1. MEP and planetary climates: insights from a two-box climate model containing atmospheric dynamics Tim Jupp 26 th August 2010

2. For the gory detail: http://rstb.royalsocietypublishing.org/content/365/1545/1355

3. Entropy – a terminological minefield Boltzmann/2 nd lawmaximum entropy state JaynesMaxEnt PrigogineMinimum Entropy Production DewarMaximum Entropy Production Two “entropies”thermodynamic entropy S information entropy S I Two steady statesequilibrium [gas] closed non-equilibrium [convection] open

4. Thermodynamic Entropy, S [J.K -1 ] # microstates yielding macrostate Boltzmann constant [J.K -1 ] entropy of macrostate [J.K -1 ] [microscopic view] 1 macrostate, but  microstates

5. Thermodynamic Entropy, S [J.K -1 ] energy added reversibly to body at temperature T : [macroscopic view]

6. Entropy production, [W.K -1 ] rate of entropy production [W.K -1 ] flux“force”

7. Information (Shannon) Entropy, S I system is in microstate i with probability p i Scatter “quanta” of probability over microstates, retain distributions which satisfy constraints….. pipi microstates i What is a sensible way to assign p i ?

8. Information (Shannon) Entropy, S I The MaxEnt distribution (greatest S I, given constraints) is a logical way to assign probabilities to a set of microstates [Information entropy of distribution] pipi i pipi i i pipi i pipi  = # ways of obtaining distribution by throwing N quanta

9.   0  = 0 Closed, equilibrium: example 2 nd law:Equilibrium state has maximum entropy, S

10. cold sink hot source fluid temperature conduction Rayleigh-Benard convection Open, non-equilibrium: example

11. cold sink hot source convection fluid temperature Rayleigh-Benard convection Open, non-equilibrium: example

12. MEP?

13. Maximum Entropy Production (MEP): observed steady state maximises (Min? / Max?)imum Entropy Production Dewar system state (steady or non-steady) Minimum Entropy Production: all steady states are local minima of Prigogine

14. An ongoing challenge The distribution of microstates which maximises information entropy The macroscopic steady state in which the rate of thermodynamic entropy production is maximised ?link?

15. MEP and climate: overviews Science, 2003 Nature, 2005

16. Kleidon + Lorenz Jaynes Bedtime reading

17. Earth as a producer of entropy

18. Usefulness of MEP MEP can suggest numerical value for (apparently) free parameter(s) in models MEP gives observed value => model is sufficient Otherwise: model needs more physics free parameter best value?

19. Atmospheric Heat Engine (Mk 1) Physics: “hot air rises” vs. “surface friction”

20. Atmospheric Heat Engine (Mk 2) Physics : “hot air rises” + “Coriolis” vs. “surface friction”

21. Climate models invoking MEP LorenzJuppKleidon simplest model [no dynamics] simple model [minimal dynamics] numerical model [plausible dynamics]

22. Simplest model (Lorenz, GRL, 2001) Model has no dynamics ! Solve system with equator-to-pole flux F (equivalently, diffusion D) as free parameter

23. Lorenz energy balance (LEB)… blackbody (linearised) natural scale of fluxes natural scale of temperatures Maximise [entropy production] [energy conservation] …Nondimensionalise, apply MEP subject to ep (subscript) – equator-to-pole difference a (subscript) – atmosphere sa (subscript) – surface-to-atmosphere difference Notation: “LEB solution” system driven by

24. LEB solution: Earth model equatorial temperature model polar temperature Diffusion (free parameter) “candidate steady states”

25. …and Titan… model equatorial temperature model polar temperature model entropy production Diffusion (free parameter) observation “candidate steady states”

26. …and Mars… model equatorial temperature model polar temperature model entropy production observation Diffusion (free parameter) “candidate steady states”

27. Simplest model: summary MEP gives observed fluxes in a model containing no dynamics Great! But why? …surely atmospheric dynamics matter? …surely planetary rotation rate matters?

28. Numerical model (Kleidon, GRL, 2006) credit: U. Hamburg Five levels, spatial resolution ~ 5°, resolves some spatial dynamics Solve system with von Karman parameter k as free parameter

29. MEP gives right answer Surface friction (free parameter) [true value is 0.4] model entropy production “candidate steady states”

30. Numerical model: summary MEP gives observed surface friction in a model containing a lot of dynamics Great! But why? …which model parameters are important? …how does the surface friction predicted by MEP change between planets?

31. Simple model including dynamics (Jupp + Cox, Proc Roy Soc B, 2010) Solve for flow U,  with surface drag C D as free parameter

32. Energy balance (schematic)

33. conservation of energy surface-to-atmosphere flux equator-to-pole flux dynamics (quadratic surface drag, pressure gradient, Coriolis) 5 governing equations Steady state solutions obtained analytically with surface drag C D treated as free parameter

34. Fixed parameters: incoming radiation, planetary radius, rotation rate… Vary free parameter: surface friction C D Steady state solution: surface temperature, atmospheric flux, wind Which steady-state solution maximises - entropy production?(MEP solution) - atmospheric flux?(MAF solution)

35. Nondimensionalisation: 3 parameters parameters “advective capacity of atmosphere” “thickness of atmosphere” “rotation rate” What happens – as a function of (  ) - for an arbitrary planet? where “geometric constant”

36. Solar system parameters

37. Example solution: Earth N-S flow E-W flow angle E-W N-S speed “candidate steady states”

38. Example solution: Earth MEP states Simple dynamics give same flux at MEP as “no-dynamics” model of Lorenz [2001] “candidate steady states” MAF state LEB state

39. Example solution: Venus MEP states “candidate steady states” LEB state MAF state LEB state MAF state LEB state

40. Example solution: Titan MEP states “candidate steady states” MAF state LEB state

41. Example solution: Mars MEP states “candidate steady states” MAF state LEB state

42. entropy production at MEP

43. Plot planets in parameter space Rotation matters Dynamics affect MEP state

44. LEB, MEP, MAF

45. The dynamical constraint

46. Summary - Insight to numerical result of Kleidon [2006] - Confirms “no dynamics” result of Lorenz [2001] as the limit of a dynamical model - Shows how MEP state is affected by dynamics / rotation

47. My philosophy MEP can tell you when your model contains “just enough” physics