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

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MEP and planetary climates: insights from a two-box climate model containing atmospheric dynamics Tim Jupp 26th August 2010 Usefulness of MEP MEP can suggest ... – PowerPoint PPT presentation

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


1
MEP and planetary climates insights from a
two-box climate model containing atmospheric
dynamics
Tim Jupp 26th August 2010
2
For the gory detail
http//rstb.royalsocietypublishing.org/content/36
5/1545/1355
3
Entropy a terminological minefield
Boltzmann/2nd law maximum entropy
state Jaynes MaxEnt Prigogine Minimum Entropy
Production Dewar Maximum Entropy Production
Two entropies thermodynamic entropy S
information entropy SI Two steady
states equilibrium gas closed non
-equilibrium convection open
4
Thermodynamic Entropy, S J.K-1
microscopic view
1 macrostate, but W microstates
microstates yielding macrostate
Boltzmann constant J.K-1
entropy of macrostate J.K-1
5
Thermodynamic Entropy, S J.K-1
macroscopic view
energy added reversibly to body at temperature T
6
Entropy production, W.K-1
rate of entropy production W.K-1
flux
force
7
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
8
Information (Shannon) Entropy, SI
pi
pi
i
i
W ways of obtaining distribution by throwing
N quanta
Information entropy of distribution
The MaxEnt distribution (greatest SI, given
constraints) is a logical way to assign
probabilities to a set of microstates
9
Closed, equilibrium example
?? ? 0
2nd law Equilibrium state has maximum
entropy, S
?? 0
10
Open, non-equilibrium example
Rayleigh-Benard convection
11
Open, non-equilibrium example
Rayleigh-Benard convection
12
Open, non-equilibrium example
MEP?
13
(Min? / Max?)imum Entropy Production
Prigogine
Minimum Entropy Production all steady states are
local minima of
system state (steady or non-steady)
Maximum Entropy Production (MEP) observed steady
state maximises
Dewar
14
An ongoing challenge
The distribution of microstates which maximises
information entropy
?link?
The macroscopic steady state in which the rate of
thermodynamic entropy production is maximised
15
MEP and climate overviews
Science, 2003
Nature, 2005
16
Bedtime reading
Kleidon Lorenz
17
Earth as a producer of entropy
18
Usefulness of MEP
  • MEP can suggest numerical value for (apparently)
    free parameter(s) in models
  • MEP gives observed value gt model is sufficient
  • Otherwise model needs more physics

free parameter
best value?
19
Atmospheric Heat Engine (Mk 1)
Physics hot air rises vs. surface friction
20
Atmospheric Heat Engine (Mk 2)
Physics hot air rises Coriolis vs.
surface friction
21
Climate models invoking MEP
simplest model no dynamics
simple model minimal dynamics
numerical model plausible dynamics
Lorenz
Jupp
Kleidon
22
Simplest model (Lorenz, GRL, 2001)
Model has no dynamics !
Solve system with equator-to-pole flux F
(equivalently, diffusion D) as free parameter
23
Lorenz energy balance (LEB)
blackbody (linearised)
natural scale of fluxes
natural scale of temperatures
Nondimensionalise, apply MEP
LEB solution
entropy production
Maximise
subject to
energy conservation
ep (subscript) equator-to-pole difference a
(subscript) atmosphere sa (subscript)
surface-to-atmosphere difference
Notation
24
LEB solution Earth
model equatorial temperature
model polar temperature
candidate steady states
Diffusion (free parameter)
25
and Titan
model equatorial temperature
observation
observation
model polar temperature
model entropy production
candidate steady states
Diffusion (free parameter)
26
and Mars
model equatorial temperature
observation
observation
model polar temperature
model entropy production
candidate steady states
Diffusion (free parameter)
27
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?

28
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
29
MEP gives right answer
model entropy production
Surface friction (free parameter) true value is
0.4
candidate steady states
30
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?

31
Simple model including dynamics (Jupp Cox,
Proc Roy Soc B, 2010)
Solve for flow U, q with surface drag CD as free
parameter
32
Energy balance (schematic)
33
5 governing equations
conservation of energy
surface-to-atmosphere flux
equator-to-pole flux
dynamics (quadratic surface drag, pressure
gradient, Coriolis)
Steady state solutions obtained analytically with
surface drag CD treated as free parameter
34
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)

35
Nondimensionalisation 3 parameters
advective capacity of atmosphere
thickness of atmosphere
parameters
rotation rate
geometric constant
where
What happens as a function of (x,h,w) - for an
arbitrary planet?
36
Solar system parameters
37
Example solution Earth
angle
speed
E-W
N-S
candidate steady states
E-W flow
N-S flow
38
Example solution Earth
MEP states
LEB state
LEB state
MAF state
candidate steady states
Simple dynamics give same flux at MEP as
no-dynamics model of Lorenz 2001
39
Example solution Venus
MEP states
LEB state
LEB state
LEB state
LEB state
MAF state
MAF state
candidate steady states
40
Example solution Titan
MEP states
LEB state
MAF state
candidate steady states
41
Example solution Mars
MEP states
LEB state
MAF state
candidate steady states
42
entropy production at MEP
43
Plot planets in parameter space
Dynamics affect MEP state
Rotation matters
44
LEB, MEP, MAF
45
The dynamical constraint
46
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

47
My philosophy
MEP can tell you when your model contains just
enough physics
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