Climate sensitivity and dynamical systems

Transcription

1 Michel Crucifix, 01 February 2016, p. 1. Climate sensitivity and dynamical systems Michel Crucifix Université catholique de Louvain & Fonds National de la Recherche Scientifique Doctoral training school: Cross-Disciplinary Approaches to Non-Equilibrium Systems (CANES), King s College, London

2 Cross-Disciplinary Approaches to Non-Equilibrium Systems (CANES) Michel Crucifix, 01 February 2016, p. 2.

3 Michel Crucifix, 01 February 2016, p. 3. Plan The energy balance model Fluctuations Mechanistic models for the fluctuations Beyond time scale separation Phenomenological approaches to palaeoclimate variability Ice age weather? Conclusions

4 The energy balance model Michel Crucifix, 01 February 2016, p. 4.

5 Joseph Fourier ( ) Michel Crucifix, 01 February 2016, p. 5.

6 Michel Crucifix, 01 February 2016, p. 6. Surface energy balance Sun Visible IR T

7 Michel Crucifix, 01 February 2016, p. 7. Climate Sensitivity Climate sensitivity is a metric used to characterise the response of the global climate system to a given forcing. It is broadly defined as the equilibrium global mean surface temperature change following a doubling of atmospheric CO 2 concentration IPCC Report, AR4 (2007) Ch. 8 Metric: A system or standard of measurement Oxford English dictonary to be defined: forcing, global climate system, equilibrium, mean surface temperature, following (and broadly )

8 Michel Crucifix, 01 February 2016, p. 8. Climate sensitivity as T 2 T 2 = T (560 ppm) T (280 ppm), where T (C) is a global temperature function of the concentration in CO 2.

9 Michel Crucifix, 01 February 2016, p. 9. The climate greenhouse forcing: We may also define the forcing associated with an increase in GHG: F 2 = LW (560 ppm, T (280 ppm)) LW (280 ppm, T (280 ppm)), where LW (C, T ) is the amount of outgoing longwave radiation given a carbon dioxide concentration and a temperature

10 Michel Crucifix, 01 February 2016, p. 10. Climate sensitivity ratio We can also a climate sensitivity ratio (e.g. Cess, 1990): Λ = T 2 / F 2 and even provide a more general framework: Λ(C ref, C) = T (C ref + C) T (C ref ), LW (C ref + C, T (C ref )) LW (C ref, T (C ref )) (1)

11 1 J. E. Hansen et al. In: Climate Processes and Climate Sensitivity. Ed. by J. E. Hansen and T. Takahashi. 1984, M. E. Schlesinger. In: Understanding climate change. Ed. by A. Berger, R. E. Dickinson, and J. W. Kidson Michel Crucifix, 01 February 2016, p. 11. Climate feedbacks F = R (2) = λ IR T + j λ j T (3) The λ j are radiative feedbacks 1 (runaway if j λj λ IR ) ( ) 1 T = λ IR + F (4) j }{{ λj } Λ=1/λ

12 Michel Crucifix, 01 February 2016, p. 12. Application to the spherical Earth F = 1 F (lat, lon, z = toa) ds, (5) S S R = 1 R(lat, lon, z = toa) ds, (6) S S T =???. (7) The classical choice is to use a global average of the surface temperature ( global temperature ), but in fact nothing forces us a priori to do that.

13 Michel Crucifix, 01 February 2016, p. 13. Radiative relaxation c dt dt = F R (8) = F λ(t T ref ) (9) This diff. equation converges to T T ref = F /λ for λ > 0. It converges monotonously seems to play the role of an energy reservoir. c dt dt

14 Michel Crucifix, 01 February 2016, p. 14. Generalisation to multiple energy reservoirs: Sun Visible IR T 1 E 1 = c 1 T 1 T 2 E 2 = c 2 T 2

15 Michel Crucifix, 01 February 2016, p. 15. Generalisation to multiple energy reservoirs (ii) (term: E the vector of energy reservoirs) de dt = F R (10) = F λ[t T ref ] (11) A classical system of linear equations resolved by diagonalisation. Given, say, a forcing of the form F 1 = H(t) (Heaviside function) and T(0) = 0: T 1 (t) = i Λ i (1 e t/ξ i ), (12) where the ξ i are the eigenvalues of λ.

16 Michel Crucifix, 01 February 2016, p. 16. Multiple relaxation time scales Linear scale in time T 1 Λ ξ 1 ξ 2 Logarithmic scale in time T 1 t Λ 1 Λ 2 1/ξ 1 1/ξ 2 log(t) Λ 1

17 Michel Crucifix, 01 February 2016, p. 17. Plan The energy balance model Fluctuations Mechanistic models for the fluctuations Beyond time scale separation Phenomenological approaches to palaeoclimate variability Ice age weather? Conclusions

18 Michel Crucifix, 01 February 2016, p. 18. Accounting for fluctuations in the linear regime For an unforced, stationary regime, it is tempting to simply write: de dt = λt + σε(t), (13) with ε(t) a Gaussian white noise. Mathematicians (and physicists) recognise an Orstein-Uhlenbeck process.

19 Michel Crucifix, 01 February 2016, p. 19. Linear relaxation spectrum log(ˆf 2 (ω)) 2D 1D τ 2 τ 1 log(ω)

20 Empirical spectrum (based on many data) high latitudes Energy ( C ka) tropical latitudes Frequency (cycles / ka) P. Huybers and W. Curry. In: Nature (2006). DOI: /nature04745 Michel Crucifix, 01 February 2016, p. 20.

21 Fluctuation dissipation theorem? It may be tempting to use the fluctuation dissipation theorem, which says (1-D case) cov(x t, x t+ t ) = σ2 2λ (e λ t ) (14) suggested by (Leith 1975) [applied to pulse-forings (North, 1993)], and used to analyse natural fluctuations by Cionni et al. (2004), Schwartz (2007) but this method consistently underestimates climate sensitivity ( 2 ) One problem is a too-literal interpretation of the relaxation equation as merely a radiative balance problem, where energy imbalance cause changes in energy stocks. 2 D. B. Kirk-Davidoff. In: Atmospheric Chemistry and Physics (2009). DOI: /acp Michel Crucifix, 01 February 2016, p. 21.

22 Michel Crucifix, 01 February 2016, p. 22. Plan The energy balance model Fluctuations Mechanistic models for the fluctuations Beyond time scale separation Phenomenological approaches to palaeoclimate variability Ice age weather? Conclusions

23 Michel Crucifix, 01 February 2016, p. 23. The Dynamical System model In all generality, we model state changes as caused by non-linear relationships: dx = f(x) (15) dt where X(t) might have a few... or many dimensions

24 Michel Crucifix, 01 February 2016, p. 24. Lorenz 63 has a chaotic attractor At stationary regime: the attractor is described by a probability phase space (a space measure µ(x)) preserved by the flow

25 Michel Crucifix, 01 February 2016, p. 25. Exponential dichotomy stable exponential dichotomy In essence : the time-evolution is governed by contraction along stable subspaces, and expansion along unstable subspaces

26 Michel Crucifix, 01 February 2016, p. 26. Chaotic, ergodic? Zoo of concepts, not all equivalent. In (too) short: Chaotic : sensitive dependence to initial conditions Topologically mixing (visits all points of attractor) Ergodic : The bottom line: At stationary regime: the attractor is described by a unique measure µ(x) preserved by the flow At the very least: we need systems with an exponential dichotomy... dissipative, and structurally stable

27 Michel Crucifix, 01 February 2016, p. 27. Time scale separation, as usually presented In a GCM, Ẋ = f (X) where X has many,... many dimensions fluctuations controlled by dynamics embedded in f, (weather) quasi-stationary change in measure µ(t) as forcing changes (climate)

28 Michel Crucifix, 01 February 2016, p. 28. If we can do time-scale separation, we are fine The stationary regime may be described by a summary statistics T = X T (X)µ(X) dx = lim t f 1 t f tf 0 T (X) dt (16) Again, at stationary regime, energy balance must be respected, so we should be able to write: F = R (17) = λ IR T + j λ j T (18) We can carry feedback analysis as usual. But we could also play with T 2, P, or whatever we are fancy.

29 Michel Crucifix, 01 February 2016, p. 29. Application 1 : global sensitivity analysis in a GCM Consider 5 forcing factors: ice volume, CO 2, long. perihelion, eccentricity and obliquity. call INPUT the input space perform 63 experiments with a GCM (HadCM3) following a latin-hypercube design for each experiment, a set of summary OUTPUT map INPUT OUTPUT with a statistical model (Gaussian process) Study the influence of different components of the input, separately or together see N. Bounceur, M. Crucifix, and R. D. Wilkinson. In: Earth System Dynamics (2015). DOI: /esd , P. A. Araya-Melo, M. Crucifix, and N. Bounceur. In: Climate of the Past (2015). DOI: /cp based on [7] for full theory.

30 Michel Crucifix, 01 February 2016, p. 30. Application 2 : estimates of palaeoclimates evolutions Simulated evolution of NINO34 mean and variability indices ICE CO2 [ppm] Obliquity (o) PRECESSION INDEX MEAN NINO34 SST (oc) FREQUENCY VARIABILITY NINO34 NINO34 SST (cycle/year) (oc) Crucifix and Araya-Melo, in prep Time [ka]

31 Michel Crucifix, 01 February 2016, p. 31. Plan The energy balance model Fluctuations Mechanistic models for the fluctuations Beyond time scale separation Phenomenological approaches to palaeoclimate variability Ice age weather? Conclusions

32 Michel Crucifix, 01 February 2016, p. 32. Non-homogeneous model In practice we can t always expect the quasi-stationary assumption to hold Forcing may vary at rates faster than mixing time Technically : dx = f(x) + F(t) (19) dt (note: the additive forcing is for notation convenience)

33 Response theory for the deterministic model With some caution ( 3 ), we can still define a time-evolving measure µ(x )(t), hence we can define, e.g.: T (t). and a convolution equation of the form: T (t) = F (t )G(t t ) dt (20) t <t The equation can be justified as a first-order approximation in response theory,... but not merely as an heat diffusion / relaxation phenomenon! and there is no reason why G( ) should simply be an exponential decay. Will need enough experiments to estimate T (t). 3 V. Lucarini, F. Ragone, and F. Lunkeit. In: Journal of Statistical Physics (2016). DOI: /s z. Michel Crucifix, 01 February 2016, p. 33.

34 Michel Crucifix, 01 February 2016, p. 34. Stochastic reduction We have to admit that we can t represent all scales at once, deterministically Starting from a large dynamical system, the hope is : dx = f(x, t) dt be described by a reduced order representation: dx = a(x, t) dt }{{} + b(x, t) dω }{{} deterministic drift stochastic diffusion where ω is some stochastic process and x a state-description relevant at the scale of interest

35 4 A. J. Majda, C. Franzke, and D. Crommelin. In: Proceedings of the National Academy of Sciences of the United States of America (2009). DOI: /pnas Kondrashov06ab. Michel Crucifix, 01 February 2016, p. 35. Two main approaches for stochastic reduction ab-initio : analytical treatment of deterministic system + conservation laws some have done this for idealised flows 4 empirical : find out sets of equations which match the observations / simulation 5 In addition, phenomenological : dynamical systems describing grossly known processes e.g.: models of El-Nino, DO events, ice ages.

36 Michel Crucifix, 01 February 2016, p. 36. Plan The energy balance model Fluctuations Mechanistic models for the fluctuations Beyond time scale separation Phenomenological approaches to palaeoclimate variability Ice age weather? Conclusions

37 Dansgaard-Oeshger events Insolation [W/m 2 ] log 10Ca δ 18 O ice[permil] n(t) (a) (b) (c) (d) MIS 2 MIS 3 MIS 4 MIS Age t [ka BP] δ 18 O benthic[permil] T. Mitsui and M. Crucifix. In: Climate Dynamics (2016). DOI: /s z and ref. therein for data. Michel Crucifix, 01 February 2016, p. 37.

38 Michel Crucifix, 01 February 2016, p. 38. Calibration of palaeoclimate data on simple model dx(t) = { ky αx β } 3 [(x x ) 3 + x 3 ] dt + σ 1 dw 1 (t) dy(t) = { x + γ 0 + γ 1 I (t) γ 2 V (t) } dt + σ 2 dw 2 (t)

39 Reconstructed attractor v stochastic orbit deterministic orbit fast manifold slow manifold x T. Mitsui and M. Crucifix. In: Climate Dynamics (2016). DOI: /s z (see also F. Kwasniok and G. Lohmann. In: Nonlinear Processes in Geophysics (2012). DOI: /npg for a similar study) Michel Crucifix, 01 February 2016, p. 39.

40 Michel Crucifix, 01 February 2016, p. 40. Model selection? (needs a whole lecture) We can define a likelihood function L z (parameters) assuming independent observational errors The Likelihood function is, roughly, the probability of observing what we observed, given the model, and given the parameters. Estimating the likelihood function in stochastic dynamical systems is generally very difficult (because many potential histories) - particle filter, unscented Kalman filter We can also do model selection based on Bayes Factors In the Mitsui - Crucifix paper, for example, we confirm the influence of astronomical forcing on the timing of millennial scale variability. T. Mitsui and M. Crucifix. In: Climate Dynamics (2016). DOI: /s z, J. Carson et al., in press In: Trans. of Royal. Stat. Society, available from

41 Michel Crucifix, 01 February 2016, p. 41. Plan The energy balance model Fluctuations Mechanistic models for the fluctuations Beyond time scale separation Phenomenological approaches to palaeoclimate variability Ice age weather? Conclusions

42 Michel Crucifix, 01 February 2016, p. 42. Two models (among many others) Forcing (mid June insolation at 65N) A.U. Ice volume (A. U.) Ice volume (1E15 m3) time [ka] Saltzman and Maasch time [ka] Tziperman et al., two previously published models phenomenological low order (2 or 3 dimensions) deterministic forced by astronomical forcing founded on the principle of a limit cycle time [ka]

43 Stochastic forcing Alternative histories are also excited by additive forcing : is there a palaeoclimate weather? (here : Saltzman and Maasch model, 1990)) σ x = σ y = σ z = 0 SM90 : X trajectories (scaled ice volume) σ x = σ y = σ z = 0.01 σ x = σ y = σ z = Time (ka) Michel Crucifix, 01 February 2016, p. 43.

44 Michel Crucifix, 01 February 2016, p. 44. Dynamical system theory of ice ages Ice ages viewed as a forced quasi-limit cycle (The autonomous counterpart may have a limit cycle ) The astronomical forcing is quasi-periodic (sum of harmonics) non-linear synchronisation on the astronomical forcing Small changes in parameters excite alternative histories. We believe that this feature is linked to the existence of strange non-chaotic attractors in these systems (non-robust synchronisation) Strange non-chaotic attractors occur in many ice age model, we don t know how general it is 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

45 M. Crucifix, T. Mitsui, and G. Lenoir. In: Nonlinear and Stochastic Climate Dynamics. Ed. by C. Franzke and T. J. O Kane Michel Crucifix, 01 February 2016, p. 45. Are ice ages unstable like weather? Spectrum obtained in one model (Saltzman - Maasch 90) SM90 : Spectrum (log log) Spectral density (ka) 1e 12 1e 06 1e+00 σ = 0 σ = 0.01 σ = 0.1 slope = 2 The spectrum most compatible with observations is also the one that has the most unstable ice age trajectories (in this particular model!) 100 ka 10 ka 1 ka Period

46 Michel Crucifix, 01 February 2016, p. 46. Plan The energy balance model Fluctuations Mechanistic models for the fluctuations Beyond time scale separation Phenomenological approaches to palaeoclimate variability Ice age weather? Conclusions

47 Conclusions Climate response is often framed by reference to energy balance This is fine for a gross depiction of the greenhouse forcing The principle of relaxation seems to provide a first idea of climate spectra However : Going from energy balance to relaxation is not so straightforward! because fluctuations emerge from various physical constraints At a given time scale, we accept to represent fast processes stochastically At palaeoclimate time scales: empirical approaches remain one of the few viable ways forward to study dynamical properties e.g.: we can study the instability of ice age trajectories as emerging from generic conditions the system is non-homogeneous! (astronomical forcing) We have still no theory which identifies dynamical regimes consistently across different time scales, from weather to palaeoclimates Michel Crucifix, 01 February 2016, p. 47.