湍流研究的动力系统方法 英文PDF|Epub|txt|kindle电子书版本网盘下载

湍流研究的动力系统方法 英文PDF格式电子书版下载

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Chapter 1 Turbulence and dynamical systems1

1.1 What do we mean by turbulence?1

1.2 Examples of turbulent phenomena3

1.2.1 Fluids3

1.2.2 Chemical turbulence7

1.2.3 Flame fronts9

1.3 Why a dynamical system approach?11

1.4 Examples of dynamical systems for turbulence11

1.4.1 Shell models11

1.4.2 Coupled map lattices12

1.4.3 Cellular automata13

1.5 Characterization of chaos in high dimensionality14

1.5.1 Lyapunov exponents in extended systems14

1.5.2 Lyapunov spectra and dimension densities15

1.5.3 Characterization of chaos in discrete models18

1.5.4 The correlation length18

1.5.5 Scaling invariance and chaos19

Chapter 2 Phenomenology of hydrodynamic turbulence21

2.1 Turbulence as a statistical theory21

2.1.1 Statistical mechanics of a perfect fluid22

2.1.2 Basic facts and ideas on fully developed turbulence24

2.1.3 The closure problem28

2.2 Scaling invariance in turbulence31

2.3 Multifractal description of fully developed turbulence34

2.3.1 Scaling of the structure functions34

2.3.2 Multiplicative models for intermittency36

2.3.3 Probability distribution function of the velocity gradients40

2.3.4 Multiscaling43

2.3.5 Number of degrees of freedom of turbulence44

2.4 Two-dimensional turbulence45

Chapter 3 Reduced models for hydrodynamic turbulence48

3.1 Dynamical systems as models of the energy cascade48

3.2 A brief overview on shell models49

3.2.1 The model of Desnyansky and Novikov51

3.2.2 The model of Gledzer,Ohkitani and Yamada(the GOY model)52

3.2.3 Hierarchical shell models56

3.2.4 Continuum limit of the shell models57

3.3 Dynamical properties of the GOY models58

3.3.1 Fixed points and scaling58

3.3.2 Transition from a stable fixed point to chaos60

3.3.3 The Lyapunov spectrum65

3.4 Multifractality in the GOY model68

3.4.1 Anomalous scaling of the structure functions68

3.4.2 Dynamical intermittency71

3.4.3 Construction of a 3D incompressible velocity field from the shell models73

3.5 A closure theory for the GOY model74

3.6 Shell models for the advection of passive scalars78

3.7 Shell models for two-dimensional turbulence83

3.8 Low-dimensional models for coherent structures88

Chapter 4 Turbulence and coupled map lattices91

4.1 Introduction to coupled chaotic maps92

4.1.1 Linear stability of the coherent state93

4.1.2 Spreading of perturbations94

4.2 Scaling at the critical point97

4.2.1 Scaling of the Lyapunov exponents97

4.2.2 Scaling of the correlation length99

4.2.3 Spreading of localized perturbations101

4.2.4 Renormalization group results104

4.3 Lyapunov spectra106

4.3.1 Coupled map lattices with conservation laws107

4.3.2 Analytic results110

4.4 Coupled maps with laminar states112

4.4.1 Spatio-temporal intermittency112

4.4.2 Invariant measures and Perron-Frobenius equation of CML114

4.4.3 Mean field approximation and phase diagram115

4.4.4 Direct iterates and finite size scaling118

4.4.5 Spatial correlations and hyperscaling121

4.5 Coupled map lattices with anisotropic couplings123

4.5.1 Convective instabilities and turbulent spots123

4.5.2 Coherent chaos in anisotropic systems128

4.5.3 A boundary layer instability in an anisotropic system131

4.5.4 A coupled map lattice for a convective system134

Chapter 5 Turbulence in the complex Ginzburg-Landau equation138

5.1 The complex Ginzburg-Landau equation139

5.2 The stability of the homogeneous periodic state and the phase equation143

5.3 Plane waves and their stability146

5.3.1 The stability of plane waves146

5.3.2 Convective versus absolute stability148

5.4 Large-scale simulations and the coupled map approximation148

5.5 Spirals and wave number selection150

5.6 The onset of turbulence152

5.6.1 Transient turbulence and nucleation155

5.7 Glassy states of bound vortices158

5.8 Vortex interactions161

5.8.1 Microscopic theory of shocks162

5.8.2 Asymptotic properties163

5.8.3 Weak shocks167

5.9 Phase turbulence and the Kuramoto-Sivashinsky equation168

5.9.1 Correlations in the Kuramoto-Sivashinsky equation170

5.9.2 Dimension densities and correlations in phase turbulence171

5.10 Anisotropic phase equation172

5.10.1 The shape of a spreading spot173

5.10.2 Turbulent spots and pulses178

5.10.3 Anisotropic turbulent spots in two dimensions181

Chapter 6 predictability in high-dimensional systems183

6.1 Predictability in turbulence185

6.1.1 Maximum Lyapunov exponent of a turbulent flow185

6.1.2 The classical theory of predictability in turbulence186

6.2 Predictability in systems with many characteristic times188

6.3 Chaos and butterfly effect in the GOY model191

6.3.1 Growth of infinitesimal perturbations and dynamical intermittency191

6.3.2 Statistics of the predictability time and its relation with intermittency195

6.3.3 Growth of non-infinitesimal perturbations197

6.4 Predictability in extended systems200

6.5 Predictability in noisy systems202

6.6 Final remarks209

Chapter 7 Dynamics of interfaces211

7.1 Turbulence and interfaces211

7.2 The Burgers equation212

7.3 The Langevin approach to dynamical interfaces:the KPZ equation215

7.4 Deterministic interface dynamics:the Kuramoto-Sivashinsky equation219

7.4.1 Cross-over to KPZ behaviour222

7.4.2 The Kuramoto-Sivashinsky equation in 2+1 dimensions225

7.4.3 Interfaces in coupled map lattices227

7.5 Depinning models231

7.5.1 Quenched randomness and directed percolation networks231

7.5.2 Self-organized-critical dynamics:the Sneppen model232

7.5.3 Coloured activity235

7.5.4 A scaling theory for the Sneppen model:mapping to directed percolation237

7.5.5 A geometric description of the avalanche dynamics240

7.5.6 Multiscaling241

7.6 Dynamics of a membrane243

Chapter 8 Lagrangian chaos244

8.1 General remarks244

8.1.1 Examples of Lagrangian chaos247

8.1.2 Stretching of material lines and surfaces251

8.2 Eulerian versus Lagrangian chaos253

8.2.1 Onset of Lagrangian chaos in two-dimensional flows255

8.2.2 Eulerian chaos and fluid particle motion258

8.2.3 A comment on Lagrangian chaos264

8.3 Statistics of passive fields265

8.3.1 The growth of scalar gradients265

8.3.2 The multifractal structure for the distribution of scalar gradients266

8.3.3 The power spectrum of scalar fields268

8.3.4 Some remarks on the validity of the Batchelor law270

8.3.5 Intermittency and multifractality in magnetic dynamos272

Chapter 9 Chaotic diffusion277

9.1 Diffusion in incompressible flows279

9.1.1 Standard diffusion in the presence of Lagrangian chaos279

9.1.2 Standard diffusion in steady velocity fields281

9.1.3 Anomalous diffusion in random velocity fields283

9.1.4 Anomalous diffusion in smooth velocity fields284

9.2 Anomalous diffusion in fields generated by extended systems285

9.2.1 Anomalous diffusion in the Kuramoto-Sivashinsky equation286

9.2.2 Multidiffusion along an intermittent membrane290

Appendix A Hopf bifurcation292

Appendix B Hamiltonian systems294

Appendix C Characteristic and generalized Lyapunov exponents301

Appendix D Convective instabilities and linear front propagation309

Appendix E Generalized fractal dimensions and multifractals315

Appendix F Multiaffine fields320

Appendix G Reduction to a finite-dimensional dynamical system325

Appendix H Directed percolation329

References332

Index347