Turbulence and Diffusion (eBook)

Preface 7
General References 9
Diffusion Concept 9
Correlations in Complex Systems 9
Hydrodynamics and Turbulence 9
Correlation Functions and Geophysical Turbulence 10
Plasma Physics and Magnetohydrodynamic Turbulence 10
Chaos and Mixing 10
Fractals and Percolation 11
The Fokker-Planck Equation and Kinetic Theory 11
Contents 12
Turbulent Diffusion Concepts 15
Introduction 16
1.1 Brownian Motion, Random Walks, and Correlation Scales 16
1.2 The Fick Transport Equation 20
1.3 Diffusion and the Characteristic Velocity Scale 23
1.4 Lagrangian Description of Turbulent Diffusion 26
Turbulent Diffusion and Scaling 33
2.1 Correlation Functions and Scaling 33
2.2 The Richardson Law and Anomalous Transport 35
2.3 The Kolmogorov Description of Turbulence 38
2.4 Relative Diffusion and Scalings 44
2.5 Cascade Phenomenology and Scalar Spectrum 46
Nonlocal Effects and Diffusion Equations 52
3.1 The Einstein Functional Equation 52
3.2 Nonlocality and Levy-Stable Distributions 55
3.3 Fractional Derivatives and Anomalous Diffusion 58
3.4 The Monin Nonlocal Equation 62
Correlation Effects and Scalings 65
Diffusive Renormalization and Correlations 66
4.1 The Corrsin Independence Hypothesis 66
4.2 The Correlation Function and Anomalous Diffusion 69
4.3 Seed Diffusivity and Correlations 71
4.4 Effective Diffusivity and the Peclet Number 73
4.5 Diffusive Renormalization and the Correlation Function 75
Diffusion Equations and the Quasilinear Approximation 79
5.1 The Taylor Dispersion 79
5.2 Advection and Scalar Transport 83
5.3 Zeldovich Flow and the Kubo Number 85
5.4 Quasilinear Equations 89
5.5 Short-Range and Long-Range Correlations 91
5.6 The Telegraph Equation 92
Return Effects and Random Shear Flows 95
6.1 “Returns” and Correlations 95
6.2 Superdiffusion and Return Effects 98
6.3 Random Shear Flows and Stochastic Equations 101
6.4 The “Manhattan-Grid” Flow and Turbulent Transport 103
Turbulence of Magnetic Field Lines 108
7.1 Basic Equations of Plasma 108
7.2 Magnetic Field Evolution and Magnetic Reynolds Number 111
7.3 Magnetic Diffusivity and the Quasilinear Approach 113
7.4 Stochastic Magnetic Field and Transport Scalings 117
7.5 Diffusive Renormalization and a Braded Magnetic Field 119
Stochastic Instability and Turbulence 124
8.1 Stochastic Instability and Correlations 124
8.2 Quasilinear Scaling for the Stochastic Instability Increment 126
8.3 The Rechester-Rosenbluth Model 129
8.4 Collisional Effects and Transport 131
8.5 The Quasi-Isotropic Stochastic Magnetic Field 134
Anomalous Transport and Convective Cells 138
9.1 Convective Cells and Turbulent Diffusion 138
9.2 Complex Structures and the Statistical Topography 142
9.3 Fluctuation–Dissipative Relation and Turbulent Mixing 143
9.4 Bohm Scaling and Electric Field Fluctuations 145
9.5 Diffusive Renormalization and Correlations 148
Fractals and Percolation Transport 152
Fractal and Percolation Concepts 153
10.1 Self-Similarity and the Fractal Dimension 153
10.2 Fractality and Anomalous Transport 157
10.3 Turbulence Scalings and Fractality 159
10.4 Percolation Transition and Correlations 163
10.5 Continuum Percolation and Transport 166
10.6 Finite Size Renormalization and Scaling 169
Percolation and Turbulent Transport 174
11.1 Random Steady Flows and Seed Diffusivity 174
11.2 Reorganization of Flow Topology and Percolation Scalings 179
11.3 Spatial and Temporal Hierarchy of Scales 183
11.4 Percolation in Drift Flows 186
11.5 Drift and Low-Frequency Regimes 189
11.6 Renormalization and the Stochastic Instability Increment 192
11.7 Stochastic Magnetic Field and Percolation 194
Multiscale Approach and Scalings 198
12.1 The Nested Hierarchy of Scales and Drift Effects 198
12.2 The Brownian Landscape and Percolation 201
12.3 Correlations and Transport Scalings 204
12.4 Diffusive Approximation and the Multiscale Model 206
12.5 Stochastic Instability and the Temporal Hierarchy of Scales 208
12.6 Isotropic and Anisotropic Magnetohydrodynamic Turbulence 209
Trapping and the Escape Probability Formalism 215
Subdiffusion and Trapping 216
13.1 Diffusion in the Presence of Traps 216
13.2 Trapping and Strong Turbulence 218
13.3 Comb Structures and Transport 221
13.4 Double Diffusion and Return Effects 223
Continuous Time Random Walks and Transport Scalings 226
14.1 The Montroll andWeiss Approach and Memory Effects 226
14.2 Fractional Differential Equations 229
14.3 Correlation Function andWaiting Time Distribution 231
14.4 The Klafter Blumen and Shlesinger Approximation 233
14.5 Stochastic Magnetic Field and Balescu Approach 236
14.6 Longitudinal Correlations and the Diffusive Approximation 238
14.7 Vortex Structures and Trapping 241
Correlation and Phase-Space 245
15.1 Kinetics and the Diffusion Equation 245
15.2 Phase Space and Transport Scaling 247
15.3 The One-Flight Model and Anomalous Diffusion 249
15.4 Correlations and Nonlocal Velocity Distribution 251
15.5 The Corrsin Conjecture and Phase-Space 254
References 257
Index 265