A deterministic excitation (DE) paradigm is formulated, according to which the abrupt late Pleistocene glacial terminations correspond to the excitation, by the orbital forcing, of nonlinear relaxation oscillations (ROs) internal to the climate system in the absence of any stochastic parameterization. Specific rules are derived from the DE paradigm: they parameterize internal climate feedbacks, which, when activated by the crossing of certain tipping points, excite a RO. Such rules are then applied to the output of an energy-balance model simulating the fluctuations induced by realistic orbital forcing on the glacial state. The timing of the glacial terminations, thus, obtained in a reference simulation is found to be in good agreement with proxy records. A sensitivity analysis insures the robustness of the timing. The potential irrelevance of noise allowing DE to hold is discussed, and a possible explanation of the 100-kyr cycle problem based on DE is outlined. In conclusion, the DE paradigm provides the simplest possible dynamical systems characterization of the link between orbital forcing and glacial terminations implied by the Milankovitch hypothesis.
REFERENCES
1.
L. E.
Lisiecki
and M. E.
Raymo
, “A Pliocene-Pleistocene stack of 57 globally distributed benthic records
,” Paleoceanography
20
, PA1003
, https://doi.org/10.1029/2004PA001071 (2005
). 2.
T. Y. M.
Konijnendijk
, M.
Ziegler
, and L. J.
Lourens
, “On the timing and forcing mechanisms of late Pleistocene glacial terminations: Insights from a new high-resolution benthic stable oxygen isotope record of the eastern Mediterranean
,” Quat. Sci. Rev.
129
, 308
–320
(2015
). 3.
A.
Berger
, M.
Crucifix
, D. A.
Hodell
et al., “Interglacials of the last 800,000 years
,” Rev. Geophys.
54
, 162
–219
, https://doi.org/10.1002/2015RG000482 (2016
). 4.
J.
Hansen
, M.
Sato
, G.
Russell
, and P.
Kharecha
, “Climate sensitivity, sea level and atmospheric carbon dioxide
,” Philos. Trans. R. Soc. A
371
, 20120294
(2013
). 5.
M.
Milankovitch
, Theorie Mathematique des Phenomenes Thermiques Produits par la Radiation Solaire
(Gauthier-Villars
, Paris
, 1920
).6.
M.
Milankovitch
, Canon of Insolation and the Ice-Age Problem, Narodna Biblioteka Srbije—English translation of the original publication (Textbook Publishing Company, Belgrade
, 1998
), 634 pp.7.
J. D.
Hays
, J.
Imbrie
, and N. J.
Shackleton
, “Variations in the Earth’s orbit: Pacemakers of the ice ages
,” Science
194
, 1121
–1132
(1976
). 8.
M.
Ghil
and S.
Childress
, Topics in Geophysical Fluid Dynamics: Atmospheric Dynamics, Dynamo Theory, and Climate Dynamics
(Springer Science
, New York
, 1987
).9.
R. S.
Bradley
, Paleoclimatology
, 2nd ed. (Academic Press
, San Diego, CA
, 1999
), p. 610
.10.
11.
M.
Ghil
, “Cryothermodynamics: The chaotic dynamics of paleoclimate
,” Physica D
77
, 130
–159
(1994
). 12.
M.
Ghil
, “A century of nonlinearity in the geosciences
,” Earth Space Sci.
6
, 1007
–1042
(2019
). 13.
N.
Boers
, M.
Ghil
, and T. F.
Stocker
, “Theoretical and paleoclimatic evidence for abrupt transitions in the Earth system
,” Environ. Res. Lett.
17
, 093006
(2022
). 14.
W. D.
Sellers
, “A global climatic model based on the energy balance of the Earth atmosphere
,” J. Appl. Meteorol.
8
, 392
–400
(1969
). 15.
M. I.
Budyko
, “The effect of solar radiation variations on the climate of the Earth
,” Tellus
21
, 611
–619
(1969
). 16.
I. M.
Held
and M. J.
Suarez
, “Simple albedo feedback models of the ice caps
,” Tellus
26
, 613
–629
(1974
). 17.
G. R.
North
, “Analytical solution to a simple climate model with diffusive heat transport
,” J. Atmos. Sci.
32
, 1301
–1307
(1975
). 18.
M.
Ghil
, “Climate stability for a Sellers-type model
,” J. Atmos. Sci.
33
, 3
–20
(1976
). 19.
M.
Ghil
, “Climate sensitivity, energy balance models, and oscillatory climate models
,” J. Geophys. Res. Atmos.
89
, 1280
–1284
, https://doi.org/10.1029/JD089iD01p01280 (1984
). 20.
K.
Riechers
, T.
Mitsui
, N.
Boers
, and M.
Ghil
, “Orbital insolation variations, intrinsic climate variability, and quaternary glaciations
,” Clim. Past
18
, 863
–893
(2022
). 21.
M.
Crucifix
, “Oscillators and relaxation phenomena in Pleistocene climate theory
,” Philos. Trans. R. Soc. A
370
, 1140
–1165
(2012
). 22.
D. V.
Alexandrov
, I. A.
Bashkirtseva
, M.
Crucifix
, and L. B.
Ryashko
, “Nonlinear climate dynamics: From deterministic behaviour to stochastic excitability and chaos
,” Phys. Rep.
902
, 1
–60
(2021
). 23.
K.
Hasselmann
, “Stochastic climate models: Part I. Theory
,” Tellus
28
, 473
–485
(1976
). 24.
Stochastic Climate Models, edited by P. Imkeller and J.-S. Von Storch (Birkhauser, 2001).
25.
P.
Imkeller
and A. H.
Monahan
, “Conceptual stochastic climate models
,” Stoch. Dyn.
2
, 311
–326
(2002
). 26.
D. S.
Wilks
, “Effects of stochastic parameterisation on conceptual climate models,” in Stochastic Physics and Climate Modelling, edited by T. Palmer and P. Williams (Cambridge University Press, 2010), pp. 191–206.27.
N.
Calder
, “Arithmetic of ice ages
,” Nature
252
, 216
–218
(1974
). 28.
J.
Imbrie
, A.
Berger
, E. A.
Boyle
et al., “On the structure and origin of major glaciation cycles 2. The 100,000-year cycle
,” Paleoceanography
8
, 699
–735
, https://doi.org/10.1029/93PA02751 (1993
). 29.
D.
Paillard
, “The timing of Pleistocene glaciations from a simple multiple-state climate model
,” Nature
391
, 378
–381
(1998
). 30.
D.
Paillard
, “Glacial cycles: Toward a new paradigm
,” Rev. Geophys.
39
, 325
–346
, https://doi.org/10.1029/2000RG000091 (2001
). 31.
E.
Tziperman
and H.
Gildor
, “On the mid-Pleistocene transition to 100-kyr glacial cycles and the asymmetry between glaciation and deglaciation times
,” Paleoceanography
18
, 1001
, https://doi.org/10.1029/2001pa000627 (2003
). 32.
P.
Huybers
, “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
–55
(2007
). 33.
P. D.
Dietlevsen
, “Bifurcation structure and noise-assisted transitions in the Pleistocene glacial cycles
,” Paleoceanography
24
, PA3204
, https://doi.org/10.1029/2008PA001673 (2009
). 34.
J. Z.
Imbrie
, A.
Imbrie-Moore
, and L. E.
Lisiecki
, “A phase-space model for Pleistocene ice volume
,” Earth Planet. Sci. Lett.
307
, 94
–102
(2011
). 35.
F.
Parrenin
and D.
Paillard
, “Amplitude and phase of glacial cycles from a conceptual model
,” Earth Planet. Sci. Lett.
214
, 243
–250
(2003
). 36.
F.
Parrenin
and D.
Paillard
, “Terminations VI and VIII (530 and 720 kyr BP) tell us the importance of obliquity and precession in the triggering of deglaciations
,” Clim. Past
8
, 2031
–2037
(2012
). 37.
P. C.
Tzedakis
, M.
Crucifix
, T.
Mitsui
, and E. W.
Wolff
, “A simple rule to determine which insolation cycles lead to interglacials
,” Nature
542
, 427
–432
(2017
). 38.
C. J.
Berends
, P.
Kohler
, L. J.
Lourens
, and R. S. W.
van de Wal
, “On the cause of the mid-Pleistocene transition
,” Rev. Geophys.
59
, e2020RG000727
, https://doi.org/10.1029/2020RG000727 (2021
). 39.
G.
Leloup
and D.
Paillard
, “Influence of the choice of insolation forcing on the results of a conceptual glacial cycle model
,” Clim. Past
18
, 547
–558
(2022
). 40.
A. S.
Pikovsky
and J.
Kurths
, “Coherence resonance in noise-driven excitable systems
,” Phys. Rev. Lett.
78
, 775
–778
(1997
). 41.
S.
Pierini
, “Low-frequency variability, coherence resonance, and phase selection in a low-order model of the wind-driven ocean circulation
,” J. Phys. Oceanogr.
41
, 1585
–1604
(2011
). 42.
S.
Pierini
, “A Kuroshio Extension system model study: Decadal chaotic self-sustained oscillations
,” J. Phys. Oceanogr.
36
, 1605
–1625
(2006
). 43.
S.
Pierini
, H. A.
Dijkstra
, and A.
Riccio
, “A nonlinear theory of the Kuroshio Extension bimodality
,” J. Phys. Oceanogr.
39
, 2212
–2229
(2009
). 44.
S.
Pierini
, “Kuroshio Extension bimodality and the North Pacific Oscillation: A case of intrinsic variability paced by external forcing
,” J. Clim.
27
, 448
–454
(2014
). 45.
S.
Pierini
, “Ensemble simulations and pullback attractors of a periodically forced double-gyre system
,” J. Phys. Oceanogr.
44
, 3245
–3254
(2014
). 46.
S.
Pierini
, M.
Ghil
, and M. D.
Chekroun
, “Exploring the pullback attractors of a low-order quasigeostrophic ocean model: The deterministic case
,” J. Clim.
29
, 4185
–4202
(2016
). 47.
S.
Pierini
, M.
Chekroun
, and M.
Ghil
, “The onset of chaos in nonautonomous dissipative dynamical systems: A low-order ocean–model case study
,” Nonlinear Process. Geophys.
25
, 671
–692
(2018
). 48.
S.
Pierini
and M.
Ghil
, “Tipping points induced by parameter drift in an excitable ocean model
,” Sci. Rep.
11
, 11126
(2021
). 49.
S.
Pierini
, “Stochastic tipping points in climate dynamics
,” Phys. Rev. E
85
, 027101
(2012
). 50.
H.
Gildor
and E.
Tziperman
, “Sea ice as the glacial cycles’ climate switch: Role of seasonal and orbital forcing
,” Paleoceanography
15
, 605
–615
, https://doi.org/10.1029/1999PA000461 (2000
). 51.
H.
Gildor
and E.
Tziperman
, “A sea ice climate switch mechanism for the 100-kyr glacial cycles
,” J. Geophys. Res.
106
, 9117
–9133
, https://doi.org/10.1029/1999JC000120 (2001
). 52.
J. D.
Pelletier
, “Coherence resonance and ice ages
,” J. Geophys. Res.
108
, 4645
, https://doi.org/10.1029/2002JD003120 (2003
). 53.
J.
Ji
, W.
Balsam
, X.
Chen
, J.
Chen
, Y.
Chen
, and H.
Wang
, “Rate of solar insolation change and the glacial//interglacial transition
,” Geophys. Res. Lett.
33
, L04706
, https://doi.org/10.1029/2005GL025401 (2006
). 54.
Q. Z.
Yin
, Z. P.
Wu
, A.
Berger
, H.
Goosse
, and D.
Hodell
, “Insolation triggered abrupt weakening of Atlantic circulation at the end of interglacials
,” Science
373
, 1035
–1040
(2021
). 55.
J.
Laskar
, A.
Fienga
, M.
Gastineau
, and H.
Manche
, “La2010: A new orbital solution for the long-term motion of the Earth
,” Astron. Astrophys.
532
, A89
(2011
). 56.
J.
Laskar
, P.
Robutel
, F.
Joutel
, M.
Gastineau
, A. C. M.
Correia
, and B.
Levrard
, “A long-term numerical solution for the insolation quantities of the Earth
,” Astron. Astrophys.
428
, 261
–285
(2004
). 57.
J.
Hansen
, R.
Ruedy
, M.
Sato
, and K.
Lo
, “Global surface temperature change
,” Rev. Geophys.
48
, RG4004
, https://doi.org/10.1029/2010RG000345 (2010
). 58.
J. D.
Annan
and J. C.
Hargreaves
, “A new global reconstruction of temperature changes at the Last Glacial Maximum
,” Clim. Past.
9
, 367
–376
(2013
). 59.
J. E.
Tierney
, J.
Zhu
, J.
King
, S. B.
Malevich
, G. J.
Hakim
, and C. J.
Poulsen
, “Glacial cooling and climate sensitivity revisited
,” Nature
584
, 569
–573
(2020
). 60.
R.
Benzi
, G.
Parisi
, A.
Sutera
, and A.
Vulpiani
, “Stochastic resonance in climate change
,” Tellus
34
, 10
–16
(1982
). 61.
C.
Nicolis
, “Stochastic aspects of climatic transitions—Response to a periodic forcing
,” Tellus
34
, 1
–9
(1982
). 62.
L.
Gammaitoni
, P.
Hanggi
, P.
Jung
, and F.
Marchesoni
, “Stochastic resonance
,” Rev. Mod. Phys.
70
, 223
–287
(1998
). 63.
M.
De Domenico
and A.
Vulpiani
, “Understanding climate and turbulence: The mark of Giorgio Parisi
,” Nat. Italy
(2021
). 64.
A.
Sutera
, “On stochastic perturbation and long-term climate behaviour
,” Q. J. R. Meteorol. Soc.
107
, 137
–151
(1981
). 65.
C.
Nicolis
and G.
Nicolis
, “Stochastic aspects of climate transitions—Additive fluctuations
,” Tellus
33
, 225
–234
(1981
). 66.
L.
Gammaitoni
, P.
Hanggi
, P.
Jung
, and F.
Marchesoni
, “Stochastic resonance: A remarkable idea that changed our perception of noise
,” Eur. Phys. J. B
69
, 1
–3
(2009
). 67.
Stochastic Physics and Climate Modelling, edited by T. Palmer and P. E. Williams (Cambridge University Press, 2010), p. 480.
68.
E.
Tziperman
, M. E.
Raymo
, P.
Huybers
, and C.
Wunsch
, “Consequences of pacing the Pleistocene 100 kyr ice ages by nonlinear phase locking to Milankovitch forcing
,” Paleoceanography
21
, PA4206
, https://doi.org/10.1029/2005PA001241 (2006
). 69.
D.
Flath
, H. G.
Kaper
, F.
Wattenberg
, and E.
Widiasih
, Energy Balance Models
(DIMACS: Center for Discrete Mathematics and Theoretical Computer Science, Rutgers
, 2012
).70.
V.
Lucarini
, K.
Fraedrich
, and F.
Lunkeit
, “Thermodynamic analysis of snowball earth hysteresis experiment: Efficiency, entropy production, and irreversibility
,” Q. J. R. Metereol. Soc.
136
, 2
–11
(2010
). 71.
V.
Lucarini
, L.
Serdukova
, and G.
Margazoglou
, “Lévy noise versus Gaussian-noise-induced transitions in the Ghil-Sellers energy balance model
,” Nonlinear Process. Geophys.
29
, 183
–205
(2022
). 72.
P. F.
Hoffman
, A. J.
Kaufman
, G. P.
Halverson
, and D. P.
Schrag
, “A Neoproterozoic snowball Earth
,” Science
281
, 1342
–1346
(1998
). 73.
R. T.
Pierrehumbert
, D. S.
Abbot
, A.
Voigt
, and D.
Koll
, “Climate of the Neoproterozoic
,” Annu. Rev. Earth Planet. Sci.
39
, 417
–460
(2011
). 74.
D.
Herwartz
, A.
Pack
, D.
Krylov
, Y.
Xiao
, K.
Muehlenbachs
, S.
Sengupta
, and T.
Di Rocco
, “Revealing the climate of snowball Earth from systematics of hydrothermal rocks
,” Proc. Natl. Acad. Sci. U.S.A.
112
, 5337
–5341
(2015
). © 2023 Author(s). Published under an exclusive license by AIP Publishing.
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