Preface .................................................... xvii
Acknowledgments ............................................. xix
PART I GENERAL INTRODUCTION .................................... 1
1 Introductory Remarks .......................................... 3
2 Structure Formation in Fluids and Plasmas ..................... 6
2.1 Flow in a Pipe ............................................ 6
2.1.1 Enhancement of Mixing Effects Due to Turbulence ..... 6
2.1.2 Mean-Flow Structure Formation in Pipe Flows ......... 8
2.2 Magnetic-Field Generation by Turbulent Motion ............. 9
2.3 Collimation of Jets ...................................... 13
2.4 Magnetic Confinement of Plasmas .......................... 15
2.4.1 Magnetic Confinement and Toroidal Plasmas .......... 15
2.4.2 Flows in Toroidal Plasmas .......................... 17
2.4.3 Topological Change of Magnetic Surfaces ............ 17
2.5 Nonlinearity in Transport and Structural Transition ...... 18
2.5.1 Nonlinear Gradient - Flux Relation ................. 18
2.5.2 Bifurcation in Flow ................................ 19
2.5.3 Bifurcation in Structural Formation ................ 20
References ................................................... 23
PART II FLUID TURBULENCE ...................................... 25
Nomenclature ................................................. 26
3 Fundamentals of Fluid Turbulence ............................. 28
3.1 Fundamental Equations .................................... 28
3.2 Averaging Procedures ..................................... 31
3.3 Ensemble-Mean Equations .................................. 32
3.3.1 Mean-Field Equations ............................... 32
3.3.2 Turbulence Equations ............................... 33
3.4 Homogeneous Turbulence ................................... 41
3.4.1 Fundamental Concepts ............................... 42
3.4.2 Kolmogorov's Scaling Law ........................... 48
3.4.3 Failure of KoImogorov's Sealing .................... 52
3.4.4 Two-Dimensional Turbulence ......................... 55
3.5 Production and Diffusion Characteristics of Turbulent
Energy ................................................... 56
References ................................................... 59
4 Heuristic Turbulence Modelling ............................... 60
4.1 Approaches to Turbulence ................................. 60
4.2 Algebraic Turbulence Modelling ........................... 62
4.2.1 Modelling of Reynolds Stress ....................... 62
4.2.2 Modelling of Heat Flux ............................. 65
4.2.3 Modelling of Turbulence Equations .................. 66
4.2.4 The Simplest Algebraic Model ....................... 71
4.2.5 Investigation into Some Representative Turbulent
Flows .............................................. 72
4.3 Second-Order Modelling ................................... 79
4.3.1 Modelling of Pressure-Strain Term .................. 79
4.3.2 Modelling of Dissipation and Transport Terms ....... 80
4.3.3 The Simplest Second-Order Model and its
Relationship with a Higher-Order Algebraic Model ... 81
4.4 A Variational-Method Model ............................... 82
4.4.1 Helicity and Vortical-Structure Persistence ........ 83
4.4.2 Derivation of the Vorticity Equation Using
the Variational Method ............................. 84
4.4.3 Analysis of Swirling Pipe Flow ..................... 85
4.4.4 Swirl Effect on Reynolds Stress .................... 88
4.5 Subgrid-Scale Modelling .................................. 90
4.5.1 Filtering Procedure ................................ 90
4.5.2 Filtered Equations ................................. 92
4.5.3 Fixed-Parameter Modelling .......................... 94
4.5.4 Dynamic Model ...................................... 99
References .................................................. 102
5 Statistical Theory of Fluid Turbulence ...................... 104
5.1 Mathematical Methods Necessary for Turbulence Theory .... 104
5.1.1 Partial Summation of Infinite Series .............. 104
5.1.2 Gaussian Distribution Function .................... 105
5.1.3 Solution of Differential Equation Using Method
of Partial Summation .............................. 108
5.2 Theoretical Approach to Inhomogcneous Turbulence ........ 110
5.2.1 Perturbational Method to Turbulence ............... 111
5.2.2 Introduction of Green's Function .................. 115
5.2.3 Statistical Evaluation of Reynolds Stress ......... 125
5.3 Contributions to Turbulence Modelling ................... 133
5.3.1 Modelling of the Turbulent-Energy Equation ........ 133
5.3.2 Modelling of the Mach-Number Effect ............... 135
References .................................................. 141
PART III MAGNETOHYDRODYNAMIC TURBULENCE: DYNAMO .............. 143
Nomenclature ................................................ 144
6 Fundamentals of Mean-Field Theory of Dynamo ................. 146
6.1 One-Fluid Magnetohydrodynamic Approximation ............. 146
6.1.1 Fundamental Equations ............................. 146
6.1.2 Nondimensional Parameters Characterizing Flows .... 150
6.1.3 Elsasser's Variables and Conservation
Properties ........................................ 153
6.2 Cowling's Anti-Dynamo Theorem ........................... 154
6.3 Mean-Field Equations .................................... 156
6.4 Turbulence Equations .................................... 158
References .................................................. 160
7 Theoretical Estimate of Turbulence Effects on
Magnetic-Field Equations .................................... 161
7.1 Kinematic Method ........................................ 161
7.1.1 Introduction of Two Scales and Scale-Parameter
Expansion ......................................... 161
7.1.2 Evaluation of Turbulent Electromotive Force ....... 163
7.1.3 Evaluation of Reynolds Stress ..................... 166
7.2 Counter-Kinematic Method ................................ 167
7.2.1 Scale-Parameter Expansion ......................... 167
7.2.2 Evaluation of Turbulent Electromotive Force ....... 170
7.2.3 Evaluation of Reynolds Stress ..................... 171
7.3 Discussions on Dynamo Effects from Kinematic and
Counter-Kinematic Methods ............................... 172
7.3.1 Mathematical Features of Obtained Expressions ..... 172
7.3.2 Physical Meanings of Obtained Expressions ......... 173
7.4 Magnetohydrodynamic Method .............................. 176
7.4.1 Elsasser's Variables and Two-Scale Description .... 176
7.4.2 Perturbational Solution ........................... 178
7.4.3 Evaluation of Elsasser's Reynolds Stress .......... 181
7.4.4 Comparison with Kinematic and Counter-Kinematic
Methods ........................................... 183
References .................................................. 186
8 One-Point Dynamo Modelling with Emphasis on
Self-Consistency ............................................ 187
8.1 Necessity and Significance of One-Point Modelling ....... 187
8.2 Modelling Policy and Procedures ......................... 188
8.3 Summary of Dynamo Model ................................. 191
8.3.1 System of Model Equations ......................... 192
8.3.2 Model Constants ................................... 194
8.3.3 Remarks on Characteristic Time Scales ............. 195
References ................................................. 197
9 Typical Magnetic-Field Generation Processes ................. 198
9.1 Dominant-Helicity Dynamo ................................ 198
9.1.1 Convection Columns and Helicity ................... 198
9.1.2 Mean-Field Equations .............................. 199
9.1.3 Turbulence Equations .............................. 201
9.2 Dominant/Cross-Helicity Slate ........................... 203
9.2.1 Mean-Field Equations .............................. 203
9.2.2 Turbulence Equations .............................. 205
9.3 Traditional Kinematic Dynamos ........................... 206
9.3.1 Alpha Alpha Dynamo ............................... 207
9.3.2 Alpha Omega Dynamo ............................... 208
References ................................................. 209
10 Application to Astro/Geophysical and Fusion Dynamos ........ 210
10.1 Solar Magnetic Fields ................................. 210
10.1.1 Sunspot's Magnetic Field ....................... 210
10.1.2 Relationship of Sunspot's Polarity with Polar
Field .......................................... 212
10.1.3 Lorentz Force and Meridional Flow .............. 213
10.1.4 Mean-Field-Theory Interpretation of Polarity
Reversal ....................................... 214
10.2 Geomagnetic Fields .................................... 215
10.2.1 Computer SimuIation of Geodynamo ............... 215
10.2.2 Saturation of Generated Magnetic Field ......... 216
10.2.3 Frame-Rotation Effect on Magnetic Field ........ 218
10.3 Collimation of Accretion-Disc Jets .................... 220
10.3.1 Computer Simulation and Mean-Field Theory ...... 220
10.3.2 Driving Force of Bipolar Jets .................. 220
10.3.3 Collimation Mechanism Due to Magnetic Effect ... 222
10.3.4 Sustainment of Turbulent State ................. 224
10.3.5 Physical Interpretation of Jet Collimation ..... 226
10.4 Reversed-Field Pinches of Plasmas ..................... 226
10.4.1 Magnetic Plasma Confinement in a Torus ......... 226
10.4.2 Derivation of Force-Free Field by Mean-Field
Theory ......................................... 228
10.4.3 Derivation of Force-Free Field by Variational
Method ......................................... 229
10.5 Plasma Rotation in Tokamaks ........................... 230
10.6 Transport Suppression Due to Electric-Field Effects ... 233
10.6.1 Equations with Electric-Field Effects
Supplemented ................................... 233
10.6.2 Analysis of Turbulent Transport Rate of
Thermal Energy ................................. 234
10.6.3 Effect of Radial Electric Field on
Thermal-Energy Transport ....................... 235
References ................................................. 237
PART IV PLASMA TURBULENCE .................................... 239
Nomenclature ............................................... 241
11 Equations for Plasmas ...................................... 244
11.1 Fluid Equations ....................................... 244
11.2 Reduced Set of Equations .............................. 245
11.2.1 Yagi Morton Equations ......................... 246
11.2.2 Hasegawa-Mima Equation ......................... 248
11.2.3 Hasegawa Wakalani Equations ................... 249
11.2.4 Reduced MHD Equations .......................... 250
11.3 Reduced Set of Equations and Conservation Properly .... 250
11.3.1 Hasegawa Mima Equation ......................... 251
11.3.2 Three-Field Equations .......................... 252
11.3.3 Yagi Horton Equations .......................... 253
11.3.4 Dissipation and Transport Flux ................. 253
11.4 Kinetic Equation ...................................... 255
11.4.1 Vlasov Equation ................................ 255
11.4.2 Gyro-Averaged Equations ........................ 255
Appendix 11A Relations in Thermodynamics and Mean-Field
Equation ..................................... 256
References ................................................. 256
12 Inhomogeneity and Modes in Plasmas ......................... 258
12.1 Linear Mode ........................................... 258
12.1.1 Dispersion Relation ............................ 258
12.1.2 Vlasov Equation and Linear Dielectric Tensor ... 259
12.2 Examples of Modes ..................................... 261
12.2.1 Ion Sound Wave, Drift Wave and Convective
Cell ........................................... 261
12.2.2 Shear Alfven Wave and Drift Alfven Mode ........ 263
12.2.3 Interchange Mode ............................... 263
12.2.4 Ion Temperature Gradient Mode .................. 264
12.2.5 Dissipative Drift Mode ......................... 264
12.3 Weak Turbulence Theory ................................ 264
12.3.1 Ansatz of Weak Turbulence ...................... 264
12.3.2 Wave Kinetic Equation .......................... 265
12.3.3 Integral, Lyapunov Function and
Thermodynamics ................................. 266
12.4 Transport Matrix and Symmetry ......................... 268
Appendix 12A Quasilinear Theory of Transport .............. 269
References ................................................. 272
13 Inhomogeneous Strong Turbulence ............................ 273
13.1 Regime of Strong Plasma Turbulence .................... 273
13.2 Concepts to Describe Inhomogeneous Turbulent
Plasmas ............................................... 274
13.2.1 Gradients (Magnetic Surface, Shear, etc.) ...... 274
13.2.2 Mode, Wave, and Vortex ......................... 276
13.2.3 Propagating Solitary Structure ................. 277
13.2.4 Convective Cell, Zonal Flow and Streamer ....... 277
13.2.5 Reconneclion, Island Overlapping, Braiding,
and Mixing ..................................... 278
13.2.6 Plume and Avalanche (Time Intermittence) ....... 279
13.2.7 Clumps ......................................... 279
13.3 Microscale and Mesoscale Structures and Competition ... 280
Appendix 13A Clumps ....................................... 281
References ................................................. 282
14 Method for Strong Turbulence I. Renormalization and
Statistical Method ......................................... 284
14.1 Resonance Broadening and Renormalization in
the Kinetic Propagator ................................ 284
14.1.1 Renormalization of the Propagator .............. 284
14.1.2 Strong Turbulence Limit and Fluid Model ........ 286
14.1.3 Strong Turbulence Limit and Kubo Number ........ 287
14.2 Nonlinear Response in Fluid-Like Equations ............ 288
14.2.1 Short-Wavelength Fluctuations .................. 288
14.2.2 Rapidly-Changing, Long-Wavelength
Components ..................................... 289
14.2.3 Static but Sheared Flow ........................ 292
14.2.4 On Rigorous Upper Bound ........................ 292
14.3 Renormalization in a Reduced Set of (Fluid-Like)
Equations ............................................. 292
14.4 Randomness and the Statistical Picture ................ 295
14.4.1 Estimate of Random Source Term ................. 295
14.4.2 Dynamical Equations for Correlation
Functions ...................................... 296
14.4.3 Langevin Equations ............................. 297
14.4.4 Example of Three-Field Model ................... 298
14.5 Fokker Planck Equation ................................ 300
14.5.1 Projected Variable ............................. 300
14.5.2 Fokker-Planck Equation ......................... 300
14.5.3 Equilibrium Probability Density Function ....... 300
14.5.4 H-Theorem ...................................... 300
14.5.5 Tail in Probability Density .................... 301
14.6 Memory Ellects and Non-Markovian Property ............. 301
Appendix 14A Rigorous Upper Bounds for Transport .......... 303
References ................................................. 305
15 Methods for Strong Turbulence II. Scale Invariance
Method ..................................................... 307
15.1 Fluid Models .......................................... 307
15.1.1 Reynolds Number and Drag ....................... 307
15.1.2 Spectrum ....................................... 308
15.2 Plasma Models ......................................... 309
15.2.1 Transport Coefficient .......................... 309
15.2.2 Spectrum ....................................... 311
References ................................................. 311
16 Methods for Strong Turbulence III. Model Based on Reduced
Variables .................................................. 313
16.1 Lorenz Model .......................................... 313
16.2 Shell Model ........................................... 314
16.2.1 One-Dimensional Model .......................... 314
16.2.2 Multiple-Bin Model ............................. 316
16.3 K-ε Model ............................................. 317
16.4 Mapping Models ........................................ 317
16.4.1 Standard Map ................................... 318
16.4.2 Other Maps ..................................... 318
References ................................................. 319
17 Inhomogeneity-Driven Turbulence ............................ 320
17.1 Typical Examples ...................................... 320
17.1.1 Dissipalive Interchange Mode ................... 320
17.1.2 Ion Temperature Gradient (ITG) Mode ............ 323
17.1.3 Electron Temperature Gradient (ETG) Mode ....... 324
17.1.4 Kinetic Instabilities .......................... 325
17.2 Influence of Magnetic Field Structure ................. 326
17.2.1 Drift Due to the Magnetic Field Gradient ....... 326
17.2.2 Trapped Particle Instability ................... 327
17.2.3 Toroidal Ion Temperature Gradient (ITG) Mode ... 328
17.2.4 Current-Diffusive Ballooning Mode (CDBM)
Turbulence ..................................... 329
References ................................................. 329
18 Global Flow Driven by Turbulence ........................... 331
18.1 E x B Transport and Magnetic Transport ................ 331
18.1.1 E x B Transport ................................ 331
18.1.2 Magnetic Braiding and Transport ................ 332
18.2 Heat Flux ............................................. 333
18.2.1 ITG Mode Turbulence ............................ 334
18.2.2 CDIM Turbulence ................................ 335
18.2.3 ETG Mode Turbulence ............................ 336
18.2.4 Low or Negative Magnetic Shear ................. 336
18.3 Momentum Flux and Reynolds Stress ..................... 337
18.3.1 Anomalous Viscosity and Spontaneous Torque ..... 337
18.3.2 Excitation of Convective Cell
(Zonal Flow and Streamer) ...................... 338
18.4 Resistivity and Current Diffusivity ................... 340
References ................................................. 340
19 Generation of Structure in Flow ............................ 342
19.1 Breakdown of Ambipolarity of Turbulent Flow ........... 342
19.2 Generation of Zonal Flow by Drift Wave Turbulence ..... 343
19.3 Generation of Poloidal Flow by Collisional
Processes ............................................. 344
19.4 Electric Field Domain Interface ....................... 345
19.4.1 Domain and Domain Interface .................... 345
19.4.2 Kink-Soliton-Like Structure in Zonal Flows ..... 347
19.4.3 Soliton-Like Structure ......................... 348
19.4.4 Poloidal Shock ................................. 350
19.5 Streamer Formation .................................... 350
Appendix 19A Maxwell's Construction and Domain
Interface .................................... 351
19A.1 Nonlinear Diffusion Equation of
Radial Electric Field .................. 352
19A.2 Local Solution ......................... 352
19A.3 Electric Field Domain and Domain
Interface .............................. 353
19A.4 Structure of the Domain Interface ...... 354
19A.5 Relaxation of the Interlace ............ 355
19A.6 Solitary Radial Electric Field ......... 358
References ................................................. 360
20 Flow-Shear Suppression ..................................... 362
20.1 Effect of Flow Shear on Linear Stability .............. 362
20.1.1 Linear Stability in Fluid Dynamics ............. 362
20.1.2 Linear Stability in Plasma Dynamics ............ 363
20.2 Suppression of Turbulence ............................. 365
20.2.1 Decorrelation Rate ............................. 365
20.2.2 Turbulence Level and Turbulent Transport ....... 367
Appendix 20A Effect of Radial Electric Field
Inhomogeneity on Domain Interface ............ 368
References ................................................. 370
21 Subcritical Excitation ..................................... 372
21.1 Subcritical Excitation in Neutral Fluid ............... 372
21.1.1 Nonlinear Marginal Stability Condition ......... 372
21.1.2 Self-Sustaining Mechanism ...................... 373
21.2 Subcritical Excitation in Plasma Turbulence ........... 375
21.2.1 Current-Diffusive Interchange Mode
Turbulence ..................................... 375
21.2.2 Nonlinear Drift Instabilities .................. 376
21.2.3 Tearing Mode at High Pressure Gradient ......... 378
21.2.4 Turbulence-Turbulence Transition
(M-Mode Transition) ............................ 381
21.3 Abrupt Transition ..................................... 382
21.3.1 Microscopic Turbulence and Transport
Coefficient .................................... 382
21.3.2 MHD Modes ...................................... 382
21.4 Bubble Formation and Suppression by Shear Flow ........ 383
References ................................................. 384
22 Bifurcation ................................................ 386
22.1 System with Hysteresis ................................ 387
22.1.1 Dynamical Model Equations for Structural
Transition ..................................... 388
22.1.2 Nonlincarity in Gradient-Flux Relation ......... 388
22.1.3 Simultaneous Evolution of Fluctuation, Flow
and Gradient ................................... 391
22.2 Self-Organized Dynamics ............................... 393
22.2.1 Dithering ELMs ................................. 393
22.2.2 Giant ELMs ..................................... 393
References ................................................. 395
23 Statistical Picture of Bifurcation ......................... 397
23.1 Statistical Approaches for Bifurcation of
Turbulence ............................................ 397
23.1.1 Fluctuation Dissipation Relation from
Stochastic Equation ............................ 397
23.1.2 Fokker Planck Equation for Macrovariable
(Coarse-Grained Quantity) ...................... 399
23.1.3 Steady-State Probability Density Function ...... 400
23.2 Bifurcation Between Thermodymimical and Turbulent
Fluctuations .......................................... 400
23.2.1 Example of CDIM and Extended FD Relation ....... 400
23.2.2 Phase Diagram for Thermodynamical and
Turbulent Fluctuations ......................... 402
23.2.3 Example of CDIM and Fokker-Planck Equation ..... 403
23.3 Bifurcation Between Multiple Scale Length
Turbulences ........................................... 407
23.3.1 Scale Separation ............................... 407
23.3.2 Extended Fluctuation Dissipation Relation ...... 408
23.3.3 Example of the ITG and the CDIM Turbulence ..... 410
23.3.4 Bifurcation of Turbulence with Different
Scale Lengths .................................. 411
Appendix 23A Thermodynamical Equilibrium Limit ............ 414
References ................................................. 416
24 Transition Probability ..................................... 417
24.1 Transition by Noise ................................... 417
24.1.1 Rate Equation and Transition Probability ....... 418
24.1.2 Flux of Probability and Probability of
Transition ..................................... 419
24.1.3 Transition Probability ......................... 421
24.2 Transition Between Thermodymimical and Turbulent
Fluctuations .......................................... 421
24.2.1 Transition from Thermodynamical Fluctuations ... 421
24.2.2 Thermodynamical Equilibrium Limit .............. 422
24.2.3 Transition to Turbulent Fluctuations ........... 422
24.2.4 Back Transition Probability .................... 423
24.2.5 Example of Strong Turbulence ................... 425
24.3 Phase Boundary in Statistical Theory .................. 426
24.3.1 Phase Boundary ................................. 426
24.3.2 Averaging Time and Observation of Hysteresis ... 427
24.4 Probabilistic Transition .............................. 429
References ................................................. 430
25 Transient Response and Transport ........................... 431
25.1 Long Scale Length of Fluctuations and Transient
Response .............................................. 431
25.2 Fluctuations with Long Correlation Length ............. 432
25.2.1 Statistical Noise Excitation ................... 433
25.2.2 Geometrical Effect and Long Correlation
Length ......................................... 434
25.2.3 Kubo Number and Effective Transient Transport
Coefficient .................................... 435
25.3 Memory Effects ........................................ 436
25.4 Fast Propagation of Bump .............................. 437
25.5 Plume, Avalanche and Self-Organized Critically ........ 437
Appendix 25A Nonlocal Transport and Transient Response .... 439
References ................................................. 441
26 Thermodynamical Equilibrium Fluctuations and Far
Nonequilibrium Turbulence .................................. 443
26.1 Thermodynamical Equilibrium ........................... 443
26.1.1 Neutral Fluid .................................. 443
26.1.2 Description by the Hasegawa Mima Equation ..... 444
26.1.3 Description by the Hasegawa-Wakatani
Equations ...................................... 444
26.1.4 Representation by Use of Beltrami Functions .... 445
26.1.5 Correlation Functions and Plasma Property ...... 445
26.2 Nonequilibrium Property and Intermittency ............. 446
26.3 Comparison of Cases for Strong Turbulence and
Thermodynamical Equilibrium ........................... 448
References ................................................. 450
Summary .................................................... 451
Index ...................................................... 453
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