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

Abstract

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.

Keywords

plasticity / diffusion / mantle convection / stability criteria / dimensional analysis / geometry

References

[1] Agee C. B. Phase Transformations and Seismic Structure in the Upper Mantle and Transition Zone. Reviews in Mineralogy, 1998, 37: 165-204.

[2] Anderson D. L. Theory of the Earth, 1989.

[3] Armienti P., Gasperini D. Isotopic Evidence for Chaotic Imprint in Upper Mantle Heterogeneity, 2010, 11 5 Q0AC02.

[4] Aurnou J. M., Olson P. L. Experiments on Rayleigh–Bénard Convection, Magnetoconvection and Rotating Magnetoconvection in Liquid Gallium. Journal of Fluid Mechanics, 2001, 430: 283-307.

[5] Bercovici D. Mantle Dynamics: An Introduction and Overview. Treatise on Geophysics, 2015, 7: 1-22.

[6] Birch J. M., Wilshire B. Transient and Steady State Creep Behaviour of Polycrystalline MgO. Journal of Materials Science, 1974, 9(6): 871-875.

[7] Blagoveshchenskii N., Novikov A., Puchkov A., . Self-Diffusion in Liquid Gallium and Hard Sphere Model. EPJ Web of Conferences, 2015, 83 02018

[8] Bleazard J. G., Sun T. F., Teja A. S. The Thermal Conductivity and Viscosity of Acetic Acid-Water Mixtures. International Journal of Thermophysics, 1996, 17(1): 111-125.

[9] Boresi A. P., Schmidt R. J. Advanced Mechanics of Materials, 2003.

[10] Bridgeman P. Dimensional Analysis, 1927.

[11] Brillo J., Pommrich A. I., Meyer A. Relation between Self-Diffusion and Viscosity in Dense Liquids: New Experimental Results from Electrostatic Levitation. Physical Review Letters, 2011, 107 16 165902

[12] Buckingham E. On Physically Similar Systems; Illustrations of the Use of Dimensional Equations. Physical Review, 1914, 4(4): 345-376.

[13] Bürgmann R., Dresen G. Rheology of the Lower Crust and Upper Mantle: Evidence from Rock Mechanics, Geodesy, and Field Observations. Annual Review of Earth and Planetary Sciences, 2008, 36(1): 531-567.

[14] Carslaw H. S., Jaeger J. C. Conduction of Heat in Solids, 1959.

[15] Chakraborty S. Diffusion Coefficients in Olivine, Wadsleyite and Ringwoodite. Reviews in Mineralogy and Geochemistry, 2010, 72(1): 603-639.

[16] Chudinovskikh L., Boehler R. Eutectic Melting in the System Fe-S to 44 GPa. Earth and Planetary Science Letters, 2007, 257(1/2): 97-103.

[17] Costin L. S. Damage Mechanics in the Post-Failure Regime. Mechanics of Materials, 1985, 4(2): 149-160.

[18] Coupland J. N., McClements D. J. Physical Properties of Liquid Edible Oils. Journal of the American Oil Chemists’ Society, 1997, 74(12): 1559-1564.

[19] Criss E. M., Smith R. J., Meyers M. A. Failure Mechanisms in Cobalt Welded with a Silver-Copper Filler. Materials Science and Engineering: A, 2015, 645: 369-382.

[20] Criss R. E., Hofmeister A. M. Conductive Cooling of Spherical Bodies with Emphasis on the Earth. Terra Nova, 2016, 28(2): 101-109.

[21] Cussler E. L. Diffusion: Mass Transport in Fluid Systems, 2008.

[22] Davies G. F. Mantle Convection for Geologists, 2011

[23] Davis R. O., Selvadurai A. P. S. Plasticity and Geomechanics, 2005.

[24] Cabral A. J., de Oliveira P. C., Moreira S. G. C., . Thermal Diffusivity of Palm Olein and Compounds Containing Β-Carotene. International Journal of Thermophysics, 2011, 32(9): 1966-1972.

[25] Diamante L. M., Lan T. Y. 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, 2014(3): 1-6.

[26] Doglioni C., Anderson D. L. Top Driven Asymmetric Mantle Convection. The Interdisciplinary Earth: In Honor of Don L. Anderson. GSA Special Papers, 2015, 214: 51-64.

[27] Doglioni C., Panza G. Polarized Plate Tectonics. Advances in Geophysics, 2015, 56: 1-167.

[28] Domínguez-Rodríguez A., Gómez-García D., Zapata-Solvas E., . Making Ceramics Ductile at Low Homologous Temperatures. Scripta Materialia, 2007, 56(2): 89-91.

[29] Doremus R. H. Viscosity of Silica. Journal of Applied Physics, 2002, 92(12): 7619-7629.

[30] Du Z., Vinnik L. P., Foulger G. R. Evidence from P-to-S Mantle Converted Waves for a Flat “660-km” Discontinuity beneath Iceland. Earth and Planetary Science Letters, 2006, 241(1/2): 271-280.

[31] Dziewonski A. M., Anderson D. L. Preliminary Reference Earth Model. Physics of the Earth and Planetary Interiors, 1981, 25(4): 297-356.

[32] Elder J. The Bowels of the Earth, 1976.

[33] Ertl H., Dullien F. A. L. Self-Diffusion and Viscosity of some Liquids as a Function of Temperature. AIChE Journal, 1973, 19(6): 1215-1223.

[34] Fegley B. Jr.. Practical Chemical Thermodynamics for Geoscientists, 2015.

[35] Fichtner A., Villaseñor A. Crust and Upper Mantle of the Western Mediterranean— Constraints from Full-Waveform Inversion. Earth and Planetary Science Letters, 2015, 428: 52-62.

[36] Foulger G. R. Plates vs Plumes: A Geological Controversy, 2010

[37] Foulger G. R., Panza G. F., Artemieva I. M., . Caveats on Tomographic Images. Terra Nova, 2013, 25: 259-281.

[38] Foulger G. R., Pritchard M. J., Julian B. R., . Seismic Tomography Shows that Upwelling beneath Iceland is Confined to the Upper Mantle. Geophysical Journal International, 2001, 146(2): 504-530.

[39] French S. W., Romanowicz B. Broad Plumes Rooted at the Base of the Earth’s Mantle beneath Major Hotspots. Nature, 2015, 525(7567): 95-99.

[40] Frenkel J. Zur Theorie Der Elastizitätsgrenze Und Der Festigkeit Kristallinischer Körper. Zeitschrift für Physik, 1926, 37(7/8): 572-609.

[41] Gando A., Gando Y., Ichimura K., . Partial Radiogenic Heat Model for Earth Revealed by Geoneutrino Measurements. Nature Geoscience, 2011, 4(9): 647-651.

[42] Gao S. S., Liu K. H. Imaging Mantle Discontinuities Using Multiply-Reflected P-to-S Conversions. Earth and Planetary Science Letters, 2014, 402: 99-106.

[43] Gasparik T. Evidence for the Transition Zone Origin of some [Mg, Fe]O Inclusions in Diamonds. Earth and Planetary Science Letters, 2000, 183(1/2): 1-5.

[44] Glazier J. A., Segawa T., Naert A., . Evidence against ‘Ultrahard’ Thermal Turbulence at very High Rayleigh Numbers. Nature, 1999, 398(6725): 307-310.

[45] Goes S., Agrusta R. v H. J., . Subduction-Transition Zone Interaction: A Review. Geosphere, 2017, 13(3): 644-664.

[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.. 359–410

[47] Hamilton W. B. Plate Tectonics Began in Neoproterozoic Time, and Plumes from Deep Mantle have never Operated. Lithos, 2011, 123: 1-20.

[48] Hamilton W. B. 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, 2015, 514: 123-156.

[49] Hamza V. M. Global Heat Flow without Invoking “Kelvin Paradox”. Frontiers in Geosciences, 2013, 1: 11-20.

[50] He X. M., Fowler A., Toner M. Water Activity and Mobility in Solutions of Glycerol and Small Molecular Weight Sugars: Implication for Cryo-and Lyopreservation. Journal of Applied Physics, 2006, 100 7 074702

[51] Heap M. J., Baud P., Meredith P. G., . Brittle Creep in Basalt and Its Application to Time-Dependent Volcano Deformation. Earth and Planetary Science Letters, 2011, 307(1/2): 71-82.

[52] Heep M. J. Creep: Time-Dependent Brittle Deformation in Rocks: [Dissertation], 2009.

[53] Henderson G. Inorganic Geochemistry, 1982.

[54] Hetényi G. To Conserve or not to Conserve (Mass in Numerical Models). Terra Nova, 2014, 26(5): 372-376.

[55] Hill R. The Mathematical Theory of Plasticity, 1950.

[56] Hiraga T., Miyazaki T., Tasaka M., . Mantle Superplasticity and Its Self-Made Demise. Nature, 2010, 468(7327): 1091-1094.

[57] Hirth G. Laboratory Constraints on the Rheology of the Upper Mantle. Reviews in Mineralogy and Geochemistry, 2002, 51(1): 97-120.

[58] Hofmeister A. M. 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, 2010, 180(3/4): 138-147.

[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. 584–608

[60] Hofmeister A. M., Criss R. E. Earth’s Heat Flux Revised and Linked to Chemistry. Tectonophysics, 2005, 395(3/4): 159-177.

[61] Hofmeister A. M., Criss R. E. 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, 2012, 62(1): 111-131.

[62] Hofmeister A. M., Criss R. E. How Irreversible Heat Transport Processes Drive Earth’s Interdependent Thermal, Structural, and Chemical Evolution. Gondwana Research, 2013, 24(2): 490-500.

[63] Hofmeister A. M., Criss R. E. Evaluation of the Heat, Entropy, and Rotational Changes Produced by Gravitational Segregation during Core Formation. Journal of Earth Science, 2015, 26(1): 124-133.

[64] Hofmeister A. M., Sehlke A., Avard G., . Transport Properties of Glassy and Molten Lavas as a Function of Temperature and Composition. Journal of Volcanology and Geothermal Research, 2016, 327: 330-348.

[65] Hofmeister A. M., Whittington A. G. 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, 2012, 358(8): 1072-1082.

[66] Huang L. H., Liu L. S. Simultaneous Determination of Thermal Conductivity and Thermal Diffusivity of Food and Agricultural Materials Using a Transient Plane-Source Method. Journal of Food Engineering, 2009, 95(1): 179-185.

[67] Jin Z. M., Zhang J. F., Green H. W. I., . Eclogite Rheology: Implications for Subducted Lithosphere. Geology, 2001, 29(8): 667-670.

[68] Kajihara K., Kamioka H., Hirano M., . Interstitial Oxygen Molecules in Amorphous SiO₂. III. Measurements of Dissolution Kinetics, Diffusion Coefficient, and Solubility by Infrared Photoluminescence. Journal of Applied Physics, 2005, 98 1 013529.

[69] Kavner A., Duffy T. S. Strength and Elasticity of Ringwoodite at Upper Mantle Pressures. Geophysical Research Letters, 2001, 28(14): 2691-2694.

[70] Kestin J., Knierim K., Mason E. A., . Equilibrium and Transport Properties of the Noble Gases and Their Mixtures at Low Density. Journal of Physical and Chemical Reference Data, 1984, 13(1): 229-303.

[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. 389–427

[72] Koschmieder E. L., Pallas S. G. Heat Transfer through a Shallow, Horizontal Convecting Fluid Layer. International Journal of Heat and Mass Transfer, 1974, 17(9): 991-1002.

[73] Langdon T. G. Fracture Processes in Superplastic Flow. Metal Science, 1982, 16(4): 175-183.

[74] Lodders K. An Oxygen Isotope Mixing Model for the Accretion and Composition of Rocky Planets. Space Science Review, 2000, 92: 341-354.

[75] Luca J., Mrawira D. New Measurement of Thermal Properties of Superpave Asphalt Concrete. Journal of Materials in Civil Engineering, 2005, 17(1): 72-79.

[76] Meyer R. E. Self-Diffusion of Liquid Mercury. The Journal of Physical Chemistry, 1961, 65(3): 567-568.

[77] Meyers M. A., Chawla K. K. Mechanical Behavior of Materials, 2009.

[78] Mitchell B. S. An Introduction to Materials Engineering and Science for Chemical and Materials Engineers, 2004.

[79] Moghadam R. H., Trepmann C. A., Stöckhert B., . Rheology of Synthetic Omphacite Aggregates at High Pressure and High Temperature. Journal of Petrology, 2010, 51(4): 921-945.

[80] Mukherjee A. K., Bird J. E., Dorn J. E. Experimental Correlation for High-Temperature Creep. Transactions of the American Society of Metals, 1969, 62: 155-179.

[81] Nabelek P. I., Hofmeister A. M., Whittington A. G. 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, 2012, 317: 157-164.

[82] Nguyen L. T., Balasubramaniam V. M., Sastry S. K. 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, 2012, 15(1): 169-187.

[83] Nishi T., Shibata H., Waseda Y., . Thermal Conductivities of Molten Iron, Cobalt, and Nickel by Laser Flash Method. Metallurgical and Materials Transactions A, 2003, 34(12): 2801-2807.

[84] Nishihara Y., Tinker D., Kawazoe T., . 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, 2008, 170(3/4): 156-169.

[85] Paterson M. S. Experimental Deformation and Faulting in Wombeyan Marble. Geological Society of America Bulletin, 1958, 69(4): 465-475.

[86] Paterson M. S., Weaver C. W. Deformation of Polycrystalline MgO under Pressure. Journal of the American Ceramic Society, 1970, 53(8): 463-471.

[87] Pearson D. S., Ver Strate G., Von Meerwall E., . Viscosity and Self-Diffusion Coefficient of Linear Polyethylene. Macromolecules, 1987, 20(5): 1133-1141.

[88] Prewitt C. T., Downs R. T. High-Pressure Crystal Chemistry. Reviews in Mineralogy, 1998, 37: 284-342.

[89] Rayleigh L. On Convection Currents in a Horizontal Layer of Fluid, when the Higher Temperature is on the under Side. Philosophical Magazine Series 6, 1916, 32(192): 529-546.

[90] Rees B. A., Okal E. A. The Depth of the Deepest Historical Earthquakes. Pure and Applied Geophysics, 1987, 125(5): 699-715.

[91] Reif F. Fundamentals of Statistical and Thermal Physics, 1965.

[92] Romine W. L., Whittington A. G. A Simple Model for the Viscosity of Rhyolites as a Function of Temperature, Pressure and Water Content. Geochimica et Cosmochimica Acta, 2015, 170: 281-300.

[93] Romine W. L., Whittington A. G., Nabelek P. I., . Thermal Diffusivity of Rhyolitic Glasses and Melts: Effects of Temperature, Crystals and Dissolved Water. Bulletin of Volcanology, 2012, 74(10): 2273-2287.

[94] Schriempf J. T. A Laser Flash Technique for Determining Thermal Diffusivity of Liquid Metals at Elevated Temperatures. Review of Scientific Instruments, 1972, 43(5): 781-786.

[95] Schriempf J. T. Thermal Diffusivity of Liquid Gallium. Solid State Communications, 1973, 13(6): 651-653.

[96] Schubert G., Turcotte D. L., Olson P. Mantle Convection in the Earth and Planets, 2001

[97] Sehlke A., Whittington A., Robert B., . Pahoehoe to ‘a’a Transition of Hawaiian Lavas: An Experimental Study. Bulletin of Volcanology, 2014, 76(11): 876-896.

[98] Shimada M., Cho A., Yukutake H. Fracture Strength of Dry Silicate Rocks at High Confining Pressures and Activity of Acoustic Emission. Tectonophysics, 1983, 96(1/2): 159-172.

[99] Siggia E. D. High Rayleigh Number Convection. Annual Review of Fluid Mechanics, 1994, 26(1): 137-168.

[100] Smith E. M., Shirey S. B., Nestola F., . Large Gem Diamonds from Metallic Liquid in Earth’s Deep Mantle. Science, 2016, 354(6318): 1403-1405.

[101] Soutas-Little, R., 2011. History of Continuum Mechanics. In: Meridio, J., Saccomandi, G., eds., Continuum Mechanics. Eolss Publishers, Singapore. 80–93

[102] Stacey F. D., Stacey C. H. B. Gravitational Energy of Core Evolution: Implications for Thermal History and Geodynamo Power. Physics of the Earth and Planetary Interiors, 1999, 110(1/2): 83-93.

[103] Stein C. A., Stein S. A. A Model for the Global Variation in Oceanic Depth and Heat Flow with Lithospheric Age. Nature, 1992, 359(6391): 123-129.

[104] Stengel K. C., Oliver D. S., Booker J. R. Onset of Convection in a Variable-Viscosity Fluid. Journal of Fluid Mechanics, 1982, 120: 411-431.

[105] Thern A., Lüdemann H. D. P,^T Dependence of the Self Diffusion Coefficients and Densities in Liquid Silicone Oils. Zeitschrift für Naturforschung A, 1996, 51(3): 192-196.

[106] Timoshenko S. P., Goodier J. N. Theory of Elasticity, 1970.

[107] Transtrum M. K., Machta B. B., Brown K. S., . Perspective: Sloppiness and Emergent Theories in Physics, Biology, and beyond. The Journal of Chemical Physics, 2015, 143 1 010901

[108] Tritton D. J. Physical Fluid Dynamics, 1977

[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. 283–291

[110] Wawersik W. R., Brace W. F. Post-Failure Behavior of a Granite and Diabase. Rock Mechanics and Rock Engineering, 1970, 3: 61-85.

[111] Weidner D. J., Li L. Methods for the Study of High P/T Deformation and Rheology. Treatise on Geophysics, 2015, 2: 339-358.

[112] White D. B. The Planforms and Onset of Convection with a Temperature-Dependent Viscosity. Journal of Fluid Mechanics, 1988, 191: 247-286.

[113] Whittington A. G., Hofmeister A. M., Nabelek P. I. Temperature-Dependent Thermal Diffusivity of Earth’s Crust: Implications for Crustal Anatexis. Nature, 2009, 458: 319-321.

[114] Xu Z., Morris R., Bencsik M., . Detection of Virgin Olive Oil Adulteration Using Low Field Unilateral NMR. Sensors, 2014, 14(2): 2028-2035.

[115] Yáñez-Limón J. M., Mayen-Mondragón R., Martínez-Flores O., . Thermal Diffusivity Studies in Edible Commercial Oils Using Thermal Lens Spectroscopy. Superficies y Vacio, 2005, 18: 31-37.

[116] Zemansky M. W., Dittman R. H. Heat and Thermodynamics, 1981, 6,

[117] Zener C. Internal Friction in Solids II. General Theory of Thermoelastic Internal Friction. Physical Review, 1938, 53(1): 90-99.

[118] Zhang Y., Ni H., Chen Y. Diffusion of H, C, and O Components in Silicate Melts. Reviews in Mineralogy and Geochemistry, 2010, 72(1): 171-225.

[119] Zhong S. J., Yuen D. A., Moresi L. M., . Numerical Method for Mantle Convection. In: Schubert, G., ed., Treatise on Geophysics. Mantle Dynamics, 2015, 7: 197-222.

[120] Zombeck M. V. Handbook of Space Astronomy and Astrophysics, 2007.