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MEP and planetary climates: insights from a two-box climate model containing atmospheric dynamics Tim Jupp 26 th August 2010
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For the gory detail: http://rstb.royalsocietypublishing.org/content/365/1545/1355
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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
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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
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Thermodynamic Entropy, S [J.K -1 ] energy added reversibly to body at temperature T : [macroscopic view]
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Entropy production, [W.K -1 ] rate of entropy production [W.K -1 ] flux“force”
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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 ?
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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
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0 = 0 Closed, equilibrium: example 2 nd law:Equilibrium state has maximum entropy, S
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cold sink hot source fluid temperature conduction Rayleigh-Benard convection Open, non-equilibrium: example
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cold sink hot source convection fluid temperature Rayleigh-Benard convection Open, non-equilibrium: example
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MEP?
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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
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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?
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MEP and climate: overviews Science, 2003 Nature, 2005
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Kleidon + Lorenz Jaynes Bedtime reading
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Earth as a producer of entropy
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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?
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Atmospheric Heat Engine (Mk 1) Physics: “hot air rises” vs. “surface friction”
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Atmospheric Heat Engine (Mk 2) Physics : “hot air rises” + “Coriolis” vs. “surface friction”
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Climate models invoking MEP LorenzJuppKleidon simplest model [no dynamics] simple model [minimal dynamics] numerical model [plausible dynamics]
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Simplest model (Lorenz, GRL, 2001) Model has no dynamics ! Solve system with equator-to-pole flux F (equivalently, diffusion D) as free parameter
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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
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LEB solution: Earth model equatorial temperature model polar temperature Diffusion (free parameter) “candidate steady states”
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…and Titan… model equatorial temperature model polar temperature model entropy production Diffusion (free parameter) observation “candidate steady states”
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…and Mars… model equatorial temperature model polar temperature model entropy production observation Diffusion (free parameter) “candidate steady states”
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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?
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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
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MEP gives right answer Surface friction (free parameter) [true value is 0.4] model entropy production “candidate steady states”
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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?
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Simple model including dynamics (Jupp + Cox, Proc Roy Soc B, 2010) Solve for flow U, with surface drag C D as free parameter
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Energy balance (schematic)
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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
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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)
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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”
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Solar system parameters
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Example solution: Earth N-S flow E-W flow angle E-W N-S speed “candidate steady states”
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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
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Example solution: Venus MEP states “candidate steady states” LEB state MAF state LEB state MAF state LEB state
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Example solution: Titan MEP states “candidate steady states” MAF state LEB state
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Example solution: Mars MEP states “candidate steady states” MAF state LEB state
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entropy production at MEP
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Plot planets in parameter space Rotation matters Dynamics affect MEP state
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LEB, MEP, MAF
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The dynamical constraint
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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
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My philosophy MEP can tell you when your model contains “just enough” physics
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