Scaling Analysis and Evolution Equation of the Nor

arXiv:nlin/0406068v1[nlin.CD]29Jun2004ScalingAnalysisandEvolutionEquationoftheNorthAtlanticOscillationIndexFluctuationsC.Collette(∗)andM.AusloosSUPRATECS,B5,SartTilman,B-4000Li`ege,Belgium(*)presentaddress:ActiveStructuresLaboratory,Universit´eLibredeBruxelles,ULB-CP165/42av.F.D.Roosevelt,50B-1050Brussels,Belgium(February8,2008)AbstractTheNorthAtlanticOscillation(NAO)monthlyindexisstudiedfrom1825till2002inordertoidentifythescalingrangesofitsfluctuationsupondif-ferentdelaytimesandtofindoutwhetherornotitcanberegardedasaMarkovprocess.AHurstrescaledrangeanalysisandadetrendedfluctu-ationanalysisbothindicatetheexistenceofweaklypersistentlongrangetimecorrelationsforthewholescalingrangeandtimespanherebystud-ied.SuchcorrelationsaresimilartoBrownianfluctuations.TheFokker-PlanckequationisderivedandKramers-Moyalcoefficientsestimatedfromthedata.Theyareinterpretedintermsofadriftandadiffusioncoeffi-cientasinfluidmechanics.AllpartialdistributionfunctionsoftheNAOmonthlyindexfluctuationshaveaformclosetoaGaussian,foralltimelags,inagreementwiththefindingsofthescalinganalyses.Thisindi-catesthelackofpredictivepowerofthepresentNAOmonthlyindex.Yet1therearesomedeviationsforlarge(andthusrare)events.Whencesugges-tionsforothermeasurementsaremadeifsomeimprovedpredictabilityoftheweather/climateintheNorthAtlanticisofinterest.ThesubsequentLangevinequationoftheNAOsignalfluctuationsisexplicitlywrittenintermsofthediffusionanddriftparameters,andacharacteristictimescalefortheseisgiveninappendix.PACSnumbers:05.45.Tp,05.45.Gg,93.30.Fd,89.69.+x;02.50.Le,05.40.-a,47.27.Ak,87.23.Ge2I.INTRODUCTIONInordertoestablishasoundunderstandingforanyscientificphenomenon,onehastorecordnumericaldataandfromthelattertoobtainlawswhichcanbenextderivedtheoreticallyfromsocalledfirstprinciplemodels.Thesocalledinversemodelmethod,startingfromrawdataandusingstatisticalanalysisasafirststep,isofgreatinterestsinceitismodelfree.Somedifficultyarisesinparticularinnonlineardynamicalsystemsbecauseoftheneedtosortoutnoisefrombothchaosanddeterministiccomponents1,2.Whencetoextractmeaningfulmodel-freedynamicalequationsfromchaotic-likedataisanenormouschallenge3.Practicallyoneisoftenledtoempiricalrelationships.Thisisoftenthecaseinthemeteorology/climatologyfieldwherethereisawidelymixedsetofvarious(sometimes)unknowninfluences,overdifferenttimeandspacescales.OftenthefastvariationsaretakenasnoisetermsinsomesortofLangevinequation(s)4,5asfortheelNi˜noSouthernOscillationIndex(SOI).InordertoquantifyweatherandclimateeventsinEuropeandreportlarge-scalevari-abilityanindexhasbeenimagined6thesocalledNorthAtlanticOscillation(NAO)in-dex(://://://−bprc.mps.ohio−state.edu/gpl/NAO/Naobibliography.htm).Itisthenormalizedsealevelpressure(SLP)differencebetweenastationatPontaDelgada,AzoresandoneatAkureyri,Iceland.Sinceaboutthemid-50’stheNAOindexwastrendingfromnegativetopositivevalues,butismostlypositivesince1980,avariationattributedtoglobalwarming.Itisthought7thattheinfluenceofslowchangesintheoceanandinthegreenhousegasesmaybepicked3upasthefundamentalcausesofaprolonged(upward)trend.Untilafewyearsago,theNAOwasnotreceivingintenseattention7,becauseitwasthoughtthatitsphaseandamplitudewereratherunpredictable,becausebothinvolvemany(timeandspace)scaleswhichareoftenintrinsictochaoticbehavior;seealsoreviews8–10.Yet,evidencehasbeenpresentedthatNAOexhibits’long-range’dependencehavingwintervaluesresiduallycorrelatedovermanyyears,withshort-term2-5yearvariationsanddecadaltrends.NotethatWallace11hasarguedthattheNAOisalocalexpressionforaNorthernAnnularMode(NAM),alsocalledArcticOscillation12,13.InviewoftheaboveitseemsofpertinentinteresttoconsideragaintheNAOandadoptspecificdataanalysistechniqueswhensearchingforscalingrangesandstochasticityfeatures.WestartwiththeHurst(R/S)method14,15followedbyadetrendedfluctuationanalysis(DFA)16,17ofthemonthlyaveragedNAOsignal.Suchtestssupplementclassicalanalysesbasedonfrequencyspectra18–20whicharedebatableduetothenonstationarityofthedata.Interestinglythedatahistogramhavesocalledfattails,resemblingtheL´evyflights,signaturesofself−organizingsystems,todayemerginginmanyareasofphysicsasthosementionedhereabove.AgainthesefactsseemtoexcludelowdimensionalchaosbutsupporttheconjectureofMarkovdynamicsforatmosphericevolution,asalreadysuggestedinfactmanyyearsago8,21–23.Inthefollowing,weadoptaMarkovassumptioninordertoderivetheFPEandtowritedowntheChapman-KolmogorovequationfortheconditionalprobabilityoftheincrementsΔx(oftheNAOindex)overdifferenttimeintervalsΔt.ThisleadstoanumericalderivationoftheKramers-Moyalcoefficientswhicharethemomentsofsuchprobabilitydistributions.Uptothesecondmoments,thisleadstothediffusionanddriftcoefficientsappearingintheFokker-Planckequation(FPE)andarebasictothe4Langevinequation24.ItwillbenoticedthattheanalyticalformofbothdriftD(1)anddiffusionD(2)coefficientsaresimple.Itwillbefoundthattheexperimentalprobabilitydensityfunctions(pdf)haveallaGaussianformwhenexcludingthe(rare)large(socalledextreme)events.Themethods,appl