Preface ......................................................... v
1 Mechanisma of Transverse Conductivity and Generation of
Self-Consistent Electric Fields in Strongly Ionized
Magnetized Plasma
V. Rozhansky ................................................. 1
1.1 Introduction ............................................ 1
1.2 Conductivity Tensor in Partially Ionized Plasma ......... 4
1.3 Main Mechanisms of Perpendicular Conductivity in Fully
Ionized Plasma: Currents Caused by Viscosity, Inertia,
Collisions with Neutrals, and VB, and Mass-Loading
Currents ................................................ 6
1.3.1 Inertia Currents ................................. 6
1.3.2 Currents Caused by Ion-Neutral Collisions ........ 7
1.3.3 Diamagnetic Currents ............................ 8
1.3.4 Viscosity-Driven Currents ........................ 8
1.3.5 Mass-Loading Current ............................ 10
1.4 Inertial (Polarization) and VB Currents.
Acceleration of Plasma Clouds in an Inhomogeneous
Magnetic Field ......................................... 11
1.5 Alfven Conductivity .................................... 13
1.6 Perpendicular Viscosity, Radial Current, and Radial
Electric Field in an Infinite Cylinder ................. 15
1.7 Current Systems in Front of a Biased Electrode
(Flush-Mounted Probe) and Spot of Emission ............. 16
1.7.1 Viscosity-Driven Perpendicular Currents ......... 18
1.7.2 Currents Driven by Ion-Neutral Collisions ....... 22
1.7.3 Inertia Currents ................................ 23
1.7.4 General Situation ............................... 24
1.7.5 Spot of Emission ................................ 25
1.8 Currents in the Vicinity of a Biased Electrode That
is Smaller Than the Ion Gyroradius ..................... 26
1.9 Neoclassical Perpendicular Conductivity in a Tokamak ... 30
1.9.1 Steady State Current ............................ 30
1.9.2 Time-Dependent Current .......................... 37
1.10 Transverse Conductivity in a Reversed Field Pinch ...... 38
1.11 Modeling of Electric Field and Currents in the
Tokamak Edge Plasma .................................... 39
1.12 Mechanisms of Anomalous Perpendicular Viscosity and
Viscosity-Driven Currents .............................. 41
1.13 Transverse Conductivity in a Stochastic Magnetic
Field .................................................. 43
1.13.1 Nonstochastic Magnetic Field .................... 44
1.13.2 Stochastic Magnetic Field ....................... 45
1.14 Electric Fields Generated in the Shielding Layer
between Hot Plasma and a Solid State ................... 46
References .................................................. 50
2 Correlations and Anomalous Transport Models
O.G. Bakunin ................................................ 53
2.1 Introduction ........................................... 53
2.2 Turbulent Diffusion and Transport ...................... 56
2.2.1 The Correlation Function and the Taylor
Diffusivity ..................................... 57
2.2.2 The Richardson Law .............................. 59
2.2.3 The Davydov Model of Turbulent Diffusion ........ 60
2.2.4 The Batchelor Approximation for the Diffusion
Coefficient ..................................... 62
2.3 Nonlocal Effects and Diffusion Equations ............... 63
2.3.1 The Functional Equation for Random Walks ........ 64
2.3.2 Nonlocality and the Levy Distribution ........... 65
2.3.3 The Monin Fractional Differential Equation ...... 67
2.4 The Corrsin Conjecture ................................. 69
2.4.1 The Corrsin Independence Hypothesis ............. 69
2.4.2 The Simplified Corrsin Conjecture ............... 70
2.4.3 The Correlation Function and Scalings ........... 71
2.5 Effects of Seed Diffusivity ............................ 72
2.5.1 Seed Diffusivity and Correlations ............... 73
2.5.2 "Returns" and Correlations ...................... 74
2.5.3 The Stochastic Magnetic Field and Scalings ...... 75
2.5.4 The Howells Result .............................. 77
2.6 The Diffusive Tracer Equation and Averaging ............ 79
2.6.1 The Taylor Shear Flow Model ..................... 79
2.6.2 Generalization of the Taylor Model .............. 82
2.6.3 The Zeldovich Flow and the Kubo Number .......... 83
2.6.4 Advection and Zeldovich Scaling ................. 85
2.7 The System of Random Shear Flows ....................... 86
2.7.1 The Dreizin-Dykhne Superdiffusion Regime ........ 87
2.7.2 The Matheron-de Marsily Model ................... 88
2.7.3 The "Manhattan Grid" Flow and Transport ......... 91
2.8 The Quasi-Linear Approximation ......................... 93
2.8.1 Quasi-Linear Equations .......................... 93
2.8.2 Short-Range and Long-Range Correlations ......... 95
2.8.3 The Telegraph Equation .......................... 96
2.8.4 Magnetic Diffusivity and the Kubo Number ........ 97
2.9 The Diffusive Renormalization .......................... 98
2.9.1 The Dupree Approximation ........................ 99
2.9.2 The Bupree Theory Revisited .................... 102
2.9.3 The Taylor-McNamara Correlation Function ....... 104
2.9.4 The Kadomtsev-Pogutse Renormalization and the
Stochastic Magnetic Field ...................... 106
2.10 Anomalous Transport and Convective Cells .............. 108
2.10.1 Bohm Scaling and Electric Field Fluctuations ... 108
2.10.2 The Bohm Regime and Correlations ............... 110
2.10.3 Convective Cells and Transport ................. 112
2.10.4 Complex Structures and Convective Transport .... 114
2.11 Stochastic Instability and Transport .................. 114
2.11.1 Stochastic Instability and Correlations ........ 115
2.11.2 The Rechester-Rosenbluth Model ................. 116
2.11.3 Collisional Effects and the Stix Formula ....... 119
2.11.4 The Quasi-Isotropic Stochastic Magnetic Field
and Transport .................................. 121
2.11.5 Quasi-Linear Scaling for the Stochastic
Instability Increment .......................... 123
2.12 Fractal Conceptions and Turbulence .................... 125
2.12.1 Fractality and Transport ....................... 125
2.12.2 The Richardson Law and Fractality .............. 127
2.12.3 Intermittency and the Kolmogorov Law ........... 129
2.13 Percolation and Scalings .............................. 130
2.13.1 Continuum Percolation and Transport ............ 130
2.13.2 Renormalization and Percolation ................ 132
2.13.3 Graded Percolation ............................. 134
2.14 Percolation and Turbulent Transport Scalings .......... 135
2.14.1 Random Steady Flows and Seed Diffusivity ....... 136
2.14.2 The Spatial Hierarchy of Scales and
Stochastic Instability ......................... 137
2.14.3 Low Frequency Regimes .......................... 138
2.15 The Temporal Hierarchy of Scales and Correlations ..... 140
2.15.1 The Spatial and Temporal Hierarchy of Scales ... 140
2.15.2 The Isichenko Intermediate Regime .............. 142
2.15.3 Dissipation and Percolation Transport .......... 143
2.16 The Stochastic Magnetic Field and Percolation
Transport ............................................. 146
2.16.1 The Stochastic Magnetic Field and Percolation
Transport ...................................... 146
2.16.2 Percolation and the Kadomtsev-Pogutse Scaling .. 148
2.16.3 Percolation Renormalization and the
Stochastic Instability Increment ............... 150
2.17 Percolation in Drift Flows ............................ 151
2.17.1 Graded Percolation and Drift Flows ............. 151
2.17.2 Low Frequency Regimes and Drift Effects ........ 153
2.17.3 Compressibility and Percolation ................ 155
2.18 Multiscale Flows ...................................... 156
2.18.1 The Nested Hierarchy of Scales and Drift
Effects ........................................ 157
2.18.2 The Brownian Landscape and Percolation ......... 159
2.18.3 Correlations and Transport Scalings ............ 161
2.18.4 The Diffusive Approximation and the
Multiscale Model ............................... 163
2.18.5 Stochastic Instability and Time Scales ......... 165
2.18.6 Isotropic and Anisotropic Turbulent Energy
Spectra ........................................ 166
2.18.7 The Multiscale Model of Transport in
a Tangled Magnetic Field ....................... 168
2.19 Subdiffusion and Traps ................................ 169
2.19.1 The Balagurov and Vaks Model of Diffusion
with Traps ..................................... 170
2.19.2 Subdiffusion and Fractality .................... 171
2.19.3 Comb Structures and Transport .................. 172
2.20 Continuous Time Random Walks .......................... 173
2.20.1 The Montroll and Weiss Approach and Memory
Effects ........................................ 174
2.20.2 Fractional Differential Equations .............. 176
2.20.3 The Taylor Definition and Memory Effects ....... 177
2.21 Fractional Differential Equations and Scalings ........ 180
2.21.1 The Klafter, Blumen, and Shlesinger
Approximation .................................. 180
2.21.2 The Stochastic Magnetic Field and Balescu
Approach ....................................... 181
2.21.3 Longitudinal Correlations and the Diffusive
Approximation .................................. 183
2.21.4 Vortex Structures and Trapping ................. 185
2.21.5 Correlations and Trapping ...................... 187
2.22 Correlation and Phase-Space ........................... 189
2.22.1 The Corrsin Conjecture and Phase-Space ......... 189
2.22.2 The Hamiltonian Nature of the Universal Hurst
Exponent ....................................... 191
2.22.3 The One-Flight Model and Transport ............. 193
2.22.4 Correlations and Nonlocal Velocity
Distribution ................................... 195
2.22.5 The Arrhenius Law and Phase-Space
Distribution ................................... 197
2.23 Conclusion ........................................... 198
Acknowledgements .............................................. 198
References .................................................... 198
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