Climate Dynamics

, Volume 46, Issue 1–2, pp 557–569 | Cite as

Inherent characteristics of sawtooth cycles can explain different glacial periodicities

  • Anne Willem Omta
  • Bob W. Kooi
  • George A. K. van Voorn
  • Rosalind E. M. Rickaby
  • Michael J. Follows


At the Mid-Pleistocene Transition about 1 Ma, the dominant periodicity of the glacial-interglacial cycles shifted from ~40 to ~100 kyr. Here, we use a previously developed mathematical model to investigate the possible dynamical origin of these different periodicities. The model has two variables, one of which exhibits sawtooth oscillations, resembling the glacial-interglacial cycles, whereas the other variable exhibits spikes at the rapid transitions. When applying a sinusoidal forcing with a fixed period, there emerges a rich variety of cycles with different periodicities, each being a multiple of the forcing period. Furthermore, the dominant periodicity of the system can change, while the forcing periodicity remains fixed, due to either random variations or different frequency components of the orbital forcing. Two key relationships stand out as predictions to be tested against observations: (1) the amplitude and the periodicity of the cycles are approximately linearly proportional to each other, a relationship that is also found in the \(\delta ^{18}\hbox {O}\) temperature proxy. (2) The magnitude of the spikes increases with increasing periodicity and amplitude of the sawtooth. This prediction could be used to identify one or more currently hidden spiking variables driving the glacial-interglacial transitions. Essentially, the quest would be for any proxy record, concurrent with a dynamical model prediction, that exhibits deglacial spikes which increase at times when the amplitude/periodicity of the glacial cycles increases. In the specific context of our calcifier-alkalinity mechanism, the records of interest would be calcifier productivity and calcite accumulation. We believe that such a falsifiable hypothesis should provide a strong motivation for the collection of further records.


Sawtooth cycle Glacial-interglacial Mid-Pleistocene Transition Bifurcation Emergent phenomena 

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  1. Augustin L, Barbante C, Barnes PRF, Barnola JM, Bigler M, Castellano E, Cattani O, Chappellaz J, Dahl-Jensen D, Delmonte B, Dreyfus G, Durand G, Falourd S, Fischer H, Flückiger J, Hansson ME, Huybrechts P, Jugie G, Johnsen SJ, Jouzel J, Kaufmann P, Kipfstuhl J, Lambert F, Lipenkov VY, Littot GC, Longinelli A, Lorrain R, Maggi V, Masson-Delmotte V, Miller H, Mulvaney R, Oerlemans J, Oerter H, Orombelli G, Parrenin F, Peel DA, Petit JR, Raynaud D, Ritz C, Ruth U, Schwander J, Siegenthaler U, Souchez R, Stauffer B, Steffensen JP, Stenni B, Stocker TF, Tabacco IE, Udisti R, van de Wal RSW, van den Broeke M, Weiss J, Wilhelms F, Winther JG, Wolff EW, Zuchelli M (2004) Eight glacial cycles from an Antarctic ice core. Nature 429:623–628CrossRefGoogle Scholar
  2. Beaufort L, Lancelot Y, Camberlin P, Cayre O, Vincent E, Bassinot F, Labeyrie L (1997) Insolation cycles as a major control of equatorial Indian Ocean primary production. Science 278:1451–1454CrossRefGoogle Scholar
  3. Berger A, Li XS, Loutre MF (1999) Modelling Northern hemisphere ice volume over the last 3 Ma. Quat Sci Rev 18:1–11CrossRefGoogle Scholar
  4. Berger A, Loutre MF (1991) Insolation values for the climate of the last 10 million years. Quat Sci Rev 10:297–317CrossRefGoogle Scholar
  5. Berger AL (1978) Long-term variations of daily insolation and quaternary climatic changes. J Atmos Sci 35:2362–2367CrossRefGoogle Scholar
  6. Berger WH (1982) Increase of carbon dioxide in the atmosphere during deglaciation: the coral reef hypothesis. Naturwissenschaften 69:87–88CrossRefGoogle Scholar
  7. Bintanja R, van de Wal RSW (2008) North American ice-sheet dynamics and the onset of 100,000-year glacial cycles. Nature 45:869–872CrossRefGoogle Scholar
  8. Clark PU, Pollard D (1998) Origin of the middle Pleistocene transition by ice sheet erosion of regolith. Paleoceanography 13:1–9CrossRefGoogle Scholar
  9. Crowley TJ, Hyde WT (2008) Transient nature of late Pleistocene climate variability. Nature 456:226–230CrossRefGoogle Scholar
  10. Crucifix M (2011) How can a glacial inception be predicted? The Holocene 21:831–842CrossRefGoogle Scholar
  11. Crucifix M (2012) Oscillators and relaxation phenomena in Pleistocene climate theory. Philos Trans R Soc A 370:1140–1165CrossRefGoogle Scholar
  12. Crucifix M (2013) Why could ice ages be unpredictable? Clim Past 9:2253–2267CrossRefGoogle Scholar
  13. Daruka I, Ditlevsen PD (2014) Changing climate response: a conceptual model for glacial cycles and the Mid-Pleistocene Transition. Clim Past Discuss 10:1101–1127CrossRefGoogle Scholar
  14. de Saedeleer B, Crucifix M, Wieczorek S (2013) Is the astronomical forcing a reliable and unique pacemaker for climate? A conceptual model study. Clim Dyn 40:273–294CrossRefGoogle Scholar
  15. Denton GH, Anderson RF, Toggweiler JR, Edwards RL, Schaefer JM, Putnam AE (2010) The last glacial termination. Science 328:1652–1656CrossRefGoogle Scholar
  16. Ditlevsen PD (2009) Bifurcation structure and noise-assisted transitions in the Pleistocene glacial cycles. Paleoceanography 24:PA3204CrossRefGoogle Scholar
  17. Doedel EJ, Oldeman B (2009) AUTO07P: continuation and bifurcation software for ordinary differential equations. Concordia University, MontrealGoogle Scholar
  18. Elderfield H, Ferretti P, Greaves M, Crowhurst S, McCave IN, Hodell D, Piotrowski AM (2012) Evolution of ocean temperature and ice volume through the Mid-Pleistocene climate transition. Science 337:704–709CrossRefGoogle Scholar
  19. Foster GL, Vance D (2006) Negligible glacial-interglacial variation in continental weathering rates. Nature 444:918–921CrossRefGoogle Scholar
  20. Ghil M (1994) Cryothermodynamics: the chaotic dynamics of paleoclimate. Physica D 77:130–159CrossRefGoogle Scholar
  21. Gibbs MT, Kump LR (1994) Global chemical erosion during the last glacial maximum and the present: sensitivity to changes in lithology and hydrology. Paleoceanography 9:529–543CrossRefGoogle Scholar
  22. Gildor H, Tziperman E (2000) Sea ice as the glacial cycles’ climate switch: role of seasonal and orbital forcing. Paleoceanography 15:605–615CrossRefGoogle Scholar
  23. Gildor H, Tziperman E (2001) Physical mechanisms behind biogeochemical glacial-interglacial \(\text{CO}_2\) variations. Geophys Res Lett 28:2421–2424CrossRefGoogle Scholar
  24. Guckenheimer J, Holmes P (1985) Nonlinear oscillations, dynamical systems and bifurcations of vector fields. Springer, BerlinGoogle Scholar
  25. Herbert T (1997) A long marine history of carbon cycle modulation by orbital-climatic changes. Proc Natl Acad Sci 94:8362–8369CrossRefGoogle Scholar
  26. Huybers PJ (2007) Glacial variability over the last two million years: an extended depth-derived agemodel, continuous obliquity pacing, and the Pleistocene progression. Quat Sci Rev 26:37–55CrossRefGoogle Scholar
  27. Huybers PJ (2009) Pleistocene glacial variability as a chaotic response to obliquity forcing. Clim Past 5:481–488CrossRefGoogle Scholar
  28. Huybers PJ, Curry WB (2006) Links between annual, Milankovitch and continuum temperature variability. Nature 441:329–332CrossRefGoogle Scholar
  29. Huybers PJ, Wunsch C (2005) Obliquity pacing of the late Pleistocene glacial terminations. Nature 434:491–494CrossRefGoogle Scholar
  30. Imbrie J, Berger A, Boyle EA, Clemens SC, Duffy A, Howard WR, Kukla G, Kutzbach J, Martinson DG, McIntyre A, Mix AC, Molfino B, Morley JJ, Peterson LC, Pisias NG, Prell WL, Raymo ME, Shackleton NJ, Toggweiler JR (1993) On the structure and origin of major glaciation cycles: 2. The 100,000-year cycle. Paleoceanography 8:699–735CrossRefGoogle Scholar
  31. Jaccard SL, Hayes CT, Martínez-García A, Hodell DA, Sigman DM, Haug GH (2013) Two modes of changes in Southern Ocean productivity over the past million years. Science 339:1419–1423CrossRefGoogle Scholar
  32. Jones IW, Munhoven G, Tranter M, Huybrechts P, Sharp MJ (2002) Modelled glacial and non-glacial \(\text{HCO}_3^-\), Si and Ge fluxes since the LGM: little potential for impact on atmospheric \(\text{CO}_2\) concentrations and a potential proxy of continental chemical erosion, the marine Ge/Si ratio. Glob Planet Change 33:139–153CrossRefGoogle Scholar
  33. Kuznetsov YA, Muratori S, Rinaldi S (1992) Bifurcations and chaos in a periodic predator-prey model. Int J Bifurcat Chaos 2:117–128CrossRefGoogle Scholar
  34. le Treut H, Ghil M (1983) Orbital forcing, climatic interactions, and glaciation cycles. J Geophys Res 88:5167–5190CrossRefGoogle Scholar
  35. Lisiecki LE, Raymo ME (2005) A Pliocene–Pleistocene stack of 57 globally distributed benthic \(\delta^{18}\text{O}\) records. Paleoceanography 20:PA1003Google Scholar
  36. Liu Z, Herbert TD (2004) High-latitude influence on the eastern equatorial Pacific climate in the early Pleistocene epoch. Nature 427:720–723CrossRefGoogle Scholar
  37. Lüthi D, le Floch M, Bereiter B, Blunier T, Barnola JM, Siegenthaler U, Raynaud D, Jouzel J, Fischer H, Kawamura K, Stocker TF (2008) High-resolution carbon dioxide concentration record 650,000–800,000 years before present. Nature 453:379–382CrossRefGoogle Scholar
  38. Marlowe IT, Brassell SC, Eglinton G, Green JC (1990) Long-chain alkenones and alkyl alkenoates and the fossil coccolith record of marine sediments. Chem Geol 88:349–375CrossRefGoogle Scholar
  39. Miles J (1988) Resonance and symmetry breaking for the pendulum. Physica D 31:252–268CrossRefGoogle Scholar
  40. Milliman JD, Troy PJ, Balch WM, Adams AK, Li YH, Mackenzie FT (1999) Biologically mediated dissolution of calcium carbonate above the chemical lysocline? Deep Sea Res I 46:1653–1669CrossRefGoogle Scholar
  41. Mitsui T, Aihara K (2014) Dynamics between order and chaos in conceptual models of glacial cycles. Clim Dyn 42:3087–3099CrossRefGoogle Scholar
  42. Mudelsee M, Schulz M (1997) The Mid-Pleistocene climate transition: onset of 100-ka cycle lags ice-volume buildup by 280 ka. Earth Planet Sci Lett 151:117–123CrossRefGoogle Scholar
  43. Muller RA, MacDonald GJ (1997) Spectrum of 100-kyr glacial cycle: orbital inclination, not eccentricity. Proc Natl Acad Sci 94:8329–8334CrossRefGoogle Scholar
  44. Munhoven G (2002) Glacial-interglacial changes of continental weathering: estimates of the related \(\text{CO}_2\) and \(\text{HCO}_3^-\) flux variations and their uncertainties. Glob Planet Change 33:155–176CrossRefGoogle Scholar
  45. Omta AW, van Voorn GAK, Rickaby REM, Follows MJ (2013) On the potential role of marine calcifiers in glacial-interglacial dynamics. Glob Biogeochem Cycles 27:692–704CrossRefGoogle Scholar
  46. Paillard D (1998) The timing of Pleistocene glaciations from a simple multiple-state climate model. Nature 391:378–381CrossRefGoogle Scholar
  47. Paillard D, Parrenin F (2004) The Antarctic ice sheet and the triggering of deglaciations. Earth Planet Sci Lett 227:263–271CrossRefGoogle Scholar
  48. Pelletier JD (1998) The power spectral density of atmospheric temperature from time scales of \(10^{-2}\) to \(10^6\) yr. Earth Planet Sci Lett 158:157–164CrossRefGoogle Scholar
  49. Petit JR, Jouzel J, Raynaud D, Barkov NI, Barnola JM, Basile I, Bender M, Chappellaz J, Davis M, Delaygue G, Delmotte M, Kotlyakov VM, Legrand M, Lipenkov VY, Lorius C, Pépin L, Ritz C, Saltzman E, Stievenard M (1999) Climate and atmospheric history of the past 420,000 years from the Vostok ice core, Antarctica. Nature 399:429–436CrossRefGoogle Scholar
  50. Raymo ME, Lisiecki LE, Nisancioglu KH (2006) Plio-Pleistocene ice volume, Antarctic climate and the global \(\delta^{18}\text{O}\) record. Science 313:492–495CrossRefGoogle Scholar
  51. Rial JA, Oh J, Reischmann E (2013) Synchronization of the climate system to eccentricity forcing and the 100,000-year problem. Nat Geosci 6:289–293CrossRefGoogle Scholar
  52. Ridgwell A, Watson AJ, Raymo ME (1999) Is the spectral signature of the 100 kyr glacial cycle consistent with a Milankovitch origin? Paleoceanography 14:437–440CrossRefGoogle Scholar
  53. Rinaldi S, Muratori S (1993) Conditioned chaos in seasonally perturbed predator-prey models. Ecol Model 69:79–97CrossRefGoogle Scholar
  54. Saltzman B, Maasch KA (1991) A first-order global model of late Cenozoic climatic change. II. Further analysis based on a simplification of the \(\text{CO}_2\) dynamics. Clim Dyn 5:201–210CrossRefGoogle Scholar
  55. Schefuss E, Jansen JHF, Sinninghe-Damsté JS (2005) Tropical environmental changes at the mid-Pleistocene transition: insights from lipid biomarkers. In: Head MJ, Gibbard PL (eds) Early-middle Pleistocene transitions: the land-ocean evidence. The Geological Society, Bath, pp 35–63Google Scholar
  56. Schoepfer SD, Shen J, Wei H, Tyson RV, Ingall E, Algeo TJ (2015) Total organic carbon, organic phosphorus, and biogenic barium fluxes as proxies for paleomarine productivity. Earth Sci Rev 146:49–78Google Scholar
  57. Schulz KG, Zeebe RE (2006) Pleistocene glacial terminations triggered by synchronous changes in Southern and Northern insolation: the insolation canon hypothesis. Earth Planet Sci Lett 249:326–336CrossRefGoogle Scholar
  58. Sexton PF, Barker S (2012) Onset of ‘Pacific-style’ deep-sea sedimentary carbonate cycles at the mid-Pleistocene transition. Earth Planet Sci Lett 321/322:81–94CrossRefGoogle Scholar
  59. Tian J, Pak DK, Wang P, Lea D, Cheng X, Zhao Q (2006) Late Pliocene monsoon linkage in the tropical South China Sea. Earth Planet Sci Lett 252:72–81CrossRefGoogle Scholar
  60. Tziperman E, Gildor H (2003) On the mid-Pleistocene transition to 100-kyr glacial cycles and the asymmetry between glaciation and deglaciation times. Paleoceanography 18:1001CrossRefGoogle Scholar
  61. Tziperman E, Raymo ME, Huybers PJ, Wunsch C (2006) Consequences of pacing the Pleistocene 100 kyr ice ages by nonlinear phase locking to Milankovitch forcing. Paleoceanography 21:PA4206CrossRefGoogle Scholar
  62. Walker JGC, Hays PB, Kasting JF (1981) A negative feedback mechanism for the long-term stabilization of Earth’s surface temperature. J Geophys Res 86:9776–9782CrossRefGoogle Scholar
  63. White AF, Blum AE, Bullen TD, Vivit DV, Schultz M, Fitzpatrick J (1999) The effect of temperature on experimental and natural chemical weathering rates of granitoid rocks. Geochim Cosmochim Acta 63:3277–3291CrossRefGoogle Scholar
  64. Wiggins S (1990) Introduction to applied nonlinear dynamical systems and chaos. Springer, New YorkCrossRefGoogle Scholar
  65. Wunsch C (2003) The spectral description of climate change including the 100 ky energy. Clim Dyn 20:353–363Google Scholar
  66. Zeebe RE, Westbroek P (2003) A simple model for the \(\text{CaCO}_3\) saturation state of the ocean: the ‘Strangelove’, the ‘Neritan’, and the ‘Cretan’ Ocean. Geochem Geophys Geosyst 4:1104CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  • Anne Willem Omta
    • 1
  • Bob W. Kooi
    • 2
  • George A. K. van Voorn
    • 3
  • Rosalind E. M. Rickaby
    • 4
  • Michael J. Follows
    • 1
  1. 1.EAPS DepartmentMassachusetts Institute of TechnologyCambridgeUSA
  2. 2.Faculty of Earth and Life SciencesVU UniversityAmsterdamThe Netherlands
  3. 3.BiometrisWageningen University and Research CentreWageningenThe Netherlands
  4. 4.Department of Earth SciencesOxford UniversityOxfordUK

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