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1StochastlcNaVler1StokesEquatlonandGeostrophlcTurbulence
5
Theformationoflargescalecoherentstructuresisafascinatingproblemand
anessentialpartofthedynamicsofEarth,satmosphereandoceanslThisisthe
mainmotiVationforsettingupatheoryfortheself-organizationof2Dturbulencel
li2StItźstźcIlmechInźcsoftheself-orgInźzItźonof
two-dźmensźonIlIndgeophysźcIlflows:OnsIger7s
equźlźbrźumroute
AnyturbulenceprobleminVolVesahugenumberofdegreesoffreedomcoupled
ViacompleXnonlinearinteractionslTheaimofanytheoryofturbulenceisto
understandthestatisticalpropertiesoftheVelocityfieldlItisthuseXtremely
temptingtoattacktheseproblemsfromastatisticalmechanicspointofViewl
StatisticalmechanicsisindeedaVerypowerfulsetoftheoreticaltoolsthat
allowsustoreducethecompleXityofasystemdowntoafewthermodynamic
parameterslAsaneXample,theconceptofphasetransitionallowsustodescribe
drasticchangesofthewholesystemwhenafeweXternalparametersarechangedl
StatisticalmechanicsisthemaintheoreticalapproachwedeVelopintheselecturesl
ItsucceedsineXplainingmanyofthephenomenaassociatedwithtwo-dimensional
turbulence[13]l
Thismayseemsurprisingatfirst,asitisacommonbeliefthatstatisticalme-
chanicsisnotsuccessfulinhandlingturbulenceproblemslThereasonforthis
beliefisthatmostturbulenceproblemsareintrinsicallyfarfromequilibriumlFor
instance,theforwardenergycascadeinthree-dimensionalturbulenceinVolVesa
finiteenergydissipation,nomatterhowsmalltheViscosity(anomalousdissipa-
tion)(seeforinstanceOnsager,sinsightfulconsiderationofthenon-conserVation
ofenergybythethreedimensionalEulerequations[28])lAsaresultofthisfinite
energyfluX,threedimensionalturbulentflowscannotbeconsideredclosetosome
equilibriumdistributionl
Bycontrast,two-dimensionalturbulencedoesnotsulerfromtheanomalous
dissipationoftheenergy,soequilibriumstatisticalmechanics,orclosetoequi-
libriumstatisticalmechanicsmakessensewhensmallfluXesarepresentlThe
firstattempttouseequilibriumstatisticalmechanicsideastoeXplaintheself-
organizationoftwo-dimensionalturbulencedatesfromOnsagerworkin1949[51]
(see[28]forareViewofOnsager,scontributionstoturbulencetheory)lOnsager
workedwiththepoint-VorteXmodel,amodelthatdescribesthedynamicsofsin-
gularpointVortices,firstusedbyLordKelVinandwhichcorrespondstoaspecial
classofsolutionsofthe2DEulerequationslTheequilibriumstatisticalmechanics
ofthepoint-VorteXmodelhasalongandVeryinterestinghistory,withwonderful
piecesofmathematicalachieVements[1;18;21;26;27;37;39;51]l
ThegeneralizationofOnsager,sideastothe2DEulerequationswithacontin-
uousVorticityfield,takingintoaccountallinVariants,hasbeenproposedinthe
beginningofthe1990s[45;57;58;60],leadingtotheMillerŹRobertŹSommeria
theory(MRStheory)lTheMRStheoryincludesthepreViousOnsagertheory
