湍流研究的动力系统方法

TothememoryofGiovannivii
Prefaceix
Introductionxi
Chapter1Turbulenceanddynamicalsystems1
1.1Whatdowemeanbyturbulence?1
1.2Examplesofturbulentphenomena3
1.2.1Fluids3
1.2.2Chemicalturbulence7
1.2.3Flamefronts9
1.3Whyadynamicalsystemapproach?11
1.4Examplesofdynamicalsystemsforturbulence11
1.4.1Shellmodels11
1.4.2Coupledmaplattices12
1.4.3Cellularautomata13
1.5Characterizationofchaosinhighdimensionality14
1.5.1Lyapunovexponentsinextendedsystems14
1.5.2Lyapunovspectraanddimensiondensities15
1.5.3Characterizationofchaosindiscretemodels18
1.5.4Thecorrelationlength18
1.5.5Scalinginvarianceandchaos19
Chapter2Phenomenologyofhydrodynamicturbulence21
2.1Turbulenceasastatisticaltheory21
2.1.1Statisticalmechanicsofaperfectfluid22
2.1.2Basicfactsandideasonfullydevelopedturbulence24
2.1.3Theclosureproblem28
2.2Scalinginvafianceinturbulence31
2.3Multifractaldescriptionoffullydevelopedturbulence34
2.3.1Scalingofthestructurefunctions34
2.3.2Multiplicativemodelsforintermittency36
2.3.3Probabilitydistributionfunctionofthevelocitygradients40
2.3.4Multiscaling43
2.3.5Numberofdegreesoffreedomofturbulence44
2.4Two-dimensionalturbulence45
Chapter3Reducedmodelsforhydrodynamicturbulence48
3.1Dynamicalsystemsasmodelsoftheenergycascade48
3.2Abriefoverviewonshellmodels49
3.2.1ThemodelofDesnyanskyandNovikov51
3.2.2ThemodelofGledzer,OhkitaniandYamada(theGOYmodel)52
3.2.3Hierarchicalshellmodels56
3.2.4Continuumlimitoftheshellmodels57
3.3DynamicalpropertiesoftheGOYmodels58
3.3.1Fixedpointsandscaling58
3.3.2Transitionfromastablefixedpointtochaos60
3.3.3TheLyapunovspectrum65
3.4MultifractalityintheGOYmodel68
3.4.1Anomalousscalingofthestructurefunctions68
3.4.2Dynamicalintermittency71
3.4.3Constructionofa3Dincompressiblevelocityfieldfromtheshellmodels73
3.5AclosuretheoryfortheGOYmodel74
3.6Shellmodelsfortheadvectionofpassivescalars78
3.7Shellmodelsfortwo-dimensionalturbulence83
3.8Low-dimensionalmodelsforcoherentstructures88
Chapter4Turbulenceandcoupledmaplattices91
4.1Introductiontocoupledchaoticmaps92
4.1.1Linearstabilityofthecoherentstate93
4.1.2Spreadingofperturbations94
4.2Scalingatthecriticalpoint97
4.2.1ScalingoftheLyapunovexponents97
4.2.2Scalingofthecorrelationlength99
4.2.3Spreadingoflocalizedperturbations101
4.2.4Renormalizationgroupresults104
4.3Lyapunovspectra106
4.3.1Coupledmaplatticeswithconservationlaws107
4.3.2Analyticresults110
4.4Coupledmapswithlaminarstates112
4.4.1Spatio-temporalintermittency112
4.4.2lnvariantmeasuresandPerron-FrobeniusequationofCML114
4.4.3Meanfieldapproximationandphasediagram115
4.4.4Directiteratesandfinitesizescaling118
4.4.5Spatialcorrelationsandhyperscaling121
4.5Coupledmaplatticeswithanisotropiccouplings123
4.5.1Convectiveinstabilitiesandturbulentspots123
4.5.2Coherentchaosinanisotropicsystems128
4.5.3Aboundarylayerinstabilityinananisotropicsystem131
4.5.4Acoupledmaplatticeforaconvectivesystem134
Chapter5TurbulenceinthecomplexGinzburg-Landauequation138
5.1ThecomplexGinzburg-Landauequation139
5.2Thestabilityofthehomogeneousperiodicstateamithephaseequation143
5.3Planewavesandtheirstability146
5.3.1Thestabilityofplanewaves146
5.3.2Convectiveversusabsolutestability148
5.4Large-scalesimulationsandthecoupledmapapproximation148
5.5Spiralsandwavenumberselection150
5.6Theonsetofturbulence152
5.6.1Transientturbulenceandnucleation155
5.7Glassystatesofboundvortices158
5.8Vortexinteractions161
5.8.1Microscopictheoryofshocks162
5.8.2Asymptoticproperties163
5.8.3Weakshocks167
5.9PhaseturbulenceandtheKuramoto-Sivashinskyequation168
5.9.1CorrelationsintheKuramoto-Sivashinskyequation170
5.9.2Dimensiondensitiesandcorrelationsinphaseturbulence171
5.10Anisotropicphaseequation172
5.10.1Theshapeofaspreadingspot173
5.10.2Turbulentspotsandpulses178
5.10.3Anisotropicturbulentspotsintwodimensions181
Chapter6Predictabilityinhigh-dimensionalsystoms183
6.1Predictabilityinturbulence185
6.1.1MaximumLyapunovexponentofaturbulentfiow185
6.1.2Theclassicaltheoryofpredictabilityinturbulence186
6.2Predictabilityinsystemswithmanycharacteristictimes188
6.3ChaosandbutterflyeffectintheGOYmodel191
6.3.1Growthofinfinitesimalperturbationsanddynamicalintermittency191
6.3.2Statisticsofthepredictabilitytimeanditsrelationwithintermittency195
6.3.3Growthofnon-infinitesimalperturbations197
6.4Predictabilityinextendedsystems200
6.5Predictabilityinnoisysystems202
6.6Finalremarks209
Chapter7Dynamicsofinterfaces211
7.1Turbulenceandinterfaces211
7.2TheBurgersequation212
7.3TheLangevinapproachtodynamicalinterfaces:theKPZequation215
7.4Deterministicinterfacedynamics:theKuramoto-Sivashinskyequation219
7.4.1Cross-overtoKPZbehaviour222
7.4.2TheKuramoto-Sivashinskyequationin2+1dimensions225
7.4.3Interfacesincoupledmaplattices227
7.5Depinningmodels231
7.5.1Quenchedrandomnessanddirectedpercolationnetworks231
7.5.2Self-organized-criticaldynamics:theSneppenmodel232
7.5.3Colouredactivity235
7.5.4AscalingtheoryfortheSneppenmodel:mappingtodirectedpercolation237
7.5.5Ageometricdescriptionoftheavalanchedynamics240
7.5.6Multiscaling241
7.6Dynamicsofamembrane243
Chapter8Lagrangianchaos244
8.1Generalremarks244
8.1.1ExamplesofLagrangianchaos247
8.1.2Stretchingofmateriallinesandsurfaces251
8.2EulerianversusLagrangianchaos253
8.2.1OnsetofLagrangianchaosintwo-dimensionalflows255
8.2.2Eulerianchaosandfluidparticlemotion258
8.2.3AcommentonLagrangianchaos264
8.3Statisticsofpassivefields265
8.3.1Thegrowthofscalargradients265
8.3.2Themultifractalstructureforthedistributionofscalargradients266
8.3.3Thepowerspectrumofscalarfelds268
8.3.4SomeremarksonthevalidityoftheBatchelorlaw270
8.3.5Intermittencyandmultifractalityinmagneticdynamos272
Chapter9Chaoticdiffusion277
9.1Diffusioninincompressibleflows279
9.1.1StandarddiffusioninthepresenceofLagrangianchaos279
9.1.2Standarddiffusioninsteadyvelocityfields281
9.1.3Anomalousdiffusioninrandomvelocityfields283
9.1.4Anomalousdiffusioninsmoothvelocityfields284
9.2Anomalousdiffusioninfieldsgeneratedbyextendedsystems285
9.2.1AnomalousdiffusionintheKuramoto-Sivashinskyequation286
9.2.2Multidiffusionalonganintermittentmembrane290
AppendixAHopfbifurcation292
AppendixBHamiltoniansystems294
AppendixCCharacteristicandgeneralizedLyapunovexponents301
AppendixDConvectiveinstabilitiesandlinearfrontpropagation309
AppendixEGeneralizedfractaldimensionsandmultifractals315
AppendixFMultiaffinefields320
AppendixGReductiontoafinite-dimensionaldynamicalsystem325
AppendixHDirectedpercolation329
References332
Index347