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Thermodynamics of Climate – Part 2 –
Efficiency, Irreversibility, Tipping Points
Valerio Lucarini
Meteorological Institute, University of Hamburg
Dept. Mathematics and Statistics, University of Reading
valerio.lucarini@uni-hamburg,de
Cambridge, 12/11/2013
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Scales of Motions (Smagorinsky)
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FLOW
ENERGY
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Energy & GW – Perfect GCM
NESS→Transient → NESS Applies to the whole climate and to to all climatic subdomains for atmosphere τ is small, always quasi-equilibrated
Forcing
τ
Total warming
4
L. and Ragone, 2011
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Energy and GW – Actual GCMs
Not only bias: bias control ≠ bias final state Bias depends on climate state! Dissipation
Forcing τ
5
L. and Ragone, 2011
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Non-equilibrium in the Earth system
(Kleidon, 2011)
climate
Multiscale
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Looking for the big pictureGlobal structural properties (Saltzman 2002).Deterministic & stochastic dynamical systemsExample: stability of the thermohaline circulation Stochastic forcing: ad hoc “closure theory” for noise
Stat Mech & Thermodynamic perspectivePlanets are non-equilibrium thermodynamical systemsThermodynamics: large scale properties of the climate
system; definition of robust metrics for GCMs, dataStat Mech for Climate response to perturbations
7EQ NON EQ
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Thermodynamics of the CSThe CS generates entropy (irreversibility),
produces kinetic energy with efficiency η (engine), and keeps a steady state by balancing fluxes with surroundings (Ozawa et al., 2003)
Fluid motions result from mechanical work, and re-equilibrate the energy balance.
We have a unifying picture connecting the Energy cycle to the MEPP (L. 2009);
This approach helps for understanding many processes (L et al., 2010; Boschi et al. 2013):Understanding mechanisms for climate transitions;Defining generalised sensitivities Proposing parameterisations
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Energy BudgetTotal energy of the climatic system:
ρ is the local densitye is the total energy per unit mass u, and k indicate the internal, potential
and kinetic energy components
Energy budget
KPkudVedVEkinetic
potentialstaticmoist
KPE
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Detailed BalancesKinetic energy budget
Potential Energy budget
Total Energy Budget
WDKPCdVK
),(2
WQdVP
HQ 21
HndSHdVE
ˆ
),( KPCW
WORK
DISSIPATION
FLUXES
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Johnson’s idea (2000)
Partitioning the Domain
Better than it
seems!
QdVQdVWP
0Q 0Q
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Long-Term averages
Stationarity:
Work = Dissipation
Work = Input-Output
A different view on Lorenz Energy cycle
0 KPE
0
0
WWP
DWWK
0
)( ),( )(
KDndissipatio
KACconversion
AGheatingaldifferenti
DW
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EntropyMixing neglected (small on global scale), LTE:Entropy Balance of the system:
Long Term average:
Note: if the system is stationary, its entropy does not grow balance between generation and boundary fluxes
sdVsdVT
QdV
T
QdVS
TsQ
00 S
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Carnot EfficiencyMean Value Theorem:
We have
Hot Cold reservoirs
Work:
Carnot Efficiency:
0
W
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Bounds on Entropy ProductionMinimal Entropy Production (Landau):
Efficiency:entropy production entropy fluctuations
Min entropy production is due to dissipation:
2min
W
TdV
QdV
SSin
TdVS
2
min
and the rest?
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Entropy ProductionContributions of dissipation plus heat
transport:
We can quantify the “excess” of entropy production, degree of irreversibility with α:
Heat Transport down the T gradient increases irreversibility
min
2 11S
THdV
TdV
THdVSin
11
min
BeST
HdV
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MEPP re-examinedLet’s look again at the Entropy production:
If heat transport down the temperature is strong, η is small
If the transport is weak, α is small.
MEPP: joint optimization of heat transport
and of the production of mechanical work
11minSS in
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But…Two ways to compute EP:
Direct vs Indirect
Material vs Radiative
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All in units mW m-2K-1. Hyperdiffusion in the atmosphere and mixing in the ocean each contribute about 1 unit.
GCMs entropy budget
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A 2D formula
EP from 2D radiative fields onlySeparation between effect of vertical
(convection)/horizontal processes (large scale heat transport)
Lower bounds on Lorenz Energy CycleHigh precision, very low computational
cost; planetary systems? 20
horin
vertin S
S E
TOAnet
S
S SE
surfnetin T
RdTT
RdS111
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4-box model of entropy budget
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1
3 4
2
Poleward transport
Vertical transport
2-box × 2-box
Fluid
Surface
EP & Co from 2D radiative fields onlyHigh precision, very low res needed
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Results on IPCC GCMs
Hor vs Vert EP in IPCC models
Warmer climate:Hor↓ Vert↑
Venus, Mars, Titan22
vertinS
horinS
L., Ragone, Fraedrich, 2011
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Model Starter and
Graphic User Interface
Spectral Atmospheremoist primitive equations
on levels
Sea-Icethermodynamic
Terrestrial Surface: five layer soil
plus snow
Vegetations(Simba, V-code,
Koeppen)
Oceans:LSG, mixed layer,or climatol. SST
PlaSim: Planet Simulator
Key features• portable• fast• open source• parallel• modular• easy to use• documented• compatible
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MoSt – The Model Starter
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Snowball HysteresisSwing of S* by ±10% starting from present climate hysteresis experiment!
Global average surface temperature TS
Wide (~ 10%) range of S* bistable regime -TS ~ 50 Kd TS/d S* >0 everywhere, almost linear
SB
W
L., Lunkeit, Fraedrich, 2010
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Thermodynamic Efficiencyd η /d S* >0 in SB regimeLarge T gradient due to large albedo gradient
d η /d S* <0 in W regimeSystem thermalized by efficient LH fluxes
η decreases at transitions System more stableSimilar behaviour for total Dissipation
η=0.04Δθ=10K
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Entropy Productiond Sin/d S* >0 in SB & W regimeEntropy production is “like” TS… but better than TS!
Sin is about 400% benchmark for SB vs W regime
Sin is an excellent state variable
System MUCH more irreversible in W state (Bejan)
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Generalized Sensitivities
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EnergyCycle
Efficiency
EP Irreversibility
002.0 2 06.0 WmW
-1-2K Wm0.0004inS
7.0
CO2 concentration ranging from 50 to 1750 ppm no bistability!
L., Lunkeit, Fraedrich, 2010
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d)
100 ppm CO2
1000 ppm CO2
Heating Patterns
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1000-100 ppm Differences KE @ Surface
Temperature LH Heating
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Bringing it together…Parametric Analysis of Climate ChangeStructural Properties of the system (Boschi, L.,
Pascale 2012)Lower Manifold Upper Manifold
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η
CO2
S*
CO2
S*
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Bringing it together…Parametric Analysis of Climate ChangeStructural Properties of the system (Boschi, L.,
Pascale 2012)Lower Manifold Upper Manifold
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TS
CO2
S*
CO2
S*
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Bringing it together…Parametric Analysis of Climate ChangeStructural Properties of the system (Boschi, L.,
Pascale 2012)Lower Manifold Upper Manifold
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Smat
CO2
S*
CO2
S*
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A 3D picture
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Is there a common framework?Going from a 1D to a 2D parameter
exploration we gain completeness, we lose focus
Necessarily so?Can find an overall equivalence between the
atmospheric opacity and incoming radiation perturbations
Concept of radiative forcing…If so, we gain some sort of universality
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ParametrizationsEP vs Emission Temperature
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ParametrizationsDissipation vs Emission Temperature
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ParametrizationsEfficiency vs Emission Temperature
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ParametrizationsHeat Transport vs Emission Temperature
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Now we change the LOD
Will we recover similar relations?
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Temp vs EP
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Temp vs Efficiency
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Temp vs Transport
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Phase Transition
Width bistability vs length year (L. et al. 2013)Fast rotating planets cannot be in Snowball Earth
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Planetary AtmospheresVast range of planetary atmospheresRotation rate - Orbital Phase lock Atmospheric opacities/ incoming radiation
Thermodynamics habitable super-Earths?
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ConclusionsUnifying picture connecting Energy cycle to EP;Snow Ball hysteresis experiment Mechanisms involved in climate transitions;
Analysis of the impact of [CO2] increaseGeneralized set of climate sensitivities
Simplified 2D formula for studying GCMsMany challenges ahead:Analysis of GCMs performanceThermodynamics of celestial bodiesBasic macroscopic thermodynamics
Next:Entropy Production & Coarse Graining (20/11)Tipping Points & EVT (25/11)
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Bibliography Boschi R., S. Pascale, V. Lucarini: Bistability of the climate around the
habitable zone: a thermodynamic investigation, Icarus (2013) Held, I.M., The Gap between Simulation and Understanding in Climate
Modeling. Bull. Amer. Meteor. Soc., 86, 1609–1614 (2005) Johnson D.R., Entropy, the Lorenz Energy Cycle and Climate, 659-720
in General Circulation Model Development: Past, Present and Future, D.A. Randall Ed. (Academic Press, 2002)
Kleidon, A., Lorenz, R.D. (Eds.) Non-equilibrium thermodynamics and the production of entropy: life, Earth, and beyond (Springer, 2005)
Lucarini V., Thermodynamic Efficiency and Entropy Production in the Climate System, Phys Rev. E 80, 021118 (2009)
Lucarini, V., K. Fraedrich, and F. Ragone, 2011: New results on the thermodynamical properties of the climate system. J. Atmos. Sci., 68, 2438-2458
Lucarini V., Blender R., Herbert C., Pascale S., Wouters, J., Mathematical and Physical Ideas for Climate Science, arXiv:1311.1190 [physics.ao-ph] (2013)
Lucarini V. S. Pascale, Entropy Production and Coarse Graining of the Climate Fields in a General Circulation Model, sub Clim. Dyn. (2013)
Saltzman B., Dynamic Paleoclimatology (Academic Press, 2002)