Summer school on
Dynamics, Stochastics and Predictability of the Climate System
Valsavarenche, 9-18 June 2014
Research Seminar
The Maximum Entropy Production
Hypothesis
Didier PAILLARD
LSCE - IPSL
Laboratoire des Sciences du Climat et de l Environnement
Institut Pierre-Simon Laplace
What is climate ?
Etymology:
κλιµα = tilt (height of the Sun above the horizon)
Dictionnary:
The « averaged » weather, over a period of 30 years. Quantities considered are
surface values (temperatures, precipitations, winds, …).
What you can predict:
« Climate is what you expect, weather is what you get » (R. Heinlein)
A priori
A posteriori
(moving average)
100 random numbers between 0 and 1, then 100 numbers between 0,2 and 1,2.
The « a priori » average (red line), jumps by 0,2 units instantaneously in
contrast to the moving average (black line).
Carbon cycle
Ice sheets
Tectonics
Deep Ocean
Vegetation
Surface Ocean
Atm chemistry
Atm. dynamics
10-7
10-6
101
Climate
102
103
Weather
High latitude records
Low latitude records
(Huybers and Curry, 2006)
What is complexity ?
complexity
order
“dynamical systems”
dx
= F(x,t)
dt
!
disorder
“thermodynamics”
S = "$ # Log(# )
Paltridge s model
Paltridge, QJRMS, 1975, 1978
Unknown (turbulent) flux with divergence Dj
Stationarity: local energy conservation:
dTi/dt = 0 = -Di + Ri(Ti)
Find a variational problem to compute a solution.
Cost function:
F( Di,Ti ) = ∑ g( Di,Ti ) Unique solution:
Min (∑g(Ri(Ti),Ti ) - β ∑Ri(Tk) )
Seems to work well for g( Di,Ti ) = Di/Ti :
Paltridge s model
Entropy production:
dS/dt = ∑ (-Di)/Ti
Paltridge, QJRMS, 1975, 1978
Unknown (turbulent) flux with divergence Dj
Stationarity: local energy conservation:
dTi/dt = 0 = -Di + Ri(Ti)
Find a variational problem to compute a solution.
Cost function:
F( Di,Ti ) = ∑ g( Di,Ti ) Unique solution:
Min (∑g(Ri(Ti),Ti ) - β ∑Ri(Tk) )
Seems to work well for g( Di,Ti ) = Di/Ti :
Paltridge s model
Critics on the model
-  rather empirical radiative formulation
(eg. cloud representation)
-  vertical fluxes are obeing a different principle (maximizing
the energy flux)
Critics on the physics
-  no effect of Earth s rotation (and no role for dynamics)
-  no effect of fluid properties (air, water, or wathever…)
-  no theoretical justification
Why is there a maximum ?
R1(T1)
Radiative budget:
Ri (Ti ) = Si (1" # i ) " $Ti 4
R2(T2)
Unknown flux: Q
Q
Energy balance:
!
R1 (T1 ) + R2 (T2 ) = 0
R1 (T1 ) = Q
Entropy production as a function of Q:
#
#1 1&
dS
1
1 &
= Q% " ( = Q% "1
" "1
(
dt
T
T
$ 2 1'
$ R2 ("Q) R1 (Q) '
!
0.06
0.04
0.02
0.1
- 0.02
0.2
0.3
0.4
0.5
Other planets
D = Q/∆T = “diffusivity”
Lorenz, GRL, 2001
D = Q/∆T = “diffusivity”
D ~ ρCPHv/R
D ~ (P/P0)(Cp/Cp0)(m0/m)2(Ω0/Ω)2#
Revised version of Paltridge s idea
MEP
Input:
-  surface albedo
- radiative code
MEP
IPSL
Herbert et al, QJRMS, 2011
MEP-IPSL
With the cloud effect from
the IPSL model:
- better match
Sensitivity to surface albedo changes (LGM)
MEP -2°C
Herbert et al, QJRMS, 2011
IPSL -2,5°C
From divergences to transport
MEP => div(Q) = D
Q = grad( ϕ ) + rot( ξ )
Atm
Océan
Océan
Herbert, PhD thesis, 2012
Indien
Atlantique
Pacifique
From divergences to transport
MEP => div(Q) = D
Q = grad( ϕ ) + rot( ξ )
Atm
Océan
Océan
Indien
Atlantique
Pacifique
(Trenberth & Caron, 2001)
Herbert, PhD thesis, 2012
On the vertical, grey atmosphere
Ozawa & Ohmura, 1997
Herbert et al, 2013
On the vertical, more realistic radiative code
Herbert et al, 2013
Sensitivity to 2xCO2
Herbert et al, 2013
A way to “tune” GCMs ?
Surface heat and momentum
fluxes:
FM = CM |u| u
FH = CH |u| (TA-TS)
with
CM,H = (k/Log(z/z0
))2
fM,H(Ri, z/z0)
Global entropy production
(mW m-2 s-1)
(idealized / simple GCM)
total
vertical
horizontal
k = 0.4
von Karman parameter, k
Kleidon et al. (2006) Geophys. Res. Lett. 33: L06706
BUT not so clear that it could work on the vertical
Pascale et al. Climate Dynamics, 2011
Other examples
from physics
Rayleigh-Bénard convection
g"#Td 3
Ra =
$%
Ra* = 1708
Linear regime:
F=
k"T
d
Turbulent regime:
F is maximum (Malkus 1954)
3
1/3
!
g"#T (2$ )
k"T # Ra &
Ra* =
F=
%&
Ozawa, RoG, 2003
%
(
d $ Ra *'
The “logarithmic region” in the Plane Poiseuille Flow
Maximum dissipation subject to:
1) Momentum balance at z
2) Global power balance
+
Two additional dynamic
stability criteria:
3) U ""(z )< 0 !z
U (z )
Uô
max D m
4) Boundary layer is marginally
stable
channel
centre
obs.
log(1 + z )Rô
Malkus WR. J Fluid Mech 489:185-198 (2003)
Endoreversible engines
Reversible engine
With explicit dissipation
W $ TC '
" = = &1# )
Q % TW (
Carnot efficiency :
Chambadal-Novikov-Curzon-Ahlborn efficiency :
TW
P $ T2 '
"=
= &1# )
FQ % T1 (
!
FQ = g1 (TW # T1 )
FQ
FQ # P = g2 (T2 # TC )
P
" g1g2 % TW (1) () ) TC
P=$
'(
1) (
! # g1 + g2 &
T
" MAX = 1# C
TW
FQ-P
TC
!
Power Plant!
#
#
West Thurrock (UK) coal-fired power plant
CANDU (Canada) nuclear power plant
Larderello (Italy) geothermal power plant
!
#Tc (°C)
#25
#25
#80
#Th (°C)
#565
#300
#250
#ηCARNOT
#0.64
#0.48
#0.33
#ηMAX
#0.40
#0.28
#0.178
#ηOBS#
#0.36#
#0.30#
#0.16#
The Earth as a heat engine …
The Earth as a collection of endoreversible
engines.
Radiative equilibrium:
∑ Qi = 0
Maximum power hypothesis:
max of W = ∑ Wi
The resulting total efficiency W/(Solar input) is about right at 1%
(de Vos et al., 1993)
Is there any justification ?…(wishful thinking)
Statistical physics
-  hypothesis on the phase space
(ergodicity, ..)
-  global constraints
(E conservation, …)
S1 = " % #(x)Log(#(x))
x$states
At thermodynamical equilibrium:
S1 = thermodynamical entropy!
(= Q/T)
For instance:
PV = nkT
d
U = nkT
2
Maximum entropy production
-  hypothesis on trajectories ?
-  global constraints
(Flux conservation, …)
S2 = " & #($)Log(#($))
$%trajectories
At stationary states (??):
S2 ~? thermo. entropy production
(= FQ/T)
Mihelich et al. 2014
KS-entropy and thermodynamical entropy seem to reach
their maximum for the same parameter value
σthermo
hKS
Mihelich et al. 2014
Mihelich et al. 2014
The “zero range process”
ni p
1
2
ni (1-p)
…
L
σthermo
hKS
s
+ o(s)
4
s
3s
= "
+ o(s)
1
4 2L( "1)
#
Mihelich et al., to be submitted
f max EP =
f max KS
!
Strong debates around MEP
Heavily polarized between two points of view:
1/ we know the laws of dynamics, we don’t need additional laws
-  consequence of “fundamental laws” ? (probably not !)
2/ this is THE law of nature (from physics to sociology…!!)
-  very interesting (and sometimes dangerous) speculations…
Strong debates around MEP
Heavily polarized between two points of view:
1/ we know the laws of dynamics, we don’t need additional laws
-  consequence of “fundamental laws” ? (probably not !)
2/ this is THE law of nature (from physics to sociology…!!)
-  very interesting (and sometimes dangerous) speculations…
We need to develop a more pragmatic approach:
3/ Is it a useful and reasonable approximation (in some specific cases) ?
(note that this is more or less the status of the second “law” right now !!)
Kleidon, 2012
Thank you …