# The middle Pleistocene transition as a generic bifurcation on a slow manifold

- 441 Downloads
- 10 Citations

## Abstract

The Quaternary period has been characterised by a cyclical series of glaciations, which are attributed to the change in the insolation (incoming solar radiation) from changes in the Earth’s orbit around the Sun. The spectral power in the climate record is very different from that of the orbital forcing: prior to 1000 kyr before present most of the spectral power is in the 41 kyr band while since then the power has been in the 100 kyr band. The change defines the middle Pleistocene transition (MPT). The MPT does not indicate any noticeable difference in the orbital forcing. The climate response to the insolation is thus far from linear, and appears to be structurally different before and after the MPT. This paper presents a low order conceptual model for the oscillatory dynamics of the ice sheets in terms of a relaxation oscillator with multiple levels subject to the Milankovitch forcing. The model exhibits smooth transitions between three different climate states; an interglacial (i), a mild glacial (g) and a deep glacial (G) as proposed by Paillard (Nature 391:378–381, 1998). The model suggests a dynamical explanation in terms of the structure of a slow manifold for the observed allowed and “forbidden” transitions between the three climate states. With the model, the pacing of the climate oscillations by the astronomical forcing is through the mechanism of phase-resetting of relaxation oscillations in which the internal phase of the oscillation is affected by the forcing. In spite of its simplicity as a forced ODE, the model is able to reproduce not only general features but also many of the details of oscillations observed in the climate record. A particular novelty is that it includes a slow drift in the form of the slow manifold that reproduces the observed dynamical change at the MPT. We explain this change in terms of a transcritical bifurcation in the fast dynamics on varying the slow variable; this bifurcation can induce a sudden change in periodicity and amplitude of the cycle and we suggest that this is associated with a branch of “canard oscillations” that appear for a small range of parameters. The model is remarkably robust at simulating the climate record before, during and after the MPT. Even though the conceptual model does not point to specific mechanisms, the physical implication is that the major reorganisation of the climate response to the orbital forcing does not necessarily imply that there was a big change in the environmental conditions.

## Keywords

Middle Pleistocene transition Nonlinear oscillation Ice age Slow manifold Bifurcation## Notes

### Acknowledgments

We thank Sebastian Wieczorek, Martin Krupa and Frank Kwasniok for discussions in relation to this work; PA thanks the University of Copenhagen for hospitality and EPSRC via CliMathNet EP/K003216/1 for arranging meetings that facilitated this work, and Martin Rasmussen and Jeroen Lamb for arranging a “Workshop on Critical Transitions in Complex Systems” in 2012 where this was first discussed. We thank the referees and Anna von der Heydt for their insightful commments.

## References

- Abe-Ouchi A, Saito F, Kawamura K, Raymo ME, Okuno J, Takahashi K, Blatter H (2013) Insolation-driven 100,000-year glacial cycles and hysteresis of ice-sheet volume. Nature 500:190–193CrossRefGoogle Scholar
- Ashkenazy Y, Tziperman E (2004) Are the 41 kyr oscillations a linear response to Milankovitch forcing? Quat Sci Rev 23:1879–1890CrossRefGoogle Scholar
- Benoît E, Callot JL, Diener F, Diener M (1981) Chasse au canards, i–iv. Collect Math 32:37–119Google Scholar
- Benzi R, Parisi G, Sutera A, Vulpiani A (1982) Stochasic resonance in climate change. Tellus 34:10–16CrossRefGoogle Scholar
- Berger A (1978) Long-term variations of daily insolation and quaternary climatic change. J Atmos Sci 35:2362–2367CrossRefGoogle Scholar
- Berger A (2012) A brief history of the astronomical theories of paleoclimates. In: Berger A, Mesinger F, Sijacki D (eds) Climate Change. Springer Vienna, pp 107–129. ISBN:978–3–7091–0972–4Google Scholar
- Berglund N, Gentz B (2002) Metastability in simple climate models: pathwise analysis of slowly driven langevin equations. Stoch Dyn 2:327–356CrossRefGoogle Scholar
- Clark PU, Pollard D (1998) Origin of the middle Pleistocene transition by ice sheet erosion of regolith. Paleoceanography 13:1–9CrossRefGoogle Scholar
- Clark PU, Archer D, Pollard D, Blum JD, Rial JA, Brovkin V, Mix AC, Pisias NG, Roy M (2006) The middle pleistocene transition: characteristics, mechanisms, and implication for long-term changes in atmospheric \(\text{ pCO }_2\). Quart Sci Rev 25:3150–3184CrossRefGoogle Scholar
- Crucifix M (2012) Oscillators and relaxation phenomena in Pleistocene climate theory. Philos Trans R Soc A 370:1140–1165CrossRefGoogle Scholar
- Daruka I, Ditlevsen P (2014) Changing climatic response: a conceptual model for glacial cycles and the mid-Pleistocene transition. Clim Past Discuss 10:1101–1127CrossRefGoogle Scholar
- 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 40:273–294. doi: 10.1007/s00382-012-1316-1 CrossRefGoogle Scholar
- Ditlevsen PD (2009) The bifurcation structure and noise assisted transitions in the Pleistocene glacial cycles. Paleoceanography 24:PA3204CrossRefGoogle Scholar
- Golombek D, Rosenstein R (2010) Physiology of circadian entrainment. Physiol Rev 90:1063–1102CrossRefGoogle Scholar
- Golubitsky M, Schaeffer M (1985) Singularities and groups in bifurcation theory, vol 1. Springer, New YorkCrossRefGoogle Scholar
- Hays J, Imbrie J, Shackleton N (1976) Variations in earths’s orbit: pacemaker of the ice ages. Science 194:1121–1132CrossRefGoogle Scholar
- Hilgen FJ, Lourens LJ, van Dam JA (2012) The Neogene period. In: Gradstein F, Ogg J, Schmitz M, Ogg G (eds) The geological time scale. Elsevier, Amsterdam, p 923978Google Scholar
- Huybers P (2007) Glacial variability over the last 2 ma: an extended depth-derived age model, continuous obliquity pacing, and the pleistocene progression. Quat Sci Rev 26:37–55CrossRefGoogle Scholar
- Huybers P (2009) Pleistocene glacial variability as a chaotic response to obliquity forcing. Clim Past 5:481488CrossRefGoogle Scholar
- Huybers P, Wunsch C (2004) A depth-derived pleistocene age model: Uncertainty estimates, sedimentation variability, and nonlinear climate change. Paleoceanography 19:PA1028CrossRefGoogle Scholar
- Imbrie J, Imbrie-Moore A, Lisiecki L (2011) A phase-space model for Pleistocene ice volume. Earth Planet Sci Lett 307:94–102CrossRefGoogle Scholar
- Källen E, Crafoord C, Ghil M (1979) Free oscillations in a climate model with ice-sheet dynamics. J Atmos Sci 36:2292–2303CrossRefGoogle Scholar
- Krupa M, Szmolyan P (2001) Relaxation oscillation and canard explosion. J Differ Equ 174:312–368CrossRefGoogle Scholar
- Kuznetsov Y (2004) Elements of applied bifurcation theory, 3rd edn. Springer, New YorkCrossRefGoogle Scholar
- Laskar J, Robutel P, Joutel F, Boudin F, Gastineau M, ACM C, Levrard B (2004) A long-term numerical solution for the insolation quantities of the earth. Astron Astrophys 428:261285CrossRefGoogle Scholar
- LeTreut H, Ghil M (1983) Orbital forcing, climate interactions, and glacial cycles. J Geophys Res 88:5167–5190CrossRefGoogle Scholar
- Lisiecki LE, Raymo ME (2005) A Pliocene–Pleistocene stack of 57 globally distributed benthic D18O records. Paleoceanography 20:PA1003Google Scholar
- Maasch K, Saltzman B (1990) A low-order dynamical model of global climate variability over the full pleistocene. J Geophys Res 95:1955–1963CrossRefGoogle Scholar
- McClymont E, Sodian S, Rosell-Mele A, Rosenthal Y (2013) Pleistocene sea-surface temperature evolution: early cooling, delayed glacial intensification, and implications for the mid-Pleistocene climate transition. Earth Sci Rev 123:173–193CrossRefGoogle Scholar
- Meyers SR, Hinnov L (2010) Northern hemisphere glaciation and the evolution of Plio-Pleistocene climate noise. Paleoceanography 25:PA3207CrossRefGoogle Scholar
- 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
- North GRIP members (2004) High resolution climate record of the northern hemisphere reaching into the last glacial interglacial period. Nature 431:147–151CrossRefGoogle Scholar
- Paillard D (1998) The timing of pleistocene glaciations from a simple multiple-state climate model. Nature 391:378–381CrossRefGoogle Scholar
- Pelletier JD (2003) Coherence resonance and ice ages. J Geophys Res 108(D20):4645. doi: 10.1029/2002JD003,120 CrossRefGoogle Scholar
- Rial JA, Oh J, E R (2013) Synchronization of the climate system to eccentricity forcing and the 100,000-year problem. Nat Geosci 6:289293CrossRefGoogle Scholar
- Saltzman B, Sutera A (1987) The mid-Quaternary climate transition as the free response of a three-variable dynamical model. J Atmos Sci 44:236–241CrossRefGoogle Scholar
- Shackleton NJ, Hall MA, Vincent E (2000) Phase relationships between millennial-scale events 64,000–24,000 years ago. Paleoceanography 15:565–569CrossRefGoogle Scholar
- Sima A, Paul A, Schulz M, Oerlemans J (2006) Modeling the oxygen–isotopic composition of the North American ice sheet and its effect on the isotopic composition of the ocean during the last glacial cycle. Geophys Res Lett 33(L15):706Google Scholar
- Tziperman E, Gildor H (2003) On the mid-Pleistocene transition to 100-kyr glacial cycles and the asymmetry between glaciation and deglaciation times. Paleoceanography 18:1–8CrossRefGoogle Scholar
- Wechselberger M (2012) A propos de canards. Trans AMS 364:3289–3309CrossRefGoogle Scholar