The Fractional Energy Balance Equation ShaunLovejoy1,RomanProcyk1,RaphaelHébert1,2,LenindelRioAmador1

1Physics,McGillUniversity

3600Universityst.Montreal,Que.H3A2T8

Canada

2Alfred-Wegener-Institut,HelmholtzCentreforPolarandMarineResearchTelegrafenbergA45-2N104D-14473,Potsdam

Germany

Abstract

ClassicalEnergyBalanceEquations(EBEs)aredifferentialequationsofintegerorder(h=1),herewegeneralizethistofractionalorders:theFractionalEBE(FEBE,0<h≤1). In theFEBE,whentheEarth isperturbedbya forcing, the temperaturerelaxestoequilibriumviaaslowpowerlawprocess: h=1istheexceptional(butstandard) exponential case. Our FEBE derivation is phenomenological, itcomplementsderivationsbasedontheclassicalcontinuummechanicsheatequation(thatimplyh=1/2forthesurfacetemperature)andofthemoregeneralFractionalHeatEquationwhichallows for0<h<2. Unlikesomeof theearlier “scale free”modelsbasedpurelyonscaling,theFEBEhasanextrablackbodyradiationtermthatallowsforenergybalance.Itthereforehastwoscalingregimes(notone),ithastheadvantageofbeing stable to infinitesimal step functionperturbationsand ithasafiniteEquilibriumClimateSensitivity.

WesolvetheFEBEusingGreen’sfunctions,whosehighandlowfrequencylimitsarepowerlawswitharelaxationscaletransition(severalyears).Whenstochasticallyforced, the high frequency part of the internal variability are fractional Gaussiannoisesthatcanbeusedformonthlyandseasonalforecasts;whendeterministicallyforced, the low frequency responsesdescribes the consequencesof anthropogenicforcing,ithasbeenusedforclimateprojections.

The FEBE introduces complex climate sensitivities that are convenient forhandling periodic (especially annual) forcing. The FEBE obeys Newton’s law ofCooling,buttheheat fluxcrossingasurfacenonethelessdependsonthefractionaltime derivative of the temperature. The FEBE’s ratio of transient to equilibriumclimatesensitivityiscompatiblewithGCMestimates.Asimplerampforcingmodelof the industrial epoch warming combining deterministic (external forcing) withstochastic (internal forcing), is statistically validated against centennial scaletemperatureseries.

KeyWords:Energybalance,fractionalequations,scaling,climateresponsefunctions,internalvariability,externallyforcedvariability.

1. Introduction

Theearthisaheterogeneoussystemwithadynamicallyevolvingatmosphere,oceanandlandsurface,ithasahugenumberofdegreesoffreedom.WhileGCMsareroutinelyusedtomodelitsevolution,thesearehighlycomplexandeachGCMhasitsownclimate,andthesearedifferentfromtherealworldclimate.Simplifiedmodelshavethereforebeendevelopedthatexploitwhatatfirstsightappearstobeafairlysimpleyetfundamentalconstraint:theconservationofenergy.

The corresponding Energy Balance Models (EBMs) are based on the (near)energybalanceoftheearthwiththesunandouterspace.Forexample,anthropogenicforcingsarecurrentlyaround2.4W/m2abovethelongtermmeansolarforcingof≈238W/m2 (both figuresare forglobalaverages), so that theperturbationsawayfrombalanceareoftheorderof1%.Thismakesitplausiblethattheresponsestoexternalforcingsareroughlylinear.Thiswasconfirmedbyexaminingtheevolutionof the spatial patterns of 32 CMIP5 GCM responses when forced by the variousRepresentativeCarbonPathway(RCP)scenarios,[HébertandLovejoy,2018].Evenwhen these included forcingashighas8.5W/m2,3.5%of thereferencemean, theresponse(theregionaldistribution)ofeachmodel–althoughdifferentfromtheothermodels-was(nearly)alinearprojectionof itsownpast. Theextentofthelinearregime is unclear, and it is certain that due to albedo-temperature feedbacks andothernonlinearities,itwilleventuallybreakdown.Consequently,stronglynonlinear(chaotic)modelshavebeenproposed,see[Dijkstra,2013]forareview. Thesearebased on [Hasselmann, 1976]’s rather general and now classical mathematicaldecompositionofthedynamicsintoslow/fastcomponentswherethefast(weather)scalesstochasticallydrivetheslow(climate)component(seesection4below).

TwodifferentlinearEBMmodellingapproacheshaveevolved,theearliestgoingbackto[Budyko,1969]and[Sellers,1969]whoappliedthecontinuummechanicsheatequation.Thisyielded1D(longitudinallyaveraged)modelsoftheequilibriumsurfacetemperature’ latitudinaldistribution, later extended to time [DwyersandPetersen,1975]. An attractive feature of the Budyko-Sellersmodels is that the system canbecomenonlinearifthealbedo(andhenceforcing)iscoupledwiththetemperature(e.g.viathepositionofthesnowline).Thishasenabledittobeusedformodellingpastand(possible)futureclimates.

AsecondEBMstrandwasinspiredby“boxmodels”thatrelatedCO2emissionstoatmosphericconcentrations.Inenergybalanceboxmodels,oneormoreboxesareusedtomodeltheregionalor-moreusually-thegloballyaveragedtemperature.Thesimplestsuch“zero-dimensional”boxmodels,werefirstproposedby[Hasselmannetal.,1993]andinvolveasingleuniformslabofmaterial-the“box”–representingtheEarthandthatinteractedradiativelywithouterspaceaccordingtoNewton’slawofcooling(NLC).AlthoughtheBudyko-Sellersandboxapproachesaredistinct,whenthetime-dependentversionsarereducedtozero-dimensionsandlinearized,theyaremathematically equivalent to a (single) boxmodel (for a review see [North et al.,1981]andtheupdate[Northetal.,2011],andthevaluablemonograph[NorthandKim, 2017]). In addition, a stochastically forced version of the same relaxationequation (applicable not only to the temperature)was proposed by [Hasselmann,1976] based on the weather-climate scale separation rather than on energyconsiderations.

AkeydifficultyforallEBMsisthatsomeoftheenergyreceivedfromthesuntodayisstoredandemittedtoouterspaceonlyatalatertime.IfweconsidermodelsoftheEarthaveragedover(atleast)macroweathertimescales-i.e.longerthanthetypicallifetimeofatmospheric(weather)structures[Lovejoy,2013]-thensomeofthe longwave/shortwaveradiative flux imbalancesarestored in thesubsurface(mostly intheoceans)andsomearetransportedhorizontally. Conversely,energythatwasstoredlongago–forexampleinthedeepocean–mayappearandcontributetotoday’slongwaveemissions.Thekeytoapplyingtheenergybalanceprincipleisthereforetohavearealisticmodelofthestorageprocesses.

The box and Budyko-Sellers models handle the storage problem in quitedifferentways. TheoriginalBudyko-Sellersmodelshadno storageat all, the fluximbalancewassimplyredirectedmeridionallyawayfromtheequator. Intheboxmodels,thetemperature(T)ofeachboxisspatiallyuniformandtheenergystored(S)istheproductoftemperaturewiththeboxheatcapacity(C)sothatallthestorageoccursasthermodynamicinternalenergy.Sincethemodelcorrespondstothestateoftheatmosphereaveragedoveramonthorlongerandoverspatialscalescontainingmany(energetic)meteorologicalandoceanographicdegreesoffreedom,thisseemsunnecessarilyrestrictive. Themodelalsounrealisticallyimpliesthatenergyfluxesintoandoutoftheboxinstantaneouslychangethetemperatureoftheentirebox.

A recent series of papers showed how the realism of these Budyko-Sellersmodelscouldbegreatlyimprovedbyintroducingaverticalcoordinate.Eventhoughtheaimistoimprovethemodellingofthehorizontaltemperaturedistribution,theinclusionofaverticalcoordinatehastwoadvantages.First,itallowsustoapplythecorrectconductive-radiativesurfaceboundaryconditiontodeterminethefractionoftheradiativeimbalancethatisconductedintothesubsurfaceandthefractionthatisre-emitted at longwavelengths. Second, since the subsurface stores theheat, thethirddimensionsimultaneouslyprovidesaprecisestoragemechanism. Whenthisconductive radiative boundary condition was applied to the classical 3D heatequation(theBudyko-Sellersmodelwithanextraverticalcoordinate),asurprisingconsequence was that the surface temperature obeyed the long memory, Half-orderedEnergyBalanceEquation,(HEBE,[Lovejoy,2020a;b],hereafterL1,L2).IntheHEBE,thesourceofthesurfacetemperaturelongmemoryiscompletelyclassical:itissimplyaconsequenceoftheslowdiffusionofheatintoandoutofthesubsurface.

Evenifweusethestandardinteger-orderedheatequation,wearebeforcedtodeal with fractional ordered surface temperature equations. But why restrictourselvestotheclassicalheatequation?Indeed,Budyko-Sellerstypemodelsareformacroweather Earth states i.e. they are averaged over awide range of (weather)degreesoffreedom,theyare“effectivematerial”typemodels,thereisnocompellingreason that they should be integer ordered. Indeed, it suffices tomake amodestextensiontotheclassicalheatequationtoobtaintheFractionalHeatEquation(FHE);whichisafractionaldiffusionequationthathasbeenstudiedinthestatisticalphysicsliterature.

Theconductive-radiativeboundaryconditionappliedtotheFHEimpliesthatthesurfacetemperaturesatisfiestheFractionalEBEorFEBE,generalizingtheHEBEfromh=1/2to0<h<1.LiketheHEBE,theFEBEisfundamentallya2Dmacroweathertemperature anomaly model, whose zero-dimensional version is a fractionalrelaxationequation,itselfafractionalgeneralizationoftheclassicalzero-dimensionalboxmodelsthataretheh=1specialcases.WhentheFEBEisdrivenbyaGaussianwhite noise, the result is fractional Relaxation noise (fRn) that generalizes theclassical Ornstein-Uhlenbeck process and its high frequency limit is a fractionalGaussian noise process (fGn) that generalizesGaussianwhite noise. The resultingFEBE can be used advantageously for both macroweather forecasts (the highfrequencyFEBEapproximation,[Lovejoyetal.,2015],[DelRioAmadorandLovejoy,2019]) and multidecadal projections (the low frequency approximation [Hebert,2017],[Lovejoyetal.,2017],[Hébertetal.,2020]and[Procyketal.,2020]forthefullFEBE).

Thispaper focusesonmodels thatcombine thephysicalprinciplesofenergybalanceandscaling.Thesearedistinctfromboththeclassicalmodelsthatuseonlyenergybalanceaswellasfromvariousproposalstomodelthetemperatureusingonlythescalingprinciple.Forexample([vanHateren,2013]proposeda“fractal”modelas a scaling hierarchy of (energy balancing) box models whereas [Hebert, 2017;Hébertetal.,2020])proposedamodelwhoseapproachtoequilibriumwasscaling,essentiallyalowfrequencyapproximationtotheFEBE(seesection3.3below).Thiswasachievedbytruncatingthescalingathighratherthanlowfrequencies.

Whilebothof thesescalingmodelsrespectenergybalance,andarestable tosmall perturbations, there are several scalingmodels that are not. For example,[Rypdal, 2012] proposed a model in which the temperature at decadal andmulticentennial scales responds in a “scale free” (pure power law)manner. ThismodeliseffectivelytheFEBEbutwithonlythefractionalstorageterm.Withouttheblack body emission term, energy balance is impossible. While the authoracknowledges that thismodel is unstable to infinitesimal step forcings (it has aninfinite Equilibrium Climate Sensitivity, ECS), he argues that unspecified “slowfeedbacks”will eventually truncate thescalingbehaviouratvery long timescales.Similarly[RypdalandRypdal,2014]arguethatdivergenceswillbeavoidedbecauseeventuallythelinearscalingmodelwillbe“saturatedbynonlineareffects”andbreakdown (see [Hébert and Lovejoy, 2015] for a critique). It is revealing thatwhen[Rypdaletal.,2015]madea fractionalgeneralizationof [Northetal.,2011]’sheatdiffusion model, that it was obtained precisely by removing the critical energybalanceterm.

Theuseof infiniteECSmodelshasbeenalso justifiedby the fact thatoveralimitedrangeofscales,theycanbeapproximatedbymulti-boxmodelsthathavefiniteECS([FredriksenandRypdal,2017]),sothattheymightstillbeuseful.Alternatively,sinceeveninfiniteECSmodelshavefiniteresponsesafterfinitetimes,theirTransientClimate Responses (TCRs) will be finite, prompting [Rypdal and Rypdal, 2014] toarguethattheTCRmaybeamoresignificantquantitythantheECS.Inanyevent,anunsatisfactory consequence of the absence of energy balance is that when purescaling models are used for climate projections, in order to avoid infinitetemperatures,theforcingsmustreturntozero[Rypdal,2016],[Myrvoll-Nilsenetal.,2020].

SincetheFEBEhasdifferenthighand lowfrequencyscalingregimesandthepurescalingmodelhasonlyone, the twomodelsare theoreticallyandempiricallydistinct. For example, model differences are important at poorly understoodmulticentennial,multimillennialscalessothatempiricalclarificationoftheclimatevariabilityover these scales is an important goal of theClimateVariabilityAcrossScales(CVAS)PAGESworkinggroup(since2015,[Lovejoy,2017b],forreviews,see[Franzkeetal.,2020]and[Lovejoyetal.,2020]).

RatherthanderivingtheFEBEfromathreedimensionalcontinuumheatmodel(followingL1,L2),inthispaperitisinsteadderivedphenomenologicallybyarguingthatthebasicenergystoragemechanismsrespectascalingprincipleandwetreattheconsequencesoftheFEBEfortheEarth’stemperature. Insection2,wederivethezero-dimensional FEBE and in section 3, some of its mathematical properties,consequencesandsolutions.Thisincludesthestochasticinternalvariabilitythatwasexaminedfromamoremathematicalpointofviewin[Lovejoy,2019b],(hereafterL3).Insection4,wediscussmodelparameterestimation/validation(highfrequency,lowfrequency,annualperiodicity).Wealsoproposeasimplerampandplateaumodeloftheindustrialepochthatincludesbothstochasticinternalforcinganddeterministicexternalforcing.

2. Energy Storage and the classical energy balance equation (EBE)

2.1 Multibox models and Climate Response Functions

Asingleboxwithitsuniquerelaxationtimeisaninflexiblemodel.[Hasselmannetal.,1993;Hasselmannetal.,1997]alreadynotedthatitwasdesirabletogobeyondthistousethemoregenerallinearresponsefunctionframework.Inthiscontext,theresponse functions are called “climate response functions” (CRFs) and following[Hasselmann et al., 1993]wehaveachoicebetweentheequivalentimpulseandstepCRFs,theformerbeingthederivativeofthelatter(seebelow).

Unfortunately,withoutmoreassumptionsorinformation,thelinearframeworkwith(nearly)arbitraryCRFsisunmanageablygeneral. Inordertomakeprogress,[Hasselmannetal.,1997]proposedaresponsefunctionconsistingofaCRFequaltothesumofN exponentialscorresponding toNboxes thatmutuallyexchangeheat,eachwithitsownexponentialrelaxationtimeandrelativeweight.Buta“multibox”modelwithNboxesrequires2Nparameters-itisstilltoogeneral-sothatoutofthepracticalnecessityoffittingGCMoutputs,[Hasselmannetal.,1997]ultimatelychoseN=3.Followingthemoreusualprocedureofderivingtheimpulseresponsesfrominteger-orderedlineardifferentialequations(whereCRFsare“Green’s functions”),[Li and Jarvis, 2009] recalled that nth order differential equations (with constantcoefficients and with n an integer), can quite generally be reduced to sums ofexponentials.However,intheapplicationpartoftheirpaper,theyneverthelessalsousedthevalueN=3.Similarly,[FredriksenandRypdal,2017]arguedthattwoboxeswerenotenoughanduseda threeboxmodel toapproximateawider scale range(powerlaw)CRFoverscalesrangingfromroughlyonetoathousandyears.

EventheN=3multiboxmodel isdifficult tomanageandmostauthorshavesettled for N = 1 or 2, including the IPCC AR5 (2013, section 8.SM.11.2) whorecommend the four parameterN = 2 model. Yet problems remain: to be takenseriously,theboxesandtheirtimeconstantsmusthavephysicalinterpretations.Forexample, the role of each box is to store heat, but what do the boxes physicallyrepresent? Using theatmosphere, impliesrelaxation timesofweekswhich is tooshortformostapplications.Forthemorepopulartwoboxmodel,thereisaroughconsensusthateachrepresentspartoftheocean:anupper“mixedlayer”andalower“deep”layer.Byappropriatelychoosingthelayerthicknesses,onecangetrelaxationtimestspanningyearstomillenia.Forexample,singleboxmodelshavetrangingfrom4 to40years (e.g. [Schwartz,2007], [Schwartz,2012], [ZengandGeil,2016],[Heldetal.,2010]).Twoboxmodelstypicallyhaveoneboxwithtintherange2-10yearsandanotherwithatintherange20–800years([Rypdal,2012],[Geoffroyetal.,2013])withtheIPCCAR5favouringt=8.4andt=409.5years.

2.2 Scaling storage

RatherthanjustifyingachoiceofCRFbyinvokinghypotheticalhomogeneousboxes,onecan insteadderivetheCRFdirectly fromnumericalmodeloutputs(e.g.[Hansenetal.,2011])orfromphysicalconsiderations.Thekeyistoexploitthewiderangespatialscale invarianceofgeo-processessuchastheoceanandatmosphericdynamicsassociatedwithenergystorage(seee.g.[LovejoyandSchertzer,2013]forareview).Theideaistousethefactthatfromsmalltolargespatialscales,thereisawhole scaling hierarchy of storage processes (e.g. atmospheric or ocean eddies).Sincetheheattransfertimesofeachstructureinthehierarchydependsonitsspatialscale,itisreasonabletoassumethatthecollectiveoverallheatstorageisalsoscaling.

Asmentioned in the introduction,upuntilnow the scalingprinciplehasnotbeenappliedtothestorage,butinsteaddirectlytotheoveralltemperatureresponse([Rypdal, 2012], [Rypdal and Rypdal, 2014], [van Hateren, 2013], [Hebert, 2017],[Hebertetal2020],[Myrvoll-Nilsenetal.,2020]).Scalingimpulse(Diracdfunction)responseCRFsarepowerlaws:

(1)Gδ t( )∝tHF−1

whereHFisascalingexponent.HoweversuchpurepowerlawCRFssufferfromthe“runawayGreen’s function effect”, ([Hébert andLovejoy, 2015]); the fact that theyimplynonphysicaldivergenceseitherat loworathigh frequencies (dependingonwhetherHF > 0 or < 0). This can readily be seen by considering the physicallyimportant integralofGd, the step responseCRF: . Forexample [Rypdal,

2012]proposedaCRFwithHF>0which implies that divergesas : such

CRFswouldyieldinfinitetemperatureresponsestoadoublinginCO2:aninfiniteECS.To avoid the divergence, forcingsmust be carefully restricted so as to eventuallyreturntozero.IflowfrequencytruncationsonapowerlawwithHF>0areinvoked([Rypdal,2012],[RypdalandRypdal,2014]),thenthebehaviourdependscriticallyonthedetailsofthetruncation.

Toavoiddivergences,[vanHateren,2013]insteadproposeda“fractalclimateresponse”modelthatbyusingN=6boxesbutusedthescalingprincipletolinktheamplitudesandtimeconstantsbypowerlawsresultinginafourparametermodel.As mentioned in [Hébert et al., 2020], his model featured unphysical logarithmicoscillationsandanunnecessarylowfrequencycutoff.

Recently,[Myrvoll-Nilsenetal.,2020]haveappliedtheoriginal[Rypdal,2012]infiniteECSmodelto21stcenturywarmingscenariosclaimingthattheirmodelwith“notwell-defined”ECSnevertheless“providesanaccuratedescriptionofbothforcedand unforced surface temperature fluctuations”. Be that as it may, an additionattempttojustifytheirmodelas“acostofthereductionofmodelcomplexity”cannotbesustained.Forexample,[Hebert,2017;Hébertetal.,2020]proposedascalingCRFbut instead took HF<0 combined with an explicit high frequency cutoff to avoiddivergences:

(2)

Atlongtimest>>t,thisstepresponseCRFyieldsapowerlawapproachtoaconstant(correspondingtoenergybalance),and sothatthereisnodivergenceatt=

0either.Thetruncatedscalingmodel(eq.2)ortheFEBE(below)bothshowthatitis easy to avoidmodelswith “notwell-defined” ECSwhile continuing tomaintainmodelsimplicity.

Thefundamentalcutoffscaletwasestimatedas=2yearsfromtheempiricalcouplingtimeoffluctuationsintheoceanandatmosphere([Hebert,2017],[Lovejoyetal.,2017]).Usinghistoricalforcingsandtemperatureresponsessince1880,theyempirically estimated the exponentHF (≈ -0.5) as well as the climate sensitivityparameters=2.4K/CO2doubling(with90%confidenceinterval[1.8-3.7],[Hébertetal.,2020]).UsingtheIPCCAR5scenarios(theRepresentativeCarbonPathways,RCPs), the parameterswere then used tomake temperature projections to 2100.Sincethesourceoftheprojectionuncertaintieswerefromthepastdata-notthelargeGCM“structuraluncertainties”–theyshowedthattheprojectionswerecompatiblewith-butmuchlessuncertain-thantheCMIP5alternatives.

GΘt( )∝t HF

GΘt( ) t→∞

GΘt( ) =1− 1+ t

τ

⎛

⎝⎜

⎞

⎠⎟

HF

; HF<0

GΘ 0( ) =0

Inspiteofitssuccessinmakingclimateprojections,duetotheirtruncations,thescalingCRFmodelsareonlyapproximations:theyapplythescalingprincipledirectlytotheCRFwhereasinfactitshouldonlybeappliedtothestoragetermintheenergybalance equation (EBE). It turns out that thismodel change can bemadewith asurprisingly small tweak to the usual EBE: it suffices to change the order ofdifferentialequationfromintegertofractionalorder,theFractionalEnergyBalanceEquation (FEBE) discussed below. We will show that this seemingly trivialgeneralization can account for responses to both externally forced and internalvariability, effectively explaining them both as different aspects of energyconservation.

2.3 The Earth’s Energy Balance Theearth’senergybalancecanbeexpressedas:

(3)wheretheupwardarrowindicatesenergyemittedtoouterspaceandthedownwardarrowtheenergyreceivedfromthesun(andothersourcesexternaltotheclimatesystem);thedifferenceistheenergystored,S(t);thesubscript“tot”indicatesthetotaloveralongtime,startingataconvenientreferencebaseline.

Beforecontinuing,itisworthmakingafewcomments:a)Thisequationdirectlyexpressesconservationofenergy;itisnotyetthemore

usualrateequationobtainedbytakingderivatives(seebelow).b)Equation3is“zero-dimensional” i.e.all thespatialdegreesof freedomare

averaged out, S, , are averages over the whole earth: energies per unitsurfacearea(instandardunits,J/m2).

Nowconsideranomalies i.e. thedeviations , from the long termvalues

, :

(4)

Thelongtermradiances , couldbetakenasbeingproportionaltolongterm

average rates, outgoing and incoming radiances , so that

.

Sincetheearthisnearlyinradiativebalance, = hence ,sothatthe storage S(t) is the difference between the long and short wave radiativeanomalies:

(5)(subtractthetwoequationsin4anduseeq.3).

S t( )+Etot ,↑ t( ) = Etot ,↓ t( )

Etot ,↑ E

tot ,↓

E↑ E↓

E0,↑ E0,↓Etot ,↑ t( )= E0,↑ t( )+E↑ t( )Etot ,↓ t( )= E0,↓ t( )+E↓ t( )

E0,↑ E0,↓R0,↑ R0,↓ E0,↑ t( )= tR0,↑

E0,↓ t( )= tR0,↓R0,↑ R0,↓ E0,↑ =E0,↓

S t( )+E↑ t( ) = E↓ t( )

If the storage processes are scaling in space (i.e. hierarchical), and thecharacteristicstoragetimeofeachstructuredependsonitsspatialextentinapowerlawmanner,thenitisreasonabletoassumethatthestorageatatimetdependsonthepastinapowerlawmanner:

(6)

where Ch is the (generalized) heat capacity per unit area of the earth, which in

standard units is , h is a fundamental scaling exponent and t is the

(powerlaw)relaxationtime(seebelow).Thepowerlawkernelineq.6relatesthestoragetothetemperatureviaaRiemann-Liouville fractional integraloforder1-hwith0≤h≤1;GistheusualGammafunction.t0isthereferencetimefromwhichthetotalstorageismeasured(here,takentobeinthedistantpast):SH(t0)=0,laterwewill take . Wehave added the subscripth to emphasize that except in theclassicalh=1case,Chisnotausualheatcapacity.

The storagedefined in eq. 6 is a generalization of the standard “boxmodel”storagetowhichitreduceswhenh=1.Toseethis,weintegratebypartsandtakeT(t0)=0,thisisequivalenttofixingthereferenceofouranomalies:

(7)

Takingh=1,wenowobtaintheusualh=1boxstorageresult:

(8)

Akeydifferencebetweentheh<1andh=1caseisthatonlyinthelatter(boxmodel)doesthestoragedependinstantaneouslyonthetemperature.

3. The fractional energy balance equation:

3.1 Fractional Storage

Theusualenergybalanceequationisanequationfortherates,itisobtainedbydifferentiationofeq.5:

(9)

whereFistherateofanomalousenergyinput-theusualforcing(powerperarea),sistheclimatesensitivityandTthetemperatureanomalywithrespecttothelongtermequilibriumtemperature.

Sht( )= C

h

Γ 1−h( ) T u( ) t −uτ⎛⎝⎜

⎞⎠⎟

−hduτt0

t

∫ ; 0≤h≤1

Ch

⎡⎣ ⎤⎦ =J

Km2

t0 =−∞

Sht( )= C

hτh−1

Γ 2−h( ) ′T u( ) t −u( )1−h dut0

t

∫ ; 0≤h≤1

S1 t( )=C1T t( ); h=1

dSh t( )dt

+ s−1T t( ) = F t( ); F t( ) = dE↓

dt; s−1T t( ) = dE↑

dt

The sensitivity is the temperature rise per forcing; in standard units,

. When combinedwith the generalized heat capacity, it defines a time

scale: (10)

Since this dimensional combination is unique, t is the characteristic time for thesystem to relax to equilibrium when it is perturbed by a step function forcing.Howeverwhenh≠1,Chdoesnotcorrespondtotheusualthermodynamicspecificheatwhereenergyinamaterialisstoredininternalthermodynamic(molecular)degreesof freedom. Our model is a macroweather model i.e. for the system averaged(roughly)monthlyorlongerandinspaceovertensorhundredsofkilometers.Theenergycaneffectivelybestoredininternalweatherdegreesoffreedom,itisreallyanaverage coefficient that expresses the collective heat transfer and storagecharacteristics of a hierarchy of storagemechanisms. Rather than treating CH asfundamentalandthenusingeq.10todeterminetherelaxationtime, it isbettertotreattasfundamentalandChasaderivedquantity: sothatthestoragecanbewritten:

(11)

Inthismodel,beyondthedimensionlessexponenth,t isthefundamentalphysicalquantitycharacterizingthestorageprocesses.

WenowrecalltheRiemann-Liouvillederivativeoforderhthatisdefinedastheordinaryderivativeofthe1-horderfractionalintegral:

(12)

WherethebottomequalityfollowsbyintegratingbypartsandusingT(t0)=0,thereferencetemperature.Therefore:

(13)

Puttingthisintoeq.9andtaking weobtain: (14)

s⎡⎣ ⎤⎦ =Km2

W

τ = sCh

Ch= τ/ s

Sht( )= τh

sΓ 1−h( ) T u( ) t −u( )−h dut0

t

∫ ; 0≤h≤1

t0DthT=

ddt t0

Dth−1T( )= 1

Γ 1−h( )ddt

t−u( )−hT u( )dut0

t

∫⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

=1

Γ 1−h( )t−u( )−h ′T u( )du

t0

t

∫ ;0≤h≤1

dSh

dt= τh

s t0DthT ; 0≤h≤1

t0→−∞τh −∞Dt

hT +T = sF

This is the fractionalenergybalanceequation,FEBE. Thephysicalmeaningof thefractionalderivativeisthatwhentheforcingchanges,thechangeintemperatureisnot instantaneous but rather depends on the entire past history of temperaturechanges. At low frequencies, the linear term dominates, it corresponds to theanomalous black body radiation caused by the anomalous forcing. If F is a stepfunction,thenthenewequilibriumtemperatureissF.Withoutthisterm-thepurescalingmodeldiscussed in the introduction (e.g. [Rypdal, 2012]), the temperaturewoulddiverge.

Startingwiththecontinuummechanicsheatequation,L1derivesthiswiththevalue h = ½; the full FEBE (with 0<h<1) can be analogously derived from thefractional heat equation, itself a generalization of the continuummechanics heatequation.Theuseofhalf-orderderivativesinheatproblemshasalonghistorygoingbacktoatleast[Meyer,1960],[OldhamandSpanier,1972],[Oldham,1973],[OldhamandSpanier,1974],developedalsoby[Babenko,1986],[Podlubny,1999;Sierociuketal.,2013;Sierociuketal.,2015].Interestingly,[Oldham,1973],derivesanequationmathematically identical to the h = 1/2 special case of the FEBE as a short timeapproximation to electrolyte diffusion in a spherical geometry, and [Oldham andSpanier, 1974] anticipates our present application by noting that half-orderderivatives can be applied to “not one but an entire class of boundary valueproblems…”.

The limit is convenient when the forcing is a stationary stochasticprocess (such as a white noise representing the internal variability) since thetemperatureresponsewill thenalsobestatisticallystationary. Alternatively, it isalsoconvenientiftheforcingisperiodicwithmeanzero.Inthecaseofadeterministicforcing,thatstartsatafinitetime(e.g.t=0),weneedonlyassumethatT(t)=0,F(t)=0fort≤0.Duetotheinfiniterangeofthefractionalderivative,theFEBEeq.16isa“Weyl”fractionalrelaxationequation.

We derived the FEBE for temperatures and forcing that were taken asanomalieswithrespecttoalooselydefinedreferencestate.Infact,sincetheequationis linear, one can simply add or subtract reference forcings and temperatures forconvenience. The same is true for deterministic and stochastic forcings whosesolutions can be superposed when needed (section 4.5). If the temperature andforcingarenonlinearlycoupled(e.g.viathealbedo),thenthisfreedomcanbeusedtoderiveanonlinearFEBEthatcouldbeusedtomodeltemperature/albedofeedbacksaswellastheeffectsoforbitalforcings,glacial-interglacialtransitionsandotherlowfrequency behaviours. In the future, specific models and their stability may beinvestigatedinthisway.

3.2 Newton’s law of Cooling and thermal impedance

TounderstandtheFEBE,weintegratebothsidesofeq14byorderhandrewriteitintheform:

t0 = −∞

0<h<1; (15)Thetemperatureisthusthefractionalintegraloforderhofthedifferencebetweenthetemperatureandtheequilibriumtemperature .IftheforcingisconstantatavalueF, thenat longenoughtimes,thetemperaturewill“relax”totheequilibriumvalue . Whenh≠1, thisoccurs inapower lawway(due to thepower law

weight term). In the specialH = 1 case – the conventional EBE – the

relaxationisinsteadexponential.If we interpret the forcing sF in terms of an effective external equilibrium

temperatureTeq, thentheFEBEsatisfiesNewton’s lawofcooling(NLC)thatstatesthatabody’srateofheatlossisdirectlyproportionaltothedifferencebetweenitstemperatureanditsenvironment.Inthesehorizontallyhomogeneousmodels,itistheheatflux(energyrate/area=Qs)acrossthesurfaceintothebody(seefig.1)thatisimportantsothattheNLCcanbewritten:

(16)

whereZ isa transfercoefficientsometimescalledthe“thermal impedance”(units:m2K/W),itsreciprocalYisthesurface“thermaladmittance”).SinceQsequalstherateofstoredheatloss,wecanrewriteeq.9as:

(17)

Bycomparingthiswitheq.16,weconcludea)thattheFEBEsatisfiestheNLC,b)thattheclimatesensitivitysisathermalimpedance.

Before proceeding, a word about the energy balance, radiative equilibrium,thermalequilibriumandthermodynamicequilibrium.Aslongasweconsidera“zero-dimensional”modelrepresentingacompletelyhomogeneousEarth,energybalanceis a state of radiative equilibrium, itself a state of thermal equilibrium. Thermalequilibriumisanecessaryconditionforthermodynamicequilibrium,butisgenerallynotidenticalsincethermodynamicallyirreversibleprocessesgenerallyoccur.Whenthe(fractional)timederivativeiszero,theearthisthusinasteady,energybalancestate,equivalentlyinastateofradiativeorthermalequilibrium.

T t( ) = −∞Dt−h τ−h sF −T( )( ) = 1

Γ h( ) Teq u( )−T u( )( ) t − uτ

⎛⎝⎜

⎞⎠⎟h−1 du

τ;

−∞

t

∫ Teq t( ) = sF t( )

Teq

Teq = sF

t − uτ

⎛⎝⎜

⎞⎠⎟h−1

Qs =1ZTeq −T( )

Qs =dShdt

= τ h

s t0DthT = 1

sTeq −T( )

Fig. 1: A schematic showing Newton’s law of cooling (NLC) that relates the temperature difference across a surface to the heat flux crossing the surface, Qs (into the surface). Teq is the fixed outside temperature, heat will flow across the surface as long as the surface temperature Ts ≠ Teq. Z is the thermal impedance, equal here to the climate sensitivity s. To apply the NLC, we need to relate the heat flux to the surface temperature. The lower left shows the consequence of applying heat equation with conductive – radiative BC’s, the lower right shows the phenomenological assumption made by box models. The arrows represent heat fluxes, hence the factor s in the denominators. The system is assumed to be horizontally homogeneous.

3.2 Solving the FEBE using Green’s functions

Mathematicallywhen0<h<1,theFEBEisa“fractionalrelaxationequation”,theextensionto1<h≤2,isa“fractionaloscillationequation”. Intheinitialvalueproblem,F(t)=0fort≤t0,theFEBEimpulseGreen’sfunction-theimpulseCRF-

isgivenintermsofMittag-Lefflerfunctions:

(18)

Where:

��� ���������������

���� �������

Teq

Ts

Qs = CdTsdt

Qs =1ZTeq −Ts( )

Teq / s Ts / s Teq / s Ts / s

Qs =τ h

s −∞DthTs

Gδ t( ) = G0,h t( ) = τ−1 tτ

⎛⎝⎜

⎞⎠⎟

h−1

Eh,h − tτ

⎛⎝⎜

⎞⎠⎟

h⎛

⎝⎜

⎞

⎠⎟ ; t ≥ 0; 0 ≤ h ≤ 2

G0,h t( ) = 0; t < 0; 0 ≤ h ≤ 2

(19)

isthea, borderMittag-Lefflerfunction(theseandmostofthefollowingresultsareinthenotationof[Podlubny,1999]).TheconditionG=0fort<0isneededtorespectcausality.InthefollowingthisconditionwillbeunderstoodforalltheGfunctions.Ascanbeseendirectlyfromtheseriesexpansion(eq.19),theMittag-Lefflerfunctionsareoftencalled“generalizedexponentials”.Inparticular,theclassicalh=1boxmodelisrecoveredwithhelpofthe(exceptional)ordinaryexponential:

.As usual, the Green’s function gives us the particular solution to the

inhomogeneousequationforaDiracdforcing.Thefullsolution,includingthesolutiontothehomogeneousequationis:

(20)

[ChengandChu,2011].Since wecanconfirmthatthissatisfiestheinitialconditionT(t0)=T0.

Justasintegerorderedlineardifferentialequationswithconstantcoefficientscan generally be solved with exponentials, the analogous fractional ordereddifferential equations can generally be solved with generalized exponentials.However, the latter are based on power laws so that the usual exponentialscorrespondtothespecialcasewhereall theordersofderivativesarerestrictedtointegervalues:powerlawGreen’sfunctionsarethusmoregeneral.

A convenient feature of Mittag-Leffler functions is that they can be easilyintegratedbyanypositiveorderz:

(21)([Podlubny,1999]).SinceG0,histheresponsetoaDiracfunctionforcing,withz=1weobtaintheresponsetothe firstorder integralof theDirac: theHeaviside(stepfunction) forcing. Similarly,usingz =2weobtain the response to its integral, the“ramp”(=0fort<0, =t fort≥0,section4.5),usingz=3wehavetheresponsetoaparabolicforcingetc.):

(22)

wherenisaninteger.Fig.2showstheimpulse,stepandrampresponsesforvariousH’s.Thelog-logplotisparticularlyrevealingsinceitshowsdirectlythatforh<1,therearetwopowerlawlimitsandthatfortheh=1case,weobtaintheusualexponentialCRF. Noteforthestepresponse(middlegraphs,andfig.3),theapproachestotheasymptoticvalue1(left)correspondingtoenergybalancecanbeextremelyslow.

Eα,β z( ) = zn

Γ αn +β( )n=0

∞

∑

E1,1 z( ) = ez

T t( ) = s G0,h t − u( )F u( )du +T0t0

t

∫ Eh,1 − t − t0τ

⎛⎝⎜

⎞⎠⎟h⎛

⎝⎜⎞

⎠⎟; T0 = T t0( ); t ≥ t0

Eh,1 0( ) = 1

Gζ,h t( ) = 0Dt−ζ G0,h t( )( ) = τζ−1 t

τ⎛⎝⎜

⎞⎠⎟

h−1+ζ

Eh,h+ζ − tτ

⎛⎝⎜

⎞⎠⎟

h⎛

⎝⎜

⎞

⎠⎟ ; ζ ≥ 0; 0 ≤ h ≤ 2

t0Dt

−nδ t( ) = t n−1

n−1( )!Θ t( ); n ≥1; t0 ≤ 0

The physically important step function response – the step CRF – is thusobtainedbytakingz =1:

(23)Theh=1(boxmodel)yields:

(24)

wherewehaveused .

Byintegratingeq.20byparts,andusing ,andthecondition ,wecanexpressthetemperatureresponseintermsofthestepCRFratherthantheimpulseCRF:

(25)

(wehavetakenT(t0)=0forsimplicity).Mathematically,theequivalencebetweeneq.20and25arisesbecaused(t)isthederivativeofQ(t)and isthederivativeof

.ExpressingthesolutionintermsofthestepCRFratherthantheimpulseCRFis advantageous. For example, is dimensionless so that the sensitivityparametershastheusualdimensionsofdegreesperpowerperarea. Inaddition,following [Hansen et al., 2011] and emphasized by [Marshall et al., 2014] and by[Marshalletal.,2017],thestepCRFdirectlyexpressestemperatureintermsoftheresultsofaCO2doublingexperiment.

GΘ t( ) = G1,h t( ) = G0,h ′t( )d ′t =0

t

∫tτ

⎛⎝⎜

⎞⎠⎟h

Eh,h+1 − tτ

⎛⎝⎜

⎞⎠⎟h⎛

⎝⎜⎞

⎠⎟; t ≥ 0; 0 ≤ h ≤ 2

G0,1 t( ) = τ−1e−t /τ; G1,1 t( ) =1− e−t /τ

E1,2 z( ) =ez −1z

G1 0( ) = 0 F t0( ) = 0

T t( ) = s G1,h t − u( ) ′F u( )dut0

t

∫ ; ′F t( ) = dFdt

G0,h t( )G1,h t( )

G1,h t( )

Fig. 2: The various response functions linear - linear (left column) and log - log right for h = 1/10 to 1 (exponential) in steps of 1/10. The lines bounding the envelope are h=1/10 (dashed), h = 1 (thick). The middle value (h = 1/2) is also thick. The top row shows the Dirac (d(t)) response functions (G0), the middle row the step (Q(t)) responses (G1), and the bottom row, the ramp (tQ(t)) responses (G2).

0 2 4 6 8 10t

0.51.01.52.0G 0

- 4 - 3 - 2 - 1 1 2 Log10t

- 4- 3- 2- 1

12

Log 10G 0

0 2 4 6 8 10t

0.20.40.60.81.0

- - - -

4-

- 3

- 2

- 14 3 2 1 1 2

0 2 4 6 8 10t

2468

10

Log10tLog 10G 1

- 3 - 2 - 1 1 2

- 4- 3- 2- 1

12

Log10t

Log 10G 2

G 1

G 2

d(t)

Q(t)

tQ(t)

Fig. 3: The FEBE response to a step function forcing (shown in thick black) for various values of h, using s = 1 with the response for h = 0.1, 0.3, 0.5, 0.7, 0.9 (bottom to top). The red is for h = 0.5, the value close to the empirical exponent; the blue is for the box model value h = 1. The storage is the difference between the response and the forcing; both are shown with arrows for the case h = 0.5.

3.3 Complex climate sensitivities and the annual cycle

The essential FEBE behaviour can be understood by taking its Fourier

transform(“F.T.”).Using (e.g.[Podlubny,1999])andtakingsF(t)=d(t)

weobtaintheFouriertransformoftheGreen’sfunction andsolution :

(26)

� � � � ��

���

���

���

��

�����

��������������

�������

��� ���

� �����

� �����

� �����

��τ

�� ���

���� ��� �� �

���

��

−∞Dth↔F .T .

iω( )hG! 0,H ω( ) T! ω( )

iωτ( )h +1( )G! 0,h ω( ) = 1; G! 0,h ω( ) = 11+ iωτ( )h

T! ω( ) = sG! 0,h ω( )F! ω( ) = sF! ω( )1+ iωτ( )h

wherethetildeindicatesFouriertransformwithconjugatevariablew,thefrequency.This shows that if the forcing is purely sinusoidalF =Fseiwt that the temperatureresponseisalsosinusoidalwiththesamefrequencybutdifferentphase:T=Tseiwt.Usingthenotionofthermalimpedanceidentifiedwiththeclimatesensitivity,wecanfollowengineeringpractice(usefulforestimatingdiurnaltemperature-heatinglagsin buildings and structures) and use complex thermal impedances i.e. complexclimatesensitivitiess(w):

(27)

Wherewehaveusedthenotations0= s(0)forthestaticclimatesensitivity(i.e.s0=sin thepreviousnotation). In the contextof theEarth’s energybalance, it ismoreusefultothinkintermsofsensitivitiesthanimpedancessothatwhenconvenientweuse s(w). We couldmention that in dynamical systems theory, the Fourier filterbetweentheforcingandtheresponse(theimpedances(w)=Z(w))isalsocalledthe“susceptibility”,seee.g.[Lucarinietal.,2017]forapplicationstoclimatemodels.

Deterministicscaledependentclimatesensitivitieshavebeenused inscalingcontexts in order to account for the temperature fluctuations and spectra atmulticentennial, multimillennial scales. For example, [Rypdal, 2012] proposedfrequencydependentscalingsensitivitiestoaccountforthesolarcycleand[Rypdaland Rypdal, 2014] for the temperature response to Gaussianwhite noise internalvariability. Anticipatingpossiblenonlinear feedbacksat thesescales, [LovejoyandSchertzer, 2012b] used scale dependent but stochastic sensitivities to account forresponsestostochastic(andscaling)externalnaturalforcings.Inthisframework,atmulticentennial,multimillennial scales, the deterministic temperature response toexternal forcing could be nonlinear while the statistics of the correspondingfluctuationscouldstillbelinear.

Duetothelowfrequencycutoffineq.27,FEBEsensitivitiesarecomplexclimatesensitivities butwith different high and low frequency scaling regimes. Complexsensitivities are useful in understanding the responses to periodic forcings, inparticularwhentheforcingisduetotheannualcycle.Figs.4,5comparethephasesandamplitudesofs(w)asfunctionsoftherelaxationtimet(withw=2prad/yr)forvariousvaluesofhincludingtheHEBEvalue(h=1/2)andtheEBEvalueh=1.Fromfig.4,thethinhorizontallinesshowthattakingtheempiricalvalueoftintherange1-5years,andtheapproximateempiricalvalueh≈0.4(dashed),thatthelag(thephaseofthesensitivity)isintherange25-30days,whichisintheobservedrangeforthelagbetweenthesummerforcingmaximumandmaximumtemperaturesovermostlandareas.

Ts = s ω( )Fs; s ω( ) = Z ω( ) = s01+ iωτ( )h

If the forcing was only due to conductive heating (rather than conductive-radiative forcing), then, as pointed out in L1, one obtains the classical (p/4) lag(correspondingtoh=1/2, )obtainedby[Brunt,1932]forthediurnallag.Thispureheatinglagcorrespondsto46daysandisalreadytoolongformostoftheglobe.Indeed, from the detailedmaps in [Donohoe et al., 2020]we estimate that in theextratropicalregions,overland,thesummertemperaturemaximumistypically30-40daysafterthesolstice,butonly20-30daysafterthemaximumforcing(insolation)andforocean,60-70daysafterthesolsticebutonly30-40daysafterthemaximuminsolation.Similarly,theh=1EBElagisintherange82to91days(fort>1year),alsomuchtoolong.Whileitistruethatovertheocean,thelagistypicallylongerthanoverland,thisisprobablybecauseofthestrongalbedoperiodicityassociatedwithseasonaloceancloudcover[Stubenrauchetal.,2006],[Donohoeetal.,2020].Thisdelaysthesummersolsticeforcingmaximumovertheocean,potentiallyexplainingtheextralag.

L1testedouttheimplicationsoftheHEBE(h=1/2)modelusingalatitudinallyaveragedmodelandparameterstakenfrom[Northetal.,1981].Themodelusesa2ndorder Legendre polynomial to take into account the latitudinal variations and asinusoidal annual cycle with empirically fit parameters. Since it only modelslatitudinalvariations,iteffectivelyaveragesoverlandandocean.Inworkinprogress,weusemodernsatellitedatatotestthemodelat2ospatialresolution.Futureworkwillusethesenewresultscoupledwithmodernhigherresolutiondataofoutgoinglongwaveradiationtoinvestigatethisfurtherandobtainmoreaccuratelocalclimatesensitivities.

τ = ∞

Fig. 4: The (negative) phase lag of the temperature with respect to the forcing (the negative of the argument of s) Lag in days for h = 1/10, 3/10, ½, 7/10, 9/10 (black), 4/10 (dashed middle), 1 (dashed top) as a function of t measured in years. The horizontal lines are for 25 and 30 days, we see that the dashed h = 4/10 falls in this range for between one and 5 years. This range of lags is close to the typical extra-tropical lags found in [Donohoe et al., 2020].

1 2 3 4 5τ (yrs)

20

40

60

80

θ (days)

Fig. 5: The modulus of the climate sensitivity as a function of the relaxation time t for h ranging from 1/10, 3/10, 5/10, 7/10, 9/10 (solid, top to bottom) and h = 4/10 (middle dashed) and h = 1 (bottom dashed).

4. Estimating H, t

4.1 The low frequency response and climate projections

The FEBE has distinct high and low frequency behaviours that can beunderstoodfromthecomplexclimatesensitivity:

(28)

Wheretheapproximationswereobtainedasusualbyexpandingthedenominatorofeq.27. Inthehighfrequency limit, is theFourierTransformof theHorder fractional integral of F, therefore at high frequencies the temperatureintegrates(smooths)theforcingwhereasatlowfrequencies,thelowestorderterm(unity) corresponds to a Dirac function (associated with the equilibriumtemperature) while the next order term corresponds to a fractionaldifferentiationoforderhthatdeterminestherateatwhichequilibriumisapproached.

0 1 2 3 4 5τ yrs0.0

0.2

0.4

0.6

0.8

1.0λ(ω)s ω( )

T! ω( ) = s ω( )F! ω( ) ≈s0 iωτ( )−h F! ω( ); ω >> τ−1

s0 1− iωτ( )h( )F! ω( ); ω << τ−1

iωτ( )−h F! ω( )

iωτ( )h F! ω( )

Alternativelyandequivalently,wecanusesmallandlargetexpansionsoftheGreen’sfunctions(eq.21).Let’sfirstconsiderthelargetasymptoticexpansions:

(29)

[Podlubny, 1999]. The most important cases are for z = 0, 1, 2 corresponding toimpulse,stepandrampresponses:

;

0<h<2

(30)

Wehaveused forz =0,notethatG(-h)<0for0<h<1.Fromthese,weseethatasrequiredforthestepresponse, and sothatat larget,theFEBEimpliesapowerlawrelaxationtothenewequilibriumtemperature.

From either the low frequency Fourier approximation (eq. 28) or theasymptoticexpansion(eq.30),weseethatthelowfrequencyresponseTlowcanbeapproximatedbythefirstterm: :

(31)

Tlow is expected to be dominated by the forced response to external forcings,appropriateformakingmulti-decadaltemperatureprojectionsfromanthropogenicforcings (ineq.31,weassumed that the initial temperature iszero,otherwise theextra term in eq. 20 can be used). Comparing thiswith the [Hébert et al., 2020]truncated scaling CRF (eq. 1) we see that the FEBE has identical low frequencybehaviourwith:

h=-HF. (32)Using the empirical value HF ≈ -0.5 that [Hébert et al., 2020] obtained by

(Bayesian)fittingtheglobaltemperatureseriessince1880totheIPCCCO2eqforcing,wefindh≈0.5.Similarly,[Procyketal.,2020]fittheFEBE(usingtheexactG0,HGreen’sfunctionwiththesamecentennialscaleclimateforcing)toestimateh=0.38±0.06).Inthenextsectionweconsiderthehighfrequencybehaviourandusethe internalvariability (assumed to be forced by a Gaussianwhite noise) that gives themorepreciseestimateh≈0.42±0.02[DelRioAmadorandLovejoy,2019].

Gζ ,h t( ) = τ ζ −1 ∑n=0

∞ −1( )nΓ ζ − nh( )

tτ

⎛⎝⎜

⎞⎠⎟

ζ −1−nh

; ζ ≥ 0

G0,h t( ) = −τ −1

Γ −h( )tτ

⎛⎝⎜

⎞⎠⎟

−1−h

+τ −1

Γ −2h( )tτ

⎛⎝⎜

⎞⎠⎟

−1−2h

−O tτ

⎛⎝⎜

⎞⎠⎟

−1−3h

G1,h t( ) = 1− 1Γ 1− h( )

tτ

⎛⎝⎜

⎞⎠⎟−h

+ 1Γ 1− 2h( )

tτ

⎛⎝⎜

⎞⎠⎟−2h⎛

⎝⎜⎞

⎠⎟+O t −3h( ) t >> τ

G2,h t( ) = t − τΓ 2− h( )

tτ

⎛⎝⎜

⎞⎠⎟

1−h

+ τΓ 2− 2h( )

tτ

⎛⎝⎜

⎞⎠⎟

1−2h

+O t1−3h( )Γ 0( ) =∞

G1,h 0( ) = 0 G1,h ∞( ) = 1

Glow,h t( ) ≈G0,h t( )

Tlow,h t( ) ≈ sGlow,h ∗F; Glow,h t( ) = − τ−1

Γ −h( )tτ

⎛⎝⎜

⎞⎠⎟−1−h

Wehaveseen(eq.15)thattquantifiestheratethatthetemperaturerelaxestoequilibrium following a perturbation. At the same time, according to the aboveanalysis,trepresentsthetransitiontimebetweentwopowerlawregimes(t/t)h-1to(t/t)-h-1;usingtheempiricalvalueh≈0.4thisisatransitionfrom(t/t)-0.6(t<<t)to(t/t)-1.4(t>>t)behaviourwitharelativelysmallchangeinthescaling.TheexactFEBEsolution(eq.20,fig.2,righthandcolumn)confirmsthatthetransitionisveryslow-i.e.itoccursoverawiderangeofscales.Thisveryslowtransitionmakestdifficulttoaccuratelyestimatefromobservationsof thetemperatureresponsesto forcings(seeappendixA).

Basedon the truncatedpower lawCRF(eqs.1,2), [Hébertetal.,2020]usedphysicalargumentstodeterminet.Atshortenoughtimescales,differentpartsoftheglobehave temperature fluctuations thatvarymoreor less independentlyof eachother: they are primarily responses to “internal” forcings, small scale storagemechanisms.Atlongenoughscales,thevariabilityisexpectedtobeprimarilyduetothe responses fromexternal forcings. [Hébert etal., 2020]argued thatt couldbeestimated from the scale atwhich thewhole global surface temperature starts tofluctuateasaunit.Inparticular,theatmosphereandoceanshouldhavetemperaturesthatfluctuatesynchronously:tshouldbetheatmosphere-oceancouplingtimescale.

t wasthereforeestimatedbyconsideringthecorrelationsbetweenfluctuationsin the atmospheric temperature averaged over land, and fluctuations in the SSTaveragedovertheoceans.Atscalesofmonths,thereispracticallynocorrelation:thelandandoceanfluctuatenearlyindependentlyofeachother.Howeveratlongtimescales, they are on the contrary highly correlated: they fluctuate together. Thetransition between independence and dependence occurs over a short interval;betweenabout6monthsand2years.[Hébertetal.,2020]usedthistoinferthatt≈2years,avaluethatgavereasonableclimateprojectionsthroughto2100.Thistimescaleisessentiallythesameasthelifetimeofplanetaryscaleoceangyres;theoceanweather–oceanmacroweathertransitionscale([Lovejoyetal.,2017]).

Morerecently,[Procyketal.,2020]usedaBayesianapproachandtheFEBECRFtofindtheoptimumparametersl,t,hconstrainedbytheestimatedhistoricCO2eqforcingsandtheresponsesasestimatedbyfiveobservationalseriesofglobalaveragetemperatures since 1880, they obtain a very similar value t ≈ 4.7yrs (the 90%confidenceintervalwas2.4-7.0years).SincetheFEBEisinthesamefamilyastheEBE,asaBayesianpriordistributionfort,theyused[Geoffroyetal.,2013]estimateof4.1±1.1years for thebox (h=1)modelbasedonadozendifferentGCMmodeloutputs.

4.2 The high frequency response: internal variability and macroweather forecasts

In section 4.1we saw that ifwe takeh = -HF, that to leading order, the lowfrequency FEBE and truncatedmodel impulse CRFs are both proportional to t-h-1,thesegivereasonableapproximationstothelowfrequencies.ButwiththeFEBE,wearenolongerthinkingintermsofasmallnumberofhomogeneousboxes;rathertheFEBEisaphenomenologicalmodelofahierarchyofstorageprocessesfromlargetosmall. It isthereforelogicaltoassumethattheFEBEappliestobothexternalandinternalforcings.MathematicallythisimpliesthatbothvariabilitieshavethesameGreen’sfunction,openingupthepossibilityofestimatingclimateparametersdirectlyfromtheinternalvariability. Basinghimselfontheweather-macroweatherscaleseparation andassumingthatthe internal forcingfromtheweatherregimewasaGaussianwhitenoise[Hasselmann,1976]proposedastochasticclimatemodelthatreducestothestochasticbox-model(h=1),inthezero-dimensionalcase,thelatteristhereforesometimescalledthe“Hasselmannequation”(see[Watkinsetal.,2020]forausefuldiscussion,reviewaswellasvariousfractionalextensions).

Theideaofusingtheinternalvariabilitytoinfertheresponsetoexternalforcinggoesback to at least [Leith, 1975],whoproposed that theFluctuationDissipationTheorem(FDT)beusedforthispurpose. TheFDTsaysthattheimpulseresponseCRF of a nonlinear dynamical system to (small) external forcings is equal to thenormalized covariance function. The FDT is attractive since - if it is valid – it isindependent of specific climate models. While the original FDT only applied atthermodynamicequilibrium, ithasbeenextendedusing linear response theory tononequilibriumsystems[Kubo,1966],althoughtheextensionremainscontroversial[Gottwaldetal.,2016],itisnotatallobviousthattheclimatesystemiscloseenoughtoequilibriumfortheFDTtoapply.Nevertheless,startingwithLeith,numeroussuchattemptshavebeenmade(especially[Northetal.,1993],[CionniandVisconti,2004],[Schwartz,2007]andrecently[Coxetal.,2018]).Asweshowinafuturepublication,itturnsoutthattheFEBEdoesobeytheFDT,butonlyinthelowfrequencyregimewheret≫twherethescalingstoragetermissmallenoughthatthesystemisindeedclosetoequilibriumandtheasymptoticformeq.44holds(inthislimitthenormalizedcorrelationfunctionandGreen’sfunctionG0,Hareequal).Incontrast,theEBE(h=1)specialcaseisexceptional:theFDTalwaysholds,andthetemperatureisanOrnstein-Uhlenbeckprocess,[Northetal.,1993],[Coxetal.,2018])).

Althoughathighfrequencies,theFDTisnotvalid,ifthereisinformationabouttheinternalforcing,thentheFEBEcanstillbeusedtoinfertheCRFandhencethesystemresponsetoexternalforcing.Forexample,ifwemaketheusualassumptionthattheinternalforcingisaGaussianwhitenoise,thenwecanusethetemperaturestatistics to deduce the exponent h, (see table 1, discussed here and in the nextsection).Ifthenoiseamplitudeisalsoknown,wecandeduces(seetable1,bottomrow).Beforediscussingthefullstochasticallyforcedcase,inthissectionwediscussthehighfrequencystochasticresponsedominatedbyscalingstorageprocesses.

Inordertoinvestigatethehighfrequencybehaviour,usethefirsttermoftheGreen’sfunctionexpansion(eqs.18-20)toobtain:

(33)

This is the singular high frequency FEBE limit corresponding to a fractionalintegrationoforderh(eq.28).

Since we associate the high frequency behaviour primarily with internalvariability, let’s consider the casewhere its source f(t) is awhite noise. IgnoringexternalforcingssothatThighisitsstochasticresponse:

(34)

to ensure convergence, we assume that f(t) is a statistically stationary processrepresentingthe“innovations”withmean<f(t)>=0andwetake .

Tounderstandthishighfrequencybehaviour,itisusefultocompareitwiththedefinitionofafractionalGaussiannoiseprocess(fGn):

(35)where g(t) is a “d-correlated” unitwhite noise process, the innovations satisfying

and<g>=0. g(t) isageneralized function–a “noise”– that isproperlydefinedoverfinite integrals(seee.g. [Biaginietal.,2008]whotermsthis“fractionalwhitenoise”,itiscommonlydefinedasaderivativeoffractionalBrownianmotion,seee.g.[KouandSunneyXie,2004]).Often,fGndefinedintermsofaWienerprocessW with dW = g(t)dt. KH is a standard normalization constant chosen forconvenience. The fGn process goes back to [Mandelbrot and Van Ness, 1968],[Mandelbrot,1971]whouseditasahydrologymodel(seealso[MolzandLiu,1997]for a review including hydrology applications). Explicit use of fGn processes as amacroweathertemperaturemodelswasproposedby[RypdalandRypdal,2014]and[Lovejoy,2015;Mandelbrot,1971;MandelbrotandVanNess,1968].

Comparingeq.34and35,weseethatiff(t)isawhitenoise(i.e.proportionaltog(t)),andifwetake:H=h-½thenthehighfrequencytemperatureresponseisanfGn process with exponent H. This result is easy to understand since at highfrequencies(eq.28),thederivativetermintheFEBEisdominant(i.e. )andgHisindeedthesolutionoftheresultingfractionaldifferentialequation:

(36)

Thigh ≈ sGhigh ∗F t( ); Ghigh =1

τΓ h( )tτ

⎛⎝⎜

⎞⎠⎟h−1

; t << τ

Thigh t( ) ≈ sτhΓ h( ) t − u( )h−1

−∞

t

∫ f u( )du

T −∞( ) =0

gH t( ) = KH

Γ H +1/ 2( ) t − u( )H−1/2 γ u( )du−∞

t

∫ ; −1< H < 0; KH = π2cos πH( )Γ −2 − 2H( )

⎛⎝⎜

⎞⎠⎟

1/2

γ t( )γ ʹt( ) = δ t − ʹt( )

−∞DthT >>T

−∞DtH +1/2g

Ht( )= KH

γ t( ); −1<H <0

The standard assumption about internal variability is that it is forced by aGaussianwhitenoise.Withthis,wecanusethishighfrequencybehaviourtoestimatehfromtheempiricalvaluesofH.Asmentionedabove,accordingtotheglobalscaleHaarfluctuationanalyses(seee.g.[Lovejoyetal.,2015];[DelRioAmadorandLovejoy,2019]),H≈-0.08±0.02sothath=H+1/2≈0.4whichisessentiallythesameasthe[Hébert etal., 2020], [Procyketal., 2020]valueestimated from the low frequencyresponsetoexternalforcings(eq.31).ThesearealsoclosetoestimatesofHintherange0to-0.3forvariousoceanandlandareas([FredriksenandRypdal,2015])andwithcorrespondingspectralexponentsestimatedin[LovejoyandSchertzer,2013].

We could note that [Cox et al., 2018] recently used an analogousmethod ofexploiting the internal variability to estimate the ECS. However, they made theunrealistic Ornstein-Uhlenbeck (h = 1) assumptions that the autocorrelation wasexponentialandthattheratiooftemperaturevariancetoforcingvariancefollowsthe(stronglybiased)h=1result(table1bottomrow).

fGn processes are power law smoothed white noises (eq. 35), but when-1/2<H≤0(0<h≤1/2),thesmoothingisnotenoughtoeliminatethesmallscaledivergences (i.e. mathematically, they remain “noises”). For this reason, finiteresolution fGn processes are defined as the increments of fractional Brownianmotions(fBm).Thisisequivalenttodefiningthe“trresolutionfGn”byaveragingoveratimeintervaltr :

(37)

Then,withtheabovechoiceofconstantKwehave:

(38)

Ifh<1/2,thenH=h-½<0,implyingasmallscaledivergenceastheresolutiontrisreduced:fGnprocessesthushavestrongresolutiondependencies.Animportantbutregularly overlooked practical consequence of such divergences is that inmacroweather,thisleadsto“space-timereductionfactors”thatmultiplicativelybiasmacroweatheranomaliesandareimportantforexampleinaccuratelyestimatingthetemperatureoftheearth[Lovejoy,2017a]).

ThisisperhapsanopportunemomenttocommentontherelationbetweenhandtheHurst-inspiredexponentH.AlthoughHurst’soriginalexponent[Hurst,1951]was for the “rescaled range”ofaprocess, at least formanystatistically stationaryprocesses(suchasfGn),itwasequaltothescalingexponentoffluctuationsofrunningsums.Atfirst,[MandelbrotandVanNess,1968]consideredGaussianprocessesthathad average (absolute) fluctuations growingwith scale (DtHwithH>0): fractionalBrownianmotion (fBm). Ifwe define fluctuations using differences (increments)thenthefBmfluctuationexponentH isdirectlyrelatedtotheorder(H+1/2)ofthefractionalintegrationofawhitenoise.

gH ,τr t( ) = 1τr

gH ′t( )d ′tt−τr

t

∫

gH ,τr t( )2

= τr2H ; −1< H < 0

Later, in the 1980’swavelets allowed fluctuations to be definedmuchmoregenerallythanjustasdifferences,sothatfluctuationswithexponentsH<0couldbeobtained.Forexample,wecandefinetheanomalyfluctuationatscaleDtasthemeanovertheintervalDtoftheprocesswithitslongtermaverageremoved-orwithzeroensembleaverage([LovejoyandSchertzer,2012a]).Inthiscaseeq.37showsthatforfGnwithDt=tr,thatwecanobtainfluctuationexponentsintherange-1<H<0,seeeq.38,(forfGn,theexponentoftheaverageabsoluteanomalyandRMSanomalyareequal).

Also in the 1980’s, it was realized that scaling processes in general, andatmospheric processes in particular are more generally multifractal so that theycouldnotbecharacterizedbyasingleexponent.However,[SchertzerandLovejoy,1987]showedthatmultifractalprocessescouldneverthelessalsobecharacterizedbyfluctuationexponentsandusedtheHurst-inspiredsymbol“H” forthis. BeyondsharingafluctuationexponentHthatcouldtakeonanyrealvalue,bothmultifractalandGaussianprocesscouldbegeneratedbyfractionalintegration;formultifractals,byfractionallyintegratinga“conservative”multifractalcascadeprocessbyorderH,while for quasi-Gaussian processes by order H+1/2 fractional integration of aGaussianwhite noise (the extra½ is needed because thewhite noise itself has afluctuationexponent=-1/2).

Insummary,ifweuseappropriatewaveletstodefinethefluctuationsanddefineHasafluctuationexponent,thenitappliesequallywelltobothquasi-Gaussianandmultifractalprocesses.

IntheFEBE,theorderhplaysnumerousrolesincludingseveralthatrelateittoanH.Forexample,athighfrequencies(whenthestoragetermdominates),histheorderoffractionalintegrationneededtoobtainTfromtheforcing,sothatwhenFisa Gaussianwhite noise, at high frequenciesH = h -1/2. However F can also bedeterministic in which case h characterizes the responses to deterministic stepforcingsforshorttimes(th)whileforlongtimes,equilibriumisapproachedast-h.

Beforeleavingthetopic,twomorecommentsareinorder.First,althoughthedomainoftheFEBEdiscussedhere-macroweatherintime–haslowintermittency(it isnot far frombeingquasiGaussian,wemodel itassuch), it is in fact theonlydynamical atmospheric regimewhere themultifractality is low. Yet in the spatialdomain,macroweathermultifractalityisonthecontraryveryhigh[Lovejoy,2018],sothat for regional (space-time FEBE, ([Lovejoy, 2020b], [Lovejoy, 2020a])multifractalityisimportant.Second,itisworthpointingoutthatthereisalsoalargemathematical literature on fBm and fGn processes that characterize themparametrically.Addingprimestoavoidconfusion,theydiscuss“H’parameterfBm”andthe“H’parameterfGn”processes.Forreference,forfBm,wehaveH=H’whereasforfGn,H=H’-1;thedifference(unity)isduetodifferencing/summingoforderoneneededtoobtainfGnfromfBm(andvisaversa).

4.3 The stochastic FEBE, fractional Relaxation motion (fRm), fractional Relaxation noise (fRn)

Intheprevioussectionweconsideredthehighfrequencystochasticresponse:fGn which is the solution of the FEBE at high frequencies where the (fractional)storagetermdominates.WenotedthatsincetheFEBEisalinearequation,wecouldseparately model the deterministic and stochastic (internal) variability. In thissectionwegivemore informationabout thispure stochastic casedrivenbywhitenoise innovations,we briefly summarize and expand upon some of the results in[Lovejoy,2019b](hereafterL3).WenotethatthestationaryprocessgeneratedbythestochasticFEBE–fractionalRelaxationnoise(fRn)generalizesOrnstein-UhlenbeckprocessesfromH=1to0<H<2.

Whereasthedeterministicfractionalrelaxationequationhasbeenwellstudied(see e.g. [Miller and Ross, 1993], [Podlubny, 1999]), the stochastic version hasenjoyedmuchlessattentionalthough[Westetal.,2003]reviewssomeapplicationsof the Riemann-Liouville case to nonstationary random walks. In contrast, thestochastic infinite range (Weyl version, when ) leading to stationarysolutionshasonlyrecentlybeenconsidered(L3) including theoptimumpredictorproblem(L3).Belowwediscussthemainpointsandgivethedimensionalformofthemainstatisticalproperties.

Although the stochastic fractional relaxation equation has received littleattention, a closely related equation, the Fractional Langevin equation (FLE), hasreceivedsomewhatmorestudyduetoitsapplicationsinthephysicsofrandomwalks,anddiffusion([MainardiandPironi,1996],[Coffeyetal.,2012]).Whereasthehighestorderterminthefractionalrelaxationequationisfractional,intheFLEthefractionaltermisoflowestorder.[Watkinsetal.,2020]providesausefulreviewthatcomparesandcontrastsvariosstochasticequationthatappearinclimatemodelsandpointsouttherelationshipbetweentheFLEandtheFEBE.

ThephysicalproblemthatwewishtosolveisthenoisedrivenFEBEwithnoiseamplitudesandsensitivitys:

(39)

withinitialconditions ,0≤h≤1andg(t)aunitGaussianwhitenoisesothat

,sistheamplitudeofthenoise,eq.39isafractionalLangevin

equation(seeL3).Tounderstandthestatisticalpropertiesoftheinternalvariabilityresponse,itsufficestostudythenondimensionalequation:

(40)Wherefor0<h<1/2,Nh=Khisthenormalizationconstantineq.35(for½<h<2,seeeq.42)andthenondimensionalUh(t)functioniscalledafractionalRelaxationnoise(fRn) since it generalizes fGn (see L4). It also generalizes the h = 1 Ornstein-Uhlenbeckprocess.

t0→−∞

τh −∞DthT +T = sf t( ); f t( ) = σγ t( )

T −∞( ) = 0

f 2

1/2=σ; f =0

−∞DthUh +Uh = Nhγ t( )

UsingUh and noting that ,we can obtain the solution to the

dimensionaleq.39using:

(41)

“ ”meansequalityinaprobabilitysense.SincethefRnprocessisthesolutionofthefractionalrelaxationequationwitha

stationary,Gaussian,zeromean,whitenoise forcing, it isalsostationary,Gaussianwithzeromean.Itsstatisticsarethereforefullycharacterizedbyitsautocorrelationfunction.Acomplicationinthecalculationisthatwhen0<H≤1/2,inthesmalltlimit,thefractionaltermofeq.39dominatessothatweobtainthe(fGn)limit.Thesolutionofeq.40istherefore-likeg(t)-ageneralizedfunction:toobtainsolutionswithfinitevariances,wemusttakeaveragesoverfiniteresolutionstr.

The resulting tr resolution autocorrelation function at lag Dt is:

(42)

Notethatwhen0<h<1/2,theequationfor isonlyvalidforDt≥tr,alsorecall

the classical Ornstein-Uhlenbeck process (h = 1) where G0,1(t) = e-t and theautocorrelationis: . TheaboveformulaeweregivenforDt>0butsinceG0hiscausal,G0h(t)=0fort≤0,whichimpliesRh(Dt)=Rh(-Dt).Table1givestheexplicit dimensional formulae taken fromL3 for the temperature autocorrelation.Alsogiveninthetablearethecorrespondingformulaeforthetemperature-forcing(f=sg) and temperature storage cross-correlations ( ).Physically, the model is justified as long as the resolution tr is greater than theweather-macroweather transition scaletw (≈10days). tw is the inner (smallest)scaleoverwhichtheFEBEmayexpectedtobevalid.Fromthis,wecanobtainthehigh frequency fGn approximation (valid forDt≪1 corresponding toΔt<<t in thedimensionalequation):

(43)

γ t( )=d

τ −1/2γ tτ

⎛⎝⎜

⎞⎠⎟

T t( ) = sσNhτ

1/2Uhtτ

⎛⎝⎜

⎞⎠⎟

=d

Rh,τr

Δt( )= Uh,τr

t( )Uh,τrt − Δt( ) =N

h2 G0,h Δt +u( )G0,h u( )du;0

∞

∫

Rh,τr

0( )= Uh,τr

t( )2 = τr2h−1; 0<h<1/2

Rh,τr

0( )=1; 1/2<h<2

Nh = Kh; 0 < h <1/ 2

Nh = c; c−2 =0

∞

∫G0,h u( )2 du; 1/ 2 < h < 2

Rh,τr

Δt( )

R1 Δt( ) = e−Δt / 2

!S = dSh / dt = τh−∞Dt

hT = sf −T

Rh,τr Δt( ) ≈ h 2h +1( )Δt 2h−1; τr << Δt <<1; 0 < h < 12

(seetable2for½<h<2).Atlowfrequencies,forDt>>1,weobtain:

(44)

validforall(non-negative)Dt,seeL3andtable1forthedimensionalexpressionsandnotethatG(-h)<0for0<h<1.ThislargeDtresultisindependentoftheresolutiontrandthatitholdsoverawiderrangeofhvalues(h=1istheexponentialexception,seetable1).Atechnicalpointisthatalthoughwhen0<h<1/2,thehighfrequencylimitisthefGnwithahugememory,duetotheslowDt2h-1fall-offinRh(Dt)(eq.43),thecorrespondingfRniseffectivelytruncatedwithamemoryoftheorderoft(seeL3).

Table 1 summarizes the (dimensionalized) versions of these relations alongwiththecorrespondingcross-correlationfunctions.Ofparticularnoteareparameterrangeswheretheresolution is important(0<h≤1/2forRTT,trand0<h<1 forthetemperature–forcingcrosscorrelationRTf,tr)aswellastheexceptionallyparticularlyrapidfall-offatlargeDtforthetemperature–storagecrosscorrelationRTS,tr.Otherpropertiesofsolutionstothestochasticeq.40aregiveninL3.

A final interesting property is the predictability of fRn, also discussed in L3.sincefortimes<t,fRnisclosetothelongmemoryfGnprocess,thetwocanbothbewellpredicted(forfGn,theskill–withinfinitepastdata-becomesperfectinthelimith->1/2(H->0)),however,beyondtherelaxationtimet,fRncannotbewellpredicted,itapproachesawhitenoise.

Rh Δt( ) = −Nh2

Γ −h( )Δt−1−h +O Δt−1−2h( ) : 0 < h <1

1< h < 2; Δt >>1

General Dt≪t Dt≫ t Range

TemperatureAutocorrelation

function:

Temperature-Forcingcross-correlationfunction:

VarianceRatiooftemperatureversusforcing:

Table 1: Above the thick horizontal line: Dimensional formulae summarizing auto and crosscorrelationfunctionsofthewhitenoiseforcedFEBE,theautocorrelationsaredimensionalizedversionofthose inL3. t is therelaxationtime,tr is theresolution. isaresolutiontr fRnprocess, thespecialh=1(EBE)istheOrnstein-Uhlenbeckprocess;intheh=1/2(HEBE)specialcase,therearelogarithmiccorrectionstoRTT,trseeappendixBofL3.Toensuresmallscaleconvergence,allquantitiesaretakenatresolutiontr,sothat (forRTT,tr,thisisonlyimportantforh≤½,forRTf,tr, it isimportantforh<1). Alsoshown(bottomrow)isthetemperaturetotheforcingvarianceratio.ThenumericalconstantKisfromeq.35andcisfromeq.42.Notetheexactvaluec2=2forh=1andthatG(-h)<0for0<h<1.Whenh=1/2therearesmallscalelogarithmicdivergences.

Tτ r t( )Tτ r t − Δt( ) =

RTT ,τ r Δt( )

s2σ fτ r

2

Kh2

τ rτ

⎛⎝⎜

⎞⎠⎟h 1+ 2h( ) Δt

τ⎛⎝⎜

⎞⎠⎟

2h−1

+ ...⎡

⎣⎢⎢

⎤

⎦⎥⎥; τ r << Δt

RT ,τ r 0( ) =σ T ,τ r2 =

s2σ fτ r

2

Kh2

τ rτ

⎛⎝⎜

⎞⎠⎟

2h

+ ...−λ 2σ f ,τ r

2 τ rτ

⎛⎝⎜

⎞⎠⎟

Δt / τ( )−h−1Γ −h( )

0 < h <1/ 2

s2σ f ,τ r2

c2τ rτ

⎛⎝⎜

⎞⎠⎟1−

Γ 1− 2h( ) sin πh( )π

c2 Δtτ

⎛⎝⎜

⎞⎠⎟

2h−1

+ ...⎡

⎣⎢⎢

⎤

⎦⎥⎥

1/ 2 < h <11< h < 3/ 2

s2σ f ,τ r2

c2τ rτ

⎛⎝⎜

⎞⎠⎟1− Δt

τ⎛⎝⎜

⎞⎠⎟

2c2

2′G0,h u( )2 du

0

∞

∫ + ...⎡

⎣⎢⎢

⎤

⎦⎥⎥

3/ 2 < h < 2

s2σ f ,τ r2

2τ rτ

⎛⎝⎜

⎞⎠⎟e−Δt /τ

h = 1

Tτ r t( ) fτ r t − Δt( ) =

RTf ,τ r Δt( )

sσ f ,τ r2 τ r

τ⎛⎝⎜

⎞⎠⎟G0,h

Δtτ

⎛⎝⎜

⎞⎠⎟; τ r << τ ; τ r << Δt

RTf ,τ r 0( ) = Tτ r fτ r =sσ f ,τ r

2

Γ 2+ h( )τ rτ

⎛⎝⎜

⎞⎠⎟

h

; τ r << τ

ρTf =Tτ r fτ r

σ T ,τ rσ f ,τ r

=Kh

Γ 2+ h( )

0 < h < 2

Tτ r2 / fτ r

2

s2Kh−2 τ r

τ⎛⎝⎜

⎞⎠⎟

2h

0 < h <1/ 2

s2c−2τ rτ 1/ 2 < h < 2

Tτ r t( )

fτ r2 =σ 2 / τ r

4.4 ECS and TCR

Inordertomakeaccurateclimateprojectionsthroughoutthe21stcentury,thesinglemostimportantparameteristheclimatesensitivityssinceforanygivenforcing,thisdetermines theultimate(long time) temperaturechange. It isconventional toexpresss intermsofthechangeintemperaturefollowingaCO2doubling,thissisthe“EquilibriumClimateSensitivity”(ECS).Thisiseffectivelyachangeinunitsandcanbeeffectedusing the canonical value3.71W/m2 for aCO2 doubling. However, inordertomakeaccurateprojectionsthroughthe21stcentury,itisalsoimportanttohave accurate estimates of thememory, the “warming in the pipe” [Hansen et al.,2011].ThestandardwayofquantifyingthisisbycomparingtheECStotheTransientClimateResponse(TCR)thatisdefinedasthetemperaturechangefollowingaCO2doubling with forcing linearly increasing over 70 years: since the CO2 forcing islogarithmicintheCO2concentration,this isnearlyexactlya1%increaseperyear.ThesmallertheTCR/ECSratio,thelargerthememory.Notethatinthefollowing,weonlyconsiderstaticclimatesensitivitiesthatweindicatebys(nots0).

ThedimensionlessratioTCR/ECSisindependentofs,fortheFEBEmodel,itisonlyafunctionoft,handtheoretically–sincetheTCRisdefinedintermsofa“ramp”,theratiois:

;

(45)whereDtistheperiodoverwhichthedoublingoccurs(conventionally70years).

Fig. 6 shows a contour plot of the ratio as a function of exponent H andrelaxationtimest.TheFEBEparameters(blackcircle)estimatedfromthehistoricaldataandIPCCAR5forcingby[Procyketal.,2020]with90%confidenceintervalsareshown(dashedrectangle).Onecanseethattheratioisintherange≈0.65-0.75withmean≈0.7.Oncanalsoseethatthesinglebox(h=1)modelhasratiosnear1unlesstherelaxationtimeis20yearsorlonger(Dt/t<≈3.5).

AsHapproacheszero,weseethatthetransitionfromtherampresponsetotheequilibriumresponsetakesalongerandlongertime.Inthelimit ;theratioTCR/ECS=1/2,anexactresultascanbeeasilyseenbyusing intheFEBE

(eq.14).Theh=1(exponential)casehastooshortamemorytoberealisticsothatmost

GCMoutputshavebeenfittodoubleexponential(2box)models; forexample theIPCCAR5suggestsboxrelaxationtimesof8.5and409.5yearswithTCR/ECS=0.58.Similarly,theIPCCAR5(ch.9,seealso[Yoshimorietal.,2016]),obtainedTCR/ECS=0.56±0.19(90%confidencelimits)from23CMIP5GCMs.Fig.7showsthesemodelbased estimates compared to the FEBE estimates. We can see that the two arecompatiblealthoughtheFEBEdata-basedestimatehasamuchsmalleruncertaintythanthemodelbasedone.

TCR / ECS = G2,h Δt( ) / Δt = 1− 1Γ 2− h( )

Δtτ

⎛⎝⎜

⎞⎠⎟

−h

+ 1Γ 2− 2h( )

Δtτ

⎛⎝⎜

⎞⎠⎟

−2h

+ ... Δt >> τ

h→ 0+

limH→0+

Dh = 1

Fig. 6: A contour plot of TCR/ECS for various h values as well as the dimensionless ratio of the ramp time Dt to the relaxation time t; the canonical value of Dt is 70 years. The dashed lines show the 90% confidence intervals from the analysis in Procyk et al 2020 for both t and h; (using Dt = 70 years) the black circle represents the median values.

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Fig. 7: The TCR/ECS ratio for the FEBE model as a function of h for various relaxation times using the canonical ramp time constant Dt = 70 years with relaxation time t. The black circle and rectangle shows the mean and 90% confidence interval values found by [Procyk et al., 2020]. The horizontal dashed lines show the 90% confidence limits (IPCC AR5 ch. 9) found using CMIP5 GCMs.

4.5 Modelling the internal and externally forced variability: a simple FEBE model from 1825-2100

Wehaveseenthatwithasingleexponenth,theFEBEcorrectlypredictsbothhighandlowfrequencyscalingregimes,allowsittoconvincinglymodeltheinternallyandexternally forcedvariability, the twoare linkedby theearth’smultiscale (andscaling)storagemechanisms.Letusthereforetreatthetotalforcing asthesumofdeterministicpartF(t)andastochasticpartf(t)with<f(t)>=0:

(46)andsimilarlyforthetemperatureresponse:

(47)wherethesubscripts“e”and“i”areforexternallyforcedandinternalrespectively.Weproposethat satisfiesthestochasticFEBE:

(48)

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ℑ t( )

ℑ t( ) = F t( )+ f t( )

T t( ) =Te t( )+Ti t( )

ℑ t( )τH −∞Dt

HT +T = sℑ

If we identify the deterministic part with the externally forced response and thestochasticpartwiththeinternalvariability,thenthelinearityoftheFEBEimpliesthateachsatisfies theFEBEseparately. First, takeensembleaveragesanduse<f(t)>=<Ti(t)>=0:

(49)

(since ).Now,subtractthisfromeq.48toyield:

(50)ThereforeinthestochasticFEBE,theexternallyforcedandinternalvariabilitybothsimplyrepresentsourcesorsinksofheatenergythatareexternaltotheradiatingsurfacelayer.Whetherheatcomesfrominternalstorageorfromoutsidethesystem,all that matters is that it appears at the surface where it can participate in theradiativepartoftheenergybalance.

To illustrate how the FEBE models both the deterministic external andstochasticinternalvariability,let’smakeasimplemodelthatincludesthemboth(seefigs.8,9anddetails inappendixA). Themodelhasaresolutionof theweather -macroweathertransitionscale(takenas10days)anditsolvestheFEBEover100,000daysstarting in theyear1825,upto theyear2100. Thedeterministicpartof theforcing is taken to be zeroupuntil 1880, this is followedby a linearly increasingforcing-a“ramp”-upuntil theyear2020(≈51100days)afterwhichthe forcingremainsconstantatthatvalueuntiltheyear2100.Thestochasticpartoftheforcingis due to a Gaussianwhite noisewhose amplitude is such that the RMSmonthlyanomalyis±0.14K(asobserved,e.g.[Lovejoy,2017a]).Theforcingwasadjustedsothat the forced response was 1K in 2020 (close to the observed anthropogenicwarming),accordingtothismodel,thereisanother0.24Kofwarming“inthepipe”.More details including a scale by scale statistical analysis and comparison withstatisticalanalysesofglobaltemperatureseriesaregiveninappendixA.Inplaceofthis“toymodel”,[Procyketal.,2020]usesIPCCAR5forcingsandmakesquiterealistichindprojections(1880-present)andprojections(throughto2100)thatarewithintheuncertaintylimitsoftheIPCCAR5(CMIP5)GCMs.

τh −∞DthTe( )+Te = sF t( ); F = ℑ ; Te = T

Ti −∞( ) =0τh −∞Dt

hTi( )+Ti = sf t( )

Fig. 8: This shows the nondimensional temperature response (thin lines) for nondimensional forcing

(thick line) with unit sensitivity (s = 1). Each graph shows the results for h = 0 (bottom) to h = 1 (top, exponential), at intervals of 0.1; the red curve highlights the most realistic value h = 0.4. The plots are shown for the relaxation time t =102, 103, 104 days (upper left to bottom as indicated). The resolution of the calculations were taken to be the weather – macroweather transition scale (tw = 10 days), but this is only important for the stochastic internal variability; smooth deterministic forcings such as those here are insensitive to tw as long as it is much smaller than t.

0.2

0.4

0.6

0.8

1.0

0.2

0.4

0.6

0.8

1.0τ=100 days τ=1000 days

2020

1880

T /T∞ T /T∞

1825 21001825 2100

2020

1880

0.2

0.4

0.6

0.8

1.0τ=10000 days

T /T∞

1825 2100

2020

1880

F /F∞ F /F∞

F /F∞

T /T∞

F /F∞

Fig. 9: Thin black line: A simulation of the monthly resolution temperatures with resolution tw = 10 days, the sum of the externally forced model with h = 0.4, t = 1000 days (fig. 8, upper right) with the Gaussian white noise driven stochastic internal variability model averaged over a factor of three to simulate a monthly resolution series. The amplitude of the internal variability was chosen to match observations of the monthly resolution standard

deviation of global temperatures ( = 0.14C. Thick black line: at annual resolution (obtained by

averaging the previous over an additional factor of 12.

5. Conclusions

Beyond the deterministic forecasting limit of about ten days, GCMs areeffectivelystochastic.Inaddition-atleastwiththeexternalforcingscenariosusedin the IPCC AR5 - the Representative Carbon Pathways – each GCM response toexternalforcingsisquitelinear[HébertandLovejoy,2018]:toagoodapproximation,climateprojectionsoverthenextcenturyappeartobea linearstochasticproblem[Hebert, 2017], [Hébert et al., 2020], [Procyk et al., 2020]. In addition, stochastic,linear monthly, seasonal and interannual scale (macroweather) temperatureforecastshaveskillcomparable–orbetter–thanthoseofGCMs[Lovejoyetal.,2015],[DelRioAmadorandLovejoy,2019].

20000 40000 60000 80000 100000

0.5

1.0

T oC

1880

2020

Days since 1825

Tτwt( )

2 1/2

Thesefacts–andtheurgencyofreducingforecastand(especially)projectionuncertainties – militates for the rapid development of stochastic macroweathermodelsconstructeddirectlyatmacroweatherscales.Upuntilnow,suchmodelshavebeen based on the longmemory of the Earth’s atmosphere and climate that wasexploitedbyusingthescalingprinciple:thefactthatoverwiderangesofspatialscales,atmosphericdynamicsarescaling(seetheextensivereview[LovejoyandSchertzer,2013]). Predictingsuch longmemorystochasticprocessrequiresa longseriesofhistoricaldata,thereforemathematically,thesemethodsareeffectively“pastvalue”ratherthanconventional“initialvalue”typeproblems([Lovejoyetal.,2015]).

Atfirst,scalingwasinvokedtodirectlymediatetheforcingandtheresponse;ascalingClimateResponseFunction(CRF,[Rypdal,2012],[RypdalandRypdal,2014],[vanHateren,2013],[Hebert,2017],[Hébertetal.,2020],[Myrvoll-Nilsenetal.,2020]).Recently,itwassuggestedby[Lovejoy,2019a],thatthisphenomenologicalapproachcouldbemademorephysical(andrealistic)bysuggestingthatratherthanapplyingittotheCRFdirectly,thatthescalingprincipleshouldinsteadbeappliedtotheenergystoragemechanisms.Itwaspointedoutthatifthiswasdone,thatbothhighandlowfrequencymemories and exponentswould automatically be explained by a singlemore fundamental exponent h, the (fractional) order of the resulting FractionalEnergyBalanceEquation(FEBE).

Sincethen,anotherapproachstartingwithenergybalancemodelshasreachedthesameconclusionandhasshownhowthezero-dimensional(global)modelscanbe generalized to full 2Dmodels for temperature anomalies. Perhaps the mostsurprising result [Lovejoy, 2020a; b]was that an apparentlyminor change to theclassicalBudyko-SellersEnergyBalanceModels–theextensionfrom2Dtothefull(classical,integer-ordered)3Dcontinuumheatequation–genericallyimpliesthatthesurfacetemperaturesatisfiesthehalf-orderEBE(HEBE).Finally,ageneralizationtothe (3D) Fractional Heat Equation was shown to imply that the (2D) surfacetemperaturessatisfytheFEBE.

Inthispaper,wepursuedamorephenomenologicalapproachbyderivingtheFEBE from the scaling principle applied to the energy storage processes.Mathematically, the FEBE is a fractional relaxation equation whose deterministicversionhasbeenstudiedforsomewhile[MillerandRoss,1993],[Podlubny,1999]andwhose stationary stochastic version was studied in [Lovejoy, 2019b] in order tounderstandtheresponsetointernalforcing.WeshowedhowtheFEBEcanbesolvedusingGreen’sfunctionsandwelinkedthehighandlowfrequencylimitsoftheGreen’sfunction to fractional Gaussian noise (high frequencies dominated by internalvariability)andlowfrequenciestopowerlawCRFsforclimateprojections.ThefactthatthestochasticFEBEisagoodmodelfortheinternalvariabilitymeansthattheinternalvariabilitycanbeusedtodeterminetheresponseofthesystemtoexternalforcing, a goal that has beenpursued ever since [Leith, 1975] proposedusing theFluctuationDissipationTheoremforthispurpose.

Weshowedhowtheresponsetoperiodicforcingcanbeconvenientlyhandledbyconsideringcomplexclimatesensitivitiesnotablytotakeintoaccountthephaselagbetweentheannualmaximumforcingandthetemperatureresponse.Thisopensthepossibilityofusingtheannualcycletoestimatethemodelparameters,exponents,relaxationtimesandusual(static)climatesensitivities.WeshowedthatalthoughtheFEBEobeysNewton’slawofCooling,thattheheatfluxcrossingasurfacenonethelessdependsonthefractionaltemperaturetimederivativeratherthantheusualintegerorderedone(thisisanextensionofthehalf-orderresultthatfollowsquiteclassicallyfromthecontinuummechanicsheatequation).WealsoderivedthetheoreticalFEBEratiooftransienttoequilibriumclimatesensitivity(TCR/ECS)andshowedthatitwas≈0.70,i.e.withinthe90%confidenceintervalofGCMestimates(IPCCAR5).

Finally,weputboththedeterministic(externalforcing)andstochastic(internalforcing)togetherintoasimple140yearrampforcingmodelfortheindustrialepochwarming.Themodelassumedthatbetween1880and2020therewas1Kofwarmingandpredictedthatifthecurrentforcingwasheldconstantfrom2020onwards,thattherewouldbeanother0.24K“inthepipe”.WethenstatisticallyevaluatedthemodelusingHaar fluctuationsandshowedthat theRMSfluctuationsas functionsof timescalewerealreadyfairlyrealistic,butthatpreciseparameterestimatesaredifficult.

Althoughinthispaperweonlyconsideredthezero-dimensionalFEBE,variousextensions to 2D have already been proposed [Lovejoy, 2020a]. When forced bystochastic internal anddeterministic external forcing, theFEBE thus constitutes anewclassoflowfrequencyatmosphericmodelthatuseshistoricaldataandthelongmemory to make macroweather forecasts and climate projections. There areindications[DelRioAmadorandLovejoy,2019],[Procyketal.,2020]thatthesearemore skillful and are less uncertain than conventional approaches. The futurechallengewillbetoimprovethisapproachandextenditforpredictingandprojectingotheratmosphericparameters-especiallyforprecipitation.

7. Acknowledgements Thisprojectreceivednospecificfunding,butS.Lovejoybenefittedfromasmallgrantfrom the National Science and Engineering research Council (Canada). WeacknowledgediscussionswithD.ClarkeandC.Penland.

Appendix A: To illustrate how the FEBE models both the deterministic external and

stochastic internal variability, in section 4.5 we presented a simple model thatincludesthemboth(figs.8,9).Inthisappendixwegiveafewtechnicaldetailsandastatisticalvalidationofthemodelusingglobaltemperatureseries.

First, consider the externally forced variability using a linearly increasing(ramp) model of the industrial epoch anthropogenic warming. Fig. 8 shows theforcingmodel(thickblack)nondimensionalizedwiththelong-timeforcing withtheresponsenondimensionalizedby the long time(equilibrium) temperature ;

since ,thecurvesareindependentofs.Themodelhasaresolutionoftheweather-macroweathertransitionscale(takenas10days)andstartingintheyear1825,itsolvestheFEBEupto100,000days(totheyear2100).Theforcingistakentobezeroupuntil1880,followedbyalinearlyincreasingforcing-a“ramp”-upuntiltheyear2020(≈51100days)afterwhichtheforcingremainsconstantatthatvalueuntiltheyear2100.

Inthehistoricalpart,theforcingisindeedroughlyofthisform,theamplitudeestimated for2020atclose to2.4W/m2 (CO2eq forcing, since thebeginningof theindustrial epoch). The resolution of the simulation was the typical weather-macroweathertransitiontimetw≈10days,butaslongast >>twtheresultsforthissmooth,deterministicsimulationwillbeinsensitivetothis.However,asexplainedinsection4.2(eq.42,table1),theresolutionisfundamentallyimportantforstochastic,noise driven simulations of internal variability. The future part of the modelcorrespondstoacompletehalttoallemissions(andotherforcing)in2020.

Before continuing, it’s worthmaking a quick comment about the numericaltechniquesthatwereusedforthesimulations.Numericalconvolutionalgorithmsareveryefficient,sothatweusedtheFEBEsolutioninconvolutionform.SincetheFEBEGreen’sfunctionshavepowerlawbehaviouratbothhighandlowfrequencies,careisrequiredat small and large scales. Toavoidhigh frequencyproblems, (especiallysensitivesincebelow,theFEBEisstochasticallyforcedwithaGaussianwhitenoise),wethereforeusedthestepresponseG1,HGreen’sfunctionthathasasmoothersmallt, behaviour (th) rather than the impulse response t-1+h behaviour. After theconvolution, the resulting serieswas thennumericallydifferentiated toobtain thesolution.Thelowfrequenciesalsohavesomeissuesbecauseofthelongmemory:thestorage. For simulationsofpure fGn this is anontrivial issue. Herehowever, forscalest≫t,theGreen’sfunctionthatdeterminesthememory-althoughstillpowerlaw(eq.31,t-1-h)-ismuchshorterthanforpurefGn(t-1+h)sothatitsufficestomakesimulationsoftotallengthafewmultiplesoft. Moredetailsaregivenin[Lovejoy,2019b].

Using these techniques, three plots are shown in fig. 8, one for each of therelaxation scales t = 100, 1000, 10,000, simulated days (upper left, upper right,bottom).Theupperrightsimulationwitht=1000days(≈3years)hasavalueclosetothevariousrelaxationscaleestimatesandthecurvecorrespondingtotherealisticparameterh=0.4(red)isfairlyclosetotheactualreconstructedwarmingovertheperiod.Thesecurvesillustratewhatmighthappenifemissionssuddenlystopped,thepartsbeyond2020showthethermal“inertia”inthesystem.Weseethatwithonly some exceptions at the very long t = 10,000 (≈30 years, bottom) that theresponseisverylinearovertherampregion,sothattheresponseattheendofthe140yearrampisalittlelargerthantheTCR(definedaftera70yearramp,seefig.6,7:theTCR/ECScontoursinthefigurearefairlyhorizontal,notverysensitivetoDt/t).

F∞

T∞

s =T∞ /F∞

TheH =0.4 curveon thegraphwith themost realistict (=1000daysupperright)showsthat =0.81(theTCR/ECSis≈0.7seefigs.6,7).Ifwe

takethecurrentincreaseintemperaturesincepre-industrialepochtobe =1K, then theequilibriumtemperature =ECS=1/0.81=1.24K. Ifwe take the

corresponding forcing since the pre-industrial epoch to be 2.4 W/m2, then s =1.24/2.4=0.52K/W/m2;usingthevalue3.71W/m2/CO2doubling,weobtains=1.92C/CO2doubling. ComparingthiswiththeIPCCAR5rangeof1.5-4.5K/CO2doubling(90%confidence),weseethatitisaabout4%belowthemeanandeasilywithintherange,itisclosetothe[Procyketal.,2020]estimate2.0±0.4K/W/m2.Thisis surprisingly good confirmation for a model based only on the scaling storageassumption.

We can now combine the externally forced part of themodel (fig. 8)with asimulationofthemonthlyresolutioninternallyforcedtemperatures:thesumofusingtheh=0.4,t=1000daymodel(middleupperleft)averagedoverafactorofthreetosimulateamonthlyresolutionseries.Fig.9showstheresultatmonthlyresolution(thin)andwhenthisisaveragedfurthertoannualresolution(thick);theamplitudeofthenoisecontributionwaschosentoequal±0.14Katmonthlyresolution,closetotherealglobalstatistics.Duetothe(singular)smallscalefGnbehaviour,atannualresolution,theamplitudeofthenoiseisthusreducedbythefactor12H=12-0.1=0.78(eq.38);i.e.theannualanomalyfluctuationamplitudeis0.11=12H0.14K.

T 2020( ) /T ∞( )T 2020( )

T ∞( )

Fig. 10a: The average Haar fluctuation structure function for six globally averaged

monthly resolution temperature time series (brown) compared with ramp model simulations with h = 4/10 (H = h - ½ = -1/10) analyzed over the simulated period 1880-2012 (black; these are the series analysed in [Lovejoy, 2017a]). The simulations have been calibrated so that their RMS variability at one year is the same as the data. Simulation results are shown for relaxation times varying from 10-2 days (top left) to 104 days (dashed, bottom, left) each labelled at left by log10t. The reference (red) line has the theoretical slope (HI) for fGn i.e. infinite relaxation time.

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Fig. 10b: The same as fig. 10a except for h = 1/2, hence H = 0 (red, flat reference line).

Inspiteofitssimplicity,fig.9showsthatthemodelisapparentlyquiterealistic,

yetproperstatisticalanalysisisneededtoconfirmthis.Aconvenientwaytocomparethemodelstatisticswiththoseofgloballyaveragedtemperatureseriesistoestimatetheir rootmean square Haar fluctuations. For a seriesT(t), the Haar fluctuationDT(Dt) over an intervalDt is the difference between the average of the first andsecondhalvesoftheinterval. Fig.10a,bshowstherootmeansquare(RMS)Haar

fluctuations ( , “< >” indicates averaging over all the available

seriesanddisjointintervals)forsixgloballyaveragedseriesfrom1880-2012(thickbrown)comparedtosimulatedserieswithvariousrelaxationtimestrangingfrom10-2 (top) to 104 days (bottom) forh = 4/10 (fig. 10a) andh = 1/2 (fig. 10b, theempiricalcurveisrepeatedoneach).Asmentioned,thesimulationswere“calibrated”by adjusting the amplitude of the Gaussian noise part of the forcing so that theamplitudeofthemodelandempiricalfluctuationswereequalatannualscales.Themonthlyseriesareidenticaltothoseanalysedindepthin[Lovejoy,2017a];fulldetailsandreferencesaregiventhere.

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The Haar fluctuations combine averaging and differencing so that theirinterpretation is straightforward. When the fluctuations increase with scale (i.e.regimes where the slopes in the plot are positive), and the raw fluctuations aremultipliedbyafactor2(astheyarehere,thisisthecanonicalfactor),thedifferencingdominatesandthevaluesareclosetotheRMSdifferencesthatquantifythetypicalchangeintemperatureoveragiventimeinterval.Forexample,thefigureshowthatatcentennialscales(farright),thetypicalfluctuationsareoftheorderof1K.InscalerangesoverwhichtheRMSfluctuationsdecrease,theaveragingdominatesandthevaluesareclosetotheanomalyfluctuationsi.e.thetemporalmeanoftheseriesoverDtafteritsoverallmeanhasbeenremoved.

Over the range of 2 months (the smallest Dt) to about 10 years, theirapproximateslope(indicatedinthebythestraightreferencelinesinfig.10a,b)isaboutH=-0.1.Inthisdecreasingmacroweatherregime,successivefluctuationstendto cancel eachother out, the series appears to be stable. ForDt ≈>10 years, thefluctuationsstartincreasing:theclimateregime.10yearsisthetypicalscaleatwhichtheresponsetoanthropogenicforcingexceedstypicalresponsestointernalforcing,atthesescales,fluctuationsnolongertendtocanceleachotherout,theseriesappearstobeunstable.

Accordingtothehighfrequencytheory,wedevelopedinsection4.2,fortimescalesmuchsmallerthantherelaxationscalet,thestochasticpartoftheresponseisanfGnwithRMSexponentH=h-1/2,sothatwhenDt≪tislargeenough .Fig.10ashowsthesimulationresultsfortheempiricallyestimatedH=4/10andthetheoretical(larget)H=-1/10referencelineisshowninred.Evenwhent=104days(≈30years,dashed) thesmallDt slopeof thesimulationS(Dt) isstilla littlehigh;although overall the fit is not bad. When we compare the simulation structurefunctionsfordecreasingt,wenoteasurprisingfeature:thelog10S(Dt)versuslog10Dtplots are remarkably linear over several orders ofmagnitude butwith (absolute)slopesthatsystematicallyincreaseastdecreases.Withoutthedeterministicrampthatdominatesthelowerfrequencies,this“pseudo-scaling”wasinvestigatedfurtherinL3.Ratherthanbeingatruescalingregime,itisinrealityaveryslowtransitionalregimefromhighfrequencyDt-HtoDt-1/2(independentofh,H).Fig.10bshowsthesamesimulationsbutwithh =1/2, (H =0). In this case, theeffect is evenmorepronouncedwiththedatabeingmorecompatiblewitht≈103days(3years).Fromthesesimulations,weseethatwhenonlyafactorof100or1000inscaleisavailableforanalysis,itismaybedifficulttoaccuratelymeasurehandt.

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