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Melancholiastatesintheclimatesystem:exploringglobalinstabili5esandcri5cal
transi5onsValerioLucarini
UniversityofReading/UniversityofHamburg
Thanks:T.Bodai,R.Boschi,H.Dijkstra,B.Eckhardt,C.Grebogi,A.Gritsun,F.Lunkeit,E.OF,F.Ragone,A.Tantet,J.Yorke
LesHouchesAugust3rd2017
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Whatcomesnext• Reading,Aug-Sept2017
– CliMathNetConference(withAshwin,Bodai,Broecker,Fowler,Freitag,Kuna,Neves,ScoF,Shepherd,Williams)
• ICTP,May2018– AdvancedWorkshoponNonequilibriumSystemsinPhysics,Geosciences,andLifeSciences(withBouchet,Ruffo,Gallavo^,Gambassi)
• LesHouches,Feb-March2019– PhysicsandMathema`csofTurbulentFlowsatDifferentScales(withDubrulle,Faranda,Wouters,GoFwald)
• Inst.Poincare,Autumn2019– Themathema`csofclimateandtheenvironment(withGhil,Chekroun,Klein,LeTreut,Speich)
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Melancholia(2011,VonTrier)
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• CloseencounterbtwEarthandbrowndwarf–Tensiongrows
• Veryclosetothe`ppingpoint-Tensionpeaks
• UnavoidableImpact–Tensionisreleased
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Mo`va`ons:Basics• Understandinghowasystemrespondstoperturba`onsisacentralareaofresearchinnaturalandmathema`calsciences– Robustnessofthesystem?– Smoothresponse?– Cri`calTransi`ons?
• GroundbreakingworkbyKubo(1957)– Responsetheoryforsta`s`calphysicalsystems– Onlyfornear-equilibrium(canonicalensemble)– Mathema`callyandphysicallynon-rigorous,manycri`cisms– Acous`cs,Op`cs,etc.basedonKubo’sresults– Fluctua`on-Dissipa`onTheorem:dic`onarybetweenforcedandfreefluctua`ons
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FromSmoothResponse…• Ruelle(‘90s):rigorousresponsetheoryforsmoothobservablesofAxiomAsystems(eq.&noneq.!)– UsualFDTdoesnotapplyfornonequilibriumsystems:unstablevsstabledirec`onsintangentspace
– Cri`calTransi`onsaslossofsmoothnessinresponse– Theoryusefultoperformpredic`onsbuthardtoconstructresponseoperator!
• Liverani,Baladi,Dolgopyat,..(‘00s)– Responsetheoryderivedusingtransferoperatorapproach– Changeintheinvariantmeasurevschangeinobservables– BeyondAxiomAsystems
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…toCri`calTransi`ons• Catastrophetheory(’60s,Thom,Arnold):comprehensiveviewofbifurca`onsin“rela`velysimplesystems’
• Mul`stabilityincomplexsystems&hysteresis• Definingtheboundariesbetweenthebasins• Ashwinetal.2012:Parameter-,rate-,andnoise-induced`pping
• Freidlin-Wentzelltheory(’70s)basedonLargeDevia`onsTheory(‘60s):generallawsfornoise-inducedescapefrombasinsofaFrac`on
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WhyClimateisrelevant• Theclimateisanonequilibriumsystemwhoseevolu`onisdrivenbyinhomogeneousabsorp`onofsolarradia`on
• Theclimatefeaturesvariabilityofavastrangeofspa`alandtemporalscales
• Understandingtheclimateresponsetoperturba`onisgreatscien`ficchallenge– Anthropogenicclimatechange– Paleoclimate→Life– PlanetaryScience→Habitability
• Smoothvs.nonsmoothresponse– Climatechange,climatesurprises,climate`ppingpoints
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• Ver`cal
• Horizontal
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Nonequilibrium
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Anextremelynon-idealengine
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T1 T2
Qin Qout
dissipation
Q1 Q2
W
Irreversible heat transport
• Efficiency• Energytransforma`on• EntropyProduc`on
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Scales of Motions Smagorinsky/Stommel
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ClimateResponse
A. Smoothresponse–ResponseTheoryConstruc`ngthesensi`vityoftheclimateandthemeasureofthepullbackaFractor.Linkbetweenclimatevariabilityandclimatechange?
B. Highsensi`vity–Ruelle-PollicoFResonancesRoughdependenceonsystem’sparameters
C. Cri`calTransi`onsCrisisofthehigh-dimensionalaFractor
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A
J.Stat.Phys.166,1036-1064(2017)
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• Perturba`ontoAxiomA:• Changeinexpecta`onvalueofasmoothΦ:
• Linearterm:
• LinearGreen:• Linearsuscept:obeysKKrela`ons
Construc5ngthe5me-dependentmeasureviaRuelleResponseTheory
Φ1( ) t( ) = dσ∫ GΦ
(1) σ( )e t −σ( )GΦ(1) t( ) = ρ0 dx( )∫ Θ t( )X ⋅∇StΦ
χΦ(1) ω( ) = dt∫ exp iωt[ ]GΦ
(1) t( )
Φ1( ) ω( ) = χΦ
(1) ω( )e ω( )
!x = !F x, t( )= F x( )+εe t( )X x( )
Φ t( ) = Φ0+ ε k Φ
(k ) t( )k=1
∞
∑ PullbackaFractor
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ModelStarterand
GraphicUserInterface
SpectralAtmospheremoistprimi`veequa`ons
onσlevels
Sea-Icethermodynamic
TerrestrialSurface:fivelayersoilplussnow
Vegeta`ons(Simba,V-code,
Koeppen)
Oceans:LSG,mixedlayer,orclimatol.SST
PLASIM: An efficient Climate Model
Keyfeatures• portable• fast• opensource• parallel• modular• easytouse• documented• compa`ble
O(105)d.o.f.16
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• Observable:globallyaveragedTS• Forcing:increaseofCO2concentra`on• Linearresponse:• Weperformensembleexperiments
– Concentra`onatt=0• Fantas`c,wees`mate
• …andwepredict:
Step1
T S f
(1) t( ) = dσ∫ GTS(1) σ( ) f t −σ( )
ddt
T S f
(1) t( ) = εGTS(1) t( )
f t( ) = εΘ t( )
T S g
(1) t( ) = dσ∫ GTS(1) σ( )g t −σ( )
f t( )
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ClimateChangePredic5on-TS
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[CO2]360ppm→720ppmat1%peryear2Xayerτ≈70years,constantayerthat
T S gτ
(1) t( ) = dσ∫ GTS(1) σ( )gτ t −σ( )
Predic5on
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(Transient)ClimateSensi5vity
• ΔTattheendoftheramp(τ=70ys)• SmallerthanECS(4.1Kvs4.8K)• Wanttodefineiner`aatallvaluesofτ
– Instantaneousvsquasista`c19
TCR τ( ) = T S gτ
(1)τ( ) = dσ∫ GTS
(1) σ( )gτ τ −σ( )
= ECS −P fCO2
2 x χTs(1)(ω)1+ sinc(ωτ / 2)e−iωτ /2
2πiω−∞
+∞
∫ dω
ECS =ℜ χTS(1) 0( ){ }= 2
πdω∫ Re[ Ts
(1) (ω)] “EQUILIBRIUM”
“TRANSIENT”
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TSProjec5on
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Error
NonlinearProcess
Ice-Albedo
BeyondGlobalIndicators
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CommonSense• Forcedfluctua`onswillprojectonthefreeones
– FDTwillwork• …unlessyouareinlowdimensionand/orusecooked-upobservables&forcings
• Pastexperiments:some`mesFDTworks,some`mesitdoesnotwork.Whyso?
• ..Butsome`mescowsarenotreallyspherical...
Gritsun&Lucarini,2017,PhysicaD21
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A
PhysicaD349,62-76(2017)
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Simplemodelofthemid-la`tudes
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• Ψisstreamfunc`on,ΔΨvor`city• Rota`on,Orographicforcing,diffusion,fric`on• Externaldriving• Madetolooklikewinteratmosphere• Verychao`c(#(λj,>0)=28,231dof)
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ResponsetoForcings• Responseorographicforcingvsnaturalvariability• ResonanceinexplicablewithFDT
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• ResonancecomesfromagroupofUPOs• UPOsrarelyvisitedbysystembutresonant• Possibleparadigmforclima5csurprises
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EffectofOrographicforcing
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“Tippingelements”
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• “Highlysensi`ve”regionstoclimatechange,“irreversible”responsetoforcings
Lentonetal.2007
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HighSensi`vityvsTippingPoint
28StolenfromLenton,somewhere…
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Effec5vePoten5alplusnoise
Cold
Warm
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• Asimplemodeloftheform• Transi`ons:noiseac`ngontheeffec`vepoten`al• Prob≈exp[-2ΔV/ε2]• Timeseriesanalysisvsdynamics:Procedureisnotrobust
• Unless:EVT-Faranda,Lucarini,Manneville,Wouters2014
e.g.Lentonetal.2008
dY = −dV dY +αdW
ΔV
ΔV
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ClimateResponse
A. Smoothresponse–ResponseTheoryConstruc`ngthesensi`vityoftheclimateandthemeasureofthepullbackaFractor.Linkbetweenclimatevariabilityandclimatechange?
B. Highsensi`vity–Ruelle-PollicoFResonancesRoughdependenceonsystem’sparameters
C. Cri`calTransi`onsCrisisofthehigh-dimensionalaFractor
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WhathappensNearCri`calTransi`ontwoseparateprocesses:• Cri`calSlowingdown:thedecayofcorrela`onsbecomesslowerandslower– PropertyoftheaFractor– Thesystemhaslongermemory– Theresponsetoperturba`ondiverges– RadiusofexpansionofRuelle’stheory
• Convergenceofensembles:theaFractoraFractslessefficientlynearbytrajectories– PropertyofneighborhoodoftheaFractor– CannotbeflaggedbydynamicsontheaFractor
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ClimateResponse
A. Smoothresponse–ResponseTheoryConstruc`ngthesensi`vityoftheclimateandthemeasureofthepullbackaFractor.Linkbetweenclimatevariabilityandclimatechange?
B. Highsensi`vity–Ruelle-PollicoFResonancesRoughdependenceonsystem’sparameters
C. Cri`calTransi`onsCrisisofthehigh-dimensionalaFractor
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Snowball/SnowfreeTransi5ons
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• …andthereservedtransi`on
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HistoricalNote
• ThebistabilityoftheEarthsystemwasdiscoveredwhenstudyingthepossibleeffectsofthenuclearwinter
• Budyko,Sellersinthelate‘60realizedthataprolongednuclearwintermightleadtoaglobalglacia`on
• Ghil(1976)extendedtheanalysis• Peoplelaughedatthispossibility…butinearly’90spaleoevidencesemerged!– Bewarecri`caltransi`ons!
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Feedbacks(1)
• Radia`vefeedback• Warmerbodiesemitmore
• Coolerbodiesemitless
• Nega`vefeedback
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Feedbacks(2)• Ice-albedofeedback:• Awarmersurfacehaslesssnowcover
• Albedodecreases• Moreradia`onisabsorbed
• Temperatureanomalyisstrengthened
• Posi`vefeedback38
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0-DEnergybalance
Ice/AlbedoFeedback
• Energybalance:
• With:
39 100 150 200 250 300 350 4000.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
T (K )
α p T( )
α1, T < T1
α1 +T −T1
T2 −T1
α2 −α1( ), T2 < T < T1
α2 T > T2
"
#$$
%$$
C dTdt
= I −O
C dTdt
= 1−α p T( )( ) S4 − A+BT( )
I0
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Bifurca5ons
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Solar Irradiance
Surface
Tem
peratu
re
Bistability
TIPPINGPOINT
TIPPINGPOINT
STABLE
STABLE
UNSTABLE
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1DEnergyBalanceModel–Ghil‘76
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DiffusionparameterGreenhouseeffect
Ice-albedofeedback
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TemperatureProfiles
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Grebogi,O^,Yorke(1983):
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A
?
Fromthis..
…tothis!
?
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Clima5cEdgeStates
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CO2 S*
Boschietal.2013
DIFF
NonDIFF
EDGESTATES
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C Clim.Dyn.44,3361-3381(2015)
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C
Nonlinearity30,R32-R66(2017)
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SpectralAtmospheremoistprimi`veequa`ons
onσlevels
Sea-Icethermodynamic
TerrestrialSurface:fivelayersoilplussnow
Vegeta`ons(Simba,V-code,
Koeppen)
Ghil-SellersDiffusiveOceanModelwithAlbedo
PUMA-GS
• SimplifiedClimateModel
• Primi`veEqua`onsAtmosphere
• SimpleDiffusiveGhil‘76Ocean
• Slowandfastclimatevariability
• Posi`ve/nega`vefeedbacks
O(105)d.o.f.
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EdgeState:“rela5vea^ractor”onthebasinboundary
481
2
3
4
5
W
C
E
ixw,0,0
xc,0,0
"1
• AyerGOY1983:Eckhardtetal.,mul`stablefluids– Pipeflow,PlaneCoueFeFlowwithfixedpointvs(transient)turbulentregimeforsuitableRe
• Dynamicsonthebasinboundary
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TrackingtheEdgeState
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• Dynamicsofanorbitnearthethebasinboundary– First,itrelaxesVERYrapidlytowardstheedgestate;– Second,itdecidestowardswhichaFractoritshouldheadto;– Third,itreachesthefinaldes`na`on.
• Reitera`ngtheprocedureweenduponthe..“Melancholiastates”(ayerL.VonTrier’smovie)
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Acloserlookattheboundary• Pikovsky:“isthebasinboundarysmooth?”
– Itisfolded,indeedfractal.– Resultof1024integra`onsbetweentwotrajectoriesneartheboundary,0.5KdifferenceinTsurf
– Resultofdifferent`mescalesofinstabilityontheedgestatesvsacrossit
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Mul5pleSteadyStates
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CHAOTICMELANCHOLIASTATE
CHAOTICWARMCLIMATESTA
TE
STATIONARYCOLDCLIMATESTATE
SYMMETRYBREAK
SYMM.BREAK
CHAOTICTRANSIENTSTATE
3CLIMATES
3STABLESTATES
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Mul5stability
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ColdState MelancholiaState WarmState
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SymmetryBreakoftheMelancholiaState
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Conclusions• Climateasnonequilibriumsta`s`calmechanicalsystem• Beyondinvariantmeasure:pullbackaFractor• Responsetheoryforsmoothresponse
– Predictclimatechange,• Highsensi`vityandmixingrate
– Transferoperatorapproach• Mul`stabilityandTippingpoints
– MelancholiaState,gateforthetransi`ons• WecanconstructtheMelancholiastate,whichseparatesthewarm
fromthesnowballclimatestate,alsoinrealGCMs• Theedgestateisthegateallowingfornoise-inducedtransi`ons
betweenthetwobasinsofaFrac`on• Note:ProximitytoTippingPointscanbedetectedusingEVT(Faranda,
Lucarini,Manneville,2014)
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BewareCri`calTransi`ons!