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
MEP and planetary climates: insights from a two-box climate model containing atmospheric dynamics
Tim Jupp
26th August 2010
For the gory detail:
http://rstb.royalsocietypublishing.org/content/365/1545/1355
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
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
Thermodynamic Entropy, S [J.K-1]
E
T
ES
T
energy added reversiblyto body at temperature T:
[macroscopic view]
Entropy production, [W.K-1]
1T 2TE
dV
TTT
TTES
1
21
21 Q
rate of entropy production [W.K-1]
S
flux “force”
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 ?
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
0
= 0
Closed, equilibrium: example
2nd law: Equilibrium state has maximum entropy, S
0S
cold sink
hot source
fluid temperature
conduction
Rayleigh-Benard convection
0S
1760 cRaRa
TRa
Open, non-equilibrium: example
cold sink
hot source
convection
fluid temperature
0S
0S
0S
Rayleigh-Benard convection 1760 cRaRa
Open, non-equilibrium: example
S cRa
Ra
Open, non-equilibrium: example
MEP?
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
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?
MEP and climate: overviewsScience, 2003
Nature, 2005
Kleidon + LorenzJaynes
Bedtime reading
Earth as a producer of entropy
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
Atmospheric Heat Engine (Mk 1)
Physics: “hot air rises” vs. “surface friction”
Atmospheric Heat Engine (Mk 2)
Physics : “hot air rises” + “Coriolis” vs. “surface friction”
Climate models invoking MEP
Lorenz Jupp Kleidon
simplest model
[no dynamics]
simple model
[minimal dynamics]
numerical model
[plausible dynamics]
Simplest model (Lorenz, GRL, 2001)
Model has no dynamics !
Solve system with equator-to-pole flux F (equivalently, diffusion D) as free parameter
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
LEB solution: Earth
model equatorial
temperature
model polar temperature
Diffusion (free parameter) “candidate steady states”
…and Titan…
model equatorial
temperature
model polar temperature model
entropy production
Diffusion (free parameter)
observation
observation
“candidate steady states”
…and Mars…
model equatorial
temperature
model polar temperature
model entropy
production
observation
observation
Diffusion (free parameter)
“candidate steady states”
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?
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
MEP gives right answer
Surface friction (free parameter)
[true value is 0.4]
model entropy
production
“candidate steady states”
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?
Simple model including dynamics
(Jupp + Cox, Proc Roy Soc B, 2010)
Solve for flow U, with surface drag CD as free parameter
Energy balance (schematic)
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
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)
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”
Solar system parameters
Example solution: Earth
N-S flowE-W flow
angle
E-WN-S
speed
“candidate steady states”
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
Example solution: VenusMEP states
“candidate steady states”
LEB state
MAF state
LEB stateLEB state
MAF state
LEB state
Example solution: Titan MEP states
“candidate steady states”
MAF state
LEB state
Example solution: Mars MEP states
“candidate steady states”
MAF state
LEB state
entropy production at MEP
Plot planets in parameter space
Rotation matters
Dyn
amic
s af
fect
ME
P
stat
e
LEB, MEP, MAF
The dynamical constraint
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
My philosophy
MEP can tell you when your model contains “just enough” physics