Skip to main content
×
×
Home

Convection in the earth's mantle: towards a numerical simulation

  • D. P. Mckenzie (a1), J. M. Roberts (a1) and N. O. Weiss (a2)
Abstract

Plate tectonics provides a remarkably accurate kinematic description of the motion of the earth's crust but a fully dynamical theory requires an understanding of convection in the mantle. Thus the properties of plates and of the mantle must be related to a systematic study of convection. This paper reviews both the geophysical information and the fluid dynamics of convection in a Boussinesq fluid of infinite Prandtl number. Numerical experiments have been carried out on several simple two-dimensional models, in which convection is driven by imposed horizontal temperature gradients or else by heating either internally or from below. The results are presented and analysed in terms of simple physical models. Although the computations are highly idealized and omit variation of viscosity and other major features of mantle convection, they can be related to geophysical measurements. In particular, the external gravity field depends on changes in surface elevation; this suggests an observational means of investigating convection in the upper mantle.

Copyright
References
Hide All
Akimoto, S. & Fujisawa, H. 1966 Olivine–Spinel transition in system Mg2SiO4–Fe2SiO4 at 800°C. Earth Planet. Sci. Lett. 1, 237240.
Akimoto, S. & Fujisawa, H. 1968 Olivine–Spinel solid solution equilibrium in the system Mg2SiO4–Fe2SiO4. J. Geophys. Res. 73, 14671479.
Allan, D. W., Thompson, W. B. & Weiss, N. O. 1967 Convection in the Earth's Mantle. In Mantles of the Earth & Terrestrial Planets (ed. S. K. Runcorn), p. 507. Wiley.
Anderson, D. L., Sammis, C. & Jordan, T. 1972 Composition of the Mantle and Core. In The Nature of the Solid Earth (ed. E. C. Robertson), p. 41. McGraw-Hill.
Anderson, R. N., McKenzie, D. P. & Sclater, J. G. 1973 Gravity, bathymetry and convection within the earth Earth Planet. Sci. Lett. 18, 391407.
Archambeau, C. B., Flinn, E. A. & Lambert, D. G. 1969 Fine structure of the upper mantle J. Geophys. Res. 74, 58255865.
Ave'Lallemant, H. G. & Carter, N. L. 1970 Syntectonic recrystallization of Olivine and modes of flow in the Upper Mantle Bull. Geol. Soc. Am. 81, 22032220.
Banks, R. J. 1969 Geomagnetic variations and the electrical conductivity of the upper mantle Geophys. J. Roy. Astr. Soc. 17, 457487.
Barazangi, M. & Dorman, J. 1969 World seismicity maps compiled from ESSA Coast and Geodetic survey epicenter data, 1961–1967. Bull Seism Soc. Am. 59, 369.
Beardsley, R. C. & Festa, J. F. 1972 A numerical model of convection driven by a surface stress and non-uniform horizontal heating J. Phys. Ocean. 2, 444.
Birch, F. 1952 Elasticity and constitution of the Earth's Interior J. Geophys. Res. 57, 227289.
Brindley, J. 1967 Thermal convection in horizontal fluid layers J. Inst. Maths. Applies. 3, 313343.
Brune, J. & Dorman, J. 1963 Surface waves and earth structure in the Canadian Shield Bull. Seism. Soc. Am. 53, 167200.
Bullard, E. C., Everett, J. E. & Smith, A. G. 1965 The fit of the Continents around the Atlantic. Phil. Trans. Roy. Soc. A 258, 41.
Busse, F. H. 1967 On the stability of two-dimensional convection in a layer heated from below J. Math. & Phys. 46, 140.
Busse, F. H. & Schubert, G. 1971 Convection in a fluid with two phases J. Fluid Mech. 46, 801.
Busse, F. H. & Whitehead, J. A. 1971 Instabilities of convection rolls in a high Prandtl number fluid J. Fluid Mech. 47, 305.
Carter, N. L. & Ave'Lallemant, H. G. 1970 High temperature flow of Dunite and Peridotite Bull. Geol. Soc. Am. 81, 21812202.
Chandrasekhar, S. 1961 Hydrodynamic and Hydromagnetic Stability, chap. 2. Clarendon Press.
Chase, C. G. 1972 The N plate problem of plate tectonics Geophys. J. Roy. Astr. Soc. 29, 117.
Cleary, J. R. 1967 Azimuthal variation to the Longshot source term Earth. Planet. Sci. Lett. 3, 2937.
Davies, D. & Julian, B. R. 1972 A study of short period P-wave signals from Longshot Geophys. J. Roy. Astr. Soc. 29, 185.
Davies, D. & McKenzie, D. P. 1969 Seismic travel-time residuals and plates Geophys. J. Roy. Astr. Soc. 18, 5163.
Dewey, J. F. & Bird, J. M. 1971 Origin and emplacement of the Ophiolite suite: Appalachian Ophiolites in Newfoundland. J. Geophys. Res. 76, 31793206.
Elsasser, W. M. 1969 Convection and stress propagation in the Upper Mantle. In The Applications of Modern Physics to the Earth and Planetary Interiors (ed. S. K. Runcorn), p. 223. Interscience.
Engel, A. E. J. & Engel, C. G. 1970 Mafic and Ultramafic rocks. In The Sea, vol. 4 (ed. A. E. Maxwell), p. 465. Interscience.
Foster, T. D. 1969 The effect of initial conditions and lateral boundaries on convection J. Fluid Mech. 37, 81.
Foster, T. D. 1971 Intermittent convection Geophys. Fluid Dyn. 2, 201.
Fromm, J. E. 1965 Numerical solution of the nonlinear equations for a heated fluid layer Phys. Fluids, 8, 1757.
Fujisawa, H., Fujii, N., Mizutani, H., Kanamori, H. & Akimoto, S. 1968 Thermal diffusivity of Mg2SiO4, Fe2SiO4, and NaCl at high pressures and temperatures. J. Geophys. Res. 73, 47274733.
Fukao, Y. 1969 On the radiative heat transfer and the thermal conductivity in the Upper Mantle Bull. Earth. Res. Inst. 47, 54969.
Fukao, Y., Mizutani, H. & Uyeda, S. 1968 Optical absorption spectra at high temperatures and radiative thermal conductivity of Olivines Phys. Earth Planet. Interiors, 1, 5762.
Gaposchkin, E. M. & Lambeck, K. 1971 Earth's gravity field to sixteenth degree and station coordinates from satellite and terrestrial data J. Geophys. Res. 76, 48444883.
Gast, P. W. 1968 Trace element fractionation and the origin of tholeiitic and alkaline magma types Geoch. Cosmochim. Acta, 32, 1057.
Gast, P. W. 1972 The chemical composition of the Earth, the Moon and Chondritic meteorites. In The Nature of the Solid Earth (ed. E. C. Robertson), p. 19. McGraw-Hill.
Gilbert, F. 1972 Inverse problems for the Earth's normal modes. In The Nature of the Solid Earth (ed. E. C. Robertson), p. 125. McGraw-Hill.
Goetze, C. 1971 High temperature rheology of westerly granite J. Geophys. Res. 76, 1223.
Goldreich, P. & Toomre, A. 1969 Some remarks on polar wandering J. Geophys. Res. 74, 2555.
Goldstein, H. 1950 Classical Mechanics. Addison-Wesley.
Goldstein, R. J. & Graham, D. J. 1969 Stability of a fluid layer with zero shear boundaries Phys. Fluids, 12, 1133.
Gordon, R. B. 1965 Diffusion creep in the Earth's Mantle J. Geophys. Res. 70, 24132417.
Gordon, R. B. 1971 Observations of crystal plasticity under high pressure with applications to the earth's mantle J. Geophys. Res. 76, 12481254.
Griggs, D. T. 1972 The sinking lithosphere and the focal mechanism of deep earthquakes. In The Nature of the Solid Earth (ed. E. C. Robertson), p. 361. McGraw-Hill.
Gutenberg, B. 1959 Physics of the Earth's Interior. Academic.
Hales, A. L. 1936 Convection currents in the Earth Mon. Not. Roy. Astr. Soc., Geophys. Suppl. 3, 372.
Hales, A. L. & Herrin, E. 1972 Travel times of seismic waves. In The Nature of the Solid Earth (ed. E. C. Robertson), p. 172. McGraw-Hill.
Hanks, T. C. 1971 The Kuril Trench – Hokkaido Rise system: large shallow earthquakes and simple models of deformation. Geophys. J. Roy. Astr. Soc. 23, 173.
Hendershott, M. C. 1972 The effects of solid Earth deformation on global ocean tides Geophys. J. Roy. Astr. Soc. 29, 389.
Hendershott, M. C. & Munk, W. 1970 Tides Ann. Rev. Fluid Mech. 2, 205224.
Hess, H. H. 1962 History of the ocean basins. Petrologic Studies: Buddington Memorial Volume, pp. 599620. Geol. Soc. Am.
Howard, L. N. 1966 Convection at high Rayleigh numbers. In Proc. 11th Int. Congr. Appl. Mech. Munich, 1964 (ed. H. Görtler), p. 1109. Springer.
Howard, L. N., Malkus, W. V. R. & Whitehead, J. A. 1970 Self-convection of floating heat sources: a model for Continental Drift. Geophys. Fluid Dyn. 1, 123142.
Hoyle, F. 1972 The history of the Earth Quart. J. Roy. Astr. Soc. 13, 328.
Hoyle, F. & Narlikar, J. V. 1972 Cosmological models in a conformably invariant gravitational theory. II. A new model Mon. Not. Roy. Astr. Soc. 155, 323.
Hsui, A. T., Turcotte, D. L. & Torrance, K. E. 1972 Finite-amplitude thermal convection within a self-gravitating fluid sphere Geophys. Fluid Dyn. 3, 3544.
Huppert, H. E. 1971 A note on the Howard–Malkus–Whitehead floating heat sources Geophys. Fluid Dyn. 2, 317322.
Isacks, B. L. & Molnar, P. 1969 Mantle earthquake mechanisms and the sinking of the Lithosphere Nature, 223, 1121.
Isacks, B. L. & Molnar, P. 1971 Distribution of stresses in the descending lithosphere from a global survey of focal-mechanism solutions of Mantle earthquakes Rev. Geophys. 9, 103174.
Isacks, B. L., Oliver, J. & Sykes, L. R. 1968 Seismology and the new global tectonics J. geophys. Res. 73, 58555899.
Jeffreys, H. 1963 On the hydrostatic theory of the figure of the Earth, Geophys. J. Roy. Astr. Soc. 8, 196.
Jeffreys, H. 1971 The Earth, 5th ed. Cambridge University Press.
Johnson, L. R. 1967 Array measurements of P velocities in the upper mantle J. Geophys. Res. 72, 63096325.
Johnson, L. R. 1969 Array measurements of P velocities in the lower mantle Bull. Seism. Soc. Am. 59, 9731008.
Jolley, L. B. W. 1961 Summation of Series. Dover.
Kanamori, H., Fujii, N. & Mizutani, H. 1968 Thermal diffusivity measurement of rock-forming minerals from 400°K to 1100°K J. Geophys. Res. 73, 595605.
Kaula, W. M. 1971 Dynamical aspects of lunar origin Rev. Geophys. Space Phys. 9, 217238.
Knopoff, L. 1964 The convection current hypothesis Rev. Geophys. 2, 89.
Krishnamurti, R. 1968a Finite amplitude convection with changing mean temperature. Part 1. Theory J. Fluid Mech. 33, 445.
Krishnamurti, R. 1968b Finite amplitude convection with changing mean temperature. Part 2. An experimental test of theory J. Fluid Mech. 33, 457.
Krishnamurti, R. 1970a On the transition to turbulent convection. Part 1. The transition from two- to three-dimensional flow J. Fluid Mech. 42, 295.
Krishnamurti, R. 1970b On the transition to turbulent convection. Part 2. The transition to time-dependent flow J. Fluid Mech. 42, 309.
Kulacki, F. A. & Goldstein, R. J. 1972 Thermal convection in a horizontal fluid layer with uniform volumetric energy sources J. Fluid Mech. 55, 271.
Lambeck, K. 1972 Gravity anomalies over oceanic ridges Geophys. J. Roy. Astr. Soc. 30, 37.
Langseth, M. G., Le Pichon, X. & Ewing, M. 1966 Crustal structure of the mid-ocean ridges, 5. Heat flow through the Atlantic Ocean floor and convection currents J. Geophys. Res. 71, 53215355.
Le Pichon, X. 1968 Sea-floor spreading and Continental Drift J. Geophys. Res. 73, 3661.
Lipps, F. B. & Somerville, R. C. J. 1971 Dynamics of variable wavelength in finiteamplitude Bénard convection Phys. Fluids, 14, 759765.
Lliboutry, L. 1969 Sea floor spreading, Continental drift and lithosphere sinking with an asthenosphere at melting point J. Geophys. Res. 74, 6525.
Macdonald, G. J. F. 1963 The deep structure of oceans and continents Rev. Geophys. 1, 587665.
Mckenzie, D. P. 1966 The viscosity of the Lower Mantle J. Geophys. Res. 71, 39954010.
Mckenzie, D. P. 1967a Some remarks on heat flow and gravity anomalies J. Geophys. Res. 72, 6171.
Mckenzie, D. P. 1967b The viscosity of the Mantle Geophys. J. Roy. Astr. Soc. 14, 297.
Mckenzie, D. P. 1968a The influence of the boundary conditions and rotation on convection in the Earth's Mantle Geophys. J. Roy. Astr. Soc. 15, 457500.
Mckenzie, D. P. 1968b The geophysical importance of high temperature creep. In The History of the Earth's Crust, Proc. NASA Conf., p. 28. Princeton University Press.
Mckenzie, D. P. 1969 Speculations on the causes and consequences of plate motions Geophys. J. Roy. Astr. Soc. 18, 1.
Mckenzie, D. P. 1970 Temperature and potential temperature beneath island arcs Tectonophys. 10, 357.
Mckenzie, D. P. 1972 Plate tectonics. In The Nature of the Solid Earth (ed. E. C. Robertson), p. 323. McGraw-Hill.
Mckenzie, D. P. & Parker, R. L. 1967 The North Pacific: an example of tectonics on a sphere. Nature, 216, 12761280.
Mckenzie, D. P. & Sclater, J. G. 1969 Heat flow in the Eastern Pacific and sea floor spreading Bull. Volcanologique, 33, 101118.
Mckenzie, D. P. & Sclater, J. G. 1971 The evolution of the Indian Ocean since the late Cretaceous Geophys. J. Roy. Astr. Soc. 25, 437528.
Malkus, W. V. R. 1954 Discrete transitions in turbulent convection. Proc. Roy. Soc. A 225, 185.
Melson, W. G. & Thompson, G. 1971 Petrology of a transform fault zone and adjacent ridge segments. Phil. Trans. Roy. Soc A 268, 423441.
Mitronovas, W., Isacks, B. & Seeber, L. 1969 Earthquake locations and seismic wave propagation in the upper 250 km of the Tonga island arc Bull. Seism. Soc. Am. 59, 11151135.
Mizutani, H., Hamano, Y., Ida, Y. & Akimoto, S. 1970 Compressional wave velocities of Fayalite, Fe2SiO4 Spinel, and Coesite. J. Geophys. Res. 75, 27412747.
Moore, D. R., Peckover, R. S. & Weiss, N. O. 1974 Difference methods for timedependent two-dimensional convection. Comp. Phys. Comm. 7, in press.
Moore, D. R. & Weiss, N. O. 1973 Two-dimensional Rayleigh–Bénard convection J. Fluid Mech. 58, 289312.
Morgan, W. J. 1968 Rises, trenches, great faults and crustal blocks. J. Geophys. Res. 73, 1959–1982.
Morgan, W. J. 1971 Convection plumes in the lower mantle Nature, 230, 42.
Munk, W. 1968 Once again – tidal friction Quart. J. Roy. Astr. Soc. 9, 352375.
Munk, W. H. & MacDonald, G. J. F. 1960 The Rotation of the Earth. Cambridge University Press.
O'Connell, R. J. 1971 Pleistocene glaciation and the viscosity of the lower mantle Geophys. J. Roy. Astr. Soc. 23, 299327.
Oldenburg, D. W. & Brune, J. N. 1972 Ridge transform fault spreading pattern in freezing wax Science, 178, 301.
Oliver, J. & Isacks, B. 1967 Deep earthquake zones, anomalous structures in the upper mantle and the lithosphere J. Geophys. Res. 72, 42594275.
Oxburgh, E. R. & Turcotte, D. L. 1968 Mid ocean ridges and geotherm distribution during mantle convection J. Geophys. Res. 73, 2643.
Parker, R. L. 1971 The inverse problem of electrical conductivity in the mantle Geophys. J. Roy. Astr. Soc. 22, 121138.
Peckover, R. S. 1972 Convection in the presence of magnetic fields. Ph.D. thesis, University of Cambridge.
Pekeris, C. L. 1935 Thermal convection in the interior of the Earth Mon. Not. Roy. Astr. Soc., Geophys. Suppl. 3, 343.
Plows, W. H. 1968 Some numerical results for two-dimensional steady laminar Bénard convection Phys. Fluids, 11, 1593.
Prager, W. 1961 Introduction to Mechanics of Continua. Ginn.
Rayleigh, C. B. & Kirby, S. H. 1970 Creep in the Upper Mantle Am. Mineral., Spec. Pap. 3, 113.
Read, W. T. 1953 Dislocations in Crystals McGraw-Hill.
Ringwood, A. E. 1972a Mineralogy of the deep mantle: current status and future developments. In The Nature of the Solid Earth (ed. E. C. Robertson), p. 67. McGraw-Hill.
Ringwood, A. E. 1972b Phase transformations and mantle dynamics Earth. Planet. Sci. Lett. 14, 233.
Ringwood, A. E. & Major, A. 1970 The system Mg2SiO4–Fe2SiO4 at high pressures and temperatures. Phys. Earth. Planet. Interiors, 3, 89108.
Roberts, K. V. & Weiss, N. O. 1966 Convective difference schemes Math. Comp. 20, 272.
Roberts, P. H. 1967 Convection in horizontal layers with internal heat generation. Theory J. Fluid Mech. 30, 33.
Rossby, H. T. 1969 A study of Bénard convection with and without rotation J. Fluid Mech. 36, 309335.
Runcorn, S. K. 1965 Changes in the convection pattern in the Earth's mantle and continental drift: evidence for a cold origin of the Earth. Phil Trans. Roy. Soc. A 258, 228.
Schilling, J. G. 1971 Sea-floor evolution: rare-earth evidence. Phil. Trans. Roy. Soc A 268, 661706.
SchlÜter, A., Lortz, D. & Busse, F. 1965 On the stability of steady finite amplitude convection J. Fluid Mech. 23, 129.
Schubert, G. & Turcotte, D. L. 1971 Phase changes and Mantle convection J. Geophys-Res. 76, 1424.
Schwiderski, E. W. & Schwab, H. J. A. 1971 Convection experiments with electrolytically heated fluid layers J. Fluid Mech. 48, 703.
Sclater, J. G., Anderson, R. N. & Bell, M. L. 1971 The elevation of ridges and the evolution of the central eastern Pacific J. Geophys. Res. 76, 78887915.
Sclater, J. G. & Francheteau, J. 1970 The implications of terrestrial heat flow observations on current tectonic and geochemical models of the crust and upper Mantle of the Earth Geophys. J. Roy. Astr. Soc. 20, 509.
Sherby, O. D. & Burke, P. M. 1968 Mechanical behaviour of crystalline solids at elevated temperatures. In Progress in Materials Science (ed. B. Chalmers & W. Hume-Rothery), p. 325. Pergamon.
Skinner, B. J. 1966 Thermal expansion. In Handbook of Physical Constants (ed. S. P. Clark), p. 75. Geol. Soc. Am. Memoir no. 97.
Slater, J. C. 1939 Introduction to Chemical Physics. McGraw-Hill.
Somerville, R. C. J. 1967 A nonlinear spectral model of convection in a fluid unevenly heated from below J. Atmos. Sci. 24, 665676.
Sparrow, E. M., Goldstein, R. J. & Jonsson, V. K. 1964 Thermal instability in a horizontal fluid layer: effect of boundary conditions and nonlinear temperature profile. J. Fluid Mech. 18, 513.
Spiegel, E. A. 1971 Convection in Stars. I. Basic Boussinesq convection Ann. Rev. Astron. Astrophys. 9, 323.
Straus, J. M. 1972 Finite amplitude doubly diffusive convection J. Fluid Mech. 56, 353.
Sykes, L. R. 1966 The seismicity and deep structure of island arcs J. Geophys. Res. 71, 29813006.
Talwani, M. & Le Pichon, X. 1969 Gravity field over the Atlantic Ocean. In The Earth's Crust and Upper Mantle (ed. P. J. Hart), p. 341. Am. Geophys. Un. Monograph 13.
Talwani, M., Windisch, C. C. & Langseth, M. G. 1971 Rekyjanes Ridge crest: a detailed geophysical study. J. Geophys. Res. 76, 473517.
Thirlby, R. 1970 Convection in an internally heated layer J. Fluid Mech. 44, 673.
ToksÖz, M. N., Minear, J. W. & Julian, B. R. 1971 Temperature field and geophysical effects of a downgoing slab J. Geophys. Res. 76, 11131138.
Torrance, K. E. & Turcotte, D. L. 1971 Thermal convection with large viscosity variations J. Fluid Mech. 47, 113.
Tozer, D. C. 1965 Heat transfer and convection currents. Phil. Trans. Roy. Soc A 258, 252271.
Tritton, D. J. & Zarraga, M. N. 1967 Convection in horizontal layers with internal heat generation. Experiments J. Fluid Mech. 30, 21.
Turcotte, D. L. & Oxburgh, E. R. 1967 Finite amplitude convection cells and continental drift J. Fluid Mech. 28, 2942.
Turcotte, D. L. & Oxburgh, E. R. 1969 Convection in a mantle with variable physical properties J. Geophys. Res. 74, 1458.
Turcotte, D. L. & Oxburgh, E. R. 1972 Mantle convection and the new global tectonics Ann. Rev. Fluid Mech. 4, 33.
Turner, J. S. 1973 Convection in the mantle: a laboratory model with temperature-dependent viscosity. Earth Planet. Sci. Lett. 17, 369.
Vening Meinesz, F. A. 1962 Thermal convection in the Earth's Mantle. In Continental Drift, (ed. S. K. Runcorn), pp. 145176. Academic.
Verhoogen, J. 1965 Phase changes and convection in the Earth's Mantle. Phil. Trans. Roy. Soc A 258, 276283.
Veronis, G. 1966 Large amplitude Bénard convection J. Fluid Mech. 26, 49.
Von Herzen, R. P. & Lee, W. H. K. 1969 Heat flow in oceanic regions. In The Earth's Crust and Upper Mantle (ed. P. J. Hart), p. 88. Am. Geophys. Un. Monograph 13.
Weertman, J. 1970 Creep strength in the Earth's Mantle Rev. Geophys. 8, 145168.
Wetherill, G. W. 1966 Radioactive decay constants and energies. In Handbook of Physical Constants (ed. S. P. Clark), p. 513. Geol. Soc. Am. Memoir no. 97.
Whitehead, J. A. 1972 Moving heaters as a model of continental drift Phys. Earth Planet. Interiors, 5, 199212.
Willis, G. E. & Deardorff, J. W. 1970 The oscillatory motions of Rayleigh convection J. Fluid Mech. 44, 661.
Willis, G. E., Deardorff, J. W. & Somerville, R. C. J. 1972 Roll-diameter dependence in Rayleigh convection and its effect upon heat flux J. Fluid Mech. 54, 351.
Wilson, J. T. 1963 A possible origin of the Hawaiian Islands Can. J. Phys. 41, 863.
Wilson, J. T. 1965 A new class of faults and their bearing on continental drift Nature, 207, 343347.
Woollard, G. P. 1969 Regional variations in gravity. In The Earth's Crust and Upper Mantle (ed. P. J. Hart), p. 320. Geophys. Un. Monograph no. 13.
Zhivago, A. V. 1966 Bathymetric chart (Indian sector of the southern ocean). Atlas of the Antarctic, p. 190. Moscow: Akad. Nauk.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Journal of Fluid Mechanics
  • ISSN: 0022-1120
  • EISSN: 1469-7645
  • URL: /core/journals/journal-of-fluid-mechanics
Please enter your name
Please enter a valid email address
Who would you like to send this to? *
×
MathJax

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed