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Chapter2.Newworldsversusscaling:fromvanLeeuwenhoektoMandelbrot

2.1ScaleboundthinkingandthemissingquadrillionWe just tookavoyage throughscales,noticing structures in cloudphotographsand

wiggles on graphs. Collectively these spanned ranges of scale over factors of billions inspaceandbillionsofbillions in time. Weare immediately confrontedwith thequestion:howcanweconceptualizeandmodelsuchfantasticvariation?

Twoextremeapproacheshavedeveloped,forthemomentIwillcallthedominantonethe “new worlds” view after Antoni van Leeuwenhoek (1632-1723), who developed apowerfulearlymicroscope,theother,theself-similar(scaling)viewbyBenoitMandelbrot(1924-2010)thatIdiscussinthenextsection.Myownview-scalingbutwiththenotionofscaleitselfanemergentproperty-isdiscussedinch.3.

WhenvanLeeuwenhoekpeeredthroughhismicroscopea,inhisamazementheissaidto have discovered a “new world in a drop of water”: “animalcules”, the first micro-organismsb(fig.2.1).Sincethen,theideathatzoominginwillrevealsomethingtotallynewhas become second nature: in the 21st century atom-imagingmicroscopes are developedprecisely because of the promise of such newworlds. The scale-by-scale “newness” ideawas graphically illustrated by K. Boeke’s highly influential book “Cosmic View” (1957)which starts with a photograph of a girl holding a cat, first zooming away showing thesurroundingvastreachesofouterspace,andthenzoominginuntilreachingthenucleusofan atom. The book was incredibly successful, and was included in Mortimer Adler's“Gateway to the Great Books” (1963), a 10 volume series featuring works by Aristotle,Shakespeare,Einsteinandothers.In1968,twofilmswerebasedonBoeke’sbook:“CosmicZoom”cand“PowersofTen”(1968d,re-releasedin1977e)whichencouragedtheideathatnearly every power of ten in scale hosted different phenomena. More recently (2012),there’seventheinteractiveCosmicEye,appfortheiPad,iPhone,oriPod.Ina1981paper,Mandelbrot coined the term “scalebound” for this “New Worlds” view, a convenientshorthandfthatIusefrequentlybelowg.

While“PowersofTen”wasproselytizingthenewworldsviewtoanentiregeneration,therewereotherdevelopmentsthatpushedscientificthinkinginthesamedirection.Inthe1960’s,longiceandoceancoreswererevolutionizingclimatesciencebysupplyingthefirstquantitativedataatcentennial,millennialandlongertimescales. Thiscoincidedwiththe

aTheinventorofthefirstmicroscopeisnotknown,butvanLeuwenhoek’swaspowerful,uptoabout300timesmagnification.bRecenthistoricalresearch indicatesthatRobertHookemayin facthaveprecededvanLeeuwenhoek,butthelatterisusuallycreditedwiththediscovery.cProducedbytheNationalFilmBoardofCanada.dByCharlesandRayEames.eThere-releasehadthesubtitle:“AFilmDealingwiththeRelativeSizeofThingsintheUniverseandtheEffectofAddingAnotherZero”andwasnarratedbyP.Morrison.Morerecently,thesimilar“CosmicVoyage”(1996),appearedinIMAXformat.fHewroteitashere,asoneword,asasingleconcept.gHewaswritinginLeonardo,toanaudienceofarchitects:“Iproposethetermscaleboundtodenoteanyobject,whether innatureoronemadebyanengineeroranartist, forwhichcharacteristicelementsofscale,suchaslengthandwidth,arefewinnumberandeachwithaclearlydistinctsize”:1 Mandelbrot,B.Scaleboundorscalingshapes:ausefuldistinctioninthevisualartsandinthenaturalsciences.Leonardo14,43-47(1981).

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development of practical techniques to decompose a signal into oscillating components:“spectral analysis”. While ithadbeenknownsince JosephFourier (1768-1830) thatanytime series may be written as a sum of sinusoids, applying this idea to real data wascomputationally challenging and in atmospheric science had been largely confined to thestudyofturbulenceh. Thebreakthroughwasthedevelopmentoffastcomputerscombinedwiththediscoveryofthe“FastFourierTransform”(FFT)algorithmi(1968).

ThebeautyofFourierdecompositionisthateachsinusoidhasanexact,unambiguoustimescale: itsperiod(theinverseof its frequency)isthelengthoftimeittakestomakeafull oscillation (fig. 2.2a, upper left for examples). Fourier analysis thus provides asystematicwayofquantifyingthecontributionofeachtimescaletoatimeseries.Fig.2.2aillustrates this for the Weierstrass function which in this example, is constructed bysummingsinusoidswithfrequenciesincreasingbyfactorsoftwosothatthenthfrequencyisω=2n. Theamplitudesdecreasebyfactorsof2-H(here=0.79)sothatthenthamplitudeis2-nH.Fig2.2a(upperleft)showstheresultforH=1/3withallthetermsupuntil128cyclesper second (upper row); eliminating n, we find the power law relation A = ω-H. Moregenerallyforascalingprocess,wehave:

Spectrum=(frequency)-β

Whereβistheusualnotationforthe“spectralexponent”j.Thespectrumisthesquareoftheamplitude, so that in this (discrete) examplekwe have β =2H. The spectrum of theWeierstrassfunctionisshowninfig.2.2abottomrow(left)asadiscreteseriesofdots,onefor each of the 8 sinusoids in the upper left construction. On the bottom row (right)weshowthesamespectrumbutonalogarithmicplotonwhichpowerlawsarestraightlines.Ofcourse,intherealworld-unlikethisacademicexample-thereisnothingspecialaboutpowersof2sothatallfrequencies–acontinuum-arepresent.

TheWeierstrassfunctionwascreatedbyaddingsinusoids:Fouriercomposition.Nowtakeamessypieceofdata–forexamplethemultifractalsimulationofthedataseries(lowerleftinfig.1.3):ithassmall,mediumandlargewiggles.Toanalyzeitweneedtheinverseofcomposition,andthisiswheretheFFTishandy.Inthiscase,byconstruction,weknowthatall the wiggles are generated randomly by the process; that they are unimportant l .However, ifwe hadno knowledge – or only a speculation - about of themechanism thatproduced it,wewouldwonder: do thewiggles hide signatures of important processes ofinterest,oraretheysimplyuninterestingdetailsthatshouldbeaveragingoutandignored?

hEvenhere,spectrawereoftenbyusingspecializedcircuitryinvolvingnumerousnarrowbandfilters.iThe speed-up due to the invention of the FFT is huge: even for the relatively short series in fig. 1.3 (2048points)itisaboutafactorofonehundred.InGCM’sitacceleratescalculationsbyfactorsofmillions.jThenegativesignisusedbyconventionsothatintypicalsituations,βispositive.kInthemoreusualcaseofcontinuousspectra,wehaveβ =1+2Hpossiblywithcorrectionswhenintermittencyisimportant.lWemeanthattheydon’timplyanyspecialorigin,mechanism.Howeverinvariousapplications,wemightonlybesensitivetoanarrowrangeoffrequencies-forexamplewindblowingagainstaswing.

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Fig.2.1:AntonivanLeuwenhoekdiscovering“animalcules”(micro-organisms),circa1675.Fig.2.2bshowsthespectrumofthemultifractalsimulation(fig.1.3lowerleft)forall

periodslongerthan10milliseconds.Howdoweinterprettheplot?Oneseesthreestrongspikes,atfrequenciesof12,28and41cyclespersecond(correspondingtoperiodsof1/12,1/28 and 1/41 of a second, about 83, 35, 24 milliseconds). Are they signals of someimportantfundamentalprocessoraretheyjustnoise?

Naturally,thisquestioncanonlybeansweredifwehaveamentalmodelofhowtheprocessmightbegenerated,andthisiswhereitgetsinteresting. Firstofall,considerthecasewherewehaveonlyasingleseries.Ifweknewthesignalwasturbulent(asitwasforthetopdataseries),thenturbulencetheorytellsusthatwewouldexpectallthefrequenciesin awide continuumof scales tobe important, and furthermore, that at leastonaverage,that their amplitudes should decay in a power law manner (as with the Weierstrassfunction). But the theory only tells us the spectrum that wewould expect to find if weaveragedovera largenumberof identicalexperimentsm(eachonewithdifferent“bumps”andwiggles,butfromthesameoverallconditions).Infig.2.2b,thisaverageisthesmoothbluecurve.

Butinthefigure,weseethatthereareapparentlylargedeparturesfromthisaverage.Arethesedeparturesreallyexceptionalorarethesejust“normal”variationsexpectedfromrandomlychosenpiecesofturbulence?Beforethedevelopmentofcascademodelsandthediscovery ofmultifractals in the1970’s and80’s, turbulence theorywouldhave ledus toexpect that theupanddownvariationsaboutasmooth line throughthespectrumshouldroughlyfollowthe“bellcurve”. Ifthiswasthecase,thenthespectrumshouldnotexceedthe bottom red curvemore than1%of the time and the top curvemore than one in tenbilliontimes. Yet,weseethateventhis1/10,000,000,000curveisexceededtwiceinthissinglebutnonexceptionalsimulationn. Thisturnsouttobeanexampleofextreme“black

mAn“ensemble”or“statistical”average,ch.4.nIadmitthattomakemypoint,Imade500simulationsofthemultifractalprocessinfig.1.3andthensearchedthroughthefirst50tofindtheonewiththemoststrikingvariation.Butthiswasbynomeansthemostextremeofthe5000andifthestatisticshadbeenfromthebellcurve,thentheextremepointinthespectruminfig.2.3wouldhavecorrespondedtoaprobabilityofonein10trillion,sothatmyslightcheatingintheselectionprocesswouldstillhavebeenextremelyunlikelytohavecausedtheresult!

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swan”variabilitythatwediscussinch.?Hadweencounteredthisseriesinanexperiment,turbulencetheory itselfwouldprobablyhavebeenquestioned–as it indeed itrepeatedlywas(andstillis).Failuretofullyappreciatethehugevariabilitythatisexpectedinturbulentprocesses and continued embrace of inappropriate bell curve type paradigms hasspuriouslysheddiscreditonmanyattemptsatestablishingturbulent lawsandhasbeenamajorobstacleintheirunderstanding.

Fig. 2.2a: Upper left: The first eight contributions to the Weierstrass function (displaced in thevertical for clarity). Sinusoidswith frequenciesof1, 2, 4, 8, 16, 32,64,128 cyclesper second (thetimetisinseconds).Upperright:Sinusoidswithfrequenciesof2,4,8,16,32,64,128,cyclespersecond,stretchedbyafactor2H=1.25intheverticalandafactoroftwointhehorizontal.Thesum(top)isthesameasthatontheleftbutismissingthehighestfrequencydetail(seethediscussionalittlelater).Lower left: The spectrumon a linear-linear scale each point indicating the contribution (squared)andthefrequency.Lowerright:Thesameaslowerleftbutonalogarithmicplot(itisnowlinear).

Inconclusion,untilthe1980’swiththedevelopmentofmultifractals,evenifweknewthat the series came from an apparently routine turbulentwind trace on the roof of thephysicsbuilding,wewouldstillhaveconcludedthatthebumpswereindeedsignificant.

Butwhatwouldbeourinterpretationifinsteadfig.2.2bwasthespectrumofaclimateserieso?Wewouldhavenogoodtheoryofthevariabilityandwewouldtypicallyonlyhaveasingletrace.

Let’staketheexampleofanicecorerecord.TheseriesitselfwaslikelytheproductofoObviouslywithtotallydifferenttimescales!

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a near heroic scientific effort, possibly involving months in freezing conditions near thesouthpole. Thesamplewouldfirstbecoredandthentransportedtothelab. Thiswouldhave been followed by a painstaking sampling and analysis of the isotopic compositionusingamassspectrometer,thenadigitizationoftheresult.Carefulcomparisonwithothercoresorwith ice flowmodelswouldeventuallyestablisha chronology. At thispoint, theresearcherwouldbeeagerforaquantitativelookatwhatshehadfound.Iftheblackcurvein fig. 2.2b was the spectrum of such a core, how would she react to the bumps in thespectrum? Unlike the turbulence situationwhere therewas some theory, an early corewouldhavehad littlewithwhich to compare it. This is thepointwhere thenewworldsview could easily influence the researcher’s resultsp. She would be greatly tempted toconclude that the spikes were so strong, so far from the bell curve theory that theyrepresented real physical oscillations occurring over a narrow range of time scales. Shewould also remark that the twomain bumps in the spectrum involve several successivefrequencies,andaccordingtousualstatisticalassumptions,“backgroundnoise”shouldnotbecorrelatedinthisway. Thiswidebumpwouldstrengthentheinterpretationthattherewasahiddenoscillatoryprocessatworkq.Armedwiththeseriesofbumps,shemightstarttospeculateaboutpossiblephysicalmechanismstoexplainthemr.

Fig.2.2b:Black:theFourierspectrumofthechangesinwindspeedinthe1secondlongsimulationshownat thebottom left of fig. 1.3, showing the amplitudessof the first 100 frequencies (ω). Theupperleftisthusforonecycleoverthelengthofthesimulation,i.e.onecyclepersecond,aperiodofonesecond.Thefarrightshowsthevariabilityat100cyclespersecondgivingtheamplitudeofthewigglesat10milliseconds (higher frequencieswerenot shown for clarity). Thebrownshows theaverageover500randomserieseachidenticaltothatinfig.1.3:asexpected,itisnearlyexactlythetheoretical(scaling)powerlaw(blue,thetwovirtuallyontopofeachother).Thethreeredcurvesshowthetheoretical1%,oneinamillionandoneintenbillionextremefluctuationlimits(bottomtotop)determinedbyassumingthatthespectrumhasbellcurve(Gaussian)probabilities.

pAlternatively, theremight be incorrect theories that could be spuriously supported by unfortunately placedrandomspectralbumps,andmuchtimewouldbewastingchasingblindalleys.qAccording to standard assumptions that –if only demonstrated by this example are clearly inappropriate -successivefrequenciesshouldbestatisticallyindependentofeachother.rIn ch. 4? We give some examples of spurious periodicities that emerged from inappropriate probabilityassumptionsintheuseofthemulti-tapermethod(MTM)andSingularSpectralAnalysis(SSA)method.sThespectrumisactuallytheensembleaverageofthesquaresoftheabsoluteamplitudes.Itwaswindowed”inordertoavoidspurious“spectralleakage”thatcouldartificiallysmearoutthespectrum.

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Weshouldthusnotbesurprisedtolearnthatthe1970’switnessedarashofpapers

basedonspectraresemblingthatoffig.2.2b:oscillatorsweresuddenlyubiquitoust.ItwasinthiscontextthatMurrayMitchell5(1928-1990)famouslymadethefirstexplicitattempttoconceptualizetemporalatmosphericvariability(fig2.3a).Mitchell’sambitiouscompositespectrum ranged from hours to the age of the earth (≈4.5x109 to 10-4 years, bottom, fig.2.3a). In spite of his candid admission that this was mostly an “educated guess”, andnotwithstanding the subsequent revolution in climate and paleoclimate data, over fortyyears later it has achieved an iconic status and is still regularly cited and reproduced inclimatepapersandtextbooks6,7,8. Itscontinuing influence isdemonstratedby theslightlyupdatedversionshowninfig.2.3bthat(until2015)adornedNOAA’sNationalClimateDataCenter (NCDC) paleoclimate web siteu. The site was surprisingly forthright about thefigure’sideologicalcharacter.Whileadmittingthat“insomerespectsitovergeneralizesandover-simplifiesclimateprocesses”,itcontinued:“…thefigureisintendedasamentalmodeltoprovideageneral"powersoften"overviewofclimatevariability,andtoconveythebasiccomplexitiesofclimatedynamicsforageneralsciencesavvyaudience.”Noticetheexplicitreferencetothe“powersoften”mindsetoverfiftyyearsafterBoeke’sbookv.

Certainly the continuing influence of Mitchell’s figure has nothing to do with itsaccuracy.Withinfifteenyearsofitspublication,twoscalingcomposites(closetoseveralofthose shown in fig. 2.3 a), over the ranges 1 hr to 105 yrs, (see fig. 2.10 for the relatedfluctuations) and 103 to 108 yrs, already showed astronomical discrepancies9, 10. In thefigure, we have superposed the spectra of several of the series analysed in ch. 1; thedifferencewithMitchell’soriginalisliterallyastronomical.Whereasovertherange1hrto109yrs,Mitchell’sbackgroundvariesbyafactor≈150,thespectrafromrealdataimplythatthe true range isa factorofaquadrillionw(1015),NOAA’s fig.2.3bextends thiserrorbyafurtherfactoroftenx.

Writing a decade and a half after Mitchell, leading climatologists Shackelton andImbrie10laconicallynotedthattheirownspectrumwas“muchsteeperthanthatvisualisedbyMitchell”,aconclusionsubsequentlyreinforcedbyseveralscalingcomposites11,12.Overat leastasignificantpartof thisrange,Wunsch13 furtherunderlined itsmisleadingnatureby demonstrating that the contribution to the variability from specific frequencies

tI could alsomention the contribution of “Box-Jenkins” techniques (1970) to bolstering scalebound blinkers.Thesewereoriginallyengineeringtools foranalyzingandmodelingstochasticprocessesbasedontheaprioriscaleboundassumptionthatthecorrelationsdecayedinanexponentialmanner.Thisespeciallycontributedtoscaleboundthinkinginprecipitationandhydrologyseeforexampletheinfluentialpublications:2 Zawadzki,I.Statisticalpropertiesofprecipitationpatterns.JournalofAppliedMeteorology12,469-472(1973).3 Bras,R. L.&Rodriguez-Iturbe, I.Rainfall generation: anonstationary timevaryingmultidimensionalmodel.WaterResourcesResearch12,450-456(1976);4Bras,R.L.&Rodriguez-Iturbe,I.RandomFunctionsandHydrology.(Addison-WesleyPublishingCompany,1985).uThesiteexplicitlyacknowledgesMitchell’sinfluence.vIf this were not enough, the site adds a further gratuitous interpretation: assuring any sceptics that just“because a particular phenomenon is called an oscillation, it does not necessarilymean there is a particularoscillator causing the pattern. Some prefer to refer to such processes as variability.” Since any time serieswhether produced by turbulence, the stock market or a pendulum can be decomposed into sinusoids: thedecompositionhasnophysicalcontentperse,yetwearetoldthatvariabilityandoscillationsaresynonymous.wIn fig. 2.11a,weplot the same informationbut in real space and find thatwhereas theRMS fluctuations at5.53x108yearsare≈±10KsothatextrapolatingGaussianwhitenoiseovertherangeimpliesavalue≈10-6K,i.e.itisinerrorbyafactor≈107.xIf we attempt to extend Mitchell’s picture to the dissipation scales (at frequencies a million times higher,correspondingtomillisecondvariability),thespectralrangewouldincreasebyanadditionalfactorofabillion.

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associatedwithspecific“spikes”(presumedtooriginateinoscillatoryprocesses)wasmuchsmallerthanthecontributionduetothecontinuum.

JustasvanLeuwenhookpeeredthroughthefirstmicroscopeanddiscoveredanewworld,today,weautomaticallyanticipatefindingnewworldsbyzoominginoroutofscale.Itisascientificideologysopowerfulthatevenquadrillionsdonotshakeit.

Mitchell’s scalebound view led to a framework for atmospheric dynamics thatemphasized the importanceof numerousprocesses occurring atwell defined time scales,thequasiperiodic“foreground”processesillustratedasbumps–thesignals-onMitchell’snearly flat background that was considered to be an unimportant noisey. Although inMitchell’soriginalfigure,theletteringisdifficulttodecipher,fig.2.3bspellsthemoutmoreclearlywithnumerousconventionalexamples.Forexample,theQBOisthe“Quasi-BiennalOscillation”, ENSO is the “El Nino Southern Oscillation”, the PDO is the “Pacific DecadalOscillation” and the NAO is the “North Atlantic Oscillation”. At longer time scales, theDansgaards-OescherandMilankovitchandtectonic“cycles”zwillbediscussedinch.4.Thepointhereisnotthattheseprocesses,mechanisms,arewrongorinexistent,itisratherthattheyonlyexplainasmallfractionoftheoverallvariability.

Even the nonlinear revolution was affected by scalebound thinking. This includedatmosphericapplicationsof lowdimensionaldeterministicchaos. Whenchaostechniqueswere applied to weather and climate, the spectral bumps were associated with specificchaosmodels,analysedwiththehelpof thedynamicalsystemsmachineryofbifurcations,limit cycles and the likeaa. Of course – as discussed below - from the alternative scaling,turbulence view,wide range continuum spectra are generic results of systemswith largenumbersofinteractingcomponents(“degreesoffreedom”)-“stochasticchaos”15–andareincompatiblewith theusualsmallnumberof interactingcomponents (“lowdimensional”)deterministicchaos.Incredibly,afamouspaperpublishedinNatureevenclaimedthatfourinteractingcomponents(!)wereenoughtodescribeandmodeltheclimate16.Similarly,thespectrawill be scaling - i.e. power laws –whenever there are no dynamically importantcharacteristicscalesorscalebreaksbb(ch.3).

yMitchellactuallyassumedthathisbackgroundwaseitherawhitenoiseorovershortranges,sums(integrals)ofawhitenoise.zThefigurereferstotheseas“cycles”ratherthanoscillations,perhapsbecausetheyarebroader.aaMorerecentlyupdatedwiththehelpofstochastics:the“randomdynamicalsystems”approach,seee.g.:14 Chekroun,M.D., Simonnet,E.&Ghil,M. StochasticClimateDynamics:RandomAttractorsandTime-dependentInvariantMeasuresPhysicaD240,1685-1700(2010).8 Dijkstra,H.NonlinearClimateDynamics.(CambridgeUniversityPress,2013).bbAlthough in themorerecentrandomdynamicalsystemsapproach, thedrivingnoisemaybeviewedas theexpressionofalargenumbersofdegreesoffreedom,thisinterpretationisonlyjustifiedifthereisasignificantscalebreakbetweenthescalesofthenoiseandoftheexplicitlymodelleddynamics,itisnottriviallycompatiblewithscalingspectra.

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Fig.2.3a:AcomparisonofMitchell’srelativescale,“educatedguess”ofthespectrum(grey,bottom5)with modern evidence from spectra of a selection of the series displayed in fig. 1.4 (the plot islogarithmic in both axes). There are three sets of red lines; on the far right, the spectra from the1871-2008 20CR (at daily resolution) quantifies the difference between the globally averagedtemperature(bottom)andlocalaverages(2ox2o,top).

Thespectrawereaveragedoverfrequencyintervals(10perfactorofteninfrequency),thus“smearing out” the daily and annual spectral “spikes”. These spikes have been re-introducedwithout this averaging, and are indicated by green spikes above the red daily resolution curves.Using thedaily resolutiondata, the annual cycle is a factor≈1000 above the continuum,whereasusinghourlyresolutiondata,thedailyspikeisafactor≈3000abovethebackground.Alsoshownistheotherstrikingnarrowspectralspikeat(41kyrs)-1(obliquity;≈afactor10abovethecontinuum),this is shown in dashed green since it is only apparent over the period 0.8 - 2.56MyrBP (beforepresent).

The blue lines have slopes indicating the scaling behaviours. The thin dashed green linesshowthetransitionperiodsthatseparateouttheregimesdiscussedindetailinch.3;theseareat20days,50yrs,80,000yrs,and500,000yrs. Mitchell’soriginal figurehasbeen faithfullyreproducedmanytimes(withthesameadmittedlymediocrequality).Itisnotactuallyveryimportanttobeabletoreadtheletteringnearthespikes,ifneededtheycanseeninfig.2.3bwhichwasinspiredbyit.

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Fig.2.3b:TheupdatedversionofMitchell’s spectrumreproduced fromNOAA’sNCDCpaleoclimateweb sitecc. The “background” on this paleo site is perfectly flat; hence in comparison with theempiricalspectruminfig.2a,itisinerrorbyanoverallfactor≈1016.

***

At weather scales, and at virtually the same time as Mitchell’s scaleboundframework for temporal variability, Isidoro Orlanski proposed a scalebound spatialclassification of atmospheric phenomena by powers of ten (fig. 2.4)17. The figure is areproduction of Orlanski’s phenomenological space-time diagramddwith eight differentdynamical regimes indicated on the right according to their spatial scales. The diagramdoesmorethanjustclassifyphenomenaaccordingtotheirsize,italsorelatestheirsizestotheir lifetimesee. Along the diagonal, various pre-existing conventional phenomena areindicated including fronts, hurricanes, tornadoes and thunderstorms. The straight lineembellishmentwasaddedbycolleaguesandIin199718andshowsthatthefigureactuallyscaling,notscaleboundbehaviour!Thisisbecausestraightlinesonlogarithmicplotssuchasthisarepowerlawsandeventheslopeofthelineturnsouttobetheoreticallypredictedusingtheenergyratedensity;moreonthisbelow.

AtthetimeofOrlanski’sclassification,meteorologywasalreadylargelyscalebound.This was partly due to its near total divorce from turbulence theory that was largelyscalingff, and also itwas due to its heritage from the oldermore qualitative traditions of“synoptic”meteorologyandoflinearizedapproaches-thesebeingtheonlyonesavailableinthe pre-computer era. Orlanski’s classification therefore rapidly became popular as asystematicrationalizationofanalreadystronglyphenomenologicallyscaleboundapproach.It is ironic that just as Orlanski tried to perfect the old scalebound approach, and

ccThepageishassincebeentakendown.ddSometimescalled“Stommeldiagrams”afterHenryStommelwhoproducedsuchdiagramsinoceanography.eeNoticethatheindicatesthattheclimatestartsatabouttwoweeks(bottomrow).ffAlltheturbulencetheorieswerescaling,thequestionwaswhetheroneortwo–orinsomecasesthree-rangeswererequired;wediscussthisindetailinch.3.

Lovejoy� 2017-4-11 6:49 PMComment [1]: CommentaboutcloudclassificationgoingbacktoLukeHoward’slatinnamingsystemin1802,twelvecategoriesofclouds.Thiswasnothoweverdirectlyrelatiedtosize,itwasrathermorphology.

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unbeknownst to the modellers, the computer revolution was ushering in the oppositescalingapproach:theGeneralCirculationModelgg.

Fig.2.4:Orlanski’sspace-timediagramwitheightdifferentdynamicalregimesindicatedontherightaccordingtotheirspatialscales.Noticethatheindicatesthattheclimatestartsatabouttwoweeks(bottomrow).Thestraightshowsthatthefigureisactuallyscaling(straightonthislogarithmicplot).Reproducedfrom18.

2.2Scaling:BigwhirlshavelittlewhirlsandlittlewhirlshavelesserwhirlsScaleboundthinkingisnowsoentrenchedthatitseemsobviousthat“zoomingin”to

practically anything will disclose hidden secrets. Indeed, we would likely express morewonderifwezoomedinonlytofindthatnothinghadchanged,ifthesystem’sstructurewasscaling(fig.2.5)!Yetinthelastthirtyyearsantiscalingprejudiceshavestartedtounravel;much of this is thanks to Mandelbrot’s path breaking “Fractals, form chance anddimension”19hh(1977) and “Fractal Geometry of Nature”20 (1982). His books made animmediatevisualimpactthankstohisstunningavant-gardeuseofcomputergraphicsthatwerethefirsttodisplaytherealismofscaling.Onewasalsostruckbytheword“geometry”inthetitle:thelasttimescientistshadcaredaboutgeometrywaswhenD’ArcyThompson21

ggAtthetimetheGCMsweremuchtosmalltoallowforproperstatisticalscalinganalysesoftheiroutputs,andthereigningturbulencetheoryturnedouttobeseriouslyunrealistic,seech.3. Eventoday,thefactthatGCMsare scaling is practically unknown; the absence of a scale break is even seen as amodel deficiency, see thediscussioninch.4!hhThiswasactuallyatranslationandextensionoftheearlierFrenchbook“Fractals”,1975.

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brilliantlyusedittounderstandtheshapesofdiatomsandotherbiomorphologies. WhileMandelbrot’s simulations, imagery and scaling ideas sparked the fractal strand of thenonlinearrevolution-andcontinuetotransformourthinking-hisinsistenceongeometryis nownearly forgotten. The basic reason is that scientists are – inmy opinion rightly -moreinterestedinstatisticsthaningeometry.Thereisalsoalessobviousreason:themostinteresting thing to come from the scaling revolution was arguably not fractals, butmultifractals,and–despiteattempts-thesecannotgenerallybereducedtogeometryii.

In contrast with a “scalebound” object, Mandelbrot counterposed his new scaling,fractalone:

“A scaling object, by contrast, includes as its defining characteristic thepresence ofverymanydifferentelementswhosescalesareofany imaginablesize.Therearesomany different scales, and their harmonics are so interlaced and interact soconfusingly that they are not really distinct from each other, but merge into acontinuum. For practical purposes, a scaling object does not have a scale thatcharacterizesit.Itsscalesvaryalsodependingupontheviewingpointsofbeholders.Thesamescalingobjectmaybeconsideredasbeingofahuman'sdimensionorofafly'sdimension.”1

Fig.2.5:Thescalingapproach:lookingthroughthemicroscopeattheMandelbrotset(theblackintheupperleftsquare),Mandelbrotnoticesoneofaninfinitenumberofreducedscaleversions.

I had the good fortune to begin my own graduate career in 1976, just as the

iiThemathematicalissueistheirsingularsmallscalenature.Thebasicmultifractalprocessiscascades(box2.1)thatdonotconvergetomathematicalpointsbutonlyconvergeintheneighbourhoodofpoints.Thisprecludesthemfrombeingrepresentedasageometricsetofpoints.

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scalebound weather and climate paradigms were ossifying but before the nonlinearrevolutionreallytookoff(seetheboxonuniversality). Iwasthustotallyunpreparedandcan vividly remember the epistemic shock when shortly after it appeared, I firstencountered “Fractals, form chance and dimension”. Revealingly, it was neithermy PhDsupervisorGeoffAustin noranyotherscientificcolleaguewhointroducedmetothebook,butrathermymotherjj-anartist–whowasawedbyMandelbrot’simageryandfascinatedby its artistic implications. At the time, my thesis topic was the measurement ofprecipitationfromsatelliteskkandIhadbecomefrustratedbecauseoftheenormousspace-time variability of rain that was way beyond anything that conventional methods couldhandlell.Theproblemwasthattherewereseveralcompetingtechniquesforestimatedrainfromsatellitesandeachonewasdifferent,yettherewasessentiallynowaytovalidateanyof them: scientific progress in the field was essentially blocked. Fortunately, this didn’tpreventradarandsatelliteremotesensingtechnologyfromcontinuingtoadvance.

Not long after reading Mandelbrot’s book, I started working on developing fractalmodelsofrain,sothatwhenIfinallysubmittedmythesisinNovember1980,abouthalfofitwas on conventional remote sensing topics while the other half was an attempt tounderstandprecipitationvariabilitybyusingfractalanalysesandfractalmodelsofrainmm.Given that three of themore conventional thesis chapters had already been published injournals - and had thus passed peer review - I was confident of a rubber stamp by theexternal thesisexaminer. Since Ihadalreadybeenawardedofapost-doctoral fellowshipfinancedbyCanada’sNationalScienceandEngineeringResearchCouncil(NSERC)IhappilybeganpreparingforamovetoParistotakeitupattheMétéorologieNationale(theFrenchweatherservice).

Butratherthangettinganodandawink,theunimaginablehappened:mythesiswasrejected!TheexternalexaminerDavidAtlas(1924-2015)thenatNASA,wasapioneeringradarmeteorologist,whowas involved in the - then fashionable - scaleboundmeso-scaletheorizing(ch.3).Atlaswasclearlyuncomfortablewiththefractalmaterialbutratherthanattacking it directly, he instead claimed that while the content of the thesis might beacceptable,thatitsstructurewasnot.Tohiswayofthinking,therewereinfacttwothesesnotone.Thefirstwasaconventionalonethathadalreadybeenpublished,whilethesecondwasathesisonfractalprecipitationwhichaccordingtohimwasunrelatedtothefirst.Thelastpointpiquedmesinceitseemedobviousthatthefractalswerethereinanattempttoovercome longstandingproblemsof untamed space-timevariability: on the contrary theywereveryrelevanttoaremotesensingthesis.

At thatpoint, Ipanicked. According to theMcGill thesis regulations, Ihadonly twooptions:eitherIacceptthereferee’sstrictures,amputatetheoffendingfractalmaterialandresubmit, or I could refuse to bend. In the latter case, the thesiswould be sentwithoutchange to two external referees, both of which would have to accept it, a highly risky

jjShewas a pioneering electronic artist and had beenworkingwith early colour Xeroxmachines to developelectronicimageryyearsbeforethedevelopmentofpersonalcomputersandcheapcomputergraphics:22 Lovejoy,M.PostmodernCurrents:ArtandArtistsintheAgeofElectronicMedia. (PrenticeHallCollegeDivision,1989).kkMythesis(1981)wasentitled“Theremotesensingofrain”,Physics,dept.McGillUniversity.llConventionalmethodsarestillinvogue,butoverthelasttenyearsourunderstandingofprecipitationhasbeenrevolutionizedbytheapplicationofthefirstsatelliteborneweatherradar(theTropicalRainfallMeasurementMission), that has unequivocally demonstrated that - like the other atmospheric variables - precipitation is aglobal scale cascade process that is distinctive primarily because its intermittency parameter ismuch largerthanfortheotherfields.Moreonthisbelow.mmMy approach to rainfall modeling followed the method that Mandelbrot had used to make cloud andmountain models in his book, except that I used a variant that was far more variable (based on “Levy”distributionsratherthanthebellcurve).

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proposition.AlthoughIwasreadytodefendthefractalmaterial,Iknewfullwellithadnotreceivedseriouscriticalattention. Theremightbeerrorsthatwouldsinkthewholething.AsecondrejectionwouldbedisastrousbecauseMcGillwouldnotpermitmetoresubmitathesisonthesametopic. However,beforemakingadecisionandwiththeencouragementofAustin,IcontactedMandelbrot,visitinghimathisYorktownheightsIBMofficeinJanuary1981.

Mandelbrotwasverypositiveaboutthematerialinthedraftthesis. Notbeingveryfamiliarwith atmospheric sciencenn, andwanting to giveme the best possible advice, hecontactedhisfriend,theoceanographerEricMollo-Christensen(1923-2009)atMIT.Mollo-Christensenadvisedmetosimplyremovethefractalmaterial,andgetthethesisoutoftheway.Icouldthentrytopublishitintheusualscientificliterature.Beyondthat,Mandelbrotadvisedmetomakeashortpublicationoutoftheanalysispart,hintingthatwecouldlaterstartacollaborationtodevelopanimprovedfractalmodelofrain.

Withthefractalsexcised,thethesiswasacceptedwithoutahitchoo,andattheendofJune,aweekafterdefendingmythesis,IwentofftomyParispost-docattheMétéorologieNationale, toworkwith a radar specialist,MarcGiletpp. In literallymy firstweek in theFrenchcapital, Iwroteup theanalysispart - theempirical rainandcloudarea-perimeterrelation, fig. 2.8 and submitted it toScience23qq. A fewmonths later, I started a seriesofthree week visits to Mandelbrot in Yorktown heights. This collaboration eventuallyspawnedtheFractalsSumsofPulses(FSP)model24rrclosetothe“H–model”thatIdescribelater.

***An object is scaling if when blown up, a (small) part in someway resembles the

(large) whole. An unavoidable example is the icon that now bears his name – theMandelbrotset,theblacksilhouetteinfig.2.5.Itcanbeseenthatafteraseriesofblow-upswe find reduced scale copies of the original (largest version) of the setss. While theMandelbrot set has been termed “the most complex object in mathematics”25 it issimultaneously one of the simplest, being generated by simply iterating the algorithm: “Itakeanumber,squareit,addaconstant,squareit,addaconstant…”tt.Preciselybecauseofthis algorithmic simplicity, it is now the subject of a small cottage industry of computergeekswhosuperblycombinenumerics,graphics,andmusic.TheYouTubeisrepletewithexamples; the last time I looked, the record-holder displayed a zoom by a factor of over104000(aonewithfourthousandzeroes)!

TheMandelbrotsetmaybeeasytogenerate,butitishardlyeasytounderstand.Tounderstandthescaling,fractalidea,considerinsteadthesimplest(andhistoricallythefirst)fractal,the“perfect”Cantorset(fig.2.6).Startwithasegmentoneunitlong(infinitelythin:this ismathematics!);the“base”. Thenremovethemiddle1/3,this isthe“motif”(second

nnThe“FractalGeometryofNature”containedsimulationsofcloud“surfaces”basedonturbulencetheory(theCorrsin-Obhukhovlaw)notdirectlyonmeteorology.ooTwenty five years later, Imet upwithAtlas, by then in his 80’s but still occupying an office atNASA. Hisrejectionofmythesishadbeenafatherlyactintendingtosteermebacktowardsmainstreamscience.Duringourdiscussion,hewasmostlyintriguedthatIwasstillpursuingthematerialhehadrejectedsolongago!ppWithin two months of the start of my post-doc, Gilet was given a high level administrative position andessentiallywithdrewfromresearch. Asa freeagent, I soonstartedcollaboratingwithDanielSchertzer in thenewlyformedturbulencegroup.qqThepapersparkedastir;sincethenithasbeencitednearlyathousandtimes.rrTheFSPmodelwasanextensionandimprovementovertheLevyfaultmodelthatIhaddevelopedduringmyPhDthesis,butwasneverthelessstillmono–notmulti–fractal.ssThesmallversionsareactuallyslightlydeformedversionsofthelargerones.ttTogetaninterestingresult,theconstantshouldbeacomplexnumber(i.e.onethatinvolvesthesquarerootofminusone).

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fromthe top in the figure). Then iteratebyremoving themiddle thirdofeachof the two1/3longsegmentsfromtheprevious.Continueiteratingsothatateverylevel,oneremovesallthemiddlesegmentsbeforemovingtothenextlevel.Whenthisisrepeatedtoinfinitelysmallsegments,theresultistheCantorset26uu. Fromthefigure,wecanseethatifeithertheleftorrighthalfofthesetisenlargedbyafactorofthree,thenoneobtainsthesameset.

This property - that a part is in some way similar to the whole - is for obviousreasons called “self-similarity”. In this case, the left or right halves are identical to thewhole, in atmospheric applications, the relationship between a part and the whole willgenerallybestatistical,smallpartsareonlythesameasthewholeonaverage. TheCantorset hasmany interesting properties, for our purposes themain one being its fractality, aconsequenceofitsself-similarity.

Let’s consider it a little more closely. After n construction levels, the number ofsegments is N = 2n, and the length of each segment is L = (1/3)n. Therefore, N and Lthemselvesarerelatedbyapower law:eliminatingthe leveln,we findN=L-DwhereD=log(2)/log(3) = 0.63… D is the fractal dimension (“log” for logarithm). In this case, it iscalledthe“boxcounting”dimensionsince-ifweconsideredafullyformedCantorset-thenumberofsegmentsL(onedimensional“boxes”)thatwewouldneedtocoverthesetwouldbethesamevvN.Ifthisfractaldimensionseemsabitweird,considerwhatwouldhappenifweappliedittotheentireinitial(onedimensional)linesegment?Wecancheckthatwedoindeed recoverD =1. To see this, considerwhatwouldhappen ifwedidnot remove themiddle third(wekept theoriginalsegment)butanalysed itusing thesamereasoning. Inthiscasewouldhavestilldivideby3ateveryiterationsothatasbefore,L=(1/3)nbutnowthenumberofsegmentsissimplyN=3ninsteadof2n.ThiswouldleadtoD=log3/log3=1,simplyconfirmingthatthesegmentdoesindeedhavetheusualdimensionofaline.

WhenaquantitysuchasNchangesinapowerlawmannerwithscaleL,itiscalled“scaling”sothatN=L-Disascalinglaw. Contrarytoascaleboundprocessthatchangesitsmechanism,(its“laws”)everyfactorof10orso,auniquescalinglawmayholdoverawiderangeofscales;fortheCantorsetandothermathematicalfractals,overaninfiniterange.Ofcourse,realphysicalfractalscanonlybescalingoverfiniterangesofscale,thereisalwaysasmallestandlargestscalebeyondwhichthescalingwillnolongerbevalid.

Whydoesapowerlawimplyscaling(andvisaversa)?TheanswerissimplythatifN=L-Dandwezoominbyafactorλ(sothatL->L/ λ),thenweseethatN->N λD;sothattheform of the law is unchanged. In contrast, for a scalebound process, changing scales byzoomingwouldgiveussomethingquitedifferent.Wheneverthereisascalinglaw,thereissomethingthatdoesn’tchangewithscale,somethingthatisscaleinvariant:inthepreviousexample it is the fractaldimensionD. NomatterhowfarwezoomintotheCantorsetwewillalwaysrecoverthesamevalueD.Self-similarityisaspecialcaseofscaleinvarianceandoccurswhen-asitsnamesuggests-someaspectofthesystemisunchangedunderausualblow-up.Inphysics,quantitiessuchasenergythatareinvariant(conserved)undervarioustransformationsareoffundamentalimportance,hencethesignificanceofexponentssuchasfractaldimensionsthatareinvariantunderscaletransformations.

Moregenerally,asystemcanbeinvariantundermoregeneralized“zooms”i.e.blow-ups combined with stretchings, rotations or other transformations. As an example, let’sreturn to theWeierstrass functionwhich is scale invariant butnot self-similar. To showthat it is indeed scale invariant, we must combine a blow- up with a squashing - oralternativelyblowupbydifferentfactorsineachofthecoordinatedirections.Thisproperty

uuItwasapparentlydiscoveredabitearlierbyH.J.S.Smithin1874.vvHere - as almost always - the box counting dimension is the same as the Hausdorff dimension that issometimesusedinthiscontext.

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isshownin(fig.2.2a)bycomparingthefullWierstrassfunctionontheintervalbetween0and 1, with the upper right that shows the left half (omitting the lowest frequencyww),stretchedinthehorizontaldirectionbyafactor2andstretchedintheverticaldirectionbythefactor2H=1.26.Objectsthatarescaleinvariantonlyafterbeingblownupbydifferentfactorsinperpendiculardirectionsarecalled“self-affine”;theWeierstrassfunctionisthusself-affine.Scaleinvarianceisstillmoregeneralthanthisaswediscussatlengthinthenextchapter.

Ontheotherhand,intheinfinitelysmalllimit,theCantorsetissimplyacollectionofdisconnectedpoints(Mandelbrotcallssuchsets“dusts”)xx,andamathematicalpointhasadimensionzeroyy.TheCantorsetisthusanexampleofsetwhosefractaldimension0.63…isbetween0and1andDthisquantifiestheextenttowhichitfillstheline.Setsofpointswithsuchin-betweendimensions(theyareusuallynoninteger)arefractalszz.Moregenerally,forthepurposesofthisbook,afractalisageometricsetofpointsthatisscaleinvariantaaa.

Asanothermathematicalexample,considernextfig.2.7b,theSierpinskicarpet27bbb.The figureshows thebase (upper left),motif (upperright)obtainedbydividingan initialsquareintosquaresonethirdthesizeandthenremovingthemiddleone;thebottomrightshowstheresultafter6iterations.Usingthesameapproachasabove,afternconstructionsteps(levels),thenumberofsquaresisN=8n,andthesizeofeachisL=(1/3)n.ThusN=L-DwithD = log8/log3 = 1.89… Indeed, the Cantor set, the Sierpinski square and the unitsegmentillustratethegeneralresult:

Numberofboxes≈(scale)-D

Wherethescaleisidentifiedwiththelengthofthesideofasquare.JustastheCantorsethasafractaldimensionD=0.63…between0and1-betweena

pointandaline-thevalueofDfortheSierpinskisquareisbetween1and2i.e.betweenalineandaplaneanditquantifiestheextentthattheSierpinskisquareexceedsalinewhilepartiallyfillingtheplane.Theseexamplesshowabasicfeatureoffractalsets:duetotheirhierarchicalclusteringofpoints, theyare“sparse”, their fractaldimensionquantifies theirsparseness.

While the number of boxes gives us information about the absolute frequency ofoccurrenceofpartsofthesetofsizeL,itisoftenmoreusefultocharacterizethedensityoftheboxesofsizeLobtainedbydividingthenumberofboxesneededtocoverthesetbythetotalnumberofpossibleboxes:forexampletheCantorsetbyL-1,theSierpinskisquarebyL-2sincetheyaresetsontheline(d=1)andplane(d=2)respectivelyccc.ThisratioistheirwwIfwedon’tremovethelowestfrequencyintheupperleftconstruction,thentheresultisonlyapproximatelyself-affine,however,theconstructionmechanismitselfisneverthelessself-affine.xxAtsomestageintheconstructionanyconnectedsegmentwouldhavebeencutbytheremovalofamiddlethird.yyThefamiliargeometricshapesstudiedbyEuclid-points,lines,planes,volumeshave“topologicaldimensions”0, 1, 2, 3. Topological dimensions have to do with the connectedness of a set. For fractal sets, the fractaldimensionandthetopologicaldimensionaregenerallydifferent.zzDue to nontrivial mathematical issues, there are numerous mathematical definitions of dimension, a fulldiscussionwouldtakeustoofarafield.aaaOfcourse,thelineintheaboveexampleisscaleinvariantwithD=1soaccordingtothisdefinitionitisalsoafractal.However,wegenerallyreservetheterm“fractal”forlesstrivialscaleinvariantsets.bbbThis construction and the analogous construction based on removing middle triangles is credited to W.Sierpinskiin1916.MobilephoneandwifiantennaehavebeenproducedusingafewiterationsoftheSierpinskicarpet,exploitingtheirscaleinvariancetoaccommodatemultiplefrequencies.TheSierpinskitrianglegoesbacktoatleastthe13thcenturywhereithasbeenfoundinchurchesasdecorativemotifs.cccThenegativesignisbecausethesmallerthesegment,themoretheremustbe,indeedNL=totallength,(ind=1),NL2=totalarea(ind=2)etc.

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relativefrequency,i.e.itistheprobabilitythatarandomlyplacedsegment(d=1)orsquare(d=2)willhappentolandonpartoftheset;theratioisL-D/L-d=Ld-D=LCwhereC=d-Disthe codimension of the set. WhereasDmeasuresabsolute sparseness and frequencies ofoccurrence,Cmeasuresrelativesparsenessandprobabilitiesofoccurence. FortheCantorset,C=1-log2/log3=0.36…and for theSierpinskisquare,C=2-log8/log3=0.11…so thattheirrelativesparsenessesarenotsodifferent.IfIputacircle(orsquare)sizeLatrandomontheSierpinskisquare(iteratedtoinfinitelysmallscales),theprobabilityofitlandingonpartof thesquare isL0.11,whereas for theCantorset,puttingarandomsegment lengthL,wouldhavealmost thesameprobability -L0.36 -of landingontheset. Notice that inbothcases, the probability gets smaller as the scale L is reduced. This is because smallsegments/boxes are more likely to land in “holes” than larger ones. In science, we’reusually interested inprobabilities, so that fractal codimensions are generallymoreusefulthanfractaldimensions.

Thisexampleillustratesthegeneralresult:probability≈(scale)C

Sinceinphysicswearegenerallyinterestedinprobabilities–statistics–codimensionsaremoreusefulthandimensions.

Fig.2.6a: TheCantorset. Startingat the top(the “base”),a segmentoneunit long - the “motif” isobtainedbyremovingthemiddlethird.Theoperationofremovingmiddlethirdsistheniteratedtoinfinitelysmallscales.Theredellipsesshowthepropertyofself-similarity:thelefthandhalfofonelevelwhenblownupbyafactorofthreegivesthenextlevelup.

X3

Mo&fBase

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Fig.2.6b:TheconstructionoftheSierpinskicarpet.Thebase(upperleft),istransformedtothemotif(upper middle) by dividing the square into nine subsquares each one third the size and thenremoving themiddle square. The construction proceeds left to right top to bottom to the sixthiteration.

Anexampleof fractal sets thatare relevant toatmospheric scienceare thepoints

where meteorological measurements are taken, fig. 2.7c. In this case, the set is sparsebecausethemeasurementstationsareconcentratedoncontinentsandinrichernations.Toestimate its fractal dimension, one can place circles of radiusL on each stationddd(one isshown in the figure) and determine the average number of other stations within thedistanceL. IfonerepeatsthisoperationforeachradiusL,averagingoverall thestations,one finds that on averageeeethere are LD stations in a radius L, and that this behaviorcontinuesdowntoascaleof1kmfff. Forthemeasuringnetwork(fig.2.7d),wefoundD=1.75.

Eventoday,muchofourknowledgeoftheatmospherecomesfrommeteorologicalstations,forclimatepurposes-suchasestimatingtheevolutionoftheatmosphereoverthelastcentury-wemustalsoconsidershipmeasurements,placethedataonagrid,(typically5oX5oinsize)andaveragethemoveramonth(e.g.fig.2.7e).Itturnsoutthatforanygivenmonth, the set of grid points having some temperature data, is similarly sparsegggso that

dddThis technique actually estimates the “correlation dimension” of the set. If instead one centres circles atpointschosenatrandomontheearth’ssurface(notonlyonstations),thenoneinsteadobtainsthebox-countingdimensiondiscussedabove.Itturnsoutthatingeneral,thetwoareslightlydifferent,thedensityofpointsisanexample of a multifractal measure. Indeed, one can introduce an infinite hierarchy of different exponentsassociatedwiththedensityofpoints.eeeTheruleLDforthenumberofstationsinacircleisaconsequenceofthefactthatthenumberofboxesatscaleLdecreaseswithLasL-Dsinceonaverage,thenumberofpointsperboxisindependentofL:L-DxLD=constant.fffThegeographicallocationsofthestationswereonlyspecifiedtothenearestkilometer,soitispossiblethatthecurveactuallyextendstoevensmallerscales.ForlargeLitisvaliduptoseveralthousandkilometerswhichisaboutasmuchasistheoreticallypossiblegiventhatwereonly9963stations.gggBothinspaceand–duetodataoutagesandshipmovements,alsointime,thefractaldimensions,codimenionsarenearlythesameasforthemeteorologicalnetwork:28 Lovejoy,S.&Schertzer,D.TheWeatherandClimate:EmergentLawsandMultifractalCascades.(CambridgeUniversityPress,2013).

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bothinsituweatherandclimatedataaretakenonfractalsets.Animmediateconsequenceof a fractal network is that it will not detect sparse fractal phenomena, for example, theviolentcentresofstormsthataresosparsethattheirfractaldimensionsarelessthanC,(inthis example, 0.25). By systematicallymissing these rare but violent events, the statisticsendupbeingbiasedhhh,asubjectthatwediscussinthecaseofmacroweatherinch.3.

This analysis shows that as we use larger and larger circles, they typicallyencompass larger and larger voids so that the number of stations per square kilometersystematically decreases: themeasuring network effectively has holes at all scales. Thismeansthattheusualwayofhandlingmissingdatamustberevised.Atpresent,onethinksofthemeasuringnetworkasatwodimensionalspatialarrayorgrid(threedimensional ifweincludetime)althoughwithsomegridpointsempty.Accordingtothiswayofthinking,sincetheearthhasasurfaceareaofabout500millionsquarekilometers,eachofthe10,000stationsrepresentsabout50thousandsquarekilometers.Thiscorrespondstoaboxabout220kilometersonasidesothatatmosphericstructures(e.g.storms)smallerthanthiswilltypically not be detected. Although it is admitted to be imperfectiii, the grid is thereforesupposedtohaveaspatialresolutionof220km.Ouranalysisshowsthatonthecontrary,theproblemisoneofinadequatedimensionalresolutionjjj.

2.6c:Thegeographicaldistributionofthe9962stationsthattheWorldMeteorologicalOrganizationlistedasgivingatleastonemeteorologicalmeasurementper24hours(in1986);itcanbeseenthatitcloselyfollowsthedistributionoflandmassesandisconcentratedintherichandpopulouscountries.ThemainvisibleartificialfeatureistheTrans-SiberianRailroad.Alsoshownisanexampleofacircleusedintheanalysis.Adaptedfromref30.

hhhInitselfthiseffectisnotthemostimportantproblem.iii The techniques for filling the “holes” such as “Kriging” typically also make scalebound assumptions(exponentialdecorrelationsandthelike).jjjWhen estimating global temperatures over scales up to decades, the problem ofmissing data does indeeddominate the other errors (although this is not the same as dimensional resolution). It dominates thoseassociatedwithinstrumentalsiting(e.g.“heatislandeffect”),changingtechnologyandotherpotentialbiasesduetohumaninfluence:29 Lovejoy, S. How accurately do we know the temperature of the surface of the earth? . Clim. Dyn.,doi:doi:10.1007/s00382-017-3561-9(2017).

L

n L( )� LD

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Fig.2.6d:Theaveragenumberofstations(verticalaxis)withinacircleradiusLhorizontalaxis (inkilometers).ThetopstraightlineslopeisD=1.75.Adaptedfromref30.

Fig.2.6e:Blackindicatesthe5ox5ogridpointsforwhichthereissomedatainthemonthofJanuary,1878(20%ofthe2560gridpointswerefilled). This is fromtheHadCRUTdatasetusinganequalangle projection that greatly distorts high latitudes31. Although highly deformed by this mapprojection,wecanalmostmakeoutthesouthAmericancontinent(whitesurroundedbyblack,dataarefromshipmeasurementslowerleft)andEurope,thecentralupperblackmassofgridpoints.

***Theprecedingexamplesoffractalsetsweredeliberatelychosensothatonecouldget

an intuitive feel for the sparseness that the dimension quantifies. In many cases, oneinsteaddealswithsetsmadeupof“wiggly”linessuchastheKochcurve32(1904),showninfig.2.7akkk;thefractaldimensioncanoftenquantifywiggliness.Theconstructionproceedsfromtoptobottombyreplacingeachstraightsegmentbysegmentstheshapeofthesecondcurvefromthetop,i.e.madeofpieces,eachoftheoriginalsize.Again,afterniterations,wehaveN=4nandL=(1/3)n,hencethe fractaldimension isD = log4/log3=1.26… In this

kkkIfthreeKochcurvesarejoinedintheshapeofatriangle,oneobtainstheKoch“snowflake”whichisprobablymorefamiliar.

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curve, the“wiggles”haveadimensionbetween1and2, thewiggliness isquantifiedbyD.Note that aswe proceed tomore andmore iterations, the length of the curve increases.Indeed, after each iteration, the length increases by the factor 4/3 since each segment isreplaced four segments each 1/3 the previous length. Therefore after n iterations, thelengthis(4/3)nwhichbecomesinfiniteasngrows.IfacompletedKochcurveismeasuredwitharuleroflengthL(sucharulerwillbeinsensitivetowigglessmallerthanthislll),thenintermsofthefractaldimension,thelengthoftheKochcurvewouldbeL1-D. Astherulergets shorter and shorter, it can measure more and more details, the length increasesaccordingly. SinceD (=1.26>1), the length grows asL-0.26 andbecomes infinite for rulerswithsmallenoughL.

Howfarcanwetakewiggliness?In1890,GiuseppePeano(1858-1932)proposedthefractal construction thatbearshisname (fig. 2.6d). ThePeano curve ismade froma linethatissowigglythat-bysuccessiveiterations–itendsupliterallyfillingpartoftheplane!At thetime, thisstunnedthemathematicalcommunitysince itwasbelievedthatasquarewastwodimensionalbecauseitrequiredtwonumbers(coordinates)tospecifyapointinit.Peano’s curve allows a point to be specified instead by a single coordinate specifying itspositiononan(infinite)linewigglingitswayaroundthesquaremmm.

Wehavealreadyseenanotherexampleofwiggliness,theWeierstrassfunction,fig.1.3,2.2a), constructed by adding sinusoids with geometrically increasing frequencies andgeometricallydecreasingamplitudes.TheWeierstrassfunctionwasoriginallyproposedasthe first exampleof a functionwhosevalue is everywherewelldefined (it is continuous),butdoesnothaveatangentanywhere(itis“nowheredifferentiable”). Avisualinspection(fig.2.2a)showswhythisisso:todeterminethetangent,wemustzoomintofindasmoothenoughpartoverwhichtoestimatetheslope,sincetheWeierstrassfunctionisafractal,weinzoomforeverwithoutfindinganythingsmooth.

Anatmosphericexampleofawigglycurveistheperimeterofacloudasdefinedbyacloud photograph separating lines that are brighter or darker than a fixed brightnessthreshold23.Inordertoestimatethefractaldimensionofacloudperimeter,wecouldtrytomeasure it with rulers of different lengths and use the fact that the perimeter lengthincreasesasL1-D(sinceD>1).Itturnsoutthatitismoreconvenienttousefixedresolutionsatelliteorradarimagesandusemanycloudsofdifferentsizes. Ifweignoreanyholesinthecloudsnnnandif theirperimetersallhavethesamefractaldimensions,thentheirareas(A)turnouttoberelatedtotheperimeteroooasP=AD/2. Fig.2.8showsanexamplewhenthistechniqueisappliedtorealcloudandrainareas.Althoughvarioustheorieswerelaterdevelopedtoexplaintheempiricaldimensionppp(D=1.35)themostimportantimplicationofthis figure isthat itgavethefirstmodernevidenceofthecompletefailureofOrlanski’sscalebound classification. Had Orlanski’s classification been based on real physicalphenomenaeachdifferentandactingovernarrowrangesofscales,thenwewouldexpectaseriesofdifferentslopes,oneforeachofhisranges.

Theexpectation that thebehaviourwouldbe radicallydifferentoverdifferent scalerangeswasespeciallystrongasconcernsthemeso-scale,thehorizontalrangefromabout1

lllThismethodissometimescalledthe“Richardsondividersmethod”afterL.F.Richardsonwhofirstusedittoestimatethelengthofcoastlinesandothergeographicfeatures,seebelow.mmmNotethatintheinfinitelysmalllimit,ateachpoint,thePeanocurvetouchesitself.Thismeansthatwhileitismappingofthelineontotheplane,themappingisnotonetoone.nnnIfthecloudareaisitselfafractalset,theP=AD/DcwhereDc<2isthefractaldimensionoftheclouds.oooThearea-perimeterrelationwasproposedin:19 Mandelbrot,B.B.Fractals,form,chanceanddimension.(Freeman,1977).pppTheseresultswereinmyoriginalthesis;inthefinalversiontheywereexcisedinordertosatisfytheexternalreferee,theyweresubsequentlypublishedinScience.

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to100kilometerswhere itwasbelieved that theatmospheric thicknesswouldplayakeyroleinchangingthebehaviour,the“meso-scalegap”,seech.3.Beforethis,theonlyotherquantitativeevidenceforwiderangeatmosphericscalingwasfromvariousempiricaltestsofRichardson’s4/3lawofturbulentdiffusion33;fig.2.8(left)showshisoriginalverificationusingnotablydatafrompilotballoonsandvolcanicashqqq.Throughthe1950’sand1960’s,Richardson’s atmospheric 4/3 power law was repeatedly confirmed with theoristsinvariably complaining that it extends beyond the range for “which it can be justifiedtheoretically”rrr. However the story didn’t end there: in the 1970’s, in the wake of twodimensionalisotropicturbulencetheory,alargescaleballoonversion,theEOLEexperimentwasundertakenthatclaimedtoinvalidatehislaw.Inch.4,wedescribethesagaofhowthisconclusionwas based on an erroneous analysis, and how (partially) in 2004 and fully in2013,Richardsonwasvindicateduptoatleast2000km.

Fig.2.7a:Left:A fractalKochcurve32(1904),reproducedfromWelander39(1955)whoused itasamodeloftheinterfacebetweentwopartsofaturbulentfluid.

qqqThe ocean is also an example of a stratified turbulent system and the 4/3 law holds fairly accurately,Richardson tested his law in the ocean using imaginative means includes bags of parsnips that he watcheddiffusingfromabridge:34 Richardson,L.F.&Stommel,H.Noteoneddydiffusivityinthesea.J.Met.5,238-240(1948).Stilllater,itwasverifiedintheoceanovertherange10mto10,000kmin:35 Okubo,A.&Ozmidov,R.V.Empiricaldependenceofthehorizontaleddydiffusivityintheoceanonthelengthscaleofthecloud.Izv.Akad.NaukSSSR,Fiz.Atmosf.iOkeana6(5),534-536(1970).rrr Meaning that it cannot be accounted for by the dominant three dimensional isotropic homogeneousturbulencetheory,seee.g.p.557of:38 Monin,A.S.&Yaglom,A.M.StatisticalFluidMechanics.(MITpress,1975).

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Fig.2.7b:Left,thefirstthreestepsoftheoriginalPeanocurve,showinghowaline(dimension1)canliterallyfilltheplane(dimension2).Right: A variant reproduced from Steinhaus40 (1960) who used it as a model for a hydrographicnetwork,illustratinghowstreamscanfillasurface.

Fig. 2.8: The left showsRichardson’s proposed4/3 lawof turbulent diffusion33ssswhich includes afewestimateddatapoints.

sssThequalityofthefigureislow,butthankstoMonin,itisalreadyimproved:41 Monin,A.S.Weatherforecastingasaprobleminphysics.(MITpress,1972).

(1km)2

(1000km)2

(10km)2

(100km)2

1km 104km

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Right:theareaperimeterrelationforradardetectedrainareas(black)andInfraredsatellitecloudimages (open circles), the perimeter is the horizontal axis, the area, the vertical axis. The slopecorrespondstoD=1.35.Themesoscale(roughly1to100km)isshownintheredbrackets:nothingspecial.Adaptedfromref.23

***Wehavediscussedseveralofthefamous19thcenturyfractals: theCantorset(figs.

2.7a), the first setwithanonintegerdimension; thePeanocurve (figs.2.7b), the first linethat could pass through every point in the unit square (a plane); and the Weierstrassfunction(figs.2.2a), thefirstcontinuouscurvethatdoesn’thaveatangentanywhere. Butthese were considered to be essentially mathematical constructions: academic odditieswithoutphysicalrelevance.Mandelbrotprovocativelycalledthem“monsters”.

Mandelbrotnotonlycoinedtheterm“fractal”butwithhisindefatigableenergyputthem squarely on the scientific map. Although he made numerous mathematicalcontributionsttt,hismostimportantonewasasatoweringpioneerinapplyingfractalsandscalingtotherealworld.Inthisregard,hisonlyseriousscientificprecursorwasLewisFryRichardsonuuu(1881-1953). DuetohisQuakerbeliefs,Richardsonwasapacifistandthismadehiscareerdifficult,essentiallydisqualifyinghimfromacademicpositions.HeinsteadjoinedtheMeteorologyOfficebuttemporarilyquititinordertodriveanambulanceduringthe firstworldwar. Afterwards,he rejoined theMeteorologyOfficebut in1920resignedwhenitwasmilitarizedbybeingmergedintotheAirMinistry.

RichardsonworkedonarangeoftopicsandisrememberedforthenondimensionalRichardsonnumber that characterizes atmospheric stability, theRichardson4/3 law (fig.2.8a), the Modified Richardson Iteration and Richardson Acceleration techniques ofnumerical analysis and the Richardson divider’smethod. The latter is a variant on box-counting (discussedabove) thathenotablyused toestimate the lengthof thecoastlineofBritain,demonstratingthatitfollowedapowerlaw.Mandelbrot’sfamous1967paperthatinitiated fractals: “How long is the coastline of Britain? Statistical self-similarity andfractionaldimension”45vvvtookRichardson’sgraphsandinterpretedtheexponentintermsof a fractional dimensionwww. Fully aware of the problem of conceptualizingwide rangeatmosphericvariability,hewasthefirsttoexplicitlyproposethattheatmospheremightbefractal.Aremarkablesubheadinginhis1926paperonturbulentdiffusionisentitled“Doesthewindpossessavelocity” this followedwith thestatement: “thisquestion,at firstsightfoolish,improvesuponacquaintance”.Hethensuggestedthataparticletransportedbythewind might have a Weierstrass function-like trajectory that would imply that its speed(tangent)wouldnotbewelldefinedxxx.

tttI will let the mathematicians judge his contributions to mathematics. However, there is no question thatMandelbrot’s contribution to science has been monumental and underrated. In any case (and in spite ofMandelbrot’s efforts!), it is still early to evaluate his place in the history of science. Interested readersmayconsulthisautobiography,“TheFractalist”thatwaspublishedposthumously:42 Mandebrot,B.B.TheFractalist.(FirstVintageBooks,2011).uuuOthernotableprecursorswereJeanPerrin(1870-1942),whoquestionedthedifferentiabilityofthecoastofBrittany:43 Perrin,J.LesAtomes.(NRF-Gallimard,1913).andHugoSteinhaus(1887-1972)whoquestionedtheintegrabilityofthelengthoftheriverVistula:44 Steinhaus,H.Length,ShapeandArea.ColloquiumMathematicumIII,1-13(1954).Lackofdifferentiabilityandintegrabilityaretypicalscalingfeaturesandarediscussedagainin(ch.7).vvvThiswasstillnearlyadecadebeforeMandelbrotcoinedtheword“fractal”.wwwAbovewesawthatthelengthofthecloudperimetervariesasL1-DwhereListhelengthoftherulerandDisthefractaldimension.xxxItturnedoutthattheproblemwasnotthevelocity,buttheacceleration.

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Richardsonisuniqueinthathestraddledthetwomain–andsuperficiallyopposing- threads of atmospheric science: the low level deterministic approach and thehigh levelstatisticalturbulenceapproach.Remarkably,hewasafoundingfigureforboth.Hisseminalbook “Weather forecasting by numerical process46” yyy (1922) inaugurated the era ofnumericalweatherprediction.Init,Richardsonnotonlywrotedownthemodernequationsofatmosphericdynamics,buthepioneerednumericaltechniquesfortheirsolution,heevenlaboriouslyattemptedamanualintegrationzzz. Yetthisworkalsocontainedtheseedofanalternative: buried in the middle of a paragraph, he slyly inserted the now iconic poemdescribing thecascade idea: “Bigwhirlshave littlewhirls that feedon theirvelocity, littlewhirls have smaller whirls and so on to viscosity (in the molecular sense)”aaaa. Soonafterwards,thiswasfollowedbythefirstturbulentlaw,theRichardson4/3lawofturbulentdiffusion33, which today is celebrated as the starting point for modern theories ofturbulenceincludingthekeyideaofcascadesandscale invariance,andaswementionnedwasfinallyvindicatedin2013.Unencumberedbylaternotionsofmeso-scalebbbb,andwithremarkableprescience,heevenproposed thathisscaling lawcouldhold fromdissipationup to planetary scales (fig. 2.8, left), a hypothesis confirmed 35 years ago by the areaperimeteranalysis,andsincethenbyalargebodyofresultsdiscussedinthechaptersbelow.Today, he is honoured both as father of numerical weather prediction by the RoyalMeteorologicalSociety’sRichardsonprizeandasgrandfatherofturbulencebytheEuropeanGeosciencesUnion’sRichardsonmedalcccc.

Asahumanist,Richardsonworked topreventwar,withhisbook “Theproblemofcontiguity:anappendixofstatisticsofdeadlyquarrels”48 foundingthemathematical(andnonlinear!) study of war. He was also anxious that his research be applied to directlyimprove the situation of humanity and proposed the construction of a vast “Weatherfactory”.Thiswouldemploytensofthousandsofhuman“computers”andwouldmakerealtime weather forecasts. Recognizing (from personal experience) the tedium of manualcomputation,heforesawtheneedforthefactorytoincludesocialandculturalamenities.

Letme now explain a deep consequence of Richardson’s cascade idea that didn’tfullymatureuntilthenonlinearrevolutioninthe1980’s.Wehaveseenthatthealternativetoscaleboundthinkingisscalingthinkingandthatfractalsembodythisideaforgeometricsetsofpoints.ForexampletheKochcurvewasamodelofaturbulentinterface,thesetofpoints bounding two different regions, the Peano curve as a model of a hydrographicnetwork. However, inorder toapply fractalgeometry to thesetofbounding (perimeter)points,wewerealreadyfacedwithaproblem:wehadtoreducethegreyshadestowhiteorblack(cloudornocloud).Sinceatmosphericsciencedoesnotoftendealwithblack/whitesets,butratherwithfieldssuchasthecloudbrightnessortemperaturethathavenumericalvalueseverywhereinspaceandthatvaryintime,somethingnewisnecessary.

(Re)consider fig. 1.5, the aircraft temperature transect. One could repeat thetreatmentoftheWeierstrassfunctiontotrytofitthetransectintotheframeworkoffractalsetsbysimplyconsideringthepointsonthetopgraphasthesetofinterest.Butthisturns

yyyLackingsupport,hepaidforthepublicationoutofhisownpocket.zzzNearthewar’send,hesomehowfoundsixweekstoattemptamanualintegrationoftheweatherequations.His estimate of the pressure tendency at a single grid point in Europe turned out to be badlywrong (as headmitted),butthesourceoftheerrorwasonlyrecentlyidentified,seethefascinatingaccountbyLynch:47 Lynch,P.Theemergenceofnumericalweatherprediction:Richardson'sDream. (CambridgeUniversityPress,2006).aaaaThispoemwasaparodyofanurseryrhyme,the“Siphonaptera”:“Bigfleashavelittlefleas,Upontheirbackstobite'em,Andlittlefleashavelesserfleas,andso,adinfinitum.bbbbThatpredictedastrongbreakinthescaling,seech.3.ccccThehighesthonouroftheNonlinearProcessesdivision.

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out to be a bad idea because aswe also saw in fig. 1.5 (bottom), the figurewas actuallyhidingsomeincrediblyvariablespiky, (intermittent)changes,andthisbehaviourrequiressomething new to handle it:multifractalsdddd. Indeed,multifractalswere first discoveredpreciselyasmodelsofsuchturbulentintermittencyeeee.

Focusonthebottomoffig.1.5,the“spikes”. Ratherthantreatingall thepointsonthe graph as a wiggly fractal set, instead consider the set of points that exceed a fixedthreshold,forexamplethoseabovethelevelofonestandarddeviationasindicatedbythehorizontallineinthefigureasakindofCantorset. Ifthespikesarescaleinvariant,thenthissetwillbea fractalwithacertainfractaldimension. Now,movethehorizontal linealittlehighertoconsideradifferentset. Wefindthatthefractaldimensionofthisdifferentset is lower. Indeed, in thisway,movingtohigherandhigher levelswecouldspecify thefractal dimension of all the different level sets, thus completely characterizing the set ofspikesbyaninfinitenumberoffractaldimensions.Theabsolutetemperaturechanges(thespikes)-andindeedthetemperaturetransectitself-arethusmultifractals.Itturnsoutthatmultifractalsarenaturallyproducedbycascadeprocesses thatarephysicalmodelsof theconcentration of energy and other turbulent fluxes into smaller and smaller regions.Interestedreaderscanfindmoreinformationaboutthisinbox2.1.

Mathematically,whereas fractals are scale invariant geometric sets of points theyareblackorwhite,youareeitheronoroffthesetofpoints. Incontrast,multifractals–atleastwhenaveragedoversmallsegmentsandtimeintervals-arescaleinvariantfields:likethetemperature,theyhavenumericalvaluesateachpointinspace,ateachinstantintime.

2.3FluctuationsasamicroscopeWhenconfrontedwithvariabilityoverhugerangeofspaceandtimescales,wehave

argued that there are two extreme opposing ways of conceptualizing it. We can eitherassumethateverythingchangesaswemovefromonerangeofscaletoanother–everfactoroftenorso–oronthecontrary,wecanassumethat–atleastoverwiderange(factorsofhundred, thousands ormore), that blowing up gives us something that is essentially thesame. But this is science; it shouldn’t be a question of ideology. If we are given atemperatureseriesoracloudimage,howcanweanalysethedatatodistinguishthetwo,toddddMathematically,thenontrivialpointisthatwhereastheWeierstrassfunctioniscontinuousi.e.welldefinedateachinstantt,(amathematicalpointonthetimeaxis),amultifractalonlyconvergesintheneighborhoodoftheinstant,inordertoconverge,themultifractalmustbeaveragedoverafiniteinterval.Thisistheoriginofthe“dressed”propertiesthatarerelatedtotheextremeeventsdiscussedinch.6.eeeeItwasactuallyalittlemorecomplicatedthanthat:thekeymultifractalformulaindependentlyappearedinthreepublicationsin1983,oneinturbulence,andtheothertwointhefieldofdeterministicchaos:49 Schertzer, D. & Lovejoy, S. in IUTAM Symp. on turbulence and chaotic phenomena in fluids. (ed T.Tasumi)141-144.50 Grassberger,P.Generalizeddimensionsofstrangeattractors.PhysicalReviewLetterA97,227(1983).51 Hentschel,H.G.E.&Procaccia,I.Theinfinitenumberofgeneralizeddimensionsoffractalsandstrangeattractors.PhysicaD8,435-444(1983).Whiletheturbulentpublicationwasadmittedlyonlyinaconferenceproceeding,thedebateaboutthepriorityofdiscoverywassoonovershadowedbyMandelbrot’sclaimtobethe“fatherofmultifractals”:52 Mandelbrot, B. B. Multifractals and Fractals. Physics Today 39, 11, doi:http://dx.doi.org/10.1063/1.2815135(1986).Soonaftertheinitialdiscoveryofmultifractals,amajorcontributionwasmadebyParisiandFrischwhowerealsothefirsttocointheterm“multifractal”:53 Parisi, G. & Frisch, U. in Turbulence and predictability in geophysical fluid dynamics and climatedynamics(edsM.Ghil,R.Benzi,&G.Parisi)84-88(NorthHolland,1985).Recognizingtheimportanceofmultifractals,Mandelbrotsubsequentlyspentahugeeffortclaimingitspaternity.Ironically, Steven Wolfram in his review of Mandelbrot’s posthumous autobiography “The Fractalist”complainedthatMandelbrothad“diluted”thefractalsconceptbyinsistingonmultifractals:54 Wolfram,S.inWallStreetjournal(2012).

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tellwhichiscorrect?Wehavealreadyintroducedtwomethods:spectralanalysiswhichisquitegeneral, and thearea-perimeter relationwhich is rather specialized. While spectralanalysis is a powerful technique, its interpretation is not so simple – indeed, had theinterpretations been obvious, we would never have missed the quadrillion and thedistinctionbetweenmacroweatherandtheclimatewouldhavebeenclarifiedlongago!

Itisthereforeimportanttouseananalysistechniquethatisbotheasytoapplyandeasy to understand, a kind of analytic microscope that allows us to zoom in and tosystematically compare a time series or a transect at different scales, to directly test thescaleboundorscalingalternative:fluctuationanalysis.

Weprobablyallhaveanintuitiveideaaboutwhatafluctuationis. Inatimeseriesit’s about the change in the value of a series over an interval in time. Consider atemperatureseries.Weareinterestedinhowmuchthetemperaturehasfluctuatedoveraninterval of time Δt. The simplest fluctuation is simply the difference between thetemperature now and at a time Δt earlier (fig. 2.9a top). This is indeed the type offluctuationthathas traditionallybeenused in turbulence theoryandthatwasused in thefirstattempttotestthescalinghypothesisonclimatedata.Inordertomakefigure2.10,Istartedwith two instrumental series: theManley series from central England starting in1659(opencircles)andanearlynorthernhemisphereseriesfrom1880(blackcircles);theformerbeingessentially local (regional), that latterglobal inscale. Theotherserieswerefromearlypaleoisotopeseriesasindicatedinthecaption,usingtheofficialcalibrationstotransformthemintotemperaturevaluesffff.

Inordertomakethegraph,foragiventimeintervalΔt,onesystematicallycalculatesall the nonoverlapping differences in each series and averaged the squares of thesedifferences,the“typicalvalue”shownintheplotisthesquarerootofthis(itisthestandarddeviationofthedifferences).Onethenplotstheresultsonlogarithmiccoordinatessinceinthatcase,scalingappearsasstraightlinesandcaneasilybeidentified. Readingthegraph,one can see for example that at 10 year intervals, the typical northern hemispheretemperature change is about 0.2 oC and that over about 50,000 years, that the typicaltemperaturedifferenceisroughly6oC(±3oC),thiscorrespondstothetypicaldifferenceoftemperature between glacials and interglacials, hence the box (which allows for someuncertainty) is the “glacial-interglacial window”. These fluctuations are thereforestraightforwardtounderstand.

Onfig.2.10,areferencelinewithslopeH=0.4isshowncorrespondingtothescalingbehaviour ΔT ≈ ΔtH, linking hemispheric temperature variations at ten years to paleovariations at hundreds of thousands of years. Although this basic picture is essentiallycorrect,laterworkprovidedanumberofnuancesthathelptoexplainwhythingswerenotfullyclearedupuntilmuchlater.Noticeinparticular,thetwoessentiallyflatsetsofpointsin the figure, one from the local central England temperate up to roughly three hundredyears,andtheotherfromanoceancorethatisflatfromscales100,000yearsandlonger.Itturns out that the flatness is an artefact of the use of differences in the definition offluctuations:weneedsomethingabitbetter.

Before continuing, let us recall the scaling laws thatwe have introduced up untilnow:

Spectrum≈(frequencyω)-β≈(scale) βNumberofboxes≈(sizeL)-D≈(scale)-D

ffffLongbefore the internet, scanners andpublicly accessible data archives, as a post-doc at theMétéorologieNationale in Paris, I recall taking the published graphs,making enlargedphotocopies, and then using tracingpapertopainstakinglydigitizethem.

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Probability≈(scale)CFluctuations≈(intervalΔt)H≈(scale)H

Whereβisthespectralexponent,Disthefractaldimensionofaset,CthecodimensionandH the fluctuation exponentgggg . A nonobvious problem with defining fluctuations asdifferencesisthatonaverage,differencescannotdecreasewithincreasingtimeintervalshhhh.Thismeans that nomatterwhat the value ofH -whether positive or negative, that theycannotdecreasesothatwheneverHisnegative,thedifferencefluctuationswillsimplygiveaconstantresultiiii,theflatpartsoffig.2.10.ButdoregionsofnegativeHexist?Onewaytoinvestigate this is to try to infer H from the spectrum (which does not suffer from ananalogous restriction: its exponent β can take any value). In this case there is anapproximateformulajjjj:β=1+2H.ThisformulaimpliesthatnegativeHcorrespondstoβ<1,andacheckonthespectrum(fig.2.3a)indicatesthatseveralregionsareindeedflatenoughto imply negativeH. How dowe fix the problem and estimate the correctH when it isnegative?

It took a surprisingly long time to clarify this issue. To start with, the turbulencecommunityhadmanyconvenienttheoreticalresultsfordifferencefluctuations.Turbulencetheorists had been the first to use fluctuations as differences and a decade beforeHurst,effectivelyintroducedthefirstH=1/3astheexponentinKolmogorov’sfamouslawkkkk,(seech.3,4).InclassicalturbulencealltheH’sarepositivesothattherestrictiontopositiveHwas not a problem. Later, in the wake of the nonlinear revolution in the 1980’s,mathematicians invented an entire mathematics of fluctuations called “wavelets” llll .Although technically, difference fluctuations are indeed wavelets, mathematicians mockthem calling them the “poor man’s wavelet” and promoting more sophisticated ones.Wavelets turned out to have many beautiful mathematical properties and often havingcolourfulnamessuch“MexicanHat”,“HermitianHat”,orthe“Cohen-Daubechies-Feauveauwavelet”. For mathematicians, it was irrelevant that the corresponding physicalinterpretationswerenotevident.Themasteryofwaveletmathematicsalsorequiredafairintellectualeffortandthisfurtherlimitedtheirscientificapplications.

Thiswas the situation in the1990’swhen scaling started tobe applied to geo timeseries involving negativeH (essentially to anymacroweather series, although at the timethiswasnotatallclear).ItfelluponastatisticalphysicistChung-KangPengtodevelopanH<0 technique that he applied to biological series; the Detrended Fluctuation Analysis(DFA)methodmmmm56.Alsoatthistime,anotherpartofthescalingcommunity(includingmycolleaguesandI)werefocusingonmultifractalityandintermittency,andtheseissuesdidn’t

ggggThe symbolH is used in honour of Edwin Hurst who discovered the “Hurst effect”: long range memoryassociatedwithscalinginhydrology.HedidthisbyexaminingancientrecordsofNileflooding.ItturnsoutthatthefluctuationexponentisingeneralnotthesameasHurst’sexponent,thattheyareonlythesameifthedatafollowthebellcurve…whichtheyonlydorarely!Thisdistinctionhascausedmuchconfusion.55 Hurst, H. E. Long-term storage capacity of reservoirs. Transactions of the American Society of CivilEngineers116,770-808(1951).hhhhThisistrueforanyseriesthathascorrelationsthatdecreasewithΔt(asphysicallyrelevantseriesalwaysdo).iiiiDandCcannotbenegativesothatthisproblemdoesnotariseforthem.jjjjValidifweignoreintermittency,otherwisethereare“intermittencycorrections”.kkkkKolmogorov’slawwasactuallyveryclosetoRichardson’s4/3law,the4/3wasH+1.llllAlthoughwaveletscanbetracedbacktoAlfredHaar(1909,seebelow),itreallytookoffstartingintheearly1980’swiththecontinuouswavelettransformationbyAlexGrossmanandJeanMorlet.mmmmThekeyinnovation–althougheventhiswasnotclearlyrealizedasthebreakthrough-wassimplytofirstsumtheseries,effectivelyaddingonetothevalueofHsothatinmostcases(aslongasH>-1),theresultbecamepositiveallowingformoreusualdifferenceanddifference-likefluctuationstobeapplied.

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involvenegativeHsothattheproblemwasignored.Overthefollowingnearlytwodecades,there were thus several more or less independent strands of scaling analysis, each withtheir ownmathematical formalism and interpretations. The wavelet community dealingwithfluctuationsdirectly,butunconcernedaboutthesimplicityofphysicalinterpretations;theDFAcommunitynnnnwieldingasomewhatcomplexmethodbutonethatcouldbereadilyimplemented numerically and didn’t require much theoretical baggage oooo ; and theturbulencecommunityfocusedonmultifractalintermittency.Inthemeantime,mainstreamgeoscientistscontinuedtousespectralanalysis,focusingonspectralpeaksthatsupposedlyrepresentedquasi-oscillatingprocesses, not on the scalingoron the interpretationof theamplitudesofthespectra,mostofwhichwastreatedasuninterestingbackgroundnoise.

Ironically, the impasse was broken by the first wavelet, the one that Alfréd Haar(1885-1933) had introduced in 1910 even before the wavelet formalism had beeninvented57.TheHaarfluctuationisbeautifulfortworeasons:thesimplicityofitsdefinitionandcalculationand the simplicityof interpretation58. Todetermine theHaar fluctuationoveratimeintervalΔt,onetakestheaverageofthefirsthalfoftheintervalandsubtractstheaverageofthesecondhalf(fig.2.9a,b).That’sitpppp!Asfortheinterpretation,itiseasyto show that when H is positive, that it is (nearly) the same as a difference, whereaswheneverH isnegative,wenotonlyrecover itscorrectvalueqqqq,butthe fluctuation itselfcanbeinterpretedasan“anomalyrrrr.”

nnnnAtlastcount,Peng’soriginalpaperhadmorethan2000citations,anastoundingnumberforsuchahighlymathematicalpaper.ooooTheDFAmethodestimatesfluctuationsbythestandarddeviationoftheresidualsofapolynomialfittotherunningsumoftheseries.Theinterpretationissoobscurethattypicalplotsdonotbothertoevenuseunitsforthefluctuationamplitudes.ppppIcanrecallacommentofarefereeofapaperinwhichIexplainedtheHaarfluctuationusingthesamewords.Expecting a complicatedwavelet expression, he complained thathedidn’t understand thewords and insteadwantedanequation!qqqqTheHaarfluctuationisonlyusefulforH intherange-1to1,butthisturnsouttocovermostoftheseriesthatareencounteredingeoscience.rrrrIn this contextananomaly is simply theaverageovera segment lengthΔt of theserieswith its long termaveragefirstremoved.

20 80 100 120

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Fig. 2.9a: Schematic illustration of difference (top) and anomaly (bottom) fluctuations for amultifractal simulation of the atmosphere in the weather regime (0≤H≤1), top, and in the lowerfrequencymacroweatherregime(bottom).Noticethe“wandering”and“cancellingbehaviours.

Fig.2.9b: Schematic illustrationofHaarfluctuations(useful forprocesseswith-1≤H≤1). TheHaarfluctuationovertheintervalΔt isthemeanofthefirsthalfsubtractedfromthemeanofthesecondhalfoftheintervalΔt.

Fig. 2.10: The RMS difference structure function estimated from local (Central England)temperaturessince1659(opencircles,upperleft),northernhemispheretemperature(blackcircles),and from paleo temperatures from Vostok (Antarctic, solid triangles), Camp Century (Greenland,opentriangles)andfromanoceancore(asterixes).Forthenorthernhemispheretemperatures,the(powerlaw,linearonthisplot)climateregimestartsatabout10years.Therectangle(upperright)is the “glacial-interglacial window” through which the structure function must pass in order toaccountfortypicalvariationsof±2to±3Kforcycleswithhalfperiods≈50kyrs.Reproducedfrom9.

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Fig. 2.11a: A composite of typicalHaar fluctuationsssssfrom (daily and annually detrended) hourlystation temperatures (left), 20CR temperatures (1871-2008 averaged over 2o pixels at 75oN) andpaleo-temperatures from EPICA ice cores (right) over the last 800kyrs. The reference lines areindicated,theirslopesareestimatesofH.Adaptedfrom59.

Fig.2.11b:Representativeseriesfromeachofthefivescalingregimestakenfromfig.1.4a-dwiththeaddition of the hourly surface temperatures from LanderWyoming, (bottom, detrended daily andannually). Inordertofairlycontrasttheirappearances,eachserieshadthesamenumberofpoints(180)andwasnormalizedbyitsoverallrange(themaximumminustheminimum),andeachserieswasoffsetby1oCintheverticalforclaritytttt.Theseriesresolutionswere1hour,1month,400years,

ssssThesearerootmeansquareHaarfluctuations.tttt From top to bottom the ranges used for normalizing are: 10.1, 4.59, 1.61 (Veizer, Zachos, Huybersrespectively,allδ18O),6.87K,2.50K,25K(Epica,Berkeley,Lander).

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ClimateEpica:25-97BPkyrs(400yrs)

MacroweatherBerkeley:1880-1895AD(1month)

WeatherLanderWy.:July4-July11,2005(1hour)

T/ΔT m

ax

MegaclimateVeizer:290Mys-511MyrsBP(1.23Myr)

t

ΔT ≈ ΔtH

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14kyrs,370kyrsand1.23Myrsbottomtotoprespectively.TheblackcurveshaveH>0,thered,H<0,reproducedfrom59.

When Haar fluctuations are substituted for difference fluctuations, and using theclimateseriesdiscussed inch.1weobtain thecomposite fig.2.11a thatcoversnearly thesame range asMitchell. One can clearlymake out five distinct regions, each –with theexception of macro climateuuuu- are scaling over ranges of roughly a thousand in timescalevvvv. We clearly see the alternation of the sign of the slope (H) from positive (theweatherregimetoabout10days,left),throughthelongermacroweatherregimewwwwwithdecreasing fluctuations and negative H, to the increasing climate regime, decreasingmacroclimate, increasingmegaclimate. Thenumberson theverticalaxisof fig.2.11aallmakeperfect sense and give a precise idea of typical fluctuations at the correspond timescale. We can compare this to the amplitude of the fluctuations implied by Mitchell’sscaleboundspectrum. Todo this,note thathisbackgroundwasmostly “whitenoise”andthis would have a slope -1/2 on fig. 2.10. Using Haar fluctuations, Mitchell would havepredicted that at scales of millions of years that changes of only millionths of a degreewouldoccur,morethanamilliontimestoosmallxxxx!

BycomparingthedifferenceandHaarfluctuations(fig.2.10,2.11arespectively),wecan now understand the limitations of the difference-based analysis (fig. 2.10), andunderstandwhymacroweatherwasnotclearlydiscerneduntilsorecently.Asexpected,theincreasingpartsof the two figuresarequite similar,whereas the flatpartsof fig.2.10doindeedcorrespond tonegativeH –bothmacroweatherandmacroclimate. TheremainingapparentdivergencebetweenthedifferencesandHaarfluctuations(figs.2.10,2.11a)hastodowiththedifferencebetweenlocalandgloballyaveragedtemperaturesandthedifferencebetween industrial and pre-industrial temperatures (due to anthropogenicwarming),wedeferdiscussionofthisuntilch.5.

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