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IN

DEGREE PROJECT ELECTRICAL ENGINEERING, SECOND CYCLE, 30 CREDITS

,

STOCKHOLM SWEDEN 2016

Analysis of ICRH of H and He-3

minorities in D and D-T plasmas in

JET

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Analysis of ICRH of H and He-3 minorities in

D and D-T plasmas in JET

Kirill Amelin

Supervisor: Thomas Jonsson

Co-supervisor: Pablo Vallejos Olivares

Examiner: Jan Sche↵el

Master of Science Thesis Fusion Plasma Physics School of Electrical Engineering KTH Royal Institute of Technology

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TRITA EE 2016:102 ISSN 1653-5146

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Abstract

Nuclear fusion is potentially a source of practically endless energy. It is one of the candidates to replace fossil fuels in the future. Fusion is also a source of clean energy, which should make

a positive environmental impact. It can decrease the CO2 emission without creating any

long-lived radioactive waste. The most promising fusion reaction is between deuterium and tritium that produces a helium atom and a highly energetic neutron. However, tritium is complicated to manufacture and handle, so its supply on Earth is currently limited. For this reason, only a small number of deuterium-tritium plasma experiments have been performed. Therefore, it is important to study the physics of the deuterium-tritium plasma to be able to make predictions for the future experiments. The topic addressed in this project is radio frequency heating, which is one of the main plasma heating methods. This method will also be applied in the future ITER experiment. The goal is to study how ion cyclotron resonance heating of hydrogen and helium-3 minorities performs in deuterium-tritium plasma. An analytical model based on previous research is presented in the report. The results from this analytical model are compared with the results from the numerical simulations performed using SELFO code to evaluate the validity of the analytical model. It is shown that the approximate analytical model provides viable estimations for power partition, fast ion population and their energy content. Collisional power transfer from minority to bulk plasma ions and electrons can be estimated to a certain extent, although a deviation from the numerical simulations are found when heating the helium-3 minority. Nevertheless, helium-3 minority fundamental resonance heating is shown to result in the strongest bulk plasma heating. The conclusion is that the analytical model can be used to recreate several important results of the simulations.

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Abstrakt

K¨arnfusion ¨ar en i praktiken outt¨omlig energik¨alla och en av kandidaterna till att ers¨atta

fos-sila br¨anslen. Det ¨ar en ren energik¨alla som kan hj¨alpa oss att minska CO2 utsl¨app utan att

skapa l˚anglivat radioaktivt avfall. Den mest lovande fusionsreaktionen ¨ar mellan deuterium och

tritium som producerar en heliumk¨arna och en mycket energetisk neutron. Dock ¨ar tritium

kom-plicerat att producera och hantera. I nul¨aget finns bara mycket sm˚a m¨angder tritium p˚a

jor-den. Av denna anledning har endast ett litet antal experiment utf¨orts med deuterium och tritium

plasman. D¨arf¨or ¨ar det viktigt att studera fysik hos deuterium-tritium plasma teoretiskt f¨or att

kunna g¨ora prognoser f¨or de kommande experiment. RF uppv¨armning ¨ar en metod som anv¨ands

i plasmauppv¨armning. Denna metod kommer att ocks˚a finnas p˚a ITER. Detta arbete behandlar

uppv¨armning av deuterium-tritium plasman med jon-cyklotron resonans p˚a ett minoritetsjonslag

som t.ex. v¨ate eller helium-3 . I arbetet anv¨ands en analytisk modell f¨or jon-cyklotron uppv¨armning

baserad p˚a tidigare forskning. Resultaten fr˚an den analytiska modellen har j¨amf¨orts med resultaten

fr˚an numeriska simuleringar som utf¨orts med SELFO-koden f¨or att utv¨ardera giltigheten av

ana-lytiska modellen. Framf¨orallt kan den anv¨andas f¨or att uppskatta hur v˚agenergin f¨ordelas mellan

de olika jonslagen, och volymsintegrerade f¨ordelningsfunktionerna av snabba partiklar och deras

energiinneh˚all. Energi¨overf¨oringen fr˚an snabba minoritetsjoner till elektroner och termiska joner

via Coulomb-kollisioner kan uppskattas till viss del. F¨or fallet med helium-3 minoritet s˚a finns det

stora avvikelser mellan den analysiska och numeriska modellen f¨or energi¨overf¨oringen. B˚ada

mod-ellerna visar att den b¨asta jon-uppv¨armningen erh˚alls vid absorption p˚a helium-3 joner. Slutsatsen

av j¨amf¨orelsen ¨ar att analytiska modellen kan anv¨andas f¨or att ˚aterskapa flera viktiga resultat fr˚an

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Acknowledgements

I would like to sincerely thank everyone who I had a pleasure to meet during my MSc studies. Every single person I met had an influence on me in one way or another, perhaps without realizing it. I would especially like to thank my closest friends for cheering me up and making this time truly exciting. I thank those from my home country who I have not seen for most of this time, but who I know support me anyway. I am very grateful to my family for unconditionally supporting me at all times. I would like to thank my colleagues for helping me through with my studies. I especially

would like to mention H˚akon Sandven, who helped a lot both morally and with his contributions

in many course projects, including this very thesis. I would like to express the gratitude to the all the teachers of KTH Royal Institute of Technology for their excellent work. It is worth to point out that most of them have been very supportive and friendly, which is a wonderful thing about KTH. I wish to thank Nickolay Ivchenko for all the help and guidance during my stay at KTH. Finally, I would like to thank my supervisor Thomas Jonsson for excellent supervision, motivation and positive attitude, co-supervisor Pablo Vallejos Olivares for all the valuable advice and examiner Jan Sche↵el for a fair assessment of my work.

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Contents

Contents

1 Introduction 1

1.1 Fusion . . . 1

1.2 Social and ethical aspects . . . 1

1.3 Plasma heating . . . 2 1.4 JET results . . . 2 1.5 ITER . . . 2 1.6 Objectives . . . 3 1.7 Outline . . . 3 2 Background 4 2.1 Charged particle motion . . . 4

2.2 Tokamak . . . 5

2.3 Setup . . . 6

3 ICRH theory 8 3.1 Fundamental and second harmonic resonance . . . 8

3.2 Wave equation . . . 9

3.3 Dispersion relation . . . 10

3.4 Fast magnetosonic wave . . . 11

3.5 Polarization . . . 12

3.6 Susceptibility tensor . . . 13

3.7 Damping . . . 14

3.8 Power partition . . . 16

3.9 The evolution of the distribution function . . . 17

3.10 Energy transfer . . . 18 4 SELFO 20 4.1 FIDO . . . 20 4.2 LION . . . 21 5 Results 22 5.1 Wave propagation . . . 22

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5.4 Bulk plasma heating . . . 34

6 Discussion 36 6.1 Polarization . . . 36

6.2 Absorption rate and partition . . . 36

6.3 Distribution function and energy content . . . 37

6.4 Collisional power transfer . . . 38

6.5 Di↵erence between D-D and D-T plasma . . . 39

7 Conclusion 40

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Chapter 1

Introduction

The population of Earth has been growing rapidly during the last 50-60 years [1] and together with this grows the demand for energy. Although considerable progress has been made in developing alternative energy sources, fossil fuels still provide most of the energy in the world, thus being the only source able to meet the demand. According to the International Energy Agency, coal, oil and natural gas formed 66.5% out of total energy consumption in the world in 2013. This does not include the production of electricity, 64.4% of which was produced from fossil fuels [2]. Consequently, humanity faces a number of serious problems, like the global climate change due to

the high CO2 emission and the perspective of soon running out of fuel.

1.1

Fusion

Nuclear fusion is considered to be one of the main candidates in replacing fossil fuels [3]. Fusion is the process of merging two light atomic nuclei into one that is accompanied by a release of energy. Potentially it is an inexhaustible energy source, since the fuel for a fusion reactor is produced from hydrogen, the amount of which is practically unlimited on our planet. Another main advantage of fusion is that it requires a very small amount of fuel compared to other energy sources. It is estimated that a fusion power plant operating on Deuterium-Tritium (D-T) fuel would require about 250 kilograms of fuel per year to provide the same amount of energy as a coal-powered power plant does using 2.7 million tons of coal per year [4].

1.2

Social and ethical aspects

While a vast amount of funds is spent on the fusion research, no functioning reactor able to generate electricity on a constant basis has yet been developed. Moreover, even in the most favorable research scenario fusion power production will not be established for at least a few more decades. This obviously rises the question of whether fusion research is worth the expenses.

At the same time, there is no denial that humanity will soon have to deal with the energy crisis. Fusion has the potential to not only meet the existing energy demand, but to provide energy for many more generations to come. As long as such opportunity exists at least in theory, leaving it unexplored would account as inaction. Although the number of challenges fusion research has to

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analysis is relevant to the experiments that are to be carried out in the future and should contribute to the fusion energy research and, hopefully, the development of the fusion power plants.

1.3

Plasma heating

To achieve fusion, extremely hot plasma at temperatures of the order of hundreds of millions of Kelvins has to be created. There are di↵erent methods to heat up the plasma inside of a fusion reactor. Those are, for example, ohmic heating, Neutral Beam Injection (NBI) heating or Ion Cyclotron Resonance Heating (ICRH).

The ohmic heating mechanism is based on the fact that plasma is conductive and has a finite resistivity. Hence, it can be heated by driving current through it, just like a conductor. How-ever, plasma resistivity decreases with temperature [5]. As the temperature rises, ohmic heating becomes more and more inefficient. Therefore, it is impossible to heat up the plasma to the desired temperature by ohmic heating only.

In NBI highly energetic neutrals are injected into the plasma. These neutrals create ener-getic ions due to inelastic collisions with plasma particles. This can be a result of a momentum transfer and/or ionization. A disadvantage of this heating system is highly complicated and bulky equipment [6].

ICRH is a plasma heating process which is based on the wave-particle interactions. Radio frequency waves are sent into plasma from antennas. They propagate through plasma and get absorbed at wave-particle resonances. The electromagnetic energy is then transformed into the kinetic energy of plasma particles [7]. ICRH is one of the main mechanisms of plasma heating in existing fusion reactors, like Joint European Torus (JET) or upcoming International Thermonuclear Experimental Reactor (ITER). For this reason, ICRH is the main topic of this work.

1.4

JET results

In 1997, a series of experiments were carried out at JET with D-T plasma, as a result of which 675 MJ of fusion energy and 16 MW of fusion power were produced [8]. This result still holds as the record of produced fusion power. In these experiments ICRH was used as one of the main methods of plasma heating. In spite of the promising results, no further experiments with D-T plasma were performed, because tritium production and handling is extremely complicated and expensive.

1.5

ITER

ITER is the experimental fusion reactor which is being built in the south of France. Its goal is to be able to produce 500 MW of fusion power from a total input of 50 MW [9]. The idea is to provide plasma of similar parameters to those of a future conventional power plant. It should be able to create a D-T plasma and confine it long enough for fusion reactions of long duration to occur. Finally, it should address the challenge of producing tritium within a blanket surrounding the plasma.

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1.6

Objectives

Since there has only been a limited number of D-T plasma fusion experiments, there are many aspects of ICRH of D-T plasma that still have to be studied. Therefore, the purpose of this work is to study the heating process at the cyclotron resonance frequency of minority species in the plasma. Previous studies show that heating at the helium-3 minority resonance with 5-10% of helium-3 produces a strong ion bulk heating [8], [10].

Here the heating of hydrogen and helium-3 minorities in pure deuterium (D-D) and deuterium-tritium plasma is analyzed. It is of most interest to make predictions of how certain parameter change when the plasma composition is changed from pure deuterium into 50:50 deuterium-tritium, which will in the future help in preparations for upcoming D-T campaigns. It is also of interest to study how the minority concentration influences ICRH and what the di↵erences are between hydrogen and helium-3 minorities.

1.7

Outline

Chapter 2 of this report presents some key concepts of the topic, like a charged particle motion in a magnetic field. It also describes the setup of the experiment along with the geometry and characteristic parameters of the simulated fusion reactor. Chapter 3 explains the physical principles behind ICRH and presents the developed theoretical model. The chapter includes the theory used for calculating wave absorption and power transfer. Chapter 4 introduces the numerical code SELFO and shows the equations used in the numerical simulations. The results from the theoretical model and SELFO code are presented in Chapter 5 and later discussed in Chapter 6. Chapter 7 concludes the report and summarizes the results.

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Chapter 2

Background

2.1

Charged particle motion

According to the definition by Gurnett and Bhattacharjee ”A plasma is an ionized gas consisting of positively and negatively charged particles with approximately equal charge densities” [11]. As all charged particles react to electromagnetic forces, the concept of magnetic confinement of plasma has been developed. This concept is based on the fact that electrically charged particles react to magnetic forces. Hence, the right magnetic field configuration allows plasma to be confined inside a reactor.

The motion of a charged particle is described by Newton’s equation of motion

mdv

dt = q(E + v⇥ B), (2.1.1)

where v is the particle velocity, E is electric field, B is magnetic field, m is the particle mass and q = Ze is its charge with Z being the atomic number and e - elementary charge. In the absence of electric fields, a charged particle gyrates around magnetic field lines as illustrated in Figure 2.1 at a cyclotron or gyrofrequency

!c=±Ze|B|

m . (2.1.2)

Figure 2.1: Ion gyro-motion in a magnetic field.

If magnetic field lines are closed without intersecting with any obstacles, particles moving along them stay confined inside the magnetic field configuration. However, in the presence of an electric

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field and a non-uniform magnetic field certain drifts occur [11]. The presence of an electric field causes the guiding center of a particle to drift in the direction perpendicular to both electric and magnetic fields with a velocity

vE⇥B = E⇥ B

B2 . (2.1.3)

A non-uniform magnetic field causes the guiding center to drift with a velocity

vrB = WqB? ˆ B⇥ rB B , (2.1.4) where W? = mv 2 ?

2 is the perpendicular to the magnetic field kinetic energy of the particle, ˆB is a

unit vector in the direction of the magnetic field andrB is the magnetic field gradient.

In a magnetic field with curved field lines, the drift of the guiding center caused by the curvature

radius Rc is found by vc= 2Wk qB ˆ Rc⇥ ˆB Rc , (2.1.5) where Wk = mv 2 k

2 is the parallel to the magnetic field kinetic energy of the particle.

2.2

Tokamak

Following this idea of magnetic confinement inside a magnetic field with closed magnetic field lines, fusion reactors of toroidal shape (tokamaks) are being developed. An example of such experimental reactor is JET, shown in Figure 2.2, which is located in Culham, England. JET has a major radius

R0 = 3 m and a minor radius a = 1 m. The magnetic field at the center of the reactor is in the

range of 2 4 T.

To cancel out the drifts the additional poloidal magnetic field is generated. Consequently, particles following the magnetic field lines drift half of the time outwards and half of the time inwards a magnetic flux surface.

In this work the analysis is carried out in the simplified geometry where the cross-section of the tokamak has a circular shape as shown in Figure 2.3. It is assumed that there are no poloidal

currents J✓ generated by the toroidal magnetic field. Hence, there is no magnetic field from these

currents. The toroidal magnetic field is then the one generated by external coils only. These assumptions simplify the analysis and correspond to the assumptions of the SELFO numerical code. In this case, the toroidal magnetic field has a dependence on the major radius R as

B = B0R0

R , (2.2.1)

where B0 is the magnetic field at the distance R0 from the major axis. As a result, a resonant

surface has the same distance R from a major axis at every vertical coordinate marked in Figure 2.3

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Figure 2.2: The inside of Joint European Torus (JET). The right hand side shows a photo of plasma inside [12].

The drawback of these simplifications is that the shapes of the flux surfaces do not correspond to the ones in reality. This may lead to wrong total particle number and energy estimations. Additional mapping is needed to adjust results for the circular-shaped cross-section to those of an actual tokamak shape. Without the poloidal magnetic field, the drifts are not canceled out and plasma cannot stay confined. However, certain parameters and dependencies can still be studied. With the right mapping, many useful results can be obtained using this simplified configuration. Moreover, particle orbits are not a part of this work, which also allows for many simplifications.

2.3

Setup

In this work a tokamak with a circular cross-section is considered. The parameters of the tokamak

are close to those of JET. The major axis is R0 = 3.058 m, the minor axis a = 0.78 m and the

magnetic field at the magnetic axis (r = 0 in Figure 2.3) is B0 = 3.31 T. The frequencies are chosen

is such a way that the resonance surface position is at the distance of 0.035R0 from the magnetic

axis.

Two di↵erent types of plasma are considered: pure deuterium and 50:50 deuterium-tritium plasma. In both cases, the heating is performed at the minority cyclotron resonance frequency. Hy-drogen and helium-3 minorities are studies. The corresponding wave frequencies are then 48.75 GHz and 32.50 GHz. The wave is launched into the plasma from the low-field side (LFS), that is from the outer part of the torus. Di↵erent cases are studies with minority concentrations of 2%, 4% and

6%. A small amount of beryllium impurity⇠ 1% was introduced in the numerical simulations and

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Figure 2.3: Scheme of the simplified configuration of a tokamak [13].

In this report a theoretical analysis of ICRH is presented. The results from both the analytical model and SELFO simulations are presented and compared in order to evaluate the validity of the approximations of the analytical model. The tendencies are studied and discussed. A side note has to be made that the equations presented in this work are expressed in SI units, unless otherwise stated. At the same time, the temperature in the equations is expressed in Joules, which

corresponds to T (J) = kBT (K), where kB is the Boltzmann constant.

The results on the analysis include:

• the incoming wave power partition between di↵erent plasma species, • the evolution of the distribution functions of minorities,

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Chapter 3

ICRH theory

Ion Cyclotron Resonance Heating (ICRH) of plasma with electromagnetic waves is considered to be a promising technique to heat plasmas to sufficiently high temperatures. In a magnetized hot plasma, the energy is transferred from an antenna to the plasma using the fast magnetosonic wave, also know as compressional Alfv´en wave. In order for the energy to be transferred from the wave to the plasma, there must exist a wave-particle resonance

! = kkvk+ n!c, (3.0.1)

where ! is the wave frequency, kkvk is the Doppler shift and n = 0, 1, 2... is a harmonic number.

Here n = 0 corresponds to pure Landau damping, n = 1 to the fundamental cyclotron resonance and n = 2 to the second harmonic. According to Eq. (2.2.1) magnetic field in a tokamak decays

as R 1 (R is the distance from the major axis of the torus), so it follows from Eq.(2.1.2) that

the cyclotron frequency has the same R 1 dependence. Hence, the position of the wave-particle

resonance can be chosen by configuring the magnetic field or/and the frequency of the wave.

3.1

Fundamental and second harmonic resonance

Since atomic number and mass ratios of hydrogen, deuterium, tritium and helium-3 follow the

relations ZH mH = 2 ZD mD and Z3He m3He = 2 ZT

mT, it follows from Eq. (2.1.2) that

!c,H= 2!c,D

!c,3He= 2!c,T.

(3.1.1) Therefore, heating hydrogen or helium-3 at the fundamental resonance corresponds to heating respectively deuterium or tritium at the second harmonic. This is demonstrated in Figure 3.1. Cyclotron frequencies at second harmonics of deuterium and tritium match the ICRH frequency at the same position as the cyclotron frequencies of hydrogen and helium-3 at the fundamental resonance.

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R, [m] 2 2.5 3 3.5 4 f, [Hz] ×107 0 5 10 15 H minority, harmonic 1 H D ICRH R, [m] 2 2.5 3 3.5 4 f, [Hz] ×107 0 5 10 15 H minority, harmonic 2 H D ICRH R, (m) 2 2.5 3 3.5 4 f, (Hz) ×107 0 5 10 15 3He minority, harmonic 1 He3 T ICRH R, [m] 2 2.5 3 3.5 4 f, [Hz] ×107 0 5 10 15 3He minority, harmonic 1 He3 T ICRH

Figure 3.1: Cyclotron frequencies as a function of distance from major axis. Red dotted line marks the

position of the wave-particle resonance.

3.2

Wave equation

The propagation of an electromagnetic wave in a medium is described by dispersion relations, which are the relations between the wave vector k with the frequency !. The derivation of a dispersion relation starts from the Maxwell’s equations.

Combining Ampere’s law and Faraday’s law

r ⇥ B = µ0J + 1 c2 @E @t (3.2.1) r ⇥ E = @B @t (3.2.2)

the wave equation in a plasma is found

r ⇥ r ⇥ E + 1 c2 @2E @t2 = µ0 @J @t, (3.2.3)

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the external and internal induced current J = Jext+ Jint, the induced current density inside the plasma can be expressed as

Jint= (E), (3.2.4)

where is electrical conductivity tensor of plasma [14]. The wave equation then reads

r ⇥ r ⇥ E + µ0@ (E)@t + 1 c2 @2E @t2 = µ0 @Jext @t . (3.2.5)

3.3

Dispersion relation

In the Fourier space the wave equation Eq. (3.2.5) is

k⇥ k ⇥ E + i!µ0 · E +!

2

c2E = i!µ0J

ext, (3.3.1)

where k is the wave vector, ! is the frequency and is the imaginary unit. A refractive index is introduced

n = c

!k. (3.3.2)

With the introduced refractive index and in the tensor notation the wave equation reads n2(ˆniˆnj ij)Ej + ( ij+ i !✏0 ij)Ej = i !✏0J ext i , (3.3.3)

where ✏0 is the dielectric constant. Finally, the dielectric tensor is defined as

Kij = ij+ i !✏0 ij, (3.3.4) or equivalently Kij= ij+ X s (s) ij , (3.3.5)

where (s)ij is the susceptibility tensor for species s. The wave equation then takes the form

n2(ˆninjˆ ij)Ej+ KijEj = i

!✏0J

ext

i . (3.3.6)

The eigenmodes of the wave equation are found by setting external current density Jiext = 0. The

solution to the equation

n2(ˆniˆnj ij) + Kij⇤Ej = 0 (3.3.7)

exists only if

det⇤ij = det⇥n2(ˆniˆnj ij) + Kij⇤= 0, (3.3.8)

where ⇤ij is the wave operator. Solutions to equation (3.3.8) give dispersion relations, which define

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3.4

Fast magnetosonic wave

If the magnetic field is chosen to be Bk ˆz and the refractive index n ? ˆy, the response tensor takes

the form K = 0 @iKAK? KiKA? 00 0 0 Kk 1 A , (3.4.1)

and equation (3.3.8) can be rewritten as

K? n2

k iKA n?nk

iKA K? n2 0

n?nk 0 Kk n2?

= 0, (3.4.2)

where nk and n? mean parallel and perpendicular to the magnetic field directions [15]. In a cold

plasma, neglecting the thermal motion of the plasma particles, the components of the dielectric tensor are K?= 1 +X s (s) ? = 1 X s !2 ps !2 !2 cs KA=X s (s) A = X s !cs ! !2ps !2 !2 cs Kk = 1 +X s (s) k = 1 X s !2 ps !2 , (3.4.3) where !ps2 = nq2

✏0m is the square of the plasma frequency of species s [7]. It can be noted that the

tensor Kij with components given by (3.4.3) is hermitian, i.e. its conjugate transpose is equal to

the tensor itself. This means that the tensor in this form does not describe the wave damping.

For ICRH the absolute value of the element Kk is very large compared to K? and KA. Setting

Kk ! 1, the dispersion relation corresponding to the fast magnetosonic wave can be

approxi-mated as n2? ⇡ (K?+ KA n 2 k)(K? KA n2k) K? n2k = K? n 2 k KA2 K? n2k. (3.4.4)

At the fundamental resonance the wave frequency is ! = !c,res, which means that both (res)?

and (res)A terms corresponding to the resonance species become infinitely large, according to the cold plasma model. To include the thermal e↵ects and allow for the absorption by the resonant species, the resonant terms are modified to be

(res) ? = 1 !2 p,res 2p2!kkvT,res[Z(z+) + Z(z )] , (3.4.5) where vT,res= ✓ 2Tres mres ◆1/2

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is the plasma dispersion function and the arguments are z+= ! + !c,resp

2kkvTres

and z = p! !c,res

2kkvTres

[6].

In this expression for the susceptibility, there is no singularity in K? at ! = !c,res.

Similarly, (res) A = !c,res ! " 1 ! 2 p,res 2p2!kkvTres (Z(z+) + Z(z )) # . (3.4.6)

As a result, in the warm plasma description the components of the response tensor (3.4.1) read as K? = ! 2 p,res 2p2!kkvTres (Z(z+) + Z(z )) X s !2 ps !2 !2 cs KA= !c,res ! " 1 ! 2 p,res 2p2!kkvTres (Z(z+) + Z(z )) # +X s !cs ! !ps2 !2 !2 cs Kk ! 1, (3.4.7)

where the summation is made over all species, except for the resonant particles. This modification also allows to calculate the refractive index from (3.4.4) and the polarization of the wave, as it is shown in the next section.

3.5

Polarization

The polarization of a wave is described by decomposing the electric field into two circularly polarized

components E± = Ex±iEy. The left hand circularly polarized component E+is the one that rotates

in the same direction as ions gyrate around the magnetic field lines, see Figure 2.1. Hence, it is the

E+ that allows for the power transfer from the wave to the ions. Right-hand circularly polarized

component E rotates in the opposite direction and does not contribute to the wave damping by

ions. The polarization of the wave is given by [16] E+ E = Ex+ iEy Ex iEy = Ex/Ey+ i Ex/Ey i. (3.5.1)

From equation (3.3.7) the ratio between Ex and Ey can be obtained

Ex Ey = iKA K? n2 k . (3.5.2)

Therefore, the polarization is

E+ E = K?+ KA n2 k K? KA n2 k . (3.5.3)

It can be shown that in a very simplified single-species plasma model, for a wave propagating in a

cold plasma purely perpendicularly to the magnetic field nk = 0, the polarization is

E+ E = KA+ K? KA K? = ! !c ! + !c. (3.5.4)

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This means that at the fundamental resonance E+

E = 0, which indicates that the wave is purely

right hand circularly polarized. Therefore, with purely left hand circular polarization no absorption occurs.

Although the ratio E+

E is not exactly zero at the resonance if the kinetic e↵ects are taken into

account, it is still very low. For that reason, the idea of introducing a minority species into the plasma has been developed. In that case the wave frequency corresponds to the cyclotron frequency of a minority, which can be, for example, hydrogen or helium-3. The analysis presented in [7] shows that even though the polarization is wrong exactly at the minority resonance, it becomes favorable

E+

E ⇠ 1 a short distance away from it, which results in a relatively high absorption rate.

3.6

Susceptibility tensor

In the detailed analysis described in [16] the susceptibility tensor for species s in the for a Maxwellian plasma is given by s= " ˆ ekˆek 2! 2 p !kkv2 T? hvki + !ps2 ! 1 X n= 1 e sYn( ) # , (3.6.1)

where in a two-dimensional configuration

Yn( s) = n2In s An in(In I 0 n)An in(In In0)An ⇣ n2 sIn+ 2 sIn 2 sI 0 n ⌘ An ! , (3.6.2)

where n is the mode number, In= In( s) is the modified Bessel function of the first kind with the

argument s= k2 ?T m!2 cs (3.6.3) and An is given by An= 1 kkvTk Z0(⇣n). (3.6.4)

Z0 is the plasma dispersion function, and since the absorption coefficient has to be a real quantity,

only the imaginary part of Z0 is of interest, which is

Z0(⇣n) = ip⇡e ⇣n2 (3.6.5)

with the argument

⇣n= ! n!cs

kkvTk

. (3.6.6)

Here the analysis is carried out in a geometry shown in Figure 3.2, where the wave propagates through the magnetic axis of a torus (center of the circular cross-section) at the equatorial plane,

y = 0. In this case, the k? corresponds to kx and kk corresponds to kz.

From the definition of the cyclotron frequency (2.1.2) it follows that this frequency depends on the distance from the major axis of the tokamak as

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For a region close to the resonance surface,|R Rres| ⌧ 1, the Taylor expansion leads to 1 1 + R Rres Rres ⇡ 1 R Rres Rres . (3.6.8)

Therefore, as the frequency of the wave is chosen to be ! = n!c,res, the argument for the plasma function can be expressed as

⇣n(R) = !(R Rres)

kkvTkRres . (3.6.9)

Figure 3.2: Quarter of a torus, the view from the top (above) and from the front (below).

3.7

Damping

In the absence of the temporal damping, the wave absorption coefficient is defined by the imaginary part of the wave vector

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such that k = kR+ ikI.

In case of pure spatial damping, that is !I = 0, the absorption is described by [15]

kI· vgM(k) = i!RM(k)e⇤M i(k)eM j(k)KijA(!, k) (3.7.2)

with

RM(k) =

ss(!, k)

!@⇤(!, k)/@! !=!M(k), (3.7.3)

where eM is the polarization vector of the mode M, e⇤M is its complex conjugate, KA

ij is the

antihermitian part of the response tensor, ssis the trace of a cofactor matrix to the wave operator

and ⇤ is the determinant of the wave operator. From the form of the response tensor components in (3.4.3), (3.4.5) and (3.4.6), we can conclude that the our plasma is not a spatially dispersive medium for the wave propagation perpendicular to the magnetic field, since there is no dependence

on k?. Therefore, an alternative form for RM can be applied

RM = 1

[1 | · eM(k)|2]· 2nM(!, )@[!nM(!, )]/@!, (3.7.4)

where  is the unit vector in the direction of a wave vector.

For the fast magnetosonic wave propagating in the direction perpendicular to the magnetic

field, i.e. group velocity vg k k?, the imaginary part of the wave vector becomes

kI = i1

vg

!KA

M

[1 | · eM(k)|2]· 2nM(!, )@[!nM(!, )]/@!, (3.7.5)

where KMA = e⇤M ieM jKijA. With the definition of the group velocity

vg = @!M(k)

@k (3.7.6)

and the refractive index given by (3.3.2), the equation reads as

kI = i ! 2KA M 2c2k ?[1 | · eM|2] . (3.7.7)

From the relation between susceptibility and response tensors (3.3.5) it follows that KijA=X

s A(s) ij . The imaginary component of the wave vector then becomes

kI = i! 2P se⇤M ieM j A(s) ij 2c2k ?[1 | · eM|2] . (3.7.8)

Considering the heating in only one mode and inserting the explicit form of the susceptibility tensor from Section 3.6, the damping rate by species s in a mode number n is

n,s= p ⇡!!2 ps c2k k vT [1 | · eM|2]e se ⇣n2

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where ✏xx = n 2In

s

, ✏xy = n(In In0) and ✏yy = n

2 s

In+ 2 s(In In0). Introducing a value h, such

that Ex = ihEy or, equivalently, eM x = iheM y, and noting that for a wave propagating in

x-direction [1 | · eM|2] =|eM y|2, the damping coefficient can be expressed as

n,s = p ⇡!!2ps c2k ?kkvTk e se ⇣n2(|h|2✏xx+ 2hR✏xy+ ✏yy). (3.7.10)

The value of h can be found according to the Section 3.5

h = KA

K? n2k. (3.7.11)

For the electrons, the absorption coefficient is found to be e = p ⇡!!2peI0 ee ⇣ 2 0 k?kkv?ec2 , (3.7.12)

with v?e being thermal electron velocity perpendicular to the magnetic field, e= k

2 ?T? me!2 ce , ⇣0 = !/kkvke [17], [18].

3.8

Power partition

As the wave travels through plasma, it is damped and its power is absorbed by the plasma. In reality, the wave is not only absorbed via the wave-particle resonance of a minority, but by all plasma species, including di↵erent ions and electrons. It is therefore important to estimate how much power is absorbed by each of the species.

The rate at which the power of a wave changes as the wave propagates through space if found from

d

dRP = P. (3.8.1)

The solution yields the power dependence on the coordinate R

P (R) = P0e

RR

Redge dR, (3.8.2)

where P0 is the initial power of the incoming wave.

With the contribution from electrons and all ion species, the damping rate is = e+

X s

s,n

and the power of the incoming wave after passing the distance Redge R is given by

P (R) = P0e

RR Redge( e+

P

s s,n)dR. (3.8.3)

Hence, the fraction of power absorbed in the plasma in calculated by

A = P0 P0e RR Redge( e+ P s s,n)dR P0 = 1 e RR Redge( e+ P s s,n)dR. (3.8.4)

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Eq. (3.8.1) can be rewritten as dP (R) dR = ( e+ X s s,n)P (R). (3.8.5)

Then the power absorbed per unit length by each of the species is dP (R)

dR s= s,nP (R) dP (R)

dR e= eP (R),

(3.8.6)

where P (R) is given by Eq. (3.8.3).

Finally, the power absorbed by each of the species is Ps(R) = Z R Redge P (R) s,ndR Pe(R) = Z R Redge P (R) edR. (3.8.7)

3.9

The evolution of the distribution function

In a thermal equilibrium plasma species have a Maxwellian distribution function. In the process of heating, the distribution function changes. The evolution of the distribution function under the influence of ICRH in a uniform magnetic field is described by a modified Fokker-Planck equation

@f

@t = C(f ) + Q(f ), (3.9.1)

where C(f ) represents Coulomb collisions and Q(f ) is a quasi-linear term that describes the particle-field interactions [6]. The Coulomb scattering term describes collisional slowing down of fast ions, or in other words drag. C(f ) can be written in the form

C(f ) = rv(h vif), (3.9.2)

whererv is the nabla operator in velocity space. v is the ensemble average of the velocity change

that a particle experiences due to collisions with the background particles per unit time.

Eq. (3.9.1) is a complicated nonlinear partial di↵erential equation that can be simplified by assuming for a heated minority distribution that the Coulomb scattering term C(f ) is only depen-dent on the bulk ion and electron distribution functions. These can be assumed to be Maxwellian. Finally, the radio-frequency pulse is assumed to be long enough to produce a steady state, thus

setting @f

@t = 0. The solution for f is then obtained from [16]

ln f (r, v) = E

Te(1 + ⇣) 

1 +R0(Te T0+ ⇣Te)

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with l0 = ✓ m0 2T0 ◆1/2 , le= ✓ me 2Te ◆1/2 E = mv2/2, E0 = mT0 m0  3p⇡ 4 1 + R0+ ⇣ 1 + ⇣ 2/3 K(x) = 1 x Z x 0 du 1 + u3/2 and ⇣ = hP?its 3nTe , (3.9.4)

in which ts is the slowing down time expressed by

ts= 6.27· 108A(Te⇤)

3/2

Z2n

eln ⇤

, (3.9.5)

where A is the atomic mass of the test particles, Z is the atomic number and ln ⇤ is the Coulomb

logarithm. In Eq. (3.9.4)hP?i is the magnetic flux surface averaged absorbed radio frequency (RF)

power per unit volume. Here it is implied that the densities and the temperatures are the functions

of the coordinates in space, i.e. ns ⌘ ns(r) and Ts ⌘ Ts(r). In Eq. (3.9.5) the electron density n⇤e

is in cm 3 and electron temperature Te⇤ is in eV. The subscript 0 denotes the background species

in the plasma. The density n and mass m are those of the test species. Eq. (3.9.3) provides an estimation for the tail temperature

Ttail= Te(1 + ⇣). (3.9.6)

The high rate of absorption of the energy by the resonant minority is generally desirable. However, high absorption rate may lead to the formation of the energetic tail of the distribution function. This, in turn, may obstruct the further energy transfer from the fast particles to the bulk plasma ions. The reason for it is that particles with energy higher than a certain critical energy collide more with electrons than bulk ions, thus transferring more energy to the electrons. This critical energy is defined as

E↵ = 14.8Te⇤ A3/2 ne X s nsZ2 s As !3/2 , (3.9.7)

where A is the atomic weight of fast ions, Asis that of species s, Zs is the atomic number of species

s and Te⇤ is the electron temperature in eV [16].

3.10

Energy transfer

Since the frequency of the wave is set to resonate with the minority ions, a considerable fraction of the power is expected to be transferred from the wave to the minority. The minority concentration is relatively low, in the range of 2-10%, therefore a large amount of power is delivered to a small amount of particles and thus fast ions with high energies are created after the wave power is ab-sorbed. Their energy is further transferred to the bulk plasma ions through collisions. Additionally, some of the fast ions have high enough energies that they collide with electrons.

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The energy transfer rate from the fast ions due to collisions is found from PC = Z mv2 2 @f (v) @t Cd 3v, (3.10.1)

where the time derivative only includes changes due to collisions. Alternatively, following the Fokker-Planck equation (3.9.1), it can be written as

PC =

Z mv2

2 C(f )d

3v. (3.10.2)

The collisional relaxation of the fast ions is described by @f (v) @t C = 1 v2 @ @v  ↵v2f + 1 2 @ @v( v 2f ) , (3.10.3)

where ↵ and are the Coulomb di↵usion coefficients [16]. The equation can be simplified by

neglecting the second term, since ⌧ ↵. Then the power transferred due to the collisions is

PC =

Z

mv↵f (v)d3v. (3.10.4)

The approximation for the di↵usion coefficient ↵ for velocities vth,i⌧ v ⌧ vth,e is given in [16] as

↵ = v ts ✓ 1 + V 3 ↵ v3 ◆ = v ts 1 + E↵3/2 E3/2 ! , (3.10.5)

where E↵ is the critical energy, at which there is equal amount of energy transferred from the fast

ions to the bulk ions and electrons defined by (3.9.7). The first term corresponds to the energy transfer from fast ions to the electrons, while the second term is dominating on lower energies

E E↵ and thus describes the energy transfer to the bulk ions.

At low velocities v ⇠ vth,i, however, Eq. (3.10.5) does not hold. To obtain a more accurate

estimation of power transfer, only collisional power transfer to electrons is considered. A more accurate form of the first term in Eq. (3.10.5) that only describes collisions with electrons at all energies is ↵e = v ts ✓ 1 Te E ◆ , (3.10.6)

which is derived from the expression for ↵ given in [16].

Thus, the total collisional power transferred between fast particles and electrons is found from

PC,e=

Z

mv↵ef (v)d3v. (3.10.7)

Knowing the total power absorbed by the fast ions PC, the collisional power transfer to ions is found

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Chapter 4

SELFO

The theoretical results in this work are compared to the results of numerical simulations in SELFO. SELFO is a code used for calculating the distribution functions of the resonant ion species and the evolution of the wave field in a self-consistent manner. It combines FIDO code, which solves the orbit-averaged Fokker-Planck equation [14]

df

dt = C(f ) + Q(f, E, k?)

by a Monte Carlo method, and LION code, which solves the wave equation [17]

r ⇥ r ⇥ E? ⇣!c⌘2✏(f, k?)· E?= iµ0!Jant

using the finite element method. In the Fokker-Planck equation C(f ) is a term that describes

collisions between particles and Q(f, E, k?) is a quasilinear term that describes wave-particle

in-teractions. In the wave equation ✏(f, k?) is the dielectric tensor and Jant is the antenna current.

Solving these equations in a self-consistent manner is necessary because the equations are coupled. This means that both of them have to be solved simultaneously. If a parameter changes in one of the equations, this change influences the other equation instantly.

The susceptibilities of the resonant species are calculated from an initial distribution function and perpendicular wave vector. The susceptibilities are then used as input for LION code, which calculates the electric wave fields, a new perpendicular wave vector and the power absorbed by di↵erent species. The FIDO code then uses the calculated fields to initialize the new distribution function, which then gives new susceptibilities. These susceptibilities are then a new input for FIDO, so the iterative process continues in a time loop [19]. The scheme of the process is illustrated in Figure 4.1.

4.1

FIDO

In the Fokker-Planck equation, the collisional operator C(f ) includes both collisional scattering and slowing down processes. These tend to change the distribution function towards a Maxwellian. The quasilinear operator Q(f ) describes acceleration of resonant particles by the wave. This, in term, results in a non-Maxwellian distribution with a possible fast ion tail.

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Figure 4.1: Illustration of the calculation process in SELFO.

An important factor is that FIDO includes wide orbits in the calculations. This means that particles are allowed to absorbed power near the resonance and then travel to outer regions. An illustration of such orbits is presented in Figure 4.2.

2.4 2.6 2.8 3 3.2 3.4 3.6 3.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 R [m] (a) n = +15 φ Z [m] 2.8 3 3.2 3.4 3.6 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 R [m] (b) n = −15 φ Z [m]

Figure 4.2: Wide orbits of fast ions. The solid vertical line represents the resonance position and the

dotted surfaces are flux surfaces. Di↵erent orbits correspond to di↵erent ion energies [14].

4.2

LION

LION solves the global wave field of the fast magnetosonic wave. It has a model of an antenna outside the plasma. The wave from this antenna propagates through the plasma to the resonance, where it is absorbed by the resonance species. The electron absorption is also included in the simulations.

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Chapter 5

Results

A total amount of 12 cases was studied. Those included either H or3He minority with the

concen-tration of 2%, 4% or 6% in both D-D and D-T plasma. In this section, the results of the analytical model are presented together with the SELFO results. For a better comparison, some SELFO data, like temperature and density profiles was used as an input for the analytical model. The calculations and simulations included heating minority at the fundamental resonance and heating either deuterium or tritium at the second harmonic.

5.1

Wave propagation

The wave is launched into the plasma from the LFS at the equatorial plane. The refractive index in

the direction perpendicular to the magnetic field is found from Eq. (3.4.4). Here nkis approximated

as nk ⇡ nR, where n is toroidal mode number. All the results in this chapter are presented for

the toroidal mode number n = 23, which is a reasonable mode number for the JET antennas [20].

The plots of n2? are shown in Figure 5.1 for di↵erent cases.

Figure 5.1 shows quite good correspondence of the calculated from the analytical model n2? to

the one obtained from SELFO simulations. The formation of the cuto↵-resonance pair is clearly seen at ion-ion hybrid resonance closer to the major axis, which becomes more pronounce the higher

is the minority concentration [6]. The high positive values of n2

?show that the wave propagates all

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(R - R 0)/a -1 -0.5 0 0.5 1 n ⊥ 2 -500 0 500 1000 1500 2000 2500 3000 3500 4000 4% of H, D-D Real part Imaginary part SELFO result, real

(R - R 0)/a -1 -0.5 0 0.5 1 n ⊥ 2 -500 0 500 1000 1500 2000 2500 3000 3500 4000 4% of H, D-T Real part Imaginary part SELFO result, real

(R - R 0)/a -1 -0.5 0 0.5 1 n ⊥ 2 -500 0 500 1000 1500 2000 2500 3000 3500 4000 4% of 3He, D-D Real part Imaginary part SELFO result, real

(R - R 0)/a -1 -0.5 0 0.5 1 n ⊥ 2 -500 0 500 1000 1500 2000 2500 3000 3500 4000 4% of 3He, D-T Real part Imaginary part SELFO result, real

Figure 5.1: Calculated n2

? compared with the SELFO simulation results with 4% of H or3He minority in

D (left) or D-T (right) plasma.

5.2

Damping and power partition

For the power to be transferred from the wave to the plasma, there has to be a left-hand circularly polarized component of the electric field at the position of the resonance. The absolute value of the polarization calculated from Eq. (3.5.3) for two cases is presented in Figure 5.2.

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(R - R 0)/a -1 -0.5 0 0.5 1 E + /E -0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 6% of H, D-D (R - R 0)/a -1 -0.5 0 0.5 1 E + /E -0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 6% of H, D-T (R - R 0)/a -1 -0.5 0 0.5 1 E + /E -0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 6% of 3He, D-D (R - R 0)/a -1 -0.5 0 0.5 1 E + /E -0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 6% of 3He, D-T

Figure 5.2: Polarization of the wave in D-D and D-T plasma with 6% of H (above) or 6% of3He (below)

according to the analytical model.

The polarization defined as the ratio of the components E+/E had the lowest absolute value

at the resonance of about 0.1 in the case of 6% of 3He minority. Decreasing 3He concentration

slightly improves the polarization. However, the right-hand side component is always present thus

allowing for the energy transfer. It is also worth noting that with the3He minority, the calculated

polarization value is always lower than the one in case of H.

The amount of power transferred to each species is calculated from Eq. (3.8.7). The results of the calculated power partition between di↵erent species after multiple passes (such that all of the wave power is absorbed) together with the SELFO simulation results are presented in Table 5.1. According to the theory described in Section 3.7, there is no damping by deuterium at helium-3 fundamental resonance. The absorption by tritium at the third harmonic resonance is very small compared to hydrogen and deuterium and is neglected in the analytical calculations. Therefore, these are not included in the table, although they are among the SELFO results.

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Table 5.1: Power partition between di↵erent species. C as e n o. C om p os it ion M in or it y ion s M in or it y con ce n tr at ion

Absorbed power fractions according to calculations

Absorbed power fractions according to SELFO Minority D T e Minority D T e 1 D-D H 2% 0.40 0.46 - 0.14 0.43 0.41 - 0.16 2 4% 0.56 0.30 - 0.14 0.59 0.25 - 0.15 3 6% 0.65 0.21 - 0.14 0.68 0.18 - 0.15 4 3He 2% 0.66 - - 0.34 0.62 0.07 - 0.30 5 4% 0.76 - - 0.24 0.74 0.04 - 0.21 6 6% 0.78 - - 0.22 0.79 0.03 - 0.18 7 D-T H 2% 0.52 0.33 - 0.15 0.53 0.29 0.01 0.16 8 4% 0.67 0.19 - 0.14 0.68 0.16 0.00 0.15 9 6% 0.73 0.13 - 0.14 0.74 0.10 0.00 0.15 10 3He 2% 0.62 - 0.16 0.21 0.60 0.03 0.16 0.21 11 4% 0.74 - 0.08 0.17 0.73 0.01 0.09 0.17 12 6% 0.76 - 0.06 0.18 0.78 0.00 0.06 0.16

It is seen from the table that, in spite of the simplifications in the model, the final results are still very accurate. The numbers are very close to those obtained from more complex SELFO simulations. As the concentration of a minority increases, the power obtained by minority species increases, while the power absorbed by bulk ions decreases. The power absorbed by electrons stays almost at a constant level in case of hydrogen and slightly decreases in the case of helium-3 with the minority concentration increase.

The change of the power partition with the minority concentration is visualized in Figure 5.3. It appears that in the analytical model and in SELFO the absorbed power fractions follow similar patterns as the concentration of the minority species is changed.

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Minority concentration, % 2 4 6 ∆ P s /P 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 H in D-D plasma H D e -Minority concentration, % 2 4 6 ∆ P s /P 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 H in D-T plasma H D e -Minority concentration, % 2 4 6 ∆ P s /P 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 3 He in D-D plasma 3 He e -Minority concentration, % 2 4 6 ∆ P s /P 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 3 He in D-T plasma 3He T e

-Figure 5.3: Power fraction absorbed by di↵erent species as a function of the minority concentration in

D-D (left) and D-T (right) plasma. The results of the analytical model (solid line) are compared with the results from SELFO (dashed line).

The amount of power absorbed by each of the species per unit length is calculated from Eq.(3.8.6). Figure 5.4 shows the power absorption per unit length at the equatorial plane,

normal-ized to the input wave power P0 for the case of H minority. The peak of the absorption is located

o↵ the resonance layer at the LFS. The reason for that is, according to the calculations, all of the wave power gets absorbed before it even reaches the resonance. This also corresponds to the SELFO predictions, marked by the green dashed line in the plot. The distinct feature is that the analytical model predicts a more localized absorption than SELFO.

In case of the 3He minority the situation is di↵erent. Not all the power is absorbed at a single

pass, according to the calculations and the maximum absorption rate is at the resonance position. This is illustrated in Figure 5.5. SELFO also predicts a spike of the absorption o↵ the resonance at the high-field side (HFS), i.e. closer to the major axis of the tokamak. This spike corresponds to the polarization spike shown in Figure 5.2. The analytical model, however, does not predict such a distinct spike and the absorption profile is quite smooth. Still, a small trace of this peak is still seen even from the analytical predictions in Figure 5.5 for the cases of 6% of concentration.

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(R - R 0)/a -1 -0.5 0 0.5 1 dP/dR normalized to P 0 [1/m] 0 1 2 3 4 5 6 7 8 2% of H, D-D Total H D e -Total, SELFO (R - R 0)/a -1 -0.5 0 0.5 1 dP/dR normalized to P 0 [1/m] 0 1 2 3 4 5 6 7 8 2% of H, D-T Total H D e -Total, SELFO (R - R 0)/a -1 -0.5 0 0.5 1 dP/dR normalized to P 0 [1/m] 0 1 2 3 4 5 6 7 8 4% of H, D-D Total H D e -Total, SELFO (R - R 0)/a -1 -0.5 0 0.5 1 dP/dR normalized to P 0 [1/m] 0 1 2 3 4 5 6 7 8 4% of H, D-T Total H D e -Total, SELFO (R - R 0)/a -1 -0.5 0 0.5 1 dP/dR normalized to P 0 [1/m] 0 1 2 3 4 5 6 7 8 6% of H, D-D Total H D e -Total, SELFO (R - R 0)/a -1 -0.5 0 0.5 1 dP/dR normalized to P 0 [1/m] 0 1 2 3 4 5 6 7 8 6% of H, D-T Total H D e -Total, SELFO

Figure 5.4: Power partition in the equatorial plane as a function of normalized major radius. Results for

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(R - R 0)/a -1 -0.5 0 0.5 1 dP/dR normalized to P 0 [1/m] 0 1 2 3 4 5 2% of 3 He, D-D Total 3 He e -Total, SELFO (R - R 0)/a -1 -0.5 0 0.5 1 dP/dR normalized to P 0 [1/m] 0 1 2 3 4 5 2% of 3 He, D-T Total 3 He T e -Total, SELFO (R - R 0)/a -1 -0.5 0 0.5 1 dP/dR normalized to P 0 [1/m] 0 1 2 3 4 5 4% of 3He, D-D Total 3 He e -Total, SELFO (R - R 0)/a -1 -0.5 0 0.5 1 dP/dR normalized to P 0 [1/m] 0 1 2 3 4 5 4% of 3He, D-T Total 3 He T e -Total, SELFO (R - R 0)/a -1 -0.5 0 0.5 1 dP/dR normalized to P 0 [1/m] 0 1 2 3 4 5 6% of 3 He, D-D Total 3 He e -Total, SELFO (R - R 0)/a -1 -0.5 0 0.5 1 dP/dR normalized to P 0 [1/m] 0 1 2 3 4 5 6% of 3 He, D-T Total 3 He T e -Total, SELFO

Figure 5.5: Power partition in the equatorial plane as a function of normalized major radius. Results for

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5.3

The distribution function

The approximate shape of the distribution function is found by taking the exponent of Eq. (3.9.3). Since the temperatures and concentrations are dependent on the position in plasma, the distribution

function obtained from Eq. (3.9.3) is f (r, v) with the dimension m 6s3. The minority distribution

functions integrated over the whole space domain f (v) =Rf (r, v)d3r are shown in Figures 5.7 and

5.8, where they are compared to the corresponding results from SELFO simulations.

It can be seen that despite being a quite rough approximation, Eq. (3.9.3) gives a quite accurate result, compared with SELFO, especially on low energies. On high energies, however, the result slightly di↵ers. Even though the amount of particles at these high energies is small, the energy contribution from them is significant. Therefore, this approximation may provide a quite good estimation, but should still be used with caution.

The total energy content of the fast minority particles is calculated from the shape of the distribution function by

Eminority=

Z

f (v)Ed3v. (5.3.1)

The calculated total fast particle energy content for di↵erent cases is presented in Table 5.2.

Table 5.2: Total energy content of fast ions.

C as e n o. C om p os it ion M in or it y ion s M in or it y con ce n tr at

ion Total energy

according to the analytical model [kJ] Total energy according to SELFO [kJ] Maximum energy according to Eq. (5.3.2) [kJ] 1 D-D H 2% 362 353 517 2 4% 509 491 717 3 6% 591 557 798 4 3He 2% 269 285 604 5 4% 314 298 657 6 6% 349 346 676 7 D-T H 2% 438 428 608 8 4% 576 548 802 9 6% 633 597 867 10 3He 2% 300 272 563 11 4% 347 307 633 12 6% 372 364 662

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if the background species only include electrons. The latter is calculated from

Emax = 1

2P ts, (5.3.2)

which is found from Eq. (3.10.4) by ignoring the second term in ↵e. The slowing down time ts is

taken at the position of the resonance and P is the total RF power absorbed from the wave. The maximum energy estimation is almost 2 times larger than the calculated or simulated in SELFO total energy values in some cases. This means that the assumption of fast ions being slowed down by electrons only is quite far from reality. However, this value is relatively easy to calculate and studying how much it diverges from the total energy calculations, one can make a good quick approximation. The numbers from Table 5.2 are illustrated in Figure 5.6.

Minority concentration, % 2 4 6 Total energy [kJ] 200 300 400 500 600 700 800 900 H in D-T plasma Analytical SELFO E max Minority concentration, % 2 4 6 Total energy [kJ] 200 300 400 500 600 700 800 900 H in D-T plasma Analytical SELFO E max Minority concentration, % 2 4 6 Total energy [kJ] 200 300 400 500 600 700 800 900 3He in D-D plasma Analytical SELFO Emax Minority concentration, % 2 4 6 Total energy [kJ] 200 300 400 500 600 700 800 900 3He in D-T plasma Analytical SELFO Emax

Figure 5.6: Fast particle energy content as a function of minority density in D-D plasma (left) and D-T

plasma (right).

The shapes of the distribution functions seem to be more accurate in case of the3He minority,

while in case of H, the distribution at high energies appears to be significantly underestimated. At the same time, the distribution at lower energies is slightly overestimated, which in the end result in relatively accurate energy content approximation, as it is seen from Table 5.2 and Figure 5.6.

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E [MeV] 0 0.5 1 1.5 f(E) [m -3 /s -3 ] 10-6 10-4 10-2 100 102 2% of H, D-D SELFO Analytical E [MeV] 0 0.5 1 1.5 f(E) [m -3 /s -3 ] 10-8 10-6 10-4 10-2 100 102 2% of H, D-T SELFO Analytical E [MeV] 0 0.5 1 1.5 f(E) [m -3 /s -3 ] 10-6 10-4 10-2 100 102 4% of H, D-D SELFO Analytical E [MeV] 0 0.5 1 1.5 f(E) [m -3 /s -3 ] 10-8 10-6 10-4 10-2 100 102 4% of H, D-T SELFO Analytical E [MeV] 0 0.5 1 1.5 f(E) [m -3 /s -3 ] 10-6 10-4 10-2 100 102 6% of H, D-D SELFO Analytical E [MeV] 0 0.5 1 1.5 f(E) [m -3 /s -3 ] 10-8 10-6 10-4 10-2 100 102 6% of H, D-T SELFO Analytical

Figure 5.7: Distribution function of H minority in deuterium (left) and deuterium-tritium (right) plasma

according to the analytical model and SELFO simulations. The critical energy E↵ is of the order of

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E [MeV] 0 0.5 1 1.5 f(E) [m -3 /s -3 ] 10-6 10-4 10-2 100 102 2% of 3He, D-D SELFO Analytical E [MeV] 0 0.5 1 1.5 f(E) [m -3 /s -3 ] 10-6 10-4 10-2 100 102 2% of 3He, D-T SELFO Analytical E [MeV] 0 0.5 1 1.5 f(E) [m -3 /s -3 ] 10-6 10-4 10-2 100 102 4% of 3He, D-D SELFO Analytical E [MeV] 0 0.5 1 1.5 f(E) [m -3 /s -3 ] 10-6 10-4 10-2 100 102 4% of 3He, D-T SELFO Analytical E [MeV] 0 0.5 1 1.5 f(E) [m -3 /s -3 ] 10-6 10-4 10-2 100 102 6% of 3He, D-D SELFO Analytical E [MeV] 0 0.5 1 1.5 f(E) [m -3 /s -3 ] 10-6 10-4 10-2 100 102 6% of 3He, D-T SELFO Analytical

Figure 5.8: Distribution function of3He minority in deuterium (left) and deuterium-tritium (right)

plasma according to the analytical model and SELFO simulations. The critical energy E↵ is of the order of

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An estimation for the tail temperature is calculated from Eq. (3.9.6). Two examples of this estimation are shown in Figure (5.9). Eq. (3.9.6) provides a rough estimation of the tail temperature that is constant at all energies. Although the value is in the right range, it still di↵ers significantly from the tail temperatures predicted by SELFO.

E [MeV]

0 1 2 3 4

Tail temperature [MeV]

0 0.1 0.2 0.3 0.4 0.5 0.6 4% of H, D-T SELFO T e(1+ζ) E [MeV] 0 1 2 3 4 5

Tail temperature [MeV]

0 0.1 0.2 0.3 0.4 0.5 0.6 4% of 3 He, D-T SELFO T e(1+ζ)

Figure 5.9: Tail temperature as a function of energy for 4% of hydrogen (left) and helium-3 (right)

minority in D-T plasma at the resonance surface.

It should be noted that the power hP?i in the calculation of ⇣ in Eq. (3.9.4) was taken as a

sum of the collisional heating powers for ions and electrons from SELFO. Using the absorption power instead results in considerably overestimated fast particle energy content and a distribution function with higher values. The total energy content for 6% of H in D-D plasma would change

from 591 to 615 kJ. The energy content for 6% of3He in D-T plasma would rise from 372 to 495 kJ,

which is much above the SELFO prediction. The distribution functions then look like shown in Figure 5.10. E [MeV] 0 0.5 1 1.5 f(E) [m -3 /s -3 ] 10-6 10-4 10-2 100 102 6% of H, D-D SELFO Analytical E [MeV] 0 0.5 1 1.5 f(E) [m -3 /s -3 ] 10-6 10-4 10-2 100 102 6% of 3He, D-T SELFO Analytical

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5.4

Bulk plasma heating

As the main goal of ICRH is to heat the bulk plasma ions, it is necessary to study how the power is transferred to them from the fast particles. The collisional power transfer to ions and electrons is calculated from Eq. (3.10.7) and Eq. (3.10.8). The results, compared to the SELFO simulation results, are presented in Table 5.3.

Table 5.3: Collisional power transfer from fast ions.

C as e n o. C om p os it ion M in or it y ion s M in or it y con ce n tr at ion

Collisional power transfer according to calculations

[MW]

Collisional power transfer according to SELFO

[MW]

Ions Electrons Ions Electrons

1 D-D H 2% 0.58 1.52 0.50 1.56 2 4% 0.99 1.94 0.83 2.04 3 6% 1.29 2.06 1.13 2.15 4 3He 2% 2.08 1.28 1.62 1.66 5 4% 2.54 1.16 2.16 1.43 6 6% 2.87 1.01 2.28 1.48 7 D-T H 2% 0.53 1.98 0.49 1.99 8 4% 0.95 2.37 0.88 2.38 9 6% 1.21 2.35 1.08 2.42 10 3He 2% 1.69 1.44 1.55 1.52 11 4% 2.22 1.34 1.98 1.45 12 6% 2.60 1.14 2.04 1.56

The table shows a very good correspondence of the results for H minority. The power transfer rises to both ions and electrons with the increased H concentration. This is seen from both SELFO results and the analytical model.

In case of 3He, however, the power transfer to electrons is significantly lower than expected

from SELFO, when the minority concentration is increased. According to the analytical model,

as the concentration of 3He is increased, the collisional power transfer to electrons decreases. In

SELFO simulations, in contrast, the collisional power transfer to electrons stays almost constant, no matter what the minority concentration is. This results in high discrepancy in case of 6% of

3He for both D-D and D-T plasma.

If the absorbed RF heating power is used in calculations instead of the collisional heating power, the total power content of the fast particles is overestimated. At the same time, the collisional power transfer to electrons and ions seems to become more accurate. For example, the values change from 1.29 and 2.06 MW into 1.17 and 2.10 MW for ions and electrons respectively in the case of 6% of H

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in D-D plasma. In case of 6% of3He in D-T plasma, the change is more drastic. Thus, instead of 2.60 and 1.14 MW, the result becomes 1.91 and 1.74 MW for ions and electrons respectively, which is much closer to 2.04 and 1.56 MW predicted by SELFO.

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Chapter 6

Discussion

6.1

Polarization

In Section 5.1 it is shown that the polarization of the wave is always more favorable in case of the H minority. The reason for that are the relations

!c,H = 2!c,D, !c,H = 3!c,T,

!c,3He = 4

3!c,D, !c,3He = 2!c,T.

From Eq. (3.5.4) it follows that for a single species plasma consisting of deuterium ions, the

polarization of the wave with frequency satisfying ! = !c,H is

E+ E = ! !c,D ! + !c,D = !c,H 12!c,H !c,H+ 12!c,H = 1 3.

If the frequency of the wave is adjusted to3He minority cyclotron frequency ! = !

c,3He, the same equation yields E+ E = ! !c,D ! + !c,D = !c,3He 32!c,3He !c,3He+ 3 2!c,3He = 1 7.

This implies that the closer the frequency of the wave is to the cyclotron frequency of main plasma

species, the smaller is the left-hand circularly polarized component of the wave. Since3He cyclotron

frequency is closer to both D and T cyclotron frequencies, the polarization of the wave is always

less favorable in case of3He minority.

6.2

Absorption rate and partition

The less favorable polarization in case of3He minority results in a lower power absorption rate per

unit length than is case of H, as it is seen from Figure 5.4 and Figure 5.5. Consequently, all of the wave power is absorbed even before the the wave reaches the resonance in case of H minority. With

3He, it takes more than one pass for the wave to get fully absorbed.

At the same time, heating H at the fundamental resonance corresponds to heating D at the second harmonic resonance. Therefore, at minority concentrations of 2%-4% there is a significant competition between H and D absorptions. Table 5.1 shows that D absorbs practically the same

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amount of power as hydrogen does at 2% concentration in D-D plasma. The numbers are close to those reported previously in [21]. This, of course, has a big contribution to the fact that all of the wave power is absorbed at a single pass.

Since concentration of D is lower in D-T plasma than it is in D-D plasma, the absorption by D decreases when the composition of plasma is changed into D-T. Consequently, the competition is smaller and power absorbed by the minority H increases.

In case of 3He minority, on the other hand, the competition between species is practically

absent. If there is T in plasma, it gets heated at the second harmonic resonance at the fundamental

cyclotron frequency of 3He. However, the absorption rate of T is quite small, less than that of

electrons. This is a consequence of the (|h|2✏xx+ 2hR✏xy+ ✏yy) factor in Eq. (3.7.10). The latter

is one order of magnitude smaller for T than for D at a second harmonic resonance. As a result,

although not all of the power is absorbed at a single pass in a plasma with 3He, most of it is

absorbed by 3He. This result is consistent with the previous research described in [8] and [10].

Table 5.1 shows that up to 80% of power gets delivered to the minority ions. It is also the reason

why in Figure 5.3 the power fraction absorbed by3He does not increase so rapidly with minority

concentration as it does in case of H.

6.3

Distribution function and energy content

In calculations of the distribution function presented in Section 3.9, the quasilinear operator Q(f ) in Eq. (3.9.1) is assumed to be a di↵usion operator with a constant di↵usion coefficient. This is a rather gross approximation, considering that it may change more than an order of magnitude [22]. For this reason the calculated using the analytical model tail temperature is very di↵erent from the one obtained from SELFO, as it is seen in Figure 5.9. The tail temperature in Eq. (3.9.6) is only calculated at one flux surface that corresponds to the resonance surface. This suggests that Eq. (3.9.3) is only an approximate formula and for better physical picture, a volume integrated distribution function should be used.

A quite good agreement in the shapes of the volume integrated distribution functions can be seen in Figure 5.7 and Figure 5.8. This is, in fact, a surprising result. It indicates that the assumptions made by Stix [16] indeed are reasonable and the balance between ICRH and slowing down is well described. Even such an assumption as a constant di↵usion coefficient provides a relatively accurate result, both in the distribution function and the total energy content.

Table 5.2 and Figure 5.6 show that the fast particle energy content increases together with concentration. For instance, the total energy of H in D-T plasma increases from 438 to 633 kJ according to the analytical model (the corresponding numbers in SELFO are 428 and 594 kJ), when

the concentration is changed from 2% to 6%. For the3He minority, the total energy rises from 300 to

372 kJ in D-T plasma according to the analytical model (or from 272 to 364 kJ according to SELFO). This is consistent with the fact that the absorbed power fraction increases with concentration too. The fact that H absorbs more in D-T plasma than it does in D-D is also reflected in Table 5.2, as the total energy content for H is larger in D-T plasma.

Although 3He absorbs a larger fraction of power than H does, it can still be seen that the

total energy of 3He is lower than that of H. The reason for this is that the slowing down time is

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profile of the power density hP?i in Eq. (3.9.4) was taken as a sum of the collisional heating powers for ions and electrons from SELFO. Choosing the wave absorption profile instead leads to an overestimated energy content. The di↵erence between the two power profiles is illustrated in Figure 6.1. r/a 0 0.2 0.4 0.6 0.8 1 P [MW/m 3 ] 0 0.5 1 1.5 2 2.5 3 3.5 4 4% of 3 He, D-T

Collisional heating power Absorbed wave power

Figure 6.1: Collisional heating power and absorbed wave power radial profiles from SELFO as a function

of the normalized distance from the magnetic axis. The dotted line corresponds to the position of the resonance at the equatorial plane.

The reason for the total energy content di↵erence may be that choosing the absorbed RF power profile does not include e↵ects of wide orbits and does not account for the time that particles spend at the resonance. The result suggests that when a non-local power absorption is studied, the collisional power gives a better representation of the fast particle acceleration. The collisional power profile used in the analytical model is obtained from SELFO, where wide orbits are taken into account.

6.4

Collisional power transfer

When the minority concentration is increased, the size of the energetic tail in the distribution function decreases. This could suggest that the collisional power transfer to electrons decreases

then too. This is indeed what can be observed in the analytical model for the case of3He minority.

However, according to SELFO , the value stays almost constant at all concentrations. This causes

a high discrepancy between the analytical model and SELFO results for 6% of 3He in both D-D

and D-T plasma. Additionally, it appears that the analytical model overestimates the collisional

ion heating power with3He in D-D plasma at all concentrations. The reason for this is not obvious

and requires further study.

With H minority, however, collisional power transfer increases to both ions and electrons with the increase of concentration. This can be a result of power partition that is shown in Table 5.1. The amount of power that H absorbs increases more rapidly with concentration than in the case of

3He, thus resulting is an even higher amount of particles with energies above the critical energy.

Another feature that can be observed is that the collisional power transfer to electrons dominates

References

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