Mathematics of the Not-So-Solid Solid Earth

  • Scott D. KingEmail author
Part of the Mathematics of Planet Earth book series (MPE, volume 5)


As a result of climatic variations over the past 700,000 years, large ice sheets in high-latitude regions of the Earth formed and subsequently melted, loading and unloading the surface of the Earth. This chapter introduces the mathematical analysis of the vertical motion of the solid Earth in response to this time-varying surface loading. This chapter focuses on two conceptual models: the first, proposed by Haskell [Physics, 6, 265–269 (1935)], describes the return to equilibrium of a viscous half-space after the removal of an applied surface load; the second, proposed by Farrell and Clark [Geophys. J. Royal Astr. Soc., 46, 647–667 (1976)], illustrates the changes in sea level that occur when ice and water are rearranged on the surface of the Earth. The sea level equation proposed by Farrell and Clark accounts for the fact that sea level represents the interface between two dynamic surfaces: the sea surface and the solid Earth, both of which are changing with time.


Gravitational potential Sea level Stokes equation Viscous relaxation 



The author acknowledges support from NSF Grant EAR-1250988.


  1. 1.
    Backus, G.E.: Converting vector and tensor equations to scalar equations in spherical coordinates. Geophys. J. 13, 71–79 (1967)zbMATHCrossRefGoogle Scholar
  2. 2.
    Batchelor, G.K.: An Introduction to Fluid Dynamics. Cambridge University Press, Cambridge (1967)zbMATHGoogle Scholar
  3. 3.
    Bean, W.B.: Nail growth. Thirty-five years of observation. Arch. Intern. Med. 140, 73–76 (1980)Google Scholar
  4. 4.
    Braun, J.: The many surface expressions of mantle dynamics. Nature Geosc. 3, 825–833 (2010)CrossRefGoogle Scholar
  5. 5.
    Cathles, L.M.: The Viscosity of the Earth’s Mantle. Princeton Univ. Press, Princeton (1975)Google Scholar
  6. 6.
    Cazenave, A., Nerem, R.S.: Present-day sea level change: observations and causes. Rev. Geophys. 42(RG3001) (2004)Google Scholar
  7. 7.
    Ekman, M.: The world’s longest continued series of sea-level observations. Pure Appl. Geophys. 127, 73–77 (1988)CrossRefGoogle Scholar
  8. 8.
    England, P.C., Houseman, G.A.: Finite strain calculations of continental deformation II. Comparison with the India-Asia collision zone. J. Geophys. Res. 91, 3664–3676 (1986)Google Scholar
  9. 9.
    Farrell, W.E., Clark, J.A.: On postglacial sea level. Geophys. J. R. Astron. Soc. 46(3), 647–667 (1976)CrossRefGoogle Scholar
  10. 10.
    Forte, A.M., Peltier, W.R., Dziewoński, A.M.: Inferences of mantle viscosity from tectonic plate velocities. Geophys. Res. Lett. 18, 1747–1750 (1991)CrossRefGoogle Scholar
  11. 11.
    Haskell, N.A.: The motion of a viscous fluid under a surface load. Physics 6, 265–269 (1935)zbMATHCrossRefGoogle Scholar
  12. 12.
    Haskell, N.A.: The motion of a viscous fluid under a surface load, part 2. Physics 7, 56–61 (1936)zbMATHCrossRefGoogle Scholar
  13. 13.
    Johansson, J.M., Davis, J.L., Scherneck, H.G., et al.: Continuous GPS measurements of postglacial adjustment in fennoscandia – 1. Geodetic results. J. Geophys. Res. 107, 2157 (2002)CrossRefGoogle Scholar
  14. 14.
    Johnson, A.M., Fletcher, R.C.: Folding of Viscous Layers. Columbia University Press, New York (1994)Google Scholar
  15. 15.
    King, S.D.: Archean cratons and mantle dynamics. Earth Planet. Sci. Lett. 234, 1–14 (2005)CrossRefGoogle Scholar
  16. 16.
    King, S.D.: Reconciling laboratory and observational models of mantle rheology in geodynamic modeling. J. Geodyn. 100, 33–50 (2016)CrossRefGoogle Scholar
  17. 17.
    King, S.D., Anderson, D.L.: Edge driven convection. Earth Planet. Sci. Lett. 160, 289–296 (1998)CrossRefGoogle Scholar
  18. 18.
    Lambeck, K., Yokoyama, Y., Johnston, P., et al.: Global ice volumes at the Last Glacial Maximum and early lateglacial. Earth Planet. Sci. Lett. 181, 513–527 (2000)CrossRefGoogle Scholar
  19. 19.
    Landerer, F.W., Swenson, S.C.: Accuracy of scaled GRACE terrestrial water storage estimates. Water Resource Res. 48(W04531) (2012).
  20. 20.
    Love, A.E.H.: The stress produced in a semi-infinite solid by pressure on part of the boundary. Phil. Tran. Roy. Soc. London, Ser. A 228, 377–379 (1929)Google Scholar
  21. 21.
    Mazzotti, S., Lambert, A., Henton, J., et al.: Absolute gravity calibration of GPS velocities and glacial isostatic adjustment in mid-continent North America. Geophys. Res. Lett. 38, L24311 (2011). CrossRefGoogle Scholar
  22. 22.
    Milne, G.A., Davis, J.L., Mitrovica, J.X., et al.: Space-geodetic constraints on glacial isostatic adjustment inFennoscandia. Science 291, 2381–2385 (2001)CrossRefGoogle Scholar
  23. 23.
    Milne, G.A., Mitrovica, J.X., Scherneck, H.G.: Estimating past continental ice volume from sea-level data. Quat. Sci. Rev. 21, 361–376 (2002)CrossRefGoogle Scholar
  24. 24.
    Mitrovica, J.X., Milne, G.A.: On post-glacial sea level: I. general theory. Geophys. J. Int. 154, 253–267 (2003)Google Scholar
  25. 25.
    Mitrovica, J.X., Tamisiea, M.E., Davis, J.L., et al.: Recent mass balance of polar ice sheets inferred from patterns of global sea-level change. Nature 409, 1026–1029 (2001)CrossRefGoogle Scholar
  26. 26.
    Morgan, W.J.: Rises, trenches, great faults, and crustal blocks. J. Geophys. Res. 73, 1959–1982 (1968)CrossRefGoogle Scholar
  27. 27.
    Nocquet, J.M., Calais, E., Parsons, B.: Geodetic constraints on glacial isostatic adjustment in Europe. Geophys. Res. Lett. 32, L06308 (2005)CrossRefGoogle Scholar
  28. 28.
    Nygård, A., Sejrup, H.P., Haflidason, H., et al.: The glacial North Sea fan, southern Norwegian Margin: architecture and evolution from the upper continental slope to the deep-sea basin. Mar. Pet. Geol. 22, 71–84 (2005)CrossRefGoogle Scholar
  29. 29.
    Olver, F.W.J., Olde Daalhuis, A.B., Lozier, D.W., et al. (eds.): NIST Digital Library of Mathematical Functions. Release 1.0.19 of 2018-06-22
  30. 30.
    Park, K.D., Nerem, R.S., Davis, J.L., et al.: Investigation of glacial isostatic adjustment in the northeast US using GPS measurements. Geophys. Res. Lett. 29, 1509–1512 (2002)CrossRefGoogle Scholar
  31. 31.
    Parsons, B., Sclater, J.G.: An analysis of the variation of ocean floor bathymetry and heat flow with age. J. Geophys. Res. 82, 803–827 (1977)CrossRefGoogle Scholar
  32. 32.
    Paulson, A., Zhong, S., Wahr, J.: Modelling post-glacial rebound with lateral viscosity variations. Geophys. J. Int. 163, 357–371 (2005)CrossRefGoogle Scholar
  33. 33.
    Peltier, W.: Impulse response of a Maxwell Earth. Rev. Geophys. 12, 649–669 (1974)CrossRefGoogle Scholar
  34. 34.
    Peltier, W.R., Argus, D.F., Drummond, R.: Space geodesy constrains ice-age terminal deglaciation: The global ICE-6G_C (VM5a) model. J. Geophys. Res. 120, 450–487 (2015)CrossRefGoogle Scholar
  35. 35.
    Peltier, W.R., Tushingham, A.M.: Influence of glacial isostatic-adjustment on tide-gauge measurements of secular sea-level change. J. Geophys. Res. 96, 6779–67,960 (1991)CrossRefGoogle Scholar
  36. 36.
    Ramillien, G., Bouhours, S., Lombard, A., et al.: Land water storage contribution to sea level from GRACE geoid data over 2003–2006. Global Planet. Change 60, 381–392 (2008)CrossRefGoogle Scholar
  37. 37.
    Schubert, G., Turcotte, D.L., Olson, P.: Mantle Convection in the Earth and Planets. Cambridge University Press, Cambridge (2001)CrossRefGoogle Scholar
  38. 38.
    Sella, G.F., Dixon, T.H., Mao, A.: REVEL: A model for recent plate velocities from space geodesy. J. Geophys. Res. 107(B4), ETG 11-1–ETG 11-30 (2002). CrossRefGoogle Scholar
  39. 39.
    Sella, G.F., Stein, S., Dixon, T.H., et al.: Observation of glacial isostatic adjustment in “stable” North America with GPS. Geophys. Res. Lett. 34, L02306 (2007). CrossRefGoogle Scholar
  40. 40.
    Spada, G., Antonioli, A., Cianetti, S., et al.: Glacial isostatic adjustment and relative sea-level changes: the role of lithospheric and upper mantle heterogeneities in a 3-d spherical earth. Geophys. J. Int. 165, 692–702 (2006)CrossRefGoogle Scholar
  41. 41.
    Tamisiea, M.E., Mitrovica, J.X., Milne, G.A., et al.: Global geoid and sea level changes due to present-day ice mass fluctuations. J. Geophys. Res. 106, 30,849–30,863 (2001)Google Scholar
  42. 42.
    Tamisiea, M.E., Mitrovica, J.X., Davis, J.L.: GRACE gravity data constrain ancient ice geometries and continental dynamics over Laurentia. Science 5826, 881–883 (2007)CrossRefGoogle Scholar
  43. 43.
    van Veen, J.: Bestaat er een geologische bodemdaling te Amsterdam sedert 1700? Tijdschrift Koninklijk Nederlandsch Aardrijkskundig Genootschap 2: LXII (1945)Google Scholar
  44. 44.
    Wang, H., Wu, P.: Effects of lateral variations in lithospheric thickness and mantle viscosity on glacially-induced surface motion on a spherical, self-gravitating Maxwell Earth. Earth Planet. Sci. Lett. 244, 576–589 (2006)CrossRefGoogle Scholar
  45. 45.
    Wilson, J.T.: Did the Atlantic close and then reopen? Nature 211, 676–681 (1966)CrossRefGoogle Scholar
  46. 46.
    Wu, P.: Mode coupling in a viscoelastic self-gravitating spherical earth induced by axisymmetric loads and lateral viscosity variations. Earth Planet. Sci. Lett. 197, 1–10 (2002)CrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of GeosciencesVirginia TechBlacksburgUSA

Personalised recommendations