Skip to main content

Advertisement

SpringerLink
Log in

A model for Dansgaard–Oeschger events and millennial-scale abrupt climate change without external forcing

  • Published:
Climate Dynamics Aims and scope Submit manuscript

Abstract

We propose a conceptual model which generates abrupt climate changes akin to Dansgaard–Oeschger events. In the model these abrupt climate changes are not triggered by external perturbations but rather emerge in a dynamic self-consistent way through complex interactions of the ocean, the atmosphere and an intermittent process. The abrupt climate changes are caused in our model by intermittencies in the sea-ice cover. The ocean is represented by a Stommel two-box model, the atmosphere by a Lorenz-84 model and the sea-ice cover by a deterministic approximation of a correlated additive and multiplicative noise (CAM) process. The key dynamical ingredients of the model are given by stochastic limits of deterministic multi-scale systems and recent results in deterministic homogenisation theory. The deterministic model reproduces statistical features of actual ice-core data such as non-Gaussian \(\alpha \)-stable behaviour. The proposed mechanism for abrupt millenial-scale climate change only relies on the existence of a quantity, which exhibits intermittent dynamics on an intermediate time scale. We consider as a particular mechanism intermittent sea-ice cover where the intermittency is generated by emergent atmospheric noise. However, other mechanisms such as freshwater influxes may also be formulated within the proposed framework.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10

Similar content being viewed by others

Notes

  1. See Gottwald and Melbourne (2013a) for a definition of what constitutes strong and weak chaos.

References

  • Andersen KK, Azuma N, Barnola JM, Bigler M, Biscaye P, Caillon N, Chappellaz J, Clausen HB, Dahl-Jensen D, Fischer H, Flückiger J, Fritzsche D, Fujii Y, Goto-Azuma K, Grønvold K, Gundestrup NS, Hansson M, Huber C, Hvidberg CS, Johnsen SJ, Jonsell U, Jouzel J, Kipfstuhl S, Landais A, Leuenberger M, Lorrain R, Masson-Delmotte V, Miller H, Motoyama H, Narita H, Popp T, Rasmussen SO, Raynaud D, Rothlisberger R, Ruth U, Samyn D, Schwander J, Shoji H, Siggard-Andersen ML, Steffensen JP, Stocker T, Sveinbjörnsdóttir AE, Svensson A, Takata M, Tison JL, Thorsteinsson T, Watanabe O, Wilhelms F, White JWC, members NGICP (2004) High-resolution record of Northern Hemisphere climate extending into the last interglacial period. Nature 431(7005):147–151

    Google Scholar 

  • Andersen KK, Svensson A, Johnsen SJ, Rasmussen SO, Bigler M, Röthlisberger R, Ruth U, Siggaard-Andersen ML, Steffensen JP, Dahl-Jensen D, Vinther BM, Clausen HB (2006) The Greenland ice core chronology 2005, 15–42 ka. Part 1: constructing the time scale. Quaternary Science Reviews 25(23):3246–3257 (critical quaternary stratigraphy)

    Google Scholar 

  • Applebaum D (2009) Lévy processes and stochastic calculus, Cambridge Studies in Advanced Mathematics, vol 116, 2nd edn. Cambridge University Press, Cambridge

    Google Scholar 

  • Banderas R, Álvarez-Solas J, Montoya M (2012) Role of CO\(_{2}\) and Southern Ocean winds in glacial abrupt climate change. Clim Past 8(3):1011–1021

    Google Scholar 

  • Banderas R, Alvarez-Solas J, Robinson A, Montoya M (2015) An interhemispheric mechanism for glacial abrupt climate change. Clim Dyn 44(9):2897–2908

    Google Scholar 

  • Boers N, Ghil M, Rousseau DD (2018) Ocean circulation, ice shelf, and sea ice interactions explain Dansgaard–Oeschger cycles. Proc Natl Acad Sci 115(47):E11005–E11014

    Google Scholar 

  • Cessi P (1994) A simple box model of stochastically forced thermohaline flow. J Phys Oceanogr 24(9):1911–1920

    Google Scholar 

  • Chechkin A, Pavlyukevich I (2014) Marcus versus Stratonovich for systems with jump noise. J Phys A Math Theor 47(34):342001

    Google Scholar 

  • Chechkin AV, Metzler R, Klafter J, Gonchar VY (2008) Introduction to the theory of Lévy flights. In: Klages R, Radons G, Sokolov IM (eds) Anomalous transport. Wiley, New York, pp 129–162

    Google Scholar 

  • Chevyrev I, Friz PK, Korepanov A, Melbourne I (2019) Superdiffusive limits for deterministic fast–slow dynamical systems. arXiv:1907.04825

  • Crucifix M (2012) Oscillators and relaxation phenomena in Pleistocene climate theory. Philos Trans R Soc Londo A Math Phys Eng Sci 370(1962):1140–1165

    Google Scholar 

  • Dansgaard W, Johnsen S, Clausen HB, Dahl-Jensen D, Gundestrup N, Hammer H, Oeschger H (1984) North Atlantic climate oscillations revealed by deep Greeland ice cores. Clim Processes Clim Sensit Geophys Mongogr 5:288–298

    Google Scholar 

  • Deser C, Holland M, Reverdin G, Timlin M (2002) Decadal variations in Labrador sea ice cover and North Atlantic sea surface temperatures. J Geophys Res Oceans 107(C5):31–312

    Google Scholar 

  • Ditlevsen PD (1999) Observation of \(\alpha \)-stable noise induced millennial climate changes from an ice-core record. Geophys Res Lett 26(10):1441–1444

    Google Scholar 

  • Ditlevsen PD, Kristensen MS, Andersen KK (2005) The recurrence time of Dansgaard–Oeschger events and limits on the possible periodic component. J Clim 18(14):2594–2603

    Google Scholar 

  • Ditlevsen PD, Andersen KK, Svensson A (2007) The DO-climate events are probably noise induced: statistical investigation of the claimed 1470 years cycle. Clim Past 3(1):129–134

    Google Scholar 

  • Dokken TM, Nisancioglu KH, Li C, Battisti DS, Kissel C (2013) Dansgaard–Oeschger cycles: interactions between ocean and sea ice intrinsic to the Nordic seas. Paleoceanography 28(3):491–502

    Google Scholar 

  • Drijfhout S, Gleeson E, Dijkstra HA, Livina V (2013) Spontaneous abrupt climate change due to an atmospheric blocking–sea-ice–ocean feedback in an unforced climate model simulation. Proc Natl Acad Sci 110(49):19713–19718

    Google Scholar 

  • Fang Z, Wallace JM (1994) Arctic sea ice variability on a timescale of weeks and its relation to atmospheric forcing. J Clim 7(12):1897–1914

    Google Scholar 

  • Friedrich T, Timmermann A, Menviel L, Elison Timm O, Mouchet A, Roche DM (2010) The mechanism behind internally generated centennial-to-millennial scale climate variability in an earth system model of intermediate complexity. Geosci Model Dev 3(2):377–389

    Google Scholar 

  • Fuhrer K, Neftel A, Anklin M, Maggi V (1993) Continuous measurements of hydrogen peroxide, formaldehyde, calcium and ammonium concentrations along the new GRIP ice core from Summit, Central Greenland. Atmos Environ 27A:1873–1880

    Google Scholar 

  • Ganopolski A, Rahmstorf S (2001) Rapid changes of glacial climate simulated in a coupled climate model. Nature 409(6817):153–158

    Google Scholar 

  • Ganopolski A, Rahmstorf S (2002) Abrupt glacial climate changes due to stochastic resonance. Phys Rev Lett 88(3):153–158

    Google Scholar 

  • Gardiner CW (2003) Handbook of stochastic methods for physics, chemistry, and the natural sciences, 3rd edn. Springer, New York

    Google Scholar 

  • Gaspard P, Wang XJ (1988) Sporadicity: between periodic and chaotic dynamical behaviours. Proc Natl Acad Sci 85:4591–4595

    Google Scholar 

  • Gildor H, Tziperman E (2003) Sea-ice switches and abrupt climate change. Philos Trans R Soc Lond Ser A Math Phys Eng Sci 361(1810):1935–1944

    Google Scholar 

  • Givon D, Kupferman R, Stuart A (2004) Extracting macroscopic dynamics: model problems and algorithms. Nonlinearity 17(6):R55–127

    Google Scholar 

  • Gottwald GA, Melbourne I (2020) Simulation of non-Lipschitz stochastic differential equations driven by \(\alpha \)-stable noise: a method based on deterministic homogenisation. arXiv:2004.09914

  • Gottwald GA, Melbourne I (2013a) A Huygens principle for diffusion and anomalous diffusion in spatially extended systems. Proc Natl Acad Sci USA 110:8411–8416

    Google Scholar 

  • Gottwald GA, Melbourne I (2013b) Homogenization for deterministic maps and multiplicative noise. Proc R Soc A Math Phys Eng Sci 469:2156

    Google Scholar 

  • Gottwald GA, Melbourne I (2016) On the detection of superdiffusive behaviour in time series. J Stat Mech Theory Exp 12:123205

    Google Scholar 

  • Gottwald G, Crommelin D, Franzke C (2017) Stochastic climate theory. In: Franzke CLE, O’Kane TJ (eds) Nonlinear and stochastic climate dynamics. Cambridge University Press, Cambridge, pp 209–240

    Google Scholar 

  • Gouëzel S (2004) Central limit theorem and stable laws for intermittent maps. Probab Theory Relat Fields 128:82–122

    Google Scholar 

  • Greenland Ice-core Project (GRIP) Members (1993) Climate instability during the last interglacial period recorded in the GRIP ice core. Nature 364(6434):203–207

    Google Scholar 

  • Grootes PM, Stuiver M (1997) Oxygen 18/16 variability in Greenland snow and ice with \(10^{-3}\) to \(10^5\)-year time resolution. J Geophys Res Oceans 102(C12):26455–26470

    Google Scholar 

  • Haarsma RJ, Opsteegh JD, Selten FM, Wang X (2001) Rapid transitions and ultra-low frequency behaviour in a 40-kyr integration with a coupled climate model of intermediate complexity. Clim Dyn 17(7):559–570

    Google Scholar 

  • Hasselmann K (1976) Stochastic climate models. Part 1: theory. Tellus 28(6):473–485

    Google Scholar 

  • Hein C, Imkeller P, Pavlyukevich I (2009) Limit theorems for \(p\)-variations of solutions of SDEs driven by additive stable Lévy noise and model selection for paleo-climatic data. In: Duan J, Luo S, Wang C (eds) Recent development in stochastic dynamics and stochastic analysis, interdisciplinary mathematics and sciences, vol 8. World Scientific, Singapore, pp 137–150

    Google Scholar 

  • Hoff U, Rasmussen TL, Stein R, Ezat MM, Fahl K (2016) Sea ice and millennial-scale climate variability in the Nordic seas 90 kyr ago to present. Nat Commun 7(1):12247

    Google Scholar 

  • Jensen MF, Nilsson J, Nisancioglu KH (2016) The interaction between sea ice and salinity-dominated ocean circulation: implications for halocline stability and rapid changes of sea ice cover. Clim Dyn 47(9):3301–3317

    Google Scholar 

  • Kelly D, Melbourne I (2016) Smooth approximation of stochastic differential equations. Ann Probab 44(1):479–520

    Google Scholar 

  • Kelly D, Melbourne I (2017) Deterministic homogenization for fast–slow systems with chaotic noise. J Funct Anal 272(10):4063–4102

    Google Scholar 

  • Kleppin H, Jochum M, Otto-Bliesner B, Shields CA, Yeager S (2015) Stochastic atmospheric forcing as a cause of Greenland climate transitions. J Clim 28(19):7741–7763

    Google Scholar 

  • Kuske R, Keller JB (2001) Rate of convergence to a stable law. SIAM J Appl Math 61(4):1308–1323

    Google Scholar 

  • Kwasniok F, Lohmann G (2009) Deriving dynamical models from paleoclimatic records: application to glacial millennial-scale climate variability. Phys Rev E 80(6):066104

    Google Scholar 

  • Leith CE (1975) Climate response and fluctuation dissipation. J Atmos Sci 32(10):2022–2026

    Google Scholar 

  • Li C, Born A (2019) Coupled atmosphere-ice-ocean dynamics in Dansgaard–Oeschger events. Quatern Sci Rev 203:1–20

    Google Scholar 

  • Li C, Battisti DS, Schrag DP, Tziperman E (2005) Abrupt climate shifts in Greenland due to displacements of the sea ice edge. Geophys Res Lett 32:19

    Google Scholar 

  • Lohmann J, Ditlevsen PD (2019) A consistent statistical model selection for abrupt glacial climate changes. Clim Dyn 52(11):6411–6426

    Google Scholar 

  • Lorenz EN (1984) Irregularity: a fundamental property of the atmosphere. Tellus A 36A(2):98–110

    Google Scholar 

  • Lorenz EN (1990) Can chaos and intransitivity lead to interannual variability? Tellus A 42(3):378–389

    Google Scholar 

  • Magdziarz M, Klafter J (2010) Detecting origins of subdiffusion: \(p\)-variation test for confined systems. Phys Rev E 82:011129

    Google Scholar 

  • Magdziarz M, Weron A, Burnecki K, Klafter J (2009) Fractional Brownian motion versus the continuous-time random walk: a simple test for subdiffusive dynamics. Phys Rev Lett 103:180602

    Google Scholar 

  • Majda AJ, Franzke C, Crommelin D (2009) Normal forms for reduced stochastic climate models. Proc Nat Acad Sci 106(10):3649–3653

    Google Scholar 

  • Manabe S, Stouffer R (2011) Are two modes of thermohaline circulation stable? Tellus A 51(3):400–411

    Google Scholar 

  • Marcus S (1981) Modeling and approximation of stochastic differential equations driven by semimartingales. Stochastics 4:223–245

    Google Scholar 

  • Meissner KJ, Eby M, Weaver AJ, Saenko OA (2008) CO\(_2\) threshold for millennial-scale oscillations in the climate system: implications for global warming scenarios. Clim Dyn 30(2–3):161–174

    Google Scholar 

  • Melbourne I, Nicol M (2005) Almost sure invariance principle for nonuniformly hyperbolic systems. Commun Math Phys 260:131–146

    Google Scholar 

  • Melbourne I, Nicol M (2009) A vector-valued almost sure invariance principle for hyperbolic dynamical systems. Ann Prob 37:478–505

    Google Scholar 

  • Melbourne I, Stuart A (2011) A note on diffusion limits of chaotic skew-product flows. Nonlinearity 24:1361–1367

    Google Scholar 

  • Monahan AH, Alexander J, Weaver AJ (2008) Stochastic models of the meridional overturning circulation: time scales and patterns of variability. Philos Trans R Soc Lond A Math Phys Eng Sci 366(1875):2525–2542

    Google Scholar 

  • Penland C, Sardeshmukh PD (2012) Alternative interpretations of power-law distributions found in nature. Chaos Interdiscip J Nonlinear Sci 22(2):023119

    Google Scholar 

  • Petersen SV, Schrag DP, Clark PU (2013) A new mechanism for Dansgaard–Oeschger cycles. Paleoceanography 28(1):24–30

    Google Scholar 

  • Rasmussen SO, Andersen KK, Svensson AM, Steffensen JP, Vinther BM, Clausen HB, Siggaard-Andersen ML, Johnsen SJ, Larsen LB, Dahl-Jensen D, Bigler M, Röthlisberger R, Fischer H, Goto-Azuma K, Hansson ME, Ruth U (2006) A new Greenland ice core chronology for the last glacial termination. J Geophys Res Atmos 111:D6

    Google Scholar 

  • Roebber PJ (1995) Climate variability in a low-order coupled atmosphere-ocean model. Tellus A 47(4):473–494

    Google Scholar 

  • Sadatzki H, Dokken TM, Berben SMP, Muschitiello F, Stein R, Fahl K, Menviel L, Timmermann A, Jansen E (2019) Sea ice variability in the southern Norwegian Sea during glacial Dansgaard-Oeschger climate cycles. Sci Adv 5:3

    Google Scholar 

  • Sardeshmukh PD, Penland C (2015) Understanding the distinctively skewed and heavy tailed character of atmospheric and oceanic probability distributions. Chaos Interdiscip J Nonlinear Sci 25(3):036410

    Google Scholar 

  • Sardeshmukh PD, Sura P (2009) Reconciling non-Gaussian climate statistics with linear dynamics. J Clim 22(5):1193–1207

    Google Scholar 

  • Schüpbach S, Fischer H, Bigler M, Erhardt T, Gfeller G, Leuenberger D, Mini O, Mulvaney R, Abram NJ, Fleet L, Frey MM, Thomas E, Svensson A, Dahl-Jensen D, Kettner E, Kjaer H, Seierstad I, Steffensen JP, Rasmussen SO, Vallelonga P, Winstrup M, Wegner A, Twarloh B, Wolff K, Schmidt K, Goto-Azuma K, Kuramoto T, Hirabayashi M, Uetake J, Zheng J, Bourgeois J, Fisher D, Zhiheng D, Xiao C, Legrand M, Spolaor A, Gabrieli J, Barbante C, Kang JH, Hur SD, Hong SB, Hwang HJ, Hong S, Hansson M, Iizuka Y, Oyabu I, Muscheler R, Adolphi F, Maselli O, McConnell J, Wolff EW (2018) Greenland records of aerosol source and atmospheric lifetime changes from the Eemian to the Holocene. Nat Commun 9(1):1476

    Google Scholar 

  • Siegert S, Friedrich R, Peinke J (1998) Analysis of data sets of stochastic systems. Phys Lett A 243(5–6):275–280

    Google Scholar 

  • Singh HA, Battisti DS, Bitz CM (2014) A heuristic model of Dansgaard–Oeschger cycles. Part I: description, results, and sensitivity studies. J Clim 27(12):4337–4358

    Google Scholar 

  • Stemler T, Werner JP, Benner H, Just W (2007) Stochastic modeling of experimental chaotic time series. Phys Rev Lett 98(4):044102

    Google Scholar 

  • Stommel H (1961) Thermohaline convection with two stable regimes of flow. Tellus 13(2):224–230

    Google Scholar 

  • Sura P, Sardeshmukh PD (2008) A global view of non-Gaussian SST variability. J Phys Oceanogr 38(3):639–647

    Google Scholar 

  • Svensson A, Andersen KK, Bigler M, Clausen HB, Dahl-Jensen D, Davies SM, Johnsen SJ, Muscheler R, Parrenin F, Rasmussen SO, Röthlisberger R, Seierstad I, Steffensen JP, Vinther BM (2008) A 60,000 year Greenland stratigraphic ice core chronology. Clim Past 4(1):47–57

    Google Scholar 

  • Thompson WF, Kuske RA, Monahan AH (2017) Reduced \(\alpha \)-stable dynamics for multiple time scale systems forced with correlated additive and multiplicative Gaussian white noise. Chaos Interdiscip J Nonlinear Sc 27(11):113105

    Google Scholar 

  • Timmermann A, Gildor H, Schulz M, Tziperman E (2003) Coherent resonant millennial-scale climate oscillations triggered by massive meltwater pulses. J Clim 16(15):2569–2585

    Google Scholar 

  • Venegas SA, Mysak LA (2000) Is there a dominant timescale of natural climate variability in the Arctic? J Clim 13(19):3412–3434

    Google Scholar 

  • Vinther BM, Clausen HB, Johnsen SJ, Rasmussen SO, Andersen KK, Buchardt SL, Dahl-Jensen D, Seierstad IK, Siggaard-Andersen ML, Steffensen JP, Svensson A, Olsen J, Heinemeier J (2006) A synchronized dating of three Greenland ice cores throughout the Holocene. J Geophys Res Atmos 111:D13

    Google Scholar 

  • Weaver AJ, Hughes TMC (1994) Rapid interglacial climate fluctuations driven by North Atlantic ocean circulation. Nature 367(6462):447–450

    Google Scholar 

  • Wolff E, Chappellaz J, Blunier T, Rasmussen S, Svensson A (2010) Millennial-scale variability during the last glacial: the ice core record. Quatern Sci Rev 29(21):2828–2838

    Google Scholar 

  • Wouters J, Gottwald GA (2019a) Edgeworth expansions for slow-fast systems with finite time-scale separation. Proc R Soc A Math Phys Eng Sci 475(2223):20180358

    Google Scholar 

  • Wouters J, Gottwald GA (2019b) Stochastic model reduction for slow-fast systems with moderate time scale separation. Multisc Model Simul 17(4):1172–1188

    Google Scholar 

  • Wunsch C (2006) Abrupt climate change: an alternative view. Quatern Res 65(2):191–203

    Google Scholar 

  • Yiou R, Fuhrer K, Meeker LD, Jouzel J, Johnsen S, Mayewski PA (1997) Paleoclimatic variability inferred from the spectral analysis of Greenland and Antarctic ice-core data. J Geophys Res Oceans 102(C12):26441–26454

    Google Scholar 

  • Zhang X, Lohmann G, Knorr G, Purcell C (2014) Abrupt glacial climate shifts controlled by ice sheet changes. Nature 512(7514):290–294

    Google Scholar 

Download references

Acknowledgements

The ice core data were generously provided by Peter Ditlevsen. I am grateful to Armin Köhl, Johannes Lohmann, Marisa Montoya and Xu Zhang for many interesting and helpful discussions. I would like to thank Cameron Duncan, Nathan Duingan and Eric Huang who explored the p-variation test and suitable parameter ranges of the Lorenz-84 system in a summer project in 2014 at an early stage of this work. I would like to thank Peter Ditlevsen and an anonymous referee for their valuable comments.

Author information

Authors and Affiliations

  1. School of Mathematics and Statistics, The University of Sydney, Sydney, NSW, 2006, Australia

    Georg A. Gottwald

Authors
  1. Georg A. Gottwald
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Georg A. Gottwald.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Gottwald, G.A. A model for Dansgaard–Oeschger events and millennial-scale abrupt climate change without external forcing. Clim Dyn 56, 227–243 (2021). https://doi.org/10.1007/s00382-020-05476-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00382-020-05476-z

Keywords

Search

Navigation