Contents - Max-Planck-Institut für Physik komplexer Systeme
Contents - Max-Planck-Institut für Physik komplexer Systeme
Contents - Max-Planck-Institut für Physik komplexer Systeme
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Forecasting atmospheric states. State-of-the-artmodelsof theatmosphereareroutinelyusedin<br />
weatherforecasts. Formally,theyaredynamicalsystems,andaforecastisthesolutionofaninitial<br />
valueproblemforafinitetimeinterval. Bytechniquescalleddataassimilationoneobtainstheinitial<br />
modelstatevectorwithabout 10 8 componentsfromasetofabout 10 5 measurementsofthecurrent<br />
atmosphericstate.Thisresemblestheproblemindynamicalsystemstheorycalledshadowing.Weemploynovelvariationalapproachesinordertofindbetterinitialconditionswithlesscomputationaleffort<br />
comparedtoestablishedmethods.<br />
Knowingthatthemodelinitialconditionisonlyaroughestimate,weatherservicesperformensemble<br />
forecasts: Theygeneratemodeltrajectoriesforabout50-100differentinitialconditions, whichare<br />
obtainedfromthebestguessbysmallsystematicperturbations. Thissetofdifferentforecastsforthe<br />
samequantityatthesameplaceandtimeallowsonetoassesstheuncertaintyofaforecast. We<br />
investigatewhattheproperinterpretationofsuchanensembleis. Wecouldshowthatthespreadof<br />
suchensembles,i.e.,thediversityoftheoutcomes,isonaveragetoosmall,sincerealmeasurements<br />
(theverifications)lieoutsidetheensembletoofrequently.Suchover-confidentforecastsareaparticular<br />
problemforthepredictionofextremeweather,whoseprobabilityisgenerallyunderestimated.Ouranalysis<br />
isdoneon5yearsofoperationalforecastdatafromtheECMWFinReading.<br />
InSummer2009,jointlywithProf. FraedrichfromthedepartmentofmeteorologyatUniversityof<br />
Hamburgweorganisedampipksinternationalworkshopon”DynamicsandStatisticsofClimateand<br />
Weather”,duringwhichopenissuesofatmosphericfluctuationswerediscussed.<br />
Nonlinearstochasticprocesses.Stochasticprocessesareveryflexiblemodelsfornon-equilibriumphenomena,hencetheyplayaprominentroleinseveralprojectsofourgroup.Inlowdimensions,theinterplay<br />
ofnonlinearityandstochasticitycanproduceinterestingphenomena. Wewereparticularlyinterestedin<br />
hoppingbetweenmetastablestateswhenthereisnopotentialbarrier. Quitegenerally,thewellestablishedKramershoppingrateisdefinedonlyforsystemswithapotential,whichareaverysmallsub-class<br />
ofinterestingmulti-stablesystems.Onealternativemechanismformeta-stabilityisofdynamicalorigin:<br />
regularislandsinchaoticHamiltoniansystemscanbeunderstoodasconstraintsduetoconservationlaws<br />
whicharevalidonlyinapartofthephasespace. Beingbarrierstotransportacrossthem,theyalso<br />
inducemulti-stabilitywhenHamiltoniansystemsaredrivenbynoise. Wearestudyingsuchphenomena<br />
throughtheconceptofrandomsymplecticmapsanditeratedfunctionsystems. Wecouldshowthat<br />
anothernon-potentialsourceformetastability,namelystatedependentdiffusioncoefficientswith“hot”<br />
areas,canbetranslatedintohoppingdynamics.<br />
Longrangecorrelations.Quitegenerally,longrangetemporalcorrelationsareaninterestingissue,since<br />
theymightbeeasilymisinterpretedastrends. Correlationscanevenbesostrongthatempiricaltime<br />
averageswillneverconverge.Wewerestudyingsuchphenomenainhighlyintermittentdynamicalsystems,<br />
wheretheconversionintocontinuoustimerandomwalks(CTRW)leadtoacompletecharacterisation<br />
ofthelongtimebehaviour. Insummer2011,togetherwithcolleaguesfromBarIlan,London,Vienna,<br />
wewillorganiseaseminarandworkshopat mpipksentitled”Weakchaos,infiniteergodictheory,and<br />
anomalousdynamics”whichalsocoversthedynamicaloriginoflongrangetemporalcorrelations.<br />
Extremeevents. Wecontinueoureffortstounderstandpropertiesofextremeevents. Onthepath<br />
towardstheclassificationofnaturalphenomena,wearecurrentlytestingwhetheritmakessenseto<br />
distinguishbetween”turbulencelike”extremeeventsand”SOC-like”ones.SOC,selforganisedcriticality<br />
isamechanismsuggestedin1987byBaketal.asanexplanationofpowerlawsinnature.Thetailsof<br />
powerlawsareanevidentmanifestationofextremeeventswhichareordersofmagnitudeslargerthan<br />
thebulkofevents,butnotallextremesareofthistype.<br />
Inpredictionofextremes,wedistinguishbetweendetailedmodels(suchasweatherforecastingsystems)<br />
andcoarsegrainedmodels.Sincethepredictionofeventsisaclassificationproblem(theoccurrenceof<br />
theeventisabinaryvariable),coarsegrainingmightbepossiblewithoutadramaticlossofinformation.<br />
Indeed,ourdatabasedpredictionschemesperformquitewellandinthecaseoftheAbeliansandpile<br />
modeltheyareasgoodaspredictionsusingknowledgeabouttheinternalmicro-stateofthesystem.In<br />
thecontextofdynamicalsystems,symbolicdynamicsisawellestablishedconcept,whichmightleadto<br />
aformaljustificationofthesuccessofcoarsegrainedforecast.However,thisalsoshowsthedifficulties:<br />
CoarsegrainingmightdestroyMarkovpropertiesandtherebyintroducelongrangecorrelations.<br />
WithfundsoftheVolkswagenstiftungandtogetherwithProf. KurthsfromthePotsdam<strong>Institut</strong>efor<br />
ClimateImpactResearchPIKweorganisedaninternationalworkshoponextremeeventsinPotsdamlast<br />
22 ScientificWorkanditsOrganizationatthe<strong>Institut</strong>e–anOverview
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