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