upper Mantle Heterogeneity. Geochemistry, Geophysics, Geosystems, 11(5): Q0AC02. Scholar
  4. Aurnou, J. M., Olson, P. L., 2001. Experiments on Rayleigh–Bénard Convection, Magnetoconvection and Rotating Magnetoconvection in Liquid Gallium. Journal of Fluid Mechanics, 430: 283–307. Scholar
  5. Bercovici, D., 2015. Mantle Dynamics: An Introduction and Overview. In: Schubert, G., ed., Treatise on Geophysics, 7: 1–22Google Scholar
  6. Birch, J. M., Wilshire, B., 1974. Transient and Steady State Creep Behaviour of Polycrystalline MgO. Journal of Materials Science, 9(6): 871–875. Scholar
  7. Blagoveshchenskii, N., Novikov, A., Puchkov, A., et al., 2015. Self-Diffusion in Liquid Gallium and Hard Sphere Model. EPJ Web of Conferences, 83: 02018. Scholar
  8. Bleazard, J. G., Sun, T. F., Teja, A. S., 1996. The Thermal Conductivity and Viscosity of Acetic Acid-Water Mixtures. International Journal of Thermophysics, 17(1): 111–125. Scholar
  9. Boresi, A. P., Schmidt, R. J., 2003. Advanced Mechanics of Materials. John Wiley and Sons, Hoboken, NJGoogle Scholar
  10. Bridgeman, P., 1927. Dimensional Analysis. Yale University Press, New HavenGoogle Scholar
  11. Brillo, J., Pommrich, A. I., Meyer, A., 2011. Relation between Self-Diffusion and Viscosity in Dense Liquids: New Experimental Results from Electrostatic Levitation. Physical Review Letters, 107(16): 165902. Scholar
  12. Buckingham, E., 1914. On Physically Similar Systems; Illustrations of the Use of Dimensional Equations. Physical Review, 4(4): 345–376. Scholar
  13. Bürgmann, R., Dresen, G., 2008. Rheology of the Lower Crust and Upper Mantle: Evidence from Rock Mechanics, Geodesy, and Field Observations. Annual Review of Earth and Planetary Sciences, 36(1): 531–567. Scholar
  14. Carslaw, H. S., Jaeger, J. C., 1959. Conduction of Heat in Solids, 2nd Edition. Oxford University Press, New YorkGoogle Scholar
  15. Chakraborty, S., 2010. Diffusion Coefficients in Olivine, Wadsleyite and Ringwoodite. Reviews in Mineralogy and Geochemistry, 72(1): 603–639. Scholar
  16. Chudinovskikh, L., Boehler, R., 2007. Eutectic Melting in the System Fe-S to 44 GPa. Earth and Planetary Science Letters, 257(1/2): 97–103. Scholar
  17. Costin, L. S., 1985. Damage Mechanics in the Post-Failure Regime. Mechanics of Materials, 4(2): 149–160. Scholar
  18. Coupland, J. N., McClements, D. J., 1997. Physical Properties of Liquid Edible Oils. Journal of the American Oil Chemists’ Society, 74(12): 1559–1564. Scholar
  19. Criss, E. M., Smith, R. J., Meyers, M. A., 2015. Failure Mechanisms in Cobalt Welded with a Silver-Copper Filler. Materials Science and Engineering: A, 645: 369–382. Scholar
  20. Criss, R. E., Hofmeister, A. M., 2016. Conductive Cooling of Spherical Bodies with Emphasis on the Earth. Terra Nova, 28(2): 101–109. Scholar
  21. Cussler, E. L., 2008. Diffusion: Mass Transport in Fluid Systems. Cambridge University Press, CambridgeGoogle Scholar
  22. Davies, G. F., 2011. Mantle Convection for Geologists. Cambridge University Press, CambridgeCrossRefGoogle Scholar
  23. Davis, R. O., Selvadurai, A. P. S., 2005. Plasticity and Geomechanics. Cambridge University Press, Cambridge de FreitasGoogle Scholar
  24. Cabral, A. J., de Oliveira, P. C., Moreira, S. G. C., et al., 2011. Thermal Diffusivity of Palm Olein and Compounds Containing Β-Carotene. International Journal of Thermophysics, 32(9): 1966–1972. Scholar
  25. Diamante, L. M., Lan, T. Y., 2014. Absolute Viscosities of Vegetable Oils at Different Temperatures and Shear Rate Range of 64.5 to 4 835 s-1. Journal of Food Processing, 2014(3): 1–6. Scholar
  26. Doglioni, C., Anderson, D. L., 2015. Top Driven Asymmetric Mantle Convection. In: Foulger, G. R., Lustrino, M., King, S. D., eds., The Interdisciplinary Earth: In Honor of Don L. Anderson. GSA Special Papers, 214: 51–64Google Scholar
  27. Doglioni, C., Panza, G., 2015. Polarized Plate Tectonics. Advances in Geophysics, 56: 1–167CrossRefGoogle Scholar
  28. Domínguez-Rodríguez, A., Gómez-García, D., Zapata-Solvas, E., et al., 2007. Making Ceramics Ductile at Low Homologous Temperatures. Scripta Materialia, 56(2): 89–91. Scholar
  29. Doremus, R. H., 2002. Viscosity of Silica. Journal of Applied Physics, 92(12): 7619–7629. Scholar
  30. Du, Z., Vinnik, L. P., Foulger, G. R., 2006. Evidence from P-to-S Mantle Converted Waves for a Flat “660-km” Discontinuity beneath Iceland. Earth and Planetary Science Letters, 241(1/2): 271–280. Scholar
  31. Dziewonski, A. M., Anderson, D. L., 1981. Preliminary Reference Earth Model. Physics of the Earth and Planetary Interiors, 25(4): 297–356. Scholar
  32. Elder, J., 1976. The Bowels of the Earth. Oxford University Press, Oxford. ISBN 0-19-854413-8Google Scholar
  33. Ertl, H., Dullien, F. A. L., 1973. Self-Diffusion and Viscosity of some Liquids as a Function of Temperature. AIChE Journal, 19(6): 1215–1223. Scholar
  34. Fegley, B. Jr., 2015. Practical Chemical Thermodynamics for Geoscientists. Academic Press/Elsevier, Waltham, MassachusettsGoogle Scholar
  35. Fichtner, A., Villaseñor, A., 2015. Crust and Upper Mantle of the Western Mediterranean— Constraints from Full-Waveform Inversion. Earth and Planetary Science Letters, 428: 52–62. Scholar
  36. Foulger, G. R., 2010. Plates vs Plumes: A Geological Controversy. Wiley-Blackwell, ISBN 978-1-4443-3679-5. 328CrossRefGoogle Scholar
  37. Foulger, G. R., Panza, G. F., Artemieva, I. M., et al., 2013. Caveats on Tomographic Images. Terra Nova, 25: 259–281CrossRefGoogle Scholar
  38. Foulger, G. R., Pritchard, M. J., Julian, B. R., et al., 2001. Seismic Tomography Shows that Upwelling beneath Iceland is Confined to the Upper Mantle. Geophysical Journal International, 146(2): 504–530. Scholar
  39. French, S. W., Romanowicz, B., 2015. Broad Plumes Rooted at the Base of the Earth’s Mantle beneath Major Hotspots. Nature, 525(7567): 95–99. Scholar
  40. Frenkel, J., 1926. Zur Theorie Der Elastizitätsgrenze Und Der Festigkeit Kristallinischer Körper. Zeitschrift für Physik, 37(7/8): 572–609. Scholar
  41. Gando, A., Gando, Y., Ichimura, K., et al., 2011. Partial Radiogenic Heat Model for Earth Revealed by Geoneutrino Measurements. Nature Geoscience, 4(9): 647–651. Scholar
  42. Gao, S. S., Liu, K. H., 2014. Imaging Mantle Discontinuities Using Multiply-Reflected P-to-S Conversions. Earth and Planetary Science Letters, 402: 99–106. Scholar
  43. Gasparik, T., 2000. Evidence for the Transition Zone Origin of some [Mg, Fe]O Inclusions in Diamonds. Earth and Planetary Science Letters, 183(1/2): 1–5. Scholar
  44. Glazier, J. A., Segawa, T., Naert, A., et al., 1999. Evidence against ‘Ultrahard’ Thermal Turbulence at very High Rayleigh Numbers. Nature, 398(6725): 307–310. Scholar
  45. Goes, S., Agrusta, R., van Hunen, J., et al., 2017. Subduction-Transition Zone Interaction: A Review. Geosphere, 13(3): 644–664. Scholar
  46. Hamilton, W. B., 2002. The Closed Upper-Mantle Circulation of Plate Tectonics. In: Stein S., Freymueller, J. T., eds., Plate Boundary Zones: Geodynamics Series. American Geophysical Union, Washington, D.C.. 359–410Google Scholar
  47. Hamilton, W. B., 2011. Plate Tectonics Began in Neoproterozoic Time, and Plumes from Deep Mantle have never Operated. Lithos, 123(1/2/3/4): 1–20. Scholar
  48. Hamilton, W. B., 2015. Terrestrial Planets Fractionated Synchronously with Accretion, but Earth Progressed through Subsequent Internally Dynamic Stages whereas Venus and Mars have been Inert for more than 4 Billion Years. GSA Special Papers, 514: 123–156Google Scholar
  49. Hamza, V. M., 2013. Global Heat Flow without Invoking “Kelvin Paradox”. Frontiers in Geosciences, 1: 11–20Google Scholar
  50. He, X. M., Fowler, A., Toner, M., 2006. Water Activity and Mobility in Solutions of Glycerol and Small Molecular Weight Sugars: Implication for Cryo-and Lyopreservation. Journal of Applied Physics, 100(7): 074702. Scholar
  51. Heap, M. J., Baud, P., Meredith, P. G., et al., 2011. Brittle Creep in Basalt and Its Application to Time-Dependent Volcano Deformation. Earth and Planetary Science Letters, 307(1/2): 71–82. Scholar
  52. Heep, M. J., 2009. Creep: Time-Dependent Brittle Deformation in Rocks: [Dissertation]. University College London, LondonGoogle Scholar
  53. Henderson, G., 1982. Inorganic Geochemistry. Permagon Press, New York. ISBN 0-08-020448-1Google Scholar
  54. Hetényi, G., 2014. To Conserve or not to Conserve (Mass in Numerical Models). Terra Nova, 26(5): 372–376. Scholar
  55. Hill, R., 1950. The Mathematical Theory of Plasticity. Oxford University Press, OxfordGoogle Scholar
  56. Hiraga, T., Miyazaki, T., Tasaka, M., et al., 2010. Mantle Superplasticity and Its Self-Made Demise. Nature, 468(7327): 1091–1094. Scholar
  57. Hirth, G., 2002. Laboratory Constraints on the Rheology of the Upper Mantle. Reviews in Mineralogy and Geochemistry, 51(1): 97–120. Scholar
  58. Hofmeister, A. M., 2010. Scale Aspects of Heat Transport in the Diamond Anvil Cell, in Spectroscopic Modeling, and in Earth’s Mantle: Implications for Secular Cooling. Physics of the Earth and Planetary Interiors, 180(3/4): 138–147. Scholar
  59. Hofmeister, A. M., Branlund, J. M., 2016. Thermal Conductivity of the Earth. In: Schubert, G., ed., Treatise in Geophysics, 2nd Edition. V. 2 Mineral Physics (Price, G. D., ed.). Elsevier, The Netherlands. 584–608Google Scholar
  60. Hofmeister, A. M., Criss, R. E., 2005. Earth’s Heat Flux Revised and Linked to Chemistry. Tectonophysics, 395(3/4): 159–177. Scholar
  61. Hofmeister, A. M., Criss, R. E., 2012. A Thermodynamic and Mechanical Model for Formation of the Solar System via 3-Dimensional Collapse of the Dusty Pre-Solar Nebula. Planetary and Space Science, 62(1): 111–131. Scholar
  62. Hofmeister, A. M., Criss, R. E., 2013. How Irreversible Heat Transport Processes Drive Earth’s Interdependent Thermal, Structural, and Chemical Evolution. Gondwana Research, 24(2): 490–500. Scholar
  63. Hofmeister, A. M., Criss, R. E., 2015. Evaluation of the Heat, Entropy, and Rotational Changes Produced by Gravitational Segregation during Core Formation. Journal of Earth Science, 26(1): 124–133. Scholar
  64. Hofmeister, A. M., Sehlke, A., Avard, G., et al., 2016. Transport Properties of Glassy and Molten Lavas as a Function of Temperature and Composition. Journal of Volcanology and Geothermal Research, 327: 330–348. Scholar
  65. Hofmeister, A. M., Whittington, A. G., 2012. Effects of Hydration, Annealing, and Melting on Heat Transport Properties of Fused Quartz and Fused Silica from Laser-Flash Analysis. Journal of Non-Crystalline Solids, 358(8): 1072–1082. Scholar
  66. Huang, L. H., Liu, L. S., 2009. Simultaneous Determination of Thermal Conductivity and Thermal Diffusivity of Food and Agricultural Materials Using a Transient Plane-Source Method. Journal of Food Engineering, 95(1): 179–185. Scholar
  67. Jin, Z. M., Zhang, J. F., Green, H. W. II, et al., 2001. Eclogite Rheology: Implications for Subducted Lithosphere. Geology, 29(8): 667–670.<0667:erifsl>;2CrossRefGoogle Scholar
  68. Kajihara, K., Kamioka, H., Hirano, M., et al., 2005. Interstitial Oxygen Molecules in Amorphous SiO2. III. Measurements of Dissolution Kinetics, Diffusion Coefficient, and Solubility by Infrared Photoluminescence. Journal of Applied Physics, 98(1): 013529. Scholar
  69. Kavner, A., Duffy, T. S., 2001. Strength and Elasticity of Ringwoodite at Upper Mantle Pressures. Geophysical Research Letters, 28(14): 2691–2694. Scholar
  70. Kestin, J., Knierim, K., Mason, E. A., et al., 1984. Equilibrium and Transport Properties of the Noble Gases and Their Mixtures at Low Density. Journal of Physical and Chemical Reference Data, 13(1): 229–303. Scholar
  71. Kohlstedt, D. L., Hansen, L. N., 2015. Constituative Behavior, Rheological Behavior, and Viscosity of Rocks. In: Schubert, G., ed., Treatise in Geophysics, 2nd Edition, Vol. 2. Elsevier, The Netherlands. 389–427Google Scholar
  72. Koschmieder, E. L., Pallas, S. G., 1974. Heat Transfer through a Shallow, Horizontal Convecting Fluid Layer. International Journal of Heat and Mass Transfer, 17(9): 991–1002. Scholar
  73. Langdon, T. G., 1982. Fracture Processes in Superplastic Flow. Metal Science, 16(4): 175–183. Scholar
  74. Lodders, K., 2000. An Oxygen Isotope Mixing Model for the Accretion and Composition of Rocky Planets. Space Science Review, 92: 341–354CrossRefGoogle Scholar
  75. Luca, J., Mrawira, D., 2005. New Measurement of Thermal Properties of Superpave Asphalt Concrete. Journal of Materials in Civil Engineering, 17(1): 72–79. Scholar
  76. Meyer, R. E., 1961. Self-Diffusion of Liquid Mercury. The Journal of Physical Chemistry, 65(3): 567–568. Scholar
  77. Meyers, M. A., Chawla, K. K., 2009. Mechanical Behavior of Materials. Cambridge University Press, CambridgeGoogle Scholar
  78. Mitchell, B. S., 2004. An Introduction to Materials Engineering and Science for Chemical and Materials Engineers. John Wiley and Sons, Inc., HobokenGoogle Scholar
  79. Moghadam, R. H., Trepmann, C. A., Stöckhert, B., et al., 2010. Rheology of Synthetic Omphacite Aggregates at High Pressure and High Temperature. Journal of Petrology, 51(4): 921–945. Scholar
  80. Mukherjee, A. K., Bird, J. E., Dorn, J. E., 1969. Experimental Correlation for High-Temperature Creep. Transactions of the American Society of Metals, 62: 155–179Google Scholar
  81. Nabelek, P. I., Hofmeister, A. M., Whittington, A. G., 2012. The Influence of Temperature-Dependent Thermal Diffusivity on the Conductive Cooling Rates of Plutons and Temperature-Time Paths in Contact Aureoles. Earth and Planetary Science Letters, 317/318: 157–164CrossRefGoogle Scholar
  82. Nguyen, L. T., Balasubramaniam, V. M., Sastry, S. K., 2012. Determination of In-Situ Thermal Conductivity, Thermal Diffusivity, Volumetric Specific Heat and Isobaric Specific Heat of Selected Foods under Pressure. International Journal of Food Properties, 15(1): 169–187. Scholar
  83. Nishi, T., Shibata, H., Waseda, Y., et al., 2003. Thermal Conductivities of Molten Iron, Cobalt, and Nickel by Laser Flash Method. Metallurgical and Materials Transactions A, 34(12): 2801–2807. Scholar
  84. Nishihara, Y., Tinker, D., Kawazoe, T., et al., 2008. Plastic Deformation of Wadsleyite and Olivine at High-Pressure and High-Temperature Using a Rotational Drickamer Apparatus (RDA). Physics of the Earth and Planetary Interiors, 170(3/4): 156–169. Scholar
  85. Paterson, M. S., 1958. Experimental Deformation and Faulting in Wombeyan Marble. Geological Society of America Bulletin, 69(4): 465–475.[465:edafiw];2CrossRefGoogle Scholar
  86. Paterson, M. S., Weaver, C. W., 1970. Deformation of Polycrystalline MgO under Pressure. Journal of the American Ceramic Society, 53(8): 463–471. Scholar
  87. Pearson, D. S., Ver Strate, G., Von Meerwall, E., et al., 1987. Viscosity and Self-Diffusion Coefficient of Linear Polyethylene. Macromolecules, 20(5): 1133–1141. Scholar
  88. Prewitt, C. T., Downs, R. T., 1998. High-Pressure Crystal Chemistry. Reviews in Mineralogy, 37: 284–342Google Scholar
  89. Rayleigh, L., 1916. On Convection Currents in a Horizontal Layer of Fluid, when the Higher Temperature is on the under Side. Philosophical Magazine Series 6, 32(192): 529–546. Scholar
  90. Rees, B. A., Okal, E. A., 1987. The Depth of the Deepest Historical Earthquakes. Pure and Applied Geophysics, 125(5): 699–715. Scholar
  91. Reif, F., 1965. Fundamentals of Statistical and Thermal Physics. McGraw-Hill Book Company, St. Louis. 651Google Scholar
  92. Romine, W. L., Whittington, A. G., 2015. A Simple Model for the Viscosity of Rhyolites as a Function of Temperature, Pressure and Water Content. Geochimica et Cosmochimica Acta, 170: 281–300. Scholar
  93. Romine, W. L., Whittington, A. G., Nabelek, P. I., et al., 2012. Thermal Diffusivity of Rhyolitic Glasses and Melts: Effects of Temperature, Crystals and Dissolved Water. Bulletin of Volcanology, 74(10): 2273–2287. Scholar
  94. Schriempf, J. T., 1972. A Laser Flash Technique for Determining Thermal Diffusivity of Liquid Metals at Elevated Temperatures. Review of Scientific Instruments, 43(5): 781–786. Scholar
  95. Schriempf, J. T., 1973. Thermal Diffusivity of Liquid Gallium. Solid State Communications, 13(6): 651–653. Scholar
  96. Schubert, G., Turcotte, D. L., Olson, P., 2001. Mantle Convection in the Earth and Planets. Cambridge University Press, CambridgeCrossRefGoogle Scholar
  97. Sehlke, A., Whittington, A., Robert, B., et al., 2014. Pahoehoe to ‘a’a Transition of Hawaiian Lavas: An Experimental Study. Bulletin of Volcanology, 76(11): 876–896. Scholar
  98. Shimada, M., Cho, A., Yukutake, H., 1983. Fracture Strength of Dry Silicate Rocks at High Confining Pressures and Activity of Acoustic Emission. Tectonophysics, 96(1/2): 159–172. Scholar
  99. Siggia, E. D., 1994. High Rayleigh Number Convection. Annual Review of Fluid Mechanics, 26(1): 137–168. Scholar
  100. Smith, E. M., Shirey, S. B., Nestola, F., et al., 2016. Large Gem Diamonds from Metallic Liquid in Earth’s Deep Mantle. Science, 354(6318): 1403–1405. Scholar
  101. Soutas-Little, R., 2011. History of Continuum Mechanics. In: Meridio, J., Saccomandi, G., eds., Continuum Mechanics. Eolss Publishers, Singapore. 80–93Google Scholar
  102. Stacey, F. D., Stacey, C. H. B., 1999. Gravitational Energy of Core Evolution: Implications for Thermal History and Geodynamo Power. Physics of the Earth and Planetary Interiors, 110(1/2): 83–93. Scholar
  103. Stein, C. A., Stein, S. A., 1992. A Model for the Global Variation in Oceanic Depth and Heat Flow with Lithospheric Age. Nature, 359(6391): 123–129. Scholar
  104. Stengel, K. C., Oliver, D. S., Booker, J. R., 1982. Onset of Convection in a Variable-Viscosity Fluid. Journal of Fluid Mechanics, 120: 411–431. Scholar
  105. Thern, A., Lüdemann, H. D., 1996. P, T Dependence of the Self Diffusion Coefficients and Densities in Liquid Silicone Oils. Zeitschrift für Naturforschung A, 51(3): 192–196. Scholar
  106. Timoshenko, S. P., Goodier, J. N., 1970. Theory of Elasticity. McGraw-Hill, New YorkGoogle Scholar
  107. Transtrum, M. K., Machta, B. B., Brown, K. S., et al., 2015. Perspective: Sloppiness and Emergent Theories in Physics, Biology, and beyond. The Journal of Chemical Physics, 143(1): 010901. Scholar
  108. Tritton, D. J., 1977. Physical Fluid Dynamics. Van Nostrand Reinhold, New YorkCrossRefGoogle Scholar
  109. Van Schmus, W. R., 1995. Natural Radioactivity of the Crust and Mantle. In: Ahrens, T. J., ed., Global Earth Physics. American Geophysical Union, Washington D.C. 283–291Google Scholar
  110. Wawersik, W. R., Brace, W. F., 1970. Post-Failure Behavior of a Granite and Diabase. Rock Mechanics and Rock Engineering, 3: 61–85CrossRefGoogle Scholar
  111. Weidner, D. J., Li, L., 2015. Methods for the Study of High P/T Deformation and Rheology. In: Schubert, G., ed., Treatise on Geophysics, 2: 339–358CrossRefGoogle Scholar
  112. White, D. B., 1988. The Planforms and Onset of Convection with a Temperature-Dependent Viscosity. Journal of Fluid Mechanics, 191: 247–286. Scholar
  113. Whittington, A. G., Hofmeister, A. M., Nabelek, P. I., 2009. Temperature-Dependent Thermal Diffusivity of Earth’s Crust: Implications for Crustal Anatexis. Nature, 458: 319–321CrossRefGoogle Scholar
  114. Xu, Z., Morris, R., Bencsik, M., et al., 2014. Detection of Virgin Olive Oil Adulteration Using Low Field Unilateral NMR. Sensors, 14(2): 2028–2035. Scholar
  115. Yáñez-Limón, J. M., Mayen-Mondragón, R., Martínez-Flores, O., et al., 2005. Thermal Diffusivity Studies in Edible Commercial Oils Using Thermal Lens Spectroscopy. Superficies y Vacio, 18: 31–37Google Scholar
  116. Zemansky, M. W., Dittman, R. H., 1981. Heat and Thermodynamics, 6th Edition. McGraw-Hill, New YorkGoogle Scholar
  117. Zener, C., 1938. Internal Friction in Solids II. General Theory of Thermoelastic Internal Friction. Physical Review, 53(1): 90–99. Scholar
  118. Zhang, Y., Ni, H., Chen, Y., 2010. Diffusion of H, C, and O Components in Silicate Melts. Reviews in Mineralogy and Geochemistry, 72(1): 171–225. Scholar
  119. Zhong, S. J., Yuen, D. A., Moresi, L. M, et al., 2015. Numerical Method for Mantle Convection. In: Schubert, G., ed., Treatise on Geophysics. Mantle Dynamics, 7: 197–222Google Scholar
  120. Zombeck, M. V., 2007. Handbook of Space Astronomy and Astrophysics. Cambridge University Press, CambridgeGoogle Scholar

Copyright information

© China University of Geosciences and Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Earth and Planetary SciencesWashington UniversitySt. LouisUSA
  2. 2.Panasonic Avionics CorporationLake ForestUSA

Personalised recommendations