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

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