Thermodynamics and Statistical Mechanics of Climate

Transcription

1 Thermodynamics and Statistical Mechanics of Climate Valerio Lucarini Institute of Meteorology, U. Hamburg Dept of Mathematics and Statistics, U. Reading 100 Congresso Nazionale Societa Italiana di Fisica Pisa, 22/09/2014

2 Introduction 2

3 Climate Science and Physics A solved problem, just some well-known equations and a lot of integrations who cares about the mathematical/physical consistency of models: better computers, better simulations, that s it! I regret to inform the author that geophysical problems related to climate are of little interest for the physical community 3 Who cares of energy and entropy? We are

4 Scales of Motions (Stommel/Smagorinsky)

5 Scales of Motions - different models for different scales - ff 5

6 Atmospheric Motions Three contrasting approaches: Those who like maps, look for features/particles Those who like regularity, look for waves Those who like irreversibility, look for turbulence Let s see schematically these 3 visions of the world 6

7 Atmospheric (macro) turbulence Energy, enstrophy cascades, 2D vs 3D Note: NOTHING is really 2D in the atmosphere 7

8 Features/Particles Focus is on specific (self)organised structures Hurricane physics/track 8

9 9 Waves in the atmosphere Large and small scale patterns

10 Waves in the atmosphere? Hayashi-Fraedrich decomposition 10

11 Waves in GCMs GCMs differ in representation of large scale atmospheric processes Just Kinematics? What we see are only unstable waves and their life cycle 11

12 Non-equilibrium A Non-equilibrium Statistical Mechanical System is in contact with several reservoirs Gallavotti,

13 Non-equilibrium in the Earth system climate Multiscale (Kleidon, 2011)

14 G O A L S O F M O D E L L I N G Local evolution in the phase space NWP vs. Statistical properties on the attractor Climate Modeling PREDICTABILITY

15 Climate Models uncertainties Uncertainties of the 1st kind Are our initial conditions correct? Not so relevant for CM, crucial for NWP Uncertainties of the 2nd kind Are we representing all the most relevant processes for the scales of our interest? Are we representing them well? (structural uncertainty) Are our heuristic parameters appropriate? (parametric uncertainty) Uncertainty on the metrics: Are we comparing properly and in a meaningful way our outputs with the observational data? 15

16 Zen and the Art of Climate Modelling Toy

17 Thermodynamics of Climate 17

18 Energy & Climate Response Perfect Model Forcing τ L. and Ragone, 2011 Total warming NESS Transient NESS 18

19 Energy and Climate Response Actual GCMs L. and Ragone, 2011 Forcing τ Not only bias: bias control bias final state Bias depends on climate state! Dissipation 19

20 Energy Budget Total energy of the climatic system: E dve dv u k P K ρ is the local density e is the total energy per unit mass moist static potential kinetic u, and k indicate the internal, potential and kinetic energy components Energy budget E P K 20

21 Detailed Balances WORK Kinetic energy budget K 2 dv C( P, K) D W Potential Energy budget P dv Q W 2 Q 1 H W C( P, K) Total Energy Budget E dv H dsnˆ H FLUXES DISSIPATION 21

22 Johnson s idea (2000) Partitioning the Domain P W dvq dvq Better than it seems! Q 0 Q 0 Q + Q - 22

23 Carnot Efficiency Lorenz (1955) Energy Cycle We have differentialheating G( A) W D 0 conversion C( A, K ) 0 dissipation D( K ) Hot Cold reservoirs Work: Carnot Efficiency : W = F+ +F - F + F + = Q+ -Q - Q + F + 23

24 Entropy Production Contributions of dissipation plus heat transport: S in ( W) = ò dv -Ñ H T + dv e 2 ò T» ò dv -Ñ H T + S W min W W We can quantify the excess of entropy production, degree of irreversibility with α: W ( ) a = ò dv -Ñ H T S W min W ( ) = Be-1 EP: S in 1 1 S min 24

25 A very imperfect engine dissipation Qin W Qout Q1 Q2 T1 T2 Something interesting to get out of this picture Work Entropy Production Irreversible Heat transport 25

26 Transport, Mixing, Adjustment Vertical Transport of Energy Convective adjustment Irreversible mixing C Horizontal Transports Baroclinic adjustment Irreversible mixing W 2-box model(s) W C 26

27 Results on IPCC GCMs vert S in T E < T E > T E < hor S in L., Ragone, Fraedrich, 2011 Hor vs Vert EP in IPCC models Warmer climate: Hor Vert Venus, Mars, Titan 27

28 Snowball Hysteresis Swing of S * by ±10% starting from present climate hysteresis experiment with full climate model Global average surface temperature T S Wide (~ 10%) range of S * bistable regime -T S ~ 50 K d T S /d S * >0 everywhere, almost linear L., Lunkeit, Fraedrich, 2010 W SB 28

29 Thermodynamic Efficiency d η /d S * >0 in SB regime Large T gradient due to large albedo gradient d η /d S * <0 in W regime System thermalized by efficient LH fluxes η decreases at transitions System more stable Similar behaviour for total Dissipation η=0.04 Δθ=10K 29

30 A 3D picture Parametric Analysis of Climate Change Structural Properties (Boschi et al. 2013) 30

31 Is there a common framework? Going from a 1D to a 2D parameter exploration we gain completeness, we lose focus Necessarily so? Can find an overall equivalence between the atmospheric opacity and incoming radiation perturbations Concept of radiative forcing If so, we gain some sort of universality 31

32 Parametrizations EP vs Emission Temperature 32

33 Parametrizations Dissipation vs Emission Temperature 33

34 Parametrizations Efficiency vs Emission Temperature 34

35 Now we reduce the length of the year 35

36 36

37 Phase Transition Width bistability vs length year (L. et al. 2013) Fast orbiting planets cannot be in Snowball Earth 37

38 Climate Change as a problem in Non-equilibrium Statistical Mechanics 38

39 IPCC Scenarios 39

40 Models Response 40

41 Climate Response IPCC scenario 1% increase p.y. 41

42 Response theory The response theory is a Gedankenexperiment: a system, a measuring device, a clock, turnable knobs. Changes of the statistical properties of a system in terms of the unperturbed system Divergence in the response tipping points Suitable environment for a climate change theory Blind use of several CM experiments We struggle with climate sensitivity and climate response Deriving parametrizations! 42

43 Ruelle ( 98) Kubo-like Response Theory Perturbed chaotic (Axiom A) flow: Change in expectation value of Φ: n th order perturbation: 43

44 This is a perturbative theory with a causal Green function: Expectation value of an operator evaluated over the unperturbed invariant measure ρ SRB (dx) where: and Linear term: Linear Green: Linear suscept: 44 ( 1) F e (t) = ò ds (1) G F G F (1) c F (1) ò ( t) = r ( 0 dx) ( s )e( t -s ) Q( t)lp( t)f ( w) = ò dt exp[ (1) iwt]g F t ( )

45 G F (1) G F (1) G F (1) ò Applicability of FDT ( t) = dxr ( 0 x) Q t ( ) ( ) X( x) ÑF x( t) FDT ( ) ( t) = -ò dxr ( 0 x) Ñ r ( 0 x) X( x) ( x) t é ë ( ) = -C s x OR ( ( )) ( )F x t If measure is singular, FDT has a boundary term Forced and Free fluctuations non equivalent Recently(Cooper, Alexeev, Branstator.): FDT is OK In fact, coarse graining sorts out the problem Parametrization by Wouters and L. 2012, 2013 has noise 45 r 0 ù û F( x( t) )

46 Linear (and nonlinear) Spectroscopy of L63 ( 1) c z ( w) e t ( ) = cos wt ( ) L Resonances have to do with UPOs 46

47 e t Therefore, We obtain: d e r F Stochastic forcing ( ) =eh( t) =edw( t) dt h t ( ) = 0 ( ) and ( ) = e 2 ò 2 dt 1 G ( F t 1,t ) 1 + o(e 4 ) = =1 2e 2 r ( 0 dx) dt 1 Q( t ) 1 The linear correction vanishes; only even orders of perturbations give a contribution No time-dependence ò Convergence to unperturbed measure ò h( t)h t ( ) = d ( t - t ) ( ) X i i X j j F f t 1 x 47

48 Correlations Power Spectra Fourier Transform , We end up with the linear susceptibility... Let s rewrite he equation: P 2,, 1 2 A P A 2 A So: difference between the power spectra square modulus of linear susceptibility Stoch forcing enhances the Power Spectrum Can be extended to general (very) noise KK linear susceptibility Green function 48

49 We choose observable A, forcing f(t) Let s perform an ensemble of experiments Linear response: Broadband forcing ( ) A f 1 ( t) = ò (1) dsg A s ( ) f t -s ( ) Fantastic, we estimate c f (1) ( w) = ( 1) A f ( w) ( ) f w and we obtain: we can predict ( 1) A g ( t) = ò (1) dsg A G A (1) s ( ) s ( )g t -s ( )

50 Lorenz 96 model Excellent toy model of the atmosphere Advection, Dissipation, Forcing Test Bed for Data assimilation schemes Popular within statistical physicists Evolution Equations x i x x x x F i1 i1 i2 i i 1,..., N x i x i N Spatially extended, 2 Parameters: N & F N e= å 2 x j 2 N m= x j N j=1 N å j=1 Properties are intensive F F +ee( t) 50

51 Broadband forcing G (1) (t) e( t) = Q( t) 51 Inverse FT of the susceptibility Response to any forcing with the same spatial pattern but with general time pattern

52 Spectroscopy Im [χ (1) (ω)] LW HF e( t) = 2cos( wt) L. and Sarno 2011 Rigorous extrapolation 52

53 (Non-)Differentiability of the measure for the climate system Boschi et al CO2 S* 53

54 A Climate Change experiment Observable: globally averaged T S Forcing: increase of CO 2 concentration Linear response: T S Let s perform an ensemble of experiments (1) Concentration at t =0 f ( t) = ò ds G TS (1) f t s ( ) f ( t -s ) ( ) =eq( t) Fantastic, we estimate d dt T S (1) f ( t) (1) = eg TS t ( ) and we predict: T S g (1) ( t) = ò ds G TS (1) s ( )g t -s ( )

55 PlaSim: Planet Simulator Sea-Ice thermodynamic Oceans: LSG, mixed layer, or climatol. SST Spectral Atmosphere moist primitive equations on levels Vegetations (Simba, V-code, Koeppen) Terrestrial Surface: five layer soil plus snow Key features portable fast open source parallel modular easy to use documented compatible Model Starter and Graphic User Interface

56 Step 1 We double instantaneously [CO 2 ] 360 ppm 720 ppm We look at response of the surface temperature T S We average over the N members of the ensemble This is how we probe the system 56

57 What we get CO 2 doubling N =

58 Linear Susceptibility (1) c TS ò dt w ( ) = exp[ (1) iwt]g TS t ( ) 58

59 Step 2 We increase the [CO 2 ] 360 ppm 720 ppm at 1% per year [CO 2 ] is doubled after τ 70 years We keep [CO 2 ] constant after that Note: radiative forcing is log[co 2 ] Our forcing amounts to a linear increase g τ (t) is a ramp function reaching 1 at τ We look at response of T S We average over the N ensemble members (1) We predict using T ( S gt t) = ò (1) dsg TS If linear response holds.. 59 s ( )g t t -s ( )

60 Climate Change Prediction - T S ECS= Â c TS { (1) ( 0) } = 2 dw ò Re[ T s (1) (w)] 60

61 Bibliography Lucarini V., Thermodynamic Efficiency and Entropy Production in the Climate System, Phys Rev. E 80, (2009) Lucarini, V., K. Fraedrich, and F. Lunkeit, Thermodynamic Analysis of Snowball Earth Hysteresis Experiment: Efficiency, Entropy Production, and Irreversibility. Q. J. R. Meterol. Soc., 136, 2-11 (2010) Lucarini, V., K. Fraedrich, and F. Ragone, New results on the thermodynamical properties of the climate system. J. Atmos. Sci., 68, (2011) Lucarini V., S. Sarno, A Statistical Mechanical Approach for the Computation of the Climatic Response to General Forcings. Nonlin. Processes Geophys., 18, 7-28 (2011) Lucarini, V., Stochastic perturbations to dynamical systems: a response theory approach. J Stat Phys. 146, (2012) Lucarini V., Modeling Complexity: the case of Climate Science, in Models, Simulations, and the Reduction of Complexity, Gähde U, Hartmann S, Wolf J. H., De Gruyter Eds., Hamburg (2013) Boschi R., S. Pascale, V. Lucarini: Bistability of the climate around the habitable zone: a thermodynamic investigation, Icarus 226, (2013) Lucarini V. and S. Pascale, Entropy Production and Coarse Graining of the Climate Fields in a General Circulation Model, Climate Dynamics DOI /s (2014)) Lucarini V., R. Blender, C. Herbert, F. Ragone, S. Pascale, J. Wouters, 61 Mathematical and Physical Ideas for Climate Science, Arxiv (2014)