Plasma and Fluid Turbulence: Theory and Modelling explains modelling methodologies in depth with regard to turbulence phenomena and turbulent transport both in fluids and plasmas. Special attention is paid to structural formation and transitions. In this detailed book, the authors examine the underlying ideas describing turbulence, turbulent transport, and structural transitions in plasmas and fluids. By comparing and contrasting turbulence in fluids and plasmas, they demonstrate the basic physical principles common to fluids and plasmas while also highlighting particular differences. The book also discusses the application of these ideas to neutral fluids.

Akira Yoshizawa Institute of Industrial Science, University of Tokyo, Japan Sanae-I Itoh Research Institute for Applied Mechanics, Kyushu University, Japan Kimitaka Itoh National Institute for Fusion Science, Toki, Japan

Preface, Acknowledgments, PART I GENERAL INTRODUCTION, 1 Introductory Remarks, 2 Structure Formation in Fluids and Plasmas, 2.1 Flow in a Pipe, 2.1.1 Enhancement of Mixing Effects Due to Turbulence, 2.1.2 Mean-Flow Structure Formation in Pipe Flows, 2.2 Magnetic-Field Generation by Turbulent Motion, 2.3 Collimation of Jets, 2.4 Magnetic Confinement of Plasmas, 2.4.1 Magnetic Confinement and Toroidal Plasmas, 2.4.2 Flows in Toroidal Plasmas, 2.4.3 Topological Change of Magnetic Surfaces, 2.5 Nonlinearity in Transport and Structural Transition, 2.5.1 Nonlinear Gradient-Flux Relation, 2.5.2 Bifurcation in Flow, 2.5.3 Bifurcation in Structural Formation, References, PART II FLUID TURBULENCE, Nomenclature, 3 Fundamentals of Fluid Turbulence, 3.1 Fundamental Equations, 3.2 Averaging Procedures, 3.3 Ensemble-Mean Equations, 3.3.1 Mean-Field Equations, 3.3.2 Turbulence Equations, 3.4 Homogeneous Turbulence, 3.4.1 Fundamental Concepts, 3.4.2 Kolmogorov's Scaling Law, 3.4.3 Failure of Kolmogorov's Scaling, 3.4.4 Two-Dimensional Turbulence, 3.5 Production and Diffusion Characteristics of Turbulent Energy, References, 4 Heuristic Turbulence Modelling, 4.1 Approaches to Turbulence, 4.2 Algebraic Turbulence Modelling, 4.2.1 Modelling of Reynolds Stress, 4.2.2 Modelling of Heat Flux, 4.2.3 Modelling of Turbulence Equations, 4.2.4 The Simplest Algebraic Model, 4.2.5 Investigation into Some Representative Turbulent Flows, 4.3 Second-Order Modelling, 4.3.1 Modelling of Pressure-Strain Term, 4.3.2 Modelling of Dissipation and Transport Terms, 4.3.3 The Simplest Second-Order Model and its Relationship with a Higher-Order Algebraic Model, 4.4 A Variational-Method Model, 4.4.1 Helicity and Vortical-Structure Persistence, 4.4.2 Derivation of the Vorticity Equation Using the Variational Method, 4.4.3 Analysis of Swirling Pipe Flow, 4.4.4 Swirl Effect on Reynolds Stress, 4.5 Subgrid-Scale Modelling, 4.5.1 Filtering Procedure, 4.5.2 Filtered Equations, 4.5.3 Fixed-Parameter Modelling, 4.5.4 Dynamic Model, References, 5 Statistical Theory of Fluid Turbulence, 5.1 Mathematical Methods Necessary for Turbulence Theory, 5.1.1 Partial Summation of Infinite Series, 5.1.2 Gaussian Distribution Function, 5.1.3 Solution of Differential Equation Using Method of Partial Summation, 5.2 Theoretical Approach to Inhomogeneous Turbulence, 5.2.1 Perturbational Method to Turbulence, 5.2.2 Introduction of Green's Function, 5.2.3 Statistical Evaluation of Reynolds Stress, 5.3 Contributions to Turbulence Modelling, 5.3.1 Modelling of the Turbulent-Energy Equation, 5.3.2 Modelling of the Mach-Number Effect, References, PART III MAGNETOHYDRODYNAMIC TURBULENCE: DYNAMO, Nomenclature, 6 Fundamentals of Mean-Field Theory of Dynamo, 6.1 One-Fluid Magnetohydrodynamic Approximation, 6.1.1 Fundamental Equations, 6.1.2 Nondimensional Parameters Characterizing Flows, 6.1.3 Elsasser's Variables and Conservation Properties, 6.2 Cowling's Anti-Dynamo Theorem, 6.3 Mean-Field Equations, 6.4 Turbulence Equations, References, 7 Theoretical Estimate of Turbulence Effects on Magnetic-Field Equations, 7.1 Kinematic Method, 7.1.1 Introduction of Two Scales and Scale-Parameter Expansion, 7.1.2 Evaluation of Turbulent Electromotive Force, 7.1.3 Evaluation of Reynolds Stress, 7.2 Counter-Kinematic Method, 7.2.1 Scale-Parameter Expansion, 7.2.2 Evaluation of Turbulent Electromotive Force, 7.2.3 Evaluation of Reynolds Stress, 7.3 Discussions on Dynamo Effects from Kinematic and Counter-Kinematic Methods, 7.3.1 Mathematical Features of Obtained Expressions, 7.3.2 Physical Meanings of Obtained Expressions, 7.4 Magnetohydrodynamic Method, 7.4.1 Elsasser's Variables and Two-Scale Description, 7.4.2 Perturbational Solution, 7.4.3 Evaluation of Elsasser's Reynolds Stress, 7.4.4 Comparison with Kinematic and Counter-Kinematic Methods, References, 8 One-Point Dynamo Modelling with Emphasis on Self-Consistency, 8.1 Necessity and Significance of One-Point Modelling, 8.2 Modelling Policy and Procedures, 8.3 Summary of Dynamo Model, 8.3.1 System of Model Equations, 8.3.2 Model Constants, 8.3.3 Remarks on Characteristic Time Scales, References, 9 Typical Magnetic-Field Generation Processes, 9.1 Dominant-Helicity Dynamo, 9.1.1 Convection Columns and Helicity, 9.1.2 Mean-Field Equations, 9.1.3 Turbulence Equations, 9.2 Dominant/Cross-Helicity State, 9.2.1 Mean-Field Equations, 9.2.2 Turbulence Equations, 9.3 Traditional Kinematic Dynamos, 9.3.1 Alpha-Alpha Dynamo, 9.3.2 Alpha-Omega Dynamo, References, 10 Application to Astro/Geophysical and Fusion Dynamos, 10.1 Solar Magnetic Fields, 10.1.1 Sunspot's Magnetic Field, 10.1.2 Relationship of Sunspot's Polarity with Polar Field, 10.1.3 Lorentz Force and Meridional Flow, 10.1.4 Mean-Field-Theory Interpretation of Polarity Reversal, 10.2 Geomagnetic Fields, 10.2.1 Computer Simulation of Geodynamo, 10.2.2 Saturation of Generated Magnetic Field, 10.2.3 Frame-Rotation Effect on Magnetic Field, 10.3 Collimation of Accretion-Disc Jets, 10.3.1 Computer Simulation and Mean-Field Theory, 10.3.2 Driving Force of Bipolar Jets, 10.3.3 Collimation Mechanism Due to Magnetic Effect, 10.3.4 Sustainment of Turbulent State, 10.3.5 Physical Interpretation of Jet Collimation, 10.4 Reversed-Field Pinches of Plasmas, 10.4.1 Magnetic Plasma Confinement in a Torus, 10.4.2 Derivation of Force-Free Field by Mean-Field Theory, 10.4.3 Derivation of Force-Free Field by Variational Method, 10.5 Plasma Rotation in Tokamaks, 10.6 Transport Suppression Due to Electric-Field Effects, 10.6.1 Equations with Electric-Field Effects Supplemented, 10.6.2 Analysis of Turbulent Transport Rate of Thermal Energy, 10.6.3 Effect of Radial Electric Field on Thermal-Energy Transport, References, PART IV PLASMA TURBULENCE, Nomenclature, 11 Equations for Plasmas, 11.1 Fluid Equations, 11.2 Reduced Set of Equations, 11.2.1 Yagi-Horton Equations, 11.2.2 Hasegawa-Mima Equation, 11.2.3 Hasegawa-Wakatani Equations, 11.2.4 Reduced MHD Equations, 11.3 Reduced Set of Equations and Conservation Property, 11.3.1 Hasegawa-Mima Equation, 11.3.2 Three-Field Equations, 11.3.3 Yagi-Horton Equations, 11.3.4 Dissipation and Transport Flux, 11.4 Kinetic Equation, 11.4.1 Vlasov Equation, 11.4.2 Gyro-Averaged Equations, Appendix 11A Relations in Thermodynamics and Mean-Field Equation, References, 12 Inhomogeneity and Modes in Plasmas, 12.1 Linear Mode, 12.1.1 Dispersion Relation, 12.1.2 Vlasov Equation and Linear Dielectric Tensor, 12.2 Examples of Modes, 12.2.1 Ion Sound Wave, Drift Wave and Convective Cell, 12.2.2 Shear Alfve¿ n Wave and Drift Alfve¿ n Mode, 12.2.3 Interchange Mode, 12.2.4 Ion Temperature Gradient Mode, 12.2.5 Dissipative Drift Mode, 12.3 Weak Turbulence Theory, 12.3.1 Ansatz of Weak Turbulence, 12.3.2 Wave Kinetic Equation, 12.3.3 Integral, Lyapunov Function and Thermodynamics, 12.4 Transport Matrix and Symmetry, Appendix 12A Quasilinear Theory of Transport, References, 13 Inhomogeneous Strong Turbulence, 13.1 Regime of Strong Plasma Turbulence, 13.2 Concepts to Describe Inhomogeneous Turbulent Plasmas, 13.2.1 Gradients (Magnetic Surface, Shear, etc.), 13.2.2 Mode, Wave, and Vortex, 13.2.3 Propagating Solitary Structure, 13.2.4 Convective Cell, Zonal Flow and Streamer, 13.2.5 Reconnection, Island Overlapping, Braiding, and Mixing, 13.2.6 Plume and Avalanche (Time Intermittence), 13.2.7 Clumps, 13.3 Microscale and Mesoscale Structures and Competition, Appendix 13A Clumps, References, 14 Method for Strong Turbulence I. Renormalization and Statistical Method, 14.1 Resonance Broadening and Renormalization in the Kinetic Propagator, 14.1.1 Renormalization of the Propagator, 14.1.2 Strong Turbulence Limit and Fluid Model, 14.1.3 Strong Turbulence Limit and Kubo Number, 14.2 Nonlinear Response in Fluid-Like Equations, 14.2.1 Short-Wavelength Fluctuations, 14.2.2 Rapidly-Changing, Long-Wavelength Components, 14.2.3 Static but Sheared Flow, 14.2.4 On Rigorous Upper Bound, 14.3 Renormalization in a Reduced Set of (Fluid-Like) Equations, 14.4 Randomness and the Statistical Picture, 14.4.1 Estimate of Random Source Term, 14.4.2 Dynamical Equations for Correlation Functions, 14.4.3 Langevin Equations, 14.4.4 Example of Three-Field Model, 14.5 Fokker-Planck Equation, 14.5.1 Projected Variable, 14.5.2 Fokker-Planck Equation, 14.5.3 Equilibrium Probability Density Function, 14.5.4 H-Theorem, 14.5.5 Tail in Probability Density, 14.6 Memory Effects and Non-Markovian Property, Appendix 14A Rigorous Upper Bounds for Transport, References, 15 Methods for Strong Turbulence II. Scale Invariance Method, 15.1 Fluid Models, 15.1.1 Reynolds Number and Drag, 15.1.2 Spectrum, 15.2 Plasma Models, 15.2.1 Transport Coefficient, 15.2.2 Spectrum, References, 16 Methods for Strong Turbulence III. Model Based on Reduced Variables, 16.1 Lorenz Model, 16.2 Shell Model, 16.2.1 One-Dimensional Model, 16.2.2 Multiple-Bin Model, 16.3 K-¿ Model, 16.4 Mapping Models, 16.4.1 Standard Map, 16.4.2 Other Maps, References, 17 Inhomogeneity-Driven Turbulence, 17.1 Typical Examples, 17.1.1 Dissipative Interchange Mode, 17.1.2 Ion Temperature Gradient (ITG) Mode, 17.1.3 Electron Temperature Gradient (ETG) Mode, 17.1.4 Kinetic Instabilities, 17.2 Influence of Magnetic Field Structure, 17.2.1 Drift Due to the Magnetic Field Gradient, 17.2.2 Trapped Particle Instability, 17.2.3 Toroidal Ion Temperature Gradient (ITG) Mode, 17.2.4 Current-Diffusive Ballooning Mode (CDBM) Turbulence, References, 18 Global Flow Driven by Turbulence, 18.1 E B Transport and Magnetic Transport, 18.1.1 E B Transport, 18.1.2 Magnetic Braiding and Transport, 18.2 Heat Flux, 18.2.1 ITG Mode Turbulence, 18.2.2 CDIM Turbulence, 18.2.3 ETG Mode Turbulence, 18.2.4 Low or Negative Magnetic Shear, 18.3 Momentum Flux and Reynolds Stress, 18.3.1 Anomalous Viscosity and Spontaneous Torque, 18.3.2 Excitation of Convective Cell (Zonal Flow and Streamer), 18.4 Resistivity and Current Diffusivity, References, 19 Generation of Structure in Flow, 19.1 Breakdown of Ambipolarity of Turbulent Flow, 19.2 Generation of Zonal Flow by Drift Wave Turbulence, 19.3 Generation of Poloidal Flow by Collisional Processes, 19.4 Electric Field Domain Interface, 19.4.1 Domain and Domain Interface, 19.4.2 Kink-Soliton-Like Structure in Zonal Flows, 19.4.3 Soliton-Like Structure, 19.4.4 Poloidal Shock, 19.5 Streamer Formation, Appendix 19A Maxwell's Construction and Domain Interface, 19A.1 Nonlinear Diffusion Equation of Radial Electric Field, 19A.2 Local Solution, 19A.3 Electric Field Domain and Domain Interface, 19A.4 Structure of the Domain Interface, 19A.5 Relaxation of the Interface, 19A.6 Solitary Radial Electric Field, References, 20 Flow-Shear Suppression, 20.1 Effect of Flow Shear on Linear Stability, 20.1.1 Linear Stability in Fluid Dynamics, 20.1.2 Linear Stability in Plasma Dynamics, 20.2 Suppression of Turbulence, 20.2.1 Decorrelation Rate, 20.2.2 Turbulence Level and Turbulent Transport, Appendix 20A Effect of Radial Electric Field Inhomogeneity on Domain Interface, References, 21 Subcritical Excitation, 21.1 Subcritical Excitation in Neutral Fluid, 21.1.1 Nonlinear Marginal Stability Condition, 21.1.2 Self-Sustaining Mechanism, 21.2 Subcritical Excitation in Plasma Turbulence, 21.2.1 Current-Diffusive Interchange Mode Turbulence, 21.2.2 Nonlinear Drift Instabilities, 21.2.3 Tearing Mode at High Pressure Gradient, 21.2.4 Turbulence-Turbulence Transition (M-Mode Transition), 21.3 Abrupt Transition, 21.3.1 Microscopic Turbulence and Transport Coefficient, 21.3.2 MHD Modes, 21.4 Bubble Formation and Suppression by Shear Flow, References, 22 Bifurcation, 22.1 System with Hysteresis, 22.1.1 Dynamical Model Equations for Structural Transition, 22.1.2 Nonlinearity in Gradient-Flux Relation, 22.1.3 Simultaneous Evolution of Fluctuation, Flow and Gradient, 22.2 Self-Organized Dynamics, 22.2.1 Dithering ELMs, 22.2.2 Giant ELMs References, 23 Statistical Picture of Bifurcation, 23.1 Statistical Approaches for Bifurcation of Turbulence, 23.1.1 Fluctuation Dissipation Relation from Stochastic Equation, 23.1.2 Fokker-Planck Equation for Macrovariable (Coarse-Grained Quantity), 23.1.3 Steady-State Probability Density Function, 23.2 Bifurcation Between Thermodynamical and Turbulent Fluctuations, 23.2.1 Example of CDIM and Extended FD Relation, 23.2.2 Phase Diagram for Thermodynamical and Turbulent Fluctuations, 23.2.3 Example of CDIM and Fokker-Planck Equation, 23.3 Bifurcation Between Multiple Scale Length Turbulences, 23.3.1 Scale Separation, 23.3.2 Extended Fluctuation Dissipation Relation, 23.3.3 Example of the ITG and the CDIM Turbulence, 23.3.4 Bifurcation of Turbulence with Different Scale Lengths, Appendix 23A Thermodynamical Equilibrium Limit, References, 24 Transition Probability, 24.1 Transition by Noise, 24.1.1 Rate Equation and Transition Probability, 24.1.2 Flux of Probability and Probability of Transition, 24.1.3 Transition Probability, 24.2 Transition Between Thermodynamical and Turbulent Fluctuations, 24.2.1 Transition from Thermodynamical Fluctuations, 24.2.2 Thermodynamical Equilibrium Limit, 24.2.3 Transition to Turbulent Fluctuations, 24.2.4 Back Transition Probability, 24.2.5 Example of Strong Turbulence, 24.3 Phase Boundary in Statistical Theory, 24.3.1 Phase Boundary, 24.3.2 Averaging Time and Observation of Hysteresis, 24.4 Probabilistic Transition, References, 25 Transient Response and Transport, 25.1 Long Scale Length of Fluctuations and Transient Response, 25.2 Fluctuations with Long Correlation Length, 25.2.1 Statistical Noise Excitation, 25.2.2 Geometrical Effect and Long Correlation Length, 25.2.3 Kubo Number and Effective Transient Transport Coefficient, 25.3 Memory Effects, 25.4 Fast Propagation of Bump, 25.5 Plume, Avalanche and Self-Organized Criticality, Appendix 25A Nonlocal Transport and Transient Response, References, 26 Thermodynamical Equilibrium Fluctuations and Far Nonequilibrium Turbulence, 26.1 Thermodynamical Equilibrium, 26.1.1 Neutral Fluid, 26.1.2 Description by the Hasegawa-Mima Equation, 26.1.3 Description by the Hasegawa-Wakatani Equations, 26.1.4 Representation by Use of Beltrami Functions, 26.1.5 Correlation Functions and Plasma Property, 26.2 Nonequilibrium Property and Intermittency, 26.3 Comparison of Cases for Strong Turbulence and, Thermodynamical Equilibrium, References, Summary