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MEP and planetary climates: insights from a two-box climate model containing atmospheric dynamics

This article explores the Maximum Entropy Production (MEP) principle and its application to climate modeling, specifically in understanding planetary climates. It discusses how MEP can be used to determine numerical values for model parameters and predict observed fluxes, such as surface friction. The article also summarizes simple and numerical models that incorporate atmospheric dynamics and surface drag as free parameters.

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MEP and planetary climates: insights from a two-box climate model containing atmospheric dynamics

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

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

  3. Entropy – a terminological minefield Boltzmann/2nd law maximum entropy state Jaynes MaxEnt Prigogine Minimum Entropy Production Dewar Maximum Entropy Production Two “entropies” thermodynamic entropyS information entropySI Two steady states equilibrium [gas] closed non-equilibrium [convection] open

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

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

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

  7. Information (Shannon) Entropy, SI system is in microstateiwith probability pi What is a sensible way to assign pi ? Scatter “quanta” of probability over microstates, retain distributions which satisfy constraints….. pi microstates i

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

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

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

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

  12. Open, non-equilibrium: example MEP?

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

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

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

  16. Jaynes Bedtime reading Kleidon + Lorenz

  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 simplest model [no dynamics] simple model [minimal dynamics] numerical model [plausible dynamics] Lorenz Jupp Kleidon

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

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

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

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

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

  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 model entropy production Surface friction (free parameter) [true value is 0.4] “candidate steady states”

  30. 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? Numerical model: summary

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

  32. Energy balance (schematic)

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

  34. Fixed parameters: incoming radiation, planetary radius, rotation rate… Vary free parameter: surface friction CD 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 “advective capacity of atmosphere” “thickness of atmosphere” parameters “rotation rate” “geometric constant” where What happens – as a function of (x,h,w) - for an arbitrary planet?

  36. Solar system parameters

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

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

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

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

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

  42. entropy production at MEP

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

  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

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