1

Fractional relaxation noises, motions and the fractional 1

energy balance equation 23

ShaunLovejoy4Physics,McGillUniversity,5

3600Universityst.6Montreal,Que.H3A2T87

Canada8

Abstract: 9Weconsider thestatisticalpropertiesofsolutionsof thestochastic fractional10

relaxation equation (a fractional Langevin equation) that has been proposed as a11model for the Earth’s energy balance. In this equation, the (scaling) fractional12derivativetermmodelstheenergystorageprocessesthatoccuroverawiderangeof13scales. Up until now, stochastic fractional relaxation processes have only been14consideredwithRiemann-Liouville fractionalderivatives in thecontextof random15walkprocesseswhereityieldshighlynonstationarybehaviour.Instead,weconsider16thestationarysolutionsoftheWeylfractionalrelaxationequationswhosedomainis17

totratherthan0tot.18WedevelopaframeworkforhandlingfractionalequationsdrivenbyGaussian19

whitenoiseforcings.Toavoiddivergences,wefollowtheapproachusedinfractional20Brownian motion (fBm). The resulting fractional relaxation motions (fRm) and21fractionalrelaxationnoises(fRn)generalizethemorefamiliarfBmandfGn(fractional22Gaussiannoise).Weanalyticallydetermineboththesmallandlargescalelimitsand23showextensiveanalyticandnumericalresultsontheautocorrelationfunctions,Haar24fluctuationsandspectra.Wedisplaysamplerealizations.25

Finally,wediscussthepredictionoffRn,fRmwhich–duetolongmemoriesisa26pastvalueproblem,notaninitialvalueproblem.Wedevelopananalyticformulafor27the fRn forecast skill and compare it to fGn. The large scalewhite noise limit is28attained inaslowpower lawmannerso thatwhenthe temporal resolutionof the29series is small compared to therelaxation time(of theorderofa fewyears in the30Earth),fRncanmimicalongmemoryprocesswitharangeofexponentswiderthan31possiblewithfGntofBm.Wediscusstheimplicationsformonthly,seasonal,annual32forecastsoftheEarth’stemperatureaswellasforprojectingthetemperatureto205033and2100.34

1. Introduction: 35Overthelastdecades,stochasticapproacheshaverapidlydevelopedandhave36

spread throughout the geosciences. From early beginnings in hydrology and37turbulence,stochasticityhasmadeinroadsinmanytraditionallydeterministicareas.38This is notably illustrated by stochastic parametrisations of Numerical Weather39

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2

Predictionmodels,e.g.[Buizzaetal.,1999],andthe“random”extensionsofdynamical40systemstheory,e.g.[Chekrounetal.,2010].41

Purestochasticapproacheshavedevelopedprimarilyalongtwodistinctlines.42Oneistheclassical(integerordered)stochasticdifferentialequationapproachbased43ontheItôorStratonivchcalculiithatgoesbacktothe1950’s(seetheusefulreview44[Dijkstra, 2013]). The other is the scaling strand that encompasses both linear45(monofractal, [Mandelbrot, 1982]) and nonlinear (multifractal) models (see the46review[LovejoyandSchertzer,2013]) thatarebasedonphenomenological scaling47models, notably cascade processes. These and other stochastic approaches have48playedimportantrolesinnonlinearGeoscience.49

Upuntilnow,thescalinganddifferentialequationstrandsofstochasticityhave50hadsurprisinglylittleoverlap. Thisisatleastpartlyfortechnicalreasons:integer51orderedstochasticdifferentialequationshaveexponentialGreen’sfunctionsthatare52incompatiblewithwiderangescaling. However,thisshortcomingcan–at least in53principle-beeasilyovercomebyintroducingatleastsomederivativesoffractional54order.Oncethe(typically)adhocrestrictiontointegerordersisdropped,theGreen’s55functionsare“generalizedexponentials”andthesearebasedinsteadonpowerlaws56(see thereview[Podlubny,1999]). The integerorderedstochasticequations that57havereceivedmostattentionarethustheexceptional,nonscalingspecialcases. In58physicstheycorrespondtoclassicalLangevinequations;ingeophysicsandclimate59modelling,theycorrespondtotheLinearInverseModelling(LIM)approachthatgoes60backto[Hasselmann,1976]laterelaboratednotablyby[PenlandandMagorian,1993],61[Penland, 1996], [Sardeshmukh et al., 2000], [Sardeshmukh and Sura, 2009] and62[Newman,2013].AlthoughLIMisnottheonlystochasticapproachtoclimate,intwo63recent representative multi-author collections ([Palmer and Williams, 2010] and64[FranzkeandO'Kane,2017]),all32paperssharedthe integerorderedassumption65(thesingleexceptionbeing[Watkins,2017]).66

Underthetitle“Fractaloperators”[Westetal.,2003],reviewsandemphasizes67that in order to yield scaling behaviours, it suffices that stochastic differential68equationscontainfractionalderivatives.However,whenitisthetimederivativesof69stochasticvariablesthatarefractional-fractionalLangevinequations(FLE)-then70therelevantprocessesaregenerallynon-Markovian[Jumarie,1993],sothatthereis71no Fokker-Planck (FP) equation describing the corresponding probabilities.72Furthermore, we expect that - as with the simplest scaling stochastic model –73fractional Brownian motion (fBm, [Mandelbrot and Van Ness, 1968]) - that the74solutionswillnotbesemi-martingalesandhencethattheItôcalculususedforinteger75orderedequationswillnotbeapplicable(see[Biaginietal.,2008]).Thismayexplain76the paucity of mathematical literature on stochastic fractional equations (see77however [Karczewska and Lizama, 2009]). In statistical physics, starting with78[MainardiandPironi,1996],[Lutz,2001]andhelpedwithnumerics,theFLE(anda79moregeneral“GeneralizedLangevinEquation”[KouandSunneyXie,2004],[Watkins80et al., 2019]) has received a little more attention as a model for (nonstationary)81particlediffusion(see[Westetal.,2003]foranintroduction,or[Vojtaetal.,2019]for82amorerecentexample).83

These technicaldifficultiesexplain theapparentparadoxofContinuousTime84RandomWalks (CTRW)andotherapproaches toanomalousdiffusion that involve85

3

fractionalequations. WhileCTRWprobabilitiesaregovernedby thedeterministic86fractional ordered Generalized Fractional Diffusion equation (e.g. [Hilfer, 2000],87[Coffeyetal.,2011]),thewalksthemselvesarebasedonspecificparticlejumpmodels88rather than (stochastic) Langevin equations. Alternatively, a (spatially) fractional89ordered Fokker-Planck equation may be derived from an integer-ordered but90nonlinearLangevinequationforadiffusingparticledrivenbyan(infinitevariance)91Levymotion[Schertzeretal.,2001].92

In nonlinear geoscience, it is all too common for mathematical models and93techniques developed primarily for mathematical reasons, to be subsequently94appliedtotherealworld. Thisapproach-effectivelystartingwithasolutionand95thenlookingforaproblem-occasionallysucceeds,yethistoricallytheconversehas96generallyprovedmorefruitful. TheproposalthatanunderstandingoftheEarth’s97energybalancerequirestheFractionalEnergyBalanceEquation(FEBE,announced98in [Lovejoy, 2019a]) is an example of the latter. First, the scaling exponent of99macroweather (monthly, seasonal, interannual) temperature stochastic variability100wasdetermined(HI≈-0.085±0.02)andshowntopermitskillfulglobaltemperature101predictions,[Lovejoy,2015],[Lovejoyetal.,2015],[DelRioAmadorandLovejoy,2019].102Then, themultidecadaldeterministic response toexternal (anthropogenic) forcing103wasshowntoalsoobeyascalinglawbutwithadifferentexponent[Hebert,2017],104[Lovejoyetal.,2017](HF≈-0.5±0.2). Itwasonlylaterthatitwasrealizedthatthe105FEBEnaturallyaccountsforboththehighandlowfrequencyexponentswithH=HI+1061/2andHF=-HwiththeempiricalexponentsrecoveredwithaFEBEoforderH≈1070.42±0.02.TherealizationthattheFEBEfitthebasicempiricalfactsmotivatedthe108presentresearchintoitsstatisticalproperties.109

The FEBE is a stochastic fractional relaxation equation, it is the FLE for the110Earth’s temperature treated as a stochastic variable. The FEBE determines the111Earth’s global temperature when the energy storage processes are scaling and112modelled by a fractional time derivative term. Whereas earlier approaches ([van113Hateren,2013],[Rypdal,2012],[Hebert,2017],[Lovejoyetal.,2017])postulatedthat114theclimateresponsefunctionitselfisscaling,theFEBEinsteadsituatesthescalingin115theenergystorageprocesses.116

TheFEBEdiffers fromtheclassicalenergybalanceequation(EBE) inseveral117ways. Whereas the EBE is integer ordered and describes the deterministic,118exponential relaxation of the Earth’s temperature to thermodynamic equilibrium119(Newton’slawofcooling),theFEBEisbothstochasticandoffractionalorder. The120FEBEunitestheforcingdueinternalandexternalvariabilities.Whereastheformer121represents the forcing and response to the unresolved degrees of freedom - the122“internal variability” - and is treated as a zero mean Gaussian noise, the latter123represents the external (e.g. anthropogenic) forcing and the forced response124modelled by the (deterministic) ensemble average of the total forcing.125Complementarywork- tobereportedshortly - focusesonthedeterministicFEBE126equationanditsapplicationtoprojectingtheEarth’stemperatureto2100.127

AnimportantbutsubtleEBE-FEBEdifferenceisthatwhereastheformerisan128initialvalueproblemwhoseinitialconditionistheEarth’stemperatureatt=0,the129FEBE is effectivelyapastvalueproblemwhoseprediction skill improveswith the130amountofavailablepastdataand -dependingon theparameters - it canhavean131

4

enormousmemory.Tounderstandthis,recallthatanimportantaspectoffractional132derivativesisthattheyaredefinedasconvolutionsovervariousdomains.Todate,133themainone thathasbeenapplied tophysicalproblems is theRiemann-Liouville134(RL) fractional derivative in which the domain of the convolution is the interval135betweenaninitialtime=0andalatertimet.Thisistheexclusivedomainconsidered136in Podlubny’s mathematical monograph on deterministic fractional differential137equations[Podlubny,1999]aswellasinthestochasticfractionalphysicsdiscussedin138[West et al., 2003], [Herrmann, 2011], [Atanackovic et al., 2014], andmost of the139papers in [Hilfer, 2000] (with thepartial exceptionsof [Schiessel etal., 2000], and140[NonnenmacherandMetzler,2000]).AkeypointoftheFEBEisthatitisinsteadbased141onWeyl fractional derivatives i.e. derivatives defined over semi-infinite domains,142herefrom tot.143

IntheEBE,energystorageismodelledbyauniformslabofmaterialimplying144thatwhenperturbed,thetemperatureexponentiallyrelaxestoanewthermodynamic145equilibrium.However,theactualenergystorageinvolvesahierarchyofmechanisms146and theassumption that this storage is scaling is justifiedby theobserved spatial147scalingofatmospheric,oceanicandsurface(e.g.topographic)structures(reviewed148in[LovejoyandSchertzer,2013]).Aconsequenceisthatthetemperaturerelaxesto149equilibriuminapowerlawmanner.150

This is a phenomenological justification for the FEBE where the fractional151derivative of order H is an empirically determined parameter with H = 1152corresponding to the classical (exponential) exception. Alternatively, in a recent153submission, [Lovejoy, 2019b] used Babenko’s operator method to show that the154specialH=1/2FEBE-theHalf-orderedEnergyBalanceEquation(HEBE)-couldbe155derived analytically from the classical Budyko-Sellers energy balance models156([Budyko,1969],[Sellers,1969]).ToobtaintheHEBE,itisonlynecessarytoimprove157the mathematical treatment of the radiative boundary conditions in the classical158energytransportequation.159

Thepurposeofthispaperistounderstandvariousstatisticalpropertiesofthe160solutionsofnoisedrivenWeylfractionaldifferentialequations.WefocusontheWeyl161fractional relaxation equation that underpins the FEBE, its infinite range of162integrationisneededinordertoobtainstatisticallystationarysolutions“fractional163Relaxationnoise”(fRn)-anditsintegral“fractionalRelaxationmotion”(fRm)with164stationaryincrements.fRn,fRmaredirectextensionsofthewidelystudiedfractional165Gaussiannoise(fGn)andfractionalBrownianmotion(fBm)processes.Wederivethe166main statistical properties of both fRn and fRm including spectra, correlation167functions and (stochastic) predictability limits needed for forecasting the Earth168temperature([Lovejoyetal.,2015],[DelRioAmadorandLovejoy,2019])orprojecting169itto2050or2100[Hébertetal.,2020].170

The choice of a Gaussianwhite noise forcingwasmade both for theoretical171simplicity but also for physical realism. While the temperature forcings in the172(nonlinear) weather regime are highly intermittent, multifractal, in the lower173frequencymacroweatherregimeoverwhichtheFEBEapplies,theintermittencyis174lowsothatthetemperatureanomaliesarenotfarfromGaussian([Lovejoy,2018]).175ResponsestomultifractalorLevyprocessFEBEforcingsarelikelyhowevertobeof176interestelsewhere.177

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5

Thispaperisstructuredasfollows.Insection2wepresenttheclassicalmodels178offractionalBrownianmotionandfractionalGaussiannoiseassolutionstofractional179Langevin equations and define the corresponding fractional Relaxation motions180(fRm)andfractionalRelaxationnoises(fRn)asgeneralizations.Wedevelopageneral181frameworkforhandlingGaussiannoisedrivenlinearfractionalWeylequationstaking182careofbothhighandlowfrequencydivergenceissues.ApplyingthistofBm,fRmwe183show that they both have stationary increments. Similarly, application of the184frameworktofGnandfRnshowsthattheyarestationarynoises(i.e.withsmallscale185divergences).Insection3wederiveanalyticformulaeforthesecondorderstatistics186including autocorrelations, structure functions, Haar fluctuations and spectra that187determineall thecorrespondingstatisticalproperties. Insection4wediscuss the188importantproblemofpredictionderivingexpressionsforthetheoreticalprediction189skillasafunctionofforecastleadtime.Insection5weconclude.190

2. Unified treatment of fBm and fRm: 191

2.1 fRn, fRm, fGn and fBm 192In the introduction, we outlined physical arguments that the Earth’s global193

energybalancecouldbewellmodelledbythe(linearized)fractionalenergybalance194equation,moredetailswillbepublishedelsewhere.TakingTasthegloballyaveraged195temperature, tr as the characteristic time scale for energy storage/relaxation196processes, F as the (stochastic) forcing (energy flux; power per area), and l the197climatesensitivity(temperature increaseperunit fluxof forcing) theFEBEcanbe198writteninLangevinformas:199

, (1)200

where(for0<H<1)thefractionalderivativesymbol isdefinedas:201

, (2)202whereGisthestandardgammafunction.Derivativesofordern>1canbeobtained203usingn=H+mwheremistheintegerpartofn,andthenapplyingthisformulatothe204mthordinaryderivative.Themaincasestudiedinapplicationsisa=0;theRiemann-205

Liouville fractionalderivative , herewewillbe interested in ; theWeyl206

fractionalderivative .207Sinceequation1islinear,bytakingensembleaverages,itcanbedecomposed208

into deterministic and random components with the former driven by the mean209forcingexternaltosystem<F>,andthelatterbythefluctuatingstochasticcomponent210F-<F>representingtheinternalforcingdrivingtheinternalvariability.Elsewhere211wewill consider thedeterministicpart, in the following,weconsider thesimplest212purely stochasticmodel inwhich <F> = 0 andF = g where g is a Gaussian “delta213correlated”whitenoise:214

τrH

aDtHT( )+T = λF

aDtH

aDtHT =

1Γ 1−H( )

t − s( )HʹT s( )ds

a

t

∫ ; ʹT =dTds

0DtH

a=−∞

−∞DtH

6

. (3)215In[Hebert,2017],[Lovejoyetal.,2017],[Hébertetal.,2020]itwasarguedonthebasis216ofanempiricalstudyofocean-atmospherecouplingthattr≈2years(recentwork217indicatesavaluesomewhathigher)andin[Lovejoyetal.,2015]and[DelRioAmador218andLovejoy,2019]thatthevalueH≈0.4reproducedboththeEarth’stemperature219both at scales >> tr as well as formacroweather scales (longer than the weather220regimescalesofabout10days)butstill<tr.221

When0<H<1,eq.1withg(t)replacedbyadeterministicforcingisafractional222generalization of the usual (H = 1) relaxation equation; when 1 < H < 2, it is a223generalization of the usual (H = 2) oscillation equation, the “fractional oscillation224equation”,seee.g.[Podlubny,1999].Thisclassificationisbasedonthedeterministic225equations; for the noise driven equations, we find that there are two critical226exponentsH=1/2andH=3/2andhencethreeranges.Althoughwefocusonthe227range0<H<3/2(especially0<H<1/2),wealsogiveresultsforthefullrange0<H228<2thatincludesthestrongoscillationrange.229

Tosimplifythedevelopment,weusetherelaxationtimettonondimensionalize230time i.e. to replace time by t/tr to obtain the canonicalWeyl fractional relaxation231equation:232

(4)233

for the nondimensional process UH. The dimensional solution of eq. 1 with234nondimensionalg= lFissimplyT(t)=tr-1UH(t/tr)sothatinthenondimensionaleq.2354, the characteristic transition “relaxation” time between dominance by the high236frequency(differential)andthelowfrequency(UHterm)ist=1.Althoughwegive237resultsforthefullrange0<H<2-i.e.boththe“relaxation”and“oscillation”ranges238–forsimplicity,werefertothesolutionUH(t)as“fractionalRelaxationnoise”(fRn)239andtoQH(t)as“fractionalRelaxationmotion”(fRm).NotethatwetakeQH(0)=0so240thatQHisrelatedtoUHviaanordinaryintegralfromtime=0totandthatfRnisonly241strictlyanoisewhenH≤1/2.242

Indealingwith fRnand fRm,wemustbecarefulofvarioussmalland larget243divergences. For example, eqs. 1 and 4 are the fractional Langevin equations244correspondingtogeneralizationsof integerorderedstochasticdiffusionequations:245thesolutionwiththeclassicalH=1valueistheOhrenstein-Uhlenbeckprocess.Since246g(t) isa“generalizedfunction”-a“noise”- itdoesnotconvergeatamathematical247instantintime,itisonlystrictlymeaningfulunderanintegralsign.Therefore,amore248standard form of eq. 4 is obtained by integrating both sides by order H (i.e. by249differentiatingby-H):250

, (5)251252

(seee.g.in[KarczewskaandLizama,2009],forthecorrespondingRiemann-Liouville253fractionalderivative relaxationequation). Thewhitenoise forcing in theabove is254

γ s( ) = 0; γ s( )γ u( ) = δ s − u( )

−∞DtH +1( )UH = γ t( ); UH = dQH

dt

UHt( ) = − −∞

Dt−HU

H+

−∞Dt−Hγ = −

1Γ H( )

t − s( )H−1UHs( )ds

−∞

t

∫ +1

Γ H( )t − s( )H−1 γ s( )ds

−∞

t

∫

7

statisticallystationary;weshowbelowthatthesolutionforUH(t)isalsostatistically255stationary.ItistemptingtoobtainanequationforthemotionQH(t)byintegratingeq.2564from tottoobtainthefractionalLangevinequation: whereW257

is Wiener process (a standard Brownian motion) satisfying .258UnfortunatelytheWienerprocessintegrated totalmostsurelydiverges,hence259werelateQHtoUHbyanintegralfrom0tot.260

fRn and fRm are generalizations of fractional Gaussian noise (fGn, FH) and261fractionalBrownianmotion (fBm,BH); this canbe seen since the latter satisfy the262simplerfractionalLangevinequation:263

(6)264

sothatFHisaWeylfractionalintegrationoforderHofawhitenoiseandifH=0,then265FHitselfisawhitenoiseandBHisit’sordinaryintegral(fromtime=0tot),astandard266Brownianmotion,itsatisfiesBH(0)=0(FHisnottobeconfusedwiththeforcingF).267

Before continuing, a comment is necessary on the use of the symbolH that268MandelbrotintroducedforfBminhonourofE.Hurstwhopioneeredthestudyoflong269memoryprocessesinNileflooding[Hurst,1951].First,notethateq.6impliesthat270the root mean square (RMS) increments of BH over intervals Dt grow as271

(see below). Since fBm is often defined by this scaling272

property,itisusualtousethefBmexponentHB=H+1/2.IntermsofHB,fromeq.6,273weseethatfGn(FH)isafractionalintegrationofawhitenoiseoforderH=HB-1/2,274whereas fBm isan integraloforderHB +1/2, the1/2beingaconsequenceof the275fundamental scaling of the Wiener measure whose density is g(t). While the276parametrizationintermsofHBisconvenientforfGnandfBm,inthispaper,wefollow277[Schertzer and Lovejoy, 1987] who more generally usedH to denote an order of278fractional integration. Thismoregeneralusage includes theuseofH asageneral279order of fractional integration in the Fractionally Integrated Flux (FIF) model280[SchertzerandLovejoy,1987]whichisthebasisofspace-timemultifractalmodelling281(see themonograph [LovejoyandSchertzer, 2013]). In theFIFgeneralization, the282densityofaWienermeasure(i.e.thewhitenoiseforcingineq.6)isreplacedbythe283density of a (conservative)multifractalmeasure. The scaling of thismultifractal284measureisdifferentfromthatoftheWienermeasuresothattheextra1/2termdoes285not appear. A consequence is that in multifractal processes, H simultaneously286characterizestheorderoffractionaldifferentiation/integration(H<0orH>0),and287has a straightforward empirical interpretation as the “fluctuation exponent” that288characterizestherateatwhichfluctuationsgrow(H>0)ordecay(H<0)withscale.289Incomparison,forfBm,thecriticalHdistinguishingintegrationanddifferentiationis290stillzero,butH>0orH<0correspondstofluctuationexponentsHB>1/2orHB<1/2;291which for these Gaussian processes is termed “persistence” and “antiperistence”.292TherearethereforeseveralH’sintheliteratureandbelow,wecontinuetodenotethe293orderofthefractionalintegrationbyHbutwerelateittootherexponentsasneeded.294

−∞ −∞DtHQ

H+Q

H=W

dW = γ t( )dt

−∞

−∞ Dt

H FH = γ t( ); FH =dBH

dt

ΔB

HΔt( )

2 1/2∝Δt H+1/2

8

2.2 Green’s functions 295Asusual,we can solve inhomogeneous lineardifferential equationsbyusing296

appropriateGreen’sfunctions:297

, (7)298

where and are Green’s functions for the differential operators299

correspondingrespectivelyto and .300

and are the usual “impulse” (Dirac) response Green’s functions301

(hencethesubscript“0”).Forthedifferentialoperator theysatisfy:302

. (8)303IntegratingthisequationwefindanequationfortheirintegralsG1,Hwhicharethus304“step”(Heaviside,subscript“1”)responseGreen’sfunctionssatisfying:305

306307

, (9)308whereQistheHeaviside(step)function.Theinhomogeneousequation:309

(10)310hasasolutionintermsofeitheranimpulseorastepGreen’sfunction:311

, (11)312

the equivalence being established by integration by parts with the conditions313andG1,H(0)=0.314

For fGn, the Green’s functions are simply the kernels of Weyl fractional315integrals:316

, (12)317

obtainedbyintegratingbothsidesofeq.6byorderH.Weconclude:318

FH t( ) = G0,HfGn( )

−∞

t

∫ t − s( )γ s( )ds

U H t( ) = G0,HfRn( )

−∞

t

∫ t − s( )γ s( )ds

G0,H

fGn( ) G0,H

fRn( )

−∞DtH

−∞DtH +1

G0,H

fGn( ) G0,H

fRn( )

ΞΞG0,H t( ) = δ t( )

ΞG1,H t( ) =Θ t( ); Θ t( ) = δ s( )ds−∞

t

∫

dG1,Hdt

=G0,H

Ξf t( ) = F t( )

f t( )= G0,H t − s( )F s( )ds =−∞

t

∫ G1,H t − s( ) ′F s( )ds−∞

t

∫ ; ′F s( )= dFds

F −∞( ) = 0

FHt( ) = 1

Γ H( )t − s( )

H−1γ s( )ds

−∞

t

∫

9

. (13)319

Similarly,appendixAshowsthatforfRn,duetothestatisticalstationarityofthewhite320noiseforcingg(t),thattheRiemann-LiouvilleGreen’sfunctionscanbeused:321

(14)322

with:323

, (15)324

sothat , aresimplythefirsttermsinthepowerseriesexpansionsofthe325corresponding fRn, fRm Green’s functions. These Green’s functions are often326equivalentlywrittenintermsofMittag-Lefflerfunctions,Ea,b:327

. (16)328

By taking integer H, the G functions reduce to factorials and G0,H, G1,H reduce to329exponentials, hence , are sometimes called “generalized exponentials”.330Finally,wenotethatattheorigin,for0<H<1,G0,HissingularwhereasG1,Hisregular331so that it isoftenadvantageous touse the latter (step) response function. These332Green’s functions are shown in figure 1. When 0 <H ≤ 1, the step response is333monotonic; in an energy balance model, this would correspond to relaxation to334thermodynamicequilibrium. When1<H <2,wesee that there isovershootand335oscillations around the long term value; it is therefore (presumably) outside the336physicalrangeofathermodynamicequilibriumprocess.337

In order to understand the relaxation process – i.e. the approach to the338asymptotic value 1 in fig. 1 for the step response G1,H - we need the asymptotic339expansion:340

G0,HfGn( ) =

t H−1

Γ H( );

G1,HfGn( ) =

t H

Γ H +1( );

−12≤ H <

12

UHt( ) = G0,H

fRn( ) t − s( )γ s( )ds−∞

t

∫

G0,HfRn( ) t( ) = −1( )

n+1 t nH−1

Γ nH( )n=1

∞

∑

G1,HfRn( ) t( ) = −1( )

n+1 t nH

Γ nH+1( )n=1

∞

∑0<H ≤2

G0,H

fGn( ) G1,H

fGn( )

G0,HfRn( ) t( ) = t H−1EH ,H −t H( ) Eα,β z( ) = zk

Γ αk +β( )k=0

∞

∑

G1,HfRn( ) t( ) = t HEH ,H+1 −t

H( ) H ≥ 0

G0,HfRn( ) G1,H

fRn( )

10

,341

wherezisthe(possiblyfractional)orderofintegrationoftheimpulseresponseG0,H.342Specifically,forz=0,1weobtainthespecialcasescorrespondingtoimpulseandstep343responses:344

345

, (17)346

(0<H<1,1<H<2)[Podlubny,1999],i.e.powerlawsint-HratherthantH.According347tothis,theasymptoticapproachtothestepfunctionresponse(bottomrowinfig.1)348isaslow,powerlawprocess.IntheFEBE,thisimpliesforexamplethattheclassical349CO2doublingexperimentwouldyieldapowerlawratherthanexponentialapproach350toanewthermodynamicequilibrium.ComparingthistotheEBE,i.e.thespecialcase351H=1,wehave:352

, (18)353

sothatwhenH=1,theasymptoticstepresponseisinsteadapproachedexponentially354fast.TherearealsoanalyticformulaeforfRnwhenH=1/2(theHEBE)discussedin355appendixCnotablyinvolvinglogarithmiccorrections.356

Gζ ,H t( ) = ∑n=0

∞ −1( )nΓ ζ − nH( ) t

ζ −1−nH ; t >>1

G0,HfRn( ) t( ) = −1( )n+1

n=1

∞

∑ t −1−nH

Γ −nH( ) ; t >>1

G1,HfRn( ) t( ) = 1+ −1( )n

n=1

∞

∑ t −nH

Γ 1− nH( ) ; t >>1

G0,1 t( ) = e−t ; G1,1 t( ) =1− e−t

11

357Fig.1:Theimpulse(top)andstepresponsefunctions(bottom)forthefractionalrelaxation358range(0<H<1,left,redisH=1,theexponential),theblackcurves,bottomtotopareforH359=1/10,2/10,..9/10)andthefractionaloscillationrange(1<H<2,redaretheintegervalues360H=1,bottom,theexponential,andtop,H=2,thesinefunction,theblackcurves,bottomto361topareforH=11/10,12/10,..19/10.362

2.3 A family of Gaussian noises and motions: 363Intheabove,wediscussedfGn,fRnandtheirintegralsfBm,fRm,buttheseare364

simply special cases of a more general theory valid for a wide family of Green’s365functionsthatleadtoconvergentnoisesandmotions.Weexpectforexamplethatour366approachalsoappliestothestochasticBasset’sequationdiscussedin[Karczewska367andLizama,2009],whichcouldberegardedasanaturalextensionofthestochastic368relaxation equation. With the motivation outlined in the previous sections, the369simplestwaytoproceedistostartbydefiningthegeneralmotionZH(t)as:370

, (19)371

whereNHisanormalizationconstantandHisanindex.Itisadvantageoustorewrite372thisinstandardnotation(e.g.[Biaginietal.,2008])as:373

, (20)374

1 2 3t

0.5

1.0

1.5

G0,HIm

pulseRe

spon

se 0<H<1

2 4 6 8 10 12 14 t

0.2

0.4

0.6

0.8

1.0

t2 4 6 8 10 12 14

0.5

1.0

1.5

2.0

G0,H

G1,H G1,H

Step

Respo

nse

2 4 6 8 10 12 14 t

-1.0

-0.5

0.5

1.0

t

1<H<2

ZH t( ) = NH G1,H t − s( )γ s( )ds−∞

t

∫ − NH G1,H −s( )γ s( )ds−∞

0

∫

ZH t( ) = NH G1,H t − s( )+ −G1,H −s( )+( )γ s( )ds!∫

12

wherethe“+”subscriptindicatesthattheargumentis>0,andtherangeofintegration375isoveralltherealaxis .Hereandthroughout,theGreen’sfunctionsneedonlybe376specifiedfort>0correspondingtotheircausalrange.377

The advantage of startingwith themotionZH is that it is based on the step378responseG1,Hwhichisfiniteatsmallt;thedisadvantageisthatintegralsmaydiverge379atlargescales.Thesecond(constant)termineq.20wasintroducedby[Mandelbrot380andVanNess,1968]forfBmpreciselyinordertoavoidlargescaledivergencesinfBm.381AsdiscussedinappendixA,theintroductionofthisconstantphysicallycorresponds382toconsideringthelongtimebehaviourofthefractionalrandomwalksdiscussedin383[KobelevandRomanov, 2000]and [Westetal., 2003]. Thephysical settingof the384randomwalkapplicationsisawalkerwithpositionX(t)andvelocityV(t).Assuming385that thewalker starts at theorigin corresponds to a fractionallydiffusingparticle386obeyingthefractionalRiemann-Liouvillerelaxationequation.387

Fromthedefinition(eq.19or20),wehave:388

. (21)389

Hence,theoriginplaysaspecialrole,sothattheZH(t)processisnonstationary.390The variance VH(t) of ZH (not to be confused with the velocity of a random391

walker)is:392. (22)393

Equivalently,withanobviouschangeofvariable:394

, (23)395

sothatVH(0)=0.ZHwillconvergeinarootmeansquaresenseifVHconverges.IfG1,H396

is a power law at large scales: thenHl < 1/2 is required for397

convergence.Similarly,ifatsmallscales ,thenconvergenceofVH398requiresHh>-1/2.WeseethatforfBm(eq.13),Hl=Hh=Hsothatthisrestriction399implies-1/2<H<1/2whichisequivalenttotheusualrange0<HB<1withHB=H+4001/2.Similarly,forfRm,usingG(fRn)1,H(t),wehaveHh=H,(eq.15)andHl=-H,(eq.17)401sothatfRmconvergesforH>-1/2,i.e.overtheentirerange0<H<2discussedin402thispaper.SincethesmallscalelimitoffRmisfBm,weseethattherange0<H<2403overlapswiththerangeoffBmandextendsitatlargeH.404

Fromeq.19wecanconsiderthestatisticsoftheincrements:405

, (24)406

wherewehaveusedthefactthat where meansequalityinaprobability407sense.Thisshowsthat:408

!

ZH 0( ) = 0; ZH 0( ) = 0

VH t( ) = ZH

2 t( ) = NH2 G1,H t − s( )+ −G1,H −s( )+( )2 ds!∫

VH t( ) = NH2 G1,H s + t( )−G1,H s( )( )2 ds +0

∞

∫ NH2 G1,H s( )2 ds0

t

∫

G1,H ∝ t Hl ; t >>1

G1,H ∝ tHh ; t <<1

ZH t( )− ZH u( ) = NH G1,H t − s( )+ −G1,H u − s( )+( )γ s( )ds!∫

=dNH G1,H t − u − ′s( )+ −G1,H − ′s( )+( )γ ′s( )d ′s

!∫ ; ′s = s − u

γ ʹs( )=dγ s( ) =

d

13

, (25)

409sothattheincrementsZH(t)arestationary.Fromthis,weobtainthevarianceofthe410increments =ZH(t)-ZH(t-Dt):411

. (26)412Since isameanzeroGaussianprocess,itsstatisticsaredeterminedbythe413

covariancefunction:414

. (27)415Thenoisesarethederivativesofthemotionsandaswementioned,dependingonH,416we only expect their finite integrals to converge. Let us therefore define the417resolutiontnoiseYH,tcorrespondingtothemeanincrementsofthemotions:418

. (28)419Thenoise,YH(t)cannowbeobtainedasthelimit :420

. (29)421

Applyingeq.26,weobtainthevariance:422

, (30)423since , couldbeconsideredastheanomalyfluctuationofYH,so424that istheanomalyvarianceatresolutiont.425

FromthecovarianceofZH(eq.27)weobtainthecorrelationfunction:426

427. (31)428

Alternatively,takingtimeinunitsoftheresolutionl=Dt/t:429

430

. (32)431RH,tcanbeconvenientlywrittenintermsofcentredfinitedifferences:432

ZH t( )− ZH u( )=dZH t −u( )− ZH 0( ) = ZH t −u( )

ΔZH Δt( )ΔZH Δt( )2 =VH Δt( ); Δt = t − u

ZH t( )

CH t,u( ) = ZH t( )ZH u( ) = 12VH t( )+VH u( )−VH t −u( )( )

YH ,τ t( ) =

ZHt( )−ZH t −τ( )

τ

τ →0

YHt( ) =

dZHt( )

dt

YH ,τ t( )2 = YH ,τ2 = τ−2VH τ( )

YH ,t 0( ) = 0 YH ,τ t( )

τ−2VH τ( )

RH ,τ Δt( ) = YH ,τ t( )YH ,τ t −Δt( ) = τ−2 ZH t( )− ZH t − τ( )( ) ZH t −Δt( )− ZH t −Δt − τ( )( )

= τ−212VH Δt − τ( )+VH Δt + τ( )−2VH Δt( )( )

Δt ≥ τ

RH ,τ 0( ) = YH ,τ t( )2= τ−2VH τ( ); Δt = 0

RH ,τ λτ( ) = YH ,τ t( )YH ,τ t −λτ( ) = τ−2 ZH t( )− ZH t − τ( )( ) ZH t −λτ( )− ZH t −λτ − τ( )( )

= τ−212VH λ −1( ) τ( )+VH λ+1( ) τ( )−2VH λτ( )( )

λ ≥1

RH ,τ 0( ) = YH ,τ t( )2= τ−2VH τ( ); λ = 0

14

. (33)433

ThefinitedifferenceformulaisvalidforDt≥t.Forfinitet,itallowsustoobtainthe434correlationbehaviourbyreplacingtheseconddifferencebyasecondderivative,an435approximationthatisverygoodexceptwhenDtisclosetot.436

Takingthelimit ineq.33toobtainthesecondderivativeofVH,andafter437somemanipulations,weobtainthefollowingsimpleformulaforthelimitingfunction438RH(Dt):439

. (34)440

If the integral forVH converges, this integral forRH(Dt)will also converge except441possiblyatDt=0(intheexamplesbelow,whenH≤1/2).442

Eq.34showsthatRHisthecorrelationfunctionofthenoise:443

. (35)444Thisresultcouldhavebeenderivedformallyfrom:445

, (36)446butourderivationexplicitlyhandlestheconvergenceissues.447

Auseful statistical characterizationof theprocesses isby the statisticsof its448Haar fluctuationsoveran intervalDt. Foran intervalDt,Haar fluctuationsarethe449differencesbetweentheaveragesofthefirstandsecondhalvesofaninterval.For450thenoiseYH,theHaarfluctuationis:451

. (37)452IntermsofZH(t):453

. (38)454Therefore: 455

. (39)456Thisformulawillbeusefulbelow.457

RH ,τ λτ( ) = 12Δτ2VH λτ( ) ≈ 1

2VH ʹ Δt( ); Δτ f t( ) =

f t + τ / 2( )− f t − τ / 2( )τ

τ→ 0

RH Δt( ) = 12d 2VH Δt( )dΔt 2

= G0,H s+Δt( )G0,H s( )ds0

∞

∫ ; G0,H =dG1,H

ds

YH t( ) = G0,H t − s( )γ s( )ds−∞

t

∫

YH t( ) = ZH′ t( ) = dZH t( )

dt= ddt

G1,H−∞

t

∫ t − s( )γ s( )ds;

= G0,H−∞

t

∫ t − s( )γ s( )ds

ΔYH Δt( )Haar

= 2Δt

YH s( )ds−t−Δt /2

t

∫2Δt

YH s( )dst−Δt

t−Δt /2

∫

ΔYH Δt( )Haar

= 2Δt

ZH t( )− 2ZH t − Δt / 2( ) + ZH t − Δt( )( )

ΔYH Δt( )Haar

2= 2

Δt⎛⎝⎜

⎞⎠⎟

2

2 ΔZH Δt / 2( )2− 2 YH ,Δt /2 t( )YH ,Δt /2 t − Δt / 2( )( )

= 2Δt

⎛⎝⎜

⎞⎠⎟

2

4VH Δt / 2( )−VH Δt( )( )

15

3 Application to fBm, fGn, fRm, fRn: 458

3.1 fBM, fGn: 459Theabovederivationswere fornoisesandmotionsderived fromdifferential460

operatorswhose impulseandstepGreen’s functionshadconvergentVH(t). Before461applyingthemtofRn,fRm,weillustratethisbyapplyingthemfirsttofBmandfGn.462

ThefBmresultsareobtainedbyusingthefGnstepGreen’sfunction(eq.13)in463eq.23toobtain:464

.

(40)465

Thestandardnormalizationandparametrisationis:466

. (41)467

ThisnormalizationturnsouttobeconvenientforbothfBmandfRmsothatweuseit468belowtoobtain:469

, (42)470

sothat:471

, (43)472

so–asmentionedearlier-HBisthefluctuationexponentforfBm.473We cannowcalculate the correlation function relevant for the fGn statistics.474

WiththenormalizationNH=KH:475

476

, (44)

477

the bottom line approximations are valid for large scale ratio l. We note the478differenceinsignforHB>1/2(“persistence”),andforHB<1/2(“antipersistence”).479When HB = 1/2, the noise corresponds to standard Brownian motion, it is480uncorrelated.481

VHfBm( ) t( ) = NH

2 −2sin πH( )Γ −1−2H( )

π

⎛

⎝⎜

⎞

⎠⎟t 2H+1; −

12≤ H <

12

NH = KH = −π

2sin πH( )Γ −1−2H( )

⎛

⎝⎜⎜

⎞

⎠⎟⎟

1/2

=π HB +1/ 2( )

2cos πHB( )Γ −2HB( )

⎛

⎝⎜⎜

⎞

⎠⎟⎟

1/2

;

HB = H +12; 0 ≤ HB <1

VHB

fBm( ) t( ) = t 2H+1 = t 2HB ; 0 ≤ HB <1

ΔBH Δt( )21/2= Δt HB ; ΔBH Δt( ) = BH t( )− BH t −Δt( )

RH ,τfGn( ) λτ( ) = 1

2τ2H−1 λ+1( )2H+1 + λ −1( )2H+1 −2λ2H+1( ); λ ≥1; −

12< H <

12

RH ,τfGn( ) 0( ) = τ2H−1

RHB ,τfGn( ) λτ( ) ≈ H 2H +1( ) λτ( )2H−1 = HB 2HB −1( ) λτ( )2 HB−1( ) ; −

12< H <

12

λ >>1

16

3.2 fRm, fRn 482Therearevariouscasestoconsider,appendixBgivessomeofthemathematical483

detailsincludingasmalltseriesexpansionsfor0<H<3/2;theleadingtermsare:484

(45)485

486

487

,488489allfort<<1.ThechangeinnormalizationforH>1/2isnecessarysinceKH2<0forthis490range. Similarly, theH >1/2 normalization cannot be used forH < 1/2 since CH491divergesforH<1/2.Seefig.2forplotsofV(fRm)H(t).Notethatthesmallt2behaviour492

forH>1/2correspondstofRmincrements i.e.toa493

smoothprocess,differentiableoforder1;seesection3.4.494Forlarget,wehave:495

,496

(46)497

whereaH isaconstant,theaboveisvalidfort≫1. Since ,the498

correctionsimplythatatlargescales

sothatthefRmprocessQH499

appearstobeanti-persistentatlargescales.500

VH

fRm( ) t( ) = t1+2 H +O t1+3H( ); N H = KH 0 < H <1/ 2

VH

fRm( ) t( ) = t2 −2Γ −1−2H( )sin πH( )

πCH2 t1+2 H +O t1+3H( ); N H =CH

−1; 1/ 2 < H < 3/ 2

VH

fRm( ) t( ) =t2− t 4

12CH2 ʹG0,H

fRm( ) s( )2ds

0

∞

∫ +O t2H+1( ); 3/2<H <2

CH2 = G0,H

fRm( ) s( )2ds

0

∞

∫

ΔQ

H2 Δt( )

1/2= V

H

fRm( ) Δt( )( )1/2≈ Δt

VHfRm( ) t( ) = N H

2 t − 2t1−H

Γ 2−H( )+ aH +O t1−2 H( )

⎡

⎣⎢⎢

⎤

⎦⎥⎥; H <1

VHfRm( ) t( ) = N H

2 t + aH −2t1−H

Γ 2−H( )+O t1−2 H( )

⎡

⎣⎢⎢

⎤

⎦⎥⎥; H >1

ΔQH Δt( )2

=VH Δt( )

ΔQH Δt( )2 1/2

< Δt1/2

17

501Fig.2:TheVHfunctionsforthevariousrangesofHforfRm(thesecharacterizethevariance502offRm).Theplotsfromlefttoright,toptobottomarefortheranges0<H<1/2,1/2<H<1,5031<H<3/2,3/2<H<2. Withineachplot,thelinesareforH increasinginunitsof1/10504startingatavalue1/20abovetheplotminimum;overall,Hincreasesinunitsof1/10starting505atavalue1/20,upperleftto39/20,bottomright(ex.fortheupperleft,thelinesareforH=5061/20,3/10,5/20,7/20,9/20).ForallH’sthelargetbehaviourislinear(slope=one,although507notetheoscillationsforthelowerrighthandplotfor3/2<H<2).Forsmallt,theslopesare5081+2H(0<H≤1/2)and2(1/2≤H<2).509

- 4 - 2 2 4Log10t

- 5

5

Log10VH

- 4 - 2 2 4

- 5

5

Log10VH

- 1.0 - 0.5 0.5 1.0 1.5 2.0

- 2

- 1

1

2

Log10VH

Log10t

Log10t

- 1.0 - 0.5 0.5 1.0 1.5 2.0

- 2

- 1

1

2

Log10VH

Log10t

18

510511Fig.3:ThecorrelationfunctionsRHforfRncorrespondingtotheVHfunctioninfig.20<H512½/2(upperleft)½/2<H<1(upperright),1<H<3/2)lowerleft,3/2<H<2lowerright.513In each plot, the curves correspond toH increasing from bottom to top in units of 1/10514starting from 1/20 (upper left) to 39/20 (bottom right). For H<1/2, the resolution is515

importantsinceRH,tdivergesatsmallt.Intheupperleftfigure,RH,t isshownwitht=10-5;516they were normalized to the value at resolution t = 10-5. For H >1/2, the curves are517normalizedwithNH=1/CH;forH<1/2,theywerenormalizedtothevalueatresolutiont=51810-5.Inallcases,thelargetslopeis–1-H.519

520TheformulaeforRHcanbeobtainedbydifferentiatingtheaboveresultsforVH521

twice(eqs.45,46,seeappendixBfordetailsandmoreaccuratePadéapproximants):522

523

524

, (47)525

- 4 - 2 2 4Log10t

- 8

- 6

- 4

- 2

Log 1

0R

- 4 - 2 2 4

- 8

- 6

- 4

- 2

- 1 1 2 3 4

- 8

- 6

- 4

- 2- 1 1 2 3 4

- 10

- 8

- 6

- 4

- 2Log10t

Log 1

0R

Log 1

0R

Log 1

0R

Log10t

Log10t

RHfRn( ) t( ) = H 1+ 2H( )t−1+2H +O t−1+3H( ); τ << t <<1; 0 < H <1/ 2

RH

fRn( ) t( ) =1−Γ 1−2H( )sin πH( )

πCH2 t−1+2 H +O t−1+3H( ); t <<1; 1/ 2 < H < 3/ 2

RH

fRn( ) t( ) =1− t2

2CH2

ʹG0,H s( )2ds

0

∞

∫ +O t−1+2 H( )...; t <<1; 3 / 2 < H < 2

19

(when0<H<1/2,fort≈twemustusetheexactresolutiontfGnformula,eq.44,top).526Forlarget:527

;t>>1. (48)528

Note that for 0<H<1, G(-H)<0 so that R>0 over this range (fig. 3). Formulae 45, 47 show 529that there are three qualitatively different regimes: 0 < H < 1/2, 1/2 < H < 3/2, 3/2 < H < 2; 530this is in contrast with the deterministic relaxation and oscillation regimes (0 < H <1 and 1 531< H <2). We return to this in section 3.4. 532

Now that we have worked out the behaviour of the correlation function, we can 533comment on the issue of the memory of the process. Starting in turbulence, there is the 534notion of “integral scale” that is conventionally defined as the long time integral of the 535correlation function. When the integral scale diverges, the process is conventionally 536termed a “long memory process”. With this definition, if the long time exponent of RH is 537> -1, then the process has a long memory. Eq. 48 shows that the long time exponent is -5381-H so that for all H considered here, the integral scale converges. However, it is of the 539order of the relaxation time which may be much larger than the length of the available 540sample series. For example, eq. 47 shows that when H <1/2, the effective exponent 2H -1 541implies (in the absence of a cut-off), a divergence at long times, so that up to the relaxation 542scale, fRn mimics a long memory process. 543

3.3 Haar fluctuations 544Using eq. 39we can determine the behaviour of the RMSHaar fluctuations.545

ApplyingthisequationtofGnweobtain withHHaar=H–1/2546

(thesubscript“Haar”indicatesthatthisisnotadifference/incrementfluctuationbut547rather a Haar fluctuation). For the motion, the Haar exponent is equal to the548

exponentsoftheincrements(eq.43)sothat withHHaar=HB549

=H+1/2(bothresultswereobtainedin[Lovejoyetal.,2015]).Therefore,froman550empiricalviewpointifwehaveascalingGaussianprocessand(uptotherelaxation551timescale)when-1/2<HHaar<0,ithasthescalingofanfGnandwhen0<HHaar<5521/2,itscalesasanfBm.553

Usingeq.39,wecandeterminetheHaarfluctuationsforfRn .554

WiththesmallandlargetapproximationsforVH(t),wecanobtainthesmallandlarge555DtbehaviouroftheHaarfluctuations.Therefore,theleadingtermsforsmallDtare:556

, (49)557

RHfRn( ) t( ) = − N H

2

Γ −H( )t−1−H +O t−1−2 H( ) : 0 < H < 2

ΔF

HΔt( )

Haar

2 1/2∝Δt HHaar

ΔB

HΔt( )

Haar

2 1/2∝Δt HHaar

ΔU

HΔt( )

Haar

2 1/2

ΔUHΔt( )

Haar

2 1/2=Δt HHaar

HHaar

=H−1/2; 0<H <3/2HHaar

=1; 3/2<H <2; Δt <<1

20

wheretheDtH-1/2behaviourcomesfromtermsinVH≈t1+2HandtheDtbehaviourfrom558

theVH≈t4termsthatarisewhenH>3/2.Note(eq.39)that depends559

on4VH(Dt/2)-VH(Dt)sothatquadratictermsinVH(t)cancel.560AsHincreasespastthecriticalvalueH=1/2,thesignofHHaarchangessothat561

when 1/2 <H < 3/2,we have 0 <HHaar < 1 so that over this range, the smallDt562behaviourmimicsthatoffBmratherthanfGn(discussedinthenextsection).563

ForlargeDt,thecorrespondingformulais:564

. (50)565

ThiswhitenoisescalingisduetotheleadingbehaviorVH(t)≈toverthefullrangeof566H(eq.47),seefig.4a.567

568Fig.4a:TheRMSHaarfluctuationplotsfor the fRn process for 0 < H < 1/2 (upper left), 1/2 < H 569< 1 (upper right), 1 < H < 3/2 (lower left), 3/2 < H < 2 (lower right). The individual curves 570correspond to those of fig. 2, 3. ThesmallDtslopesfollowthetheoreticalvaluesH-1/2upto571H=3/2(slope=1);forlargerH,thesmalltslopesall=1.Also,atlargetduetodominantV≈572tterms,inallcasesweobtainslopest-1/2.573574

ΔU

HΔt( )

Haar

2 1/2

ΔU

Haar2 Δt( )

2 1/2∝ Δt −1/2; Δt >>1; 0<H <2

- 4 - 2 2

- 1

1

2

10

Log

<Δ U

(Δt)2 >1/2

- 4 - 2 2

- 3.0

- 2.5

- 2.0

- 1.5

- 1.0

- 0.5

- 4 - 2 2

- 1.4

- 1.2

- 1.0

- 0.8

- 0.6

- 0.4

- 0.2 Log10Δt

Log10Δt

Log10Δt Log10Δt- 4 - 3 - 2 - 1 1 2

- 4

- 3

- 2

- 1

10

Log

<Δ U

(Δt)2 >1/2

10

Log

<Δ U

(Δt)2 >1/2

10

Log

<Δ U

(Δt)2 >1/2

21

3.4 fBm, fRm or fGn? 575Ouranalysishasshownthattherearethreeregimeswithqualitativelydifferent576

small scale behaviour, let us compare them in more detail. The easiest way to577comparethedifferentregimesistoconsidertheirincrements.SincefRnisstationary,578wecanuse:579

. (51)580

OverthevariousrangesforsmallDt,(t≪1istheresolution)recallthatwehave:581

, (52)582

(whenH>1/2theresolutionisnotimportant,theindexisdropped).Weseethatin583thesmallHrange,theincrementsaredominatedbytheresolutiont,theprocessisa584noisethatdoesnotconvergepoint-wise,hencethetdependence.Inthemiddle(1/2585<H<3/2)regime,theprocessispoint-wiseconvergent(takethelimitt->0)although586itcannotbedifferentiatedbyanypositiveintegerorder.Finally,thelargestHregime587

3/2<H<2), the process is smoother: , so that it is almost588

surelydifferentiableoforder1.SincethefRmaresimplyorderoneintegralsoffRn,589theirordersofdifferentiabilityaresimplyaugmentedbyone.590

Considering the first two ranges i.e. 0 <H < 3/2,we therefore have several591processeswith the same small scale statistics and thismay lead to difficulties in592interpreting empirical data that cover ranges of time scales smaller than the593relaxationtime.Forexample,wealreadysawthatovertherange0<H<1/2thatat594smallscaleswecouldnotdistinguishfRnfromthecorrespondingfGn;theybothhave595anomalies (averages after the removal of the mean) or Haar fluctuations that596decreasewith time scalewith exponentH -1/2, (eq. 49). This similitudewasnot597surprisingsincetheybothweregeneratedbyGreen’sfunctionswiththesamehigh598frequencyterm.Fromanempiricalpointofview,withdataonlyavailableoverscales599muchsmallerthantherelaxationtime,itmightbeimpossibletodistinguishthetwo;600theirstatisticscanbeveryclose.601

Theproblemiscompoundedwhenweturntoincrementsorfluctuationsthat602increasewithscale.Toseethis,notethatinthemiddlerange(1/2<H<3/2),the603exponent-1+2Hspanstherange0to2.Thisoverlapstherange1to2spannedby604fRm(QH)with0<H<1/2:605

, (53)606andwithfBm(BH)overthesameHrange(butforallDt):607

ΔU

HΔt( )

2= U

Ht( )−UH

t −Δt( )( )2=2 R

H

fRn( ) 0( )−RHfRn( ) Δt( )( )

ΔUH ,τ Δt( )2 ≈2τ −1+2H −2H 2H +1( )Δt −1+2H ; 1>> Δt >>τ ; 0<H <1/2

ΔUHΔt( )2 ≈ Δt −1+2H ; 1/2<H <3/2

ΔUHΔt( )2 ≈ Δt2; 3/2<H <2

limΔt→0

ΔUHΔt( )/Δt( )

2=1

ΔQ

HΔt( )

2=V

H

fRm( ) Δt( )∝Δt1+2H ; Δt <<1; 0<H <1/2

22

. (54)608IfweusetheusualfBmexponentHB=H+1/2,then,overtherange0<H<1/2we609maynotonlycomparefBmwithfRmwiththesameHB,butalsowithanfRnprocess610withanHlargerbyunity,i.e.withHB=H-1/2intherange1/2<H<3/2.Inthiscase,611wehave:612

(55)613

,614

615616wherea,bareconstants(section3.2).Overtheentirerange0<HB<1,weseethat617theonlydifferencebetweenfBm,andfRn,fRmistheirdifferentlargescalecorrections618tothesmallscale behaviour.Therefore,ifwefoundaprocessthatoverafinite619rangewasscalingwithexponent1/2<HB<1,thenoverthatrange,wecouldnottell620thedifferencebetweenfRn,fRm,fBm,seefig.4bforanexamplewithHB=0.95.621622

ΔB

HΔt( )

2=V

H

fBm( ) Δt( ) =Δt1+2H ; 0<H <1/2

ΔUHΔt( )

2∝Δt2HB ; Δt <<1; 0<H

B<1

∝2 1−aΔt −HB−3/2( ); Δt >>1

ΔQHΔt( )

2∝Δt2HB ; Δt <<1; 1/2<H

B<1

∝Δt −bΔt3/2−HB ; Δt >>1

ΔB

HΔt( )

2=Δt2HB ; 0<H

B<1

Δt2HB

23

623Fig.4b:AcomparisonoffRnwithH=1.45,fRmwithH=0.45andfBmwithH=0.45.For624smallDt,theyallhaveRMSincrementswithexponentHB=0.95andcanonlybedistinguished625bytheirbehavioursatDtlargerthantherelaxationtime(log10Dt=0inthisplot).626

3.5 Spectra: 627SinceYH(t) isstationaryprocess, itsspectrumis theFouriertransformof the628

correlationfunctionRH(t)(theWiener-Khintchintheorem).However,itiseasierto629determineitdirectlyfromthefractionalrelaxationequationusingthefactthatthe630Fourier transform (F.T., indicatedby the tilda)of theWeyl fractionalderivative is631

simply (e.g.[Podlubny,1999],thisissimplytheextension632

oftheusualrulefortheF.T.ofinteger-orderedderivatives).ThereforetaketheF.T.633ofeq.4(thefRn),toobtain:634

, (56)635

sothatthespectrumofYis:636

-3 -2 -1 1 2 3Log10Δt

-6

-4

-2

2

4

6

Log10<Δf2>

fBmH=0.45

fRmH=0.45

fRnH=1.45

F .T . −∞DtHY

H⎡⎣ ⎤⎦ = iω( )HYH! ω( )

iω( )H +1⎛⎝

⎞⎠UH! = γ!

24

. (57)637

(where the Gaussian white noise was normalized such that =1). The638

asymptotichighandlowfrequencybehavioursaretherefore,639

. (58)640Thiscorrespondstothescalingregimesdeterminedbydirectcalculationabove:641

. (59)642

(H≠1).Notethattheusual(Orenstein-Uhlenbeck)resultforH=1hasnowHterm,643hencenot-1-Hterm;ithasanexponentialratherthanpowerlawdecayatlarget.644

FromthespectrumofU,wecaneasilydeterminethespectrumofthestationary645DtincrementsofthefRmprocessQH:646

. (60)647

3.6 Sample processes 648ItisinstructivetoviewsomesamplesoffRn,fRmprocesses.Forsimulations,649

both the small and large scaledivergencesmustbe considered. Startingwith the650approximatemethodsdevelopedby[MandelbrotandWallis,1969],ittooksometime651for exact fBm, and fGn simulation techniques to be developed [Hipel andMcLeod,6521994],[Palma,2007].Fortunately,forfRm,fRn,thelowfrequencysituationiseasier653sincethelongtimememoryismuchsmallerthanforfBm,fGn.Therefore,aslongas654wearecarefultoalwayssimulateseriesafewtimestherelaxationtimeandthento655throw away the earliest 2/3 or 3/4 of the simulation, the remainder will have656accuratecorrelations.Withthisproceduretotakecareoflowfrequencyissues,we657canthereforeusethesolutionforfRnintheformofaconvolution(eqs.19,35,36),658andusestandardnumericalconvolutionalgorithms.659

However,westillmustbecarefulaboutthehighfrequenciessincetheimpulse660responseGreen’sfunctionsG0,HaresingularforH<1.Inordertoavoidsingularities,661

EUω( ) = U

H! ω( )

2=

γ" ω( )2

1+ −iω( )H⎛

⎝⎜

⎞⎠⎟ 1+ iω( )

H⎛⎝⎜

⎞⎠⎟=

11+ −iω( )

H⎛⎝⎜

⎞⎠⎟ 1+ iω( )

H⎛⎝⎜

⎞⎠⎟

= 1+2Cos πH2

⎛

⎝⎜

⎞

⎠⎟ωH +ω2H

⎛

⎝⎜⎜

⎞

⎠⎟⎟

−1

γ! ω( ) 2

EUω( ) =

ω−2H +O ω−3H( ); ω>>1

1−2cos πH2

⎛

⎝⎜

⎞

⎠⎟ωH +O ω2H( ) ω<<1

RHt( )∝ t −1+2H + .. t <<1

t −1−H + .. t >>1

EΔQ

ω( ) =2sinωΔt2

ω

⎛

⎝

⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟

2

EUω( ); ΔQ Δt( ) = U s( )ds

t−Δt

t

∫

25

simulationsoffRnarebestmadebyfirstsimulatingthemotionsQHusing 662(* denotes a Weyl convolution) and obtain the resolution t fRn, using663

. Numerically, this allows us to use the smoother664(nonsingular)G1 in the convolution rather than the singularG0. The simulations665shown in figs. 5, 6 follow this procedure and the Haar fluctuation statisticswere666analyzedverifyingthestatisticalaccuracyofthesimulations.667

Inordertoclearlydisplaythebehaviours,recallthatwhent>>1,weshowedthat668all the fRn converge to Gaussianwhite noises and the fRm to Brownianmotions669(albeitinaslowpowerlawmanner).Attheotherextreme,fort<<1,weobtainthe670fGnandfBmlimits(when0<H<1/2)andtheirgeneralizationsfor1/2<H<2.671

Fig. 5a shows three simulations, each of length 219, pixels, with each pixel672correspondingtoatemporalresolutionoft=2-10sothattheunit(relaxation)scaleis673210elementarypixels.EachsimulationusesthesamerandomseedbuttheyhaveH’s674increasingfromH=1/10(topset)toH=5/10(bottomset).ThefRmattherightis675fromtherunningsumofthefRnattheleft.Eachserieshasbeenrescaledsothatthe676range(maximum-minimum)isthesameforeach. Startingatthetoplineofeach677group,weshow210pointsoftheoriginalseriesdegradedbyafactor29.Thesecond678lineshowsablow-upbyafactorof8ofthepartoftheupperlinetotherightofthe679dashedvertical line. The linebelow isa furtherblownupby factorof8,until the680bottomlineshows1/512partofthefullsimulation,butatfullresolution.Theunit681scaleindicatingthetransitionfromsmalltolargeisshownbythehorizontalredline682inthemiddlerightfigure.Atthetop(degradedbyafactor29),theunit(relaxation)683scaleis2pixelssothatthetoplinedegradedviewofthesimulationisnearlyawhite684noise (left), (ordinary)Brownianmotion (right). In contrast, thebottomseries is685exactlyoflengthunitysothatitisclosetothefGnlimitwiththestandardexponent686HB=H+1/2.Movingfrombottomtotopinfig.5a,oneeffectivelytransitionsfromfGn687tofRn(leftcolumn)andfBmtofRm(right).688

Ifwe take the empirical relaxation scale for the global temperature to be27689months(≈10years,[Lovejoyetal.,2017])andweusemonthlyresolutiontemperature690anomalydata,thenthenondimensionalresolutionis2-7correspondingtothesecond691seriesfromthetop(whichisthus210months≈80yearslong).SinceH≈0.42±0.02692([DelRioAmadorandLovejoy,2019]),thesecondseriesfromthetopinthebottom693set is themost realistic,we canmakeout the low frequencyondulutions that are694mostlypresentatscales1/8oftheseries(orless).695

Fig.5bshowsrealizationsconstructedfromthesamerandomseedbutforthe696extendedrange1/2<H<2(i.e.beyondthefGnrange).Overthisrange,thetop(large697scale,degradedresolution)seriesisclosetoawhitenoise(left)andBrownianmotion698(right).Forthebottomseries,thereisnoequivalentfGnorfBmprocess,thecurves699becomesmootheralthoughtherescalingmayhidethissomewhat(seeforexample700theH=13/20set,theblow-upofthefarright1/8ofthesecondseriesfromthetop701showninthethirdline.For1<H<2,alsonotetheoscillationswithwavelengthof702orderunity,thisisthefractionaloscillationrange.703

Fig.6ashowssimulationssimilartofig.5a(fRnontheleft, fRmontheright)704exceptthatinsteadofmakingalargesimulationandthendegradingandzooming,all705

QH∝G1,H ∗γ

UH ,τ t( ) = QH t +τ( )−QH t( )( ) / τ

26

the simulations were of equal length (210 points), but the relaxation scale was706changedfrom215pixels(bottom)to210,25and1pixel(top).Againthetopiswhite707noise(left),Brownianmotion(right),andthebottomis(nearly)fGn(left)andfBm708(right),fig.6bshowstheextensionsto1/2<H<2.709

710Fig.5a: fRnandfRmsimulations(leftandrightcolumnsrespectively) forH=1/10,3/10,7115/10(toptobottomsets)i.e.theexponentrangethatoverlapswithfGnandfBm.Thereare712threesimulations,eachoflength219pixels,eachusethesamerandomseedwiththeunitscale713equalto210pixels(i.e.aresolutionoft=2-10). Theentiresimulationthereforecoversthe714rangeofscale1/1024to512units.ThefRmattherightisfromtherunningsumofthefRn715attheleft.716

Startingatthetoplineofeachset,weshow210pointsoftheoriginalseriesdegradedin717resolutionbyafactor29.Sincethelengthist=29unitslong,eachpixelhasresolutiont=7181/2).Thesecondlineofeachsettakesthesegmentoftheupperlinelyingtotherightofthe719dashedverticalline,1/8ofitslength.Itthereforespanst=0tot=29/8=26butresolution720wastakenast=2-4,henceitisstill210pixelslong.Sinceeachpixelhasaresolutionof2-4,the721unitscaleis24pixelslong,thisisshowninredinthesecondseriesfromthetop(middleset).722Theprocessoftaking1/8andblowingupbyafactorof8continuestothethirdline(lengtht723=23,resolutiont=2-7),unitscale=27pixels(shownbytheredarrowsinthethirdseries)724untilthebottomserieswhichspanstheranget=0tot=1andaresolutiont=2-10withunit725scale210pixels(thewholeseriesdisplayed).Eachserieswasrescaledintheverticalsothat726itsrangebetweenmaximumandminimumwasthesame.727

Theunitrelaxationscalesindicatedbytheredarrowsmarkthetransitionfromsmall728tolargescale.Sincethetopseriesineachsethasaunitscaleof2(degraded)itisnearlya729

H=1/10

H=3/10

H=5/10

27

whitenoise(left),or(ordinary)Brownianmotion(right).Incontrast,thebottomseriesis730exactlyof lengtht=1sothatit isclosetothefGnandfBmlimits(leftandright)withthe731standardexponentHB=H+1/2.Asindicatedinthetext,thesecondseriesfromthetopinthe732bottomsetismostrealisticformonthlytemperatureanomalies.733734

735Fig.5b:Thesameasfig.5abutforH=7/10,13/10and19/10(toptobottom).Overthis736range, the top(largescale,degradedresolution)series isclose toawhitenoise (left)and737Brownianmotion(right).Forthebottomseries,thereisnoequivalentfGnorfBmprocess,738thecurvesbecomesmootheralthoughtherescalingmayhidethissomewhat(seeforexample739themiddleH=13/20set,theblow-upofthefarright1/8ofthesecondseriesfromthetop740showninthethirdline).Alsonoteforthebottomtwosetswith1<H<2,theoscillationsthat741havewavelengthsoforderunity,thisisthefractionaloscillationrange.742743

H=7/10

H=13/10

H=19/10

28

744Fig.6a:Thissetofsimulationsissimilartofig.5a(fRnontheleft,fRmontheright)except745thatinsteadofmakingalargesimulationandthendegradingandzooming,allthesimulations746wereofequal length(210points),butresolutionst=2-15,2-10,2-5,1(bottomtotop). The747simulationsthereforespannedtherangesofscale2-15to2-5;2-10to1;2-5to25;1to210and748thesamerandomseedwasusedineachsothatwecanseehowthestructuresslowlychange749when the relaxation scale changes. The bottom fRn,H = 5/10 set is the closest to that750observedfortheEarth’stemperature,andsincetherelaxationscaleisoftheorderofafew751years,thesecondseriesfromthetopofthisset(withonepixel=onemonth)isclosetothat752ofmonthlyglobaltemperatureanomalyseries.Inthatcasetherelaxationscalewouldbe32753monthsandtheentireserieswouldbe210/12≈85yearslong.754

Thetopseries(oftotallength210relaxationtimes)is(nearly)awhitenoise(left),and755Brownianmotion(right),andthebottomis(nearly)anfGn(left)andfBm(right).Thetotal756rangeofscalescoveredhere(210x215)islargerthaninfig.5aandallowsonetomoreclearly757distinguishthehighandlowfrequencyregimes.758759

H=1/10

H=3/10

H=5/10

29

760Fig.6b:Thesamefig.6abutforlargerHvalues;seealsofig.5b.761

4. Prediction 762TheinitialvalueforWeylfractionaldifferentialequationsiseffectivelyat ,763

sothatforfRnitisnotdirectlyrelevantatfinitetimes(althoughtheensemblemean764isassumed=0;forfRm,QH(0)=0isimportant).Thepredictionproblemisthustouse765pastdata(say, fort<0) inordertomakethemostskilfulpredictionof the future766noisesandmotionsatt>0.Wearethereforedealingwithapastvalueratherthana767usualinitialvalueproblem.Theemphasisonpastvaluesisparticularlyappropriate768sinceinthefGnlimit,thememoryissolargethatvaluesoftheseriesinthedistant769pastareimportant.Indeed,predictionoffGnwithafinitelengthofpastdatainvolves770placingstrong(mathematicallysingular)weightsonthemostancientdataavailable771(see[GripenbergandNorros,1996],[DelRioAmadorandLovejoy,2019]).772

Ingeneral,therewillbesmallscaledivergences(forfRn,when0<H≤1/2)so773that it is important to predict the finite resolution fRn: . Using eq. 28 for774

,wehave:775

H=7/10

H=13/10

H=19/10

t =−∞

YH ,τ t( )YH ,τ t( )

30

. (61)776Letusdefinethepredictorfort≥0(indicatedbyacirconflex):777

. (62)778To show that it is indeed the optimal predictor, consider the error in the779predictor:780

. (63)781

Eq.63showsthattheerrordependsonlyong(s)fors>0whereasthepredictor(eq.78262)onlydependsong(s)fors<0,hencetheyareorthogonal:783

, (64)784

thisisasufficientconditionfor tobetheminimumsquarepredictorwhichis785theoptimalpredictorforGaussianprocesses,(e.g.[Papoulis,1965]).Theprediction786errorvarianceis:787

, (65)788orwithachangeofvariables:789

, (66)790wherewehaveused (theunconditionalvariance).791

Usingtheusualdefinitionofforecastskill(alsocalledtheMinimumSquareSkill792ScoreorMSSS)weobtain:793

YH ,τ t( ) = 1τ

G1,H t − s( )γ s( )ds−∞

t

∫ − G1,H −s( )γ s( )ds−∞

0

∫⎡

⎣⎢

⎤

⎦⎥ −

1τ

G1,H t − τ − s( )γ s( )ds−∞

t−τ

∫ − G1,H −s( )γ s( )ds−∞

0

∫⎡

⎣⎢

⎤

⎦⎥

= 1τ

G1,H t − s( )γ s( )ds−∞

t

∫ − G1,H t − τ − s( )γ s( )ds−∞

t−τ

∫⎡

⎣⎢

⎤

⎦⎥

Yτ! t( ) == 1

τG1,H t − s( )γ s( )ds

−∞

0

∫ − G1,H t − τ − s( )γ s( )ds−∞

0

∫⎡

⎣⎢⎢

⎤

⎦⎥⎥

Eτ t( )

Eτ t( ) =Yτ t( )−Yτ! t( ) = τ−1 G1,H t − s( )γ s( )ds−

−∞

t

∫ G1,H t − τ − s( )γ s( )ds−∞

t−τ

∫⎡

⎣⎢

⎤

⎦⎥

−τ−1 G1,H t − s( )γ s( )ds−−∞

0

∫ G1,H t − τ − s( )γ s( )ds−∞

0

∫⎡

⎣⎢

⎤

⎦⎥

= τ−1 G1,H t − s( )γ s( )ds−0

t

∫ G1,H t − τ − s( )γ s( )ds0

t−τ

∫⎡

⎣⎢

⎤

⎦⎥

Eτ t( )Yτ

! t( ) = 0

Yτ! t( )

Eτ t( )2

= τ−2 G1,H t − s( )−G1,H t − τ − s( )( )2ds+ G1,H t − s( )2

dst−τ

t

∫0

t−τ

∫⎡

⎣⎢⎢

⎤

⎦⎥⎥

Eτ t( )2

= τ−2N H−2VH τ( )− τ−2 G1,H u+ τ( )−G1,H u( )( )

2du

t−τ

∞

∫⎡

⎣⎢

⎤

⎦⎥

Yτ

2 = τ−2N H−2VH τ( )

31

. (67)794

WhenH<1/2and ,wecancheckthatweobtainthefGn795

result:796

(68)797[Lovejoyetal.,2015].Thiscanbeexpressedintermsofthefunction:798

, (69)799sothattheusualfGnresult(independentoft)is:800

. (70)801

Tosurveytheimplications,let’sstartbyshowingthetindependentresultsfor802fGn,showninfig.7whichisavariantonaplotpublishedin[Lovejoyetal.,2015].We803seethatwhenH≈1/2(HB≈1)thattheskillisveryhigh,indeed,inthelimit ,804wehaveperfectskillforfGnforecasts(thiswouldofcourserequireaninfiniteamount805ofpastdatatoattain).806

807

Sk ,τ t( ) = 1−Eτ t( )2

Eτ ∞( )2=NH2 G1,H u + τ( )−G1,H u( )( )2 dut−τ

∞

∫VH τ( )

=G1,H u + τ( )−G1,H u( )( )2 du

t−τ

∞

∫

G1,H u + τ( )−G1,H u( )( )2 du0

∞

∫ + G1,H u( )2 du0

τ

∫

G1,H t( ) = G1,H

fGn( ) t( ) = t H

Γ 1+ H( )

G1,H u + τ( )−G1,H u( )( )2du

t−τ

∞

∫ ≈ τ1+2 H

Γ 1+ H( )2 v +1( )H− vH( )2

dvλ−1

∞

∫ ; v = u / τ; λ = t / τ

ξH λ( ) = u+1( )H

−uH( )2

du0

λ−1

∫

Sk =ξH ∞( )−ξH λ( )ξH ∞( )+ 1

2H +1

H→1/2

32

808Fig.7:Thepredictionskill(Sk)forpurefGnprocessesforforecasthorizonsuptol =80910steps(tentimestheresolution).Thisplotisnon-dimensional,itisvalidfortime810stepsofanyduration.Frombottomtotop,thecurvescorrespondtoH=1/20,3/10,811…9/20(red,top,closetotheempiricalH).812813

2 4 6 8 10λ

0.2

0.4

0.6

0.8

1.0

Sk

33

814Fig.8:Theleftcolumnshowstheskill(Sk)offRnforecasts(asinfig.7forfGn)forfRn815skillwithH =1/20, 5/20, 9/20 (top tobottomset);l is the forecasthorizon, the816numberofstepsofresolutiontforecastintothefuture.Therighthandcolumnshows817theratio(r)ofthefRntocorrespondingfGnskill.818

Heretheresultdependsont;eachcurveisfordifferentvaluesincreasingfrom81910-4(top,black)to10(bottom,purple)increasingbyfactorsof10(theredsetinthe820bottomplotswitht=10-2,H=9/20areclosesttotheempiricalvalues).821

822Now consider the fRn skill. In this case, there is an extra parameter, the823

resolutionofthedata,t.Figure8showscurvescorrespondingtofig.7forfRnwith824forecasthorizonsintegermultiples(l)ofti.e.fortimest=ltinthefuture,butwith825separatecurves,oneforeachoffivet valuesincreasingfrom10-4to10byfactorsof826ten.Whentissmall,theresultsshouldbeclosetothoseoffGn,i.e.withpotentially827highskill,andinallcases,theskillisexpectedtovanishquiterapidlyfort>1sincein828this limit, fRnbecomesan (unpredictable)whitenoise (although thereare scaling829correctionstothis).830

TobetterunderstandthefGnlimit,itishelpfultoplottheratioofthefRntofGn831skill(fig.8,rightcolumn).Weseethatevenwithquitesmallvaluest=10-4(top,black832curves),thatsomeskillhasalreadybeenlost.Fig.9showsthismoreclearly,itshows833onetimestepandtentimestepskillratios.Toputthisinperspective,itishelpfulto834compare thisusingsomeof theparametersrelevant tomacroweather forecasting.835

2 4 6 8 10λ

-3.0

-2.0

-1.0

Log 10Sk

2 4 6 8 10λ

-3.0

-2.0

-1.0

Log 10Sk

2 4 6 8 10λ

-3.0

-2.0

-1.0

Log 10Sk

H=1/20

H=5/20

H=9/20

2 4 6 8 10λ

-2.0

-1.0

Log 10r

2 4 6 8 10λ

-2.0

-1.0

Log 10r

2 4 6 8 10λ

-2.0

-1.0

Log 10r

34

According to [Lovejoy et al., 2015] and [Del Rio Amador and Lovejoy, 2019], the836relevantempiricalHaarexponentis≈-0.08fortheglobaltemperaturesothatH=1/2837-0.08≈0.42.Directempiricalestimatesoftherelaxationtime,isdifficultsinceare838the response to anthropogenic forcing begins to dominate over the internal839variabilityafter≈10years,asmentionedabove,itisoftheorder5-10years.For840monthly resolution forecasts, the non-dimensional resolution is t ≈ 1/100. With841thesevalues,wesee(redcurves)thatwemayhavelost≈30%ofthefGnskillforone842monthforecastsand≈85%fortenmonthforecasts.Comparingthiswithfig.7we843seethatthisimpliesabout60%and10%skill(seealsotheredcurveinfig.8,bottom844set).845

Goingbeyondthe0<H<1/2regionthatoverlapsfGn,fig.10clearlyshowsthat846theskillcontinuestoincreasewithH.Wealreadysaw(fig.4)thattherange1/2<H847<3/2hasRMSHaarfluctuationsthatforDt<0mimicfBmandthesedoindeedhave848higherskill,approachingunityforHnear1correspondingtoaHaarexponent≈1/2,849i.e. close to an fBmwithHB =1/2, i.e. a regularBrownianmotion. Recall that for850Brownian motion, the increments are unpredictable, but the process itself is851predictable(persistence).852

Finally, in figure 11a, b, we show the skill for various H’s as a function of853resolutiont.Fig.11afortheH<3/2showsthatforallH,theskilldecreasesrapidly854fort >1. Fig.12binthefractionaloscillationequationregimeshowsthattheskill855alsooscillates.856

857

-4 -3 -2 -1 1

Log10τ

0.2

0.4

0.6

0.8

1.0

-4 -3 -2 -1 1

0.2

0.4

0.6

0.8

1.0r r

Log10τ

35

Fig.9:TheratiooffRnskilltofGnskill(left:onestephorizon,right:tenstepforecast858horizon)asafunctionofresolutiontforHincreasingfrom(atleft)bottomtotop(H859= 1/20, 2/20, 3/20…9/20); theH = 9/20 curves (close to the empirical value) is860showninred.861

862Fig.10:Theonestep(left)andtenstep(right)fRnforecastskillasafunctionofHfor863variousresolutions(t) rangingfromt=10-4(black,leftofeachset)throughtot=10864(rightofeachset,purple, fortherightsetthet=1(orange),10(purple) linesare865nearlyon topof theSk=0 line,againred is themoreempirical relevantvalue for866monthlydata,t =10-2). Recall that the regimeH <1/2 (to the leftof thevertical867dashedlines)correspondstotheoverlapwithfGn.868869

0.5 1.0 1.5 2.0H

0.2

0.4

0.6

0.8

1.0Sk

0.5 1.0 1.5H

0.2

0.4

0.6

0.8

1.0Sk

2.0

36

870Fig.11a:OnestepfRnpredictionskillsasafunctionofresolutionforH’sincreasing871from1/20(bottom)to29/20(top),every1/10.Notetherapidtransitiontolowskill,872(whitenoise)fort >1.ThecurveforH=9/20isshowninred.873

-4 -3 -2 -1 1Log10τ

0.2

0.4

0.6

0.8

1.0Sk

37

874Fig.11b: Sameas fig.11aexcept forH=37/20,39/20showingtheonestepskill875(black),andthetenstepskill(dashed).Therighthanddashedandrighthandsolid876lines,areforH=39/20,theyclearlyshowthattheskilloscillatesinthisfractional877oscillationequationregime.ThecorrespondingleftlinesareforH=37/20.878

4. Conclusions: 879Eversince[Budyko,1969]and[Sellers,1969],theenergybalancebetweenthe880

earth and outer space has beenmodelled by the Energy Balance Equation (EBE)881whichisanordinaryfirstorderdifferentialequationforthetemperature(Newton’s882lawofcooling).IntheEBE,theintegerorderedderivativetermaccountsforenergy883storage. Physically, it corresponds to storage in a uniform slab of material. To884increaserealism,onemayintroduceafewinteractingslabs(representingforexample885the atmosphere and ocean mixed layer; the Intergovernmental Panel on Climate886Changerecommendstwosuchcomponents[IPCC,2013]). Howeverduetospatial887scaling, a more realistic model involves a continuous hierarchy of storage888mechanismsandthiscaneasilybemodelledbyusingfractionalratherthaninteger889orderedderivatives: theFractionalEnergyBalanceEquation (FEBE, announced in890[Lovejoy,2019a]).891

TheFEBE isa fractionalrelaxationequationthatgeneralizes theEBE. When892forcedby aGaussianwhitenoise, it is also a generalizationof fractionalGaussian893noise(fGn)anditsintegralgeneralizesfractionalBrownianmotion(fBm).Overthe894parameterrange0<H<1/2(H is theorderof the fractionalderivative), thehigh895

-1 1 2 3Log10τ

0.2

0.4

0.6

0.8

1.0

Sk

38

frequency FEBE limit (fGn) has been used as the basis of monthly and seasonal896temperatureforecasts[Lovejoyetal.,2015],[DelRioAmadorandLovejoy,2019].For897multidecadaltimescalesthelowfrequencylimithasbeenusedasthebasisofclimate898projections through to the year 2100 [Hebert, 2017], [Lovejoy et al., 2017]. The899successofthesetwoapplicationswithdifferentexponentsbutwithvaluespredicted900bytheFEBEwiththesameempiricalunderlyingH≈0.4,iswhatoriginallymotivated901theFEBE,andtheworkreportedhere.Thestatisticalcharacterizations–correlations,902structure functions Haar fluctuations and spectra as well as the predictability903propertiesareimportantfortheseandotherFEBEapplications.904

While the deterministic fractional relaxation equation is classical, various905technicaldifficultiesarisewhenitisgeneralizedtothestochasticcase:inthephysics906literature,itisaFractionalLangevinEquation(FLE)thathasalmostexclusivelybeen907consideredasamodelofdiffusionofparticlesstartingatanorigin.Thisrequirest=9080 (Riemann-Liouville) initial conditions that imply that the solutions are strongly909nonstationary. In comparison, the Earth’s temperature fluctuations that are910associatedwithitsinternalvariabilityarestatisticallystationary.Thiscaneasilybe911modelledbyWeylfractionalderivatives,i.e.initialconditionsat .912

BeyondtheproposalthattheFEBEisagoodmodelfortheEarth’stemperature,913thekeynoveltyofthispaperisthereforetoconsidertheFEBEasaWeylfractional914Langevinequation.WhendrivenbyGaussianwhitenoises,thesolutionsareanew915stationaryprocess–fractionalRelaxationnoise(fRn).Overtherange0<H<1/2,we916showthatthesmallscalelimitisafractionalGaussiannoise(fGn)–anditsintegral-917fractionalRelaxationmotion(fRm)-hasstationaryincrementsandwhichgeneralizes918fractionalBrownianmotion(fBm).Althoughatlongenoughtimes,thefRntendstoa919Gaussian white noise, and fRm to a standard Brownian motion, this long time920convergenceisslow(itisapowerlaw).921

ThedeterministicFEBEhastwoqualitativelydifferentcases:0<H<1and1<H922<2 corresponding to fraction relaxation and fractional oscillation processes923respectively.Incomparison,thestochasticFEBEhasthreeregimes:0<H<1/2,1/2924<H<3/2,3/2<H<2,withthelowerranges(0<H<3/2)havinganomaloushigh925frequencyscaling. Forexample, itwas found that fluctuationsoverscalessmaller926thantherelaxationtimecaneitherdecayorgrowwithscale-withexponentH-1/2927(section3.5)-theparameterrange0<H<3/2hasthesamescalingasthe(stationary)928fGn(H<1/2)andthe(nonstationary)fBm(1/2<H<3/2),sothatprocessesthat929havebeenempiricallyidentifiedwitheitherfGnorfBmonthebasisoftheirscaling,930mayinfactturnouttobe(stationary)fRnprocesses;thedistinctionisonlyclearat931timescalesbeyondtherelaxationtime.932

Since the Riemann-Liouville fractional relaxation equation had already been933studied, themainchallengewasto implementtheWeyl fractionalderivativewhile934avoidingdivergenceissues. ThekeywastofollowtheapproachusedinfGn,i.e.to935startbydefiningfractionalmotions(e.g.fBm)andthenthecorrespondingnoisesas936the(ordinary)derivatives(orfirstdifferences)ofthemotions.Overtherange0<H937<1/2,thenoisesfGnandfRndivergeinthesmallscalelimit:likeGaussianwhitenoise,938theyaregeneralizedfunctionsthatarestrictlyonlydefinedunderintegralsigns;they939canbestbehandledasdifferencesofmotions.940

t = −∞

39

Althoughthebasicapproachcouldbeappliedtoarangeoffractionaloperators941corresponding to a wide range of FLEs, we focused on the fractional relaxation942equation. Much of the effortwas to deduce the asymptotic small and large scale943behaviours of the autocorrelation functions that determine the statistics and in944verifyingthesewithextensivenumericsimulations.Aninterestingexceptionwasthe945H=1/2specialcasewhichforfGncorrespondstoanexactly1/fnoise.Here,wewere946abletofindexactmathematicalexpressionsforthefullcorrelationfunctions,showing947thattheyhadlogarithmicdependenciesatbothsmallandlargescales.Theresulting948HalforderEBE(HEBE)hasanexceptionallyslowtransitionfromsmalltolargescales949(afactorofamillionormoreisneeded).950

Beyond improvedmonthly, seasonal temperature forecastsandmultidecadal951projections,thestochasticFEBEopensupseveralpathsforfutureresearch.Oneof952themorepromisingoftheseistofollowuponthespecialvalueH=1/2thatisvery953closetothatfoundempiricallyandthatcanbeanalyticallydeducedfromtheclassical954Budyko-Sellersenergytransportequationbyimprovingthemathematicaltreatment955oftheradiativeboundaryconditions[Lovejoy,2019b].Inthelattercase,oneobtains956apartial fractionaldifferentialequationforthehorizontalspace-timevariabilityof957temperature anomalies over the Earth’s surface, allowing regional forecasts and958projections. Generalizations include the nonlinear albedo-temperature feedbacks959neededformodellingoftransitionsbetweendifferentpastclimates.960

Acknowledgements: 961I thank L. Del Rio Amador, R. Procyk, R. Hébert, C. Penland, N.Watkins for962

discussions.IalsoacknowledgeanexchangewithK.Rypdal.Thisworkwasunfunded,963therewerenoconflictsofinterest.964 965

40

Appendix A: Random walks and the Weyl fractional Relaxation equation 966Theusualfractionalderivativesthatareconsideredinphysicalapplicationsare967

definedovertheintervalfrom0tot;thisincludestheRiemann-Liouville(“R-L”;e.g.968themonographsby[MillerandRoss,1993],and[Westetal.,2003])andtheCaputo969fractionalderivatives[Podlubny,1999]. Thedomain0tot isconvenient for initial970value problems and can notably be handled by Laplace transform techniques.971However,manygeophysicalapplicationsinvolveprocessesthathavestartedlongago972andaremostconvenientlytreatedbyderivativesthatspanthedomain tot, i.e.973that require the semi-infinite Weyl fractional derivatives and can be handled by974Fouriermethods.975

ItisthereforeofinteresttoclarifytherelationshipbetweentheWeylandR-L976stochasticfractionalequationsandGreen’sfunctionswhenthesystemsaredrivenby977stationarynoises.Inthisappendix,weconsiderthestochasticfractionalrelaxation978equationforthevelocityVofadiffusingparticle.Thiswasdiscussedby[Kobelevand979Romanov,2000]and[Westetal.,2003]inaphysicalsettingwhereVcorrespondsto980thevelocityofafractionallydiffusingparticle. ThefractionalLangevinformofthe981equationis:982

, (71)983wheregisawhitenoiseandwehaveusedtheR-Lfractionalderivative.Thisequation984canbewritteninamorestandardformbyintegratingbothsidesbyorderH:985

. (72)986

Theposition satisfies:987

, (73)988where isaWienerprocess.989

ThesolutionforX(t)isobtainedusingtheGreen’sfunctionG0,H:990

, (74)991whereEisaMittag-Lefflerfunction(eq.16).IntegratingbypartsandusingG1,H(0)=9920,W(0)=0weobtain:993

,994 (75)995thisyields:996

. (76) 997

−∞

0DtHV +V = γ

V t( ) = − 0Dt

−HV + 0Dt−Hγ = − 1

Γ H( ) t − s( )H−1V s( )ds0

t

∫ + 1Γ H( ) t − s( )H−1 γ s( )ds

0

t

∫

X t( ) = V s( )ds

0

t

∫ + X0

0DtHX + X =W

dW = γ s( )ds

X t( ) = G0,H t − s( )W s( )ds

0

t

∫ + X0E1,H −t H( ); G0,H t( ) = t H−1EH ,H −t H( )

G0,H t − s( )W s( )ds

0

t

∫ = G1,H t − s( )γ s( )ds0

t

∫ ; dW = γ s( )ds; G1,H t( ) = G0,H s( )ds0

t

∫

X t( ) = G1,H t − s( )γ s( )ds

0

t

∫ + X0E1,H −t H( )

41

X(t) is clearly nonstationary: its statistics depend strongly on t. The first step in998extractingastationaryprocessistotakethelimitofverylarget,andconsiderthe999processoverintervalsthataremuchshorterthanthetimesincetheparticlebegan1000diffusing.Wewillshowthattheincrementsofthisnewprocessarestationary.1001

DefinethenewprocessZt’(t)overatimeintervaltthatisshortcomparedtothe1002timeelapsedsincethebeginningofthediffusion(t’):1003

(77)1004(forsimplicitywewilltakeX0=0,butsince rapidlydecreasestozero,at1005

larget’thisisnotimportant).Nowusethechangeofvariables’=s-t’+t:1006

. (78)1007

Now,usethe fact that (equality inaprobabilitysense)andtake1008thelimit .Droppingtheprimeonswecanwritethisas:1009

, (79)1010wherewehavewrittenZ(t)forthelimitingprocess.1011

SinceZ(0)=0,Z(t) is stillnonstationary. Butnowconsider theprocessY(t)1012givenbyitsderivative:1013

(80)1014(sinceG1(0)=0).Y(t)isclearlystationary.1015

WenowshowthatY(t) satisfies theWeylversionof therelaxationequation.1016Consider the shifted function: and take Y0 as a solution to the1017Riemann-Liouvillefractionalequation:1018

, (81)1019orequivalentlyinintegralform:1020

, (82)1021withsolution:1022

(83)1023(withY0(0)=0).1024

Nowshiftthetimevariablesoastoobtain:1025

Z

ʹtt( ) = X ʹt( )− X ʹt −t( ) = G0,H ʹt − s( )γ s( )ds

0

ʹt

∫ − G0,H ʹt −t − s( )γ s( )ds0

ʹt −t

∫

E1,H − ʹt H( )

Z ′t t( ) = G1,H t − ′s( )γ ′s + ′t −t( )d ′s

− ′t +t

t

∫ − G1,H − ′s( )γ ′s + ′t −t( )d ′s− ′t +t

0

∫

γ ′s + ′t −t( )=

d

γ ′s( ) ′t →∞

Z t( ) = Z∞ t( ) = G1,H t − s( )γ s( )ds

−∞

t

∫ − G1,H −s( )γ s( )ds−∞

0

∫

Y t( ) = dZ t( )

dt= G0,H t − s( )γ s( )ds

−∞

t

∫ ; G0,H t( ) = dG1,H t( )dt

Y ′t t( ) =Y0 t + ′t( )

0DtHY0 +Y0 = γ

Y0 t( ) = − 0Dt

−HY0 + 0Dt−Hγ = − 1

Γ H( ) t − s( )H−1Y0 s( )ds0

t

∫ + 1Γ H( ) t − s( )H−1 γ s( )ds

0

t

∫

Y0 t( ) = G0,H t − s( )γ s( )

0

t

∫ ds

42

, (84)1026(withYt’(-t’)=0).Nowmakethechangeofvariables’=s–t’:1027

. (85)1028WeseethatYt’isthereforethesolutionof:1029

. (86)1030However,sinceYt’istheshiftedY0wehavethesolution:1031

. (87)1032

Again,using anddroppingtheprimes,weobtain:1033

. (88)1034Finally,takingthelimit wehavetheequationandsolutionfor :1035

, (89)1036with .1037

Theconclusionisthataslongastheforcingsarestatisticallystationarywecan1038use the R-L Green’s functions to solve the Weyl fractional derivative equation.1039Althoughwehaveexplicitlyderivedtheresultforthefractionalrelaxationequation,1040wecanseethatitisofwidergenerality.1041 1042

Y ′t t( ) = − 1

Γ H( ) t + ′t − s( )H−1Y0 s( )ds0

t+ ′t

∫ + 1Γ H( ) t + ′t − s( )H−1 γ s( )ds

0

t+ ′t

∫

Y ′t t( ) = − 1

Γ H( ) t − ′s( )H−1Y ′t ′s( )d ′s− ′t

t

∫ + 1Γ H( ) t − ′s( )H−1 γ ′s( )d ′s

− ′t

t

∫ ; γ ′s + ′t( )=d

γ ′s( )

− ′t DtHY ′t +Y ′t = γ

Y ′t t( ) =Y0 t + ′t( ) = G0 t + ′t − s( )γ s( )ds

0

t+ ′t

∫ = G0 t − ′s( )γ ′s + ′t( )− ′t

t

∫ d ′s

γ ′s + ′t( )=

d

γ ′s( )

Y ′t t( ) = G0 t − s( )γ s( )

− ′t

t

∫ ds

′t →∞Y t( ) =Y∞ t( )

−∞Dt

HY +Y = γ ; Y t( ) = G0 t − s( )γ s( )−∞

t

∫ ds; Y t( ) =Y∞ t( )

Y −∞( ) =0

43

Appendix B: The small and large scale fRn, fRm statistics: 1043

B.1 Discussion 1044Insection2.3,wederivedgeneralstatisticalformulaefortheauto-correlation1045

functionsofmotionsandnoisesdefined in termsofGreen’s functionsof fractional1046operators. SincetheprocessesareGaussian,autocorrelations fullydeterminethe1047statistics.WhiletheautocorrelationsoffBmandfGnarewellknown(anddiscussed1048insection3.1),thoseforfRmandfRnarenewandarenotsoeasytodealwithsince1049theyinvolvequadraticintegralsofMittag-Lefflerfunctions.1050

In this appendix,we derive the leading terms in the basic small and large t1051expansions, including results of Padé approximants that provide accurate1052approximationstofRnatsmalltimes.1053

B.2 Small t behaviour 1054fRnstatistics:1055a)Therange0<H<1/2:1056

Startwith:1057

, (90)1058(eq.34)andusetheseriesexpansionforG0,H:1059

(91)1060sothat:1061

. (92)1062

Thiscanbewritten:1063

.(93)1064

Evaluatingtheintegral,andchangingsummationvariables,weobtain:10651066

, (94)1067

wherewehavetakentakek=n+mandthesquarebracketsindicatetheintegerpart;1068beyondtheindicatedkrange,theintegralsdivergeatinfinity.1069

Wecannowsumoverm:1070

RH t( ) = N H

2 G0,H t + s( )G0,H s( )ds0

∞

∫

G0,H s( ) = −1( )n+1 s n+1( )H−1

Γ n+1( )n=0

∞

∑

RH t( ) = N H

2 −1( )n+m

Γ n+1( )Γ m+1( )s+ t( ) n+1( )H−1

s m+1( )H−1

0

∞

∫ dsn,m=0

∞

∑

RH t( ) = N H2 t−1+2 H Anm

n,m=0

∞

∑ t m+n( )H ; Anm =−1( )n+m

Γ n+1( )Γ m+1( )1+ξ( ) n+1( )H−1

ξ m+1( )H−1

0

∞

∫ dξ

Akm =−1( )k

Γ 1−H k +2( )( )sin Hπ m+1( )( )π

; k = m+ n; k < 1H⎡

⎣⎢

⎤

⎦⎥−2

44

, (95)1071wherewehaveused:1072

. (96)1073Finally,wecanintroducethepolynomialf(z)andwrite:1074

. (97)1075Takingthek=0 termonlyandusing theH<1/2normalizationNH=KH,wehave1076

and(asexpected),weobtainthefGnresult:1077

, (98)1078(fortlargerthantheresolutiont).1079

Sincetheseriesisdivergent,theaccuracydecreasesifweusemorethanone1080terminthesum.Theseriesisneverthelessusefulbecausethetermscanbeusedto1081determinePadéapproximants,and theycanbequiteaccurate (see fig.B1and the1082discussionbelow).Theapproximantoforder1,2wasfoundtoworkverywellover1083thewholerange0<H<3/2.10841085b)Therange1/2<H<3/2:1086

Inthisrange,notermsintheexpansioneq.97converge,however,theseries1087stillturnsouttobeuseful.Toseethis,usetheidentity:1088

1089, (99)1090wherewehaveusedtheH>1/2normalizationNH=1/CH.1091

Itturnsoutthatifweusethisidentityandsubstitutetheseriesexpansionfor1092G0,H,thattheintegralsconvergeupuntilorderm+n<[3/H]-2(ratherthan[1/H]-2),1093andthecoefficientsareidentical.Weobtain:1094

, (100)1095

wheretheBkarethesameasbefore.Thisformulaisveryclosetotheonefor0<H1096<1/2(eq.97).1097

RH t( ) = N H2 t−1+2 H Bk

k=0

1H

⎡⎣⎢

⎤⎦⎥−2

∑ t kH ; Bk = −1( )kΓ 1− H k + 2( )( )sin H k +1( )π2

⎛⎝⎜

⎞⎠⎟

sin H k + 2( )π2⎛⎝⎜

⎞⎠⎟

π sin H π2

⎛⎝⎜

⎞⎠⎟

sin Hπ m+1( )( )m=0

k+1

∑ =sin H k +1( )π2

⎛⎝⎜

⎞⎠⎟

sin H k + 2( )π2⎛⎝⎜

⎞⎠⎟

sin H π2

⎛⎝⎜

⎞⎠⎟

RH t( ) = N H

2 t−1+2 H f t H( ); f z( ) = Bk zk

k=0

1H

⎡⎣⎢

⎤⎦⎥−2

∑

KH2 B0 = H 1+ 2H( )

RH t( ) = H 1+2H( )t−1+2 H +O t−1+3H( ); t <<1; 0 < H <1/ 2

2 1− RH t( )( ) = N H

2 G0,H s+ t( )−G0,H s( )( )2

ds+ N H2

0

∞

∫ G0,H s( )2ds

0

t

∫ ; N H =CH−1; H >1/ 2

RH t( ) =1− N H2 t−1+2 H f t H( ); f z( ) = Bk zk

k=0

3H⎡

⎣⎢

⎤

⎦⎥−1

∑ ; 1/ 2 < H < 3/ 2

45

1098c)Therange3/2<H<2:1099

Again using the identity eq. 99, we can make the approximation1100

;thisisusefulsincewhenH>3/2, andwe1101

obtain:1102

. (101)1103

1104Padé:1105

Althoughtheseries(eqs.97,100)diverge,theycanstillbeusedtodetermine1106Padé approximants (see e.g. [Bender and Orszag, 1978]). Padé approximants are1107rationalfunctionssuchthatthefirstN+M+1oftheirTaylorexpansionsofarethe1108sameasthefirstN+M+1coefficientsofthefunctionftowhichtheyapproximate.1109Theoptimum(forH<1/4)istheN=1,M=2approximant(“Padé12”,denotedP12).1110Appliedtothefunctionf(z)ineq.97,itsfirstfourtermsare:1111

1112f(z)=B0+B1z+B2z2+B3z3, (102)11131114withapproximant:1115

, (103)1116

where theBk are taken from theexpansioneq.95. FiguresB1,B2show that the1117approximantsareespeciallyaccurateinthelowerrangeofHvalueswherethefirst1118termintheseries(thefGnapproximation)isparticularlypoor.1119

G0,H s+t( )−G0,H s( )≈ t ʹG0,H s( )

ʹG0,H s( )2ds

0

∞

∫ <∞

RHt( ) =1− t2

2CH2 ʹG0,H s( )

2ds

0

∞

∫ +O t2H−1( ); 3/2<H <2

P12 z( ) =

B0 B12−B0B2( )+z B13−2B0B1B2+B02B3( )

B0B2−B12+z B0B3−B1B2( )+z2 B1B3−B22( )

46

1120Fig. B1: The log10 ratio of the fRn correlation function R(fRn)H(t) to the fGn1121approximationR(fGn)H(t)(solid)andtothePadéapproximantR(Padé)H(t)(dashed)for1122H=1/20(black),2/20(red),3/20(blue),4/20(brown),5/20(purple). ThePade1123approximantisthePade12polynomial(eq.103).AsHincreasesto0.25,Padegets1124worse,fGngetsbetter(seefig.B2).1125

-12 -10 -8 -6 -4 -2 2

Log10t

-1.0

-0.5

0.5

1.0

Log 1

0r

47

1126Fig.B2:Thesameasfig.B1butforH=6/20(brown),7/20(blue),8/20(red),9/201127(black). The Pade12 approximant (dashed) is generally a bit worse than fGn1128approximation(solid).11291130fRmstatistics:1131

For the small t behaviourof themotion fRm, it is simplest to integrateRH(t)1132twice:1133

. (104)1134Usingtheexpansioneq.95,weobtain:1135

(t<<1).1136 (105)1137Theleadingtermsare:1138

-12 -10 -8 -6 -4 -2

-1.0

-0.5

0.5

1.0

Log10t

Log 1

0r

VH t( ) = 2 RH p( )dp

0

s

∫⎛

⎝⎜⎞

⎠⎟ds

0

t

∫

VH t( ) = KH2 t1+2 H Bk

H k +2( ) 1+H k +2( )( )k=0

1H⎡

⎣⎢

⎤

⎦⎥−2

∑ t kH

VH t( ) = t2 −CH−2t1+2 H Bk

H k +2( ) 1+H k +2( )( )k=0

3H⎡

⎣⎢

⎤

⎦⎥−2

∑ t kH ; 1/ 2 < H < 3/ 2

; 0 < H <1/ 2

48

, (t<<1)1139 (106)1140and:1141

(t<<1).1142

(107)1143Tofindanexpansionfortherange3/2<H<2,wesimilarlyintegrateeq.101:1144

. (108)1145

B.3 Large t behaviour: 1146Whentislarge,wecanusetheasymptotictexpansion:1147

1148

(109)1149

toevaluatethefirstintegralontherightineq.23. Usingeq.109fortheG1,H(s+t)1150termandtheusualseriesexpansionfortheG1,H(s)weseethatweobtaintermsofthe1151type:1152

, (110)1153therewillonlybetermsofdecreasingorder(theunittermhasnotdependence).1154

Nowconsiderthesecondintegralineq.23:1155

. (111)1156

AslongasH<1,bothofthesetermswillincreasewithtandwillthereforedominate1157thefirstterm:theywillthusbetheleadingterms.Wethereforeobtaintheexpansion:1158

, (112)1159

whereaHisaconstanttermfromthefirstintegral.Puttingthetermsinleadingorder,1160dependingonthevalueofH:1161

. (113)1162

VH t( ) = t1+2 H +O t1+3H( ); 0 < H <1/ 2

VH t( ) = t2 −

Γ −1−2H( )sin πH( )πCH

2 t1+2 H +O t1+3H( ); 1/ 2 < H < 3/ 2

VHt( ) =t2− t 4

12CH2 ʹG0,H s( )

2ds

0

∞

∫ +O t2H+1( ); 3/2<H <2

G1,H t( ) = 1+−1( )m

Γ 1−mH( )m=1

∞

∑ t−mH

s+ t( )−mH

snH ds ∝ t1− m−n( )H

0

∞

∫ ; m− n( )H >1

I2 = G1,H s( )2ds ≈

0

t

∫ 1− 2s−H

Γ 1−H( )+ ..

⎛

⎝⎜⎜

⎞

⎠⎟⎟ds ≈

0

t

∫ t − 2t1−H

Γ 2−H( )+O t1−2 H( ); t >>1

VH t( ) = N H

2 t − 2t1−H

Γ 2−H( )+ aH +O t1−2 H( )

⎡

⎣⎢⎢

⎤

⎦⎥⎥

VH t( ) = N H2 t − 2t1−H

Γ 2−H( )+ aH +O t1−2 H( )

⎡

⎣⎢⎢

⎤

⎦⎥⎥; H <1

VH t( ) = N H2 t + aH −

2t1−H

Γ 2−H( )+O t1−2 H( )

⎡

⎣⎢⎢

⎤

⎦⎥⎥; H >1

49

TodetermineRH(t)wesimplydifferentiatetwiceandmultiplyby½:1163

. (114)1164

Notethatfor0<H<1,G(-H)<0sothatoverthisrangeR>0.1165All the formulae for both the small and large t behaviours were verified1166

numerically;seefigs.2,3,4.1167 1168

RH t( ) = −N H2 t−1−H

Γ −H( )+O t−1−2 H( )

⎡

⎣⎢⎢

⎤

⎦⎥⎥; 0 < H < 2

50

Appendix C: The H=1/2 special case: 1169WhenH=1/2,thehighfrequencyfGnlimitisanexact“1/fnoise”,(spectrum1170

w-1)ithasbothhighandlowfrequencydivergences.Thehighfrequencydivergence1171canbetamedbyaveraging,butthenotthelowfrequencydivergence,sothatfGnis1172onlydefinedforH<1/2. However, forthefRn, the lowfrequenciesareconvergent1173(appendix B) over the whole range 0 <H < 2, and forH = 1/2 we find that the1174correlationfunctionhasalogarithmicdependenceatbothsmallandlargescales.This1175is associated with particularly slow transitions from high to low frequency1176behaviours. ThecriticalvalueH=1/2correspondstotheHEBEthatwasrecently1177proposed [Lovejoy, 2019b] where it was shown that the valueH = 1/2 could be1178derivedanalyticallyfromtheclassicalBudyko-Sellersenergybalanceequation.1179

ForfRn,itispossibletoobtainexactanalyticexpressionsforRH,VHandtheHaar1180fluctuations;wedeveloptheseinthisappendix,forsomeearlyresults,see[Mainardi1181andPironi,1996].Forsimplicity,weassumethenormalizationNH=1.1182

Thestartingpointistheexpression:1183

, (115)1184(e.g.[Podlubny,1999]).Fromthis,weobtaintheimpulseandstepGreen’sfunctions:1185

, (116)1186(seeeq.16).TheimpulseresponseG0,H(t)canbewrittenasaLaplacetransform:1187

. (117)1188Therefore,thecorrelationfunctionis:1189

(118)1190Performingthesandpintegralswehave:1191

. (119)1192Finally,thisLaplacetransformyields:1193

, (120)1194where:1195

E1/2,1/2 −z( ) = 1π− zez

2

erfc z( )

E1/2,3/2 −z( ) = 1− ez2erfc z( )z

erfc z( ) = 2π

e− s2

dsz

∞

∫

G0,1/2 t( ) =1πt− eterfc t1/2( )

G1,1/2 t( ) =1− eterfc t1/2( )

G0,1/2 t( ) = 1

πp

1+ pe− tp dp

0

∞

∫

R1/2 t( ) = G0,1/2 t + s( )G0,1/2 s( )ds

0

∞

∫ = 1π 2 dse−s p+q( ) qp

1+ p( ) 1+ q( ) e−qt dp dq0

∞

∫0

∞

∫0

∞

∫

R1/2 t( ) = 12π

11+ q( ) +

q1+ q( ) −

1

1+ q( )⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥e−qt dq

0

∞

∫

R1/2 t( ) = 1

2e− terfi t − eterfc t( )− 1

2πet Ei −t( ) + e− t Ei t( )( )

51

, (121)1196and:1197

. (122)1198ToobtainthecorrespondingVHuse:1199

. (123)1200TheexactV1/2(t)is:1201

,1202 (124)1203

where is the MeijrG function, Chi is the CoshIntegral function and Shi is the1204SinhIntegralfunction.1205

Wecanusetheseresultstoobtainsmallandlargetexpansions:1206

(125)1207

,12081209

wheregEisEuler’sconstant=0.57…and:1210

(126)1211

.1212WecanalsoworkoutthevarianceoftheHaarfluctuations:1213

1214 (127)1215

Ei z( ) = − e−u du

u− z

∞

∫

erfi z( ) = −i erf iz( )( ); erf z( ) = 2

πe−s2

ds0

z

∫

V1/2 t( ) = 2 R1/2 p( )dp

0

s

∫⎛

⎝⎜⎞

⎠⎟ds

0

t

∫

V1/2 t( ) =G3,42,2 t2, 2, 5/2

2, 2, 0, 5/2

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥+et

πShi t( )−Chi t( )( )+ e−terfi t( )−eterf t( )( )

+t 1+ γE −1π

⎛

⎝⎜⎜

⎞

⎠⎟⎟−4

tπ+1+t( )logt

π+1+ γE

π

G3,42,2

R1/2 t( ) = − 2γ E +π +2log t

2π

⎛

⎝⎜

⎞

⎠⎟+

2 tπ−

t2−

3+2γ E +π +2log t4π

⎛

⎝⎜

⎞

⎠⎟t2 +O t3/2( ); t <<1

R1/2 t( ) = 1

2 πt−3/2 −

1π

t−2 +15

8 πt−7/2 +O t−4( ); t >>1

V1/2 t( ) = − t2 log t

π+

191−156γ E −78π144π

+16

15 πt5/2 −

t3

6−

t4 log t12π

+O t3/2( ); t <<1

V1/2 t( ) = t +

π +2γ E

π+

2log tπ

−4

πt1/2 +

1

πt−1/2 −

2π

t−2 +15

4 πt−3/2 +O t−4( ); t >>1

ΔU1/2

2 Δt( ) =Δt2 logΔt

4π+

6π +12γ E − log16+960log2240π

+512 2 −2( )

240 πΔt1/2 +

Δt3+O Δt3/2( ); Δt <<1

52

.1216Figure C1 shows numerical results for the fRn with H = ½, the transition1217

betweensmalland largetbehaviour isextremelyslow;the9ordersofmagnitude1218depicted in the figure are barely enough. The extreme low (R1/2)1/2 (dashed)1219asymptotesatthelefttoaslopezero(asquarerootlogarithmiclimit,eq.125),and1220toa-3/4slopeattheright.TheRMSHaarfluctuation(black)changesslopefrom01221to-1/2(lefttoright).Thisisshownmoreclearlyinfig.C2thatshowsthelogarithmic1222derivativeoftheRMSHaar(black)comparedtoaregressionestimateovertwoorders1223ofmagnitudeinscale(blue;afactor10smallerand10largerthantheindicatedscale1224wasused).ThisfigureunderlinesthegradualnessofthetransitionfromH=0toH=1225-1/2.Ifempiricaldatawereavailableonlyoverafactorof100inscale,depending1226onwherethisscalewaswithrespecttotherelaxationtimescale(unityintheplot),1227theRMSHaarfluctuationscouldhaveanyslopeintherange0to-1/2withonlysmall1228deviations.1229

1230Fig.C1:fRnstatisticsforH=1/2:thesolidlineistheRMSHaarfluctuation,thedashed1231lineistherootcorrelationfunction(R1/2)1/2(thenormalizationconstant=1,ithasa1232logarithmicdivergenceatsmallt).1233

ΔU1/2

2 Δt( ) = 4Δt−1 −32 2

πΔt−3/2 +

3t−2 logΔtπ

+O Δt−2( ); Δt >>1

-4 -2 2 4Log10t

-3.0

-2.5

-2.0

-1.5

-1.0

-0.5

Log 1

0R1/2

53

1234Fig. C2: The logarithmic derivative of the RMSHaar fluctuations (solid) in fig. C11235comparedtoaregressionestimateovertwoordersofmagnitudeinscale(dashed;a1236factor 10 smaller and 10 larger than the indicated scale was used). This plot1237underlinesthegradualnessofthetransitionfromH=0toH=-1/2:overrangeof1001238orso inscale there isapproximatescalingbutwithexponents thatdependon the1239rangeofscalescoveredbythedata.Ifdatawereavailableonlyoverafactorof100in1240scale,theRMSHaarfluctuationscouldhaveanyslopeinthefGnrange0to-1/2with1241onlysmalldeviations.1242 1243

- 4 - 2 2 4Log10Dt

-0.5

-0.4

-0.3

-0.2

-0.1

Log

10< D

TDt(

)2 >1/

2

54

12441245

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