GEM - International Journal on Geomathematics

, Volume 4, Issue 2, pp 227–297 | Cite as

Exploration of a simple model for ice ages

Original Paper


We argue that, while Milanković variations in solar radiation undoubtedly have a major influence on the timing of the Quaternary ice ages, they are partly incidental to their underlying causes. Based on observations of the significance of CO2, we propose a conceptually simple (but complicated in detail) energy balance type model which has the ability to explain the underlying oscillatory nature of ice ages. We are led to develop a model which combines ice sheet growth and atmospheric energy balance with ocean carbon balance. In order to provide results which mimic the basic features of the observations, we develop novel hypotheses as follows. The succession of the most recent ice ages can be explained as being due to an oscillation due to the interaction of the growing northern hemisphere ice sheets and proglacial lakes which form as they migrate south. The CO2 signal which faithfully follows the proxy temperature signal can then be explained as being due to a combination of thermally activated ocean biomass production, which enables the rapid CO2 rise at glacial terminations, and enhanced glacial carbonate weathering through the exposure of continental shelves, which enables CO2 to passively follow the subsequent glacial cooling cycle. Milanković variations provide for modulations of the amplitude and periods of the resulting signals.


Ice ages Milanković Energy balance model Proglacial lakes Carbonate weathering 

Mathematics Subject Classification



Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Archer D., Kheshgi H., Maier-Reimer E.: Multiple timescales for neutralization of fossil fuel CO2. Geophys. Res. Lett. 24, 405–408 (1997)CrossRefGoogle Scholar
  2. Berger W.H.: Increase of carbon dioxide in the atmosphere during deglaciation: the coral reef hypothesis. Naturwissenschaften 69, 87–88 (1982)CrossRefGoogle Scholar
  3. Berger A.: Milankovitch theory and climate. Revs. Geophys. 26, 624–657 (1988)CrossRefGoogle Scholar
  4. Bigg, G.: The Oceans and Climate, 2nd edn. CUP, Cambridge (2003)Google Scholar
  5. Boulton G.S., Hagdorn M., Maillot P.B., Zatsepin S.: Drainage beneath ice sheets: groundwater–channel coupling, and the origin of esker systems from former ice sheets. Quat. Sci. Rev. 28, 621–638 (2009)CrossRefGoogle Scholar
  6. Broecker W.S., Peng T.-H.: Tracers in the Sea. Eldigio Press, New York (1982)Google Scholar
  7. Budyko M.I.: The effect of solar radiation variations on the climate of the Earth. Tellus 21, 611–619 (1969)CrossRefGoogle Scholar
  8. Caldeira K.: Enhanced Cenozoic chemical weathering and the subduction of pelagic carbonate. Nature 357, 578–581 (1992)CrossRefGoogle Scholar
  9. Cheng H., Edwards R.L., Broecker W.S., Denton G.H., Kong X., Wang Y., Zhang R., Wang X.: Ice age terminations. Science 326, 248–252 (2009)CrossRefGoogle Scholar
  10. Croll J.: On the physical cause of the change of climate during geological epochs. Phil. Mag. 28(4), 121–137 (1864)Google Scholar
  11. Croll J.: Climate and Time in Their Geological Relations: A Theory of Secular Changes of the Earth’s Climate. Daldy, Isbister, London (1875)Google Scholar
  12. Cuffey K., Paterson W.S.B.: The Physics of Glaciers. 4th edn. Butterworth-Heinemann, Oxford (2010)Google Scholar
  13. Emerson S.R., Hedges J.I.: Chemical Oceanography and the Marine Carbon Cycle. CUP, Cambridge (2008)CrossRefGoogle Scholar
  14. Evatt G., Fowler A.C., Clark C.D., Hulton N.: Subglacial floods beneath ice sheets. Phil. Trans. R. Soc. 364, 1769–1794 (2006)MathSciNetCrossRefGoogle Scholar
  15. Foster G.L., Vance D.: Negligible glacial-interglacial variation in continental weathering rates. Nature 444, 918–921 (2006)CrossRefGoogle Scholar
  16. Fowler A.C.: Mathematical Geoscience. Springer, London (2011)CrossRefMATHGoogle Scholar
  17. Ganopolski A., Rahmstorf S.: Rapid changes of glacial climate simulated in a coupled climate model. Nature 409, 153–158 (2001)CrossRefGoogle Scholar
  18. Ghil M., Le Treut H.: A climate model with cryodynamics and geodynamics. J. Geophys. Res. 86, 5262–5270 (1981)CrossRefGoogle Scholar
  19. Goldbeter A.: Biochemical Oscillations and Cellular Rhythms. CUP, Cambridge (1996)CrossRefMATHGoogle Scholar
  20. Hays J.D., Imbrie J., Shackleton N.J.: Variations in the Earth’s orbit: pacemaker of the ice ages. Science 194, 1121–1132 (1976)CrossRefGoogle Scholar
  21. Hodson A.: Phosphorus in glacial meltwaters. In: Knight, P.G. (ed.) Glacier Science and Environmental Change, pp. 81–82. Blackwell, Oxford (2006)CrossRefGoogle Scholar
  22. Houghton J.T.: Global Warming: The Complete Briefing. 4th edn. CUP, Cambridge (2009)CrossRefGoogle Scholar
  23. Huybers P.: Early Pleistocene glacial cycles and the integrated summer insolation forcing. Science 313, 508–511 (2006)CrossRefGoogle Scholar
  24. Jahnke R.A.: The phosphorus cycle. In: Butcher, S.S., Charlson, R.J., Orians, G.H., Wolfe, G.V. (eds) Global Biogeochemical Cycles, pp. 301–315. Academic Press, London (1992)CrossRefGoogle Scholar
  25. Källén E., Crafoord C., Ghil M.: Free oscillations in a climate model with ice-sheet dynamics. J. Atmos. Sci. 36, 2292–2303 (1979)CrossRefGoogle Scholar
  26. Kawamura K., Parrenin F., Lisiecki L., Uemura R., Vimeux F., Severinghaus J.P., Hutterli M.A., Nakazawa T., Aoki S., Jouzel J., Raymo M.E., Matsumoto K., Nakata , Motoyama H., Fujita S., Goto-Azuma K., Fujii Y., Watanabe O.: Northern Hemisphere forcing of climatic cycles in Antarctica over the past 360,000 years. Nature 448, 912–916 (2007)CrossRefGoogle Scholar
  27. Krauskopf K.B., Bird D.K.: Introduction to Geochemistry. 3rd edn. McGraw-Hill, New York (1995)Google Scholar
  28. Laskar J., Robutel P., Joutel F., Gastineau M., Correia A.C.M., Levrard B.: A long-term numerical solution for the insolation quantities of the Earth. Astron. Astrophys. 428, 261–285 (2004)CrossRefGoogle Scholar
  29. Le Treut H., Ghil M.: Orbital forcing, climatic interactions, and glaciation cycles. J. Geophys. Res. 88, 5167–5190 (1983)CrossRefGoogle Scholar
  30. Leverington D.W., Mann J.D., Teller J.T.: Changes in the bathymetry and volume of Glacial Lake Agassiz between 11,000 and 9,300 14C yr b.p. Quat. Res. 54, 174–181 (2000)CrossRefGoogle Scholar
  31. Lisiecki L.E., Raymo M.E.: A Pliocene-Pleistocene stack of 57 globally distributed benthic δ18 O records. Paleoceanography 20, PA1003 (2005). doi:10.1029/2004PA001071 Google Scholar
  32. Maasch K.A., Saltzman B.: A low-order dynamical model of global climatic variability over the full Pleistocene. J. Geophys. Res. 95, 1955–1963 (1990)CrossRefGoogle Scholar
  33. MacAyeal D.R.: A catastrophe model of the paleoclimate. J. Glaciol. 24, 245–257 (1979)Google Scholar
  34. Mackenzie F.T., Lerman A., Andersson A.J.: Past and present of sediment and carbon biogeochemical cycling models. Biogeosciences 1, 11–32 (2004)CrossRefGoogle Scholar
  35. Milanković, M.: Canon of insolation and the ice-age problem. Translated from German edition of 1941. Agency for Textbooks, Belgrade (1998)Google Scholar
  36. Munhoven G.: Glacial-interglacial changes of continental weathering: estimates of the related CO2 and HCO3 flux variations and their uncertainties. Glob. Planet. Change 33, 155–176 (2002)CrossRefGoogle Scholar
  37. Murton J.B., Bateman M.D., Dallimore S.R., Teller J.T., Yang Z.: Identification of Younger Dryas outburst flood path from Lake Agassiz to the Arctic Ocean. Nature 464, 740–743 (2010)CrossRefGoogle Scholar
  38. Oerlemans J.: Model experiments on the 100,000-yr glacial cycle. Nature 287, 430–432 (1980)CrossRefGoogle Scholar
  39. Oerlemans J.: Some basic experiments with a vertically-integrated ice sheet model. Tellus 33, 1–11 (1981)CrossRefGoogle Scholar
  40. Pagani M., Huber M., Liu Z., Bohaty S.M., Henderiks J., Sijp W., Krishnan S., DeConto R.M.: The role of carbon dioxide during the onset of Antarctic glaciation. Science 334, 1261–1264 (2011)CrossRefGoogle Scholar
  41. Petit J.R., Jouzel J., Raynaud D., Barkov N.I., Barnola J.-M., Basile I., Bender M., Chappellaz J., Davis M., Delaygue G., Delmotte M., Kotlyakov V.M., Legrand M., Lipenkov V.Y., Lorius C., Pépin L., Ritz C., Saltzman E., Stievenard M.: Climate and atmospheric history of the past 420,000 years from the Vostok ice core. Antarctica. Nat. 399, 429–436 (1999)Google Scholar
  42. Parrenin F., Paillard D.: Amplitude and phase of glacial cycles from a conceptual model. Earth Planet. Sci. Lett. 214, 243–250 (2003)CrossRefGoogle Scholar
  43. Pollard D.: A simple ice sheet model yields realistic 100 kyr glacial cycles. Nature 296, 334–338 (1982)CrossRefGoogle Scholar
  44. Pollard D., DeConto R.M.: Hysteresis in Cenozoic Antarctic ice-sheet variations. Global Planet. Change 45, 9–21 (2005)CrossRefGoogle Scholar
  45. Raymo M.E., Lisiecki L.E., Nisancioglu K.H.: Plio-Pleistocene ice volume, Antarctic climate, and the global δ 18O record. Science 313, 492–495 (2006)CrossRefGoogle Scholar
  46. Ridgwell A.J., Kennedy M.J., Caldeira K.: Carbonate deposition, climate stability, and Neoproterozoic ice ages. Science 302, 859–862 (2003)CrossRefGoogle Scholar
  47. Ridgwell A., Zeebe R.E.: The role of the global carbonate cycle in the regulation and evolution of the Earth system. Earth Planet. Sci. Lett. 234, 299–315 (2005)CrossRefGoogle Scholar
  48. Saltzman B.: Dynamical Paleoclimatology: Generalized Theory of Global Climate Change. Academic Press, San Diego (2002)Google Scholar
  49. Saltzman B., Maasch K.A.: Carbon cycle instability as a cause of the late Pleistocene ice age oscillations: modeling the asymmetric response. Global Biogeochem. Cycles 2, 177–185 (1988)CrossRefGoogle Scholar
  50. Saltzman B., Maasch K.A.: A first-order global model of late Cenozoic climatic change. II. Further analysis based on a simplification of CO2 dynamics. Clim. Dyn. 5, 201–210 (1991)Google Scholar
  51. Saltzman B., Hansen A.R., Maasch K.A.: The late Quaternary glaciations as the response of a three-component feedback system to Earth–orbital forcing. J. Atmos. Sci. 41, 3380–3389 (1984)CrossRefGoogle Scholar
  52. Sellers W.D.: A climate model based on the energy balance of the earth-atmosphere system. J. Appl. Meteorol. 8, 392–400 (1969)CrossRefGoogle Scholar
  53. Shakun J.D., Clark P.U., He F., Marcott S.A., Mix A.C., Liu Z., Otto-Bliesner B., Schmittner A., Bard E.: Global warming preceded by increasing carbon dioxide concentrations during the last deglaciation. Nature 484, 49–55 (2012)CrossRefGoogle Scholar
  54. Shreve R.L.: Esker characteristics in terms of glacier physics, Katahdin esker system. Maine. Geol. Soc. Amer. Bull. 96, 639–646 (1985)CrossRefGoogle Scholar
  55. Siddall M., Rohling E.J., Almogi-Labin A., Hemleben Ch., Melschner D., Schmelzer I., Smeed D.A.: Sea-level fluctuations during the last glacial cycle. Nature 423, 853–858 (2003)CrossRefGoogle Scholar
  56. Toggweiler J.R.: Origin of the 100,000-year timescale in Antarctic temperatures and atmospheric CO2. Paleoceanogr. 23, PA2211 (2008). doi:10.1029/2006PA001405 Google Scholar
  57. Tranter M.: Glacial chemical weathering, runoff composition and solute fluxes. In: Knight, P.G. (ed.) Glacier Science and Environmental Change, pp. 71–75. Blackwell, Oxford (2006)CrossRefGoogle Scholar
  58. Tziperman E., Raymo M.E., Huybers P., Wunsch C.: Consequences of pacing the Pleistocene 100 kyr ice ages by nonlinear phase locking to Milanković forcing. Paleoceanogr. 21, PA4206 (2006). doi:10.1029/2005PA001241 Google Scholar
  59. Walker J.C.G., Hays P.B., Kasting J.F.: A negative feedback mechanism for the long-term stabilization of Earth’s surface temperature. J. Geophys. Res. 86(C10), 9776–9782 (1981)CrossRefGoogle Scholar
  60. Weertman J.: Milanković solar radiation variations and ice age ice sheet sizes. Nature 261, 17–20 (1976)CrossRefGoogle Scholar
  61. Wingham D.J., Siegert M.J., Shepherd A., Muir A.S.: Rapid discharge connects Antarctic subglacial lakes. Nature 440, 1033–1037 (2006)CrossRefGoogle Scholar
  62. Wolff E.W., Fischer H., Röthlisberger R.: Glacial terminations as southern warmings without northern control. Nat. Geosci. 2, 206–209 (2009)CrossRefGoogle Scholar
  63. Zeebe R.E., Westbroek P.: A simple model for the CaCO3 saturation state of the ocean: the “Strangelove”, the “Neritan”, and the “Cretan” ocea. Geochem. Geophys. Geosyst. 4, 1104 (2003). doi:10.1029/2003GC000538 Google Scholar
  64. Zeebe R.E., Wolf-Gladrow D.: CO2 in Seawater: Equilibrium, Kinetics, Isotopes. Elsevier, Amsterdam (2001)Google Scholar

Copyright information

© Springer-Verlag 2012

Authors and Affiliations

  • A. C. Fowler
    • 1
    • 2
  • R. E. M. Rickaby
    • 3
  • E. W. Wolff
    • 4
  1. 1.MACSI, University of LimerickLimerickRepublic of Ireland
  2. 2.OCIAM, University of OxfordOxfordUK
  3. 3.Department of Earth SciencesUniversity of OxfordOxfordUK
  4. 4.British Antarctic SurveyCambridgeUK

Personalised recommendations