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
Jaynes
Prigogine
Dewar
Two “entropies”
maximum entropy state
MaxEnt
Minimum Entropy Production
Maximum Entropy Production
thermodynamic entropy S
information entropy SI
Two steady states equilibrium
closed
non-equilibrium
open
[gas]
[convection]
Thermodynamic Entropy, S [J.K-1]
[microscopic view]
1 macrostate, but
 microstates
S  kB ln 
entropy of
macrostate [J.K-1]
Boltzmann
constant [J.K-1]
# microstates
yielding macrostate
Thermodynamic Entropy, S [J.K-1]
[macroscopic view]
energy added reversibly
to body at temperature T:
E
S 
T
E
T
Entropy production, S [W.K-1]
T1
E
T2
 T  T 
E
1
1
2

S
  Q   dV
T1T2
T 
rate of entropy
production [W.K-1]
flux
“force”
Information (Shannon) Entropy, SI
system is in microstate i with probability pi
What is a sensible way to assign pi ?
Scatter “quanta” of probability over microstates, retain
distributions which satisfy constraints…..

pi

microstates i
Information (Shannon) Entropy, SI
pi
pi
pi
pi
i
i
i
 = # ways of obtaining distribution by throwing N quanta
1
S I  lim ln    pi ln pi
N  N
i
[Information entropy of distribution]
The MaxEnt distribution (greatest SI, given constraints) is a
logical way to assign probabilities to a set of microstates
i
Closed, equilibrium: example
S  0
  0
 = 0
2nd law:
Equilibrium state
has maximum entropy, S
Open, non-equilibrium: example
Rayleigh-Benard convection Ra  Rac  1760
cold sink
S  0
conduction
hot source
fluid temperature
Ra  T
Open, non-equilibrium: example
Rayleigh-Benard convection Ra  Rac  1760
cold sink
convection
S  0
S  0
S  0
hot source
fluid temperature
Open, non-equilibrium: example
MEP?

S
Ra c
Ra
(Min? / Max?)imum Entropy Production
Prigogine Minimum Entropy Production:
all steady states are local minima of S
S
system state (steady or non-steady)
Dewar
Maximum Entropy Production (MEP):
observed steady state maximises S
An ongoing challenge
The distribution of microstates
which maximises information
entropy
SI
?link?
The macroscopic steady state
in which the rate of
thermodynamic entropy
production is maximised

S
MEP and climate: overviews
Science, 2003
Nature, 2005
Bedtime reading
Jaynes
Kleidon + Lorenz
Earth as a producer of entropy
Usefulness of MEP
• MEP can suggest numerical value for (apparently) free
parameter(s) in models
S
best value?
free parameter
• MEP gives observed value => model is sufficient
• Otherwise: model needs more physics
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
simplest model
simple model
[no dynamics]
[minimal dynamics]
Lorenz
Jupp
numerical model
[plausible dynamics]
Kleidon
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)…
T 4  A  BT
Fep / 2
natural scale of temperatures
…Nondimensionalise, apply MEP
subject to
  4 f a tep
[entropy production]
1  f a  tep
[energy conservation]
Notation:
Fep  I 0  I1
natural scale of fluxes
Fep / B
Maximise
system driven by
blackbody (linearised)
“LEB solution”
f a  tep  1
2
  1
ep (subscript) – equator-to-pole difference
a (subscript) – atmosphere
sa (subscript) – surface-to-atmosphere difference
LEB solution: Earth
model
equatorial
temperature
model polar
temperature
Diffusion (free parameter)
“candidate steady
states”
…and Titan…
model
equatorial
temperature
observation
observation
model polar
temperature
model
entropy
production
Diffusion (free parameter)
“candidate steady
states”
…and Mars…
model
equatorial
temperature
observation
observation
model polar
temperature
model
entropy
production
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
model
entropy
production
Surface friction (free parameter)
[true value is 0.4]
“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, q with surface drag CD as free parameter
Energy balance (schematic)
5 governing equations
conservation of energy
surface-to-atmosphere flux
equator-to-pole flux
dynamics (quadratic
surface drag, pressure
gradient, Coriolis)
Fep  BTep  2 Fa
Fa  CD cUTsa
R 2 Fa  3RHcU cos q Tep  2Tsa 
R 2CDU 2 cos q


R 2CDU 2 sin q

 R 2 HU sin q
3RH Tep  2Tsa gH / T0
R 2 HU cos q
Steady state solutions obtained analytically
with surface drag CD treated as free parameter
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
 c gH 3 
x  12 

 BR 
H 
parameters
h  3 
R
1
w
12
where

 R

 gH
“advective
capacity of
atmosphere”
“thickness of
atmosphere”



3 3 
 0.218
3
“rotation rate”
“geometric
constant”
What happens – as a function of (x,h,w) for an arbitrary planet?
Solar system parameters
Example solution: Earth
angle
speed
E-W
N-S
E-W flow
“candidate steady
states”
N-S flow
Example solution: Earth
MEP states
LEB state
MAF state
“candidate steady
states”
Simple dynamics give same flux at MEP as
“no-dynamics” model of Lorenz [2001]
Example solution: Venus
MEP states
LEB state
MAF state
“candidate steady
states”
Example solution: Titan
MEP states
LEB state
MAF state
“candidate steady
states”
Example solution: Mars
MEP states
LEB state
MAF state
“candidate steady
states”
entropy production at MEP
Dynamics
affect MEP
state
Plot planets in parameter space
Rotation matters
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