PhD and MPhil Thesis Classes - Université Libre de Bruxelles

Transport Analysis in TokamakPlasmasSara MoradiSupervisors: Dr. Boris Weyssow and Dr. Daniele CaratiStatistical and Plasma PhysicsUniversité Libre de BruxellesA thesis submitted for the degree ofPhilosophiæDoctor (PhD)July 2010

AbstractIn this thesis we mainly focus on the study of the turbulent transport ofimpurity particles in the plasma due to the electrostatic drift wave microinstabilities.In a fusion reactor, the helium produced as a result of the fusionprocess is an internal source of impurity. Moreover, impurities are releasedfrom the material surfaces surrounding the plasma by a variety of processes:by radiation from plasma, or as a result of sputtering, arcing and evaporation.Impurities in tokamak plasmas introduce a variety of problems. Themost immediate effect is the radiated power loss (radiative cooling). Anothereffect is that the impurity ions produce many electrons and in view ofthe operating limits on density and pressure, this has the effect of replacingfuel ions. For example, at a given electron density, n e , each fully ionized carbonion (used in the wall materials in the form of graphite) replaces six fuelions, so that a 7% concentration of fully ionized carbon in the plasma core,would reduce the fusion power to one half of the value in a pure plasma.Therefore, for all tokamaks it become an immediate and continuing task toreduce impurities to acceptably low concentrations. However, the presenceof impurities, with control, can be beneficial for the plasma performanceand reduction of strong plasma heat loads on the plasma facing walls. Theradiative cooling effect which was mentioned above can be used at the edgeof the plasma in order to distribute the plasma heat more evenly on thewhole surface of the vessel walls and therefore, reduce significantly plasmaheat bursts on the small regions on the divertor or limiter tiles. The experimentsat TEXTOR show that the presence of the impurities at the plasmaedge can also improve the performance and reduce the turbulent transportacross the magnetic field lines. The observed behavior was explained troughthe proposed mechanism of suppression of the most important plasma driftwave microinstability in this region, namely, the Ion Temperature Gradient

mode (ITG mode) by the impurities. The impurity’s positive impact onthe plasma performance offered a possibility to better harness the fusionpower, however, it is vital for a fusion reactor to have feedback controls inorder to keep impurities at the plasma edge and limit their accumulation inthe plasma core where the fusion reactions are happening. In order to havecontrol over the impurity transport we first need to understand differentmechanisms responsible for its transport.One of the least understood areas of the impurity transport and indeedany plasma particle or heat transport in general, is the turbulent transport.Extensive efforts of the fusion plasma community are focused on thesubject of turbulent transport. Motivated by the fact that impurity transportis an important issue for the whole community and it is an area whichneeds fundamental research, we focused our attention on the developmentof turbulent transport models for impurities and their examination againstexperiments. In a collaboration effort together with colleagues (theoreticiansas well as experimentalist) from different research institutes, we triedto find, through our models, physical mechanisms responsible for experimentalobservations. Although our main focus in this thesis has been onthe impurity transport, we also tried a fresh challenge, and started lookingat the problem of drift wave turbulent transport in a different frameworkall together. Experimental observation of the edge turbulence in the fusiondevices show that in the Scrape of Layer (SOL: the layer between lastclosed magnetic surface and machine walls) plasma is characterized withnon-Gaussian statistics and non-Maxwellian Probability Distribution Function(PDF). It has been recognized that the nature of cross-field transporttrough the SOL is dominated by turbulence with a significant ballistic ornon-local component and it is not simply a diffusive process. There are studiesof the SOL turbulent transport using the 2-D fluid descriptions or basedon probabilistic models using the Levy statistics (fractional derivatives inspace). However, these models are base on the fluid assumptions whichis in contradiction with the non-Maxwellian plasmas observed. Therefore,we tried to make a more fundamental study by looking at the effect ofthe non-Maxwellian plasma on the turbulent transport using a gyro-kinetic

formalism. We considered the application of fractional kinetics to plasmaphysics. This approach, classical indeed, is new in its application. Our aimwas to study the effects of a non-Gaussian statistics on the characteristic ofthe drift waves in fusion plasmas.The results obtained during the course of this thesis are presented here.These results have been previously presented to the fusion plasma communityas conference presentations and publications.

I dedicate my thesis to maman, baba, Bart and Ehsan.

AcknowledgementsCompleting a PhD is truly a marathon event, and I would not have beenable to complete this journey without the aid and support of countless peopleover the past four years. It is a pleasure to convey my gratitude to themall in my humble acknowledgment.I must first express my gratitude towards my advisors, Professor BorisWeyssow and Professor Daniele Carati who were abundantly helpful andoffered invaluable assistance, support and guidance. Their expertises, understanding,and patience, added considerably to my graduate experience.Deepest gratitude are also due to Professor Mikhaeil Tokar, for his supervision,advice, and crucial contribution from the very early stage of thisresearch which made him a backbone of this thesis. Above all and the mostneeded, he provided me unflinching encouragement and support in variousways. His truly scientist intuition has made him as a constant oasis of ideasand passions in science, which exceptionally inspire and enrich my growthas a student, a researcher and a scientist want to be.I gratefully acknowledge Dr. Clarisse Bourdelle, without whose knowledgeand assistance this study would not have been successful. She always kindlygrants me her time even for answering some of my unintelligent questions.Her leadership, support, attention to detail and hard work have set an exampleI hope to match some day.Collective and individual acknowledgments are also owed to my fellow PhDstudents at ULB, for creating such a great friendship at the office, for our

philosophical debates, exchanges of knowledge, skills, and venting of frustrationduring my PhD, which helped enrich the experience.Appreciation also goes out to our secretaries, Mrs. Fabienne De Neyn andMrs. Marie-France Rogge for all of their administrative and technical assistancewhich helped me along the way.My parents deserve special mention for their inseparable support. My Fatherin the first place is the person who put the fundament of my learningcharacter, showing me the joy of intellectual pursuit ever since I was a child.My Mother is the one who sincerely raised me with her caring and gentlylove. Ehsan, thanks for being supportive and caring brother.Words fail me to express my appreciation to my husband Bart whose dedication,love and persistent confidence in me, has taken the load off myshoulder.I would like to thank everybody who was important to the successful realizationof this thesis, as well as expressing my apology that I could notmention them personally one by one.I would also like to convey thanks to the FNRS-FRIA of Belgium for providingthe financial means and the science faculty of the Universit Libre deBruxelles for providing the laboratory facilities.And finally, I would like to stress that I consider this thesis to be the startof a challenging research program, rather than my final say on this topic.

ContentsList of Figuresix1 Introduction 11.1 Towards a New Source of Energy . . . . . . . . . . . . . . . . . . . . . . 11.2 Fusion Reaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.3 The Necessary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . 31.4 Tokamak . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.5 Plasma Heating . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.6 Power Balance and Lawson Criteria . . . . . . . . . . . . . . . . . . . . 71.7 Plasma Operational Limits . . . . . . . . . . . . . . . . . . . . . . . . . 91.7.1 Ohmic plasma . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101.7.2 L-mode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111.7.3 H-mode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111.7.4 Global Instabilities . . . . . . . . . . . . . . . . . . . . . . . . . . 111.7.4.1 Sawtooth . . . . . . . . . . . . . . . . . . . . . . . . . . 121.7.4.2 Edge Localized Modes . . . . . . . . . . . . . . . . . . . 121.7.5 RI-mode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131.8 Mechanisms of Transport in Tokamaks . . . . . . . . . . . . . . . . . . . 151.8.1 Classical Transport . . . . . . . . . . . . . . . . . . . . . . . . . . 151.8.2 Debye Shielding . . . . . . . . . . . . . . . . . . . . . . . . . . . 161.8.3 Neo-classical Transport . . . . . . . . . . . . . . . . . . . . . . . 181.8.4 Anomalous Transport . . . . . . . . . . . . . . . . . . . . . . . . 191.9 Global Energy Confinement Scaling . . . . . . . . . . . . . . . . . . . . . 21v

CONTENTS2 Aims of the project 232.1 Aims . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232.2 Strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243 Drift Waves 253.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253.2 Hamiltonian Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . 263.2.1 Principle of Least Action and Equation of Motion . . . . . . . . 263.2.2 Maxwell-Lorentz Equations . . . . . . . . . . . . . . . . . . . . . 273.3 Charged Particle Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . 293.3.1 Equation of Motion . . . . . . . . . . . . . . . . . . . . . . . . . 303.3.2 Uniform E and B fields . . . . . . . . . . . . . . . . . . . . . . . 303.3.2.1 B = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313.3.2.2 E = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313.3.2.3 Particle Drift . . . . . . . . . . . . . . . . . . . . . . . . 333.3.2.4 Validity Limit . . . . . . . . . . . . . . . . . . . . . . . 373.3.3 Nonuniform E and B fields . . . . . . . . . . . . . . . . . . . . . 383.3.3.1 ∇B⊥B: Grad−B Drift . . . . . . . . . . . . . . . . . . 383.3.3.2 Curved B: Curvature Drift . . . . . . . . . . . . . . . . 393.3.3.3 Magnetic Mirrors . . . . . . . . . . . . . . . . . . . . . 403.3.4 Nonuniform E Field . . . . . . . . . . . . . . . . . . . . . . . . . 413.3.5 Summary of Guiding Center Drifts . . . . . . . . . . . . . . . . . 443.4 Fluid Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 443.4.1 Summary of Fluid Equations . . . . . . . . . . . . . . . . . . . . 473.4.2 Closure of Fluid Equations . . . . . . . . . . . . . . . . . . . . . 473.5 Transport Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 483.5.1 Experimental Aspects . . . . . . . . . . . . . . . . . . . . . . . . 483.6 Fluid Drifts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 493.6.1 Fluid Drifts ⊥B . . . . . . . . . . . . . . . . . . . . . . . . . . . 503.6.2 Fluid Motion ‖ B . . . . . . . . . . . . . . . . . . . . . . . . . . . 523.7 Drift Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 533.7.1 Drift Wave Propagation . . . . . . . . . . . . . . . . . . . . . . . 533.7.2 Drift Instability . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54vi

CONTENTS3.7.3 Drift Wave in Laboratory . . . . . . . . . . . . . . . . . . . . . . 553.8 Anomalous Transport due to Drift Waves . . . . . . . . . . . . . . . . . 553.8.1 Mixing Length Approximation . . . . . . . . . . . . . . . . . . . 574 Drift Instability Analysis: Linear Fluid Model 594.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 594.2 Basic Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 604.2.0.1 Collisional Effect On Trapped Electrons . . . . . . . . . 614.2.1 Compressibility of The Drift Velocity . . . . . . . . . . . . . . . . 624.2.2 Quasi-Neutrality Condition . . . . . . . . . . . . . . . . . . . . . 644.2.3 Linear Response . . . . . . . . . . . . . . . . . . . . . . . . . . . 654.2.4 Solution of Dispersion Equation . . . . . . . . . . . . . . . . . . . 664.3 Instability Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 674.3.1 ITG Instability Mechanism . . . . . . . . . . . . . . . . . . . . . 684.3.2 TE Instability Mechanism . . . . . . . . . . . . . . . . . . . . . . 694.3.3 ETG Instability Mechanism . . . . . . . . . . . . . . . . . . . . . 694.3.4 Numerical Computations . . . . . . . . . . . . . . . . . . . . . . 694.3.5 Computation of the Solutions of the Dispersion Equation . . . . 724.3.6 Impurity Effect on Drift Instabilities . . . . . . . . . . . . . . . . 785 Anomalous Transport due to Drift Wave Microinstabilities 875.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 875.2 Transport coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 875.3 Impurity Transport . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 925.3.1 Peaking Factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . 935.4 Importance of Collisions on Anomalous Transport of Impurity . . . . . . 965.5 Results of calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1005.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1066 Benchmark with a Quasi-linear Gyro-kinetic Model 1096.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1096.1.1 Dispersion Equation . . . . . . . . . . . . . . . . . . . . . . . . . 1136.2 QuaLiKiz code . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1146.3 Comparison With AFC-FL . . . . . . . . . . . . . . . . . . . . . . . . . 115vii

CONTENTS7 Transport Modeling of Impurity Seeded Experiments at JET 1177.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1177.2 Experimental observations . . . . . . . . . . . . . . . . . . . . . . . . . . 1197.3 Framework in which the fluid modeling is applied . . . . . . . . . . . . . 1247.4 Results of the modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . 1287.5 Predictive Transport Modeling . . . . . . . . . . . . . . . . . . . . . . . 1337.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1378 Study of Drift Wave Characteristics Using Fractional Kinetics 1398.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1398.2 From Brownian Motion to Levy Motion . . . . . . . . . . . . . . . . . . 1408.3 Fractional Fokker-Planck Equation . . . . . . . . . . . . . . . . . . . . . 1438.4 Dispersion Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1448.5 Solution of Dispersion Equation . . . . . . . . . . . . . . . . . . . . . . . 1488.6 Numerical Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1508.7 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1509 Discussion 153References 159viii

List of Figures1.1 Thermal reactivity values for the primary fusion reactions. This figureis taken from Ref. (1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2 Plasma confinement: charged particle are forced to follow spiral pathsabout the field lines and are prevented from striking the walls of a containingvessel. This figure is taken from Princeton plasma physics laboratorywebsite: http://www.pppl.gov/fusion basics. . . . . . . . . . . . . 41.3 Schematic representation of the tokamak set up. The figure is takenfrom JET website: http://www.jet.efda.org. . . . . . . . . . . . . . . . . 61.4 Schematic representation of heating methods for plasma. The figure istaken from JET website: http://www.jet.efda.org. . . . . . . . . . . . . 71.5 Fusion triple product. The figure is taken from Ref. (2). . . . . . . . . . 91.6 Schematic representation of tokamak temperature profiles for a numberof modes of operation. The figure is taken from Ref. (2). . . . . . . . . . 101.7 Divertor region D α intensity in a typical plasma showing the characteristicsof different types of ELMs. The figure is taken from Ref. (3). . . . 131.8 The reduction of the heat flux to the wall by the creation of the radiativemantel. Figure is taken from TEXTOR website: http://www.fz-juelich.de. 141.9 Debye shielding. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161.10 Schematic representation of the diffusion coefficient in a tokamak as afunction of the collisionallity. Here, the following notation has been usedA = ɛ −1 where ɛ is the inverse aspect ration. . . . . . . . . . . . . . . . . 183.1 Larmor orbits in a magnetic field. . . . . . . . . . . . . . . . . . . . . . . 323.2 Particle drifts in crossed electric and magnetic fields. . . . . . . . . . . . 343.3 Drift of a gyrating particle in a gravitational field. . . . . . . . . . . . . 37ix

LIST OF FIGURES3.4 Drift of a gyrating particle in a nonuniform magnetic field. . . . . . . . . 383.5 A curved magnetic field. . . . . . . . . . . . . . . . . . . . . . . . . . . . 393.6 Origin of the diamagnetic drift. . . . . . . . . . . . . . . . . . . . . . . . 513.7 The drift wave phase shift, δ > 0 measured in the Q-machine. Thisfigure is taken from Ref. (4). . . . . . . . . . . . . . . . . . . . . . . . . 554.1 ITG instability mechanism. . . . . . . . . . . . . . . . . . . . . . . . . . 684.2 A summary of the ITG/TE modes. . . . . . . . . . . . . . . . . . . . . . 694.3 Characteristic wave number and growth rates of drift instabilities. . . . 704.4 Stability Diagram-Weiland model. . . . . . . . . . . . . . . . . . . . . . 704.5 Simple Flowchart of the AFC-FL code. . . . . . . . . . . . . . . . . . . . 714.6 The k y ρ s -spectra of ITG/TE instability growth rate in deuterium plasmacontaining 2% of Ne 10+ ions with a flat density profile, ɛ n = 0, calculateddirectly in a one impurity species approximation. . . . . . . . . . . . . . 734.7 The k y ρ s -spectra of main ions ITG instability growth rate. . . . . . . . 744.8 The k y ρ s -spectra of TE instability growth rate. . . . . . . . . . . . . . . 744.9 The k y ρ s -spectra of impurity ITG instability growth rate. . . . . . . . . 754.10 The k y ρ s -spectra of TE instability growth with and without taking intoaccount collisional detrapping effects on trapped electrons. . . . . . . . . 764.11 The k y ρ s -spectra of ITG/TE instability growth rate with and withouttaking the effects of the magnetic shear into account. . . . . . . . . . . . 774.12 s-dependences of the ITG (left) and TE (right) growth rates γ max . Solidline: no shear effects taken into account, Dashed line: λ = λ t = 1 and√k ⊥ = k y 1 + (π 2 /3 − 5/2)s 2 , Dashed dotted line: λ = 2/3+s·5/9, λ t =1/4 + 2s/3 and k ⊥ = k y , and Red solid line: both effects considered. . . 774.13 θ-dependences of the instability characteristics γ max , ω max . . . . . . . . 784.14 θ-dependences of the instability characteristics γ max , ω max for differentdensity scaling lengths. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 794.15 The k y ρ s -spectra of ITG/TE instability growth rate in deuterium plasmacontaining 3% of Ne 10+ ions with a flat density profile, ɛ nNe = 0, calculateddirectly in a one impurity species approximation (symbols) anditeratively (solid lines). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81x

LIST OF FIGURES4.16 The growth rates (left column), real frequency (middle column) and dimensionlesswave vector (right column) of the most unstable modes asfunctions of the temperature gradient parameter ɛ Te,i calculated with differentconcentrations ξ Ne and density gradient parameters ɛ nNe of Ne 10+impurity ions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 824.17 The growth rates (left column), real frequency (middle column) and dimensionlesswave vector (right column) as functions of magnetic shearcalculated for different concentrations ξ Ne and density gradient parametersɛ nNe of neon impurity. . . . . . . . . . . . . . . . . . . . . . . . . . 834.18 The growth rates (left column), real frequency (middle column) anddimensionless wave vector (right column) versus temperature gradientscale in deuterium plasma with C +6 , N +7 , O +8 , Ne +10 , Ar +18 impurityions of different total concentrations: no impurity (red curves), ∑ ξ j =1% (black solid curves), 3% (black dashed curves) and 5% (black dasheddottedcurves); the impurity ion density gradient parameters are calculatedself-consistently from zero particle fluxes. . . . . . . . . . . . . . . 854.19 The growth rates (left column), real frequency (middle column) and dimensionlesswave vector (right column) versus magnetic shear computedfor ɛ Te,i = 10. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 855.1 Diffusivity and pinch velocity of deuterons and electrons, D i,e (left column)and V i,e (right column), respectively, versus ɛ Te,i calculated fors = 1.8. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 895.2 Diffusivity and pinch velocity of deuterons and electrons versus magneticshear calculated for ɛ Te,i = 10. . . . . . . . . . . . . . . . . . . . . . . . . 895.3 Heat transport coefficients κ i,e (left column) and Π i,e (right column) asfunctions of ɛ Ti,e calculated for s = 1.8. . . . . . . . . . . . . . . . . . . . 915.4 Heat transport coefficients versus magnetic shear calculated for ɛ Te,i = 10. 915.5 The main mechanisms for impurity pinch. . . . . . . . . . . . . . . . . . 925.6 The impurity peaking factor effect on the impurity density profile. . . . 935.7 The experimental measurements of the impurity peaking factor and theneoclassical predictions (this figure is taken from the presentation by C.Giroud at 21st IAEA fusion energy conference Chengdu, China 2006). . 94xi

LIST OF FIGURES5.8 The peaking factor for impurity species C +6 (red curves), Ne +10 (bluecurves) and Ar +18 (pink curves) versus ɛ Ti computed with ɛ Te = 0,∑s = 1.8 and different total impurity concentrations: ξj = 1% (solidcurves), 3% (dashed curves) and 5% (dashed-dotted curves). . . . . . . . 955.9 The peaking factor for impurity species C +6 (red curves), Ne +10 (bluecurves) and Ar +18 (pink curves) versus magnetic shear computed withɛ Te = 0, ɛ Ti = 5 and different total impurity concentrations: ∑ ξ j = 1%(solid curves), 3% (dashed curves) and 5% (dashed-dotted curves). . . . 955.10 The k y ρ s -spectrum of the instability growth rate as a function of θ calculatedfor the plasma parameters in the JET core, r/a = 0.42. . . . . . 1015.11 Impurity peaking factor as a function of the impurity charge Z andparameter θ computed without impurity ion collisions (r/a = 0.42). . . . 1015.12 Impurity peaking factor as a function of the impurity charge Z andparameter θ computed with impurity ion collisions (r/a = 0.42). . . . . 1025.13 The contribution from thermal forces, PV T Fcoll, to the pinch-velocity factorP Vz as a function of the impurity charge Z and parameter θ (r/a = 0.42).103))5.14 The values Im(α ′ i ΘN z/Y z and Im(α eΨN ′ z /Y z as functions of theimpurity charge Z and parameter θ (r/a = 0.42). . . . . . . . . . . . . . 1035.15 The k y ρ s -spectrum of the instability growth rate as a function of θ calculatedfor the plasma parameters close to the plasma edge, r/a = 0.8. . 1045.16 Impurity peaking factor as a function of the impurity charge Z andparameter θ computed without impurity ion collisions (r/a = 0.8). . . . 1055.17 Impurity peaking factor as a function of the impurity charge Z andparameter θ computed with impurity ion collisions (r/a = 0.8). . . . . . 1055.18 The k y ρ s -spectrum of the instability growth rate as a function of themagnetic shear s calculated for the plasma parameters in the plasmacore, r/a = 0.42, in the regime of TE-modes, θ = 1. . . . . . . . . . . . . 1065.19 Impurity peaking factor as a function of the impurity charge Z and magneticshear s computed with impurity ion collisions included (r/a = 0.8, θ = 1).1075.20 Charge dependence of the impurity peaking factor observed (right figure)and computed by AFC-FL code (left figure). . . . . . . . . . . . . . . . 1076.1 A local moving reference frame attached to the particle. . . . . . . . . . 111xii

LIST OF FIGURES6.2 Comparison between AFC-FL and QuaLiKiz models. . . . . . . . . . . . 1157.1 Prad tot total radiated power as a function of time (taken from JET measurementsbase on 2D tomographic reconstruction) for the three discharges:69089 (solid line), 69091 (dashed line) and 69093 (dashed dotted line). . 1207.2 P rad total radiated power inside LCFS as a function of normalized flux,Ψ, (taken from JET measurements based on Abel inversion (reconstructionunder assumption that plasma radiation is constant on the fluxsurface) for the two discharges at t ≈ 7.6s: 69091 (dashed line) and69093 (dashed dotted line). . . . . . . . . . . . . . . . . . . . . . . . . . 1217.3 T i profiles as functions of major radius (taken from JET CXFM measurements)for the three discharges at t ≈ 7.6s: 69089 (solid line), 69091(dashed line) and 69093 (dashed dotted line). . . . . . . . . . . . . . . . 1227.4 T e profiles as functions of major radius (taken from JET LIDAR measurements)for the three discharges at t ≈ 7.6s: 69089 (solid line), 69091(dashed line) and 69093 (dashed dotted line). . . . . . . . . . . . . . . . 1237.5 Electron density n e profiles as functions of major radius (taken from JETLIDAR measurements) for the three discharges at t ≈ 7.6s: 69089 (solidline), 69091 (dashed line) and 69093 (dashed dotted line). . . . . . . . . 1237.6 Neon concentration, n Ne /n e profiles as functions of major radius (takenfrom JET charge exchange recombination spectroscopy measurements,CXF6) for the three discharges at t ≈ 7.6s: 69089 (solid line), 69091(dashed line) and 69093 (dashed dotted line). . . . . . . . . . . . . . . . 1247.7 Carbon concentration, n C /n e profiles as functions of major radius (takenfrom JET charge exchange recombination spectroscopy measurements,CXFM) for the three discharges at t ≈ 7.6s: 69089 (solid line), 69091(dashed line) and 69093 (dashed dotted line). . . . . . . . . . . . . . . . 1257.8 Effective charge, Z eff profiles as functions of major radius (taken fromJET charge exchange recombination spectroscopy measurements CXFM)for the three discharges at t ≈ 7.6s: 69089 (solid line), 69091 (dashedline) and 69093 (dashed dotted line). . . . . . . . . . . . . . . . . . . . . 125xiii

LIST OF FIGURES7.9 The energy confinement time defined as the ratio of the diamagneticenergy to the total input power: τ E = W dia /P tot versus the fractionof radiated power, P rad /P tot for different tests. The reference discharge69089 with only D fueling is shown with a diamond. The other twodischarge: 69091 (D fueling and Ne injection) and 69093 (only Ne injection)are presented by a square and a circle, respectively. Other symbolsrepresent the tests where only n e (right triangle), only Z eff of 69091(plus), only Z eff of 69093 (up triangle), only P rad (left triangle) andfinally, P rad + n e (star), P rad + Z eff (down triangle) have been replacedfrom their reference value in 69089 by those from 69091 − 3 discharges. . 1267.10 ITG growth rate as a function of the normalized toroidal flux coordinate,ρ, calculated by RITM code for the three discharges: (solid line)shot 69089, (dashed line) shot 69089 with Z eff from 69091, and (dasheddotted line) shot 69089 with Z eff from 69093, shown in figure 7.9 by adiamond, a plus and an up triangle symbols, respectively. . . . . . . . . 1297.11 Electron (top) and ion (bottom) temperature profiles as functions of thenormalized toroidal flux coordinate, ρ, (solid line) for 69089, (dashedline) for 69089 with Z eff increased to that from 69091, and (dashed dottedline) for 69089 with Z eff increased to that from 69093, correspondingto the diamond, plus and up triangle symbols in figure 7.9, respectively. 1307.12 Electron (top) and ion (bottom) temperature profiles as functions of thenormalized toroidal flux coordinate, ρ, (solid line) for 69089, and (dashedline) for 69089 with P rad increased to that from 69091−3, correspondingto the diamond and left triangle symbols in figure 7.9. . . . . . . . . . . 1317.13 Electron (top) and ion (bottom) temperature profiles as functions ofthe normalized toroidal flux coordinate, ρ, (solid line) for 69089, and(dashed line) for 69089 with P rad + Z eff increased to that from 69093,corresponding to the diamond and down triangle symbols in figure 7.9. . 1327.14 Radial profiles of the ion effective charge Z eff , ion temperature T i andion heat diffusivity χ i computed with the RITM code for JET dischargeswith low (dashed curves) and high (solid curves) neon content; the fittedexperimental data of the T i profile taken from JET experimental database are shown by squares and crosses, respectively. . . . . . . . . . . . 135xiv

LIST OF FIGURES8.1 u 2 Λ(u) (Solid line), uΛ(u) (Dashed line) and Λ(u) (Dotted line) as functionsof u. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1488.2 2wJ0 2(b iw)Λ(w) as functions of w. . . . . . . . . . . . . . . . . . . . . . . 1498.3 γ from two solutions of the dispersion equation as functions of ɛ forb i = 0.1. The solution with γ < 0 (blue) gives the growth rate of theunstable mode. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151xv

LIST OF FIGURESxvi

1Introduction1.1 Towards a New Source of EnergyEnergy supplies for future are becoming more and more vulnerable while the globalconsumption is escalating dramatically, expected to increase by 70% in 2030 and continuingto rise (5). All stars in the universe are powered by nuclear fusion which similarto fission can produce huge amounts of carbon-neutral energy. However, there is onevital difference between them: the nuclear fusion does not produce dangerous and longlasting radioactive wast. Waste from nuclear fusion is only radioactive for 50-70 years,compared to the thousands of years of radioactivity that result from fission. Only afew nuclear reactions of the many reactions which occur in the stars are of practicalvalue for potential energy production on earth. All of these reactions involve isotopesof hydrogen. Three isotopes of hydrogen are known; they are hydrogen (H), deuterium(D), and tritium (T). Raw materials for nuclear fusion: isotopes of hydrogen are plentifuland widespread on Earth. The hydrogen and deuterium can be extracted fromwater and the tritium required would be produced from lithium, which is available fromland deposits or from sea water which contains thousands of years’ supply. The worldwideavailability of these materials would thus eliminate international tensions causedby imbalance in fuel supply. Nuclear fusion could also help meet international climatechange targets and the current zero-carbon technologies are unlikely to meet our energydemands this century. However, nuclear fusion could render carbon dioxide-producingfossil fuels obsolete.1

1. INTRODUCTION1.2 Fusion ReactionThe possible fusion reactions between the isotopes of hydrogen are listed below:2 D + 3 T → n(14.03MeV ) + 4 He(3.56MeV )2 D + 2 D → n(2.45MeV ) + 3 He(0.82MeV ) (1.1)The power production process which can occur at the lowest temperature and hence,the most readily attainable fusion process on earth, is the combination of a deuteriumnucleus with one of tritium (see figure 1.1). The products are energetic helium-4 (He 4 ),the common isotope of helium also called an alpha particle, and a more highly energeticfree neutron (n). The helium nucleus carries one-fifth of the total energy released andthe neutron carries the remaining four fifths.Figure 1.1: Thermal reactivity values for the primary fusion reactions. This figure istaken from Ref. (1)2

1.3 The Necessary Conditions1.3 The Necessary ConditionsFusion reactant are positively charged and must overcome their electrostatic repulsionin order to get close enough for the strong nuclear forces of attraction to dominate.Therefore, the essential condition for the fusion is the requirement of a sufficientlyhigh kinetic temperature of the reacting species in order to facilitate the penetrationof the Coulomb barrier. To reach the needed high kinetic temperatures we can startby confining a population of the deuterium and tritium atoms in some closed spaceand by heating attain both ionization and high temperatures (about 100 million degreesCelsius). The resulting ensemble of positive and negative charges then formsa plasma which is expected to reach thermodynamic equilibrium as a result of randomcollisions. The resultant spectrum of particle energies is then well described byMaxwelle-Boltzmann distribution, with the high energy part of this distribution providingfor most of the desired fusion reactions. The critical technical requirement isthe sustainment of a sufficiently stable high temperature plasma in a practical reactionvolume and for a sufficiently long period of time to render the entire process energeticallyviable. Confinement of the plasma fuel by some means is thus crucial to maintainthese conditions within the required reaction volume.One of the most effective means for the plasma confinement involves the use of themagnetic fields. In the absence of a magnetic field the charged particles in a plasmamove in straight lines and random directions. Since nothing restricts their motion thecharged particles can strike the walls of a containing vessel, thereby cooling the plasmaand inhibiting fusion reactions. In a magnetic field however, the particles are forcedto follow spiral paths about the field lines (see figure 1.2). Consequently, the chargedparticles in the high-temperature plasma are confined by the magnetic field within therequired reaction volume.1.4 TokamakApplying a magnetic field can significantly increase the confinement of this high temperatureplasma. The cyclotron motion (gyration of the charged particles around magneticfield lines) reduces the transport of the charged ions and electrons in the direction perpendicularto the magnetic field. Therefore, the magnetic field must be configured in3

1. INTRODUCTIONFigure 1.2: Plasma confinement: charged particle are forced to follow spiralpaths about the field lines and are prevented from striking the walls of a containingvessel. This figure is taken from Princeton plasma physics laboratory website:http://www.pppl.gov/fusion basics.such a way that it does not cross the vacuum vessel.In 1963 Russians Physicists (6) came up with the tokamak, “Toroidal naya KameraMagnitnaya Katushka” (Toroidal chamber with Magnetic Coil), concept based onthe ideas of Sakharov (7). In this device the vacuum vessel has a torus shape (seefigure 1.3). The largest component of the magnetic field, along the torus, is producedby the toroidal magnetic field coils and is called the toroidal magnetic field, B T . Theplasma is heated by the current induced by a transformer, where the plasma itselfserves as a secondary winding (Ohmic heating). This plasma current induces the socalledpoloidal magnetic field, which encircles the center of the tube. The toroidal andpoloidal magnetic fields together produce a helical field, whose field lines lie on thetoroidal surfaces. Apart from the toroidal field generated by the external field coils andthe field generated by the flow of the plasma, the Tokamak requires a third verticalfield (poloidal field), fixing the position of the plasma column in the vessel.One important plasma confinement indicator is the ratio of kinetic particle pressure4

1.5 Plasma HeatingP kin = n i k B T i + n e k B T e (1.2)to the magnetic pressureP mag = B22µ 0(1.3)where n i,e , T i,e are the density and temperatures of plasma ion and electron respectivelyand k B is the Boltzmann constant, B = |B| is the magnetic field strength, and µ 0 isthe permeability of free space. This ratio is defined as the beta parameter, β, and is ameasure of how effectively the magnetic field guarded the thermal motion of the plasmaparticles. A high beta would be most desirable, but it is also known that there existsa system-specific β max at which plasma fluctuations start to destroy the confinement.That is why for confinement purposes, we requireβ maxB 22µ 0≥ n i k B T i + n e k B T e (1.4)The maximum plasma pressure is thus determined by available magnetic fields.This criteria introduces the magnetic field technology as a limit on plasma confinementin Tokamaks.1.5 Plasma HeatingOhmic heating results from the plasma toroidal inductive current and therefore dependson the the resistance of the plasma. As the temperature of plasma rises however,the resistance decreases and the ohmic heating becomes less effective. The maximumplasma temperature attainable by ohmic heating in a tokamak appears to be 20-30 milliondegrees Celsius. Therefore, to reach still higher temperatures additional heatingmethods must be used (see figure 1.3).Neutral beam injection is one of these methods which involves the introduction ofhigh-energy (neutral) atoms into the ohmically heated, magnetically confined plasma.The atoms are immediately ionized and are trapped by the magnetic field. The highenergyions then transfer part of their energy to the plasma particles in repeated collisions,thus increasing the plasma temperature.5

1. INTRODUCTIONFigure 1.3: Schematic representation of the tokamak set up. The figure is taken fromJET website: http://www.jet.efda.org.6

1.6 Power Balance and Lawson CriteriaIn radio-frequency heating, high-frequency waves are generated by oscillators outsidethe torus. If the waves have a particular frequency (or wavelength), their energycan be transferred to the charged particles in the plasma in the same way that microwavestransfer heat to food in a microwave oven. These heated charged particles inturn collide with other plasma particles, thus increasing the temperature of the bulkplasma.Figure 1.4: Schematic representation of heating methods for plasma. The figure is takenfrom JET website: http://www.jet.efda.org.1.6 Power Balance and Lawson CriteriaFusion confinement requirements are qualified in terms of energy and particle confinementtimes. The energy confinement time, τ E , is defined in terms of a steady stateplasma asτ E ≡ U W in(1.5)where U is the measured plasma energy content and W in is the rate of the energy inputnecessary to sustain the steady state.The particle confinement time, τ p , is defined analogously in terms of particle contentand replacement rate.7

1. INTRODUCTIONIn a steady state self sustained fusion plasma the energy input would come fromfusion reactions. The power input is denoted by fusion reactions, P in which measuresthe rate at which fusion energy is deposited in the plasma; it is smaller than the rate offusion energy production, because only the kinetic energy of charged reaction products,such as alpha particles, can be contained. Neutron energy is too quickly lost to thewalls of the containing vessel to contribute to P in . This is due to the fact that theneutrons have no charge and are highly energetic therefore, they rarely react with theplasma particles, hence, their energies are not deposited into the plasma. We expressthe radiation losses by P rad , where the subscript refers to bremsstrahlung radiation.Cyclotron radiation losses are conventionally omitted because their longer wavelengthallows at least partial reflection at the plasma boundary. Thus W in = P in − P rad andfusion plasma power balance is expressed asP in − P rad = U (1.6)τ EIt was noted by Lawson that, while the energy content is proportional to plasmadensity, n, both quantities on the left-hand side of (1.6) vary with the square of thedensity. That P in is proportional to n 2 follows immediately from the binary nature offusion reactions; in a D-T plasma with equal amounts of D and T,P in = 1 4 n2 i 〈σv i 〉E DT (1.7)where E DT = 17.6 MeV is the mean reaction energy of two primary D-T reactions and〈σv i 〉 is the reaction cross-section averaged over the relative velocity of colliding ions.The bremsstrahlung power loss scales asP rad ∝ n 2 eT 1/2e (1.8)It follows that the density enters (1.6) only through the combination nτ E , the Lawsonparameter. For a pulsed reactor with an efficiency of 33% for conversion of heat toelectricity, Lawson gave the criteria both for the minimum temperature and for theminimum (nτ E ) c value at the optimum temperature:T ≥ 3 keV and (nτ E ) c ≥ 10 20 m −3 s at T ≈ 30 keV for D − T (1.9)8

1.7 Plasma Operational LimitsIt is evident that smaller values of nτ E correspond to energy losses too severe forself-sustained fusion; larger values could presumably be cured by artificially enhancedlosses. Thus, the Lawson criterion for self-sustained, or “ignited” fusion, states thatthe product of density and confinement time must equal or exceed (nτ E ) c .The result is sensitive to the particular fusion reaction considered, as well as tovarious plasma parameters (such as impurity content) that are not easy to predict, sothat a simple general specification is difficult.1.7 Plasma Operational LimitsFigure 1.5: Fusion triple product. The figure is taken from Ref. (2).In recent years there has been considerable development of databases and accumulationof knowledge on the behavior of tokamak plasmas around the world. A degreeof uncertainty still exists in predicting the confinement properties and plasma performancein such a device. The progress of these researches can be illustrated by the fusionproduct n DT τ E T i , which represents a figure of merit for plasma performance (see figure9

1. INTRODUCTION1.5) during four decades of research on magnetic confinement.Experimental findings are categorized according to their empirical signatures intoconfinement ”modes”. Sudden transitions between such regimes are often observed. Inthe following a general overview of the different operational regimes is given (see figure1.6).Figure 1.6: Schematic representation of tokamak temperature profiles for a number ofmodes of operation. The figure is taken from Ref. (2).1.7.1 Ohmic plasmaAn ohmic plasma is one that is resistively heated with a power given by I p V res , where I pis the plasma current and V res is the resistive portion of the loop voltage. The electronsare heated directly, while the ions are heated by the equipartition energy flow from theelectrons.10

1.7 Plasma Operational Limits1.7.2 L-modeIt was recognized early that the ohmic regime was inefficient for achieving the necessarytemperatures. As temperature increased, the resistive heating decreased, and,therefore, auxiliary heating would help to increase the plasma temperature to the levelrequired for significant fusion power production. Auxiliary heating was performed bya variety of techniques including neutral beams and Radio Frequency heating (RF).While the temperature and stored energy increased with this auxiliary heating, the incrementalincrease in stored energy was less than that expected from the ohmic scaling,resulting in a degradation of global confinement. This mode of operation with degradedenergy confinement time, is called L-, or low confinement, mode. Characteristic featuresof L-mode plasma are the low temperatures and temperature gradients near theplasma periphery.1.7.3 H-modeThe high confinement mode (H-mode), associated with a spontaneous formation of anedge transport barrier, was first discovered in ASDEX (8) and has now been seen ona wide variety of magnetic confinement devices under a wide range of conditions. TheH- mode exhibits global energy confinement values about a factor of two better thanL-mode. Part of this is due to formation of the edge transport barrier. Another part ofthis improvement is due to a reduction in local transport throughout the plasma afterthe L-H transition.The H-mode is reached above a certain threshold of power, which depends on plasmaconditions and machine size, see reference (3).1.7.4 Global InstabilitiesA number of large scale MHD phenomena can have an impact on global confinement.Two of these are the periodic sawtooth instability, which can have a significant effecton the profiles of temperature, density and impurities in the central core region, andthe ELMs, which periodically affect plasma edge region. In the following sections wewill briefly review these two instabilities.11

1. INTRODUCTION1.7.4.1 SawtoothWhen the central value of the safety factor q ≈ rB T /RB p (B T and B p are the toroidaland poloidal magnetic field components, r and R are the minor and major radius ofthe tokamak torus, respectively) falls below unity; relaxation oscillations are normallyobserved in the core of a tokamak plasma. They appear on a number of plasma parametersbut are particularly evident in the central electron temperature T e (0) (9). Theoscillation in T e (0) exhibit a time trace with a distinctive sawtooth shape consisting ofa slow rise during which the plasma inside q = 1 heats up, followed by a rapid collapsewhen the plasma energy is redistributed from the core to the region outside q = 1.This then propagates as a heat pulse to the plasma periphery. This mechanism has theeffect of degrading the global energy confinement time.1.7.4.2 Edge Localized ModesThe transition from L-mode to H-mode in magnetic confinement systems is normally accompaniedby appearance in the H-mode phase of periodic edge instability phenomena,known as ELMs. The ELM is a relaxation oscillation triggered by an MHD instability,which leads to a fast (ms) loss of particles and energy from the plasma edge. Theunderlying cause for ELMs is the onset of MHD instability in the plasma edge whenthe edge pressure gradient exceeds a critical threshold. The subsequent loss of edgeconfinement leads to a temporary reduction of the pressure gradient, and the eventualrecovery of the pressure gradient leads to recurrence of the ELM. This cycle continuesindefinitely in a sustained H-mode discharge. The energy loss due to ELMs causes areduction of the global energy confinement time.At least three major types of ELMs have been defined (10). In a given experiment,the level of the plasma heating power, P , or more directly the net power reaching theplasma edge P edge = P in − P rad (i.e. heating power minus radiation losses inside theedge region) is a key factor in determining the ELM type. A number of types of ELMwith different amplitude, frequency and power dependences can be distinguished (seefigure 1.7):• Type I ELMs (giant ELMs) with high amplitude and low frequency, which developwhen the edge power flow significantly exceeds the H-mode threshold.12

1.7 Plasma Operational LimitsFigure 1.7: Divertor region D α intensity in a typical plasma showing the characteristicsof different types of ELMs. The figure is taken from Ref. (3).• Type II ELMs (grassy ELMs) are associated with strongly shaped tokamaks athigh edge pressure.• Type III ELMs (small ELMs)with small amplitude and high frequency appearwhen the power flow to the plasma edge is only marginally above the H-modepower threshold.Other improved regimes have also been observed besides the H-modes. Regimeswith core or Internal Barriers (ITBs), see figure 1.6, have been discovered that lead tosignificant enhancements in confinement and plasma performance. Transport barriersassociated with weak or negative shear have been observed on all of the large tokamaks:TFTR, DIII-D, JET and JT-60U. Other classes of improved confinement regimes areregimes without and with edge radiation. Examples of the former are given in reference(3) and the latter include the RI-mode of TEXTOR which will be discussed in thefollowing section.1.7.5 RI-modeThe duration of a burning fusion plasma will depend largely on the properties of theedge plasma being in contact with the wall elements. The first wall has to withstandand exhaust the α-particle heating power, and the helium ash must be removed fromthe plasma. Wall erosion will affect the lifetime of wall elements and impurities arereleased into the plasma, which then can cause fuel dilution and power loss owing to13

1. INTRODUCTIONradiation from the plasma center.The integrity of plasma facing components will crucially depend on the avoidanceof overheating of plasma facing components and on the balance between erosion anddeposition of wall materials. The lifetime of plasma facing components, an importantfactor for its economy, will ultimately govern the availability of a fusion reactor. Theproblem of overheating of small areas like the divertor strike zone or the limiter edgecan be solved by distributing the power over large areas. In the high density divertorcharge-exchange processes could provide this to a certain extent. Furthermore, a distributionof the heat on the whole vessel wall can be achieved by radiation from injectedimpurities. Feed-back control of the seeded impurities is an important requirement. Upto 90% of the heating power can be radiated from a rather thin belt at the peripheryof the confined plasma. In the limiter tokamak TEXTOR it was found that the energyconfinement with seeded impurities can be substantially improved to values comparablewith ELM-free H-mode discharges in divertor tokamaks. This regime has been namedRadiative Improved Mode (RI-mode), see figure 1.8Figure 1.8: The reduction of the heat flux to the wall by the creation of the radiativemantel. Figure is taken from TEXTOR website: http://www.fz-juelich.de.By puffing of argon impurity a noticeable confinement improvement has also been14

1.8 Mechanisms of Transport in Tokamaksseen in the H-mode in the divertor tokamak JT-60U. However, earlier analogous experimentson JET (11) did not lead to the desirable result and suffer, when argonconcentration exceeded some critical level, from unwanted and uncontrollable impurityaccumulation in the plasma core with strongly peaked core radiation and flat temperatureprofile.Therefore, these impurities must be carefully controlled to maintain the radiationat the required levels without excessive plasma core accumulation. This brings us tothe problem of transport in magnetically confined plasmas. The success of the fusionprogram therefore strongly depends on understanding and controlling the transportprocesses of matter and energy in Tokamak plasmas.In the next section we will present the mechanisms of transport in Tokamaks andtheir predictions accuracy in explaining the experimental observations.1.8 Mechanisms of Transport in TokamaksUnderstanding the mechanisms of transport of matter, electric charge, energy and momentumis one of the most important goals of research in the field of plasma physics.Practically all applications of plasmas are limited in some way or other by the transportphenomena taking place under specific circumstances.1.8.1 Classical TransportSince the early days of tokamak research it has been observed that radial transportcannot be explained by the so called classical transport theory of plasmas. The classicaltheory explains the diffusion across the magnetic field with cylindrical symmetryas resulting from friction between the electrons and the ions (12). The processcan also be interpreted as a random walk of particles, where the typical step lengthis the electron gyro-radius, if the collision time is taken to be the time a particletakes, on average, to diffuse in velocity space through an angle of 90 ◦ . The classicalpredictions, when compared to the experiments, grossly underestimate the transportcoefficients in a fusion plasma: diffusion coefficient in tokamaks are typically15

1. INTRODUCTIONof the order of ∼ m 2 s −1 and the classical diffusion coefficient can be estimated byD c ≈ ν ei ρ 2 e(∼ 10 3 × (10 −3 ) 2 = 10 −3 [m 2 s −1 ] for the electrons) where ν ei is the 90 ◦collision frequency between electron and ion and ρ is the Larmor radius, ρ = V th /Ω c ,which is proportional to the ratio of a thermal speed, V th = √ 2T e /m i , to a cyclotronfrequency Ω c = eB/mc, where B = |B| is the magnetic field strength and m is theparticle mass.To explain such disagreements, one has to understand which assumptions of theclassical theory are invalid. An invalid assumption is that the collisions in the fully ionizedplasma considered as binary interactions while these collisions are of the Coulombnature and because of the long range of the Coulomb forces they can not be consideredas binary interactions. Because of the long range of the Coulomb forces, the collectivenature of the plasma dominates its behavior. An important concept characterizingcollective phenomena in plasma is the Debye shielding.1.8.2 Debye ShieldingFigure 1.9: Debye shielding.A fundamental characteristic of the behavior of a plasma is its ability to shield outelectric potentials that are applied to it. If we put an electric field inside a plasma byinserting two charged balls connected to a battery, see figure 1.9, almost immediately16

1.8 Mechanisms of Transport in Tokamaksclouds of negative and positive charge will surround the positive and negative ballsrespectively, with their density decreasing with distance from the charged balls. Thisis due to the mobility of the electrons and ions in plasma. For cold plasma with nothermal motions, as many charges will be observed in the surrounding clouds as arerequired to neutralize the inserted charges. However, plasma temperature is finite andthe plasma particles possess a substantial kinetic energy of thermal motion so some -particularly those at the edge of the cloud - will escape from the shielding cloud and theshielding is not complete. The edge of the cloud then occurs at the radius where thepotential energy is approximatively equal to the thermal energy k B T of the particles.This characteristic shielding range, is the Debye length which in non-magnetized plasmais defined (for further details see Ref. (1; 13))λ D = ( ɛ 0k B T en e e 2 )1/2 (1.10)where ɛ 0 is the primitivity of the free space, e is the electron charge and n e is taken asthe electron density far away from the shielding cloud.Transport in plasmas is dominated by the long-range collective electric field E k,ω ,part of the Coulomb interactions between the charged particles. Here the subscriptdenote the wave-number k and frequency ω of the electric field fluctuation E k,ω . For aplasma of density n and temperature T the single particle Coulomb electric field fallsoff exponentially beyond the Debye length λ D . Therefore, for kλ D ≪ 1 the electric fieldfluctuations are collective self consistent interactions while for kλ D 1 the interactionsare binary collisional. For a plasma with a large number of particles inside the Debyesphere N D = (4π/3)nλ 3 D≫ 1, the collective electric fields dominate the plasma dynamicsthrough collective modes. In magnetized plasmas the modes with low frequency,ω ≪ Ω c , dominate the transport.The effects related to the collective interactions which exist in a plasma as a resultof the long range Coulomb forces, fall outside the scope of the classical theory and arethe object of anomalous transport theory which will be discussed later in this chapter.17

1. INTRODUCTIONFigure 1.10: Schematic representation of the diffusion coefficient in a tokamak as afunction of the collisionallity. Here, the following notation has been used A = ɛ −1 where ɛis the inverse aspect ration.1.8.3 Neo-classical TransportAn effect which has to be taken into account in fusion plasmas is the effect of the magneticfield on the free motion of the particles. The magnetic fields produced in fusiondevices are spatially inhomogeneous and have a globally toroidal topology. The particleorbits in such magnetic fields are either helices encircling (but also drifting away from)the field lines, or else depending on their velocity, they may be trapped in low-fieldregions because of the inhomogeneousity (magnetic mirror effect, see chapter 3). Whenthe classical theory (based on collisions) is combined with these effects on the geometry,one finds transport coefficients that are enhanced as compared to the classical ones.This theory is called the neoclassical transport theory of plasmas (14; 15; 16). Theneoclassical transport levels exceed the classical ones by geometrical factors: q 2 ɛ −3/2 inthe low collision frequency ”banana” regimes (ν ∗ < 1.0) and q 2 in the collisional limit,as a result of toroidal geometry. Here, ɛ = r/R is the inverse aspect ratio, with r andR being the minor and major radii of the magnetic surface. The collisionallity parameteris ν ∗ ≡ ν eff /ω b , where ν eff ≡ ν ei /ɛ is the effective collision frequency for particledetrapping, and ω b ≈ ɛ 1/2 V th /(Rq) is the trapped particle average bounce frequency.18

1.8 Mechanisms of Transport in TokamaksAlthough neoclassical transport theory is a significant step forward towards describingtransport in magnetically confined plasmas, in general the neoclassical predictionsare still far from explaining the full picture (D neo for electrons is of the order10 −3 [m 2 s −1 ] and for ions of the order 10 −1 [m 2 s −1 ]).1.8.4 Anomalous TransportThe transport theory based on particle collisions can incorporate the geometry of theTokamak magnetic system, but neoclassical theory still assumes that the plasma is inequilibrium and axisymmetric. Real Tokamak plasmas always show the presence of abroad spectrum of fluctuations, e.g. in plasma density, temperature and electromagneticfields (3); thus real Tokamak plasma are turbulent. The turbulent fluctuationsgive rise to transport across the equilibrium magnetic surfaces and it is necessary toincorporate their effect in a comprehensive transport theory. From the theoretical pointof view, most of the instabilities that we think are responsible for the observed plasmaturbulence have a very small component of wave number vector parallel to the magneticfield, compared to the perpendicular component. That is, most of the turbulent eddiesare quasi-perpendicular to the toroidal magnetic field. Therefore, we can expect thatturbulence dominates perpendicular transport, in this case we are in the presence ofanomalous transport. The influence of the plasma turbulence on the parallel transportis rather small, as experiments confirms.The anomalous transport is the least understood transport process in plasma physics.In recent years tremendous progress has been made to describe the anomalous transporttheoretically, in particular with the help of complex computer codes which calculate thegrowth rates of the underlying instabilities and allow to deduce the resulting transportcoefficients. These instabilities are driven essentially by the gradient of density andtemperature.The fluctuations of the magnetic and electric fields (δE, δB) can provide the sourcesto the anomalous transport across the magnetic field lines. If the fluctuations are purelyelectrostatic in nature (δB = 0) they would lead to enhanced cross field transport byE × B drift:19

1. INTRODUCTIONδV E = δE × BB 2 (1.11)The anomalous diffusion coefficient can be approximatively expressed aswhere τ c is the fluctuation’s time scale. By replacingD ano = (δV Eτ c ) 2τ c= (δV E ) 2 τ c (1.12)δV E = k θδφB= T k θeBeδφT(1.13)where k θ is the poloidal wave number of the mode and δφ is the electrostatic potentialfluctuation, into equation (1.12) for anomalous diffusion we getD ano ≈ ( T k θeB )2 ( eδφT )2 τ c (1.14)If τ c ∼ 10 −6 [s], 1/k θ ≈ ρ i ∼ 10 −3 , we find that ∼ 1% electrostatic fluctuation cangenerate an anomalous diffusion of the order m 2 s −1 .If the fluctuations are purely magnetic in nature (δE = 0) the parallel transportalong the stochastic field lines would be responsible for the anomalous transport in theradial direction:δV B = δBV ‖B(1.15)with the diffusion coefficient asD ano = (δV Bτ c ) 2τ c≈ ( δB rB )2 V th λ c (1.16)where λ c = V th τ c is the fluctuation’s length scale.If V th,e ∼ 10 7 [m 2 s −1 ], we find that about ∼ 0.01% magnetic fluctuation is sufficientto generate an anomalous diffusion of the order m 2 s −1 .20

1.9 Global Energy Confinement Scaling1.9 Global Energy Confinement ScalingBecause of the complexity of the processes determining heat and particle transportin fusion plasmas, it is not yet possible to provide a first principle derivation of thedependence of energy confinement properties on plasma parameters. The descriptionof the global energy confinement time by empirical scaling that are based on relevantdatabases within specific operating regimes such as L-mode or H-mode has, therefore,become the key tool in extrapolating plasma performances to a next step device, suchas ITER as well as an approximate constraint on the form of theoretical models. Thesescalings connect empirical confinement times with machine and plasma parameters likemajor radius, R, minor radius, a, toroidal magnetic field strength, B T , plasma current,I, electron line average density, n e and plasma temperature, T , along with other geometricalparameters and profile functions, the ion mass and charge numbers m i andZ i . This approach is of course already well established in other areas of science andengineering - the performance of airplanes and ships can be reliably predicted usingsimilar scalings without a detailed understanding of turbulent hydrodynamics flow.The ELMy H-mode standard database provides the basis for a robust confinementpredictions for ITER. The power law scaling expression for thermal energy confinementtime can be expressed as (3):IT ERHτth,E = 0.0562I 0.93 B 0.15 P −0.69 ne 0.41 m 0.19 R 1.97 ɛ 0.58 κ 0.78a (1.17)(in s, MA, T , MW , 10 19 m −3 , AMU, m) where m = average ion mass, P = loss power,κ a = elongation and ɛ = a/R inverse aspect ratio.The study of the scaling relations promoted dimensionless scaling or similarity rules(wind tunnel experiments). Similarity rules compare plasma behavior in geometricallysimilar devices. There are dimensional constraints that follow the similarity rules andone needs to identify the relevant dimensionless parameters. However, the number ofdimensionless parameters for confined plasma is large, up to 19 have been identified(17). They include plasma physics parameters, such as β the ratio of the plasma kineticpressure to the magnetic pressure, the collisionallity ν ∗ ≡ ν eff /ω b (see sec. 1.8.3) andthe normalized Larmor radius ρ ∗ ≡ ρ i /a. There are also parameters describing the21

1. INTRODUCTIONmagnetic field geometry (safety factor q, aspect ratio A ≡ R/a, the ellipticity κ andtriangularity δ of the plasma cross section) and parameters representing the plasmacomposition (T e /T i , m e /m i , Z eff , etc.). For a local diffusion coefficient we have toinclude the parameters related to plasma profiles, such as the ratio of the scale lengths,L T /R and L n /R,etc. The diffusivity can be expressed in the following form:D = c s ρ s (ρ ∗ ) α F (ν ∗ , β, q, A, κ, δ, L T /R, L n /R, . . . , T e /T i , m e /m i , Z eff ) (1.18)where c s = √ T e /m i is the sound speed, ρ s = √ 2T e m i /eB is the ion Larmor radiusevaluated at the electron temperature and F is a function of the dimensionless parametersto be determined. The main change in plasma parameters in going from presenttokamaks to future reactors is in ρ ∗ . Therefore, determining the transport scaling withrespect to ρ ∗ is critical. When α = 1 the scaling law is called gyro-Bohm. This isthe expected scaling from most local turbulence theories for which the turbulent scalelength is proportional to ion Larmor radius, ρ i . For α = 0 the scaling is called Bohm,when the turbulence scale length involves the macroscopic size of the plasma. However,these scalings do not agree always with the experimental observations. For a fulldiscussion see Refs.(1; 3).22

2Aims of the project2.1 AimsIn this thesis, the main goal is to study the anomalous transport of impurities due todrift wave microinstabilities. To study the characteristics of the unstable drift modesresponsible for the transport of impurities under fusion plasma conditions and to analyzefeedback mechanisms of the impurities on the unstable modes (chapters 3 and 4).Encouraged by the experimental observations which suggested charge dependence ofthe impurity anomalous transport characteristics (transport coefficients), unexplainedby the existing models, we follow our study on to investigating this charge dependence,and its possible underlying physical mechanism (chapter 5). As an important goal wego on to implement our impurity transport model for the transport modeling of theJET tokamak experiments, in which impurity seeding were applied. In these experimentsplasma confinement was improved by injecting Ne into the plasma edge and noimpurity accumulation in the plasma core were observed. Our aim is to find a possibleexplanation for the observed plasma behavior (chapter 7).At the end (chapter 8), to try a fresh challenge, we start looking at the problem ofturbulent and non-diffusive transport in a different framework. Using fractional kineticsand build on Levy type stable distributions, we study the plasma as a medium awayfrom the Maxwellian equilibrium conditions. We look at the effect of non-Maxelliansystem on the characteristics of the drift wave turbulence.23

2. AIMS OF THE PROJECT2.2 StrategyDuring the course of this thesis, in collaboration with Prof. M. Z. Tokar (from FZJ,Jlich, Germany), a linear fluid model for particle and heat transport was developed, andnumerically implemented as AFC-FL (Anomalous Flux Calculation in Fluid Limit) codein the Fortran language. It solves the dispersion equation, and calculated the transportcoefficients and fluxes of electrons, main and impurity ions. The effects of impurities onthe unstable mode characteristics are taken into account in an iterative procedure, andan arbitrary number of impurity species can be taken into consideration. The model hasbeen benchmarked against the quasi-linear gyro-kinetic code QuaLiKiz developed byDr. C. Bourdelle (from CEA, Cadarache, France) (chapter 6). The AFC-FL transportmodel for impurities has been numerically implemented into the 1-D transport codeRITM, developed by Prof. M. Z. Toka,r and used in a series of transport modeling forthe JET experiments with Ne seeding.24

3Drift Waves3.1 IntroductionTo achieve the needed conditions for fusion in a tokamak it is necessary to confine theplasma for a sufficient time, as described in sec.1.6. However, confinement is limited bytransport processes of energy and particles across magnetic field lines. In the absence ofinstabilities, the confinement of a toroidally symmetric tokamak plasma is determinedby Coulomb collisions. However, the transport which actually occurs does not agreewith the calculated values from collisional models.The plasma in tokamak is inhomogeneous, therefore, the density and temperaturegradients give rise to electron and ion diamagnetic drifts vD s 1 across the magnetic fieldB. This drift will give rise to collective oscillations which are called drift waves. In thepresence of fluctuations of the plasma dynamical quantities (density, velocity, pressure,. . . ), these waves can become linearly unstable with their amplitude growing in time.These unstable modes play a crucial role in the mechanism of anomalous transport.The theoretical picture of anomalous transport is that the free energy released by aninstability drives a steady level of fluctuations in the associated perturbed quantities.In this turbulent state fluctuations result in a radial transport of particles and energy.1 The subscript s is used to label the species of plasma particle.25

3. DRIFT WAVESOur goal is to study the anomalous transport due to drift wave microinstabilities intokamak plasmas. At first, we find the physical concepts and relevant equations whichare needed in order to achieve this goal. The objective of the transport theory is thedetermination of the magnitude and radial profiles of the plasma parameters (such asdensity, temperature, etc.). To do so it is useful to formulate the transport problem ina macroscopic way. On transport time scales a fluid representation is generally used.In the present chapter we present the underlying physics, and a set of fluid plasmaequations, for the study of the anomalous transport driven by drift wave instabilitiesin magnetized plasmas, which is the subject of the following chapters.3.2 Hamiltonian MechanicsThe dynamics of most fundamental physical systems can be derived from a variationalprinciple, Hamilton’s principle. This description is obtained by representing the completemechanical configuration of the system as a single point, q, in an n dimensionalconfiguration space. Here n is called the number of degrees of freedom of the system.We assume that the dynamics of the system would be completely defined if in additionto the configuration q, we are given the velocity vector ˙q: the dynamics takes placeon the space of all configurations and velocities. The Lagrangian, L(q, ˙q, t), gives allthe information required to construct the dynamics. For a simple particle system theLagrangian is the difference between the kinetic and potential energies.3.2.1 Principle of Least Action and Equation of MotionConsidering a path q(t), t 0 < t < t 1 , in the configuration space. For each point on thepath, the velocity is defined as the time derivative of q. The action S[q], is a functionalon the space of such paths; the action of a path is defined byS[q] =∫ t1t 0L(q(t), ˙q(t), t)dt (3.1)Hamilton’s principle asserts that the dynamically allowed paths are stationarypoints of the action, for the class of paths with given initial and final conditions,q(t 0 ) = q 0 and q(t 1 ) = q 1 . Demanding that the path be stationary gives the Euler-Lagrange equations of motion26

3.2 Hamiltonian Mechanics0 = − d ∂Ldt ∂ ˙q i + ∂L∂q i (3.2)In general these are a set of second order differential equations for each configurationpoint; their solution requires that an initial condition (q(0), ˙q(0)) be given.Equations in Hamiltonian form are obtained by the Legendre transformation fromvelocity to canonical momentum:The definition of the canonical momenta isH(p, q) = p · ˙q − L(q, ˙q, t) (3.3)3.2.2 Maxwell-Lorentz Equationsp = ∂ L(q, ˙q, t) (3.4)∂ ˙qA plasma is a gas of charged and neutral particles viewed on scale lengths that arelarge compared to the Debye Length (see sec.1.8.2).Plasma usually contain several species of charged particles, typically electrons withcharge e e = −e and mass m e , and at least one species of ions with charge e i = Z i e andmass m i .Each (non-relativistic) particle can be described by the LagrangianL(q(t), ˙q(t), t) = m s2 | ˙q|2 − e s Φ(q, t) + e s ˙q · A(q, t) (3.5)cwhere Φ(q, t) is the electrostatic potential and A(q, t) is the electromagnetic potential.The electric, E, and magnetic, B, fields can be characterized by the electrostatic andelectromagnetic potentials asE = −∇Φ − 1 ∂Ac ∂tB = ∇ × A (3.6)27

3. DRIFT WAVESA Hamiltonian description of the motion is obtained by defining the canonical momentumby (3.4)and the Hamiltonian by (3.3)p = mv + e c A (3.7)H = 12m |p − e c A|2 + eΦ (3.8)This Hamiltonian has a clear form, being the sum of the kinetic and potentialenergy. The Hamiltonian yields equations of motion asṗ i = [p i , H] P = − ∂H∂q i ,˙qi = [q i , H] P = ∂H∂p i(3.9)To exactly describe the dynamics of a plasma we must follow the motion of allits particles, and the dynamics of the fields themselves.equationsThe latter obey Maxwell’s∇ · E = 4πρ c∇ × B − 1 ∂Ec ∂t = 4π c J∇ × E + 1 ∂Bc ∂t = 0∇ · B = 0 (3.10)The interactions of the fields with the particles occur only through the plasmacharge density ρ c and current J. If the plasma consists of particles of species s at thepositions (x i s, v i s) for i = 1, ..., n, then the number density isThe charge density isn s (x, t) =n∑δ(x − x i s(t)) (3.11)i=1ρ c (x, t) = ∑ sδe s n s (x, t) (3.12)The current density J is obtained from the mean velocities28

3.3 Charged Particle Motionasn s v s (x, t) =n∑vs(t)δe i s (3.13)i=1J(x, t) = ∑ sδe s v s (x, t)n s (x, t) (3.14)Charge conservation follows from the definitions (3.13) and (3.14) as well as fromthe first and second Maxwell equations∂ρ c∂t + ∇ · J = 0 (3.15)Thus, the electric and magnetic fields at any point x in physical space depend onthe instantaneous position of all particles.3.3 Charged Particle MotionUnderstanding of the motion of a set of charged particles in an electromagnetic field isthe subject of plasma physics. A single charged particle entering a region of space wherean electromagnetic field is present (produced by some unspecified external forces) willundergo a uniform acceleration in the direction of the field when the field is reducedto a constant electric field E. In the presence of a constant magnetic field B, the particlewill perform a helical motion, wrapping itself around a field line. If the fields aresimultaneously present, the motion is more complex and the particle does not remainattached to a single magnetic field line. It rather drifts through space in a directionperpendicular to both the electric and the magnetic fields. In reality fields are spatiallyinhomogeneous and non-stationary, which introduces extra drift motions and increasesthe complexity of the problem.In this section we study the aspects of the motion of the individual charged particlein electromagnetic fields.29

3. DRIFT WAVES3.3.1 Equation of MotionFollowing the Hamilton equations (3.9) the equations of motion 1 are (see Refs. (12; 18))ẋ = v˙v = q (E + v × B) (3.16)mwhere q denotes the particle charge. We clearly recognize in eq. (3.16) the Lorentzforce asF = q(E + v × B) (3.17)An interesting feature of the Hamiltonian (3.8) is that it is independent of the magneticfield B. This is physically clear: as the magnetic field in the Lorentz force isalways perpendicular to the velocity of the particle, the magnetic field does no work.formEquations of motion (3.16) can be combined and rewritten simply in the followingwhere all field are evaluated at x.m d2 xdt 2dx= q(E + × B) (3.18)dtIn tokamaks the E and B fields are nonuniform and time dependent. Therefore, it isimportant to understand the motion of a charge particle under such conditions. In thefollowing we will present an overview of a single charge particle’s motion in uniform,time independent fields, as simplified approximation and nonuniform, time dependentE and B, fields as it would be inside tokamak plasmas.3.3.2 Uniform E and B fieldsIn uniform, time independent fields the Lorentz equation (3.18) is easily solved. In thisapproximation the E and B fields are assumed to be prescribed and not affected by1 We use MKS units.30

3.3 Charged Particle Motionthe charged particles.In order to get insight into the problems of charged particle motion, we brieflyconsider the simplest cases.3.3.2.1 B = 0First, we assume that there is no magnetic field and that the electric field is uniformand directed along x axis. The equation of motion (3.18) is reduced to˙v x = q m E x (3.19)In this trivially simple case, the solution represents a uniformly accelerated motionof the particle, parallel or antiparallel to E x field according to the sign of q. Thesolution isx(t) = x 0 + v x0 t + 1 q2 m E xt 2 (3.20)where x 0 and v x0 are the initial position and velocity. The trajectory of the particle is,in general, a parabola.3.3.2.2 E = 0In this case, a charged particle has a simple cyclotron gyration. Following eq. (3.18),the equation of motion is˙v = q m v × B (3.21)where for simplicity we used the notation dx/dt = v. Taking ẑ to be the direction ofB (B = Bẑ), we have˙v x = q m Bv y ˙v y = q m Bv x ˙v z = 0 (3.22)¨v x = qB m ˙v y = −( qB m )2 v x¨v y = qB m ˙v x = −( qB m )2 v y (3.23)31

3. DRIFT WAVESThis describes a single harmonic oscillator at the cyclotron frequency which is definedasΩ c ≡ qB m(3.24)The solution of eq. (3.23) is thenv x,y = v ⊥ exp(iΩ c t + iδ x,y ) (3.25)where v ⊥ , is a positive constant denoting the speed in the plane perpendicular to B.We can choose the phase δ so thatThenv x = v ⊥ exp(iΩ c t) = ẋ (3.26)Integrating again, we havev y = m qB ˙v x = 1 Ω c˙v x = iv ⊥ exp(iΩ c t) = ẏ (3.27)x − x 0 = −i v ⊥Ω cexp(iΩ c t)y − y 0 = v ⊥Ω cexp(iΩ c t) (3.28)We define the Larmor radius asFigure 3.1: Larmor orbits in a magnetic field.32

3.3 Charged Particle Motionρ L ≡ v ⊥= mv ⊥Ω c qBTaking the real part of eq. (3.28), we have(3.29)x − x 0 = ρ L sin Ω c t y − y 0 = ρ L cos Ω c t (3.30)The eqs. (3.30) describe a circular orbit around a guiding center positioned at(x 0 , y 0 ), which is fixed (see figure. 3.1). The direction of the gyration is always suchthat the magnetic field generated by the charged particle is opposite to the externallyimposed field. Therefore, plasma particles reduce the magnetic field, and plasmas arediamagnetic. In addition to this motion there also the motion along the magnetic fieldlines with an arbitrary velocity v z which is not affected by B. The trajectory of acharged particle in space hence, is a helix.3.3.2.3 Particle DriftIn the presence of an electric field, the motion will be the sum of two motions: theusual circular Larmor gyration plus a drift of the guiding center. If we choose E to lieon the x − z plane so that E y = 0, as before, the v z is not related to the perpendicularcomponents, v x,y , and can be treated separately. The equation of motion isits z component is˙v = q (E + v × B) (3.31)morThis is straightforward acceleration along B.eq. (3.31) arev˙z = q m E z (3.32)v z = qE zm t + v z0 (3.33)The perpendicular components ofv˙x = q m E x + Ω c v yv˙y = −Ω c v x (3.34)33

3. DRIFT WAVESTaking the derivatives of ˙v x,y , we have (for constant E)¨v x = −Ω 2 cv x¨v y = −Ω 2 c( E xB + v y) (3.35)If we replace v y by v y + (E x /B) the eq. (3.35) is reduced to the previous case. Thesolution of the eqs. (3.26,3.27) are therefore,v x = v ⊥ exp(iΩ c t) = ẋIntegrating again and taking the real part we havev y = iv ⊥ exp(iΩ c t) − E xB = ẏ (3.36)x − x 0 = ρ L sin(Ω c t)y − y 0 = ρ L cos(Ω c t) − E xB t (3.37)There is now in addition to the Larmor motion same as before, there is a drift, v gc ,of the guiding center in the −y direction (for E x > 0).Figure 3.2: Particle drifts in crossed electric and magnetic fields.The origin of this guiding center drift motion can be easily understood in physicalterms. See the ion’s trajectory in figure 3.2, in the absence of E, it gyrates around afixed center. Now if an electric field, E is present, as the ion moves to the right underthe action of B it is accelerated by E, hence, the local Larmor radius increases. On the34

3.3 Charged Particle Motionsecond half of its cycle, the ion is moved to the left by B but now, it is slowed down byE. The resulting distortion of the trajectory is a gradual decrease of the instantaneousLarmor radius. After a complete orbit, the position of the particle is shifted down; thisis the origin of the electric drift.We can obtain a general formula for the guiding center velocity, v gc by replacingthe electrostatic force, qE in the equation of motion (3.18) by a general force F. Theequation of motion then isIn the parallel direction to B field, we obtain˙v = 1 (F + qv × B) (3.38)mv ˙ ‖ = 1 m F ‖ (3.39)The solution of the parallel equation of motion is simply a uniformly acceleratedmotion in the direction of the magnetic field B,Taking the cross product of eq.(3.38) with B, we havev ‖ (t) = v ‖0 + 1 m F ‖t (3.40)m ˙v × B = F q × B + B × (v × B) = F q × B + vB2 − B(v · B) (3.41)In the perpendicular direction to the magnetic field B, the above equation can besimplified as1 dv ⊥Ω c dt= F × BqB 2 + v ⊥ (3.42)If F is directed along x axis then the solutions of the above equation is (For aconstant F)v x = v ⊥ exp(iΩ c t)v y = iv ⊥ exp(iΩ c t) − F xqB(3.43)35

3. DRIFT WAVESTherefore, the motion of the particle in the plain perpendicular to the magnetic fieldis a sum of two motions: one a gyrating motion around the magnetic field lines withthe Larmor frequency Ω c , and the second motion is the drift motion of the particle inthe direction perpendicular both to the force F and magnetic field B with the constantvelocityv ⊥gc = F q × B/B2 ≡ v f (3.44)v f is the guiding center drift caused by F. If the force which is acting on the particleis of electrostatic nature, F = qE, therefore we recover the guiding center drift whichwe have obtained previously:v E = E × BB 2 (3.45)This is called the electric drift and it is important to note that v E is independentof q, m. Under the action of an electric field, the negative and positive charges drift inthe same direction. If F is the force of gravity mg, there is a driftv g = m qg × BB 2 (3.46)This is similar to the drift v E in that it is perpendicular to both the force and B,but it differs in one important respect. The drift v g changes sign with the particle’scharge. Under a gravitational force, ions and electrons drift in opposite directions, sothere is a net current density in the plasma given byj = n(M + m) g × BB 2 (3.47)The physical reason for this drift (see figure 3.3) is again the change in Larmor radiusat the particle gains and loses energy in the gravitational field. Now the electrons gyratein the opposite direction to the ions, but the force on them is in the opposite directionto the ions, but the force on them is in the same direction, so the drift is in the oppositedirection. The magnitude of v g is usually negligible, but when the lines of force arecurved, there is an effective gravitational force due to centrifugal forces. This force,which is not negligible, is independent of mass.36

3.3 Charged Particle MotionFigure 3.3: Drift of a gyrating particle in a gravitational field.3.3.2.4 Validity LimitIf we take the limit of B → 0 in the guiding center drift velocity in eq. (3.45); thedrift velocity becomes infinite! The explanation for this physical paradox can be that:somewhere we have crossed the barrier of the classical mechanics and entered the relativisticmechanics, in which the equation of motion (3.31) is not valid. To find a precisecriterion of applicability of the equation of motion we must guarantee that the absolutevalue of the drift velocity is much smaller than the speed of light:v E = E ⊥BB 2 c ≪ c (3.48)which requires E ⊥ ≪ B. Also, we have a criterion involving the parallel electric field.The increment in velocity over a Larmor radius must also be much smaller than c.From eq. (3.32) we find∆v z = q m E zc ≪ c (3.49)Therefore, in order for the equation of motion (3.31) to be valid we need the followingcondition on the absolute values of the electric and magnetic fields:E ≪ B (3.50)Whenever this condition is not satisfied, one needs to use the relativistic equationof motion for treating the problem. This condition is also necessary for the force F inthe eq. (3.38).37

3. DRIFT WAVES3.3.3 Nonuniform E and B fieldsFor uniform fields we are able to obtain exact expressions for the guiding center drifts.As soon as we introduce inhomogeneity, the problem becomes more complicated tosolve exactly.Figure 3.4: Drift of a gyrating particle in a nonuniform magnetic field.3.3.3.1 ∇B⊥B: Grad−B DriftIf we have the a magnetic field where the magnetic field lines are straight, but theirdensity increases, as ∇B⊥B, for example, with a gradient in the y direction (see figure3.4), the gradient in |B| causes the Larmor radius to be larger at the bottom of theorbit than at the top (the same mechanism as was discussed previously in the presenceof a constant electric field), and this will lead to a drift, in opposite for ions and electrons,perpendicular to both B and ∇B. Considering the Lorentz force F = qv × B,averaged over a gyration, ¯F x = 0, since the particle spends as much time moving up asdown. To find ¯F y we take the approximation ρ L /L ≪ 0 where L is the scale length of∂B z /∂y. We can Taylor expand B about the point x 0 = 0, y 0 = 0 and use eq. (3.28)After averaging over a gyration we getB = B 0 + (r · ∇)B + ...B z = B 0 + y(∂B z /∂y) + . . . (3.51)¯F y = ∓ 1 2 qv ⊥ρ L ( ∂B∂y ) (3.52)38

3.3 Charged Particle MotionThe generalized guiding center drift velocity isThe drift v ∇Bv ∇B = ± 1 2 v ⊥ρ LB × ∇BB 2 (3.53)is called the grad−B drift, it is opposite direction for ions andelectrons and causes a current transverse to B.3.3.3.2 Curved B: Curvature DriftNow if we assume the magnetic field lines to be curved with a constant radius ofcurvature R c , and we take |B| to be constant (see figure 3.5).Figure 3.5: A curved magnetic field.A guiding center drift arises from the centrifugal force felt by the particles as theymove along the field lines in their thermal motion. The average centrifugal force isF cf = mv2 ‖R cˆr (3.54)where v‖ 2 is the average square of the component of the velocity along B. According tothe eq. (3.44), this gives rise to a driftv C = 1 qF cf × BB 2The drift v C is called the curvature drift.= mv2 ‖qB 2 R c × BR 2 c(3.55)39

3. DRIFT WAVESWe need to compute the grad−B drift that accompanies this when the decrease of|B| with radius is taken into account. In the vacuum we have ∇ × B = 0. In cylindricalcoordinates we haveThus(∇ × B) z = 1 r∂∂r (rB θ) = 0B θ ∝ 1 r(3.56)Using eq. (3.53), we have|B| ∝ 1 R c∇|B||B|= − R cR 2 c(3.57)v ∇B = ∓ 1 v ⊥ ρ L2 B 2 B × ∇B R cRc2 = 1 m R c × B2 q v2 ⊥RcB 2 2 (3.58)Adding this to the v C we have the total drift in a curved vacuum field:v C + v ∇B = m qR c × BR 2 cB 2 (v2 ‖ + 1 2 v2 ⊥ ) (3.59)The physical meaning of this is that in tokamaks where the magnetic fields arecurved for the purpose of confining a thermonuclear plasma, the particles will drift outof the tokamaks.3.3.3.3 Magnetic MirrorsWe consider a magnetic field in the z direction with its magnitude varying along the zand axisymmetric, B θ = 0 and ∂/∂θ = 0. Since we have ∇ · B = 0, there is necessarilya component B r . This gives rise to a force which can trap a particle in a magnetic field.From ∇ · B = 0 we get:1r∂∂r (rB r) + ∂B z∂z = 0 (3.60)If ∂B z /∂z is given at r = 0 and does not vary much with r, we findB r ≃ − 1 2 r[∂B z∂z ] r=0 (3.61)The variation of |B| with r causes a grad−B drift of guiding centers about the axisof symmetry, but there is no radial grad−B drift, because ∂B/∂θ = 0. From the zcomponent of the Lorentz force averaged over one gyration is then40

3.3 Charged Particle MotionF z = − 1 2mv 2 ⊥B∂B z∂zWe define the magnetic momentum µ of the gyrating particle as(3.62)µ ≡ − 1 2mv 2 ⊥BSince the particle’s energy must be conserved, we have(3.63)ddt (1 2 mv2 ‖ + 1 2 mv2 ⊥ ) = d dt (1 2 mv2 ‖+ µB) = 0 (3.64)Multiplying the component of the equation of motion along B by v ‖ we get:dv ‖mv ‖dt = −µ∂B dz∂z dt = −µdB dt(3.65)where we made use of the relation v ‖ = dz/dt. Replacing the results inside eq. (3.64)we haveddt µ = 0 (3.66)This is part of the basis for plasma confinement: magnetic mirrors. As a particlemoves from a weak field region to a stronger region in the course of its thermal motion,it feels an increasing B and therefore its v ⊥ must increase in order to keep µ constant.Since its total energy must remain constant, v ‖ must necessarily decrease. If B is highenough, v ‖ eventually becomes zero; and the particle is reflected back to the weak fieldregion. Therefore, the particle is trapped.3.3.4 Nonuniform E FieldWe assume that the magnetic field is uniform while the electric field is nonuniform forexample in x direction, and to vary sinusoidallyThe equation of motion isE ≡ E 0 (cos kx)ˆx (3.67)m ˙v = q[E(x) + v × B] (3.68)41

3. DRIFT WAVESIf the electric field is weak, we can use the undisturbed orbits to evaluate E x (x).The orbit in the absence of E field is given in the eq. (3.28). From eqs. (3.68) we have¨v y = −Ω 2 cv y − Ω 2 E 0cB cos k(x 0 + ρ L sin Ω c t) (3.69)The solution is the sum of a gyration at Ω c and a steady drift v E . We are interestedin finding an expression for v E , therefore, in order to take out the gyration we averageover a cycle. It gives ¯v x = 0. In eq. (3.69) the oscillating term ¨v y averages to zero, andwe haveWe expand the cosine¯¨v y = 0 = −Ω 2 c ¯v y − Ω 2 cE 0B cos k(x 0 + ρ L sin Ω c t) (3.70)cos k(x 0 + ρ L sin Ω c t) = cos(kx 0 ) cos(kρ L sin Ω c t) − sin(kx 0 sin(kρ L sin Ω c t)) (3.71)and used the approximation of small Larmor radius, kρ L ≪ 1 with the taylor expansionscos ɛ = 1 − 1 2 ɛ2 + . . .sin ɛ = ɛ + . . . (3.72)For ¯v y we have¯v y = − E 0B (cos k(x 0)(1 − 1 4 k2 ρ 2 L) = − E x(x 0 )B (1 − 1 4 k2 ρ 2 L) (3.73)Thus the usual E × B drift is modified by the inhomogeneityv E = E × BB 2 (1 − 1 4 k2 ρ 2 L) (3.74)An ion with its guiding center at a maximum of E actually spends most of its timein the region with weaker E. Its average drift, therefore, is less than E/B evaluated atthe guiding center. For an arbitrary variation of E we need only to replace ik with ∇in eq. (3.74) asv E = (1 + 1 4 ρ2 L∇ 2 ) E × BB 2 (3.75)42

3.3 Charged Particle MotionThe second term is called the finite-Larmor-radius effect. Since ρ L is much larger forions than electrons, v E is no longer independent of species. If a density inhomogeneityoccurs in a plasma, an electric field can cause the ions and electrons to separate, generatinganother electric field. If there is a feedback mechanism that causes the secondelectric field to enhance the first one, E grows indefinitely, and the plasma in unstable.Such an instability is called drift instability. The nonuniform E field effect, is importantat relatively larger k, or smaller scale lengths of the inhomogeneity. For this reasondrift instabilities belong to a general class of microinstabilities.Now if we assume E and B to be uniform in space but E varying sinusoidally intime and let it lie along x axisE = E 0 exp(iωt)ˆx (3.76)The guiding center motion has two components. The y component, perpendicularto both E and B, is the usual E × B drift, except that v E now oscillates slowly at thefrequency ω. The x component is a new drift along the direction of E, and is calledthe polarization drift:v p = 1 dEΩ c B dt(3.77)Since v p is in opposite directions for ions and electrons, there is a polarizationcurrent: for Z = 1, this isj p = ne(v ip − v ep ) = ne (M + m)dEeB2 dt = ρ dEB 2 dt(3.78)Considering an ion at rest in a magnetic field, if a field E is suddenly applied, thefirst thing the ion does is to move in the direction of E. Only after reaching a velocity vdoes the ion feel a Lorentz force ev ×B and began to move downward. If now E is keptconstant, there is no further v p drift but only v E drift. However, if E is reversed thereis again a momentary drift, this time to the left. Thus v p is a start up drift due theinertia and occurs only in the first half-cycle of each gyration during which E changes.43

3. DRIFT WAVES3.3.5 Summary of Guiding Center DriftsGeneral force F :Electric field :v f = 1 qF × BB 2v E = E × BB 2Gravitational field :v g = m qg × BB 2Nonuniform E :Grad − B drift :Curvature drift :Curved vacuum field :v E = (1 + 1 4 ρ2 L∇ 2 ) E × BB 2v ∇B = 1 2 v ⊥ρ LB × ∇BB 2v C = mv2 ‖qR c × BR 2 cB 2v C + v ∇B = m q (v2 ‖ + 1 2 v2 ⊥ )R c × BR 2 cB 2P olarization drift : v p = 1 dEΩ c B dt3.4 Fluid DescriptionThe equations of motion (3.9) together with the Maxwell equations (3.10) give a completedescription of plasma motions. Though formally exact, these equations are almostuseless since it is neither possible, nor even desirable, to know the positions and velocitiesof each individual particle in the plasma. Much of the analysis in plasma physicsis devoted toward deriving approximative sets of equations that are traceable. Kinetictheory approximations replace the Euler-Lagrange equations of motion with an equationfor smoothed particle distributions in phase space. Fluid theory approximationsattempt to derive equations directly for the plasma density and current, usually requiringonly minimal information about the particle distribution.44

3.4 Fluid DescriptionThe evolution of the distribution function of species s is determined by the Vlasovequation, see Refs. (12; 13; 19):df sdt = ∂f s∂t + v · ∇f s + q sm s(E + v × B) · ∂f s∂v = 0 (3.79)The Vlasov equation is a non-linear equation because the electric and magnetic fieldare unknown functions along with f s and the Vlasov equation is coupled to the full setof Maxwell equations.The fluid equations are simply moments of the Vlasov equation. The distributionfunction is assumed Maxwellian:f(r, v; t) =n(2πT/m) 3/2 exp(−m/(2T )[v − u]2 ) (3.80)when averaging over velocity we separate v into a average fluid velocity u and a randomvelocity w: v = u + w and w = 0The lowest moment is obtained by integrating eq. (3.79)∫ ∂f∂t dv + ∫v · ∇fdv + q mThis yield the equation of continuity:∫(E + v × B) · ∂f dv (3.81)∂v∂n∂t+ ∇ · (nu) = 0 (3.82)where the average velocity u is the fluid velocity. We can introduce the particle flux asΓ = nu.The next moment of the Vlasov equation is obtained by multiplying eq. (3.79) bymv and integrating over dv. We have∫mv ∂f∂t dv + m ∫∫v(v · ∇)fdv + qWe obtain the fluid equation of motionv(E + v × B) · ∂f dv (3.83)∂vmn[ ∂ + u · ∇]u = qn(E + u × B) − ∇ · P (3.84)∂t45

3. DRIFT WAVESwhere P is the pressure tensor defined asP ≡ mnww = pI + π (3.85)If the distribution function for the random velocity is isotropic then w x w x = w y w y =w z w z = (1/3)ww, w x w y = w y w z = w x w z = 0, so that P = pI with the identity tensor,I. The quantity p is the scaler pressure for particles of a given species. Here π is thestress tensor. We can define a temperature T through the pressure asp = mn w23= nT (3.86)This relation is valid assuming plasma as an ideal gas in equilibrium which is a validassumption for fusion plasmas. Throughout this thesis we will always use energy unitsfor measuring temperatures: the relation between T and temperature t measured inKelvins is: T = k B t, where k B is the Boltzmann constant.The eq. 3.84 describes the flow of the momentum. To treat the flow of the energy,we can take the next moment of the Vlasov equation by multiplying by 1/2mvv andintegrating. We would then obtain the heat flow equation:∂∂t [nmu2 + 3 p] + ∇ · [nmu2 + 5 p + π]u + ∇ · q = qnE · u (3.87)2 2 2 2We have introduced the notationq = m (nwww) (3.88)2The vector q is the heat flux density transfered by particles of a given species anddenotes the transport of energy associated with the random motion in the coordinatesystem in which the plasma globally is at rest at a given point in space.The first term in the eq. (3.87) represents the change in the total energy of particlesof a given species: this consists of the kinetic energy nmu 2 /2 and the internal energy(3/2)p per unit volume. Divergence terms represent the total energy flux and consistof the macroscopic transport of the total energy with the velocity u, the microscopicenergy flux, i.e., the heat flux q, and the work done by the total pressure forces. Theterm on the right side takes account for the work done by the presence of an electric46

3.4 Fluid Descriptionfield. Using the continuity equation and the equation of motion we can eliminate thekinetic energy from the eq. (3.87). The result is then an equation for the transport ofthe internal energy, or the heat balance equation:3 ∂2 ∂t nT + ∇ · (3 nT u) + nT ∇ · u + π∇ · u + ∇ · q = 0 (3.89)23.4.1 Summary of Fluid EquationsThe set of fluid equations which we have derived by taking the moments of the VlasovequationEquation of continuity :∂n∂t + ∇ · (nu) = 0Equation of momentum balance :mn[ ∂ + u · ∇]u = qn(E + u × B) − ∇(nT ) − ∇ · π∂tEquation of heat balance :3 ∂2 ∂t nT + ∇ · (3 nT u) + nT ∇ · u + π∇ · u + ∇ · q = 023.4.2 Closure of Fluid EquationsFollowing the derivation of the fluid equations presented in sec.3.4 we find that theequations are the result of a truncation of an infinite hierarchy of moments of kineticequation. However, the equations are not closed meaning that there remain in theseequations a number of undetermined quantities: electromagnetic fields (E, B), andthe fluxes (Γ, q, π). In a self-consistent theory the electric and magnetic fields aredetermined in terms of plasma parameters by the Maxwell equations. In general thisproblem is very complex and hence, we need an approximated approach. An approximatedapproach that is often used is to consider the electromagnetic fields as either47

3. DRIFT WAVESprescribed as a given function or statistically.The most important closure relations are the so called transport equations. Theseequations relate the fluxes to the thermodynamic forces that produce them which arerelated to the gradients of the thermodynamic quantities: ∇n, ∇T , ∇φ and ∇u. Here∇φ is the electric potential: E = −∇φ.3.5 Transport EquationsGrouping all the relevant fluxes into a supervector J, and all the thermodynamic forcesinto X, the general transport equations can be written as:J = F(X) (3.90)where the constraint on F(X) is the condition F(0) = 0. This function could benonlinear, it could depend on higher order spatial derivatives of the thermodynamicfunctions, etc. In the vast majority of cases studied in plasma transport theory it isassumed that the transport equations are linear. In the linear regime, the flux vectorcomponents J are related to the force vector components X by a transport matrix L:J = L X ⇔ J a = L ab X b (3.91)The constant components of the transport matrix are called transport coefficients.In particular the diagonal elements are: the diffusion coefficient D, relating the particleflux to the density gradient, the heat conductivity (or heat diffusivity) κ = nχ, relatingthe heat flux to the temperature gradient, and the electric conductivity σ relating theelectric current density to the electric field. In general the transport matrix is nondiagonal:every flux is determined by all the gradients present in the system.3.5.1 Experimental AspectsThere are two classes of techniques for measuring transport coefficients. These yield ingeneral different results because they measure different quantities, see Ref. (20). A firstclass is called power balanced method. These are static methods acting on plasmas ina steady state. In the parallel direction to the magnetic field lines the transport in the48

3.6 Fluid Driftsequilibrium state is such that density, temperature and velocity are constant on eachmagnetic surface. In the perpendicular direction the fluid equations in sec. 3.4.1 canbe represented in a general form of:∂h∂t + ∇ · F = S (3.92)where h is the particle or the thermal energy density, F the corresponding flux, andS the source generated by external heating and fueling processes.(∂h/∂t = 0, u = 0).In steady stateBy accurate measurement of the sources integrated inside amagnetic surface one can find the flux across this magnetic surface, F. On the otherhand a detailed measurement of the density and temperature profiles provides the valueof the local gradients, ∇h. From these two measured values, one can calculate transportcoefficients by simply dividing the net local fluxes (their value at a given radial distancefrom the magnetic axis) by the corresponding gradients:F = −D∇h + Vh (3.93)here, the flux is not only diffusive and it can also be convective with D being the diffusioncoefficient and V being the convective velocity.The second experimental strategy called perturbative transport analysis, uses asmall well localized produced perturbation of the forces (i.e. the gradients) in a steadystate plasma, and follows the evolution in space and time of the effect of this perturbation.These perturbations can be at the center of the tokamak, such as sawtoothcrashes (see section 1.7.3.1), which are present near the core of a tokamak and whosepropagation on their way towards the edge are followed in time. Alternatively, localperturbations can be produced by the injection of solid pellets into the plasma. Thismethod requires very precise diagnostics, with very good spatial and temporal resolutions.The perturbative method allows a separate measurements of the influence ofeach thermodynamic force on a given flux.3.6 Fluid DriftsKnowing that a fluid element is composed of many individual particles which undergodrift motions (see sec.3.3.2.3), one would expect the fluid to have drifts as well. How-49

3. DRIFT WAVESever, in general the fluid drifts may differ from the actual particle or guiding centerdrifts. The reason for the differences is that the fluid picture averages particle velocitiesat a point, regardless of where the guiding centers are located, while the particle driftsare obtained by identifying the particle by its guiding center.3.6.1 Fluid Drifts ⊥BThe fluid equation of motion can be obtained from the fluid momentum balance equation(see second equation in sec.3.4.1)∂u∂t + (u · ∇)u = q 1(E + u × B) − ∇p − ∇ · π (3.94)m mnAssuming the electrostatic approximation (i.e., δB = 0) and B = Bẑ, we take thevector product of the eq. (3.94) with ẑ( ∂ ∂t + u · ∇)ẑ × u = q 1[ẑ × E + B(u(ẑ · ẑ) − ẑ(ẑ · u))] − × (∇p + ∇ · π) (3.95)m mnẑBy writing d/dt = ∂/∂t + u · ∇ we havev ⊥ = 1 B (E × ẑ) + 1 Bmqddt1(ẑ × u) +qnẑ × (∇p) + 1 × (∇ · π) (3.96)qnẑThe first term is the v E drift which is the same as the guiding centers. The secondterm is the polarization drift, the third term is the diamagnetic drift. The last term isdue to the stress tensor π v . It contains a viscosity part and a finite-Larmor-radius partπ l (see Ref.(21)).There appears a term in eq. (3.96) which is proportional to ∇p. This term onlyappears in the fluid equations, therefore, diamagnetic drift is a drift associated withthe fluid elements and not a particle drift. The physical reason for this drift can beseen from figure (3.6). There we have drawn the orbits of ions gyrating in a magneticfield. There is a density gradient toward the left, as indicated by the density of theorbits. Through any fixed volume element there are more ions moving downward thanupward, since the downward moving ions coming from a region with higher density.50

3.6 Fluid DriftsThere is, therefore, a fluid drift perpendicular to ∇n and B. Even the guiding centersare stationary. The diamagnetic drift has opposite signs depending on particle chargeq, because the direction of gyration changes for electrons and ions but diamagnetic driftdoes not depend on the mass. Since ions and electrons drift in the opposite directions,there is a diamagnetic current j D . In the particle picture, one would not expect tomeasure a current if the guiding centers do not drift. In the fluid picture, the currentj D flows wherever there is a pressure gradient.Figure 3.6: Origin of the diamagnetic drift.The curvature drift which appears in the single particle picture also exists in thefluid picture, since the centrifugal force is felt by all the particles in a fluid element asthey move around a bend in the magnetic field. A term ¯F cf = nmv‖ 2 /R c = nT ‖ /R c hasto be added to the right hand side of the fluid equation of motion. This is equivalentto a gravitational force which leads to a drift v g = (m/q)(g × B)/B 2 .The grad−B drift, however, does not exist for fluids. Since the Lorentz force isperpendicular to the particle velocity it can not change the energy of any particletherefore, a magnetic field does not change the Maxwellian distribution. If f remainsMaxwellian in a nonuniform B field, and there is no density gradient, then the netmomentum carried into any fixed volume element is zero. There is no fluid drift even51

3. DRIFT WAVESthough the individual guiding centers have drifts; the particle drifts in any fixed fluidelement cancel out.3.6.2 Fluid Motion ‖ BThe ẑ component of the fluid equation of motion is∂u z∂t + (u · ∇)u z = q m E z − 1mn∂p∂z(3.97)We consider a simple case in which u z is spatially uniform and the convective term(u · ∇)u z can be neglected in comparison to the term ∂u z /∂t, and finally we haveneglected the last term due to the stress tensor.∂u z∂t = q m E z − T ∂nmn ∂z(3.98)This shows that the fluid is accelerated along B under the combined q/mE z andpressure gradient forces. If we apply eq. (3.98) to the massless electrons (the limitm e → 0) and specifying q = −e and E = −∇φ we haveeE z = e ∂φ∂z = T e ∂nn e ∂z(3.99)Electrons are so mobile that their heat conductivity is almost infinite. Integratingthe above equation we getn e = n 0 exp(eφ/T e ) (3.100)This is just the Boltzmann relation for electrons. This means that electrons, beinglight, are very mobile and would be accelerated to high energies very quickly if therewere a net force on them. If large number of electrons leave a region they leave behinda large ion charge, the electrostatic and pressure gradient forces on the electrons willbe closely in balance. This condition leads to the Boltzmann relation. Because of thequasi-neutrality of the plasma, any deviation from this neutrality adjusts itself so thatthere is just enough charge to set up the E field required to balance the forces on theelectrons.52

3.7 Drift Waves3.7 Drift WavesA variety of waves can develop and propagate in a plasma as a response to a perturbationof a stationary state. In an inhomogeneous plasma the density and temperaturegradients give rise to electron and ion diamagnetic drifts v s Dacross the magnetic fieldB. This drift will give rise to collective oscillations which are called drift waves (22).These modes play a crucial role in the mechanism of anomalous transport.3.7.1 Drift Wave PropagationA simple example of propagation of drift waves in plasma: we assume a backgrounddensity gradients in the negative x direction, and a small perturbation of density varyingsinusoidally in the y direction, and constant in x direction, we can write n e = n 0 + δn ewhere we have introduced the equilibrium value n 0 of n e . By taking the Boltzmannequation (3.100) and expanding the exponential for eφ/T e we findδn en 0= eφT e(3.101)This is the linearized Boltzmann relation of (3.100) which defines the linear adiabaticelectron model with the requirement of eφ/T e ≪ 1, with φ being the electric potentialcreated by the charge imbalance.This means that the perturbation in density willgenerate an electric field E y in the y direction, which in turn will lead to a v E driftin the x direction. From the linearized continuity equation for ions (neglecting all theother drift velocities and keep only v E )∂n i∂t + v E xdn 0dx = 0 (3.102)The above equation corresponds to the incompressible motion (∇ · v=0). Now ifwe introducev Ex = − 1 B 0∂φ∂y(3.103)We have∂n i∂t + v E x∂n 0∂x = 0 (3.104)53

3. DRIFT WAVESReplacing the eq. (3.103) into eq. (3.104) we find∂n i∂t − 1 ∂φB 0 ∂ydn 0dx = 0 (3.105)Using the quasi-neutrality condition n e = n i and the linearized Boltzmann relation(3.101) we can obtainT e∂φ∂t + (− dn 0en 0 B 0 dx )∂φ ∂y = 0 (3.106)where the term in the parentheses is the electron diamagnetic drift v De in the y direction.To solve the above equation we use the Fourier descriptionCorresponding to the dispersion equation(ω − v De k y )φ = 0 (3.107)v De = ω k y(3.108)This equation shows that the velocity of propagation of the density perturbation isthe electron diamagnetic drift velocity. This is the simplest form of a drift wave.3.7.2 Drift InstabilityAs mentioned before, the electron motion along the magnetic field lines cancels thespace charge. As long as electrons are free to move along B 0 , the Boltzmann relation(3.101) is fulfilled and drift wave is stable. However, there are several effects thatcan limit the mobility of the electrons and therefore, modify the eq. (3.101). If theelectrons are not able to move freely along magnetic field lines a phase shift betweenelectric potential and density in the (3.100) will appear. We then modify the eq. (3.101)asδn en 0= (1 − iδ) eφT e(3.109)This gives the nonadiabatic behavior of electrons and if δ > 0 it corresponds toan instability. The phase shift given by iδ will cause the density maxima to lead thepotential maxima, which produces an exponential growth exp(γt) with γ/ω ∝ δ.54

3.8 Anomalous Transport due to Drift Waves3.7.3 Drift Wave in LaboratoryThe first experimental observation of the drift waves was made in low temperature(T = 2800K) steady state plasmas produced by thermal ionization of Alkali elements(K + and Cs + plasmas) in long cylindrical devices called Q-machines (the Q is forthe quiescent plasmas produced in the system). The drift wave phase shift, δ > 0 ineq. (3.109) is measured in the Q-machine. It was observed that the density δn e /nand potential waves eφT e are approximatively equal in amplitude with density leadingpotential by Ψ ≈ 30 ◦ to 45 ◦ , see figure 3.7 (4).Figure 3.7: The drift wave phase shift, δ > 0 measured in the Q-machine. This figure istaken from Ref. (4).In tokamaks, many experiments around the world have observed the universal appearanceof a broadband of drift wave fluctuations with ω/2π ≈ 50 − 500 kHz atk ⊥ = 1 − 15 cm −1 (see Ref. (22) and the references there after).3.8 Anomalous Transport due to Drift WavesWe now consider the particle transport due to the low frequency modes in magnetizedplasmas. We observe that there are similarities between the continuity equation (3.82)and the diffusion equation55

3. DRIFT WAVES∂n= −∇ · Γ (3.110)∂twhere Γ was introduced as particle flux. According to Fick’s low the flux can be writtenasΓ = −D∇n (3.111)where D is the diffusion coefficient, and the eq. (3.110) reduces to the diffusion equation(3.111)∂n= ∇ · (D∇n) (3.112)∂tWe need to average this equation over the time and spacer variation of fluctuations.W can represent a fluctuation asφ = φ(x) exp(−iω + ik · r) + CC (3.113)where as previous section we considered a density gradient in the x direction. Herethe fluctuations obtain a slow space variation of the amplitude due to inhomogeneity.The flux in the x direction due to the variation in the fluctuations with the use ofFourier representation iswithΓ x = δn k vk ∗ + CC (3.114)v k = v Ex = −i k yB 0φ k (3.115)Using the nonadiabatic relation for electrons as in eq. (3.109), we may rewrite theaveraged fluxΓ x = 2n 0T eeB 0| eφT e| 2 k y δ k (3.116)where δ k represents k dependence of the phase shift δ in eq. (3.109). If we use thedefinition of flux used in eq. (3.111) we findD e = 2 T e| eφ | 2 k y δ k (3.117)L ne eB 0 T e56

3.8 Anomalous Transport due to Drift WavesHere we introduced the density scaling length L n = −1/n 0 (dn 0 /dx). The diffusion,thus, is due to the imaginary part of the deviation from the Boltzmann distribution.This dependence is such that unstable modes cause diffusion in positive x, towards theplasma boundary. For most cases of the practical interest δ k is proportional to L n sothat D remains finite when L n → 0.3.8.1 Mixing Length ApproximationA widely used phenomenological approximation is the so called mixing length approximation,consisting of estimating the instability to saturate due to diffusion by balancingthe linear growth rate with the nonlinearity. In this picture the saturation level for electricpotential is given by Refs.(1; 21; 23)eφT e∝γρ L k x k y c s(3.118)where ρ L is the Larmor radius and c s = √ T e /m i is the sound speed is plasma and m iis the ion mass. Inserting this relation into eq. (3.117) gives:D ∼ γ k 2 x(3.119)This result can be interpreted as a balance between the linear growth of a modeand a stabilization from anomalous diffusion, k 2 xD.57

3. DRIFT WAVES58

4Drift Instability Analysis: LinearFluid Model4.1 IntroductionMicroinstabilities provide a mechanism for the generation of fine scale plasma turbulencein tokamaks. Drift instabilities are a particularly important class of microinstabilitieswhich have been invoked as the source of anomalous transport. In the previouschapter we introduced drift waves and drift wave propagation mechanism.In this chapter we will present two of the most important instabilities for anomalousheat and particle transport in the tokamak plasma core. These are the electrostatic(where only δE are important) drift modes: the trapped electron drift mode (TE mode),and the ion temperature gradient mode (ITG mode).Each of the modes is characterized (in a Fourier description) by a specific relationbetween its frequency, ω, and its wave vector, k: ω = ω(k). Such a relation is called adispersion equation. The function ω(k) can be complex (for real k); in that case it willbe represented in the standard form: ω = ω r (k) + iγ(k). The real part ω r (k) is theordinary frequency, whereas the imaginary part is either the damping rate, if γ(k) < 0,or the growth rate, if γ(k) > 0. The latter case characterizes an unstable mode.In the previous chapter we presented the derivation of a set of fluid equations by59

4. DRIFT INSTABILITY ANALYSIS: LINEAR FLUID MODELtaking the moments of the Vlasov equation. Based on the results of the previouschapter we have developed a fluid model for the the study of the TE and ITG modecharacteristics, such as mode frequency and growth rate under relevant conditions fortokamak plasmas. Different possible stabilization effects such as collisional detrappingof the trapped electrons, presence of impurity ions and magnetic shear on the unstablemodes will be shown. The presented transport results in this chapter are taken from apaper by the author published in the journal: Physics of Plasmas, see Ref. (24).4.2 Basic EquationsUsing the fluid set of equations presented in the previous chapter (see sec. 3.4) eachion species j, including the main plasma ion (deuterium ion) and impurity, with thecharge Z j and mass m j is described by the continuity, parallel momentum and energytransport equations for its density n j , parallel velocity V ‖j and temperature T j :∂n j∂t + ∇ · (n jV j ) = 0 (4.1)m j n jdV ‖jdt+ ∇ ‖ (n j T j ) + Z j n j e∇ ‖ ϕ = 0 (4.2)32 n dT jjdt − T ∂n jj∂t + ∇ · q j − T j V j · ∇n j = 0 (4.3)where V j is the sum of the parallel velocity and the perpendicular drift velocity, V ⊥j(see sec. 3.6.1);V j = V ⊥j + V ‖j (4.4)and drift velocity arises due to the electric field E= −∇φ, pressure gradient, inertiaforce and viscosity responsible for Finite-Larmor-Radius (FLR) effects (21);V ⊥j = 1 1(E × ẑ) +B Z j n j B ẑ × (∇p j) + 1 m j dB Z j dt (ẑ × V j) + 1 ẑ × (∇ · π) (4.5)Z j n j B60

4.2 Basic Equationsφ is the electrostatic potential, and the convective derivative is defined as d/dt =∂/∂t + V j · ∇; the closure parameter, q j = 2.5n j T j [B × ∇T j ] / ( Z j eB 2) is the diamagneticdrift heat flux (19) with the magnetic field B dependent on the poloidal angle ϑ,B = B 0 / (1 + r cos ϑ/R), with r and R being the minor and major radii of the magneticsurface.The electron plasma component is separated on trapped and freely circulating particles.For the former, the bounce averaged parallel velocity is zero, and the density n etand temperature T et are described by continuity and energy conservation equations,see Ref. (25):∂n et∂t+ ∇ · (n et V ⊥e ) = S t,f (4.6)32 n dT et ∂n etet − T et + ∇ · q et − T et V ⊥e · ∇n et = Q t,f (4.7)dt ∂twhere the drift velocity V ⊥e is calculated by neglecting FLR effects and inertia effects;q et = 2.5n et T et [B × ∇T et ] / ( eB 2) ; the terms S t,f and Q t,f , represent the particle andenergy exchange between trapped and freely circulating electrons due to coulomb collisions.Parallel mass velocity of freely circulating electrons is much smaller so that theirthermal one and inertia can be neglected in the force balance along the magnetic field,where the pressure gradient is balanced by the electric force:∇ ‖ (T ef n ef ) = en ef ∇ ‖ φ (4.8)4.2.0.1 Collisional Effect On Trapped ElectronsIn the eqs. (4.6 and 4.7), the terms S t,f and Q t,f , providing the particle and energyexchange between trapped and freely circulating electrons due to coulomb collisions,are taken into account according to Ref. (25). In this reference the effect of collisions onthe trapped electrons are calculated by taking the moments of the Collisional Vlasovequation:61

4. DRIFT INSTABILITY ANALYSIS: LINEAR FLUID MODELwhere∂f∂t + v · ∇f + q ∂f(E + v × B) ·m ∂v = ν eff (δf − eφ f M ) (4.9)T ef = f M + δf δf ≪ f M (4.10)f M is the Maxwellian distribution function andν eff = ν th ( v thv )3 (4.11)where ν th = ν ei R/r, ν ei being the collisional frequency between electrons and ions. f Mis the Maxwellian distribution function used for the equilibrium and δf is the perturbationaround the equilibrium distribution. φ represents the perturbed electric potential.The expressions for S t,f and Q t,f are obtained by comparison of the results fromgyro-kinetic and fluid approaches. The terms S t,f and Q t,f are therefore, defined as:wherewhere η e = d ln T e /d ln n e , andwhere β = 1.5 − 2.5Γ.S t,f = −ν th (δn et − Γn eeφT e) (4.12)ω ∗eΓ = 1 + η e(4.13)ω − ω De + iν thQ t,f = ν th T e n e (− 1 δn et+ β eφ ) (4.14)2 n e T e4.2.1 Compressibility of The Drift VelocityBefore going further, we take a look at the divergence of the drift velocities which areresponsible for the compressibility in the direction perpendicular to the magnetics fieldlines.For the drift due to the electric field, v E , we getwhere we defined∇ · v E = ∇ · ( B × ∇φB 2 ) = ∇φ · (∇ × B ) = A · ∇φ (4.15)B2 62

4.2 Basic EquationsA ≡ ∇ × B B 2 (4.16)A is a vector characterizing the curvature of the magnetic field lines in the perpendiculardirection towards the magnetic axis of the tours where B ∝ 1/R. In the low β limit,where β is the ratio of the plasma pressure to the magnetic pressure, the vacuummagnetic field lines are not affected by the equilibrium plasma pressure we can writeA ≡Therefore, the operator A · ∇ can be evaluated as2B × ∇BB 3 (4.17)A · ∇ = 2 |∇B|B 2 e y · ∇ = 2 ∇BB 2 ∂ y (4.18)In a tokamak |B| = BR/R therefore, ∇B = −BR/R 2 ,A · ∇ = −2 BRR 2 ( RBR )2 ∂ y = − 2BR (r−1 cos θ∂ θ + sin θ∂ r ) (4.19)For the trapped electron, the motion along its banana orbits is much faster thanthe correlation time of the perturbations. Therefore, one has to take an average of theoperator A · ∇ over a banana orbit which is called bounce average.To make the averaging we use the toroidal coordinates as:• r minor radius,• θ poloidal angle,• ϕ toroidal angle,and the safety factor parameter q sf ,we can then define the magnetic shear, s,q sf = r R 0B ϕB θ(4.20)s = r qdq sfdr(4.21)which represents the radial shear in the magnetic field lines.63

4. DRIFT INSTABILITY ANALYSIS: LINEAR FLUID MODELWe define, χ = ϕ − q sf θ, an angular coordinate perpendicular to the magnetic fieldlines. If we suppose that the function F to which the operator A · ∇ is applied, onlydepends on χ and r coordinates, meaning that F is uniform along the magnetic fieldlines, therefore, the average of the operator A · ∇ over the trajectories of the particlescan be evaluated asA · ∇ = − 2BR 〈r−1 cos θ∂ θ + sin θ∂ r 〉 bounce= − 2BR 〈r−1 cos θ(−q sf )∂ χ + sin θ∂ r q sf ∂ qsf χ∂ χ + sin θ∂ r 〉 particle trajectory(4.22)The third term under averaging generates zero contribution, therefore we getA · ∇ = 2 q sfBR r 〈cos θ + sθ sin θ〉 bounce ∂ χ (4.23)For trapped electrons by averaging over a banana orbit, we getA · ∇ = 2 q sfBR r (1 4 + 2 3 s)∂ χ (4.24)For the ions, this averaging is done in the approximation of the strong ballooningof the turbulence i.e. the turbulence only exists for θ ∼ 0.4.2.2 Quasi-Neutrality ConditionA · ∇ = 2 q sfBR r (2 3 + s5 9 )∂ χ (4.25)In a stationary state all particle densities and temperatures are assumed to be functionsof the minor radius r only; parallel velocities of the ions and the electrostatic potentialare zero. The plasma quasi-neutrality requires:n i + ∑ j≠iZ j n j = n e = n et + n ef (4.26)with the fractions of trapped electrons, f et ≡ n et /n e , equal to √ 2r/(r + R). Thetemperatures of trapped and freely circulating electrons are assumed the same, T et =T ef ≡ T e .64

4.2 Basic Equations4.2.3 Linear ResponseAny plasma dynamical quantity (n, v, p, φ,. . . ) can be represented in the form A 0 +δA,where A 0 is the equilibrium reference, and small perturbation from this equilibrium referencein the form of δA. As long as the perturbations remain small one can linearizethe plasma dynamical equations around the reference state, and the solution of theselinear equations allow us to identify the various possible modes, linear response, of theplasma.We consider small perturbations of parameters, varying in time with a complexfrequency ω, and spatially, along the magnetic field, direction l, and on the magneticsurface perpendicular to the field lines, direction y, with the wave numbers k ‖ and k y ,respectively: δf ∼ exp ( −iωt + ik ‖ l + ik y y ) . By linearizing the transport eqs. (4.1-4.3)we get (24; 26; 27):(¯ω + λτ j)ñ j − λ(ɛ nj − 1)Z ˜φ + λτ j ˜Tj +m jk⊥ 2 j Z j Z j m ρ2 s[¯ω + λτ j(ɛ nj + ɛ Tj )]i Z ˜φ − Ṽ‖j = 0j(4.27)(¯ω − 2 λτ j)Ṽ‖j − ξ Z jm i ˜φ − τj ξ m i(Z j m j m ˜T j + ñ j ) = 0 (4.28)j(¯ω + 5 λτ j)3 Z ˜T j − λ(ɛ Tj − 2j 3 ɛ n j) ˜φ − 2 3 ¯ωñ j = 0 (4.29)where the dimensionless fluctuations ñ j = δn j /n j , ˜Tj = δT j /T j , ˜φ = eφ/Te andṼ ‖j = k ‖ δV ‖j /ω De have been introduced, ¯ω = ω/ω De with the magnetic drift frequencyω De = 2k y T e / (eBR) , ɛ x = −d ln x/dr × R/2 are the dimensionless radial gradientsof stationary parameters, ξ = (Rk ‖ /(2k y ρ s )) 2 , ρ s = c s /ω Li is the ion Larmor radiuswith c s = √ T e /m i being the ion sound speed, and ω Li the main ion Larmor frequency;τ j = T j /T e .The components of the mode wave vector, k ‖ = 1/ √ 3/(q sf R) and k ⊥ = k y√1 + (π 2 /3 − 5/2)s 2 ,where q sf is the safety factor and s the magnetic shear and the factor λ = 2/3 + s · 5/9,65

4. DRIFT INSTABILITY ANALYSIS: LINEAR FLUID MODELby following Ref. (28) the values θ-averaged over the assumed perturbation eigenfunction(29) (see sec. 4.2.2).The linearization of eqs. (5.4-5.5) for trapped electrons results in:(¯ω − λ t )ñ et − λ t (ɛ ne − 1) ˜φ − λ t ˜Tet = ¯ν th (Γ ˜φ − ñ et ) (4.30)(¯ω − 5 3 λ t) ˜T et − λ t (ɛ Te − 2 3 ɛ n e) ˜φ − 2 3 ¯ωñ et = ¯ν th (ñ et + β ˜φ) (4.31)where λ t = 1/4 + 2s/3 arises by bounce averaging (28), ¯ν th = Rν ei / (rω De ) with ν eibeing the frequency of electron collisions with ions, Γ = 1 + ɛ Te ω ∗eɛ ne (ω−ω De +iν th ) , β =1.5 − 2.5Γ (25). Perturbations in the temperature of freely circulating electrons areneglected because of high parallel heat conduction; therefore the Boltzmann relationfollows from eq. (4.8), ñ ef = ˜φ.4.2.4 Solution of Dispersion EquationThe sets of linearized eqs. (4.27-4.31) allow to express the perturbations of the densitiesof ion species and trapped electrons through the perturbation of the electrostaticpotential, ñ j = P j ˜ϕ and ñ et = P et ˜ϕ, respectively. The coefficients P j and P et arewhereP j = ɛ nj Nj −1 {¯ω 2 [λ − m jZj 2m (λτ j )k⊥ 2 ρ2 s] − ¯ω[ 1 λ 2 τ j− 1i 3 Z j 3− [ 153 λ(λτ j) 2 + 10Z j 3m jZ 4 j m i(λτ j ) 3 k 2 ⊥ ρ2 s − 2 3 ξ m im jλτ j ]}m jZj 3m (λτ j ) 2 k⊥ 2 ρ2 s]i+ Nj−1 {¯ω 3 [− m jk⊥ 2 Z j m ρ2 s] + ¯ω 2 [− m ji Zj 2m (λτ j )k⊥ 2 ρ2 s(ɛ Tj − 1i3 )]+ ¯ω[− m jZj 3m (λτ j ) 2 k⊥ 2 ρ2 s(− 1i3 ɛ T j− 103 ) − λ(λτ j)ɛ Tj + ξ m iτ j ]Z j m j+ [− m jZj 4m (λτ j ) 3 k⊥ 2 ρ2 s(− 10i3 ɛ T j) + 2λ( λτ j) 2 ɛ Tj + ξ m iτ j ( 5 λτ j+ λɛ Tj )]}Z j m j 3 Z j(4.32)N j = ¯ω 3 + ¯ω 2 [ 4 λτ j] − ¯ω[ 153 Z j 3 (λτ j) 2 + 5 Z j 3 ξ m iτ j ] − [ 10m j 3 (λτ j) 3 + 5 Z j 3 ξ m iτ 2 λj ]m j Z j(4.33)66

4.3 Instability AnalysisandP et = {¯ω 2 [λ t (ɛ ne − 1) + ¯ν th ] +¯ω[(− 7 3 λ t − 1)λ t ɛ ne + 5 3 λ2 t + λ t + (λ 2 t + ¯ν th )ɛ Te − ¯ν th − 7 3 λ t¯ν th +i(λ t ɛ ne − λ t¯ν th + ¯ν 2 th )]+[ 7 3 λ2 t ɛ ne − 5 3 λ2 t − λ 2 t ɛ Te + 8 3 λ t¯ν th − 256 λ t¯ν th ɛ Te ) +i(− 7 3 λ2 t ¯νɛ ne + 5 3 λ2 t ¯ν th + λ 2 t ¯ν th ɛ Te ]}/N et(4.34)whereN et = ¯ω 2 + ¯ω[− 10 3 λ t + ¯ν th ] + [ 5 3 λ2 t − 8 3 λ t¯ν th ] (4.35)Together with the Boltzmann relation for freely circulating electrons, these expressionscan be substituted into the quasi-neutrality condition providing a dispersionrelation for the perturbation frequency ω:⎛⎞P i⎝1 − ∑ Z j ξ j⎠ + ∑ P j Z j ξ j = P et f et + 1 − f et (4.36)j≠ij≠iHere the contribution from the main plasma ions with j = i is separated explicitlyand the concentrations of impurity ions ξ j ≡ n j /n e are introduced.4.3 Instability AnalysisWithout thermodynamic forces (gradients of the thermodynamics quantities i.e. density,temperature) the plasma responds adiabatically to any perturbations of the magneticand electric fields. In a tokamak plasma the magnetic field is nonuniform, theelectric field is non zero, and there exists strong gradients of density and temperatures.Therefore, the plasma’s response to the perturbations of the field is non adiabatic. Ifa perturbation increases exponentially in time we have an instability. To find theseunstable modes one needs to solve the dispersion equation of the form ω(k) = 0. Inthis section the analysis of the characteristics of the unstable modes, solutions of thedispersion eq. (4.36) are presented.67

4. DRIFT INSTABILITY ANALYSIS: LINEAR FLUID MODEL4.3.1 ITG Instability MechanismFigure 4.1: ITG instability mechanism.At first we will present a qualitative analysis of the mechanism of developmentof the toroidal ITG-instability (see figure 4.1).We consider a perturbation of theion temperature varying in the poloidal direction y. This causes a pressure gradientand therefore a diamagnetic drift motion in the radial direction (the second term ineq. (4.5)). The velocity of this motion depends on the magnetic field strength B,which decreases with the distance R from the tokamak axis. Therefore, this flow isnot divergence-free conversely to the diamagnetic motion in a cylinder and leads to aperturbation of the particle density (eq. 4.1)). The latter results in a distortion ofthe plasma potential (eq. (4.8)) and a poloidal electric field is generated. This causesadditional drift motion (the first term in eq. (4.5)) and heat flow (the second term inthe eq. (4.3)) which feeds the perturbation of the temperature.68

4.3 Instability Analysis4.3.2 TE Instability MechanismBecause trapped electrons can not move freely along the magnetic field lines, they causea strong phase shift between the electron density and electric potential perturbations.This phase shift leads to so called TE instability. Collisions of trapped electrons canlead to detrapping of electrons and therefore reducing the phase shift and has a stabilizingeffect on unstable TE modes. Figure 4.2 gives a summary of the ITG/TE modesand their destabilizing triggers.Figure 4.2: A summary of the ITG/TE modes.4.3.3 ETG Instability MechanismIts mechanism is similar to that of the ITG instability with the difference that here theelectron temperature perturbations are of importance. In order to include this type ofinstability one has to include the fluid set of equations, for passing electrons as wellas trapped electrons, and the adiabatic assumption for passing electrons should notbe considered. However, because the goal of this research is mainly the study of theimpurity transport, and the ETG modes do not affect impurity transport therefore, theETG instability is not considered in our studies.Figures 4.3 and 4.4 show characteristics of the ITG/TE/ETG growth rates versuswave number and thresholds predicted by the Weiland model (21).4.3.4 Numerical ComputationsThe fortran code AFC-FL (Anomalous Flux Calculation in Fluid Limit) has been developedduring this thesis in order to perform numerical computation for the instability69

4. DRIFT INSTABILITY ANALYSIS: LINEAR FLUID MODELFigure 4.3: Characteristic wave number and growth rates of drift instabilities.Figure 4.4: Stability Diagram-Weiland model.70

4.3 Instability Analysischaracteristics and calculation of transport coefficients of the main and impurity particles.It solves the dispersion equation for a wide range of the mode wave numbers (fork y ρ s ≤ 1 which is the relevant range to ITG and TE modes). At each wave numberit keeps the solutions which have non zero imaginary parts, (because we are only interestedin the solutions which lead to non zero growth rates). It then searches for themaximum of the growth rate as a function of the wave vector k y , and this maximum isthen used for the calculation of the transport coefficients and fluxes. It can sometimeshappen that there is more than one mode with a non zero growth rate in coexistence.The code can find up to three modes in that way. However, the transport is mainlygenerated due to the most unstable mode with highest growth rate, therefore, whencalculating the transport characteristics we only take into account unstable modes withmaximum growth rate. Figure 4.5 shows a Flowchart of the AFC-FL code.Figure 4.5: Simple Flowchart of the AFC-FL code.In the following sections, some examples of the parametric study of the mode char-71

4. DRIFT INSTABILITY ANALYSIS: LINEAR FLUID MODELacteristics computed by AFC-FC code are presented. Because the aim of this section isto present a general overview of the results calculated by this code, we therefore limitourselves in exploring the mode characteristics for some simple cases.4.3.5 Computation of the Solutions of the Dispersion EquationFigure 4.6 shows the wave number spectra of the imaginary, γ(k y ρ s ), and real, ω(k y ρ s ),parts of the solutions of the dispersion equation. These results are calculated for plasmaconditions of the JET tokamak with B T = 3T with normalized radius of r/a = 0.5where a is the minor radius of JET tokamak (a = 1.2 m). At this radial position thedensity profile is almost flat leading to a zero density scaling length i.e. ɛ n = 0, and thedensity itself n e = 4 × 10 19 m −3 . The temperatures of electrons and ions can be takenequal: T e = T j = 4KeV with gradient scaling length estimated as ɛ Te = ɛ Tj = 5 where jrepresents main and impurity ions and safety factor is q sf = 2. The composition of theplasma considered is a mixture of electrons, deuterium ions and 2% of the fully ionizedneon, Ne +10 , impurity species. In order to follow the impact of different stabilizingmechanisms on the unstable modes of interest, we start by simplifying the problem athand and by neglecting the effects of the magnetic shear i.e. λ t = λ = 1 with k ⊥ = k y .However, the collisional effects on trapped electrons are included. Later on in thissection, we will examine the impact of the magnetic shear on the characteristics of theunstable modes.As it is shown in this figure there are three unstable modes in coexistence. Twomodes in the ion direction (with negative ω r ), and one more in the electron direction(with positive ω r ). The most unstable mode with the highest growth rate is the deuteriumion ITG mode. A second ITG mode is also present which is due to the presenceof neon impurities in the plasma. It has a lower growth rate because neon particles onlycompose a fraction of the plasma ions. The TE mode has a lower growth rate becausethis mode is only generated by the trapped electrons in the plasma, since they can notmove freely along magnetic field lines, and therefore one gets a phase difference betweenthe electrostatic potential perturbation and electron density. However, the trappedelectrons are only a fraction of the total number of electrons (f et = √ 2r/(r + R)),therefore the growth rate of the TE modes are lower than those of the main ion ITG72

4.3 Instability Analysismodes, where the total number of the ions participate in the generation of the ITGinstability.Figures 4.7 to 4.8 show the k y ρ s -spectra of the three modes present in the figure 4.6as a function of the temperature gradient scaling length. As is shown in these figuresthere is a critical gradient scaling length which sets the threshold of the ITG and TEmodes, where after the modes grow by increasing the gradients. The critical gradientscaling length seems to be around ɛ Te = ɛ Tj = 2. This is to be expected since theparallel dynamics can play a role of dampening the modes, and therefore stabilizingthem. However, this will not happen if the gradients are above some critical value asto cancel the stabilizing impact of parallel dynamics.3.5 x 10532.51 x 106ITG modeTE modeImpurity ITG mode0.5!21.5#010.5−0.50−10 0.5 10 0.5 1k " k " y s y sFigure 4.6: The k y ρ s -spectra of ITG/TE instability growth rate in deuterium plasmacontaining 2% of Ne 10+ ions with a flat density profile, ɛ n = 0, calculated directly in a oneimpurity species approximation.As was mentioned previously, collisional detrapping of the trapped electrons havestabilizing impact on the TE modes, since it will allow them to move freely along themagnetic field lines, and to cancel any charge separation that might have happened.By inserting ν th = 0 in the eqs. (4.30) and (4.31) we examine this effect on the growthrate of most unstable modes, and the results are shown in figure 4.10. As seen in this73

4. DRIFT INSTABILITY ANALYSIS: LINEAR FLUID MODEL6x 10 554#32100 5 10 15 0" Ti,e0.5 k y! s1Figure 4.7: The k y ρ s -spectra of main ions ITG instability growth rate.2x 10 51.5#110.5005" Ti,e101500.5k y! sFigure 4.8: The k y ρ s -spectra of TE instability growth rate.74

4.3 Instability Analysis1500010000#500000 5 10 15 00.5k y! s1" Ti,eFigure 4.9: The k y ρ s -spectra of impurity ITG instability growth rate.figure the growth rate of the TE mode is reduced strongly by collisions.Figure 4.11 shows the magnetic shear effects on the characteristics of the ITG/TEmodes computed with s = 1. This effect is due to the magnetic shear, s, dependenceof the parameter λ, λ t and the perpendicular component of the mode wave vector, k ⊥(see eqs. 5.19-5.23). As one can see in this figure the growth rates are reduced for bothITG and TE limits.To better understand the reason for the stabilizing effect of the magnetic shear onthe ITG/TE modes we examine the dependence of the parameters λ, λ t and k ⊥ onthe magnetic shear,s, which are the only parameters with dependences on the shear.First the stabilizing effect from the FLR terms with shear dependence of the perpendicularmode wave vector k ⊥ = k y 1 + (π 2 /3 − 5/2)s 2 . By increasing the shear, the√difference in the magnetic field strength between two neighboring flux surface increaseswhich forces the unstable modes to localize, and prevents them from spreading in radialdirection, perpendicular to the field lines. This will then increase the value of theperpendicular mode wave vector k ⊥ . This is then a stabilizing effect on the growth rateof the ITG/TE modes. On the other hand the shear dependence of the parameters λ,λ t act as drivers of the unstable modes which increase by increasing magnetic shear.75

4. DRIFT INSTABILITY ANALYSIS: LINEAR FLUID MODEL!14 x 104No collsions on TEWith collsions on TE1210864200 0.2 0.4 0.6 0.8 1k y" sFigure 4.10: The k y ρ s -spectra of TE instability growth with and without taking intoaccount collisional detrapping effects on trapped electrons.Figure 4.12 shows these effects on the growth rate of the ITG/TE modes. As seen inthis figure the FLR effect leads to reduction of the growth rates for both ITG and TEmodes, while the curvature effects due to the parameters λ, λ t generate a driver forceto increase the growth rates and further destabilize the modes. Both effects increasewith increase of the magnetic shear, s, and their joint impact is to stabilize the modesfor certain values of the shear, 0 < s < 1. However, due to the approximations madewhen defining these parameters the model is not valid for large and very small valuesof the magnetic shear.In the following we only keep the maxima of the k y ρ s -spectra of the growth,γ max and its corresponding mode frequency ω max and k y . By varying the ratio ofthe ion and electron temperature gradients, either ITG or TE instability dominatedtransport can be reproduced. This has been done by changing the parameter θ =(ɛ Te − ɛ Ti ) / (ɛ Te + ɛ Ti ) from −1 to +1 and keeping ɛ Te + ɛ Ti constant and equal 5. Figure4.13 shows the variation with the parameter θ of such instability characteristics asgrowth rate γ max , mode frequency ω max and wave number k y ρ s . The change of the signof ω max from negative to positive corresponds to the transition from the ITG to TEinstability mechanism. The arrows in this figure follow the ITG mode (black arrow)and the TE mode (red arrow). At the left hand side, where ɛ Te = 0 and ɛ Tj = 5, we76

4.3 Instability AnalysisFigure 4.11: The k y ρ s -spectra of ITG/TE instability growth rate with and without takingthe effects of the magnetic shear into account.3.5 x 1053 x 10532.52.52! max2! max1.51.511ITG0.5TE0.5−1 0 1 2s0−1 0 1 2sFigure 4.12: s-dependences of the ITG (left) and TE (right) growth rates γ max .Solid line: no shear effects taken into account, Dashed line: λ = λ t = 1 and k ⊥ =k y√1 + (π2 /3 − 5/2)s 2 , Dashed dotted line: λ = 2/3 + s · 5/9, λ t = 1/4 + 2s/3 andk ⊥ = k y , and Red solid line: both effects considered.77

4. DRIFT INSTABILITY ANALYSIS: LINEAR FLUID MODELhave pure ITG mode while at the right hand side we have pure TE mode.Figure 4.13: θ-dependences of the instability characteristics γ max , ω max .The results shown so far have been computed by taking the density gradient scalinglength: ɛ n = 0. In figure 4.14 the same study has been repeated for different valuesof the density gradient scaling length, ɛ n . It shows that for small ɛ n the modes canbe slightly stabilized i.e. the growth rate is reduced, but at larger ɛ n the effect is tofurther destabilize the mode in both ITG and TE directions.4.3.6 Impurity Effect on Drift InstabilitiesIt is well known that anomalous transport in hot fusion plasmas can be significantlyinfluenced by impurities. Experiments performed on several fusion devices have demonstratedthat a significant modification of particles and energy losses can be achievedwith deliberate seeding of noble gases both in the low (L) and high (H) confinementmodes (11; 30; 31; 32; 33). It is believed that such behavior originates from the impactof impurities on the ITG instability with an increase of the ion charge, since the responsibleterm in drift velocity, V ⊥j , (eq. 4.5) is inversely proportional to Z j . Several theo-78

4.3 Instability AnalysisFigure 4.14: θ-dependences of the instability characteristics γ max , ω max for differentdensity scaling lengths.retical approaches have been proposed to explain these effects (34; 35; 36; 37; 38; 39; 40).In studies cited above, an effective impurity ion is normally introduced, in spiteof the fact that a large number of different ion species from diverse charge states ofdifferent elements is present in real fusion plasmas. Already the definition of the chargeand mass of an effective impurity ion is not straightforward in such a case. However,the main uncertainty is emerging by characterizing the density gradient of an effectiveimpurity species: experimental observations show that the density profiles are qualitativelydifferent for different impurities, tending to be hollow in the case of light specieslike carbon and significantly peaked for heavy impurities like nickel (41). Since theimpurity density gradient essentially affects the instability characteristics, the impactof impurity can be studied firmly by considering realistic plasmas if all ion species areindividually taking into account.The inclusion of any additional species increases the number of transport equationsinvolved in analysis, at least by three those describing the density, temperature andparallel velocity perturbations, see, e.g., (24; 26; 27). In order to use standard subrou-79

4. DRIFT INSTABILITY ANALYSIS: LINEAR FLUID MODELtines by solving the dispersion eq. (4.36), it has to be reduced to a polynomial form.The inclusion of any new impurity species into consideration makes the derivation ofthis equation more troublesome, and increases its order by 3, thus making its numericalsolution less reliable. For this reason another approach based on a treatment of theimpurity contribution, the second term on left hand side in eq. (4.36), by iterations hasbeen applied. This procedure begins with the solving of eq. (4.36) with the impuritycontribution neglected completely, i.e., with ξ j≠i = 0. The solution ω found in thisway is used to calculate approximate values of the coefficients P j≠i . Then eq. (4.36) issolved with the approximate impurity contribution calculated with actual ξ j≠i ≠ 0, andthe described procedure is continued until the convergence is reached. This method allowsus to analyze the effect on plasma instabilities from arbitrary number of impurityspecies, e.g., with all charge states of carbon, nitrogen, oxygen, neon, argon, etc.In order to validate the proposed approach we compare the results of calculationsfor one impurity species, with those obtained when the contribution from impurityhas been taken into account explicitly. Figure 4.15 shows such a comparison for thedependences of the instability growth rate γ on the dimensionless perpendicular wavenumber k y ρ s calculated for 3% concentration of Ne +10 impurity ions for three magnitudesof ɛ Ti = ɛ Te , with ɛ ne,i = 5 and ɛ nNe = 0. Other parameters are taken fromJET experiments dedicated to impurity transport studies, see Ref. (42), at r/a = 0.8:T i = T e = 1 KeV , n e = 2.5·10 19 m −3 , q sf = 4, s = 1.8. There is a very good agreementbetween both approaches.Figures 4.16 and 4.17 show for the case of a single impurity species Ne +10 thatthe results are, however, very sensitive to the dimensionless density gradient. In theleft column of Figure 4.16 the maximum value of γ, γ max , is shown for the range of0.05 < k y ρ s < 1 as a function of ɛ Te,i computed for different values of the neon concentrationξ Ne and density gradient scale ɛ nNe . In the middle and right columns the realfrequency part ω and the dimensionless wave vector k y ρ s of the most unstable modesare, respectively, displayed. In all cases ω < 0, i.e., the modes are driven by the iontemperature gradients. Practically always the growth rate decreases with increasingimpurity concentration. For ɛ Te,i ≥ 7.5 the behavior is qualitatively similar for differentɛ nNe , however, the quantitate difference to the reference case without impurity becomes80

4.3 Instability Analysis8 x 104 !Ti=56"420IterativeDirect0.2 0.4 0.6 0.8 11.52 x 105 !Ti=10"10.500.2 0.4 0.6 0.8 13 x 105 !Ti=152"100.2 0.4 0.6 0.8 1k # y sFigure 4.15: The k y ρ s -spectra of ITG/TE instability growth rate in deuterium plasmacontaining 3% of Ne 10+ ions with a flat density profile, ɛ nNe = 0, calculated directly in aone impurity species approximation (symbols) and iteratively (solid lines).81

4. DRIFT INSTABILITY ANALYSIS: LINEAR FLUID MODEL! max! max! max3 x 1052103 x 1052103 x 105210−2 x 105" Ne=1%# rmax−2.5−35 10 15−3.5−2 x 105" Ne=2%−2.55 10 15%Ti,e# rmax−3−3.55 10 15−4−2 x 105" Ne=3%−2.5# rmax−3−3.5−4%Ti,e0.455%10Ti,e150.15 10 15No impurity0.4% =−2nNe%nNe=00.35% =2nNe0.35 10 150.255 10 150.450.40.350.35 10 150.255 10 150.50.40.30.2k y$ sk y$ sk y$ sFigure 4.16: The growth rates (left column), real frequency (middle column) and dimensionlesswave vector (right column) of the most unstable modes as functions of the temperaturegradient parameter ɛ Te,i calculated with different concentrations ξ Ne and densitygradient parameters ɛ nNe of Ne 10+ impurity ions.82

4.3 Instability Analysis! max3 x 1052.52" Ne=1%# rmax−2.2 x 105−2.4−2.6k y$ s0.50.40.3No impurity%nNe=−2%nNe=0%nNe=21.50.5 1 1.5 2 2.5−2.80.5 1 1.5 2 2.50.20.5 1 1.5 2 2.5! max3 x 1052.521.5" Ne=2%# rmax−2.2 x 105−2.4−2.6−2.8k y$ s0.80.60.410.5 1 1.5 2 2.5−30.5 1 1.5 2 2.50.20.5 1 1.5 2 2.5! max3 x 1052.521.5" Ne=3%# rmax−2 x 105−2.5−3k y$ s0.80.60.410.5 1 1.5 2 2.5s−3.50.5 1 1.5 2 2.5s0.20.5 1 1.5 2 2.5sFigure 4.17: The growth rates (left column), real frequency (middle column) and dimensionlesswave vector (right column) as functions of magnetic shear calculated for differentconcentrations ξ Ne and density gradient parameters ɛ nNe of neon impurity.83

4. DRIFT INSTABILITY ANALYSIS: LINEAR FLUID MODELsignificant for large impurity concentration. For most unstable modes k y ρ s ≈ 0.3, i.e.,close to the value typical for ITG mode due to main deuterium ions (21). For smallerɛ Te,ithe deviations from the reference case are larger, especially for non-flat densityprofiles with ɛ nNe ≠ 0. For the peaked n Ne profile with ɛ nNe = 2 a new impurity drivenmode with significantly reduced k y ρ s exists at ɛ Te,i ≤ 2.5. Since the growth rate doesnot change too much compared to the reference case without impurity, we have to expectan increase of the transport characteristics, e.g., the particle diffusion as 1/kx. 2 Fora hollow neon density profile with ɛ nNe = −2 another impurity driven mode starts todominate if ɛ Te,i ≤ 5, with significantly reduced growth rate and wave length comparedto the reference situation. Therefore, a reduction of transport has to be expected. Theobserved behavior describes a self-stabilizing effect of impurities on transport at smalltemperature gradients; an increased impurity transport by peaked impurity densitywill presumably prevent peaking and vice versa. Therefore it is important to calculatethe impurity density gradient represented by the parameter ɛ nj in a self-consistent way,as it is shown in 4.18 and 4.19 by considering plasmas with several impurity species.Figure 4.17 displays the same characteristics as functions of magnetic shear, s.The impurity density profiles are usually characterized by the peaking factor p j ≡−R∇ r n j /n j = 2ɛ nj (41). In a steady state the fluxes of impurity ions reduce to zero,and from the above definition one gets p j = −RV j /D j = −2ImP V j /ImP Dj , i.e., avalue independent of | ˜ϕ| (see chapter 5.3.1). This allows to determine p j and ɛ nj ,which is involved into the dispersion equation, in an iterative procedure. Namely sucha self-consistent approach is applied by considering mixtures of five completely strippedimpurity ion species C +6 , N +7 , O +8 , Ne +10 , and Ar +18 . The contributions of individualspecies to the total impurity concentration are defined by adopting the same contributionsZ j n j to the electron density n e . Figures 4.18 and 4.19 show the linear growthrate and real frequency of the most unstable modes.The results above show that at high enough impurity concentration the instabilitycharacteristics are sensitive, in particular, to the impurity density gradient. In plasmaswith several impurity species characterized by very different gradients it is, however,fully unclear how to define the density gradient for such an effective impurity species.84

4.3 Instability AnalysisTherefore a reliable approach to treat instabilities with all ion species taken individuallyinto account is of importance.! max2.521.510.53 x 105No impurity%& j=1%%& j=3%%& j=5%# rmax−2.4 x 105−2.6−2.8−3−3.2k y$ s0.450.40.350.305 " −3.4Ti,e 10 155" 0.25"Ti,e 10 155 Ti,e 10 15Figure 4.18: The growth rates (left column), real frequency (middle column) and dimensionlesswave vector (right column) versus temperature gradient scale in deuterium plasmawith C +6 , N +7 , O +8 , Ne +10 , Ar +18 impurity ions of different total concentrations: no impurity(red curves), ∑ ξ j = 1% (black solid curves), 3% (black dashed curves) and 5%(black dashed-dotted curves); the impurity ion density gradient parameters are calculatedself-consistently from zero particle fluxes.2.6 x 105−2.3 x 1050.72.42.2−2.35−2.4−2.450.60.5! max21.81.6No impurity$% j=1%$% j=3%$% j=5%" rmax−2.5−2.55−2.6−2.65k y# s0.40.31.40.5 1 1.5 2 2.5s−2.70.5 1 1.5 2 2.5s0.20.5 1 1.5 2 2.5sFigure 4.19: The growth rates (left column), real frequency (middle column) and dimensionlesswave vector (right column) versus magnetic shear computed for ɛ Te,i = 10.85

4. DRIFT INSTABILITY ANALYSIS: LINEAR FLUID MODEL86

5Anomalous Transport due toDrift Wave Microinstabilities5.1 IntroductionDrift instabilities are a particularly important class of microinstabilities which havebeen invoked as the source of anomalous transport. In the previous chapter we discussedthe characteristics of the unstable ITG/TE modes; two of the most importantclasses of the microinstabilities in the tokamak plasmas. Their numerical assessmentwith the use of the linear fluid code AFC-FL were presented.In this chapter a parametric study of the heat and particle anomalous transportcoefficients and fluxes generated by these microinstabilities, that are calculated underdifferent tokamak plasma conditions, will be presented. The presented results in thischapter are taken from a paper by the author, published in the journal: Nuclear Fusion,see Ref. (27).5.2 Transport coefficientsIt is customary to divide the particle flux in diffusive and convective contribution wherethe diffusive contribution is from the gradient in density, and the convective contribution,is from the temperature gradient, and other effects which could contribute to theflux. Therefore we have87

5. ANOMALOUS TRANSPORT DUE TO DRIFT WAVEMICROINSTABILITIESΓ = −D∇n + V n (5.1)where the transport coefficients are now defined, by diffusivity, D, and convective velocity,V .The transport coefficients can be assessed from the definitions of anomalous fluxesof particles and heat, see sec. 3.8. Following from the expressions used in chapter 3 forthe particle flux density, Γ e,j , one getsΓ e,j = −2n et,jT e k yeB | ˜ϕ|2 Im(P et,j ) (5.2)In the factors P et,j one can distinguish the parts proportional to ɛ net,jand independentof this, i.e., P et,j = P De,j ɛ ne,j + P V e,j . As a result the diffusivities and convectivevelocities areD e,j = − T ek y ReB | ˜ϕ|2 Im(P De,j ),As we said before in chapter 3 (see sec.V e,j = − 2T ek yeB | ˜ϕ|2 Im(P V e,j ) (5.3)3.8.1) the absolute level of the transportcoefficients is governed by the amplitude of the potential perturbation. Withoutnon-linear turbulence simulations this can be only roughly estimated in mixing lengthapproximation, | ˜ϕ| ∼ γ max / ( k 2 yρ s c s).Figures 5.1 and 5.2 show the transport coefficients D e,i and V e,i as functions ofthe plasma temperature gradient parameter ɛ Ti,e and magnetic shear s. The transportcoefficients shown here are computed by using the instability characteristics shown infigures 4.18 and 4.19 in the previous chapter. As one can see, the increase in impurityconcentration leads normally to a decrease of the particle diffusivities, and absolutevalues of their convective velocities, caused by the reduction of the growth rate of unstablemodes (see figures 4.18 and 4.19).Similarly to the particle fluxes one can calculate anomalous heat fluxes.Withthe relation between perturbations of the temperatures and electrostatic potential,88

5.2 Transport coefficients1.81.6No impurity" # j=1%01.4"# j=3%−11.2"# j=5%D i[m 2 /s]10.80.6V i[m/s]−2−30.4−40.202 4 6 8 10 12 14−52 4 6 8 10 12 14504−0.5D e[m 2 /s]321V e[m/s]−1−1.5−2−2.50−32 4 6 8 10 12 142 4 6 8 10 12 14! ! Ti,e Ti,eFigure 5.1: Diffusivity and pinch velocity of deuterons and electrons, D i,e (left column)and V i,e (right column), respectively, versus ɛ Te,i calculated for s = 1.8.D i[m 2 /s]1.41.210.80.60.40.2No impurity! " j=1%!" j=3%!" j=5%00.5 1 1.5 2 2.5V i[m/s]−0.4−0.6−0.8−1−1.2−1.4−1.6−1.8−2−2.20.5 1 1.5 2 2.5D e[m 2 /s]3.532.521.510.500.5 1 1.5 2 2.5sV e[m/s]0.50−0.5−1−1.5−2−2.50.5 1 1.5 2 2.5sFigure 5.2: Diffusivity and pinch velocity of deuterons and electrons versus magneticshear calculated for ɛ Te,i = 10.89

5. ANOMALOUS TRANSPORT DUE TO DRIFT WAVEMICROINSTABILITIES˜T et,j = Q et,j ˜ϕ, found from linearized transport eqs. (4.27)-(4.31), one gets for the heatflux contributions non-vanishing for zero particle fluxes:q e,j = −2n et,jT e k yeB | ˜ϕ|2 Im(Q et,j ) (5.4)Also, the factors Q et,j can be separated in parts, correspondingly, proportional andindependent of the temperature gradients, Q et,j = Q ∇Tet,j ɛ T et,j+Q T et,j . Therefore the heatflux densities have also conductive and pinch contributions:with the heat conduction and pinch factors:q e,j = −κ e,j∂T e,j∂r + Π e,jT e,j (5.5)κ e,j = − T ek y ReB| ˜ϕ|2 ImQ ∇Tet,j,Π e,j = − 2T ek yeB | ˜ϕ|2 ImQ T et,j (5.6)Figures 5.3 and 5.4 show the calculated κ e,j and Π i,e as functions of ɛ Te = ɛ Timagnetic shear, respectively. These transport coefficients also reduce with increasingimpurity concentration. The heat pinch factor for electrons exhibits a change of signby varying both the temperature gradient scale and the magnetic shear. The absolutevalue of Π e can decrease significantly by increasing the impurity concentration. Thisbehavior, together with the reduction of the heat diffusivity, can contribute noticeablyto the reduction of the total heat transport observed in experiments with deliberateimpurity seeding (33).andWe note that figures 5.2 and 5.4 both show an increase of transport with magneticshear, although the growth rate of the dominating mode is decreasing with s, seefigure 4.19, left column. This is because the wave number of these modes decreaseswith s and although this decrease will lead to the reduction of the growth rates as wasdemonstrated in figure 4.12, the strong decrease of the mode wave number, see figure4.19, right column, leads to the increase of the transport coefficient since in the mixinglength approximation used above we assume: | ˜ϕ| ∼ k −2⊥ .90

5.2 Transport coefficients403.53No impurity$ % j=1%$% j=3%−0.2−0.4! i2.521.5$% j=5%" i−0.6−0.81−10.5−1.202 4 6 8 10 12 14−1.42 4 6 8 10 12 1420.501.5−0.5! e1" e−1−1.50.5−20−2.52 4 6 8 10 12 142 4 6 8 10 12 14# # Ti,e Ti,eFigure 5.3: Heat transport coefficients κ i,e (left column) and Π i,e (right column) asfunctions of ɛ Ti,e calculated for s = 1.8.! i3.532.521.510.5No impurity# $ j=1%#$ j=3%#$ j=5%" i−0.1−0.2−0.3−0.4−0.5−0.6−0.7−0.8−0.900.5 1 1.5 2 2.5−10.5 1 1.5 2 2.5! e1.41.210.80.60.40.2" e0.30.20.10−0.1−0.2−0.300.5 1 1.5 2 2.5s−0.40.5 1 1.5 2 2.5sFigure 5.4: Heat transport coefficients versus magnetic shear calculated for ɛ Te,i = 10.91

5. ANOMALOUS TRANSPORT DUE TO DRIFT WAVEMICROINSTABILITIES5.3 Impurity TransportLimited life time of plasma facing wall materials is one of the most serious problems forfuture fusion reactors. In order to reduce plasma wall interaction the concept of radiatingboundary produced by seeding of low-Z impurities has been developed over the pastyears in several tokamak devices, see, e.g., (31; 43). However, an injection of impuritiesinto the plasma could also have harmful effects, in particular, if they accumulate in theplasma core, cool the plasma center, and dilute fusion fuel. In order to benefit fromthe positive effects of the impurity presence at the plasma edge, and to prevent theiraccumulation in the plasma core, one needs to know the impurity transport propertiesunder various plasma conditions.Figure 5.5: The main mechanisms for impurity pinch.Neoclassical theory predicts transport contributions due to coulomb collisions inplasma states without instabilities (44). It describes well the impurity transport insome particular situations, e.g., in the very plasma center, internal and edge transportbarriers (45), however exceptions have been found (46). Under diverse other experimentalconditions the impurity transport has been proven to be exceeding the neoclassicalvalues by orders of magnitude (45; 47; 48; 49). Therefore, in the past years significantefforts have been made in developing models for the anomalous transport of impuritiesdue to plasma turbulence. In reference (21) the drift motion due to fluctuating electricfield and diamagnetic flow, being compressible in toroidal geometry, has been taken intoaccount. The parallel ion motion, induced by the perturbed pressure gradient, has beenintroduced in references (26; 50). In good agreement with the experiment (41) theseapproaches provide impurity diffusivity D with weakly dependence on the ion charge92

5.3 Impurity TransportZ. Concerning the impurity convection, i.e., the particle flux component independentof the sign of the density gradient, terms proportional to 1/Z (thermo-diffusion), independentof Z (curvature) and proportional to Z/A Z (parallel motion effects), whereA Z is the atomic weight, have been found, see figure 5.5. However, impurity ions arenearly fully stripped in the core region of hot fusion plasmas, the ratio Z/A Z is weaklycharge dependent and close to 1/2.5.3.1 Peaking FactorThe impurity density profiles are usually characterized by the peaking factor p j−R∇ r n j /n j = 2ɛ nj(41). In a steady state the fluxes of impurity ions reduce to zero,and from the definitions: Γ j = −D j∂n j∂r + V jn j = 0, one gets p j = −RV j /D j =−2ImP V j /ImP Dj , i.e., a value independent of | ˜ϕ|, see figure 5.6. When p j < 0, thedensity profile will be hollow in the core and no accumulation will be present (the desiredsituation) while for p j > 0 there will be an accumulation of impurity ions insidethe core and this has to be avoided.≡Figure 5.6: The impurity peaking factor effect on the impurity density profile.These findings shown in figure 5.5 explained well why the measured carbon densityprofiles are often less peaked than the electron one (47). However, for heavier impurities,from Ne to Ni, the proportionality of the impurity convective velocity V to Z has93

5. ANOMALOUS TRANSPORT DUE TO DRIFT WAVEMICROINSTABILITIESbeen observed (41; 48) (see figure 5.7) which could not be interpreted in the frameworkof existing models. As seen in figure 5.7 the impurity peaking factor measured experimentally(red symbols) show a charge dependence at the plasma very core (r/a = 0.15)which has a significant disagreement with the neoclassical predictions (blue symbols).Also, at the mid radius (r/a = 0.55) the measured charge dependence is less significantwhile the neoclassical predictions remain strongly charge dependent.Figure 5.7: The experimental measurements of the impurity peaking factor and theneoclassical predictions (this figure is taken from the presentation by C. Giroud at 21stIAEA fusion energy conference Chengdu, China 2006).Using the AFC-FL code we calculated the impurity peaking factor p j by consideringmixtures of five completely stripped impurity ion species C +6 , N +7 , O +8 , Ne +10 , andAr +18 . The contributions of individual species to the total impurity concentration aredefined by adopting the same contributions Z j n j to the electron density n e . In the zeroflux approximation, the density gradient scaling lengths of each impurity ions, ɛ nj , wascalculated self-consistently in an iterative procedure.Figures 5.8 and 5.9 show the impurity peaking factor, p j , for C +6 , Ne +10 and94

5.3 Impurity Transport−RV/D3.532.52solid line: !" =1%,C +6jDashed line: !" =3%,Ne +10jDashed−dotted line: !" j=5%Ar +182 4 6#8 10 12 14Ti,eFigure 5.8: The peaking factor for impurity species C +6 (red curves), Ne +10 (blue curves)and Ar +18 (pink curves) versus ɛ Ti computed with ɛ Te = 0, s = 1.8 and different totalimpurity concentrations: ∑ ξ j = 1% (solid curves), 3% (dashed curves) and 5% (dasheddottedcurves).2.82.72.62.5solid line: !" j=1%,Dashed line: !" j=3%,Dashed−dotted line: !" j=5%C +6Ne +10Ar +18−RV/D2.42.32.22.120.5 1 1.5 2 2.5sFigure 5.9: The peaking factor for impurity species C +6 (red curves), Ne +10 (blue curves)and Ar +18 (pink curves) versus magnetic shear computed with ɛ Te = 0, ɛ Ti = 5 anddifferent total impurity concentrations: ∑ ξ j = 1% (solid curves), 3% (dashed curves) and5% (dashed-dotted curves).95

5. ANOMALOUS TRANSPORT DUE TO DRIFT WAVEMICROINSTABILITIESAr +18 impurity ions. In these calculations the ITG-dominated situation with ɛ Te = 0and R/L ne= 3 has been considered. In agreement with Ref.(40) the peaking factoris an increasing function of the ion charge Z for low impurity concentrations with∑ξj = 1%. This tendency, however, vanishes at higher impurity concentrations ifeither the ion temperature gradient is not too large or the magnetic shear is not toosmall.5.4 Importance of Collisions on Anomalous Transport ofImpurityIn the present section the description for impurity anomalous transport, which wasdiscussed in chapter 4 (see section 4.2), is developed further by taking into account impurityion collisions with the main plasma components. Collisional effects are includedinto linearized transport equations (19) as friction and thermal forces and energy exchangebetween ion plasma components. The thermal forces are calculated by takinginto account that in the hot plasma core collision frequencies of background particlescan be smaller than the frequency of parameter fluctuations due to micro-instabilities.The investigation is done for plasma parameter fluctuations driven by the ITG/TEmodes.We use the set of fluid equations for impurity ions following from the equationspresented in chapter 4 section 2. Each ion species with the charge Z and mass m z isdescribed by the continuity, parallel momentum and energy transport equations for itsdensity n z , parallel velocity V ‖z and temperature T z :∂n z∂t + ∇ · (n zV z ) = 0 (5.7)m z n zdV ‖zdt+ ∇ ‖ (n z T z ) + Zn z e∇ ‖ ϕ = m z n z(V‖i − V ‖z)/τzth,i + F z T,i + F z T,e (5.8)32 n dT zzdt − T ∂n zz∂t + ∇ · q z − T z V z · ∇n z = 3n z (T i − T z )/τth,i z (5.9)96

5.4 Importance of Collisions on Anomalous Transport of ImpurityThe terms on the right hand side (RHS) of eq. (5.8) are due to collisions with themain plasma components. The first one is the friction force which arises due to momentumexchange with the background ions of the mass m i , density n i , temperatureT i and moving with the parallel velocity V ‖i ; τth,i z 3/2≈ 3Tim z / ( 4 √ 2πm i λZ 2 e 4 )n i is thetime between coulomb collisions estimated in the limit m i /m z ≪ T i /T z . The thermalforces FT,i z and F T,e z are generated by the inhomogeneity in the temperatures of themain ions and electrons along the magnetic field; in the limit of m e,i ≪ m z and in asteady state, F z T,i ≈ 2.6Z2 ∇ ‖ T i and F z T,e ≈ 0.71Z2 ∇ ‖ T e (51). The term on the RHS ofthe eq. (5.9) is the collisional energy exchange (19).The linearized eqs. (5.7-5.9) are:(¯ω + λ zτ zZ)ñ z − λ z (ɛ nz − 1) ˜ϕ + λ zτ zZ˜T z + m zk⊥ 2 Zm ρ2 s[¯ω + λ zτ zi Z (ɛ n z+ ɛ Tz )] ˜ϕ − Ṽ‖z = 0(5.10)¯ωṼ‖z − ξ Zm im z˜ϕ − τ z ξ m i(m ˜T)z + ñ z ) = iξ v(Ṽ‖i − Ṽ‖zzm i+ iξ (δFT,i z + δFk ‖ T e n z mT,e)zz(5.11)(¯ω + 5 3where τ z = T z /(ZT e ).λ z τ zZ) ˜T z − λ z (ɛ Tz − 2 3 ɛ n z) ˜ϕ − 2 3 ¯ωñ z = i 2ν zi( T i ˜Ti −ω De T ˜T z ) (5.12)zIn order to find the perturbations of thermal forces, δFT,i z and δF T,e z , one has totake into account that they are due to deviations δf i,e in the velocity distributionfunctions f i,e of the background plasma particles from the local Maxwellian ones f i,e,0 =( ) √n/ (πV th,i,e ) 3/2 exp −V 2 /Vth,i,e2 , with V = Vl2 + V⊥ 2 being the total particle velocity,V ⊥ its component perpendicular to the magnetic field and V th,i,e ≡ √ 2T i,e /m i, e thethermal velocities.Such deviations are caused by the temperature gradients in thedirection of the magnetic field and can be assessed from the linearized kinetic equationin the “τ-approximation” (52):97

5. ANOMALOUS TRANSPORT DUE TO DRIFT WAVEMICROINSTABILITIES∂∂t δf ∂f i,e,0i,e + V l∂l= − δf i,eτ i,e(5.13)where τ i,e is the time between collisions. In the presence of fluctuations generated bymicro-instabilities, ∇ ‖ T i,e = ik ‖ ˜Ti,e , and one gets from the equation above:δf i,e ≈ f i,e,0 V l2.5 − V 2 /V 2th,i,e1/τ i,e − iω ik || ˜T i,e (5.14)By collisions with much heavier impurities the asymmetry of δf i,e with respect to V lresults in thermal forces applied in opposite directions to the background and impurityparticles, and∫δFT,i,e z =m i,e V lδf i,eτi,ez dV (5.15)One can see that in the limit of low collisionalities with 1/τ i,e ≪ |ω| the thermalforces are affected weakly by the velocity dependence of the collision times τ i,e . Thereforewe neglect this dependence by using for τ i,e the corresponding times τ th,i,e forthermal particles with V = V th,i,e . On the contrary, the velocity dependence of τ z i,eis of importance and for coulomb collisions τ z i,e = τ z th,i,e (V/V th,i,e) 3 (53). By applyingexpressions for τ i,e and τ z i,efrom references(19) and (53), one gets:δF z T,i,e = α i,eZ 2 n z T i,e1 − iωτ th,i,eik || ˜Ti,e (5.16)where α i ≈ 1.6 and α e ≈ 0.66. For stationary perturbations with ω = 0 the foundδF z T,i,ediffer only by numerical factors of unity from the exact expressions (51). Thisdifference results from the velocity dependence of τ i,e neglected above in the limitω ≫ 1/τ i,e but being of importance for ω → 0.From the linearized momentum and heat transport eqs. (5.11) and (5.12) we get:˜T z = (x 1 ɛ nz + x 2 ) ˜ϕ − x 1 ¯ωñ z (5.17)Ṽ ‖z = (x 1 τ z ɛ nz + x 3 ) M ˜ϕ + τ z (1 − x 1 ¯ω) Mñ z (5.18)98

5.4 Importance of Collisions on Anomalous Transport of Impuritywhere x 1 = − 23M z, x 2 = ɛ TzM z+i 2ν ziω DeT iT zλ zτ zZΘM z, x 3 = 1+τ z x 2 +i ξv m zξ m iΦ−(α ′ i Z2 T iT eΘ+α eZ ′ 2 Ψ),+i 2ν ziω De, α ′ i,e = α i,e/(1−iωτ th,i,e ); the coefficientsM = ξ m im z/[¯ω+iξ v ] and M z = ¯ω+ 5 3Θ, Φ and Ψ interrelate the perturbations of the temperature and parallel velocity ofthe main ion and of the electron temperature with the perturbation of the electrostaticpotential, ˜T i = Θ ˜ϕ, Ṽ‖i = Φ ˜ϕ and ˜T e = Ψ ˜ϕ. These coefficients depend on the particulartype of the micro-instability controlling anomalous transport. By substituting theserelations into the continuity eq. (5.10) we obtain:withñ z = P z ˜ϕ (5.19)In the particular case τ Z = 1 one getsP z = P Dz ɛ nz + P Vz (5.20)where(P Dz = N z λ z 1 + 2 λ z− k2 ⊥ ρ2 s3 ZM z Z 2m z− 2m i 3M z)ξ m iY z m z(5.21)N z =Y z = ¯ω + iξ v (5.22)[¯ω + λ zZ + 2 λ z ¯ω− ξ ( )]m i 2¯ω/3 −1+ 1(5.23)3 M z Y z m z M zIn the convective velocity we separate, correspondingly, the contributions providedby the effects of magnetic field curvature, thermo-diffusion, FLR and coulomb collisions,P Vz = P VC + P Vth + P VF LR+ P Vcoll , whereP VC =(−λ z + ξ )Zm iN z (5.24)Y z m z( ξ m iP Vth = λ z − λ )z ɛTzN z (5.25)Y z m z Z M zP VF LR= −k 2 ⊥ ρ2 s(¯ω + λ zZ ɛ T z)N z (5.26)99

5. ANOMALOUS TRANSPORT DUE TO DRIFT WAVEMICROINSTABILITIES⎡P Vcoll = ⎣i 2ν ziω De+i ξv ΦY z( )Θ ξ m iM z Y z m z− λzZ+− ξ Z 2 m iY z m z(α ′ i Θ + α′ eΨ)⎤⎦ N z (5.27)The diffusivity and convective velocity are:D z = − T ek y ReB| ˜ϕ|2 Im(P Dz ), V z = − 2T ek yeB | ˜ϕ|2 Im(P Vz ) (5.28)By taking into account that ¯ω is the frequency ω normalized by ω De∼ 1/R onecan see that N z ∼ 1/R. Therefore P VC ∼ 1/R whilst other contributions to P Vz areindependent of R. Concerning the charge dependence, the terms P VC , P Vthand P VF LRprovide contributions either ∼ 1/Z or independent of Z as far as m z / (Zm i ) = 2 isassumed. These contributions to the anomalous pinch-velocity have been brought tolight in previous investigations (21; 26; 50). The new term due to coulomb collisionsprovide P Vcoll ∼ Z 2 m i /m z ∼ Z.5.5 Results of calculationsThe results presented below have been obtained for JET plasmas with the magneticfield B = 3 T , major and minor radii of the separatrix R = 3 m and a = 1.2 m,respectively (11; 27). By varying the ratio of the ion and electron temperature gradients,both ITG and TE instability regimes are reproduced changing the parameterθ = (ɛ Te − ɛ Ti ) / (ɛ Te + ɛ Ti ) from −1 to 1 and keeping ɛ Te + ɛ Ti constant and equal to 4.Figure 5.10 shows the k y ρ s -spectrum of the instability growth rate as a function ofparameter θ calculated for the radial position deep in the plasma core, r/a = 0.42, withthe parameters T i = T e = 4KeV , n e = 4 × 10 19 m −3 , q sf = 2, s = 1 and ɛ ni,e = 1; thelatter corresponds to typically flat density profiles in the core of H-mode plasmas. Motivatedby the recent experiments with neon seeding (33), we assume that the plasmacontains 2% of Ne +10 impurity ions with T z = T i and ɛ nz = 0, corresponding at theposition in question to Z eff ≈ 3.The characteristic frequencies of physical processes at r/a = 0.42 are: ν zi ≈2.7 · 10 2 Z 2 , ω De ≈ 1.7 · 10 5 , and γ ≈ 1.2 · 10 5 s −1 for the most unstable modes atθ = 1. One can see sharp jumps close to θ = 1 corresponding to the transition between100

5.5 Results of calculationsx 10 41510# in s − 1105864000.5x 10 4 !12k !" s110−120Figure 5.10: The k y ρ s -spectrum of the instability growth rate as a function of θ calculatedfor the plasma parameters in the JET core, r/a = 0.42.ITG and TE regimes. They are due to numerical problems which arise by solving thedispersion equation because at θ = 1 the coefficient at the highest degree of ¯ω reducesto zero.3.5−RV/D4321032.521.51−1−10!110203040Z0.50Figure 5.11: Impurity peaking factor as a function of the impurity charge Z and parameterθ computed without impurity ion collisions (r/a = 0.42).Figures 5.11 and 5.12 show the impurity peaking factor as a function of the instabilityparameter θ and impurity charge Z, with m z / (Zm i ) = 2 assumed, withoutand with the collisional effects taken into account, correspondingly. One can see that101

5. ANOMALOUS TRANSPORT DUE TO DRIFT WAVEMICROINSTABILITIES−RV/D432103.532.521.51−1−10!110203040Z0.50Figure 5.12: Impurity peaking factor as a function of the impurity charge Z and parameterθ computed with impurity ion collisions (r/a = 0.42).with collisions neglected the impurity peaking factor is practically independent of theimpurity charge for Z ≥ 5. With the collisional effects included the charge dependencebecomes very significant in the range of positive θ where the TE-mechanism of instabilityis dominant. In order to stress this we demonstrate separately in figure 5.13 thebehavior of the last term in P Vcoll due to the thermal forces, PV T Fcoll, which is of themost importance in P Vcoll and explains the modification of its charge dependence with)θ. In figure 5.14, which shows the Z, θ-dependences of the values Im(α ′ i ΘN z/Y z and)Im(α eΨN ′ z /Y z , involved into the P Vcoll part in question, one can see that the perturbationof the electron temperature is much more effective in the generation of impuritypinch-velocity than that in the ion temperature. This is because the collision time is∣much smaller for electrons than for ions and, thus, ∣α ′ ∣e∣ ≫ ∣α ′ ∣i∣. Since TE instabilitiesgenerate much stronger T e -perturbation than ITG-modes, the charge dependence inthe former regime with θ → 1 is significantly more pronounced than in the latter one,corresponding to θ → −1.By comparing figures 5.11 and 5.12 one can see that collision effects result predominantlyan inward convection increasing with the impurity charge. This is in aqualitative agreement with the majority of experimental observations. The found peakingfactor agrees also by the order of magnitude with that interpreted from observations(41). Nonetheless, we keep away here from a detailed comparison of theory predictions102

5.5 Results of calculations0.35PVColl−TF0.40.30.20.100.30.250.20.150.10.05−0.1−10!110203040Z0−0.05Figure 5.13: The contribution from thermal forces, P T FV coll, to the pinch-velocity factorP Vz as a function of the impurity charge Z and parameter θ (r/a = 0.42).x 10 −42x 10 −40−10−2Im(" i# N z/Y z)−2−4−6−3−4−5−8 −10!110203040Z−600.05−0.02Im(" e$ N z/Y z)0−0.05−0.1−0.04−0.06−0.08−0.15−10!110203040Z−0.1))Figure 5.14: The values Im(α ′ i ΘN z/Y z and Im(α eΨN ′ z /Y z as functions of the impuritycharge Z and parameter θ (r/a = 0.42).103

5. ANOMALOUS TRANSPORT DUE TO DRIFT WAVEMICROINSTABILITIESwith experiment because of their sensitivity to the instability characteristics. The latterare essentially controlled by the gradients of the plasma parameters which are notmeasured with high enough accuracy yet.x 10 54332.5# in s − 1221.51100x 10 5 !0.5110−10.5k !" sFigure 5.15: The k y ρ s -spectrum of the instability growth rate as a function of θ calculatedfor the plasma parameters close to the plasma edge, r/a = 0.8.As next we examine the effects of the collisions on the impurity ions for plasmaparameters relevant for a position closer to the plasma edge, r/a = 0.8: T i = T e =T z = 2KeV , n e = 3 × 10 19 m −3 , s = 3, q sf = 5, and ɛ ni,e = 3; 3.5% of Ne +10 ,corresponding to Z eff ≈ 4.5, and ɛ nz = 0 are assumed. Figure 5.15 shows the thecorresponding k y ρ s -spectrum of the instability growth rate calculated as a function ofθ. Characteristic frequencies at r/a = 0.8 are ν zi ≈ 5.6 · 10 2 Z 2 , ω De ≈ 4.9 · 10 4 , andγ ≈ 5.7 · 10 4 s −1 for the most unstable mode at θ = 1.Figures 5.16 and 5.17 demonstrate the dependence of the impurity peaking factor onZ and θ without and with collisional effects, respectively. On the one hand, collisionscontribute to the peaking factor much less in this case than for the core conditions. Thisis explained by the fact that the part of P Vcoll due to the thermal forces is proportionalto the factor ξ ∼ 1/ (k y ρ s q sf ) 2 and by going from the core to the edge conditions thisis decreasing by 25 times due to the growth both of k y ρ s for the most unstable modesand of q sf . On the other hand, the curvature contribution becomes more important at104

5.5 Results of calculations33.52.832.6−RV/D2.522.42.221.51.81−10!110203040Z1.61.4Figure 5.16: Impurity peaking factor as a function of the impurity charge Z and parameterθ computed without impurity ion collisions (r/a = 0.8).33.52.832.6−RV/D2.522.42.221.51.81−10!110203040Z1.61.4Figure 5.17: Impurity peaking factor as a function of the impurity charge Z and parameterθ computed with impurity ion collisions (r/a = 0.8).105

5. ANOMALOUS TRANSPORT DUE TO DRIFT WAVEMICROINSTABILITIESthe edge because this is essentially dependent on the precession factor λ Z , increasinglinearly with the magnetic shear.x 10 421.5x 10 5141210# in s − 11860.5000.511.542s042k !" sFigure 5.18: The k y ρ s -spectrum of the instability growth rate as a function of the magneticshear s calculated for the plasma parameters in the plasma core, r/a = 0.42, in theregime of TE-modes, θ = 1.In order to underline the role of magnetic shear for the relative importance of thecollisional effects in the impurity transport, calculations have been done for the coreplasma conditions considered above, with θ = 1 and s varying from 1 to 4. Figure 5.18shows the calculated k y ρ s -spectrum of the instability growth rate as a function of s.In figure 5.19 the Z, s-dependences of the impurity density peaking factor is displayed.One can see that with the increasing magnetic shear the relative weight of the collisioneffects in the impurity pinch-velocity decreases and the charge dependence of thepeaking factor becomes weaker.5.6 DiscussionThe relative importance of collisional effects in the impurity anomalous transportchanges significantly with the instability mechanism, safety factor and magnetic shear.For the plasma core conditions these effects generate a significant charge dependence106

5.6 Discussion43.533−RV/D22.5211.5002s410203040Z1Figure 5.19: Impurity peaking factor as a function of the impurity charge Z and magneticshear s computed with impurity ion collisions included (r/a = 0.8, θ = 1).of the impurity pinch-velocity and peaking factor while for the plasma edge their contributionsare not as significant. From the new effects considered the thermal forcesacting on impurity ions from electrons are of the most importance since electron collisionfrequency is higher than that of the main ions. Therefore, the charge dependenceof the impurity peaking factor is more pronounced in regions where TE modes arethe dominant instabilities. The impurity peaking factor is nearly proportional to theimpurity ion charge for Z ≥ 5 and magnetic shear s less than 2. With increasing s thisdependence becomes weaker.Figure 5.20: Charge dependence of the impurity peaking factor observed (right figure)and computed by AFC-FL code (left figure).107

5. ANOMALOUS TRANSPORT DUE TO DRIFT WAVEMICROINSTABILITIESFigure 5.20 shows a qualitative comparison between impurity peaking factor computedby AFC-FL code where the collisions with the background plasma particles havebeen taken into account and the experimental observation taken from (41) (the plasmaparameters used for computations are taken from the experiment). In this figure theresults are shown versus impurity charge Z at r/a = 0.55 (black lines on the left andright figures), at r/a = 0.35 (blue lines) and at r/a = 0.15 (red lines). There is aqualitative agreement between the observed and computed results.The found charge dependence of impurity anomalous transport, arising due to collisionswith the background plasma particles, can have important consequences. Undercertain conditions significantly peaked density profiles of high-Z impurity ions can beexpected, and the relation of this behavior to the experimental phenomenon of spontaneousimpurity accumulation has to be investigated in future. For this purpose theeffects discussed here have to be included into transport codes in order to predict reliablythe behavior and impacts of impurity on the plasma in future reactor devices. Inparticular, the found dependence on the plasma turbulence characteristics and magneticshear have to be validated by studying various plasma conditions.108

6Benchmark with a Quasi-linearGyro-kinetic Model6.1 IntroductionThe linear fluid approach presented in the previous chapters are formulated by makingsome rather strong approximations. The obtained growth rates and thresholds are overestimated. However, the simplicity of the fluid model allows for a broad and fast studyof the different plasma conditions composed of many particle species.Another model which is a step further into a more rigorous examination of theplasma with instabilities is the so called gyro-kinetic model. The gyro kinetic model isa Vlasov-Maxwell eq. (3.79) on which the gyro ordering is imposed.dfdt = [ ∂ ∂t + v · ∇ + q m (E + v × B) · ∂]f(r, v, t) = 0 (6.1)∂vwhere m and q the mass and charge. In order to simplify this equation we can removethe fast gyro motion by use the following assumption on the fluctuation frequency ωand the gyro frequency of ions Ω i (because Ω e ≫ Ω i we use the ion Larmor frequencyas the lower limit)ω/Ω i ∼ ɛ g ≪ 1 (6.2)Here, ɛ g is the gyro ordering parameter. With this assumption the charged rings ofgyro motion will be approximated by a quasiparticle at position of their gyro center co-109

6. BENCHMARK WITH A QUASI-LINEAR GYRO-KINETIC MODELordinates (54). Another assumption in the gyro ordering is that the spatial equilibriumscale is much larger than the Larmor radius:ρ/L n ∼ ρ/L T ∼ ɛ g ≪ 1 (6.3)where L n and L T are the density and temperature gradient scaling length. We assumestrong anisotropy, i.e. only perpendicular gradients of the fluctuation quantities can belarge (k ⊥ ρ ∼ 1, k ‖ ρ ∼ ɛ g ):k ⊥ /k ‖ ∼ ɛ g ≪ 1 (6.4)We also consider that the perturbation amplitude is small, i.e. energy of perturbationmuch smaller than the thermal energy:eφ/T e ∼ ɛ g ≪ 1 (6.5)In order to derive the gyro-kinetic equation we start from the Lagrangian of thesingle particleL = ( e c A(x) + mv) · ẋ − m 2 v2 + eφ(x, t) (6.6)We change the coordinates from a fixed reference on the particle position (x, v) toa reference fixed at the position of the guiding center (R, v ‖ , µ, ϕ) asx = R + ρ ≡ R + v ⊥ ˆn(R, ϕ)Ωµ = v2 ⊥2B(R)v ‖ = v · bϕ = tan −1 ( v · e 1v · e 2) (6.7)where R is the guiding center position; ˆn ≡ cos(ϕ)e 1 +sin(ϕ)e 2 and ϕ is the gyro phaseangle. e 1 (R, ϕ) and e 2 (R, ϕ) are orthogonal unity vectors in the plane perpendicularto b ≡ B/B, see figure 6.1.Replacing the set of new coordinates in eq. (6.7) into the single particle Lagrangian(6.6), and averaging over the gyro phase angle with averaging operator defined as110

6.1 IntroductionFigure 6.1: A local moving reference frame attached to the particle.we find〈. . .〉 = 1 ∫ 2π. . . dϕ (6.8)2π 0L g = (mv ‖ b + e c A(R)) · Ṙ + µB Ω ˙ϕ − m 2 v2 ‖− µB − qφ(R) (6.9)Therefore the equations of motion are:R = v ‖ b + B B‖∗ (v E + v ∇B + v C )µ = 0Ṙv ‖ = (−µ∇B + eE) ·mv ‖ϕ = Ω (6.10)where B ∗ ‖ ≡ B + (mc/e)v ‖∇ × b = B(1 + O(ρ ‖ /L B )) with L B being the scaling lengthof changes in the magnetic field strength. Drift velocities are defined asv E =cB 2 E × Bv ∇B = µmΩ b × ∇B111

6. BENCHMARK WITH A QUASI-LINEAR GYRO-KINETIC MODELv C = v2 ‖b × (b · ∇)b (6.11)ΩWe split the distribution function and the electrostatic potential into an equilibrium(F 0 , φ 0 ) plus fluctuations asf = F 0 + δfφ = φ 0 + δφ (6.12)We assume that equilibrium electrostatic potential is zero φ 0 = 0. In Fourierrepresentation we have∫δf(r, v, t) =∫δφ(r, v, t) =∫dω∫dωdkei(−ωt+k·r)(2π) 3/2 δf kω (v)dkei(−ωt+k·r)(2π) 3/2 δφ kω (v) (6.13)Using eqs. (6.12), (6.13), the Vlasov-Maxwell eq. (6.1), and the equations of motionfrom (6.10) we find an equation for the fluctuating distribution function δf.equation is the so called gyro-kinetic equation.ThisHowever, the gyro-kinetic equationstill remains highly nonlinear which is extremely difficult to deal with. Therefore, anadditional approximation is made, the so called Quasi-linear approximation, which isbased on the perturbation theory. It is applicable to weak turbulence defined asδf∼ eδφF 0 T ≪ 1 (6.14)This approximation is implemented by neglecting all terms in the gyro-kinetic equationthat are quadratic in fluctuations.Finally, the linearized gyro-kinetic equation is written asδf ωk = −e F 0T (1 − ω − ω ∗)J 0 (k ⊥ ρ L )δφ ωk (6.15)ω − ω D − k ‖ v ‖J 0 is the Bessel function of the zeroth order and it is present due to the fact that weaveraged over the gyro phase angle.112

6.1 IntroductionThe equilibrium distribution function is assumed Maxwellian:F 0 =n(πV T ) 3/2 exp−v2 /V 2 T (6.16)where V T = √ 2T/m is the thermal velocity defined as the mean energy∫V T =v 2 F 0 dv (6.17)ω ∗ is called the diamagnetic drift frequency:ω ∗ (v) = cTeB k ∂ln n v2θ [1 + (∂rV 2 T− 3 )η] (6.18)2where k θ is the poloidal wave vector and ω D is called the magnetic drift frequency:ω D (v) = − cTeBR k θf κv 2V 2 T(6.19)where f κ depends on the curvature effects (39).η =∂ln n/∂r∂ln T/∂r(6.20)6.1.1 Dispersion EquationFor fluctuating scales greater than the Debye length the Poisson equation becomes:4π ∑ se s δn s = 0 (6.21)where s represents the particle species and we have, see Ref. (55)δn s,ωk (t) = −n se s δφT s∫ ∞+ 2π−∞∫ ∞dv ‖ dv ⊥ v ⊥ J 0 (k ⊥ ρ Ls )δfωk s (v ‖, v ⊥ , t) (6.22)0Combining the eqs. (6.15) and (6.22), we obtain the equation deriving the electrostaticpotential perturbations, from which we want to deduce the growth rates.However, using the linearized gyro kinetic equation we have no information on thefluctuating electrostatic potential amplitude |δφ k,ω | nor its spectral shape versus the113

6. BENCHMARK WITH A QUASI-LINEAR GYRO-KINETIC MODELwave number k and the frequency ω. This is a common problem for fluid as well as gyrokineticmodels and generates some degree of uncertainty on the estimated anomalousflux. In the fluid model we used the mixing length assumptions for the amplitude ofthe fluctuating electrostatic potential (see section 4.8.1). In the linearized gyro kineticmodel one can define the amplitude and the spectral of |δφ k,ω | by using approximativeassumptions.6.2 QuaLiKiz codeThe QuaLiKiz code was developed by C. Bourdelle (56) at CEA association in CadaracheFrance. It is based on a quasi-linear model for calculating the heat and particle fluxesbased on an electrostatic linear gyro-kinetic code, KINEZERO (57) which evaluatesthe growth rates γ by solving the dispersion equation of the following formD(ω) = ∑ sn s Z 2 sT s[1 − L ts (ω) − L ps (ω)] = 0 (6.23)The species considered are electrons and two types of ion, typically the main ionsand one kind of impurity ion. The functional expressions for trapped, L ts , and passing,L ps , particles are given in detail in (57).The assumptions used for the fluctuating electrostatic potential amplitude |δφ k,ω |are based on both nonlinear simulations results, and on turbulence measurements ofwave number and frequency spectra:|δφ k,ω | 2 ∝ e 4k θρ i −8k maxρ i(6.24)where ρ i is the ion Larmor radius and k max corresponds to the maximal in the k theta -spectral of the potential fluctuations. The maximum value of |δφ k,ω | 2 at k max is chosensuch that the effective diffusivity, D eff , follows the mixing length rule:max(D eff ≈ RΓn )| k max= Rk θBeT |δφ k,ω| 2 | kmax =γ〈k 2 ⊥ 〉| k max(6.25)The choice for 〈k⊥ 2 〉 is based on both experimental measurements and nonlinearsimulation results. It should lead to maximum |δφ k,ω | 2 around k θρi ≈ 0.2 (56).114

6.3 Comparison With AFC-FLWe would like to point out that because the development of QuaLiKiz code was nota subject of this thesis we do not look in further detail at the formulation of the code,and limit ourselves as to show only the results. However, the reader can find detaildescription of the code in the references (56; 57).6.3 Comparison With AFC-FLIn a collaboration with C. Bourdelle, using the QuaLiKiz code, a cross examination withthe AFC-FL code has been performed. See figure 6.2 where the ITG/TE growth rateshave been compared. As one can see there is a qualitative agreement between two codes.However, quantitatively there are disagreements in the level and thresholds which isexpected since a linear fluid model is used in AFC-FL code. A Linear fluid model ismore accurate under conditions of fully developed turbulence where the system is farfrom the thresholds. Close to the thresholds the kinetic approach is more appropriate.However, because of the simplicity and the speed of computations in the fluid modelsthey are widely used as the first estimates of the plasma properties under differentconditions. The underlying physics can be found using the simple models, and later ona more robust models taking further physical dynamics into account is needed to finda more quantitative picture.2.53 x 1052ITG−TE mode AFC−FLITG−TE mode QuaLiKiz! max1.510.50"−0.5Ti,e0 5 10 15 20 25Figure 6.2: Comparison between AFC-FL and QuaLiKiz models.115

6. BENCHMARK WITH A QUASI-LINEAR GYRO-KINETIC MODEL116

7Transport Modeling of ImpuritySeeded Experiments at JET7.1 IntroductionTo reach regimes compatible with reasonable heat loads to the wall, several constraintsshould be overcome simultaneously (33); namely,1. ’mild’ ELM activity;2. a high level of radiated power in the divertor and3. avoiding impurity accumulation in the core.One of the most severe problems for fusion reactors is to satisfy the technologicalconstrain imposed by the divertor target PFCs. Technically only power loads ofless than 25MW m −2 are acceptable (58). However, strong ELM activity can lead topower loads in excess of 800MW m −2 . In order to reduce ELM activity, different techniquesare used. By increasing the radiated fraction, the pedestal density degrades andtherefore the ELMs activity reduces. Increase in D fueling will lead to increase of theelectron density in the SOL, and therefore the radiation from intrinsic impurities suchas carbon. Up to 40% (59; 60) of the total injected power can be radiated. To goabove 40% extrinsic impurities have to be seeded. Radiation cooling experiments withextrinsic impurities have been performed on several tokamaks (30; 31; 32; 61; 62) inorder to reduce the heat flux to the plasma facing components among which limitersand divertors. The seeded impurity is chosen to optimize the radiation in the SOL.117

7. TRANSPORT MODELING OF IMPURITY SEEDEDEXPERIMENTS AT JETHowever, some penetrate in the plasma core, which can lead to radiation from coreand also dilution. These are not desirable for achievement of high plasma performanceneeded for a tokamak reactor.Extensive studies have been performed at JET to radiated as much power as possiblenear the divertor plates by seeding nitrogen, neon or argon (58; 63). It was possibleto radiate up to 90% of the heating power mainly from the divertor. At the high radiationlevels, the ELMs are of type III and the heat flux density is reduced under theseconditions. However, the energy confinement time is reduced by about 25% by thedegradation of the edge pedestal. These results show that optimum plasma parametersto solve the problem of heat exhaust may not be optimum for core confinement. As aconsequence the non-linear interplay between edge and core plasma has been identifiedas a key issue in present tokamak experiments.During 2006-2007 JET experimental campaigns, operation at high power in configurationsrelevant to ITER steady-state operation has been explored; by injecting high-Zradiative gas, such as neon, to increase the edge radiation. It was observed that higheredge radiation due to the injected neon led to an overall degradation of the plasma confinementfrom the reference discharge without neon seeding. However, at some pointin the increase of the neon content the confinement degradation appears to stop andrevert to better confinement with increase of the neon content (33; 59; 60), as was previouslyobserved in the so called RI modes (Radiative Improved modes) (64; 65; 66; 67).An example of such phenomena observed in JET was shown in Reference (33) (seefigure 7.3 in this reference). These discharges were part of a series of experiments whereneon seeding has been explored to increase the radiated power fraction (up to 60%),providing significant reduction of target tile power fluxes (and hence temperatures) andmitigation of ELM activities. The neon was injected only between t=5 and 9 s. Whenincreasing the neon seeding in shot: 69093, the amplitude of the ELM perturbation wasstrongly reduced (59) and the diamagnetic stored energy was also increased by 25%.This performance improvement is seen on the ion temperature profiles which show atemperature increase both at the top of the H-mode pedestal and in the core (from 6to 12 keV) and also the electron density increased by ∼10 % in the core. The neon118

7.2 Experimental observationsconcentration profile, as measured by the charge exchange recombination spectroscopy,is hollow with a concentration of about 1% in the core rising to ∼ 2.5 % at the normalizedradius of 0.7. No sign of impurity accumulation is found in these discharges.The cause of the improvement was not identified.These experiments have been studied here, and discussion on the results of ourtransport modeling are presented in this chapter. We restrict our modeling inside theLCFS, mostly because in the SOL, modelling of the turbulent transport coupled toradiative processes is extremely difficult and at present there is no satisfactory wayto do it. To model the heat transport inside the LCFS we use the fluid code RITM(68; 69; 70; 71). The boundary conditions are given by the temperatures at ρ = 1. Theparticle transport is not modeled. Since the cause of the loss of the pedestal electrondensity is not well understood, we will focus on the balance between the increasedradiation level inside the LCFS, the increased Ne concentration and the modified confinement.At the end, we performed a predictive transport modeling of JET plasmas seededwith Ne (same discharges as mentioned above) with the use of RITM code. In thistype of study, a self-consistent description of heat and particle transport over the entirecross section of the plasma from the axis to the separatrix, is used. The results of thisstudy are presented at the end of this chapter.The presented results are part of a paper prepared for publication.7.2 Experimental observationsIn this study, the JET discharges 69089, 69091 and 69093 produced in the ITERsteady-state scenario configuration with respectively, only D fueling, D fueling withNe injection and only Ne injection (33) are taken into account. A summary of plasmaparameters is given in the table 7.1 where the values of Hy,therm 98 have been calculated,also shown in figure 12 in the Ref. (59). In the two shots were neon seeding wereapplied, the neon puffs were different only during the initial phase with a short strongburst in shot 69093. Nearly the same waveform as in pulse 69091 was used thereafter.119

7. TRANSPORT MODELING OF IMPURITY SEEDEDEXPERIMENTS AT JETThe stronger neon puffing in discharge 69093 has been performed in order to mitigateELM affecting detrimentally the internal transport barrier to develop an AdvancedTokamak (AT) scenario at high triangularity. The deuterium puff intensity has been6 times higher for the discharge 69091 than for 69093. All discharges in question havebeen heated with the high NBI power of 22 MW and the total power, including theohmic heating, of 30MW ; the magnetic field B 0 ∼ 3.1T , plasma current I P ∼ 1.9MA,triangularity δ ≥ 0.3. These AT scenarios are characterized by q 95 of about 5, operationat relatively high normalized β N ∼ 2, and a flat q profile (sometimes hollow) with aminimum q of about 1.5 − 2.5.2 x 107 t (s)1.8#69089#69091#690931.6[W]totP rad1.41.210.845 46 47 48 49 50Figure 7.1: Prad tot total radiated power as a function of time (taken from JET measurementsbase on 2D tomographic reconstruction) for the three discharges: 69089 (solid line), 69091(dashed line) and 69093 (dashed dotted line).The total radiated power Prad tot , from LCFS + SOL regions is shown in table 7.1 andfigure 7.1 shows the Prad tot versus time taken from JET 2D tomographic reconstruction,for the three considered discharges. The seeding of neon has increased the radiatedpower from 20% in shot 69089 to 60% for both shots 69091 − 3. Figure 7.2 shows thetotal radiated power profiles, P rad , inside LCFS as a function of normalized flux, Ψ,taken from measurements based on Abel inversion (reconstruction under assumptionthat plasma radiation is constant on the flux surface). The radiated power is calculatedby integrating with the time step δt = 0.1s (between ELMs) from t = 5s until 9.9s.120

7.2 Experimental observationsAs seen in this figure the radiated power inside LCFS is almost identical for both neonseeded discharges. The ratio between the total radiated power and radiated power fromthe main plasma (inside LCFS), P rad /Prad tot are given in table 7.1.6 x 106 !5#69089#69091#690934P rad[w]32100 0.2 0.4 0.6 0.8 1Figure 7.2: P rad total radiated power inside LCFS as a function of normalized flux, Ψ,(taken from JET measurements based on Abel inversion (reconstruction under assumptionthat plasma radiation is constant on the flux surface) for the two discharges at t ≈ 7.6s:69091 (dashed line) and 69093 (dashed dotted line).D [el/s] Ne [el/s] Pradtot P rad /Prad tot Hy,therm9869089 0.5 × 10 22 - 20% ∼ 50% 0.8569091 3 × 10 22 3.8 × 10 21 60% ∼ 50% 0.7269093 0.5 × 10 22 6.3 × 10 21 60% ∼ 50% 0.79Table 7.1:A summary of plasma parameters.Figure 7.3 shows the experimental obtained T i profiles taken from the charge exchange(CXFM) diagnostic measurements and, figures 7.4, 7.5 show T e , n e profilestaken from the Thomson scattering (LIDAR) diagnostic measurements. As seen infigures 7.3 and 7.4 the ion and electron temperature are almost identical for two shots69089 and 69091 but they increased significantly (almost doubled for T i ) in shot 69093for the whole plasma radius.121

7. TRANSPORT MODELING OF IMPURITY SEEDEDEXPERIMENTS AT JET141210#69089#69091#69093Ti [KeV]864202.8 3 3.2 3.4 3.6 3.8R (m)Figure 7.3: T i profiles as functions of major radius (taken from JET CXFM measurements)for the three discharges at t ≈ 7.6s: 69089 (solid line), 69091 (dashed line) and69093 (dashed dotted line).Although the D fueling intensity has been 6 times higher for the discharge 69091than for 69093 as seen in figure 7.5, the electron density profiles differ from only 20%in the core and are almost identical in the pedestal. However, comparing the pedestalelectron density profiles there is a significant decrease from the shot 69089 (withoutneon seeding) to the two seeded shots. The results reported by Y. Corre (60) showsthat the impact of impurities (N in this case) on the confinement is observed even whenthe same level of D fueling is applied.Changes on the pedestal temperatures are different and only a small decrease(69091) and a high increase (69093) of the pedestal temperatures, in comparison tothe shot 69089, have been observed.Figures 7.6 and 7.7 illustrate Ne concentration profiles for the three shots and figure7.8 shows the associated Z eff profiles calculated by taking into account both carbonand neon (when present) impurities. These data are taken from CXFM measurements.As seen in these three figures the concentration of neon is the highest in shot 69093122

7.2 Experimental observations108#69089#69091#69093Te [KeV]64202.8 3 3.2 3.4 3.6 3.8R (m)Figure 7.4: T e profiles as functions of major radius (taken from JET LIDAR measurements)for the three discharges at t ≈ 7.6s: 69089 (solid line), 69091 (dashed line) and69093 (dashed dotted line).7 x 1019 R (m)6ne [m −3 ]543#69089#69091#6909322.8 3 3.2 3.4 3.6 3.8Figure 7.5: Electron density n e profiles as functions of major radius (taken from JETLIDAR measurements) for the three discharges at t ≈ 7.6s: 69089 (solid line), 69091(dashed line) and 69093 (dashed dotted line).123

7. TRANSPORT MODELING OF IMPURITY SEEDEDEXPERIMENTS AT JET32.5#69089#69091#690932n Ne/ne1.510.502.8 3 3.2 3.4 3.6 3.8R (m)Figure 7.6: Neon concentration, n Ne /n e profiles as functions of major radius (takenfrom JET charge exchange recombination spectroscopy measurements, CXF6) for the threedischarges at t ≈ 7.6s: 69089 (solid line), 69091 (dashed line) and 69093 (dashed dottedline).while the concentration of the carbon is the highest in the shot 69089.7.3 Framework in which the fluid modeling is appliedIn order to investigate the underlying mechanism responsible for the observed confinementimprovement in the discharges mentioned above we perform a qualitativetransport modeling in a series of tests. For transport modeling the quasi-linear 1-Dfluid transport code RITM (68; 69; 70; 71), has been applied. The goal of the presentstudy is to model the plasma behavior inside the separatrix because we have no informationon how the SOL parameters have been modified. Although the modificationsin SOL, from one discharge to another can be of great importance on the behavior ofthe plasma, we do not take them into account here, because of the limited ability tomodel turbulence and the radiative processes in the SOL.In general, RITM code allows a self-consistent description of heat and particletransport over the entire cross section of the plasma from the axis to the separatrix.124

7.3 Framework in which the fluid modeling is applied43#69089#69091#69093n C/ne2102.8 3 3.2 3.4 3.6 3.8R (m)Figure 7.7: Carbon concentration, n C /n e profiles as functions of major radius (takenfrom JET charge exchange recombination spectroscopy measurements, CXFM) for thethree discharges at t ≈ 7.6s: 69089 (solid line), 69091 (dashed line) and 69093 (dasheddotted line).43.5#69089#69091#690933Z eff2.521.512.8 3 3.2 3.4 3.6 3.8R (m)Figure 7.8: Effective charge, Z eff profiles as functions of major radius (taken fromJET charge exchange recombination spectroscopy measurements CXFM) for the threedischarges at t ≈ 7.6s: 69089 (solid line), 69091 (dashed line) and 69093 (dashed dottedline).125

7. TRANSPORT MODELING OF IMPURITY SEEDEDEXPERIMENTS AT JET0.240.220.2P rad& n e: 69089Z eff: 69093P rad& n e: 69089Z eff: 69091n e: 69089P rad& Z eff: 69093! E=W dia/P tot0.180.160.14P rad& Z eff: 69089n e: 69091−369089 6909369091Z eff: 69089P rad& n e: 69091−3n e& Z eff: 69089P rad: 69091−30.120 0.05 0.1 0.15 0.2 0.25 0.3P /P rad totFigure 7.9: The energy confinement time defined as the ratio of the diamagnetic energyto the total input power: τ E = W dia /P tot versus the fraction of radiated power, P rad /P totfor different tests. The reference discharge 69089 with only D fueling is shown with adiamond. The other two discharge: 69091 (D fueling and Ne injection) and 69093 (onlyNe injection) are presented by a square and a circle, respectively. Other symbols representthe tests where only n e (right triangle), only Z eff of 69091 (plus), only Z eff of 69093 (uptriangle), only P rad (left triangle) and finally, P rad + n e (star), P rad + Z eff (down triangle)have been replaced from their reference value in 69089 by those from 69091 − 3 discharges.126

7.3 Framework in which the fluid modeling is appliedContinuity equations are solved separately for electrons and impurity ions in all ionizationstages and the density of the background ions is calculated under the assumptionof plasma quasi-neutrality. However, the pedestal degradation observed on density isnot presently well understood, therefore the particle transport is not modeled. Thepedestal degradation is taken as an input information. Only the heat transport is computedby RITM. The impact of the impurities on the transport is taken into accountby assuming that there are two plasma species: electrons and an effective ion with aneffective charge, Z eff . Therefore, the only way that the presence of impurities affect theplasma is through the modification of Z eff which is prescribed from the experimentalprofiles (see figure 7.8), and through their radiations.The radiation profile is computed only inside the LCFS and by taking all impurityspecies (carbon and/or neon) in all their ionization stages into account. Radiation fromD α is not considered. To match the experimental levels on the radiated power insideLCFS, P rad , the influx of impurity neutrals through separatrix have been modified.In this simplified picture no matter what is the actual carbon and/or neon concentrationlevel, we only adjust the influx so that we obtain the necessary radiation profile,because we are not taking into account impurity’s contribution either to the electrondensity (see above) or the effective charge (they are fixed by experimental values).We have not put any boundary constraint on the value of the P rad (ρ = 1). We havereproduced the same level of the radiated power as was observed experimentally forthe three considered discharges. The radiated power profiles are used together withthe experimental profiles of auxiliary heating power density from the NBI and highfrequency radio-waves in the ion and electron heat transport equations to calculate theradial temperature profiles.Despite a 15% difference in density profiles (see figure 7.5) between the two shots69091 and 69093 in our computations we assumed the same density profile as in theshot 69091 for the shot 69093 as well. The boundary conditions for temperatures, T i,eat ρ = 1 are defined using the experimental values (see figure 7.3 and 7.4. The safetyfactor, q sf , profiles are fixed by the calculated values taken from the EFIT.127

7. TRANSPORT MODELING OF IMPURITY SEEDEDEXPERIMENTS AT JET7.4 Results of the modelingStarting from the discharge with only D fueling (69089), towards the discharges withthe neon seeding, the expected impacts on the confinement are: a degradation due tothe density reduction, a degradation due to the P rad increase, and an improvement dueto the Z eff increase. In order to balance these three effects, the impact of each is firstmodeled separately, namely:1. The impact of density reduction inside LCFS from shot 69089 to 69091 − 3 isgiven, see figure 7.9, by looking at diamond to right triangle. Since the particletransport is not modeled, hence it is simply the density reduction impact on theconfinement time due to Energy = 3/2 ∫ dV nT . About 25% decrease in τ E isobserved as one would expect for a density decrease seen in figure 7.5.2. The impact of an increase of Z eff from shot 69089 to the shots 69091 and 69093on the confinement time is shown in figure 7.9 from diamond to the plus and uptriangle, respectively. A more pronounce increase of about 25% is observed fromthe shot 69089 to the shot 69093. The causes for this increase can be understoodby analyzing the figure 7.10 where the radial profiles of the corresponding ITGgrowth rates are shown. One can interpret this evolution by taking into accountthat the main contribution to the anomalous part in heat diffusivity χ i,e is dueto ITG instability (72). This contribution is determined by the ITG growth rateasγ IT G ∼√1Z eff L Ti− 1L crT i(7.1)where L Ti and L crT iare the e-folding length of the ion temperature and its criticalvalue, correspondingly. In the plasma core the T i -profile is stiff, i.e., Z eff L Ti ≈L crT iand for relatively flat density L crT i≈ αR. With a constant Z eff we getT i (0) /T i (r b ) ≈ exp (Z eff r b /αR) where r b is the minor radius of correspondingto the plasma boundary i.e., at ρ = 1. This temperature ratio grows up withincreasing Z eff . Therefore, due to the profile stiffness the increase in the temperatureat the top of the pedestal is amplified in the core. Figure 7.11 showsthe the T e and T i profiles corresponding to Z eff tests shown in figure 7.9 as adiamond, plus and up triangle symbols.128

7.4 Results of the modeling2 x 105 !1.5" ITG[s]10.569089 (only D)69089 with Z effof 6909169089 with Z effof 6909300 0.2 0.4 0.6 0.8 1Figure 7.10: ITG growth rate as a function of the normalized toroidal flux coordinate,ρ, calculated by RITM code for the three discharges: (solid line) shot 69089, (dashed line)shot 69089 with Z eff from 69091, and (dashed dotted line) shot 69089 with Z eff from69093, shown in figure 7.9 by a diamond, a plus and an up triangle symbols, respectively.3. The increase of radiated power, P rad , inside the LCFS on the confinement timeis estimated, see figure 7.9 from diamond to left triangle. A degradation of theconfinement due to the cooling of the electrons is expected, however, a very weakdegradation is found. This weak degradation is explained by the very smallreduction of the T e and T i profiles shown in figure 7.12. It seems that in ourmodel such an increase of radiated power (about 20% increase) does not generatestrong effect on the electron temperature.After investigating each individual effect on the confinement time now, we willbalance them. We know that both the density reduction and P rad increase lead to confinementdegradation whereas Z eff increase leads to confinement improvement. Eventhough in figure 7.9 by going from star symbol (only the collective degradation effectsof P rad + n e considered) to square symbol (69091) we see an improvement, if P rad + n edegradation is balanced against Z eff increase from 69089 to 69091, the confinementtime still degrades as seen in figure 7.9 from diamond to square symbol. Whereas ifP rad + n e degradation is balanced against Z eff increase from 69089 to 69093, an improvementis modeled, the confinement is recovered and its value reaches to that seen129

7. TRANSPORT MODELING OF IMPURITY SEEDEDEXPERIMENTS AT JETTe [KeV]864269089 (only D)69089 with Z effof 6909169089 with Z effof 6909300 0.2 0.4 0.6 0.8 186Ti [KeV]4200 0.2 0.4 0.6 0.8 1!Figure 7.11: Electron (top) and ion (bottom) temperature profiles as functions of thenormalized toroidal flux coordinate, ρ, (solid line) for 69089, (dashed line) for 69089 withZ eff increased to that from 69091, and (dashed dotted line) for 69089 with Z eff increasedto that from 69093, corresponding to the diamond, plus and up triangle symbols in figure7.9, respectively.130

7.4 Results of the modelingfor 69089 (see figure 7.9 going from diamond to circle symbol). Therefore, on the onehand, degradation of the confinement due to the increased radiated power is weak andif balanced against the increase of the Z eff the impact of Z eff is stronger and it leadsto an improved confinement, see figure 7.9 from left triangle to down triangle symbol.Figure 7.13 shows the T e and T i profiles corresponding to the left triangle and down trianglesymbols in figure 7.9. On the other hand, the degrading impact of the decreasedpedestal density compared to the improvement due to the increased Z eff shows thatthe impact of density reduction is stronger and to some extent it can be compensatedby the increase of the Z eff . These trends are qualitatively coherent with experimentalobservations, see figure 12 in the Ref. (59).Te [KeV]86426908969089 with P radof 69091−300 0.2 0.4 0.6 0.8 186Ti [KeV]4200 0.2 0.4 0.6 0.8 1!Figure 7.12: Electron (top) and ion (bottom) temperature profiles as functions of thenormalized toroidal flux coordinate, ρ, (solid line) for 69089, and (dashed line) for 69089with P rad increased to that from 69091 − 3, corresponding to the diamond and left trianglesymbols in figure 7.9.131

7. TRANSPORT MODELING OF IMPURITY SEEDEDEXPERIMENTS AT JETTe [KeV]864269089 (only D)69089 with Z eff& P radof 6909300 0.2 0.4 0.6 0.8 186Ti [KeV]4200 0.2 0.4 0.6 0.8 1!Figure 7.13: Electron (top) and ion (bottom) temperature profiles as functions of thenormalized toroidal flux coordinate, ρ, (solid line) for 69089, and (dashed line) for 69089with P rad + Z eff increased to that from 69093, corresponding to the diamond and downtriangle symbols in figure 7.9.132

7.5 Predictive Transport ModelingThe model predicts a 10% confinement degradation from shot 69089 to 69091 andthe measured degradation is 10 − 20%. The confinement improvement from shot 69091to 69093 predicted by the model is 10% and the measured value is 2 − 25% (59).Despite some quantitative disagreements, the modeled behavior agrees qualitativelywith the experimental observation. Hence, one can expect that there exist a criticalvalue of Z eff above which the degradation of the pedestal is counter balanced bythe improvement due to a higher Z eff . The critical Z eff will vary depending on thebackground dimensionless parameters. In the studied case here, this value is around theZ eff from 69093. In such a picture, the results reported by M. Beurskens (59) showingmostly degradation of the confinement with impurity seeding can be compatible byother, more seldom, reported results showing an improvement of the confinement inH-mode with impurity seeding (33; 60; 73).7.5 Predictive Transport ModelingFor predictive transport modeling of JET plasmas seeded with Ne (same discharges asmentioned in the previous sections) the quai-linear 1-D transport code RITM has beenapplied. In this type of study, a self-consistent description of heat and particle transportover the entire cross section of the plasma from the axis to the separatrix, is used.The plasma shaping is taken into account by using the Shafranov shift calculated fromGrad-Shafranov equation with inductive and bootstrap contributions to the currentdensity and prescribed elongation and triangularity of magnetic surfaces. Continuityequations are solved separately for electrons and impurity ions in all ionization stages,including diffusive and convective components of the particle fluxes. The charged particlesources are determined by taking into account the ionization of neutrals from NBIand those recycling into the confined volume across the separatrix. The behavior ofrecycling hydrogen atoms in the plasma is described in a fluid approximation. Theflux and density of the background ions is calculated under the assumption of plasmaquasi-neutrality. The external fueling rates of background neutrals, carbon and neonatoms are adapted to match the experimental volume average electron density and thetotal radiated power inside the confined plasma. The radiation profile is computed bytaking all impurity ionization stages into account and used together with the experimentalprofiles of auxiliary heating power density from the NBI and high frequency133

7. TRANSPORT MODELING OF IMPURITY SEEDEDEXPERIMENTS AT JETradio-waves in the ion and electron heat transport equations to calculate the radialtemperature profiles. The boundary conditions at the separatrix are defined by theexperimental decay lengths of the plasma parameters. The transport model in RITMaccounts for the most important drift modes due to toroidal ITG, dissipative trappedelectron (DTE), drift-Álfven (DA) and drift resistive ballooning (DRB) instabilitiesand the transport coefficients computed in mixing length limit (21).For impurity ions the same anomalous diffusivity as for the main particles and theconvection velocity calculated in reference (39) by accounting for ITG and TE modes inthe collisionless limit are used. In the present study the neoclassical convective velocitycomputed according to reference (74) has been added to the anomalous one excludingthe very central plasma region.Figure 7.14 shows the radial profiles of several calculated plasma characteristics.The symbols in figure 7.14a and 7.14c show the experimental data taken from JETexperimental data base. According to figure 7.14a RITM can reproduce the effectiveion charge in good enough agreement with the experimentally measured ones. In shot69091 the main contribution to Z eff is from carbon. Since a model for the scrape-offlayer is needed to calculate self-consistently the influx of neutral carbon atoms throughthe separatrix, this has been selected in order to reproduce the experimentally measuredtotal radiation losses. In shot 69093 the carbon content is significantly reducedand calculations with neon only result in the best agreement with the experiment, inparticular in the plasma core. Computations with the neoclassical impurity transporttaken into account in the whole plasma resulted in a neon concentration profile verypeaked in the near-axis region, first 20% of normalized radius (71). This resulted insome flattening of the temperature profile here, in contradiction to experimental observation.For this reason in the present calculations the neoclassical convection, butnot diffusion, has been neglected in the very plasma core. This shows that a consistentdescription of impurity transport is a problem which has to be tackled in future. Nevertheless,the self-consistent approach realized in RITM to model the impurity transporthas been applied to underline these difficulties, instead of using, e.g., the experimentalimpurity density profiles. Moreover, the main focus of the present study is not impuritytransport model but rather the impact of impurities on the plasma transport leading134

7.5 Predictive Transport ModelingFigure 7.14: Radial profiles of the ion effective charge Z eff , ion temperature T i and ionheat diffusivity χ i computed with the RITM code for JET discharges with low (dashedcurves) and high (solid curves) neon content; the fitted experimental data of the T i profiletaken from JET experimental data base are shown by squares and crosses, respectively.135

7. TRANSPORT MODELING OF IMPURITY SEEDEDEXPERIMENTS AT JETto confinement improvement. Figure 7.14b shows the calculated density of radiationlosses. They provide the total radiated power of 15% and 60% of the heating power,correspondingly.In figure 7.14c the computed ion temperature is compared with the experimentallymeasured one. Calculations reproduce the significant temperature increment in theplasma core by strong neon seeding however there are significant uncertainties in theexperimental T i data at the edge region. The causes for the temperature rise in thedischarge with high neon content can be understood by analyzing the radial profilesof the ion heat diffusivity χ i , figure 7.14d. First, the width of the ETB, where χ iis reduced to the neoclassical level, is noticeably increased. Second, χ i is decreasedthrough the whole plasma core maintaining the stiffness of the temperature profile, i.e.,the enhancement of the central T i proportionally to the pedestal temperature.In stationary states RITM modeling predicts that both deuterium and neon fluxesare strongly reduced in the plasma with higher neon content. This offer an explanationof the experimental observations that it is necessary to inject much more deuteriumin shot 69091 in order to sustain roughly the same plasma density as in shot 69093with higher neon concentration. The time evolution demonstrates that after the initialstrong burst of neon in the discharge 69093 the edge plasma undergoes the transitioninto the state of better confinement, neon particle transport reduces and it is unnecessaryto puff much more neon than in the case of the lower confinement discharge 69091in order to maintain a significantly higher neon content.Since the deuterium puff intensity has been 6 times higher in the discharge 69091than in 69093, for the RITM computations the corresponding deuterium puff intensitiesare taken equal to 3 · 10 22 and 5 · 10 21 atoms per second, respectively. In this figure, thesymbols represent the experimental data taken from JET experimental data base (fromcharge exchange recombination spectroscopy diagnostics (CXFM) database). The computedresults are in agreements with the experimental observations. In particular, theincrease in the central ion temperature by strong neon seeding and the broadening ofthe pedestal are well reproduced. As illustrated in this figure RITM can reproducethe effective ion charge in good enough agreement with the experimentally measured136

7.6 Discussionones. Calculations reproduce the significant temperature increment in the plasma coreby strong neon seeding however there are significant uncertainties in the experimentalT i data at the edge region. The causes for the temperature rise in the discharge withhigh neon content can be understood by analyzing the radial profiles of the ion heatdiffusivity χ i . First, the width of the pedestal, where χ i is reduced to the neoclassicallevel, is noticeably increased. Second, χ i is decreased through the whole plasma coremaintaining the stiffness of the temperature profile, i.e., the enhancement of the centralT i proportionally to the pedestal temperature.7.6 DiscussionWe have performed a qualitative transport modelling of the JET discharges whereneon seeding has been explored to increase the radiated power fraction (up to 60%),providing a significant reduction of target tile power fluxes (and hence temperatures),and mitigation of ELM activities. For transport modeling the one-dimensional fluidtransport code RITM has been applied. We have investigated the observed plasmabehavior in a series of tests and the obtained results show that the interplay betweenthe edge and core confinement is very delicate, and by decreasing the plasma pressureat the edge to reduce the ELM activities the edge confinement always degrades. Thedegradation of the confinement due to the increased radiated power alone is weak, andif balanced against the increase of the Z eff the impact of Z eff is stronger, therefore theoverall confinement can be compensated by the increase in the core confinement. Thiscore confinement improvement can be achieved by an increase in Z eff , and thereforereducing the core ITG instability. However, the degrading impact of the decreasedpedestal density balanced against the improvement of the core confinement due to theincreased Z eff indicates that the impact of density reduction is stronger [than improvementwith Z eff ] when going from 69089 to 69091, and it can only be compensated tosome extent by the increase of the Z eff . These results, while staying in the range ofthe experiment, can reproduce qualitatively the observed behavior.From the presented results here one would expect that the overall confinement canimprove if only the Z eff increase compensates the pedestal degradation. This does notalways occur, for example in the case of the discharge 69091, and also discharges shown137

7. TRANSPORT MODELING OF IMPURITY SEEDEDEXPERIMENTS AT JETin the reference (59) with lower input power, P tot = 20 − 25MW , where the increasein the Z eff is not able to generate high enough core confinement so as to compensatefor the edge degraded confinement. While the overall confinement recovery can occurin the discharge 69093 where the Z eff is higher, and it can even begin to improveby further increasing the Z eff . Some examples where the improved confinement byincrease of Z eff is seen: at JET in the reference (60) where nitrogen seeding for ELMmitigation in hybrid discharges were performed, in RI mode at TEXTOR (64; 65), andthe improved L-mode regime of ASDEX-Upgrade (73).In conclusion, it appears that there exists a non-monotonic variation of the confinementtime with impurity seeding. If the impurity seeding does not generate a highenough effective charge (below a certain critical value which needs to be evaluated) theresultant confinement is always degraded, in comparison to the not seeded shot, due tothe increased radiation. However, if the effective charge generated by seeding is higherthan the critical value the confinement will improve because of the reduction of thecore ITG instability and therefore increased core confinement.138

8Study of Drift WaveCharacteristics Using FractionalKinetics8.1 IntroductionExperimental observation of the edge turbulence in the fusion devices (75; 76) showthat in the Scrape of Layer (SOL) plasma is characterized with non-Gaussian statisticsand non-Maxwellian Probability Distribution Function (PDF). It has been recognizedthat the nature of cross-field transport trough the SOL is dominated by turbulence witha significant ballistic or non-local component and it is not simply a diffusive process(77). There are studies of the SOL turbulent transport using the 2-D fluid descriptions(78) or based on probabilistic models using the Levy statistics (fractional derivativesin space) (79; 80). However, these models are base on the fluid assumptions which isin contradiction with the non-Maxwellian plasma observed. Therefore, in the presentwork we try to make a more fundamental study by looking at the effect of the non-Maxwellian plasma on the anomalous transport using a gyro-kinetic formalism. Here,we consider the application of fractional kinetics to plasma physics. This approach,classical indeed, is new in its application. Our aim is to study the effects of a non-Gaussian statistics on the characteristic of the drift waves in fusion plasmas.In order to calculate an equilibrium PDF we use a model based on the motion of139

8. STUDY OF DRIFT WAVE CHARACTERISTICS USINGFRACTIONAL KINETICSa charged Levy particle in a constant external magnetic field obeying non-Gaussian,Levy statistics (81). This assumption is the natural generalization of the classical exampleof the motion of a charged Brownian particle with the usual Gaussian statistics(82). Kinetic equations from the fractional generalization of the Liouville equation havebeen developed previously (83; 84; 85). It has been proven that the chaotic dynamicscan be described by using the Fokker-Planck equation with coordinate fractionalderivatives as a possible tool for the description of anomalous diffusion (86; 87). Muchwork has been devoted on investigation of the Langevin equation with Levy white noise,see References (88; 89; 90; 91; 92), or related fractional Fokker-Planck equation (88; 93).We use the solution of the Fokker-planck equation with a collisional operator consistingof a constant, uniform friction and a stochastic field modeled by the Levy typestatistics represented by a fractional derivative in the velocity space, see Ref. (94).In general, the only thermodynamic constraints imposed on any collisional operator isto produce entropy ( ∫ f Lnfdxdv ≤ 0) and C[f, f] = 0. These constraints are satisfiedfor this collisional operator. Also, it has not yet been shown that in a direct wayfrom one of the classical form of collision operator (95) one can derive the alpha-stableequilibrium distribution. The solution of the Fokker-Planck equation with fractionalvelocity derivatives in shear less slab geometry and the stationary state is then pluggedin the linearized gyro-kinetic dispersion equation. The the dispersion equation is solvednumerically and the solutions are presented. The results shown here are part of a paperby the author, submitted to journal: Physical Review Letters.8.2 From Brownian Motion to Levy MotionThe theory of Brownian motion is concerned with the irregular, perpetual motions ofcolloidal particles in suspension in a liquid. These motions are originated from thecollisions of the colloidal particle with the surrounding particles in the fluid. Undernormal conditions in a liquid, a Brownian particle will suffer about 10 21 collisions persecond. This means that it is impossible to follow the path of a particle in any greatdetail; the details are impossibly fine for our senses.140

8.2 From Brownian Motion to Levy MotionThe theory of Brownian motion as initiated by Einstein derives from the followingset of assumptions: the motion of a free particle (i.e. one in the absence of an externalfield of force) is assumed to be governed by an equation of the formdvdt= −νv + A(t) (8.1)where v denotes the instantaneous velocity of the particle. Here, we have made theassumption that the influence of the surrounding medium can be split up to two parts:a symmetric part, −νv, which represents the operation of a dynamical friction and afluctuating part, A(t), which is characteristic of Brownian motion (the so called whitenoise).For the part A(t) the following principal assumptions are made:• A(t) is independent of v, and• A(t) varies extremely rapidly compared with v.The second assumption implies that time interval ∆t exists such that during ∆tthe changes in v to be expected are very small, while during the same interval A(t)may undergo a very large number of fluctuations. Alternatively, we may express thisassumption by the statement that though v(t) and v(t+∆t) are expected to differ by anegligible amount, no correlation between A(t) and A(t + ∆t) is expected. Consideringthen the net increments in velocity,B(∆t) =∫ t+∆ttA(ξ)dξ (8.2)which a particle experiences (due to the random fluctuations) during an interval ∆t,we assert (i) that the increments between the successive intervals (t, t + ∆t 1 ) and (t, t +∆t 1 , t, t + ∆t 1 + ∆t 2 ) have no correlation, and (ii) that the probability of occurrence ofdifferent net increments during an interval ∆t is given byP [B(∆t)] =1(4πD∆t) 3/2 exp[−|B∆t)|2 /4D∆t] (8.3)where D is certain diffusion coefficient (in velocity space) related to the frictional coefficient,η, by141

8. STUDY OF DRIFT WAVE CHARACTERISTICS USINGFRACTIONAL KINETICSD = ηk B T/m (8.4)where k B is the Boltzmann constant and T is the absolute temperature. The onlyphysically meaningful time-asymptotic (t → ∞ ) distribution of velocities which derivesfrom eq. (8.3) has always been thought to be Maxwellian:mP [v] = (2πk B T )3/2 exp[−|v| 2 /2k B T ] (8.5)A Markovian statistical description of the dynamics which consequently assumesnormal transition probabilities for the increments in the phase space results in theFokker-Planck equation for the particle density, F (r, u, t) (see Ref. (82)),∂F∂t + v ∂F∂r + F ∂Fm ∂v = ν ∂∂(vF ) + D[∂v ∂v · ∂]F (8.6)∂vThe Maxwellian distribution is but one of the broader class of time-asymptotic distributionspermitter by the Langevian equation (92), the symmetric cases of the stabledistributions named after Levy (81).Now, we make the following assumption on the fluctuating part, A(t) in the eq.(8.1): A(t) is a stationary Levy stable process. Therefore as a generalization to the eq.(8.3) we assume that the probability of occurrence of different net increments duringinterval ∆t is given by (in the Fourier representation)˜P [B(∆t)] k = exp[−D|k| α ∆t] (8.7)where α is called Levy index, 0 ≤ α ≤ 2. For α = 2 we recover the Fokker-Planck eq.(8.6) and we get as the consequence, the Maxwellian stationary probability distributionfunction over the velocity eq. (8.7) and the normal diffusion law for the particle motionin the real space. Instead, the non-Gaussian Levy statistics of the fluctuation for A(t) iseq. (8.1) provides us with a simple and straight forward, at least from the mathematicalviewpoint, possibility to consider abnormal diffusion and non-Maxwellian stationarystates, both properties are inherent to strongly non-equilibrium plasmas of solar systemand thermonuclear devices. Thus, if α ≤ 2 then, by applying the procedure describedin detail in Refs. (82; 93), we arrive at the fractional Fokker-Planck equation142

8.3 Fractional Fokker-Planck Equation∂F∂t + v ∂F∂r + F ∂Fm ∂v = ν ∂∂v (vF ) + D ∂α F∂|v| α (8.8)In the following we will use the method proposed in the Ref. (94) for the derivationof the Fokker-Planck equation (Fractional Fokker-Planck equation) for the stationaryLevy stable processes discussed above in the case of a charge particle under Lorentzforce.8.3 Fractional Fokker-Planck EquationFollowing the Ref. (94) the Fokker-Planck equation with fractional velocity derivativesfor shear less slab geometry in the presence of a constant external magnetic field and azero-averaged electric field is∂F s∂t + v ∂F s∂r + F m s∂F s∂v = ν ∂∂v (vF s) + D ∂α F s∂|v| α (8.9)where s(= e, i) represents the plasma particle species and last term on the right handside is white Levy noise with α denotes the fractional derivative and 1 ≤ α ≤ 2. F isthe Lorentz force acting on the particles of species s with mass m s , D is a constant tobe defined latter, ν is the friction coefficient and it is independent of v.asTo find the solution we make use of the fourier representation of the above equation∂F s∂t + (−k + Ω s(k v × ˆb) + νk v ) ∂F s∂k = −D|kv | α F s (8.10)where Ω s is the Larmour frequency of species s, ˆb = B/B is the unit vector in thedirection of magnetic field and F s is the characteristic function∫ ∫F s (k, k v ; t) =dr dv exp(ik · r + ik v · v)F s (r, v; t) (8.11)Equation (8.10) is solved by the method of characteristics. From Ref. (94) thesolution corresponding to the homogenous and steady state system is143

8. STUDY OF DRIFT WAVE CHARACTERISTICS USINGFRACTIONAL KINETICS∫F s (r, v) = C(r)dk v(2π) 3/2 exp(−ikv · v) exp (− D αν |kv | α ) (8.12)Since by taking α = 2 we should recover the Maxwellian distribution, by comparisonwe can find expressions for the coefficients D and C(r). We find for C(r) =n(r)/(2 √ 2Dπ 3/2 ) where D = V 2 T s /2 and V T s = √ 2T s /m s is the thermal velocity ofspecies s with the particle density n. Here we have defined D = D/ν and considerconstant temperatures.8.4 Dispersion EquationThe particle distribution function, averaged over gyro-phase is of the form (96)f s (r, v) = F s (r, v) + (2π) −4 ×∫ ∫dk dω exp(ik · r − iωt)δfk,ω s (v) (8.13)We assume that the turbulence is purely electrostatic and neglect magnetic fieldfluctuations (δB = 0), if the deviation from local equilibrium is not too large, it obeysthe well-known linearized gyro-kinetic equation(−ω + k ‖ v ‖ )δf s k,ω (v ‖, v ⊥ ) +(ω − ω ∗s ) e sT sJ 0 (|Ω s | −1 k ⊥ v ⊥ )F s (x, v)δφ k,ω = 0 (8.14)d ln n(x)dxwhere ω ∗s = cTse k sB y · is the drift wave frequency of species s, and we assumedthat the space dependence of F s is only in the x direction perpendicular to the magneticfield and so is the density gradients. J 0 is the Bessel function of order zero. Here, v ‖is in the parallel velocity, v ⊥ ≡ (v 2 x + v 2 y) 1/2 in the absolute value of the perpendicularvelocity, and v = (v 2 ⊥ + v2 ‖ )1/2 . Inserting the expression for F s from the equation (8.12)and rearranging the terms we get for the δf k,ωδf s k,ω (v ‖, v ⊥ ) = e sT s[ ω − ω ∗sk ‖ v ‖ − ω ]J 0(|Ω s | −1 k ⊥ v ⊥ )δφ k,ω ×∫n√2VT s π 3/2dk v(2π) 3/2 exp(−ikv · v) exp (− 2αVT 2 |k v | α ) (8.15)s144

8.4 Dispersion EquationHere, the wave vector perpendicular to magnetic field lines is k ⊥ = (kx 2 + ky) 2 1/2 andin the following we use the assumption of k y = k x . The gyro-kinetic equation (8.15) iscompleted with the Poisson equation for the electric potential. For fluctuations withwave vectors much smaller than the Debye wave vector, the Poisson equation becomesthe quasi-neutrality condition∑e s δn s k,ω = 0 (8.16)swhere the density fluctuation is related to the distribution function throughδn s k,ω = −n e ∫ ∞ ∫ ∞sδφ k,ω + 2π dvT ‖ dv ⊥ v ⊥ ×s −∞ 0J 0 (|Ω s | −1 k ⊥ v ⊥ )δfk,ω s (v ‖, v ⊥ ) (8.17)Here we have used the customary splitting of the density fluctuation into an adiabaticresponse and a non adiabatic response.From the quasi-neutrality condition (8.15) we find the dispersion equation which determinesthe eigenfrequencies as functions of the wave vector, ω = ω(k) = ω r (k)+iγ(k).In the simplest case we consider a plasma consisting of electrons and a single speciesof singly charged ions with the equal temperatures. In evaluating equation (8.17) forthe electrons, we consider the zero-Larmor radius approximation, J 0 (|Ω e | −1 k ⊥ v ⊥ ) = 1.For the dispersion equation (8.16) we find as in the Ref. (96)where1 + M e k,ω = −1 − M i k,ω (8.18)∫ ∞ ∫ ∞Mk,ω s = 2π dv ‖∫1√2VT s π 3/2−∞0dv ⊥ v ⊥ [ ω − ω ∗sk ‖ v ‖ − ω ]Ψ s(b s v ⊥ /V T s ) ×dk v(2π) 3/2 exp(−ikv · v) exp (− 2αVT 2 |k v | α ) (8.19)i145

8. STUDY OF DRIFT WAVE CHARACTERISTICS USINGFRACTIONAL KINETICSwith b s = k ⊥ V T s /Ω s , and Ψ e = 1, Ψ i = J 2 0 (b sv ⊥ /V T s ).If we take α = 2 in the equation (8.19) we recover the usual dispersion equation fora Maxwellian distribution as in the Ref. (96).In order to solve the general dispersion equation of (8.18) we use the method proposedin Ref. (96) with the difference that here we have extra integrations over k v .Plasma dispersion function Z(ξ s ) can now be written asZ(ξ s ) = 1 √ πLim σ→0∫ ∞−∞where ξ s = ω/(|k ‖ |V T s ) and the function Φ(v ‖ ) isΦ(v ‖ )dv ‖ [v ‖ /V T s − ξ s − iσ ] (8.20)∫Φ(v ‖ ) =dk‖v(2π) 1/2 exp(−ikv ‖ v ‖) exp (− 2αVT 2 |k‖ v |α ) (8.21)sThe integral over v ⊥ can be written in a general way aswhereΓ(b s ) = 2∫ ∞0dv ⊥ ( v ⊥V T s)Ψ s (b s v ⊥ /V T s )Φ(v ⊥ ) (8.22)∫ dkvΦ(v ⊥ ) = ⊥(2π) exp(−ikv ⊥ v ⊥) exp (− 2αVT 2 |k⊥ v |α ) (8.23)sFinding an analytical solution for the last integral over k v with an arbitrary αin the equation (8.18) requires extremely tedious calculations. Therefore, we use aninfinitesimal deviation of the form α = 2 − ɛ, where 0 ≤ ɛ ≪ 2 and expand the lastexponential in the equation (8.18) around ɛ = 0 as followsexp (− 2|kv | (2−ɛ)(2 − ɛ)VT 2 ) = exp (− V T 2 ss4 |kv | 2 ) +18 V T 2 s exp (− V T 2 s4 |kv | 2 )(−1 + 2Log[k v ])ɛ + O[ɛ 2 ] (8.24)146

8.4 Dispersion EquationWe only keep terms of the first order of ɛ. Simply inserting relation (8.24) intoequations (8.21) and (8.23) we find in a general formΦ(v) =√2V T sexp (−v 2 /V 2 T i) + ɛΛ(v) (8.25)Here Λ(v) gives the non-Gaussian contribution from the second term in the equation(8.24) is:Λ(u) =e−u2√22V 2 T(0.4228 + 3.1544u 2 − 2e u2 √ πu Erf[u] +(−2u 2 + 4u 4 )HypergeometricPFG[{1, 1}, { 3 2 , 2}, u2 ] −Log[VT 2 ] + 2u 2 Log[VT 2 ]) (8.26)With u ≡ v/V T and HypergeometricPFG[a 1 , . . . , a p b 1 , . . . , b q , z] the Mathematicageneralized hypergeometric function p F q (a; b; z) defined aspF q (a; b; z) =∞∑(a 1 ) k . . . (a p ) k /(b 1 ) k . . . (b q ) k z k /k! (8.27)k=0Replacing the above expression in the functions Γ and Z and noting that the driftwaves are defined in the frequency range |k ‖ |V T i ≪ ω ≪ |k ‖ |V T e . In evaluating equation(12) for electrons we expand the terms in the brackets in powers of ξ e = ω/(|k ‖ |V T e ) ≪ 1and for ions we expand it in powers ξ −1i= (|k ‖ |V T i )/ω ≪ 1, respectively (96).The dispersion function Z(ξ s ) now includes two parts one from usual Gaussian in(8.25) and one part which includes velocity integral over∫1 ∞√ π−∞dv ‖Λ(v ‖ )(v ‖ /V T s − ξ s )(8.28)Using the expansion in powers of ξ −1iwe can rewrite the above integral as∫1 ∞√ π−∞du(− u2ξ 3 i− u ξ 2 i− 1 ξ i)Λ(u) (8.29)147

8. STUDY OF DRIFT WAVE CHARACTERISTICS USINGFRACTIONAL KINETICSwhere we defined u = v ‖ /V T i . Figure 8.1 shows the three terms u 2 Λ(u), uΛ(u) and Λ(u)under the integral as functions of u. The second term is always an even function ofu therefore its contribution to the integral over −u, u is zero therefore, we can simplyneglect this term under the integral.Figure 8.1: u 2 Λ(u) (Solid line), uΛ(u) (Dashed line) and Λ(u) (Dotted line) as functionsof u.The integral over the v ⊥ has also two parts corresponding to two terms in equation(8.25) and for the non-Gaussian contribution of Λ(v ⊥ ) we have the integral of the form2∫ ∞0dwwJ 2 0 (b i w)Λ(w) (8.30)where w = v ⊥ /V T i . Figure 8.2 shows the function under integral as function of w.8.5 Solution of Dispersion EquationLet us consider an explicitly limiting case where adiabatic assumption for electrons isused,δn e k,ω = n e T eδφ k,ω (8.31)148

8.5 Solution of Dispersion EquationFigure 8.2: 2wJ 2 0 (b i w)Λ(w) as functions of w.where e denotes the electron charge. For ions, we insert equations (8.20) and (8.22)into equation (8.19),where we have ¯ω ∗i = ω ∗i /(|k ‖ |V T i ).M i k,ω = V 2 T i2 (ξ i − ¯ω ∗i )Γ(b i )Z(ξ i ) (8.32)Here the only gradient we have is the densitygradient imbedded into definition of ¯ω ∗i . The dispersion equation for ɛ = 0, i.e., α = 2as in Ref. (96),2 = −I 0 ( b2 i2 )e−b2 i /2 1(1 + )(1 + λ21¯ω 2¯ω ) (8.33)where I 0 denotes modified Bessel function of the zeroth order, λ = (|k ‖ |V T e )/ω ∗e andthe eigenfrequency ω is made dimensionless by introducing the complex quantity ¯ω,ω = ω ∗e¯ω (8.34)For ɛ ≠ 0 the equation (8.18) can be simply written in terms of ɛ as2 = −I 0 ( b2 i2 )e−b2 i /2 1(1 + )(1 + λ21¯ω 2¯ω ) − ɛχ i(¯ω) (8.35)149

8. STUDY OF DRIFT WAVE CHARACTERISTICS USINGFRACTIONAL KINETICSwhere χ i is a function of ¯ω and generates the contribution from non-Gaussian part ofthe distribution function and it can be written asχ i (ξ i ) = V T 2 i2 (ξ i − ω ∗i ) ×∫2 ∞√ dwwJ 2π0 (b i w)Λ(w)0∫ ∞−∞du(− u2ξ 3 i− 1 ξ i)Λ(u) (8.36)Note, that here χ i is a function of ξ i . However, we restrict ourselves to show onlythe results of our numerical calculations which were performed with the use of computercodes for different values of ɛ.8.6 Numerical SimulationWe carry out numerical simulation in order to solve the dispersion equation. Thefollowing plasma parameters of a typical fusion plasma have been used in the computations(97): V T i = 1.38 × 10 8 [m/s], T e = T i , d ln n/dx = 1, and k ‖ = 0.001[m −1 ].The dispersion equation is of the 3rd order of ¯ω, therefore there are three possiblesolutions. However, here we are interested in the solutions with non-zero imaginaryvalue, γ ≠ 0, which give rise to growth rate (if γ < 0) or dampening (if γ > 0) of theunstable mode. Therefore, we only show here the solution with such feature. Figure8.3 shows the values of the γ, for such solutions as a function of ɛ for b i = 0.1. As onecan see from this figure even a small deviation from the Gaussian statistics will lead toa significant increase of the growth rate of the linearly unstable density gradient mode.Such an increase, consequently, will lead to a significant increase of the anomalous flux.Experimental observations have shown that the nature of cross-field transport troughthe SOL is dominated by turbulence with a significant ballistic or non local componentand it is not simply a diffusive process (75; 77). The increase of the anomalous fluxshown in this work seems in line with these observations.8.7 DiscussionWe have studied the characteristics of the unstable modes in plasmas with non-Gaussianstatistics. We have used the solution of the Fokker-Planck equation with fractional velocityderivatives (94) into the plasma dispersion equation in the most simplest case150

8.7 DiscussionFigure 8.3: γ from two solutions of the dispersion equation as functions of ɛ for b i = 0.1.The solution with γ < 0 (blue) gives the growth rate of the unstable mode.of a plasma with adiabatic electrons and one singly charges ions. In the Fourier representationthe fractional derivatives are represented by the fractional exponent whichwe expand around a small deviation from Gaussian where the derivatives are of secondorder. This small deviation generates the non-Gaussian contribution to the usualGaussian. The impact of such a deviation on the imaginary contribution of the modefrequency, γ, of the plasma dispersion equation solutions, have been studied. The resultsshow that for α < 2 this contribution increases significantly with increase in thedeviation factor ɛ and therefore can lead to a significant increase in anomalous flux.The obtained results show the importance of taking into account possible deviations ofthe plasma distribution function from the Maxwellian when computing the anomalousflux. However, more in depth studies should be performed in order to further developmodels based on the fractional Liouville equation with a more robust implementationof the fractional derivatives.151

8. STUDY OF DRIFT WAVE CHARACTERISTICS USINGFRACTIONAL KINETICS152

9DiscussionUnderstanding the mechanisms of transport of matter, electric charge, energy and momentumis one of the most important goals of research in the field of plasma physics.Practically all applications of plasmas are limited in some way or other by the transportphenomena taking place under specific circumstances. Therefore, it is vital for afusion reactors to be able to control and limit cross-filed transport. The main focusin this thesis was on the study of the impurity anomalous transport. Based on a fluiddescription we have developed a linear model for the study of the transport due to thedrift wave microinstabilities: the trapped electron drift mode (TE mode), and the iontemperature gradient mode (ITG mode). The fortran code AFC-FL (Anomalous FluxCalculation in Fluid Limit) has been developed during this thesis in order to performnumerical computation for the instability characteristics and calculation of transportcoefficients for the main and impurity particles in tokamak plasmas. The code solvesthe dispersion equation for a wide range of the mode wave numbers (for k y ρ s ≤ 1 whichis the relevant range to ITG and TE modes). The TE and ITG mode characteristics,such as mode frequency and growth rate under relevant conditions for tokamak plasmaswere computed, and different possible stabilization effects such as collisional detrappingof the trapped electrons, presence of impurity ions and magnetic shear on the unstablemodes were studied. A summary of the results obtained is as followed:1. It was observed experimentally that the density profiles for different impuritiesare qualitatively different, tending to be hollow in the case of light species likecarbon and significantly peaked for heavy impurities like nickel. It is also observed153

9. DISCUSSIONthat the impurity density gradient essentially affects the instability characteristics.Therefore, we developed a complete model so that the impact of impuritycan be studied firmly by considering realistic plasmas with all ion species, individually,taken into account. The results of our numerical computations using thismodel confirmed that indeed, at high enough impurity concentration the instabilitycharacteristics are sensitive, in particular, to the impurity density gradient.2. It was observed experimentally that there exists a charge dependence of the impurityanomalous transport characteristics (transport coefficients) which was unexplainedby the existing models. We followed our study on to investigating thischarge dependence, and its possible underlying physical mechanism. The obtainedresults showed that the observed charge dependence may be explain through theimpurity ion collisions with the main plasma components. Collisional effects wereincluded into linearized transport equations as friction and thermal forces and energyexchange between ion plasma components. The thermal forces are calculatedby taking into account that in the hot plasma core collision frequencies of backgroundparticles can be smaller than the frequency of parameter fluctuations dueto micro-instabilities. The investigation was done for plasma parameter fluctuationsdriven by the ITG/TE modes. It was found that the relative importance ofcollisional effects in the impurity anomalous transport changes significantly withthe instability mechanism, safety factor and magnetic shear. For the plasma coreconditions these effects generate a significant charge dependence of the impuritypinch-velocity and peaking factor while for the plasma edge their contributionsare not as significant. From the new effects considered, the thermal forces actingon impurity ions from electrons are of the most importance since electroncollision frequency is higher than that of the main ions. Therefore, the chargedependence of the impurity peaking factor is more pronounced in regions whereTE modes are the dominant instabilities. With increasing magnetic shear thisdependence becomes weaker. The found charge dependence of impurity anomaloustransport, arising due to collisions with the background plasma particles,can have important consequences. Under certain conditions significantly peakeddensity profiles of high-Z impurity ions can be expected, and the relation of thisbehavior to the experimental phenomenon of spontaneous impurity accumulation154

has to be investigated in future. For this purpose the effects discussed here haveto be included into transport codes in order to predict reliably the behavior andimpacts of impurity on the plasma in future reactor devices.3. During 2006-2007 JET experimental campaigns, operation at high power in configurationsrelevant to ITER steady-state operation has been explored; by injectinghigh-Z radiative gas, such as neon, to increase the edge radiation. Itwas observed that higher edge radiation due to the injected neon led to an overalldegradation of the plasma confinement from the reference discharge withoutneon seeding. However, when increasing the neon content at some point the confinementdegradation appears to stop and starts to increase with increase of theneon content, as was previously observed in the so called RI modes (RadiativeImproved modes). An example of such phenomena observed in JET was shownin Ref. (33). These discharges were part of a series of experiments where neonseeding has been explored to increase the radiated power fraction (up to 60%),providing significant reduction of target tile power fluxes (and hence temperatures)and mitigation of ELM activities. When increasing the neon seeding inshot: 69093, the amplitude of the ELM perturbation was strongly reduced andthe diamagnetic stored energy was also increased by 25%. This performance improvementwas seen on the ion temperature profiles which showed a temperatureincrease both at the top of the H-mode pedestal and in the core (from 6 to 12keV) and also the electron density increased by ∼10 % in the core. The neonconcentration profile, as measured by the charge exchange recombination spectroscopy,was hollow with a concentration of about 1% in the core rising to ∼ 2.5% at the normalized radius of 0.7. No sign of impurity accumulation was foundin these discharges. The cause of the improvement was not identified.We investigated the underlying mechanism responsible for the observed confinementimprovement in the discharges mentioned above and performed a qualitativetransport modeling in a series of tests. For transport modeling the quasi-linear1-D fluid transport code RITM has been applied. The goal of the present studywas to model the plasma behavior inside the separatrix because we have no informationon how the Scrape of Layer (SOL: the layer between last closed magnetic155

9. DISCUSSIONsurface and machine walls) parameters have been modified. Although the modificationsin SOL, from one discharge to another can be of great importance onthe behavior of the plasma, we did not take them into account here, because ofthe limited ability to model turbulence and the radiative processes in the SOL.Starting from the reference discharge with only D fueling (69089), towards thedischarges with the neon seeding, the expected impacts on the confinement are: adegradation due to the density reduction, a degradation due to the P rad increase,and an improvement due to the Z eff increase. To balance these three effects,the impact of each is first modeled separately. Also, a predictive transport modelingfor the same discharges as mentioned above with the use of RITM codewas performed. In this type of study, a self-consistent description of heat andparticle transport over the entire cross section of the plasma from the axis to theseparatrix, was implemented.The obtained results show that the interplay between the edge and core confinementis very delicate, and by decreasing the plasma pressure at the edge to reducethe ELM activities the edge confinement always degrades. The degradation of theconfinement due to the increased radiated power alone is weak, and if balancedagainst the increase of the Z eff the impact of Z eff is stronger, therefore theoverall confinement can be compensated by the increase in the core confinement.This core confinement improvement can be achieved by an increase in Z eff , andtherefore suppressing the core ITG instability. However, the degrading impactof the decreased pedestal density balanced against the improvement of the coreconfinement due to the increased Z eff indicates that the impact of density reductionis stronger, and it can only be compensated to some extent by the increaseof the Z eff . These results, while staying in the range of the experiment, couldreproduce qualitatively the observed behavior.From the presented results here one would expect that the overall confinementcan improve if only the Z eff increase compensates the pedestal degradation. Thisdoes not always occur, for example in the case of the discharge 69091 the increasein the Z eff was not able to generate high enough core confinement so as tocompensate for the edge degraded confinement. While the overall confinementrecovery could occur in the discharge 69093 where the Z eff was higher, and it156

can even begin to improve by further increasing the Z eff .At the end, we considered a new approach for the study of the drift wave microinstabilities.The application of fractional kinetics to plasma physics was examined. Thisapproach, classical indeed, is new in its application. Our aim was to study the effectsof a non-Gaussian statistics on the characteristic of the drift waves in fusion plasmas.Experimental observation of the edge turbulence in the fusion devices show that in theSOL plasma is characterized with non-Gaussian statistics and non-Maxwellian ProbabilityDistribution Function (PDF). It has been recognized that the nature of cross-fieldtransport trough the SOL is dominated by turbulence with a significant ballistic or nonlocalcomponent and it is not simply a diffusive process. There are studies of the SOLturbulent transport using the 2-D fluid descriptions or based on probabilistic modelsusing the Levy statistics (fractional derivatives in space). However, these modelsare base on the fluid assumptions which is in contradiction with the non-Maxwellianplasma observed. Therefore, we tried to make a more fundamental study by lookingat the effect of the non-Maxwellian plasma on the turbulent transport using a gyrokineticformalism. We use the solution of the Fokker-planck equation with a collisionaloperator consisting of a constant, uniform friction and a stochastic field modeled by thealpha-stable statistics represented by a fractional derivative in the velocity space. Ingeneral, the only thermodynamic constraints imposed on any collisional operator is toproduce entropy ( ∫ f Lnfdxdv ≤ 0) and C[f, f] = 0. These constraints are satisfied forthis collisional operator. Also, it has not yet been shown that in a direct way from oneof the classical form of collision operator one can derive the alpha-stable equilibriumdistribution.The solution of the Fokker-Planck equation with fractional velocity derivatives inshear less slab geometry and the stationary state is then plugged in the linearizedgyro-kinetic dispersion equation. The the dispersion equation was solved numericallyand was shown that a small deviation from Maxwellian PDF can lead to a significantincrease in anomalous flux. The increase of the anomalous flux shown in this workseems in line with the observations. We have used a simplified model based on theFokker-Planck equation with fractional velocity derivatives to study a simplified case.The obtained results show the importance of taking into account possible deviations of157

9. DISCUSSIONthe plasma distribution function from the Maxwellian when computing the anomalousflux. However, ultimately a more in depth studies should be performed in order tofurther develop models based on the fractional Liouville equation with a more robustimplementation of the fractional derivatives (maybe going through a revisited derivationof the collision operator adapted to power law tails).158

[10] Surko C. M. et al. Ecole Polytechnique,Palaiseau pp 21, 1986. 12References[1] Wesson J. Tokamak. Third Ed.Clarendon Press-Oxford, 2004. ix, 2, 17,22, 57[2] R. Hawryluk. Rev. Mod. Phys 70 pp537, 1998. ix, 9, 10[3] ITER Expert Groups on Confinementand Transport. Nuclear Fusion39 pp 2175, 1999. ix, 11, 13, 19, 21, 22[4] Hendel H. W. et al. Physics of Fluids,11 pp 2426, 1968. x, 55[5] European CommissionDirectorate-General for Energyand Transport. Reporton the Green Paper on Energy.http://www.jet.efda.org/wpcontent/uploads/2005-green-paperreport-en.pdf,2005. 1[6] Artsimovich L. A. et al. PlasmaPhysics 7 pp 305, 1965. 4[7] Sakharov A. D. et al. in Proc. Of the1957 Geneva Conf. On Peaceful Appl. OfAtomic Energy, Pergamon Press Vol 1pp 21-34, 1961. 4[8] Wagner F. et al. Physical Review Letters49 pp 1408, 1982. 11[9] Campbell D. J. et al. Nuclear Fusion26 pp 1085, 1986. 12[11] Matthews G. F. et al. Nuclear Fusion39 pp 19, 1999. 15, 78, 100[12] Balescu R. Transport processesin Plasmas, Vol. 1 Classical Transport.North Holland, Amsterdam, ElsevierScience Publishers B. V., 1988. 15,30, 45[13] Chen F. F. Plasma Physics andControlled Fusion Vol. 1: PlasmaPhysics. Published by Springer secondEd., 1974. 17, 45[14] Galaeev A. A. and Sagdeev R. Z.Sov. Phys. JETP, 26 pp 233, 1968. 18[15] Hinton F. L. and Hazeltine R. D.Review Mod. Phys., 48 pp 239, 1976. 18[16] Balescu R. Transport processesin Plasmas, Vol. 2 Neo-classicalTransport. North Holland, Amsterdam,Elsevier Science Publishers B. V.,1988. 18[17] Perkins F. W. et al. Physics of Fluids,B 5 pp 477, 1993. 21[18] Goldstein H. Classical Mechanics.revised Ed. Addison-Wesley, New York,1980. 30[19] Braginski S. I. Reviews of PlasmaPhysics 1 pp 205, 1965. 45, 61, 96, 97,98[20] Lopes Cardozo N. J. Plasma Physicsand Controlled Fusion 37 pp 799, 1995.48159

REFERENCES[21] Weiland J. Collective Modes in InhomogeneousPlasmas. Kinetic andAdvanced Fluid Theory. Bristol: Instituteof Physics Publishing, 2000. 50,57, 60, 69, 84, 92, 100, 134[22] Horton W. Reviews of ModernPhysics, Vol. 71, No. 3, 1999. 53, 55[23] Horton W. Phys. Rev. Lett. 19 pp1269., 1976. 57[24] Moradi S. et al. Physics of Plasmas17 pp 012101, 2010. 60, 65, 79[25] Nilsson J. and Weiland J. NuclearFusion 34 pp 803, 1994. 61, 66[26] Angioni C. and Peeters A. G. PhysicalReview Letters 96 pp 095003, 2006.65, 79, 92, 100[27] Moradi S. et al. Nuclear Fusion 49pp 085007, 2009. 65, 79, 87, 100[28] Nordman H. et al. Plasma Physicsand Controlled Fusion 49 pp 985, 2007.66[29] Hirose A. et al. Physics of Plasmas2 pp 859, 1995. 66[30] Lazarusa E. A. et al. Journal of NuclearMatter 121 pp 61, 1984. 78, 117[31] Ongena J. et al. Physics of Plasmas8 pp 2188, 2001. 78, 92, 117[32] Kubo H. et al. Nuclear Fusion 41 pp227, 2001. 78, 117[33] Liutadon X. et al. Plasma Physicsand Controlled Fusion 49 pp B529-B550,2007. 78, 90, 100, 117, 118, 119, 133, 155[34] Paccagnella R. et al. Nuclear Fusion30 pp 545, 1990. 79[35] Frojdh M. et al. Nuclear Fusion 32pp 419, 1992. 79[36] Dominguez R.R. and StaeblerG.M. Nuclear Fusion 33 pp 51, 1993.79[37] Tokar M. Z. et al. Plasma Physicsand Controlled Fusion 41 pp L9, 1999.79[38] Fülöp T. and Weiland J. Physics ofPlasmas 13 pp 112504, 2006. 79[39] Nordman H. and Fülöp T. Physicsof Plasmas 14 pp 052303, 2007. 79, 113,134[40] Fülöp T. and Nordman H. Physicsof Plasmas 16 pp 032306, 2009. 79, 96[41] Giroud C. et al. Progress in understandingimpurity transport atJET. FEC 2006, Chengdu, paper EX/8-3, 2006. 79, 84, 92, 93, 94, 102, 108[42] Giroud C. et al. Study of Zdependence of impurity transportat JET. 34th EPS on plasmaphysics, Warsaw, Poland 2007)http://www.iop.org/Jet/fulltext/EFDC070339.pdf,2007. 80[43] Samm U. et al. Plasma Physics andControlled Fusion 35 pp B167, 1993. 92[44] Hirshman S. P. and Sigmar D. J.Nuclear Fusion pp 1079, 1981. 92[45] Guirlet R. et al. Plasma Physics andControlled Fusion 48 pp B63-B74, 2006.92[46] West W. P. et al. Physics of Plasmas9 pp 1970, 2002. 92160

REFERENCES[47] Weisen H. et al. in Fusion Energy2004 (Proc. 20th Int. Conf. Vilamoura)(Vienna: IAEA) CD-ROM file EX/P6-31, 2004. 92, 93[48] Giroud C. et al. (31th EPS onplasma physics, London) ECA 28G P-1.144, 2004. 92, 94[49] Puiatti M. E. et al. Physics of Plasmas13 pp 042501, 2006. 92[50] Dubuit N. et al. Physics of Plasmas14 pp 042301, 2007. 92, 100[51] Stangeby P. C. The Plasma Boundaryof Magnetic Fusion Device. IoPpublishing, 2000. 97, 98[52] Smirnov B. M. Physics of IonizedGases. John Wiley & Sons, 2001. 97[53] Trubnikov B. A. Reviews of PlasmaPhysics 1 pp 105, 1965. 98[54] Bizard A. Physics of Plasmas 7 pp4816. IoP publishing, 2000. 110[55] Balescu R. Aspects of AnomalousTransport in Plasmas. IoP Publishing,2005. 113[56] Bourdelle C. et al. Physics of Plasmas14 pp 112501, 2007. 114, 115[57] Bourdelle C. et al. Nuclear Fusion42 pp 892-902, 2002. 114, 115[58] Rapp J. et al. Plasma Phys. Control.Fusion 44 pp 639, 2002. 117, 118[59] Beurskens M. N. A. et al. Nucl. Fusion48 pp 095004, 2008. 117, 118, 119,131, 133, 138[60] Corre Y. et al. Plasma Phys. Control.Fusion 50 pp 115012, 2008. 117,118, 122, 133, 138[61] Dominguez R. R. and Waltz R. E.Nucl. Fusion 27 pp 65, 1987. 117[62] Matthews G. F. et al. Nucl. Fusion39 pp 19, 1999. 117[63] Monier-Garbet P. et al. Nucl. Fusion45 pp 1404, 2005. 118[64] Ongena J. et al. Nuclear Fusion 33pp 283, 1993. 118, 138[65] Ongena J. et al. Plasma Physicsand Controlled Nuclear Fusion Research(Proc. 14th Int. Conf. W´’urzburg, 1992)IAEA, Vienna, 1 pp 725, 1992. 118, 138[66] Messiaen A. M. et al. Nuclear Fusion34 pp 825, 1994. 118[67] Vandenplas P. E. et al. J. PlasmaPhys. 59 pp 587, 1998. 118[68] Tokar M. Z. Plasma Phys. Control.Fusion 36 pp 1819, 1994. 119, 124[69] Unterberg B. et al. Plasma Phys.Control. Fusion 46 pp A241, 2004. 119,124[70] Kalupin D. et al. Nucl. Fusion 45 pp468, 2005. 119, 124[71] Moradi S. et al. Modeling of confinementimprovement and impuritytransport in high power JETH-mode discharges with neon seeding.Proceedings of the 35th EPSConference on Plasma Physics, Crete,Greece, 2008. 119, 124, 134[72] Tokar M. Z. et al. Plasma Phys.Control. Fusion 41 L9, 1999. 128161

REFERENCES[73] Neuhauser J. et al. Plasma Phys.Control. Fusion 37 pp A37., 1995. 133,138[74] Wenzel K. W. and Sigmar D. J.Nucl. Fusion 30 pp 1117, 1989. 134[75] Sweben S. J. et al. Plasma Phys.Control. Fusion 49 pp S1-S23, 2007. 139,150[76] Carreras B. A. et al. Phys. Rev.Lett. 80 pp 4438, 1998. 139[77] Naulin V. Journal of Nucl. Mater. 363-365 pp 24-31, 2007. 139, 150[78] Garcia O.E. et al. Phys.Rev. Lett. 92pp 165003, 2004. 139[79] Gavnholt J. et al. Phys. of Plasmas12 pp 084501, 2007. 139[80] Sanchez R. van Milligen B. Ph.and Carreras B. A. Phys. of Plasmas11 pp 2272, 2004. 139[81] Levy P. Theorie del Addition desVariables. Gauthier-Villiers, Paris,1937. 140, 142[82] Chandrasekhar S. Rev. Modern Phys.21 pp 383, 1949. 140, 142[83] Zaslavsky G. M. Phys. Rep. 371 pp461, 2002. 140[84] Tarasov V. E. J. Phys. Conference series7 pp 17, 2005. 140[85] Skorrkhod A. V. Random Processeswith Independent Increments.Nauka, Moscow, (engl. transl.),Kluwer, Dordrecht, (1991), 1964. 140[86] Zaslavsky G. M. Physica D 76 pp 110,1994. 140[87] Barkai E. J. Statistical Phys. 115 pp1537, 2004. 140[88] West B. J. and Seshadri V. PhysicaA 113 pp 203, 1994. 140[89] Barkai E. Metzler R. and KlafterJ. Phys. Rev. Lett. 82 pp 3563, 1994. 140[90] Fogedby H. C. Phys. Rev. Lett. 73 pp2517, 1994. 140[91] Ross J. Vlad M. O. and SchneiderF. W. Phys. Rev. E 62 pp 1743,2000. 140[92] Doob L. Ann. Math. 43 pp 351, 1942.140, 142[93] Peseckis F. E. Physical Review A 36pp 892, 1987. 140, 142[94] Chechkin A. V. and Gonchar V.Yu. Phys. of Plasmas 9 pp 78, 2002.140, 143, 150[95] Gatto R. and Mynick H. E. Theoryand application of the generalizedBalescu-Lenard transport formalism.Nova Publishers, 2008. 140[96] Barkai E. Metzler R. and KlafterJ. Physics of Fluids B 4 pp 0899, 1992.144, 145, 146, 147, 149[97] Balescu R. Aspects of AnomalousTransport in Plasmas. IoP publishing,2005. 150162