1

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

Scales of Motions (Smagorinsky)

FLOW

ENERGY

3

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

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

Non-equilibrium in the Earth system

(Kleidon, 2011)

climate

Multiscale

7

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

8

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

9

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

10

Detailed BalancesKinetic energy budget

Potential Energy budget

Total Energy Budget

WDKPCdVK

),(2

WQdVP

HQ 21

HndSHdVE

ˆ

),( KPCW

WORK

DISSIPATION

FLUXES

11

Johnson’s idea (2000)

Partitioning the Domain

Better than it

seems!

QdVQdVWP

0Q 0Q

12

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

13

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

14

Carnot EfficiencyMean Value Theorem:

We have

Hot Cold reservoirs

Work:

Carnot Efficiency:

0

W

15

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?

16

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

17

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

18

But…Two ways to compute EP:

Direct vs Indirect

Material vs Radiative

All in units mW m-2K-1. Hyperdiffusion in the atmosphere and mixing in the ocean each contribute about 1 unit.

GCMs entropy budget

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

4-box model of entropy budget

21

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

Results on IPCC GCMs

Hor vs Vert EP in IPCC models

Warmer climate:Hor↓ Vert↑

Venus, Mars, Titan22

vertinS

horinS

L., Ragone, Fraedrich, 2011

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

MoSt – The Model Starter

26

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

27

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)

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

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*

Bringing it together…Parametric Analysis of Climate ChangeStructural Properties of the system (Boschi, L.,

Pascale 2012)Lower Manifold Upper Manifold

33

TS

CO2

S*

CO2

S*

Bringing it together…Parametric Analysis of Climate ChangeStructural Properties of the system (Boschi, L.,

Pascale 2012)Lower Manifold Upper Manifold

34

Smat

CO2

S*

CO2

S*

A 3D picture

35

36

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?

42

Temp vs EP

43

Temp vs Efficiency

44

Temp vs Transport

45

46

Phase Transition

Width bistability vs length year (L. et al. 2013)Fast rotating planets cannot be in Snowball Earth

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)

49

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)