Ice age challenges. Michel Crucifix. Université catholique de Louvain & Fonds National de la Recherche Scientifique

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1 Michel Crucifix, Dynamics Days Exeter 2015, p. 1. Ice age challenges Michel Crucifix Université catholique de Louvain & Fonds National de la Recherche Scientifique Dynamics Day Europe, Exeter September 10, 2015

2 If [... ] we look in the 21st century and make an optimistic forecast on the type of computer which will be available [... ] We may construct a super-model [... ] When we integrate the equations, if they are correct, we shall necessarily obtain changes in climate, including the great ice ages. E. N. Lorenz. In: J. Appl. Meteor. (1970). DOI: / (1970)009<0325:CCAAMP>2.0.CO;2 Michel Crucifix, Dynamics Days Exeter 2015, p. 2.

3 Michel Crucifix, Dynamics Days Exeter 2015, p. 3. Overview Historical Background Phenomenology The deterministic framework Going statistical The great challenges

4 Section 1 Historical Background E. Bard / C. R. Geoscience 336 (2004) what is so clearly shown for this last catastrophe is hardly less [valid] for those preceding it. Geology also went on to contribute to the first thinking on ancient climates. In the 19th century, many naturalists and travellers were intrigued by the presence, in broad Alpine valleys, of many large boulders of rock, called erratic blocks, as well as numerous hillocks composed of rock debris, called moraines. These objects occur isolated in the midst of plains and are often of very different nature from the local bedrock (Fig. 3). In particular, enormous blocks of granite are located right in the middle of vast limestone terrains. Among others [8], thegenevan Horace Bénédict de Saussure ( ) was astonished by these blocks, and remarked ironically that the granites are not formed in the ground like truffles, and do not grow like fir trees on the limestone rocks. De Saussure also noticed that erratic blocks were located along the axes of Alpine valleys, which led him to assume that they had been carried by currents of incredible violence and magnitude. Observing that erratic blocks are made up of the same rocks as the highest peaks of the Alps, the naturalists concluded that these blocks had been transported over tens or even hundreds of kilometres. At the time, the theories about this transport still invoked the effects of the Flood, either directly by the mechanical effect of water, or indirectly by the transport of the rock blocks on ice rafts. This theory was formalized by the Scottish geologist Charles Lyell ( ), and is based on the observations of explorers in polar regions (Figs. 4 and 5). Lyell writes: in countries situated in high northern latitudes, like Spitzbergen, be- Fig. 3. Plate representing an erratic block of a Swiss glacier [15]. In 1841, Jean de Charpentier describes it in these terms: a very large block, called the Pierre à Dzo, irregular in shape and polyhedral, is perched on another; but it is only kept in place by a extremely small third block split vertically by the fall of the first; without this support, it would fall down onto the small town of Monthey. It is absolutely impossible that a horizontal shock would have produced similar chance occurrences, and he further adds that to claim to explain these facts of detail, these accidents, by hypotheses other than that of diluvian glaciers, is to go well beyond the bounds of probability. Fig. 3. Gravure représentant un bloc erratique d un glacier suisse [15]. En1841JeandeCharpentierledécritencestermes:«un très gros bloc, appelé la «pierre à Dzo», d une forme irrégulière, polyédrique, est perché sur un autre ; mais il n y est retenu que par un troisième bloc fort petit et fendu verticalement par la chute du premier ; sans cet appui il se précipiterait sur le bourg de Monthey. Il est absolument impossible qu un choc horizontal ait produit de Michel Crucifix, Dynamics Days Exeter 2015, p. 4.

5 Michel Crucifix, Dynamics Days Exeter 2015, p. 5. Timeline (I) 1820 : Moraines identified as ice age remains 1896: Greenhouse theory of ice ages 1941 : Milankovitch s astronomical theory of ice ages 1840 onwards: speculations about an astronomical origin 1920 : recurrent character of ice ages established

6 Michel Crucifix, Dynamics Days Exeter 2015, p. 6. Timeline (II) 1955 : Ice ages reconstructed from 18 O of deep sea sediments : first dynamical ice sheet models (ODE, then PDE) 1980 : first CO 2 records form Antarctica 1970 : ice ages are 100,000 years long : Astronomical forcing unambiguously identified palaeoclimate records

7 Michel Crucifix, Dynamics Days Exeter 2015, p. 7. Timeline (III) 1982 onwards: conceptual climate models of ice ages 1990 onwards: snap-shot simulations with General circulation models 2010 : Ice ages with 3-D ice sheet models 1990 onwards: Earth System Model of Intermediate complexity (EMIC) 2015 : last 800 ka with an EMIC with coupled carbon and ice sheet dynamics (not quite great, though!)

8 Michel Crucifix, Dynamics Days Exeter 2015, p. 8. Section 2 Phenomenology

9 Ice ages since 3 Ma ago1 Homo erectus Homo sapiens ardapithecus closure of tropical sea-ways (Panama and Indonesia) first major glaciation of the NH Present 1 L. E. Lisiecki and M. E. Raymo. In: Paleoceanography (2005). DOI: /2004PA Michel Crucifix, Dynamics Days Exeter 2015, p. 9.

10 Michel Crucifix, Dynamics Days Exeter 2015, p. 10. Climate system CO 2 data CO2 [ppmv] Time [ka]

11 Michel Crucifix, Dynamics Days Exeter 2015, p. 11. Orbital elements e : eccentricity ϖ: long. perihelion ε : obliquity

12 Michel Crucifix, Dynamics Days Exeter 2015, p. 12. The Milankovitch diagram Insolation 500 ka ago Latitude 90S 60S 30S Eq. 30N 60N 90S J F M A M J J A S O N D

13 Timing of deglaciations corresponding to high insolation Summer Insolation 65N [W/m2] Time [ka] Sea level [m] 2 Data from K. M. Grant et al. In: Nat Commun (Sept. 2014) Michel Crucifix, Dynamics Days Exeter 2015, p. 13.

14 3 A. Berger and M. Loutre. In: Quaternary Science Reviews (1991). DOI: / (91)90033-Q. Michel Crucifix, Dynamics Days Exeter 2015, p. 14. Orbital spectrum 3 Amplitude (W/m 2 ) Summer Insolation spectrum (65 N) Obliquity terms Precession terms Period (ka)

15 4 Guillaume Lenoir, unpublished, based on K. M. Grant et al. In: Nat Commun (Sept. 2014) Michel Crucifix, Dynamics Days Exeter 2015, p. 15. Sea-level spectrum 4 RSL : Spectrum (semilog) RSL : Spectrum (log log) Spectral density (m 2 ka) 0e+00 4e+04 8e+04 interpolated Pmax Spectral density (m 2 ka) 1e 01 1e+01 1e+03 interpolated Pmax slope = ka 41 ka 19 ka 100 ka 10 ka 1 ka Period Period

16 Michel Crucifix, Dynamics Days Exeter 2015, p. 16. Section 3 The deterministic framework

17 Michel Crucifix, Dynamics Days Exeter 2015, p. 17. Dynamical systems for the natural scientist ( modeller") Definition of the state vector (x t ) Definition of the system boundaries Development of the equations based on first principles (connect with other theories) conceptual and heuristic

18 5 M. Crucifix. In: The Holocene (2011). DOI: / , B. De Saedeleer, M. Crucifix, and S. Wieczorek. In: Climate Dynamics (2013). DOI: /s Michel Crucifix, Dynamics Days Exeter 2015, p. 18. A forced van-der-pol oscillator (2 ODE) 5 dx d t dy d t = 1 (F (t) + β + y), τ = α τ (y y 3 /3 + x).

19 6 M. Crucifix. In: The Holocene (2011). DOI: / , B. De Saedeleer, M. Crucifix, and S. Wieczorek. In: Climate Dynamics (2013). DOI: /s Michel Crucifix, Dynamics Days Exeter 2015, p. 19. A forced van-der-pol oscillator (2 ODE) 6 Forcing X Time [ka]

20 Long transient excited by noise Forcing Time [ka] X 7 T. Mitsui and M. Crucifix. In: INdAM Springer Series (accepted). Michel Crucifix, Dynamics Days Exeter 2015, p. 20.

21 Michel Crucifix, Dynamics Days Exeter 2015, p. 21. The Saltzman-Maasch (1990) model (3 ODE) started from long ago [8] X X Time [ka]

22 Michel Crucifix, Dynamics Days Exeter 2015, p. 22. Long transients in the Saltzman-Maasch (1990) model Forcing X Time [ka]

23 8 M. Crucifix. In: Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences (2012). DOI: /rsta M. Crucifix. In: Climate of the Past (2013). DOI: /cp T. Mitsui and K. Aihara. In: Climate Dynamics (2014). DOI: /s x, T. Mitsui, M. Crucifix, and K. Aihara. In: Physica D: Nonlinear Phenomena (2015). DOI: /j.physd Michel Crucifix, Dynamics Days Exeter 2015, p. 23. Conceptual deterministic models Many determinstic low-order models of ice ages rely more or less explicitly on the concept of an oscillator (forced or synchronised) 8 No sensitive dependence on initial conditions but one or several pullback attractors and/or long transients 9 strong (sensitive?) dependence on parameters These properties can be understood following the theory of strange non-chaotic attractors 10

24 Michel Crucifix, Dynamics Days Exeter 2015, p. 24. Beware the semantics! Astronomical forcing may be accounted for in an: Non-autonomous framework pullback attractor and repeller multiple attractors with basins of attraction (p, T )-attractors (for long transients) a conditionnal Lyapunov exponent ready for generalisation to stochastic forcing t a M. Rasmussen. Lecture Notes in Mathematics. Berlin Heidelberg: Springer, Autonomous framework forcing embedded in system attractors... which can be a torus (quasiperiodic), or strange

25 Michel Crucifix, Dynamics Days Exeter 2015, p. 25. Section 4 Going statistical

26 Michel Crucifix, Dynamics Days Exeter 2015, p. 26. Dynamical systems for the Statistician (Bayesian) θ x 0 x(t) z w(t) x(t) : state, as a function of time x 0 = x(t 0 ) : initial state θ : parameter P(x(t) θ) : path probability w(t) : stochastic dynamical process L z (θ) = P(z θ) : Likelihood P(θ z) P(θ)L z (θ) : Parameter estimation P(x t+1 z) : Prediction

27 Michel Crucifix, Dynamics Days Exeter 2015, p. 27. Stastician s task Write-up statistical model (priors, observation model etc.) Calibration : Estimate θ (max. likelihood, posterior) Prediction : Estimate x t+1 based on obs. up to time t Ratios of model Evidence 11 PM1 (z θ) dp(θ) PM2 (z θ) dp(θ) 11 J. Carson et al

28 12 M. Crucifix and J. Rougier. In: European Physics Journal - Special Topics (2009). DOI: /epjst/e Michel Crucifix, Dynamics Days Exeter 2015, p. 28. Predicting climate based on CO 2 before 6,000 years ago Ice volume anomaly (1e18 kg) CO2 (ppm) k 80k 40k 0k 20k 60k 100k Time

29 Michel Crucifix, Dynamics Days Exeter 2015, p. 29. Reality = model discrepancy z i x(t) t dx = f (x)dt + ds

30 Michel Crucifix, Dynamics Days Exeter 2015, p. 30. Modelling discrepancy Explicit (stochastic model) Process Model Discrepancy = Truth Warning If discrepancy model hasn t the right structure we could make ourselves more ignorant than we actually are (artificially reduced predictability horizon) Implicit (Feature Matching) Reduced output to statistical descriptor (example : spectrum) x(t) z F (z) Use of an extended likelihood" (Xia and Tong, 2011)

31 Michel Crucifix, Dynamics Days Exeter 2015, p. 31. Sequential likelihood estimator x(t) x(t 1) z i x(t + 1) z i P(x(t-1)) L(z) = obs (P(z i x(t 1), θ))dp(x(t 1))

32 Michel Crucifix, Dynamics Days Exeter 2015, p. 32. Sequential likelihood estimator x(t : t + n) x(t 1) z t P(x(t-1)) L ext (z) = obs (P(z i... z i+n x(t 1), θ))dp(x(t 1))

33 Michel Crucifix, Dynamics Days Exeter 2015, p. 33. Section 5 The great challenges

34 Michel Crucifix, Dynamics Days Exeter 2015, p. 34. System physical" dynamics Paradigm Climate system is made of components interacting, submitted to forcings How does the climate system listen to the astronomical forcing? What are the respective roles of ocean carbon cycle and ice-sheet instability (or others?) in the deglaciation? What are the physical changes associated with the Mid-Pleistocene transition (if any) What are the important interactions for the glacial inception process? etc.

35 13 P. Huybers and W. Curry. In: Nature (2006). DOI: /nature Michel Crucifix, Dynamics Days Exeter 2015, p. 35. Climate spectrum 13 high latitudes Energy ( C ka) tropical latitudes Frequency (cycles / ka)

36 Michel Crucifix, Dynamics Days Exeter 2015, p. 36. System stability and scale interactions Paradigm Climate system is a complex dynamical system How do we understand the spectrum of climate system? What is the stability of the ice age sequence? What is the origin of centennial, millennial variability?

37 Michel Crucifix, Dynamics Days Exeter 2015, p. 37. Modelling frameworks Low order models : Easy to analyse Easier to include in a statistical framework (never easy in palaeo) Low physical identifiability (what is X )? Geographic explicit models with simplified flow dynamics : GCMs : Can be designed to run thousands of years or more More physically explicit Missing weather (and often decadal/centennial variability); scale poorly Simulate weather dynamics Not designed / calibrated for the long scales No chance to get an ice age by simply integrating forward

38 Michel Crucifix, Dynamics Days Exeter 2015, p. 38. Some final thoughts The climate system is complex, any model of it will be simple in some respects We can t make real life experiments; and the future is far ahead for testing predictions Palaeoclimates are our only source of information about the slow dynamics (climate regime)... and it turns out that we are still poor at understanding the spectrum We need to organise our thinking across different modelling frameworks Statistics, numerical simulation, and dynamical system analysis need to be combined