Coarse-graining and Entropy Production in a Climate Model - Part 3-

1. Coarse-graining and Entropy Production in a Climate Model- Part 3- Valerio Lucarini Klimacampus, MeteorologicalInstitute, Universityof Hamburg Department of Mathematics & Statistics, University of Reading Email: valerio.lucarini@uni-hamburg.de 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

3. Scales of Motions (Smagorinsky)

4. Update • 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

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

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?

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

9. Main Formulas for Climate Thermodynamics

10. Entropy Production • Contributions of dissipation plus heattransport: • 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 vsRadiative

12. E NERGY TR A N S P O R T E NERGY T RANSPORT

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

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

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

17. We obtain: • Where is the entropy produced at space scale kl and t scale ωm

18. Coarse Graining • We perform a coarse graining on our data cut-off for |ωm| > |ωMAX| and |kl |> |kMAX| • 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

19. Local Madness • 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

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

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

23. Temporal Coarse Graining Time averaging kernel

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

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

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

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

28. Transport, Mixing – Direct Formula • Horizontal Transports • Hydrological cycle • Dissipation of KE • 18 mWK-1m-2 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 = SKE • Concluding we get: • 13 mWK-1m-2SKE • 6 mWK-1m-2Shor • 33 mWK-1m-2Sver

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

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

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

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

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-1m-2SKE • 6 mWK-1m-2Shor • 33 mWK-1m-2Sver • 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)

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) • BoschiR., 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 GruyterEds., 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)