Transport Analysis in Tokamak Plasmas

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Transport Analysis in Tokamak Plasmas

Sara Moradi

Supervisors: Dr. Boris Weyssow and Dr. Daniele Carati Statistical and Plasma Physics

Universit´e Libre de Bruxelles

A thesis submitted for the degree of PhilosophiæDoctor (PhD)

July 2010

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In this thesis we mainly focus on the study of the turbulent transport of impurity particles in the plasma due to the electrostatic drift wave microinstabilities. In a fusion reactor, the helium produced as a result of the fusion process is an internal source of impurity. Moreover, impurities are released from the material surfaces surrounding the plasma by a variety of processes:

by radiation from plasma, or as a result of sputtering, arcing and evapora- tion. Impurities in tokamak plasmas introduce a variety of problems. The most immediate effect is the radiated power loss (radiative cooling). An- other effect is that the impurity ions produce many electrons and in view of the operating limits on density and pressure, this has the effect of replacing fuel ions. For example, at a given electron density,ne, each fully ionized carbon ion (used in the wall materials in the form of graphite) replaces six fuel ions, 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 to reduce impurities to acceptably low concentrations. However, the presence of impurities, with control, can be beneficial for the plasma performance and reduction of strong plasma heat loads on the plasma facing walls. The radiative cooling effect which was mentioned above can be used at the edge of the plasma in order to distribute the plasma heat more evenly on the whole surface of the vessel walls and therefore, reduce significantly plasma heat bursts on the small regions on the divertor or limiter tiles. The experiments at TEXTOR show that the presence of the impurities at the plasma edge can also improve the performance and reduce the turbulent transport across the magnetic field lines. The observed behavior was explained trough the proposed mechanism of suppression of the most important plasma drift wave microinstability in this region, namely, the Ion Temperature Gradient

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mode (ITG mode) by the impurities. The impurity’s positive impact on the plasma performance offered a possibility to better harness the fusion power, however, it is vital for a fusion reactor to have feedback controls in order to keep impurities at the plasma edge and limit their accumulation in the plasma core where the fusion reactions are happening. In order to have control over the impurity transport we first need to understand different mechanisms responsible for its transport.

One of the least understood areas of the impurity transport and indeed any plasma particle or heat transport in general, is the turbulent transport. Extensive efforts of the fusion plasma community are focused on the subject of turbulent transport. Motivated by the fact that impurity transport is an important issue for the whole community and it is an area which needs fundamental research, we focused our attention on the development of turbulent transport models for impurities and their examination against experiments. In a collaboration effort together with colleagues (theoreti- cians as well as experimentalist) from different research institutes, we tried to find, through our models, physical mechanisms responsible for experimental observations. Although our main focus in this thesis has been on the impurity transport, we also tried a fresh challenge, and started looking at the problem of drift wave turbulent transport in a different framework all together. Experimental observation of the edge turbulence in the fusion devices show that in the Scrape of Layer (SOL: the layer between last closed magnetic surface and machine walls) plasma is characterized with non-Gaussian statistics and non-Maxwellian Probability Distribution Func- tion (PDF). It has been recognized that the nature of cross-field transport trough the SOL is dominated by turbulence with a significant ballistic or non-local component and it is not simply a diffusive process. There are stud- ies of the SOL turbulent transport using the 2-D fluid descriptions or based on probabilistic models using the Levy statistics (fractional derivatives in space). However, these models are base on the fluid assumptions which is in contradiction with the non-Maxwellian plasmas observed. Therefore, we tried to make a more fundamental study by looking at the effect of the non-Maxwellian plasma on the turbulent transport using a gyro-kinetic

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was to study the effects of a non-Gaussian statistics on the characteristic of the 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 community as conference presentations and publications.

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

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Completing a PhD is truly a marathon event, and I would not have been able to complete this journey without the aid and support of countless peo- ple over the past four years. It is a pleasure to convey my gratitude to them all in my humble acknowledgment.

I must first express my gratitude towards my advisors, Professor Boris Weyssow and Professor Daniele Carati who were abundantly helpful and offered 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 super- vision, advice, and crucial contribution from the very early stage of this research which made him a backbone of this thesis. Above all and the most needed, he provided me unflinching encouragement and support in various ways. His truly scientist intuition has made him as a constant oasis of ideas and passions in science, which exceptionally inspire and enrich my growth as a student, a researcher and a scientist want to be.

I gratefully acknowledge Dr. Clarisse Bourdelle, without whose knowledge and assistance this study would not have been successful. She always kindly grants me her time even for answering some of my unintelligent questions.

Her leadership, support, attention to detail and hard work have set an example I hope to match some day.

Collective and individual acknowledgments are also owed to my fellow PhD students at ULB, for creating such a great friendship at the office, for our

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philosophical debates, exchanges of knowledge, skills, and venting of frus- tration during my PhD, which helped enrich the experience.

Appreciation also goes out to our secretaries, Mrs. Fabienne De Neyn and Mrs. Marie-France Rogge for all of their administrative and technical assistance which helped me along the way.

My parents deserve special mention for their inseparable support. My Fa- ther in the first place is the person who put the fundament of my learning character, 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 gently love. Ehsan, thanks for being supportive and caring brother.

Words fail me to express my appreciation to my husband Bart whose ded- ication, love and persistent confidence in me, has taken the load off my shoulder.

I would like to thank everybody who was important to the successful re- alization of this thesis, as well as expressing my apology that I could not mention them personally one by one.

I would also like to convey thanks to the FNRS-FRIA of Belgium for providing the financial means and the science faculty of the Universit Libre de Bruxelles for providing the laboratory facilities.

And finally, I would like to stress that I consider this thesis to be the start of a challenging research program, rather than my final say on this topic.

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List of Figures

1.1 Thermal reactivity values for the primary fusion reactions. This figure

is taken from Ref. (1) . . . 2

1.2 Plasma confinement: charged particle are forced to follow spiral paths about the field lines and are prevented from striking the walls of a containing vessel. This figure is taken from Princeton plasma physics laboratory website: http://www.pppl.gov/fusion basics. . . 4

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

1.4 Schematic representation of heating methods for plasma. The figure is taken from JET website: http://www.jet.efda.org. . . 7

1.5 Fusion triple product. The figure is taken from Ref. (2). . . 9

1.6 Schematic representation of tokamak temperature profiles for a number of modes of operation. The figure is taken from Ref. (2). . . 10

1.7 Divertor regionDα intensity in a typical plasma showing the characteristics of different types of ELMs. The figure is taken from Ref. (3). . . . 13

1.8 The reduction of the heat flux to the wall by the creation of the radiative mantel. Figure is taken from TEXTOR website: http://www.fz-juelich.de. 14 1.9 Debye shielding. . . 16

1.10 Schematic representation of the diffusion coefficient in a tokamak as a function of the collisionallity. Here, the following notation has been used A=⁻¹ where is the inverse aspect ration. . . 18

3.1 Larmor orbits in a magnetic field. . . 32

3.2 Particle drifts in crossed electric and magnetic fields. . . 34

3.3 Drift of a gyrating particle in a gravitational field. . . 37

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3.4 Drift of a gyrating particle in a nonuniform magnetic field. . . 38

3.5 A curved magnetic field. . . 39

3.6 Origin of the diamagnetic drift. . . 51

3.7 The drift wave phase shift, δ > 0 measured in the Q-machine. This figure is taken from Ref. (4). . . 55

4.1 ITG instability mechanism. . . 68

4.2 A summary of the ITG/TE modes. . . 69

4.3 Characteristic wave number and growth rates of drift instabilities. . . . 70

4.4 Stability Diagram-Weiland model. . . 70

4.5 Simple Flowchart of the AFC-FL code. . . 71

4.6 Thekyρs-spectra of ITG/TE instability growth rate in deuterium plasma containing 2% ofN e¹⁰⁺ions with a flat density profile,_n= 0, calculated directly in a one impurity species approximation. . . 73

4.7 The k_yρ_s-spectra of main ions ITG instability growth rate. . . 74

4.8 The k_yρ_s-spectra of TE instability growth rate. . . 74

4.9 The k_yρ_s-spectra of impurity ITG instability growth rate. . . 75

4.10 Thek_yρ_s-spectra of TE instability growth with and without taking into account collisional detrapping effects on trapped electrons. . . 76

4.11 The kyρs-spectra of ITG/TE instability growth rate with and without taking the effects of the magnetic shear into account. . . 77

4.12 s-dependences of the ITG (left) and TE (right) growth ratesγmax. Solid line: no shear effects taken into account, Dashed line: λ= λ_t = 1 and k⊥=k_yp 1 + (π²/3−5/2)s², 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

4.13 θ-dependences of the instability characteristics γmax,ωmax. . . 78

4.14 θ-dependences of the instability characteristics γmax,ωmax for different density scaling lengths. . . 79

4.15 Thekyρs-spectra of ITG/TE instability growth rate in deuterium plasma containing 3% of N e¹⁰⁺ ions with a flat density profile, _n_{N e} = 0, calculated directly in a one impurity species approximation (symbols) and iteratively (solid lines). . . 81

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LIST OF FIGURES

4.16 The growth rates (left column), real frequency (middle column) and dimensionless wave vector (right column) of the most unstable modes as functions of the temperature gradient parameter_T_e,icalculated with different concentrationsξN eand density gradient parametersnN eofN e¹⁰⁺

impurity ions. . . 82 4.17 The growth rates (left column), real frequency (middle column) and di-

mensionless wave vector (right column) as functions of magnetic shear calculated for different concentrations ξN e and density gradient param- eters_n_{N e} of neon impurity. . . 83 4.18 The growth rates (left column), real frequency (middle column) and

dimensionless wave vector (right column) versus temperature gradient scale in deuterium plasma with C⁺⁶, N⁺⁷, O⁺⁸, N e⁺¹⁰, Ar⁺¹⁸ impurity ions of different total concentrations: no impurity (red curves), P

ξ_j = 1% (black solid curves), 3% (black dashed curves) and 5% (black dashed- dotted curves); the impurity ion density gradient parameters are calculated self-consistently from zero particle fluxes. . . 85 4.19 The growth rates (left column), real frequency (middle column) and di-

mensionless wave vector (right column) versus magnetic shear computed for_T_e,i = 10. . . 85 5.1 Diffusivity and pinch velocity of deuterons and electrons, D_i,e (left col-

umn) and V_i,e (right column), respectively, versus _T_e,i calculated for s= 1.8. . . 89 5.2 Diffusivity and pinch velocity of deuterons and electrons versus magnetic

shear calculated for_T_e,i = 10. . . 89 5.3 Heat transport coefficientsκ_i,e (left column) and Π_i,e (right column) as

functions of_T_i,e calculated fors= 1.8. . . 91 5.4 Heat transport coefficients versus magnetic shear calculated for_T_e,i = 10. 91 5.5 The main mechanisms for impurity pinch. . . 92 5.6 The impurity peaking factor effect on the impurity density profile. . . . 93 5.7 The experimental measurements of the impurity peaking factor and the

neoclassical predictions (this figure is taken from the presentation by C.

Giroud at 21st IAEA fusion energy conference Chengdu, China 2006). . 94

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5.8 The peaking factor for impurity species C⁺⁶ (red curves), N e⁺¹⁰ (blue curves) and Ar⁺¹⁸ (pink curves) versus Ti computed with Te = 0, s = 1.8 and different total impurity concentrations: P

ξ_j = 1% (solid curves), 3% (dashed curves) and 5% (dashed-dotted curves). . . 95 5.9 The peaking factor for impurity species C⁺⁶ (red curves), N e⁺¹⁰ (blue

curves) and Ar⁺¹⁸ (pink curves) versus magnetic shear computed with _T_e = 0,_T_i = 5 and different total impurity concentrations: P

ξ_j = 1%

(solid curves), 3% (dashed curves) and 5% (dashed-dotted curves). . . . 95 5.10 Thek_yρ_s-spectrum of the instability growth rate as a function of θcal-

culated for the plasma parameters in the JET core,r/a= 0.42. . . 101 5.11 Impurity peaking factor as a function of the impurity charge Z and

parameterθ computed without impurity ion collisions (r/a= 0.42). . . . 101 5.12 Impurity peaking factor as a function of the impurity charge Z and

parameterθ computed with impurity ion collisions (r/a= 0.42). . . 102 5.13 The contribution from thermal forces,P_V^{T F}

coll, to the pinch-velocity factor P_V_z as a function of the impurity chargeZ and parameterθ (r/a= 0.42). 103 5.14 The values Im

α⁰_iΘN_z/Y_z

and Im

α⁰_eΨN_z/Y_z

as functions of the impurity charge Z and parameterθ(r/a= 0.42). . . 103 5.15 Thek_yρ_s-spectrum of the instability growth rate as a function of θcal-

culated for the plasma parameters close to the plasma edge, r/a= 0.8. . 104 5.16 Impurity peaking factor as a function of the impurity charge Z and

parameterθ computed without impurity ion collisions (r/a= 0.8). . . . 105 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 5.18 The k_yρ_s-spectrum of the instability growth rate as a function of the

magnetic shear s calculated for the plasma parameters in the plasma core,r/a= 0.42, in the regime of TE-modes,θ= 1. . . 106 5.19 Impurity peaking factor as a function of the impurity chargeZ and mag-

netic shearscomputed with impurity ion collisions included (r/a= 0.8, θ= 1).107 5.20 Charge dependence of the impurity peaking factor observed (right figure)

and computed by AFC-FL code (left figure). . . 107 6.1 A local moving reference frame attached to the particle. . . 111

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LIST OF FIGURES

6.2 Comparison between AFC-FL and QuaLiKiz models. . . 115 7.1 P_rad^tot total radiated power as a function of time (taken from JET measure-

ments base on 2D tomographic reconstruction) for the three discharges:

69089 (solid line), 69091 (dashed line) and 69093 (dashed dotted line). . 120 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 assumption that plasma radiation is constant on the flux surface) for the two discharges at t ≈ 7.6s: 69091 (dashed line) and 69093 (dashed dotted line). . . 121 7.3 Ti profiles as functions of major radius (taken from JET CXFM mea-

surements) for the three discharges att≈7.6s: 69089 (solid line), 69091 (dashed line) and 69093 (dashed dotted line). . . 122 7.4 Te profiles as functions of major radius (taken from JET LIDAR mea-

surements) for the three discharges att≈7.6s: 69089 (solid line), 69091 (dashed line) and 69093 (dashed dotted line). . . 123 7.5 Electron densityn_eprofiles as functions of major radius (taken from JET

LIDAR measurements) for the three discharges att≈7.6s: 69089 (solid line), 69091 (dashed line) and 69093 (dashed dotted line). . . 123 7.6 Neon concentration, n_{N e}/n_e profiles as functions of major radius (taken

from 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). . . 124 7.7 Carbon concentration,n_C/neprofiles as functions of major radius (taken

from 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). . . 125 7.8 Effective charge, Zef f profiles as functions of major radius (taken from

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). . . 125

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7.9 The energy confinement time defined as the ratio of the diamagnetic energy to the total input power: τE = Wdia/Ptot versus the fraction of radiated power, Prad/Ptot for different tests. The reference discharge 69089 with only D fueling is shown with a diamond. The other two discharge: 69091 (Dfueling and Ne injection) and 69093 (only Ne injection) are presented by a square and a circle, respectively. Other symbols represent the tests where only n_e (right triangle), only Z_{ef f} of 69091 (plus), only Z_{ef f} of 69093 (up triangle), only P_rad (left triangle) and finally, P_rad+n_e (star),P_rad+Z_{ef f} (down triangle) have been replaced from their reference value in 69089 by those from 69091−3 discharges. . 126 7.10 ITG growth rate as a function of the normalized toroidal flux coordi-

nate, ρ, calculated by RITM code for the three discharges: (solid line) shot 69089, (dashed line) shot 69089 withZ_{ef f} from 69091, and (dashed dotted line) shot 69089 with Z_{ef f} from 69093, shown in figure 7.9 by a diamond, a plus and an up triangle symbols, respectively. . . 129 7.11 Electron (top) and ion (bottom) temperature profiles as functions of the

normalized toroidal flux coordinate, ρ, (solid line) for 69089, (dashed line) for 69089 withZ_{ef f} increased to that from 69091, and (dashed dotted line) for 69089 withZ_{ef f} increased to that from 69093, corresponding to the diamond, plus and up triangle symbols in figure 7.9, respectively. 130 7.12 Electron (top) and ion (bottom) temperature profiles as functions of the

normalized toroidal flux coordinate,ρ, (solid line) for 69089, and (dashed line) for 69089 withP_radincreased to that from 69091−3, corresponding to the diamond and left triangle symbols in figure 7.9. . . 131 7.13 Electron (top) and ion (bottom) temperature profiles as functions of

the normalized toroidal flux coordinate, ρ, (solid line) for 69089, and (dashed line) for 69089 with P_rad +Z_{ef f} increased to that from 69093, corresponding to the diamond and down triangle symbols in figure 7.9. . 132 7.14 Radial profiles of the ion effective charge Zef f, ion temperature Ti and

ion heat diffusivityχi computed with the RITM code for JET discharges with low (dashed curves) and high (solid curves) neon content; the fitted experimental data of the Ti profile taken from JET experimental data base are shown by squares and crosses, respectively. . . 135

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LIST OF FIGURES

8.1 u²Λ(u) (Solid line),uΛ(u) (Dashed line) and Λ(u) (Dotted line) as functions of u. . . 148 8.2 2wJ₀²(b_iw)Λ(w) as functions ofw. . . 149 8.3 γ from two solutions of the dispersion equation as functions of for

b_i = 0.1. The solution with γ < 0 (blue) gives the growth rate of the unstable mode. . . 151

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1

Introduction

1.1 Towards a New Source of Energy

Energy supplies for future are becoming more and more vulnerable while the global consumption is escalating dramatically, expected to increase by 70% in 2030 and continuing to rise (5). All stars in the universe are powered by nuclear fusion which similar to fission can produce huge amounts of carbon-neutral energy. However, there is one vital difference between them: the nuclear fusion does not produce dangerous and long lasting 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 a few nuclear reactions of the many reactions which occur in the stars are of practical value for potential energy production on earth. All of these reactions involve isotopes of 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 plen- tiful and widespread on Earth. The hydrogen and deuterium can be extracted from water and the tritium required would be produced from lithium, which is available from land deposits or from sea water which contains thousands of years’ supply. The world- wide availability of these materials would thus eliminate international tensions caused by imbalance in fuel supply. Nuclear fusion could also help meet international climate change targets and the current zero-carbon technologies are unlikely to meet our energy demands this century. However, nuclear fusion could render carbon dioxide-producing fossil fuels obsolete.

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1.2 Fusion Reaction

The possible fusion reactions between the isotopes of hydrogen are listed below:

2D+³T →n(14.03M eV) +⁴He(3.56M eV)

2D+²D→n(2.45M eV) +³He(0.82M eV) (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 deuterium nucleus with one of tritium (see figure 1.1). The products are energetic helium-4 (He⁴), the common isotope of helium also called an alpha particle, and a more highly energetic free neutron (n). The helium nucleus carries one-fifth of the total energy released and the neutron carries the remaining four fifths.

Figure 1.1: Thermal reactivity values for the primary fusion reactions. This figure is taken from Ref. (1)

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1.3 The Necessary Conditions

Fusion reactant are positively charged and must overcome their electrostatic repulsion in 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 sufficiently high kinetic temperature of the reacting species in order to facilitate the penetration of the Coulomb barrier. To reach the needed high kinetic temperatures we can start by confining a population of the deuterium and tritium atoms in some closed space and by heating attain both ionization and high temperatures (about 100 million degrees Celsius). The resulting ensemble of positive and negative charges then forms a plasma which is expected to reach thermodynamic equilibrium as a result of random collisions. The resultant spectrum of particle energies is then well described by Maxwelle-Boltzmann distribution, with the high energy part of this distribution providing for most of the desired fusion reactions. The critical technical requirement is the sustainment of a sufficiently stable high temperature plasma in a practical reaction volume and for a sufficiently long period of time to render the entire process energeti- cally viable. Confinement of the plasma fuel by some means is thus crucial to maintain these conditions within the required reaction volume.

One of the most effective means for the plasma confinement involves the use of the magnetic fields. In the absence of a magnetic field the charged particles in a plasma move in straight lines and random directions. Since nothing restricts their motion the charged particles can strike the walls of a containing vessel, thereby cooling the plasma and inhibiting fusion reactions. In a magnetic field however, the particles are forced to follow spiral paths about the field lines (see figure 1.2). Consequently, the charged particles in the high-temperature plasma are confined by the magnetic field within the required reaction volume.

1.4 Tokamak

Applying a magnetic field can significantly increase the confinement of this high temperature plasma. The cyclotron motion (gyration of the charged particles around magnetic field lines) reduces the transport of the charged ions and electrons in the direction perpendicular to the magnetic field. Therefore, the magnetic field must be configured in

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Figure 1.2: Plasma confinement: charged particle are forced to follow spiral paths about the field lines and are prevented from striking the walls of a containing vessel. 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 Kam- era Magnitnaya Katushka” (Toroidal chamber with Magnetic Coil), concept based on the ideas of Sakharov (7). In this device the vacuum vessel has a torus shape (see figure 1.3). The largest component of the magnetic field, along the torus, is produced by the toroidal magnetic field coils and is called the toroidal magnetic field, BT. The plasma is heated by the current induced by a transformer, where the plasma itself serves as a secondary winding (Ohmic heating). This plasma current induces the so- called poloidal magnetic field, which encircles the center of the tube. The toroidal and poloidal magnetic fields together produce a helical field, whose field lines lie on the toroidal surfaces. Apart from the toroidal field generated by the external field coils and the field generated by the flow of the plasma, the Tokamak requires a third vertical field (poloidal field), fixing the position of the plasma column in the vessel.

One important plasma confinement indicator is the ratio of kinetic particle pressure

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1.5 Plasma Heating

P_kin=n_ik_BT_i+n_ek_BT_e (1.2) to the magnetic pressure

Pmag = B²

2µ₀ (1.3)

whereni,e,Ti,eare the density and temperatures of plasma ion and electron respectively and kB is the Boltzmann constant, B =|B| is the magnetic field strength, and µ0 is the permeability of free space. This ratio is defined as the beta parameter,β, and is a measure of how effectively the magnetic field guarded the thermal motion of the plasma particles. A high beta would be most desirable, but it is also known that there exists a system-specificβ_max at which plasma fluctuations start to destroy the confinement.

That is why for confinement purposes, we require βmax

B²

2µ₀ ≥nikBTi+nekBTe (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 confinement in Tokamaks.

1.5 Plasma Heating

Ohmic heating results from the plasma toroidal inductive current and therefore depends on the the resistance of the plasma. As the temperature of plasma rises however, the resistance decreases and the ohmic heating becomes less effective. The maximum plasma temperature attainable by ohmic heating in a tokamak appears to be 20-30 million degrees Celsius. Therefore, to reach still higher temperatures additional heating methods must be used (see figure 1.3).

Neutral beam injection is one of these methods which involves the introduction of high-energy (neutral) atoms into the ohmically heated, magnetically confined plasma.

The atoms are immediately ionized and are trapped by the magnetic field. The high- energy ions then transfer part of their energy to the plasma particles in repeated collisions, thus increasing the plasma temperature.

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Figure 1.3: Schematic representation of the tokamak set up. The figure is taken from JET website: http://www.jet.efda.org.

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1.6 Power Balance and Lawson Criteria

In radio-frequency heating, high-frequency waves are generated by oscillators outside the torus. If the waves have a particular frequency (or wavelength), their energy can be transferred to the charged particles in the plasma in the same way that mi- crowaves transfer heat to food in a microwave oven. These heated charged particles in turn collide with other plasma particles, thus increasing the temperature of the bulk plasma.

Figure 1.4: Schematic representation of heating methods for plasma. The figure is taken from JET website: http://www.jet.efda.org.

1.6 Power Balance and Lawson Criteria

Fusion confinement requirements are qualified in terms of energy and particle confinement times. The energy confinement time, τ_E, is defined in terms of a steady state plasma as

τE ≡ U Win

(1.5) whereU is the measured plasma energy content andWinis the rate of the energy input necessary to sustain the steady state.

The particle confinement time, τ_p, is defined analogously in terms of particle content and replacement rate.

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In a steady state self sustained fusion plasma the energy input would come from fusion reactions. The power input is denoted by fusion reactions,Pin which measures the rate at which fusion energy is deposited in the plasma; it is smaller than the rate of fusion 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 the walls of the containing vessel to contribute to Pin. This is due to the fact that the neutrons have no charge and are highly energetic therefore, they rarely react with the plasma particles, hence, their energies are not deposited into the plasma. We express the radiation losses by Prad, where the subscript refers to bremsstrahlung radiation.

Cyclotron radiation losses are conventionally omitted because their longer wavelength allows at least partial reflection at the plasma boundary. Thus Win =Pin−Prad and fusion plasma power balance is expressed as

P_in−P_rad = U

τ_E (1.6)

It was noted by Lawson that, while the energy content is proportional to plasma density, n, both quantities on the left-hand side of (1.6) vary with the square of the density. That P_in is proportional to n² follows immediately from the binary nature of fusion reactions; in a D-T plasma with equal amounts of D and T,

Pin= 1

4n²_ihσv_iiE_DT (1.7)

whereE_DT = 17.6M eV is the mean reaction energy of two primary D-T reactions and hσv_ii is the reaction cross-section averaged over the relative velocity of colliding ions.

The bremsstrahlung power loss scales as

P_rad ∝n²_eT_e^1/2 (1.8)

It follows that the density enters (1.6) only through the combinationnτE, the Lawson parameter. For a pulsed reactor with an efficiency of 33% for conversion of heat to electricity, Lawson gave the criteria both for the minimum temperature and for the minimum (nτE)cvalue at the optimum temperature:

T ≥3keV and (nτ_E)_c≥10²⁰m⁻³s at T ≈30keV f or D−T (1.9)

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1.7 Plasma Operational Limits

It is evident that smaller values of nτ_E correspond to energy losses too severe for self-sustained fusion; larger values could presumably be cured by artificially enhanced losses. Thus, the Lawson criterion for self-sustained, or “ignited” fusion, states that the 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 to various plasma parameters (such as impurity content) that are not easy to predict, so that a simple general specification is difficult.

1.7 Plasma Operational Limits

Figure 1.5: Fusion triple product. The figure is taken from Ref. (2).

In recent years there has been considerable development of databases and accumulation of knowledge on the behavior of tokamak plasmas around the world. A degree of uncertainty still exists in predicting the confinement properties and plasma performance in such a device. The progress of these researches can be illustrated by the fusion productn_DTτ_ET_i, which represents a figure of merit for plasma performance (see figure

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1.5) during four decades of research on magnetic confinement.

Experimental findings are categorized according to their empirical signatures into confinement ”modes”. Sudden transitions between such regimes are often observed. In the following a general overview of the different operational regimes is given (see figure 1.6).

Figure 1.6: Schematic representation of tokamak temperature profiles for a number of modes of operation. The figure is taken from Ref. (2).

1.7.1 Ohmic plasma

An ohmic plasma is one that is resistively heated with a power given byI_pV_res, whereI_p is the plasma current andVresis the resistive portion of the loop voltage. The electrons are heated directly, while the ions are heated by the equipartition energy flow from the electrons.

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1.7.2 L-mode

It was recognized early that the ohmic regime was inefficient for achieving the necessary temperatures. As temperature increased, the resistive heating decreased, and, therefore, auxiliary heating would help to increase the plasma temperature to the level required for significant fusion power production. Auxiliary heating was performed by a variety of techniques including neutral beams and Radio Frequency heating (RF).

While the temperature and stored energy increased with this auxiliary heating, the in- cremental increase in stored energy was less than that expected from the ohmic scaling, resulting in a degradation of global confinement. This mode of operation with degraded energy confinement time, is called L-, or low confinement, mode. Characteristic fea- tures of L-mode plasma are the low temperatures and temperature gradients near the plasma periphery.

1.7.3 H-mode

The high confinement mode (H-mode), associated with a spontaneous formation of an edge transport barrier, was first discovered in ASDEX (8) and has now been seen on a wide variety of magnetic confinement devices under a wide range of conditions. The H- mode exhibits global energy confinement values about a factor of two better than L-mode. Part of this is due to formation of the edge transport barrier. Another part of this improvement is due to a reduction in local transport throughout the plasma after the L-H transition.

The H-mode is reached above a certain threshold of power, which depends on plasma conditions and machine size, see reference (3).

1.7.4 Global Instabilities

A 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 effect on the profiles of temperature, density and impurities in the central core region, and the ELMs, which periodically affect plasma edge region. In the following sections we will briefly review these two instabilities.

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1.7.4.1 Sawtooth

When the central value of the safety factor q≈rB_T/RB_p (B_T andB_p are the toroidal and poloidal magnetic field components, r and R are the minor and major radius of the tokamak torus, respectively) falls below unity; relaxation oscillations are normally observed in the core of a tokamak plasma. They appear on a number of plasma parameters but are particularly evident in the central electron temperature T_e(0) (9). The oscillation inTe(0) exhibit a time trace with a distinctive sawtooth shape consisting of a slow rise during which the plasma insideq = 1 heats up, followed by a rapid collapse when 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 the effect of degrading the global energy confinement time.

1.7.4.2 Edge Localized Modes

The transition from L-mode to H-mode in magnetic confinement systems is normally ac- companied by 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. The underlying cause for ELMs is the onset of MHD instability in the plasma edge when the edge pressure gradient exceeds a critical threshold. The subsequent loss of edge confinement leads to a temporary reduction of the pressure gradient, and the eventual recovery of the pressure gradient leads to recurrence of the ELM. This cycle continues indefinitely in a sustained H-mode discharge. The energy loss due to ELMs causes a reduction 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 the plasma edge P_edge =Pin−P_rad (i.e. heating power minus radiation losses inside the edge region) is a key factor in determining the ELM type. A number of types of ELM with different amplitude, frequency and power dependences can be distinguished (see figure 1.7):

• TypeIELMs (giant ELMs) with high amplitude and low frequency, which develop when the edge power flow significantly exceeds the H-mode threshold.

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Figure 1.7: Divertor regionD_α intensity in a typical plasma showing the characteristics of different types of ELMs. The figure is taken from Ref. (3).

• TypeII ELMs (grassy ELMs) are associated with strongly shaped tokamaks at high edge pressure.

• Type III ELMs (small ELMs)with small amplitude and high frequency appear when the power flow to the plasma edge is only marginally above the H-mode power threshold.

Other improved regimes have also been observed besides the H-modes. Regimes with core or Internal Barriers (ITBs), see figure 1.6, have been discovered that lead to significant enhancements in confinement and plasma performance. Transport barriers associated 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 are regimes 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 the following section.

1.7.5 RI-mode

The duration of a burning fusion plasma will depend largely on the properties of the edge plasma being in contact with the wall elements. The first wall has to withstand and exhaust the α-particle heating power, and the helium ash must be removed from the plasma. Wall erosion will affect the lifetime of wall elements and impurities are released into the plasma, which then can cause fuel dilution and power loss owing to

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radiation from the plasma center.

The integrity of plasma facing components will crucially depend on the avoidance of overheating of plasma facing components and on the balance between erosion and deposition of wall materials. The lifetime of plasma facing components, an important factor for its economy, will ultimately govern the availability of a fusion reactor. The problem of overheating of small areas like the divertor strike zone or the limiter edge can be solved by distributing the power over large areas. In the high density divertor charge-exchange processes could provide this to a certain extent. Furthermore, a distribution of the heat on the whole vessel wall can be achieved by radiation from injected impurities. Feed-back control of the seeded impurities is an important requirement. Up to 90% of the heating power can be radiated from a rather thin belt at the periphery of the confined plasma. In the limiter tokamak TEXTOR it was found that the energy confinement with seeded impurities can be substantially improved to values comparable with ELM-free H-mode discharges in divertor tokamaks. This regime has been named Radiative Improved Mode (RI-mode), see figure 1.8

Figure 1.8: The reduction of the heat flux to the wall by the creation of the radiative mantel. Figure is taken from TEXTOR website: http://www.fz-juelich.de.

By puffing of argon impurity a noticeable confinement improvement has also been

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1.8 Mechanisms of Transport in Tokamaks

seen in the H-mode in the divertor tokamak JT-60U. However, earlier analogous experiments on JET (11) did not lead to the desirable result and suffer, when argon concentration exceeded some critical level, from unwanted and uncontrollable impurity accumulation in the plasma core with strongly peaked core radiation and flat temperature profile.

Therefore, these impurities must be carefully controlled to maintain the radiation at the required levels without excessive plasma core accumulation. This brings us to the problem of transport in magnetically confined plasmas. The success of the fusion program therefore strongly depends on understanding and controlling the transport processes of matter and energy in Tokamak plasmas.

In the next section we will present the mechanisms of transport in Tokamaks and their predictions accuracy in explaining the experimental observations.

1.8 Mechanisms of Transport in Tokamaks

Understanding the mechanisms of transport of matter, electric charge, energy and mo- mentum is 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 transport phenomena taking place under specific circumstances.

1.8.1 Classical Transport

Since the early days of tokamak research it has been observed that radial transport cannot be explained by the so called classical transport theory of plasmas. The classical theory explains the diffusion across the magnetic field with cylindrical symme- try as resulting from friction between the electrons and the ions (12). The process can also be interpreted as a random walk of particles, where the typical step length is the electron gyro-radius, if the collision time is taken to be the time a particle takes, on average, to diffuse in velocity space through an angle of 90^◦. The classical predictions, when compared to the experiments, grossly underestimate the transport coefficients in a fusion plasma: diffusion coefficient in tokamaks are typically

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of the order of ∼ m²s⁻¹ and the classical diffusion coefficient can be estimated by Dc ≈ νeiρ²_e(∼ 10³ ×(10⁻³)² = 10⁻³[m²s⁻¹] for the electrons) where νei is the 90^◦ collision frequency between electron and ion and ρ is the Larmor radius, ρ = Vth/Ωc, which is proportional to the ratio of a thermal speed, V_th =p

2T_e/m_i, to a cyclotron frequency Ωc = eB/mc, where B = |B| is the magnetic field strength and m is the particle mass.

To explain such disagreements, one has to understand which assumptions of the classical theory are invalid. An invalid assumption is that the collisions in the fully ionized plasma considered as binary interactions while these collisions are of the Coulomb nature and because of the long range of the Coulomb forces they can not be considered as binary interactions. Because of the long range of the Coulomb forces, the collective nature of the plasma dominates its behavior. An important concept characterizing collective phenomena in plasma is the Debye shielding.

1.8.2 Debye Shielding

Figure 1.9: Debye shielding.

A fundamental characteristic of the behavior of a plasma is its ability to shield out electric potentials that are applied to it. If we put an electric field inside a plasma by inserting two charged balls connected to a battery, see figure 1.9, almost immediately

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clouds of negative and positive charge will surround the positive and negative balls respectively, with their density decreasing with distance from the charged balls. This is due to the mobility of the electrons and ions in plasma. For cold plasma with no thermal motions, as many charges will be observed in the surrounding clouds as are required to neutralize the inserted charges. However, plasma temperature is finite and the 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 the shielding is not complete. The edge of the cloud then occurs at the radius where the potential energy is approximatively equal to the thermal energy k_BT of the particles.

This characteristic shielding range, is the Debye length which in non-magnetized plasma is defined (for further details see Ref. (1; 13))

λ_D = (₀k_BT_e

nee² )^1/2 (1.10)

where₀ is the primitivity of the free space,eis the electron charge andn_e is taken as the 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 subscript denote the wave-numberkand frequencyω of the electric field fluctuation E_k,ω. For a plasma of density nand temperature T the single particle Coulomb electric field falls off exponentially beyond the Debye lengthλ_D. Therefore, forkλ_D 1 the electric field fluctuations are collective self consistent interactions while forkλ_D >1 the interactions are binary collisional. For a plasma with a large number of particles inside the Debye sphereN_D = (4π/3)nλ³_D 1, the collective electric fields dominate the plasma dynam- ics through collective modes. In magnetized plasmas the modes with low frequency, ωΩ_c, dominate the transport.

The effects related to the collective interactions which exist in a plasma as a result of the long range Coulomb forces, fall outside the scope of the classical theory and are the object of anomalous transport theory which will be discussed later in this chapter.

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Figure 1.10: Schematic representation of the diffusion coefficient in a tokamak as a function of the collisionallity. Here, the following notation has been usedA=⁻¹where is the inverse aspect ration.

1.8.3 Neo-classical Transport

An effect which has to be taken into account in fusion plasmas is the effect of the magnetic field on the free motion of the particles. The magnetic fields produced in fusion devices are spatially inhomogeneous and have a globally toroidal topology. The particle orbits 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-field regions because of the inhomogeneousity (magnetic mirror effect, see chapter 3). When the 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). The neoclassical transport levels exceed the classical ones by geometrical factors: q²^−3/2 in the low collision frequency ”banana” regimes (ν∗<1.0) and q² in the collisional limit, as a result of toroidal geometry. Here, =r/R is the inverse aspect ratio, with r and R being the minor and major radii of the magnetic surface. The collisionallity parameter is ν^∗ ≡νef f/ωb, where νef f ≡νei/ is the effective collision frequency for particle detrapping, andω_b ≈^1/2V_th/(Rq) is the trapped particle average bounce frequency.

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Although neoclassical transport theory is a significant step forward towards describing transport in magnetically confined plasmas, in general the neoclassical predictions are still far from explaining the full picture (Dneo for electrons is of the order 10⁻³[m²s⁻¹] and for ions of the order 10⁻¹[m²s⁻¹]).

1.8.4 Anomalous Transport

The transport theory based on particle collisions can incorporate the geometry of the Tokamak magnetic system, but neoclassical theory still assumes that the plasma is in equilibrium and axisymmetric. Real Tokamak plasmas always show the presence of a broad spectrum of fluctuations, e.g. in plasma density, temperature and electromag- netic fields (3); thus real Tokamak plasma are turbulent. The turbulent fluctuations give rise to transport across the equilibrium magnetic surfaces and it is necessary to incorporate their effect in a comprehensive transport theory. From the theoretical point of view, most of the instabilities that we think are responsible for the observed plasma turbulence have a very small component of wave number vector parallel to the magnetic field, compared to the perpendicular component. That is, most of the turbulent eddies are quasi-perpendicular to the toroidal magnetic field. Therefore, we can expect that turbulence dominates perpendicular transport, in this case we are in the presence of anomalous transport. The influence of the plasma turbulence on the parallel transport is 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 transport theoretically, in particular with the help of complex computer codes which calculate the growth rates of the underlying instabilities and allow to deduce the resulting transport coefficients. These instabilities are driven essentially by the gradient of density and temperature.

The fluctuations of the magnetic and electric fields (δE,δB) can provide the sources to the anomalous transport across the magnetic field lines. If the fluctuations are purely electrostatic in nature (δB = 0) they would lead to enhanced cross field transport by E×B drift:

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δV_E= δE×B

B² (1.11)

The anomalous diffusion coefficient can be approximatively expressed as

D_ano= (δV_Eτ_c)²

τ_c = (δV_E)²τ_c (1.12)

whereτc is the fluctuation’s time scale. By replacing

δVE= k_θδφ

B = T k_θ eB

eδφ

T (1.13)

wherek_θ is the poloidal wave number of the mode andδφis the electrostatic potential fluctuation, into equation (1.12) for anomalous diffusion we get

D_ano≈(T k_θ eB )²(eδφ

T )²τ_c (1.14)

If τ_c ∼10⁻⁶[s], 1/k_θ ≈ρ_i ∼ 10⁻³, we find that∼1% electrostatic fluctuation can generate an anomalous diffusion of the order m²s⁻¹.

If the fluctuations are purely magnetic in nature (δE = 0) the parallel transport along the stochastic field lines would be responsible for the anomalous transport in the radial direction:

δVB= δBV_k

B (1.15)

with the diffusion coefficient as

Dano= (δVBτc)²

τ_c ≈(δBr

B )²V_thλc (1.16)

whereλ_c=V_thτ_c is the fluctuation’s length scale.

IfVth,e ∼10⁷[m²s⁻¹], we find that about∼0.01% magnetic fluctuation is sufficient to generate an anomalous diffusion of the order m²s⁻¹.

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1.9 Global Energy Confinement Scaling

Because of the complexity of the processes determining heat and particle transport in fusion plasmas, it is not yet possible to provide a first principle derivation of the dependence of energy confinement properties on plasma parameters. The description of the global energy confinement time by empirical scaling that are based on relevant databases 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, such as ITER as well as an approximate constraint on the form of theoretical models. These scalings connect empirical confinement times with machine and plasma parameters like major 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 geometrical parameters and profile functions, the ion mass and charge numbersmi and Zi. This approach is of course already well established in other areas of science and engineering - the performance of airplanes and ships can be reliably predicted using similar scalings without a detailed understanding of turbulent hydrodynamics flow.

The ELMy H-mode standard database provides the basis for a robust confinement predictions for ITER. The power law scaling expression for thermal energy confinement time can be expressed as (3):

τ_th,E^{IT ERH} = 0.0562I^0.93B^0.15P^−0.69n_e^0.41m^0.19R^1.97^0.58κ^0.78_a (1.17) (ins,M A,T,M W, 10¹⁹m⁻³,AM U,m) wherem= average ion mass,P = loss power, κa= elongation and=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 geometrically similar devices. There are dimensional constraints that follow the similarity rules and one needs to identify the relevant dimensionless parameters. However, the number of dimensionless parameters for confined plasma is large, up to 19 have been identified (17). They include plasma physics parameters, such asβthe ratio of the plasma kinetic pressure to the magnetic pressure, the collisionallityν^∗ ≡ν_{ef f}/ω_b (see sec. 1.8.3) and the normalized Larmor radius ρ^∗ ≡ ρ_i/a. There are also parameters describing the

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magnetic field geometry (safety factor q, aspect ratio A ≡ R/a, the ellipticity κ and triangularity δ of the plasma cross section) and parameters representing the plasma composition (T_e/T_i, m_e/m_i, Z_{ef f}, etc.). For a local diffusion coefficient we have to include the parameters related to plasma profiles, such as the ratio of the scale lengths, L_T/Rand L_n/R,etc. The diffusivity can be expressed in the following form:

D=c_sρ_s(ρ^∗)^αF(ν^∗, β, q, A, κ, δ, L_T/R, L_n/R, . . . , T_e/T_i, m_e/m_i, Z_{ef f}) (1.18) where cs = p

Te/mi is the sound speed, ρs = √

2Temi/eB is the ion Larmor radius evaluated at the electron temperature andF is a function of the dimensionless parameters to be determined. The main change in plasma parameters in going from present tokamaks to future reactors is inρ^∗. Therefore, determining the transport scaling with respect to ρ^∗ is critical. When α = 1 the scaling law is called gyro-Bohm. This is the expected scaling from most local turbulence theories for which the turbulent scale length 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. How- ever, these scalings do not agree always with the experimental observations. For a full discussion see Refs.(1; 3).

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2

Aims of the project

2.1 Aims

In this thesis, the main goal is to study the anomalous transport of impurities due to drift wave microinstabilities. To study the characteristics of the unstable drift modes responsible for the transport of impurities under fusion plasma conditions and to ana- lyze feedback mechanisms of the impurities on the unstable modes (chapters 3 and 4).

Encouraged by the experimental observations which suggested charge dependence of the impurity anomalous transport characteristics (transport coefficients), unexplained by 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 we go on to implement our impurity transport model for the transport modeling of the JET tokamak experiments, in which impurity seeding were applied. In these experiments plasma confinement was improved by injecting Ne into the plasma edge and no impurity accumulation in the plasma core were observed. Our aim is to find a possible explanation for the observed plasma behavior (chapter 7).

At the end (chapter 8), to try a fresh challenge, we start looking at the problem of turbulent and non-diffusive transport in a different framework. Using fractional kinetics and build on Levy type stable distributions, we study the plasma as a medium away from the Maxwellian equilibrium conditions. We look at the effect of non-Maxellian system on the characteristics of the drift wave turbulence.

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2.2 Strategy

During 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, and numerically implemented as AFC-FL (Anomalous Flux Calculation in Fluid Limit) code in the Fortran language. It solves the dispersion equation, and calculated the transport coefficients and fluxes of electrons, main and impurity ions. The effects of impurities on the unstable mode characteristics are taken into account in an iterative procedure, and an arbitrary number of impurity species can be taken into consideration. The model has been benchmarked against the quasi-linear gyro-kinetic code QuaLiKiz developed by Dr. C. Bourdelle (from CEA, Cadarache, France) (chapter 6). The AFC-FL transport model for impurities has been numerically implemented into the 1-D transport code RITM, developed by Prof. M. Z. Toka,r and used in a series of transport modeling for the JET experiments with Ne seeding.

Transport Analysis in Tokamak Plasmas