mep and planetary climates: insights from a two-box climate model containing atmospheric dynamics
DESCRIPTION
MEP and planetary climates: insights from a two-box climate model containing atmospheric dynamics. Tim Jupp 26 th August 2010. For the gory detail:. http://rstb.royalsocietypublishing.org/content/365/1545/1355. Entropy – a terminological minefield. Boltzmann/2 nd lawmaximum entropy state - PowerPoint PPT PresentationTRANSCRIPT
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MEP and planetary climates: insights from a two-box climate model containing atmospheric dynamics
Tim Jupp
26th 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/2nd law maximum entropy stateJaynes MaxEntPrigogine Minimum Entropy ProductionDewar Maximum Entropy Production
Two “entropies” thermodynamic entropy S information entropy SI
Two steady states equilibrium [gas]closed
non-equilibrium [convection]open
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Thermodynamic Entropy, S [J.K-1]
lnBkS# microstates
yielding macrostateBoltzmann
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]
E
T
ES
T
energy added reversiblyto body at temperature T:
[macroscopic view]
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Entropy production, [W.K-1]
1T 2TE
dV
TTT
TTES
1
21
21 Q
rate of entropy production [W.K-1]
S
flux “force”
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Information (Shannon) Entropy, SI
system is in microstate i with probability pi
Scatter “quanta” of probability over microstates, retain distributions which satisfy constraints…..
pi
microstates i
What is a sensible way to assign pi ?
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Information (Shannon) Entropy, SI
The MaxEnt distribution (greatest SI, given constraints) is a logical way to assign probabilities to a set of microstates
iii
NI pp
NS lnln
1lim
[Information entropy of distribution]
pi
i
pi
ii
pi
i
pi
= # ways of obtaining distribution by throwing N quanta
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0
= 0
Closed, equilibrium: example
2nd law: Equilibrium state has maximum entropy, S
0S
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cold sink
hot source
fluid temperature
conduction
Rayleigh-Benard convection
0S
1760 cRaRa
TRa
Open, non-equilibrium: example
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cold sink
hot source
convection
fluid temperature
0S
0S
0S
Rayleigh-Benard convection 1760 cRaRa
Open, non-equilibrium: example
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S cRa
Ra
Open, non-equilibrium: example
MEP?
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Maximum Entropy Production (MEP): observed steady state maximises
(Min? / Max?)imum Entropy Production
S
SS
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
SThe macroscopic steady state in which the rate of thermodynamic entropy production is maximised
IS?link?
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MEP and climate: overviewsScience, 2003
Nature, 2005
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Kleidon + LorenzJaynes
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 parameterbest value?
S
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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
Lorenz Jupp Kleidon
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)…
BTAT 4
epatf4
2/epF
BFep /
epa tf 1
blackbody (linearised)
natural scale of fluxes
natural scale of temperatures
Maximise [entropy production]
[energy conservation]
…Nondimensionalise, apply MEP
21 epa tf
1subject to
ep (subscript) – equator-to-pole differencea (subscript) – atmospheresa (subscript) – surface-to-atmosphere difference
Notation:
“LEB solution”
10 IIFep 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
observation
“candidate steady states”
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…and Mars…
model equatorial
temperature
model polar temperature
model entropy
production
observation
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 CD 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 CD treated as free parameter
aepep FBTF 2
saDa cUTCF
saepa TTcURHFR 2cos32
cossin
/23
sincos
2220
222
UHRUCR
TgHTTRH
UHRUCR
D
saep
D
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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)
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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?
BR
gHc 3
12
R
H3
gH
R
12
1
218.03
33
where
“geometric constant”
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Solar system parameters
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Example solution: Earth
N-S flowE-W flow
angle
E-WN-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 stateLEB state
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Example solution: VenusMEP states
“candidate steady states”
LEB state
MAF state
LEB stateLEB 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
Dyn
amic
s af
fect
ME
P
stat
e
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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