1. Coarse-graining and Entropy Production in a Climate Model - Part 3- Valerio Lucarini Klimacampus, Meteorological Institute, University of Hamburg Department of Mathematics & Statistics, University of Reading Email: valerio.lucarini@uni-hamburg.devalerio.lucarini@uni-hamburg.de 1 Cambridge, 20/11/2013

2. Today Interplay between order and chaos in the climate system From space, mid-latitude cyclones look like Von Karman’s vortices But atmosphere is NOT 2D at all Dynamical/thermodynamical processes We’ll focus on entropy production and coarse graining: irreversibility at various scales! How to account for them? EP  dissipation  parametrizations 2

3. Scales of Motions (Smagorinsky)

4. Update 4 ff

5. 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 5

6. Comments None of the 3 approaches is fully satisfactory The atmosphere has indeed features, but organised motions are just part of the story Particles have a life-cycle: what generates them, what dissipates them? The waves we observe result from unstable processes – energy budgets are crucial Nonlinearities  turbulent mixing and dissipation Scaling properties are observed at larger scales: turbulence is macro-turbulence Baroclinic (3D) / barotropic (2D) interplay Lorenz Energy Cycle 6

7. Other peculiarities of the atmosphere Negative viscosity phenomena: Turbulent momentum fluxes can be directed towards the region of high momentum (Jets) Turbulent: fast, non-zonally symmetric (but far from microscopic, infinitesimal!) Discovered by V. Starr The atmosphere is not a “simple”, Onsager- like dissipative system – eddy viscosity? Such phenomena are due to long-range correlations (travelling waves!) … but how can this be sustained? 7

8. Disequilibrium in the Earth system (Kleidon, 2011) climate Disequilibrium Work Irreversibility Multiscale

9. Main Formulas for Climate Thermodynamics 9

10. 10 Entropy Production Contributions of dissipation plus heat transport: Note: If heat transport along T is strong, η is small If the transport along T is weak, α is small

11. But… Two ways to compute EP: Direct vs Indirect Material vs Radiative 11

12. T R A N S P O R T E N E R G Y T R A N S P O R T

13. Transport, Mixing Vertical Transport of Energy Convective adjustment Irreversible mixing Horizontal Transports Baroclinic adjustment Irreversible mixing 13 W C W C 2-box model(s)

14. 4-box model of entropy budget 14 1 3 4 2 Poleward transport Vertical transport Fluid Surface 2×2 box model

15. Results on IPCC GCMs Hor vs Vert EP in IPCC models Warmer climate: Hor↓Vert↑ Venus, Mars, Titan 15 L., Ragone, Fraedrich, 2011

16. A step backwards Let’s consider a quasi-equilibrium system where Onsager relations apply with A positive definite quadratic form EP is locally positive! Let’s use extensively the Parseval Theorem for 16

17. We obtain: 17 Where is the entropy produced at space scale k l and t scale ω m

18. Coarse Graining We perform a coarse graining on our data  cut-off for |ω m | > |ω MAX | and |k l |> |k MAX | Estimate of EP: And: the coarser the graining (lower values for the cut-off), the lower the value of the estimate of the EP Each scale contributes with a positive term Different scales tell about different processes 18

19. Local Madness 19 In the stratosphere eddy (high frequency/wavenumber ) heat fluxes are locally against the T gradient NEGATIVE EP How it possible? Work on the system Local vs Global? Scales?

20. Entropy Production in the CS It is NOT a diffusive system, as we have seen It is a non-equilibrium, multiphase system What are our expectations on the properties of ? Climate is VERY heterogeneous (horizontal vs vertical), and seasonally and daily forced Feedbacks, re-equilibration processes 20

21. Computing entropy production Direct Formula Indirect Formula We use high resolution fields 21

22. Coarse graining the fields We perform coarse graining with a given temporal kernel τ and spatial kernel v Coarse grained estimates 22

23. Temporal Coarse Graining 23 Time averaging kernel

24. Time and Space Coarse Graining 24 Averaging in time and along the pressure levels Direct Method Indirect Method 1 D COLUMN CLIMATOLOGY

25. EP – 1 d resolution – Indirect Method 25 We use the indirect formula Ver Av Hor Av 1 D COLUMN 2 D HOR FLOW 0 D

26. EP - 1d resolution – Direct Method 26 Spatial coarse graining does not kill EP! Ver Av Hor Av 1 D COLUMN 2 D HOR FLOW 0 D

27. Transport, Mixing – Indirect Formula Horizontal Transports Baroclinic adjustment Irreversible mixing 6 mWK -1 m -2 27 W C 2-box model

28. Transport, Mixing – Direct Formula Horizontal Transports Hydrological cycle Dissipation of KE 18 mWK -1 m -2 28 W C 2-box model

29. Comments Full Coarse graining Indirect formula gives 0: the system is reduced to a 0D body, which absorbs and emits radiation at a given temperature Direct formula gives min EP = S KE Concluding we get: 13 mWK -1 m -2  S KE 6 mWK -1 m -2  S hor 33 mWK -1 m -2  S ver 29

30. Comment The coarser the graining, the lower the estimated value of entropy production We lose information about transfer of energy across domains at different temperatures From indirect method: can separate vertical/ horizontal only transfer processes Losing info on the vertical structure is very bad removing all resolution gives 0 Direct Method: we retain the effect of KE dissipation also at lowest resolution 30

31. EP 1d res: Direct-Indirect Method 31 Difference depends on effective resolution Difference >0,  0 with full resolution N VER Hor Av

32. Interesting Let’s compare the effect of coarse graining Note that 32

33. Interpretation System is forced, fluctuates and dissipates The correlation btw radiative heating rates & temperature >0 at all scales Energy input  temperature increase The correlation btw material heating rates & temperature <0 at all scales T fluctuation  relaxation System driven by radiative forcing, T fluctuations dissipated by material fluxes 33

34. What we have learned Positive contribution to EP comes from all time and space scales Not so locally! Waves allow for this. 13 mWK -1 m -2  S KE 6 mWK -1 m -2  S hor 33 mWK -1 m -2  S ver This looks like being a general result It may be a way to generalize Onsager-like properties to far from equilibrium systems Should be used in other, simpler systems as well (Aquaplanet, Rayleigh-Benard) 34

35. Bibliography Lucarini V. and S. Pascale, Entropy Production and Coarse Graining of the Climate Fields in a General Circulation Model, submitted to Clim. Dyn. (2013) Boschi R., S. Pascale, V. Lucarini: Bistability of the climate around the habitable zone: a thermodynamic investigation, Icarus (2013) Johnson D.R., Entropy, the Lorenz Energy Cycle and Climate, 659-720 in General Circulation Model Development: Past, Present and Future, D.A. Randall Ed. (Academic Press, 2002) Kleidon, A., Lorenz, R.D. (Eds.) Non-equilibrium thermodynamics and the production of entropy: life, Earth, and beyond (Springer, 2005) Lucarini V., Thermodynamic Efficiency and Entropy Production in the Climate System, Phys Rev. E 80, 021118 (2009) 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) Lucarini, V., K. Fraedrich, and F. Ragone, 2011: New results on the thermodynamical properties of the climate system. J. Atmos. Sci., 68, 2438-2458 Saltzman B., Dynamic Paleoclimatology (Academic Press, 2002) 35