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Table of Contents
Intro
Preface
Organization of this book
Acknowledgements
References
Author biographies
Boudewijn van Milligen
Raul Sanchez
Chapter 1 Introduction
1.1 Energy policy options
1.2 Fusion
1.3 Magnetic confinement
1.4 Fusion plasmas: a special class of systems
1.5 Transport in fusion plasmas
1.6 How important is turbulence?
1.7 Unusual transport phenomena
1.7.1 Power degradation and size scaling
1.7.2 Profile consistency and stiffness
1.7.3 Rapid transport phenomena and uphill transport
1.7.4 Zonal flows
1.7.5 The impact of rational surfaces on transport
1.8 Summary
References
Chapter 2 Characterization of turbulence in plasmas
2.1 Probability distribution functions and statistical moments
2.1.1 The PDF of particle transport
2.1.2 Self-similarity of the flux PDF
2.2 Correlation
2.2.1 Long range correlations
2.2.2 Filaments and zonal flows
2.3 Conditional averages
2.4 Spectra
2.4.1 Turbulent frequency spectra
2.4.2 Turbulent wavenumber spectra
2.5 Structure detection: the biorthogonal decomposition
2.5.1 Covariance and the biorthogonal decomposition
2.5.2 Correlation and the biorthogonal decomposition
2.5.3 Mode identification with the biorthogonal decomposition
2.5.4 Zonal flow detection in the TJ-II stellarator
2.6 Wave interactions: the bicoherence
2.6.1 Fourier bicoherence
2.6.2 Bicoherence and confinement transitions at TJ-II
2.6.3 Wavelet bicoherence
2.6.4 Bicoherence and L-H transitions
2.6.5 Non-linear energy transfer
2.7 Summary
References
Chapter 3 Complex features of plasma turbulence
3.1 Introduction
3.2 Self-similarity in random processes
3.2.1 Self-similar time processes
3.2.2 Fractional Brownian motion
3.2.3 Fractional Lévy motion.
3.2.4 Stationary versus self-similar, time-invariant processes
3.2.5 Determining self-similarity from moments
3.2.6 Multifractal analysis
3.3 The search for self-similarity in fusion plasmas
Illustration using probe data from the edge of the W7-AS stellarator
Applications in fusion plasmas
3.4 Long-term memory in time-series
3.4.1 Memory and the autocorrelation function
3.4.2 Memory and the power spectrum
3.4.3 Memory in self-similar random processes
3.4.4 The rescaled range method
3.5 The search for long-term memory in fusion plasmas
Illustration using probe data measured at the edge of the W7-AS stellarator
Selected investigations on fusion plasmas
3.6 Characterization of self-similarity and memory via event burst sampling
3.6.1 Self-similarity of events and criticality
3.6.2 Event correlation: waiting times
3.7 Burst sampling in fusion plasmas
Selected studies in fusion plasmas
3.8 Summary
References
Chapter 4 Causality
4.1 Antecedents
4.2 The quantification of information
4.3 The transfer entropy
4.4 Tests of the transfer entropy
4.4.1 Van der Pol oscillators
4.4.2 Simplified predator-prey model
4.5 Unravelling relations between turbulent variables
4.5.1 Zonal flows in a low field stellarator
4.5.2 Magnetic fluctuations and L-H transitions
4.5.3 The flux-gradient relation
4.6 Understanding radial transport
4.6.1 Heat transport in the TJ-II stellarator
4.6.2 Power degradation in the TJ-II stellarator
4.6.3 Heat transport in the W7-X stellarator
4.6.4 Heat transport in the JET tokamak
4.7 Modelling
4.8 Summary
References
Chapter 5 Intermittence
5.1 Antecedents
5.2 The intermittence parameter
5.3 Numerical study of intermittence in a turbulence model
5.4 Intermittence in the W7-X stellarator.
5.5 Intermittence in the TJ-II plasma edge
5.6 Intermittence in the TJ-II plasma core
5.6.1 Spatiotemporal mapping of turbulent structures
5.6.2 Topology of the turbulent structures
5.6.3 Comparison with model calculations
5.6.4 Summary
5.7 Intermittence and causality during confinement transitions at TJ-II
5.8 Intermittence and biasing at TJ-II
5.9 Summary
References
Chapter 6 Closing
6.1 Lessons learned
6.1.1 The need for data
6.1.2 Advanced analysis techniques
6.1.3 Turbulence and structures
6.1.4 The interaction between turbulence and transport
6.1.5 Remarks on the analysis and interpretation of turbulence simulations
6.2 Implications
6.2.1 The global picture of turbulence in fusion plasmas
6.2.2 Implications for our understanding of transport
6.3 Open questions and future perspectives
6.3.1 Expansion of studies
6.3.2 Turbulence control
References
Chapter
A.1 TJ-K
A.2 TJ-II
A.3 W7-AS
A.4 W7-X
A.5 JET
References
Chapter
References.
Preface
Organization of this book
Acknowledgements
References
Author biographies
Boudewijn van Milligen
Raul Sanchez
Chapter 1 Introduction
1.1 Energy policy options
1.2 Fusion
1.3 Magnetic confinement
1.4 Fusion plasmas: a special class of systems
1.5 Transport in fusion plasmas
1.6 How important is turbulence?
1.7 Unusual transport phenomena
1.7.1 Power degradation and size scaling
1.7.2 Profile consistency and stiffness
1.7.3 Rapid transport phenomena and uphill transport
1.7.4 Zonal flows
1.7.5 The impact of rational surfaces on transport
1.8 Summary
References
Chapter 2 Characterization of turbulence in plasmas
2.1 Probability distribution functions and statistical moments
2.1.1 The PDF of particle transport
2.1.2 Self-similarity of the flux PDF
2.2 Correlation
2.2.1 Long range correlations
2.2.2 Filaments and zonal flows
2.3 Conditional averages
2.4 Spectra
2.4.1 Turbulent frequency spectra
2.4.2 Turbulent wavenumber spectra
2.5 Structure detection: the biorthogonal decomposition
2.5.1 Covariance and the biorthogonal decomposition
2.5.2 Correlation and the biorthogonal decomposition
2.5.3 Mode identification with the biorthogonal decomposition
2.5.4 Zonal flow detection in the TJ-II stellarator
2.6 Wave interactions: the bicoherence
2.6.1 Fourier bicoherence
2.6.2 Bicoherence and confinement transitions at TJ-II
2.6.3 Wavelet bicoherence
2.6.4 Bicoherence and L-H transitions
2.6.5 Non-linear energy transfer
2.7 Summary
References
Chapter 3 Complex features of plasma turbulence
3.1 Introduction
3.2 Self-similarity in random processes
3.2.1 Self-similar time processes
3.2.2 Fractional Brownian motion
3.2.3 Fractional Lévy motion.
3.2.4 Stationary versus self-similar, time-invariant processes
3.2.5 Determining self-similarity from moments
3.2.6 Multifractal analysis
3.3 The search for self-similarity in fusion plasmas
Illustration using probe data from the edge of the W7-AS stellarator
Applications in fusion plasmas
3.4 Long-term memory in time-series
3.4.1 Memory and the autocorrelation function
3.4.2 Memory and the power spectrum
3.4.3 Memory in self-similar random processes
3.4.4 The rescaled range method
3.5 The search for long-term memory in fusion plasmas
Illustration using probe data measured at the edge of the W7-AS stellarator
Selected investigations on fusion plasmas
3.6 Characterization of self-similarity and memory via event burst sampling
3.6.1 Self-similarity of events and criticality
3.6.2 Event correlation: waiting times
3.7 Burst sampling in fusion plasmas
Selected studies in fusion plasmas
3.8 Summary
References
Chapter 4 Causality
4.1 Antecedents
4.2 The quantification of information
4.3 The transfer entropy
4.4 Tests of the transfer entropy
4.4.1 Van der Pol oscillators
4.4.2 Simplified predator-prey model
4.5 Unravelling relations between turbulent variables
4.5.1 Zonal flows in a low field stellarator
4.5.2 Magnetic fluctuations and L-H transitions
4.5.3 The flux-gradient relation
4.6 Understanding radial transport
4.6.1 Heat transport in the TJ-II stellarator
4.6.2 Power degradation in the TJ-II stellarator
4.6.3 Heat transport in the W7-X stellarator
4.6.4 Heat transport in the JET tokamak
4.7 Modelling
4.8 Summary
References
Chapter 5 Intermittence
5.1 Antecedents
5.2 The intermittence parameter
5.3 Numerical study of intermittence in a turbulence model
5.4 Intermittence in the W7-X stellarator.
5.5 Intermittence in the TJ-II plasma edge
5.6 Intermittence in the TJ-II plasma core
5.6.1 Spatiotemporal mapping of turbulent structures
5.6.2 Topology of the turbulent structures
5.6.3 Comparison with model calculations
5.6.4 Summary
5.7 Intermittence and causality during confinement transitions at TJ-II
5.8 Intermittence and biasing at TJ-II
5.9 Summary
References
Chapter 6 Closing
6.1 Lessons learned
6.1.1 The need for data
6.1.2 Advanced analysis techniques
6.1.3 Turbulence and structures
6.1.4 The interaction between turbulence and transport
6.1.5 Remarks on the analysis and interpretation of turbulence simulations
6.2 Implications
6.2.1 The global picture of turbulence in fusion plasmas
6.2.2 Implications for our understanding of transport
6.3 Open questions and future perspectives
6.3.1 Expansion of studies
6.3.2 Turbulence control
References
Chapter
A.1 TJ-K
A.2 TJ-II
A.3 W7-AS
A.4 W7-X
A.5 JET
References
Chapter
References.