Chapter 1 Turbulence and dynamical systems 1
1.1 What do we mean by turbulence? 1
1.2 Examples of turbulent phenomena 3
1.2.1 Fluids 3
1.2.2 Chemical turbulence 7
1.2.3 Flame fronts 9
1.3 Why a dynamical system approach? 11
1.4 Examples of dynamical systems for turbulence 11
1.4.1 Shell models 11
1.4.2 Coupled map lattices 12
1.4.3 Cellular automata 13
1.5 Characterization of chaos in high dimensionality 14
1.5.1 Lyapunov exponents in extended systems 14
1.5.2 Lyapunov spectra and dimension densities 15
1.5.3 Characterization of chaos in discrete models 18
1.5.4 The correlation length 18
1.5.5 Scaling invariance and chaos 19
Chapter 2 Phenomenology of hydrodynamic turbulence 21
2.1 Turbulence as a statistical theory 21
2.1.1 Statistical mechanics of a perfect fluid 22
2.1.2 Basic facts and ideas on fully developed turbulence 24
2.1.3 The closure problem 28
2.2 Scaling invariance in turbulence 31
2.3 Multifractal description of fully developed turbulence 34
2.3.1 Scaling of the structure functions 34
2.3.2 Multiplicative models for intermittency 36
2.3.3 Probability distribution function of the velocity gradients 40
2.3.4 Multiscaling 43
2.3.5 Number of degrees of freedom of turbulence 44
2.4 Two-dimensional turbulence 45
Chapter 3 Reduced models for hydrodynamic turbulence 48
3.1 Dynamical systems as models of the energy cascade 48
3.2 A brief overview on shell models 49
3.2.1 The model of Desnyansky and Novikov 51
3.2.2 The model of Gledzer,Ohkitani and Yamada(the GOY model) 52
3.2.3 Hierarchical shell models 56
3.2.4 Continuum limit of the shell models 57
3.3 Dynamical properties of the GOY models 58
3.3.1 Fixed points and scaling 58
3.3.2 Transition from a stable fixed point to chaos 60
3.3.3 The Lyapunov spectrum 65
3.4 Multifractality in the GOY model 68
3.4.1 Anomalous scaling of the structure functions 68
3.4.2 Dynamical intermittency 71
3.4.3 Construction of a 3D incompressible velocity field from the shell models 73
3.5 A closure theory for the GOY model 74
3.6 Shell models for the advection of passive scalars 78
3.7 Shell models for two-dimensional turbulence 83
3.8 Low-dimensional models for coherent structures 88
Chapter 4 Turbulence and coupled map lattices 91
4.1 Introduction to coupled chaotic maps 92
4.1.1 Linear stability of the coherent state 93
4.1.2 Spreading of perturbations 94
4.2 Scaling at the critical point 97
4.2.1 Scaling of the Lyapunov exponents 97
4.2.2 Scaling of the correlation length 99
4.2.3 Spreading of localized perturbations 101
4.2.4 Renormalization group results 104
4.3 Lyapunov spectra 106
4.3.1 Coupled map lattices with conservation laws 107
4.3.2 Analytic results 110
4.4 Coupled maps with laminar states 112
4.4.1 Spatio-temporal intermittency 112
4.4.2 Invariant measures and Perron-Frobenius equation of CML 114
4.4.3 Mean field approximation and phase diagram 115
4.4.4 Direct iterates and finite size scaling 118
4.4.5 Spatial correlations and hyperscaling 121
4.5 Coupled map lattices with anisotropic couplings 123
4.5.1 Convective instabilities and turbulent spots 123
4.5.2 Coherent chaos in anisotropic systems 128
4.5.3 A boundary layer instability in an anisotropic system 131
4.5.4 A coupled map lattice for a convective system 134
Chapter 5 Turbulence in the complex Ginzburg-Landau equation 138
5.1 The complex Ginzburg-Landau equation 139
5.2 The stability of the homogeneous periodic state and the phase equation 143
5.3 Plane waves and their stability 146
5.3.1 The stability of plane waves 146
5.3.2 Convective versus absolute stability 148
5.4 Large-scale simulations and the coupled map approximation 148
5.5 Spirals and wave number selection 150
5.6 The onset of turbulence 152
5.6.1 Transient turbulence and nucleation 155
5.7 Glassy states of bound vortices 158
5.8 Vortex interactions 161
5.8.1 Microscopic theory of shocks 162
5.8.2 Asymptotic properties 163
5.8.3 Weak shocks 167
5.9 Phase turbulence and the Kuramoto-Sivashinsky equation 168
5.9.1 Correlations in the Kuramoto-Sivashinsky equation 170
5.9.2 Dimension densities and correlations in phase turbulence 171
5.10 Anisotropic phase equation 172
5.10.1 The shape of a spreading spot 173
5.10.2 Turbulent spots and pulses 178
5.10.3 Anisotropic turbulent spots in two dimensions 181
Chapter 6 predictability in high-dimensional systems 183
6.1 Predictability in turbulence 185
6.1.1 Maximum Lyapunov exponent of a turbulent flow 185
6.1.2 The classical theory of predictability in turbulence 186
6.2 Predictability in systems with many characteristic times 188
6.3 Chaos and butterfly effect in the GOY model 191
6.3.1 Growth of infinitesimal perturbations and dynamical intermittency 191
6.3.2 Statistics of the predictability time and its relation with intermittency 195
6.3.3 Growth of non-infinitesimal perturbations 197
6.4 Predictability in extended systems 200
6.5 Predictability in noisy systems 202
6.6 Final remarks 209
Chapter 7 Dynamics of interfaces 211
7.1 Turbulence and interfaces 211
7.2 The Burgers equation 212
7.3 The Langevin approach to dynamical interfaces:the KPZ equation 215
7.4 Deterministic interface dynamics:the Kuramoto-Sivashinsky equation 219
7.4.1 Cross-over to KPZ behaviour 222
7.4.2 The Kuramoto-Sivashinsky equation in 2+1 dimensions 225
7.4.3 Interfaces in coupled map lattices 227
7.5 Depinning models 231
7.5.1 Quenched randomness and directed percolation networks 231
7.5.2 Self-organized-critical dynamics:the Sneppen model 232
7.5.3 Coloured activity 235
7.5.4 A scaling theory for the Sneppen model:mapping to directed percolation 237
7.5.5 A geometric description of the avalanche dynamics 240
7.5.6 Multiscaling 241
7.6 Dynamics of a membrane 243
Chapter 8 Lagrangian chaos 244
8.1 General remarks 244
8.1.1 Examples of Lagrangian chaos 247
8.1.2 Stretching of material lines and surfaces 251
8.2 Eulerian versus Lagrangian chaos 253
8.2.1 Onset of Lagrangian chaos in two-dimensional flows 255
8.2.2 Eulerian chaos and fluid particle motion 258
8.2.3 A comment on Lagrangian chaos 264
8.3 Statistics of passive fields 265
8.3.1 The growth of scalar gradients 265
8.3.2 The multifractal structure for the distribution of scalar gradients 266
8.3.3 The power spectrum of scalar fields 268
8.3.4 Some remarks on the validity of the Batchelor law 270
8.3.5 Intermittency and multifractality in magnetic dynamos 272
Chapter 9 Chaotic diffusion 277
9.1 Diffusion in incompressible flows 279
9.1.1 Standard diffusion in the presence of Lagrangian chaos 279
9.1.2 Standard diffusion in steady velocity fields 281
9.1.3 Anomalous diffusion in random velocity fields 283
9.1.4 Anomalous diffusion in smooth velocity fields 284
9.2 Anomalous diffusion in fields generated by extended systems 285
9.2.1 Anomalous diffusion in the Kuramoto-Sivashinsky equation 286
9.2.2 Multidiffusion along an intermittent membrane 290
Appendix A Hopf bifurcation 292
Appendix B Hamiltonian systems 294
Appendix C Characteristic and generalized Lyapunov exponents 301
Appendix D Convective instabilities and linear front propagation 309
Appendix E Generalized fractal dimensions and multifractals 315
Appendix F Multiaffine fields 320
Appendix G Reduction to a finite-dimensional dynamical system 325
Appendix H Directed percolation 329
References 332
Index 347