IN DEGREE PROJECT ENGINEERING PHYSICS,SECOND CYCLE, 30 CREDITS

, STOCKHOLM SWEDEN 2019

Helical Magnetic Fields in the FEMIC Code for RF Heating of Fusion Plasmas

HENRIK K.M. JÄRLEBLAD

KTH ROYAL INSTITUTE OF TECHNOLOGYSCHOOL OF ENGINEERING SCIENCES

Royal Institute of Technology

Master’s Thesis

Helical Magnetic Fields in the FEMIC Code

for RF Heating of Fusion Plasmas

Project Report

Henrik Jarleblad

examined byUlrich Vogt, School of Engineering Sciences

supervised byThomas Jonsson, School of Electrical Engineering and Computer Science

co-supervised byPablo Vallejos, School of Electrical Engineering and Computer Science

andBjorn Ljungberg, School of Electrical Engineering and Computer Science

June 26, 2019

Abstract

This master’s thesis implements an addition of a poloidal magnetic field to the FEMIC-code. Thenecessary pertaining measures are implemented, such as a generic coordinate transformation of theelectric field and the dieletric response tensor between the Stix and cylindrical coordinate systems usedby FEMIC. This work also analyzes the e↵ect of a poloidal magnetic field in radio-frequency (RF)simulations of the Joint European Torus (JET) and ITER tokamaks.

One e↵ect of adding a poloidal magnetic field is FEMIC predicting a bending of the wavefront.This bending e↵ect is quantized. For the investigated cases, the total absorbed power by the plasmais decreased while the power partition shifts in favor of the ions. The results obtained in this work arealso benchmarked against the work by Bilato et al [1]. The results are found to be in good agreementwith Bilato et al.

An important finding of this work is that running RF-heating simulations with FEMIC with apoloidal magnetic field requires a subcentimeter grid and mesh size in the plasma to produce accurateand satisfactory results. Furthermore, to avoid complications the scrape-o↵ layer (SOL) was modelledusing a vacuum. This resulted in accurate coordinate transformations. Future work could assess thepossibility of using a dieletric tensor model in the SOL together with a poloidal magnetic field, whilestill maintaining coordinate transformational accuracy. Future work should also implement an up- anddownshift of the parallel wave number and benchmark the results against Bilato et al [1].

Sammanfattning

Denna masteruppsats implementerar ett tillagg av ett poloidalt magnetfalt till FEMIC-koden. Denodvandiga tillhorande atgarderna implementeras, sasom en allman koordinattransformation av detelektriska faltet och den dielektriska tensorn mellan Stix- och cylinderkoordinater, som bada anvands iFEMIC. Detta projekt analyserar ocksa den inverkan som ett poloidalt magnetfalt har pa simuleringarav radiovagsuppvarmning i de tva tokamakerna the Joint European Torus (JET) och ITER.

Ett tillagg av ett poloidalt magnetfalt resulterar i att FEMIC forutspar en bojning av vagfronten.Bojningse↵ekten kvantifieras. Den totala absorberade e↵ekten i plasmat minskar samtidigt som e↵ektenomfordelas till forman for jonerna for de undersokta simuleringsfallen. Resultaten i detta projektbenchmarkas ocksa gentemot data i Bilato et al [1]. Resultaten overenstammer val med denna data.

En viktig upptackt ar att det kravs subcentimeter grid och mesh i plasmat for att erhalla godaresultat fran simuleringar av radiovagsuppvarmning med FEMIC med poloidalfalt. Vidare sa mod-ellerades scrape-o↵ omradet (SOL) med ett vakuum for att undvika komplikationer. Detta resulteradei goda koordinattransformationer. Framtida arbeten foreslas utforska mojligheten att modellera SOLmed dielektriska tensorn samtidigt som ett inkluderande av ett poloidalt magnetfalt, samt goda koor-dinattransformationer, kan bibehallas. Framtida arbeten bor ocksa implementera ett upp- och nedskiftav det parallella vagtalet och benchmarka resultaten mot Bilato et al [1].

1

Preface

To think that I will be starting my PhD studies in plasma physics and fusion energy in Copenhagenthis upcoming fall would have been impossible less than a year ago. My journey into the research areaof fusion plasma physics feels truly remarkable and I feel grateful to have stumbled upon the field. Theduration of this master’s thesis project coincides with a very stressful time for me on a personal level.To have had the opportunity of working with something that, for me, feels purposeful and meaningfulhas been highly appeasing and reposeful. To learn about everything from the very notion of a toroidaland poloidal magnetic field to how the FEMIC code actually works has been a great challenge andthere were many moments when I thought I would not finish this project in time. The fact that Ialmost did still surprises me and I cannot help but feeling joyful. Not only given the circumstancesbut also because I did not know any fusion plasma physics prior to October last year. However, I nowhave a hard time imagining when I did not know what a tokamak was.

This project was created based on the need for an inclusion of a poloidal magnetic field in theRF-heating simulations performed by the FEMIC code. That need has been met and I intend to makemyself available should any complications arise in the future with the scripts and modifications createdduring this project.

Acknowledgments

I would like to start by expressing my deepest gratitude to my supervisor Thomas Jonsson and my co-supervisors Pablo Vallejos and Bjorn Ljungberg. Without their support, feedback and encouragementthis project and its completion would not have been possible. Everything from the o�cial master’sthesis meetings to the lunch discussions and all in between has made my time at the Division of FusionPlasma Physics a very pleasurable and memorable experience.

Being a student of the Engineering Physics Master’s programme, I would like to say thank youto Magnus Andersson for helping me set up the practicalities and helping me navigate through thebureaucracy at the start of this project. Without his help, the process would have taken much longerI am sure. I am also thankful for the frictionless handling by, and communications with, my examinerUlrich Vogt. Thanks to him having replied swiftly via email and answering my questions thouroughly,the confusion regarding the examination process has been virtually non-existent.

Many thanks also to my opponent Felicia Leander Zaar who gave valuable feedback and challengingquestions at the project presentation. Finally I would like to thank my friends and family for their loveand support. Not only during this project but also during my time at KTH as a whole. A time which,after all these years since 2012, is now coming to an end.

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Contents

1 Introduction 41.1 Project Goals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2 Theory & Background 62.1 The Wave Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.2 The Dielectric Tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.3 Ion Cyclotron Resonance Heating (ICRH) . . . . . . . . . . . . . . . . . . . . . . . . . . 72.4 Coordinate Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.4.1 The Cylindrical Coordinate System . . . . . . . . . . . . . . . . . . . . . . . . . . 82.4.2 The Stix Coordinate System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.5 Coordinate Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.6 Euler Angles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.7 The Dispersion Relation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.8 Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

3 Implementation in FEMIC 123.1 Generic Magnetic Field Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . 123.2 The Rotation Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133.3 Validity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153.4 Mesh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163.5 Quantification of Poloidal Bending E↵ect . . . . . . . . . . . . . . . . . . . . . . . . . . 16

4 Results 174.1 Ion Cyclotron Resonance Layers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184.2 The Electric Wave Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

4.2.1 JET . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184.2.2 ITER . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

4.3 Power Absorption . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214.3.1 JET . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214.3.2 ITER . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

4.4 Quantification of Poloidal Bending E↵ect . . . . . . . . . . . . . . . . . . . . . . . . . . 224.5 Numerical Precision . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234.6 Benchmark . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

5 Discussion 275.1 The Electric Wave Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275.2 Power Absorption . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285.3 Numerical Precision . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285.4 Benchmark . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

6 Conclusion 30

7 References 31

3

1 Introduction

To minimize global warming, a transition to sustainable and renewable energy sources must take placeas soon as possible [2]. However, renewable energy sources alone will not be enough to sustain theglobal energy demand [3][4][5]. In addition, the global energy demand is rising and future energysources need to be able to scale and match the increase. One possible source of future energy withpromising potential is fusion energy. In its simplest and broadest terms, fusion is the joining of twolighter atomic nuclei to form one heavier nucleus. Proton-proton reactions produce the bulk energy inthe sun and stars but many other fusion reactions are possible [2]. Fusion reactions could potentiallybe used to produce energy here on Earth and the energy density compared to conventional energysources would then be several orders of magnitude greater [4]. To fuel fusion reactors, deuterium-tritium reactions would be used in the first generation of reactors. Deuterium exists in vaste quantitieson Earth while tritium does not. However, the relatively great energy output from a small fuel inputrenders energy production from fusion sustainable. Fusion energy would then be able to form the bulkof future sustainable energy production [2]. The deuterium-tritium reaction can be written as

D + T ! He4 (3.5MeV) + n (14.1MeV)

However, if fusion reactions are to be used for energy production, several problems immidiatelybecome apparent. The Coulomb forces are strongly repelling the positively charged atomic nuclei fromeach other and have to be overcome to achieve fusion. Because of the Sun’s size and gravitationalpressure, it is able to produce fusion reactions at 15 million degrees Celcius. Unfortunately, humanitydoes not currently have the possibility to create those kinds of volumes and pressures. Since thebeginning of fusion research, several methods of mastering this challenge have been explored to makethe particles come close enough together for fusion to take place. Three main ideas can be identified: 1)the particles are confined and heated together, known as magnetic confinement fusion 2) the particlesare accelerated and brought together at high speeds, known as inertial confinement fusion, and 3)the positive nuclei are ’tricked’ into approaching each other by shielding them with negative charges[2]. The work of this master’s thesis has been performed within the area of magnetic confinementfusion. Throughout fusion research history, many confinement configurations have been explored suchas reversed-field pinches (RFPs), stellarators and Z-configurations. One of the most successful andwidespread configurations is the tokamak. At the temperatures needed to achieve fusion via magneticconfinement fusion, the medium is a plasma which is essentially an ionized gas. The tokamak confinesthe plasma in a torus chamber via a strong toroidal magnetic field and a weak poloidal magnetic fieldas can be seen in Figure 1.

Figure 1: Schematic of a tokamak confinement configuration. Source: [6]

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The ultimate question to attain fusion energy is ’How do we make the particles come close enoughtogether for fusion to take place?’. If the particles are confined, then an important step on the way toa viable fusion reactor is to achieve ignition. Ignition is, as with fossil fuels, when the burning processbecomes self-sustaining without the need for further applied heating. The requirement for ignition canbe expressed approximately as

n⌧E T > 5⇥ 1021 m�3s keV

where n and T are the peak ion density and temperature in the fusion plasma and ⌧E is the energyconfinement time (a measure of how quickly energy dissipates out from the plasma). The temperaturesneeded to achieve ignition is around 10 keV (or 100 million degrees Celcius). The fusion reactionsachieved via heating confined fusion fuel (deuterium-tritium, deuterium-deuterium, etc) to su�cientlyhigh thermal velocities is called thermonuclear fusion [7].

To heat a fusion plasma, several options are available. Neutral beam injection (NBI) injects high-energy neutral particle beams into the plasma. Ohmic heating utilizes the toroidal current that flowsin a tokamak to ohmically heat the plasma. Radio frequency (RF) heating emits radio frequency wavesinto the plasma to heat the plasma via several natural resonant frequencies present in the plasma.RF-heating can be performed in several ways and the work of this project has been within the areaof ion cyclotron resonance heating (ICRH) [8]. ICRH utilizes the natural cyclotron frequency of theions to heat the plasma. The natural cyclotron frequency is f = qB/2⇡m where q is the charge, B isthe magnitude of the magnetic field and m is the mass. The left-hand circularly polarized electric fieldrotates in resonance with the gyro motion and accelerates the particles.

RF-heating simulations are necessary to provide an understanding of the physics and mechanismsinvolved, prior to designing and operating a fusion plasma confinement configuration. Several RF-codes have been produced such as the EVE code [9] which utilizes a variational approach to simulateRF-heating and the AORSA code [10], which is a spectral algorithm. There is also the LION code[11], the LEMan code [12] and the 3D LEMan code [13], as well as the ANTITER code [14] and theTOPICA code [15].

This project has been working with an alternative approach to model the wave fields. Finite ElementModel for ICRH (FEMIC) is a wave code for simulating RF-heating of fusion plasmas. It uses the finiteelement method (FEM) to solve the wave equation for the electric field. The pre-processing is done inMATLAB® where all the necessary quantities such as the dielectric tensor elements, the temperatureand density profiles are calculated. The wave equation is then solved in COMSOL Multiphysics® andthe result sent back to MATLAB® for post-processing and analysis [16]. Currently, FEMIC neglectsthe presence of a poloidal magnetic field in its simulations.

1.1 Project Goals

The project goals for the work of this master’s thesis are to

• Add a poloidal magnetic field component to FEMIC

• Implement a generic rotation of the dielectric response tensor in FEMIC

• Develop and run suitable tests to ensure

– The correctness of the implemented rotation

– That FEMIC is still converging with the implemented rotation

• Run simulations of ICRH scenarios in the JET and ITER tokamaks

• Analyze the impact of a poloidal magnetic field

– On the resonances

– On the electric wave fields

– On the power absorptions

– Examine the JET and ITER simulations with the goal of quantifying the bending of thewavefront

• Benchmark the obtained results against suitable, available previous work within the researcharea.

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2 Theory & Background

An overview of all the specific areas necessary to understand the project is presented below. Eacharea is given a brief introduction, enough for conceptualization. The wave equation for the electricfield has been solved in all ICRH simulations and the response of the electric field to currents in theplasma is governed by the dielectric tensor. In FEMIC, two coordinate systems are used: the Stixcoordinate system and the cylindrical coordinate system. A coordinate transformation between thetwo is necessary and can be e�ciently respresented via Euler angles. All of the above concepts arepresented in this section, together with the dispersion relation and a geometry overview. The dispersionrelation is necessary for calculating the elements of the dielectric tensor as well as the Euler angles.

2.1 The Wave Equation

To derive the electromagnetic wave equation, we start with Maxwell’s equations

r⇥B = µ0J+1

c2@E

@t. (1)

r⇥E = �@B

@t. (2)

r ·B = 0. (3)

r ·E =⇢

✏0. (4)

By taking the time derivative of (1), inserting (2), taking the Fourier transform in time and using ageneralized Ohm’s law J = � · E (where � is the conductivity), one obtains the electromagnetic waveequation

r⇥r⇥E� !2

c2K ·E = i!µ0Jant, (5)

where ! is the wave frequency, Jant the antenna current and K the dielectric tensor given by

K = I+i�

!✏0.

2.2 The Dielectric Tensor

A magnetized plasma is an anisotropic medium [17], so the current response to an electric field isdescribed by a tensor (rather than by a scalar as in an isotropic medium). The plasma is also gyrotropic

which means the particles can move freely in the direction parallel to the magnetic field, but areconstrained in the perpendicular directions due to the gyration around the magnetic field lines. This isbecause charged particles moving in a magnetic field will rotate around the field lines with an angularfrequency equal to the cyclotron frequency ⌦ = qB/m. The properties of the plasma are heavilydi↵erent in the direction parallel to the magnetic field lines (||) compared to perpendicular (?). Onecan model the plasma using either a cold plasma model or a hot plasma model depending on need. Ina cold plasma model, the particles are assumed to have no thermal motion and zero Larmor radius. Ina hot plasma (necessary for ICRH-modelling) both are included. A general anisotropic tensor can beexpressed as

K =

2

4Kxx Kxy Kxz

Kyx Kyy Kyz

Kzx Kzy Kzz

3

5 . (6)

Using the Stix coordinate system (see section 2.4.2), with B||z and k?y, the dielectric tensor for a hotplasma is given by [18]

K =

2

4K1 K2 K4

�K2 K1 +K0 �K5

K4 K5 K3

3

5 (7)

where

6

K0 = 2X

j

!2pje

��j

!kkvkj

1X

n=�1�j(In � I 0n)

✓1�

kkv0j!

◆Z(⇣nj)+

+kkvkj!

✓1� T?j

Tkj

◆Z 0(⇣nj)

2

� . (8)

K1 = 1 +X

j

!2pje

��j

!kkvkj

1X

n=�1

n2In�j

✓1�

kkv0j!

◆Z(⇣nj)+

+kkvkj!

✓1� T?j

Tkj

◆Z 0(⇣nj)

2

� . (9)

K2 = iX

j

✏j!2pje

��j

!kkvkj

1X

n=�1n(In � I 0n)

✓1�

kkv0j!

◆Z(⇣nj)+

+kkvkj!

✓1� T?j

Tkj

◆Z 0(⇣nj)

2

� . (10)

K3 = 1�X

j

!2pje

��j

!kkvkj

1X

n=�1In

✓! + n⌦j

kkvkj

◆⇥

⇥⇢

1 +n⌦j

!

✓1�

Tkj

T?j

◆�Z 0(⇣nj) +

2n⌦jTkjv0j!T?jvkj

Z(⇣nj) +

kkvkj! + n⌦j

��. (11)

K4 =X

j

k?!2pje

��j

kk!⌦j

1X

n=�1

nIn�j

⇢n⌦jv0j!vkj

Z(⇣nj)+

+

T?j

Tkj� n⌦j

!

✓1� T?j

Tkj

◆�Z 0(⇣nj)

2

� . (12)

K5 = iX

j

k?✏j!2pje

��j

kk!⌦j

1X

n=�1(In � I 0n)

⇢n⌦jv0j!vkj

Z(⇣nj)+

+

T?j

Tkj� n⌦j

!

✓1� T?j

Tkj

◆�Z 0(⇣nj)

2

� . (13)

Here j indicates the particle species, !pj and ⌦j are the plasma and cyclotron frequencies respectively,k|| is the wave number parallel to the magnetic field, k? is the wave number perpendicular to themagnetic field, v||j is the parallel thermal velocity, v?j is the perpendicular velocity, v0j is the netparallel flow velocity, T||j is the parallel temperature, T?j is the perpendicular temperature and ✏j isthe sign of charge. Lastly, In = In(�j) is the nth order modified Bessel function evaluated at �j =k2?v

2?/�2⌦2

j

�and Z(⇣nj) is the plasma dispersion function evaluated at ⇣nj = (!+n⌦j�k||v0j)/(k||v||j).

Prime denotes derivative. The plasma dispersion function is defined by [18]

Z(⇣) ⌘ 1p⇡

Z 1

�1

e�⇠2

⇠ � ⇣d⇠. (14)

K2 decribes the gyrotropy of the plasma, while K0 and the nondiagonal terms K4 and K5 come fromthe fact that the plasma is hot.

2.3 Ion Cyclotron Resonance Heating (ICRH)

One successful method of heating fusion plasmas is via ion cyclotron resonance heating. Radio frequencywaves are launched from an antenna on the reactor wall and propagate into the center of the plasma.At resonance, the particles gain kinetic energy via acceleration from the electric field rotating with theions. However, to obtain heating, the wave field needs to have a left-hand polarized component, samedirection as the gyro motion of the ions. The resonance condition is given by

! = n⌦i + v||k|| (15)

where n = 0, 1, 2, ... is the harmonic number, ⌦i is the cyclotron frequency and v||k|| is the Doppler shift.v|| is the particle velocity parallel to the magnetic field. If n = 0, then we have Landau damping. n = 1

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is called the fundamental cyclotron resonance. n = 2 corresponds to the second harmonic cyclotron

resonance [8].Now, the first naıve choice would be to choose ! = ⌦i + v||k||. However, at this frequency the

left-hand polarization is zero. This can be shown by employing cold plasma model and utilize the ratioof the left-hand polarized wave (E+) to the right-hand polarized wave (E�)

E+

E�=

! � ⌦i

! + ⌦i. (16)

The Doppler shift is neglected in a cold plasma model, so the wave would therefore be completelyright-hand polarized, opposite to what we want [8]. There are two possible ways to circumvent this.First, one might use the second harmonic n = 2 instead, which gives (neglecting Doppler shifts)

E+

E�=

1

3(17)

It would also be possible to introduce a minority species with a di↵erent cyclotron frequency than thebulk ions. This way the bulk ions would be heated indirectly through collisions [8].

2.4 Coordinate Systems

This project has been heavily focused on coordinate transformation between the cylindrical and Stixcoordinate systems. The Stix coordinate system follows the magnetic field lines and greatly simplifiesthe expressions of the dielectric tensor. In the purely toroidal case, the magnetic field is aligned withthe toroidal coordinate and the coordinate transformation becomes relatively simple. Adding a poloidalfield requires a non-trivial coordinate transformation.

2.4.1 The Cylindrical Coordinate System

Figure 2: The cylindrical coordinate system

The familiar cylindrical coordinate system constitutes the radial unit vector R, the azimuthal unitvector ✓ and the vertical unit vector Z.

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2.4.2 The Stix Coordinate System

Figure 3: The Stix coordinate system

The Stix coordinate system follows the magnetic field lines. The z axis points along the total magneticfield lines. y ||k ⇥ B where k is the electric field wave vector. x is simply the cross product betweenthe two.

2.5 Coordinate Transformations

To transform between two coordinate systems, a rotation matrix is required. Consider two coordinatesystems with base vectors ei and e0i. The general rotation element for the rotation matrix R can thenbe written as

ei · e0j = Rij .

Note that the i’s and j’s here are not referring to the three di↵erent components of a vector, butto di↵erent vectors (nine di↵erent dot products in total). The general rotation matrix between thecylindrical and Stix coordinate systems can thus be written as

R =

2

4ex · eR ex · e' ex · eZey · eR ey · e' ey · eZez · eR ez · e' ez · eZ

3

5 . (18)

The coordinate transformations of the electric field and dielectric tensor are written as

EStix = REcyl.

KStix = RKcylRT .

For any n-dimensional rotation matrix R acting on Rn we have R�1 = RT . That is, the rotation isan orthogonal matrix.

2.6 Euler Angles

Any rotation matrix R can be converted into three consecutive rotation matrices each with theircorresponing Euler angles

R = Rz(↵)Ry(�)Rx(�) (19)

where the subscript indicates a principal axis [19]. To clarify, perform the first rotation Rx(�) aroundthe x-axis. In the new intermediate coordinate system, perform the rotation Ry(�) around the (inter-mediate) y-axis. Finally, in the second intermediate coordinate system, perform the rotation Rz(↵)around the (second intermediate) z-axis.

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A generic rotation matrix can be expressed in Euler angles as

R =

2

4cos� cos↵ sin � sin� cos↵� cos � sin↵ cos � sin� cos↵+ sin � sin↵cos� sin↵ sin � sin� sin↵+ cos � cos↵ cos � sin� sin↵� sin � cos↵� sin� sin � cos� cos � cos�

3

5 (20)

Thus any rotation matrix can be distilled into three Euler angles for minimized memory requirement.

2.7 The Dispersion Relation

The wave vector of the electric field can be divided into a component parallel to the magnetic field anda componenet perpendicular to the magnetic field. In a tokamak configuration with only a toroidalmagnetic field present, they can be approximated as follows

k|| =n�

R(21)

k2? =!2

c2

K1 +K0 � n2

|| +K2

2

K1 � n2||

!(22)

where n|| = ck||/! is the refractive index in the direction parallel to the magnetic field, and K0, K1 andK2 are components of the dielectric tensor, as discussed in section 2.2. Equation (22) is the dispersionrelation for the so called fast magnetosonic wave. An example of its shape for JET can be observed inFigure 4.

Figure 4: The fast wave dispersion relation for a cold plasma, plotted in the equatorial plane of JET.Deuterium plasma with a small hydrogen concentration of 4 %.

In this project, when a poloidal magnetic field is added to the simulations, it is assumed that

k|| = k�|B�||B| (23)

in place of equation (21). The actual expression for k|| in a fusion plasma with the presence of both atoroidal and a poloidal magnetic field is more complex.

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2.8 Geometry

To heat a fusion plasma in a tokamak via ICRH, RF-waves are emitted by an antenna located onthe so called low-field side of the tokamak. The plasma is confined via the helical magnetic field andthe confinement extends only out to the separatrix. Outside the separatrix there is approximately avacuum and it is called the scrape-o↵ layer (SOL).

In this project, the models used to perform the FEMIC simulations are assumed to be toroidallysymmetric. The simulation quantities (such as the dielectric tensor and the electric field) have com-ponents in three dimensions. Because FEMIC only simulates the electric wave field propagation for asingle toroidal mode number n�, only a 2D cross-section is necessary for visualization and simulation.

In Figure 5 and Figure 6, realistic drawings and the 2D cross-sections upon which the simulationswere performed are shown for JET and ITER respectively. The models used were the geometriesprovided in the FEMIC code.

Figure 5: Left: A realistic drawing of JET. Source: [20]. Right: A 2D cross-section of JET upon whichthe FEMIC simulations are performed. The antenna is visible on the low-field side (far right).

Figure 6: Left: A realistic drawing of ITER. Source: [21]. Right: A 2D cross-section of ITER uponwhich the FEMIC simulations are performed. The antenna is visible on the low-field side (far right).

In the SOL, the waves are evanescent which means they have to tunnel from the antenna to theplasma. The SOL is often modelled as a vacuum. However, it can also be modelled using the dieletrictensor. In this case, the fast-wave dispersion relation needs to be solved in the SOL. The fast-wavedispersion relation in FEMIC is not as reliable outside the plasma and, in addition, the dielectrictensor is hard to resolve. Modeling the SOL with a dielectric tensor treatment requires handelingvarying refractive index, possible resonances and mode conversion [22].

In this project, due to all the increased complexity involed when introducing a poloidal field (requir-ing high resolution and precise transformation), the SOL was modeled with the vacuum permittivityand not the dielectric tensor. That is, instead of the dielectric tensor K, Kij = �ij (Kronecker delta)was used.

11

3 Implementation in FEMIC

The addition of a poloidal magnetic field to FEMIC, the necessary coordinate transformations, theprecautionary measures needed to ensure the implementation was correct and the quantification of thepoloidal e↵ect all incorporates many steps and substeps. An overview of the process and its main partsare presented in this section.

3.1 Generic Magnetic Field Implementation

A generic helical magnetic field was implemented using the equilibrium data given in FEMIC. Thetotal magnetic field can be calculated as

Btot =q

B2� +B2

✓ (24)

where B� is the toroidal magnetic field and B✓ is the poloidal magnetic field. The poloidal field wascalculated from equilibrium data as

B✓ =qB2

R +B2Z

where BR and BZ are given as equilibrium data. In figure 7, an example of the di↵erence betweenthe total magnetic field being used by FEMIC before and after this project implementation can beobserved.

Figure 7: Left: The solely toroidal magnetic field with a 1/R dependence. Right: The more complexhelical magnetic field implemented during this project. The white line is the separatrix of the plasma.The magnetic fields are calculated from JET equilibrium data.

In figure 8, the toroidal and poloidal magnetic fields can be observed, as filled contours and arrowsrespectively. The center point of the plasma where the poloidal field tends to zero is called the magneticaxis, preferably labelled by (R0, Z0). The area of the plasma where R > R0 is called the low-field sideof the plasma, and the area where R < R0 is called the high-field side.

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Figure 8: B� and B✓ plotted as filled contours and arrows respectively. The magnetic fields arecalculated from JET equilibrium data. The negative values are due to convention in the equilibriumfiles. The equilibrium data used was the same as in Vallejos et al [16]

3.2 The Rotation Matrix

In this section, the algorithm used to compute the rotation matrix between the Stix coordinate systemand the cylindrical coordinate system is presented. To calculate the rotation matrix between thecylindrical and Stix coordinate systems (see equation (18)), the unit vectors from both coordinatesystems (expressed in cylindrical coordinates) need to be calculated. The cylindrical unit vectors aretrivial

eR = [1, 0, 0]

e� = [0, 1, 0]

eZ = [0, 0, 1].

The Stix unit vector ez can be directly calculated via the magnetic field amplitudes

ez =1q

B2R +B2

� +B2Z

[BR, B�, BZ ]. (25)

The Stix unit vector ex is calculated via a Graham-Schmidt process. However, the wave vector

k = [kR, k�, kZ ]. (26)

must be obtained first. The toroidal wave vector component k� is simply equal to the purely toroidalcase: k� = n�/R. To acquire the radial and vertical wave vector components, the poloidal wave vectorcomponent k✓ defined as

|k✓|2 = |kR|2 + |kZ |2

needs be acquired first. kR and kZ are then be calculated via a predetermined angle ✓k as

kR = k✓ cos ✓k (27)

kZ = k✓ sin ✓k. (28)

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We have that|k?| = |k� (k · ez) ez|. (29)

Note that ✓k is chosen. This is to be able to solve equation (29) for k✓. If ✓k was not chosen, thenequation (29) would have two unknowns. ✓k is the angle between the vertical and radial wave vectorin the poloidal plane. That is tan ✓k = kZ/kR. Using equation (25) and (26), as well as after someextensive algebra, the following second order polynomial equation in k✓ from (29) can be obtained as

0 =�C2

1 + C22 + C2

3

�k2✓ � 2k� (C7 + C8 + C9) k✓ + k�

�C2

4 + C25 + C2

6

�� |k?|2 (30)

where

C1 = cos ✓k � B2R cos ✓k|B|2 � BZBR sin ✓k

|B|2 ,

C2 = �BRB� cos ✓k|B|2 � BZB� sin ✓k

|B|2 ,

C3 = sin ✓k � BRBZ cos ✓k|B|2 � B2

Z sin ✓k|B|2 ,

C4 =B�BR

|B|2 ,

C5 =

B2

�

|B|2 � 1

!,

C6 =B�BZ

|B|2 ,

C7 = C1C4,

C8 = C2C5,

C9 = C3C6.

Since k✓ is the length of the wave vector in the poloidal plane, the algorithm was designed to alwayspick the positive of the two roots at every point (r, z). Note that |k?|2 is obtained from (22). Using(27) and (28), the complete wave vector can then be obtained as

k = [ k✓ cos ✓k, k�, k✓ sin ✓k ].

A new parallel wave vector component k|| is then calculated as k|| = k · ez. The perpendicular wavevector is then given by the Graham-Schmidt operation

k? = k� k||ez.

The Stix unit vector ex is given by

ex =k?|k?|

The Stix unit vector ey is simply the cross-product between the other two unit vectors

ey = ez ⇥ ex

With all the unit vectors for the two coordinate systems, the rotation matrix can then be calculatedwith (18). Without a poloidal field, the rotation matrix is simply

R =

2

41 0 00 0 10 �1 0

3

5 .

This simple form leads to transformations where the quantities in one coordinate system only dependson one corresponding quantitiy in the other system. To illustrate, without a poloidal field the cylindricaldieletric tensor component KRR is simply

KRR = Kxx. (31)

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With a poloidal field however, all of the elements of the rotation matrix are non-zero and a completecoordinate transformation is required

KRR = R11 ⇤ (Kxx ⇤R11 +Kxy ⇤R21 +Kxz ⇤R31)

+R21 ⇤ (Kyx ⇤R11 +Kyy ⇤R21 +Kyz ⇤R31)

+R31 ⇤ (Kzx ⇤R11 +Kzy ⇤R21 +Kzz ⇤R31)

(32)

where Rij is the corresponding rotation matrix element.

3.3 Validity

The validity of the implementations was checked via numerous tests. One of the tests performed toensure the correctness of the implemented rotation matrix was to validate the limit when the poloidalmagnetic field tends to zero. This is illustrated in figure 9 for the magnetic axis limit where this is thecase.

To ensure that the calculated Stix unit vectors were correct, their orthonormality was validated bytaking the dot product between the unit vectors to confirm that the results were approximately equalto zero for all points. Sanity checks were also carried out to ensure that the implementation of therotation matrix was correct, by validating that the determinant of the rotation matrix was equal to1, | detR| = 1. The errors for the orthonormality of the Stix unit vectors and the determinant of therotation matrix were both within machine precision.

The angles between the cylindrical and Stix unit vector have been carefully studied, to ensure smalldeviations from the non-poloidal case, as can be observed in Figure 10. In the purely toroidal case, wehave cos�1(ex · eR) = 0°, cos�1(ey · eZ) = 0° and cos�1(ez · e�) = 180°.

In addition, to further validate the correctness of the implementation, FEMIC simulations were runwithout a poloidal field to ensure the results from the non-poloidal case were reproduced.

Figure 9: The non-poloidal rotation matrix was recovered in the non-polidal limit of the implementedgeneric rotation matrix.

15

Figure 10: The angles between the cylindrical and Stix unit vectors. The deviations from the purelytoroidal case are as expected.

3.4 Mesh

As will be further elaborated upon in the discussion section, running simulations with FEMIC witha poloidal field requires a much finer mesh size compared to the purely toroidal case. This is due tothe large di↵erence in magnitude between the dielectric tensor elements, originating from the highlyanisotropic nature of the plasma. For example, in a typical fusionplasma |Kxx| ⇠ 1000 and |Kyy| ⇠ 1000but |Kzz| ⇠ 106. Therefore, performing the full tensor rotations such as equation (32) without a well-resolved mesh will result in inaccurate rotated tensor elements.

Figure 11: Mesh size of 0.5 cm in plasma, antenna and separatrix. 1 cm in SOL. To enhance the qualityof the results, a relatively large amount of boundary layer elements were used on the plasma side ofthe separatrix. JET.

3.5 Quantification of Poloidal Bending E↵ect

There are many ways in which it is possible to quantify the bending of the electric field wavefront as aresult of a poloidal magnetic field. The method of choice for this project was as follows. The quantityof interest was chosen to be the norm of the electric wave field |E|, since this quantity incorporates allthe electric field componenets into one.

16

The Z-position of the mean of |E| was then calculated as follows for a specific R-coordinate

< Z >=

R Zmax

Z=Zmin|ER,Z | · Z

R Zmax

Z=Zmin|ER,Z |

. (33)

The process can be seen as integrating along the white lines depicted in Figure 12. To quantify thee↵ect, the following measure was used

� < Z >=< Z >��R=l

� < Z >��R=r

.

where (l, r) = (2.5, 3.5) and (l, r) = (5, 7) for JET and ITER respectively.

Figure 12: Schematic depiction of the R-columns with which 33 was calculated.

4 Results

The impacts of a poloidal magnetic field on the ion cyclotron resonance layers, the electric wave fieldsand the power absorption have been analyzed. Simulations were performed with and without a poloidalmagnetic field respectively, with all other input parameters held constant. This enabled us to observethe impact of the poloidal magnetic field on the RF-heating simulations. The poloidal magnetic fieldstrength used were |B✓|/|B�| ⇠ 0.1 for both the JET and ITER RF-heating simulations. When addinga poloidal magnetic field to the FEMIC RF-heating simulations, the ion cyclotron resonance layerswere curved compared to the purely toroidal magnetic field case where they were straight, vertical lines(given a vacuum dependence of B ⇠ 1/R). JET and ITER simulations have been examined and theelectric field wavefront bends as a result of the addition of a poloidal magnetic field. The poloidalbending e↵ect of the electric field wavefront has been quantized for RF-heating simulations of the twotokamaks with the measure described in section 3.5. The poloidal bending e↵ect is most prominent onthe high-field side of the tokamak. With the addition of a poloidal magnetic field, the power absorptionsshifted in favor of the ions at the expense of the electrons and the total absorbed power by the plasmadecreased for RF-heating simulations of both JET and ITER.

Furthermore, with the addition of a poloidal magnetic field it was found that the resulting electricwave fields from RF-heating simulations of JET and ITER experienced numerical noise. The numericalnoise was found to seemingly decrease exponentially with the mesh size of the plasma.

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Lastly, the project results were benchmarked against Bilato et al [1]. The quantities chosen tobenchmark were the cyclotron resonances, the normalized absorped power density and the absorbedpower partition as a function of helium-3 concentration. This was because these quantities give com-prehensive information about the results from a RF-heating simulation.

4.1 Ion Cyclotron Resonance Layers

RF-heating simulations were performed with FEMIC with and without a poloidal magnetic field forthe JET and ITER tokamaks. The impact of a poloidal magnetic field on the ion cyclotron resonancelayers was analyzed. The input parameters used for the RF-heating simulations of JET were set asin Table 1. The input parameters for the RF-heating simulations of ITER were set as in Table 2.When including a poloidal magnetic field, the ion cyclotron resonance layers (! = n⌦i + v||k|| where⌦ = qB/m) are expected to curve. This curving e↵ect can be observed for quantities dependent onthe magnitude of the total magnetic field B such as the dispersion relation and the dielectric tensorelements. This e↵ect was observed for both the JET and ITER RF-heating simulations. In Figure 13,the curving e↵ect can be observed for the real part of the Kxx element of the dielectric tensor for theJET tokamak. This quantity was chosen for depiction due to the curving e↵ect being clearly visible.

Figure 13: The resonance curvature e↵ect observed for the real part of the dieletric tensor elementKxx. Left: No poloidal field. Right: With poloidal field. JET simulation with on-axis magnetic fieldstrength of B0 = 2.7T

4.2 The Electric Wave Fields

For the RF-heating simulations of the JET and ITER tokamaks, the impact of a poloidal magneticfield on the electric wave field was analyzed. It was found that the electric wave field experiences abending e↵ect of the wavefront as a result of the addition of a poloidal magnetic field. The shape of theelectric wave field components was also a↵ected for RF-heating simulations of both JET and ITER.Furthermore, it was found that the addition of a poloidal magnetic field decreases the coupling of theelectric wave field to the plasma.

4.2.1 JET

For JET, the impact of a poloidal magnetic field on the electric wave field was analyzed for RF-heatingsimulations. The input parameters of interest can be seen in Table 1. The JET equilibrium wasthe same as the JET equilibrium used in Vallejos et al [16]. The electric wave field results are forsimulations of RF-heating in JET with the ITER-like antenna (ILA). In Figure 14, the e↵ect of thepoloidal magnetic field can be observed. The wavefront is shifted clockwise. The bending e↵ect canbe observed by comparing the high-field side of the tokamak for the cases with and without a poloidalmagnetic field. Note how the shape is also a↵ected on the low-field of the tokamak. The surfaceexcitations of E+, visible on the top of the high-field side of the plasma, are more distinct in thenon-poloidal case. In Figure 15, in addition to the e↵ects previously described, we can also observe

18

how a high amplitude of the electric field extends further around the plasma outside the separatrix onthe low-field side of the tokamak compared to the purely toroidal case.

Table 1: The input parameters used to obtain figures 14 - 15.

Parameter Value UnitFreq. 41.1 MHzn� 27 –B0 2.6996 TR0 3.0509 mSOL Vacuum –

Equilibrium file JET EQ 92436 50.3s.mat –Plasma D(H) X[H] = 4 %

Temperature on-axis 9 keVDensity on-axis 8 · 1019 m�3

Triangular mesh elements 2148775 –Quadrilateral mesh elements 44618 –

Figure 14: The absolute value of E+. Left: Without B✓. Right: With B✓.

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Figure 15: The absolute value of E||. Left: Without B✓. Right: With B✓.

4.2.2 ITER

For ITER, the impact of a poloidal magnetic field on the electric wave field was analyzed analogouslyto the analysis performed for the electric wave field for JET in section 4.2.1. For ITER, the inputparameters of interest can be seen in Table 2.

Table 2: The input parameters used to obtain Figure 16.

Parameter Value UnitFreq. 52.5 MHzn� 27 –B0 5.3 TR0 6.417 mSOL Vacuum –

Equilibrium file EQUI ITERscenario2 sim2Bilato2013 benchmarkSmooth scaled new.mat –Plasma (D,T,He3) = (48,46,3) %

Temperature on-axis (ions) 21 keVTemperature on-axis (electrons) 24.8 keV

Density on-axis 10.3 · 1019 m�3

Triangular mesh elements 2680658 –Quadrilateral mesh elements 50932 –

The equilibrium data used for the ITER simulations was obtained from [1]. The original equilibriumdata was modified to include a radial magnetic field BR and a vertical magnetic field BZ and rescaledto have the magnetic field strength on-axis as B0 = 5.3T. In addition, only the three bottom antennaswere used for the ITER simulations. This was done in order to have the input parameters and setupfully match the setup used in [1] with which to benchmark the obtained project results. In Figure16, the shape of the electric field can be seen a↵ected by the addition of a poloidal magnetic field.For ITER, the electric field also experienced a bending e↵ect as a result of the addition of a poloidalmagnetic field, similarly to the bending of the electric field wavefront in JET presented in section 4.2.1.

20

Figure 16: The real part of E+. Left: Without B✓. Right: With B✓.

4.3 Power Absorption

The impact of a poloidal magnetic field on the power absorption was analyzed for RF-heating simula-tions of JET and ITER. The analysis was performed analogously to the analyses used to obtain theresults in section 4.2.1 and 4.2.2, namely the RF-heating simulations were performed with and withouta poloidal magnetic field respectively with all other input parameters held constant as in Table 1 andTable 2. The obtained results were compared and analyzed. The addition of a poloidal magnetic fieldshould alter the total absorbed power to the plasma as well as the partition of the absorbed powerbetween particle species. This was indeed observed for the RF-heating simulations of both JET andITER. The total absorbed power by the plasma and the power partition are calculated by FEMIC inpost-processing after each RF-heating simulation.

4.3.1 JET

For RF-heating simulations of JET, the power partition and the total absorbed power to the plasmachanged as a result of the addition of a poloidal magnetic field. The results are based on a unit currentdensity in the antenna straps of 1 W/m. As can be observed in Table 3, the power partition shiftedin favor of the ions at the expense of the electrons. The total absorbed power to the plasma can beobserved to decrease with the addition of a poloidal magnetic field.

Table 3: The power partition and total absorbed power for FEMIC JET simulation with and withoutB✓. Parameters as in Table 1.

Particle species (on-axis density) Hydrogen, H (4%) Deuterium, D (96%) Electrons, e (100%) Ptot [W]Without B✓ 39% 24% 37% 6.62With B✓ 42% 24% 34% 5.92Di↵erence 3% 0% �3% �0.7

4.3.2 ITER

For RF-heating simulations of ITER, the impact of a poloidal magnetic field on the power absorptionwas similar to the results obtained for JET presented in section 4.3.1. As for JET, these results arebased on a unit current density in the antenna straps of 1 W/m.

21

Table 4: The power partition and total absorbed power for FEMIC ITER simulation with and withoutB✓. Parameters as in Table 2.

Particle species(on-axis density)

Tritium, T(46%)

Deuterium, D(48%)

Helium-3, He3(3%)

Electrons, e(100%)

Ptot [W]

Without B✓ 6.98% 0.02% 58.3% 34.7% 15.52With B✓ 7.19% 0.02% 59.18% 33.61% 14.76Di↵ 0.21% 0% 0.88% �1.09% �0.76

4.4 Quantification of Poloidal Bending E↵ect

The JET and ITER RF-heating simulations were examined and the poloidal bending e↵ect was quan-tified with the method described in section 3.5. For JET, the plots obtained can be seen in Figure 17.The bending e↵ect of the poloidal field can clearly be observed. Note that the impact of a poloidalmagnetic field is an increase in the Z-coordinate of the mean of |E| for R / R0 and a decrease in theZ-coordinate of the mean of |E| for R ' R0.

Figure 17: The Z-coordinate for the mean of |E|, equation (33), plotted for JET with and without B✓.

Without a poloidal field, the bending factor � < Z > was calculated to be 0.15. With a poloidal field,the bending factor � < Z > was calculated to be 0.34.

The process of quantification for the poloidal bending e↵ect was analogously performed for ITER.However, in the equilibrium files used to calculate the poloidal magnetic field in ITER, the orientation ofthe poloidal magnetic field is clockwise (as opposed to JET, were it was counter-clockwise). This shouldresult in a poloidal bending e↵ect in the opposite direction. The resulting plots for the quantificationof the poloidal bending e↵ect for ITER can be observed in Figure 18. Without a poloidal field, thebending factor � < Z > was calculated to be 0.57. With a poloidal field, the bending factor � < Z >was calculated to be 0.40.

22

Figure 18: The Z-coordinate for the mean of |E|, equation (33), plotted for ITER with and withoutB✓.

4.5 Numerical Precision

It was found that solving the wave equation with a coarse mesh grid would result in inaccurate andincorrect solutions. This problem only emerged when a poloidal field was applied. It manifested asnumerical noise, depicted in Figure 19.

By performing a 1D cutline on the most heavily a↵ected electric field component, E||, it was possibleto investigate how this noise scaled with the mesh size of the plasma. As can be seen in Figure 20, amesh size of 0.02 m results in heavy noise in the solution.

Figure 19: Electric field solution a↵ected by noise. Presented for illustrative purposes only.

23

Figure 20: 1D cutline from JET simulation with a mesh size of 0.02 m. Red line is 100 point movingaverage. Green lines are standard deviation from that average. The standard deviation was approxi-mately 20 %.

As can be seen in Figure 20, the noise increases closer to the separatrix. It was found that the noiseis greatly reduced when a finer mesh size is applied. This can be observed in Figure 21, where a 1Dcutline has been performed for a mesh size of 0.0025 m.

Figure 21: 1D cutline from JET simulation with a meshsize of 0.0025 m. Red line is 100 pointmoving average. Green lines are standard deviation from that average. The standard deviation wasapproximately 4 %.

24

It was found that to acquire accurate and satisfactory results with a poloidal magnetic field present, themesh size within the plasma, as well as at the separatrix, was required to be of subcentimeter size. Byinvestigating the solution for various plasma mesh sizes, it was found that the noise seems to decreaseexponentially with mesh size as can be seen in Figure 22.

Figure 22: The normalized L2-norm of the noise suggests an exponential decrease with plasma meshsize.

4.6 Benchmark

The obtained results were benchmarked. As intended, this was done for ITER with the data in Bilato etal [1]. A depiction of the cyclotron resonances can be observed in Figure 23. These were benchmarkedagainst [1] with good agreement.

Figure 23: A depiction of the cyclotron resonances obtained with FEMIC after the poloidal magneticfield implementation.

25

In addition, the absorbed power density for the di↵erent particle species was benchmarked against [1].The acquired normalized absorbed power density can be observed in Figure 24. In Figure 24, there isa sharp peak at ⇢tor ⇡ 0.04. At ⇢tor ⇡ 0.06 there is a small peak and at ⇢tor > 0.3 the normalizedabsorbed power density is very low. Note the sharp decrease in normalized absorbed power densitytowards ⇢tor = 0. This is because there is absorption just below the magnetic axis but not exactlyon-axis. Note also that the plotted quantity is power density and that the volume increases with ⇢tor.The electrons absorb power closer to the separatrix compared to the ions. The ions absorb powermainly near the magnetic axis. Hence, the absorbed power density for the electrons is larger than theaborbed power density for the ions for ⇢tor ' 0.2.

Figure 24: The normalized absorbed power density per species for ITER with B✓ with input parametersas in Table 2. T=tritium, He3=helium-3 and e=electron. ⇢tor is a generalized radius with ⇢tor = 0corresponding to the magnetic axis and ⇢tor = 1 corresponding to the separatrix.

A final measure used to benchmark the project results was the dependence of the absorbed powerpartition on the helium-3 concentration with all other input parameters held constant as in Table 2.The plot can be observed in Figure 25. The aborbed power can be seen peaking for He3 at X[He3] = 3%.This is in agreement with the EVE [9] code and the LION [11] code. The minimum for the electronscan be observed at X[He3] = 2%. The post-peak intersection of the absorbed power for He3 and theelectrons can be observed at X[He3] ⇡ 5.4%. Note also that at X[He3] = 0% the aborbed power forthe electrons is higher than the absorbed power for tritium, T. This is in agreement with the LION[11] but not the EVE [9] code. For X[He3] ' 1%, FEMIC is in agreement with both the EVE code [9]and the LION code [11].

26

Figure 25: The absorbed power partition plotted against He3 concentration. T=tritium, He3=helium-3and e=electron. ITER. Other input parameters held constant as in Table 2.

With Figure 25, the di↵erence compared to the purely toroidal magnetic field ([16]) case is as follows.With the addition of a poloidal magnetic field, the absorbed power for the electrons is lower for allX[He3] and the absorbed power for Helium-3 is higher for all X[He3] (except for X[He3] = 0%). AtX[He3] = 0%, the absorbed power for tritium, T is slighly larger with a poloidal magnetic field. Theminimum of the absorbed power for the electrons is more distinct at X[He3] = 2% compared to thepurely toroidal case. With a poloidal magnetic field, the post-peak intersection between the curve forthe absorbed power for the electrons and the curve for the absorbed power for Helium-3 no longeroccurs at X[He3] ⇡ 5% but rather at X[He3] ⇡ 5.4% as stated above.

5 Discussion

5.1 The Electric Wave Fields

The addition of a poloidal magnetic field clearly has an e↵ect on the electric wave field propagationin the FEMIC tokamak simulations. There are two main e↵ects which can be identified. First, theaddition of a poloidal magnetic field causes a bending of the electric field wavefront as can be observedin Figure 14. The e↵ect is most prominent on the high-field side of the tokamak. The e↵ect wasrelatively greater for the JET simulations compared to the ITER simulations. This bending e↵ectwas not observed in the old version of FEMIC [16] without a poloidal magnetic field. The bendinge↵ect observed in the new version of FEMIC with a poloidal magnetic field is similar to the bendinge↵ect observed in R.J. Dumont [9]. A possible explanation for the bending e↵ect is the asymmetryintroduced in the parallel wave number with the addition of a poloidal magnetic field. In the purelytoroidal magnetic field case, the parallel wave number is perfectly symmetric about the magnetic axis(constant). With the addition of a poloidal magnetic field, this symmetry is broken. As can thenbe deduced from the dispersion relation for the fast magnetosonic wave (Eq. 22), this will result inshorter wavelengths above (or below, depending on the orientation of the poloidal magnetic field) themagnetic axis and longer wavelengths below (or above) the magnetic axis. This will cause the electricfield wavefront to bend as a result.

Second, the addition of a poloidal magnetic field will result in a decrease of the coupling of theelectric wave field to the plasma from the SOL, as shown in Figure 15. This decrease in coupling isprobably related to the tilting e↵ect which arises when the antenna straps are no longer fully alignedwith the magnetic field lines compared to the purely toroidal case. With the addition of a poloidalmagnetic field, the electric field produced by the antenna straps are less aligned with the allowed fast

27

magnetosonic wave in the plasma when trying to couple to the plasma. Compared to the purely toroidalcase, a larger portion of the electric field will now be reflected instead of coupling to the plasma andtransmitted. This tilting e↵ect has been discussed in Messiaen et al [23].

5.2 Power Absorption

The power absorptions are a↵ected by the addition of a poloidal magnetic field. A lower partition ofthe absorbed power for the electrons was observed, as can be seen for RF-heating simulations of bothJET and ITER in Table 3 and Table 4 respectively. The addition of a poloidal magnetic field generallydecreases the magnitude of the parallel wave vector k|| on which the electron absorption depends. In thisproject, when a poloidal magnetic field is used in the simulations, it is assumed that k|| = k�|B�|/|B|.In the purely toroidal case we will always have that k|| = k�|B�|/|B| = k�|B�|/|B�| = k�. Withthe addition of a poloidal magnetic field, we will have a decrease in magnitude of k|| from the purelytoroidal magnitude of k� since |B�| |B|, as compared to the purely toroidal case where we have that|B�| = |B|. A decrease in magnitude of k|| would thus explain the lowered power absorption for theelectrons.

With the addition of a poloidal magnetic field, a decrease in the total absorbed power by the plasmawas observed. This can be seen in Table 3 and Table 4. The decrease in total absorbed power can beexplained by the decrease in coupling discussed in section 5.1.

5.3 Numerical Precision

The numerical noise presented in section 4.5 (depicted in Figure 19 with 1D cutlines in Figure 20and Figure 21) is most likely due to the great absolute di↵erence in magnitude between the di↵erentdielectric tensor components. With a poloidal magnetic field, a coordinate transformation of the tensorcomponents to a new coordinate system requires a combination of all the tensor components fromthe old coordinate system. If the mesh size is too large, the tensor components may not be sampledaccurately and the rotations could become inaccurate. This results in numerical noise in the waveequation solved by COMSOL and thus noise in the electric wave field solutions.

In Figure 20 and Figure 21, it can be observed how the numerical noise seems to increase towards theseparatrix. One possible explanation for this is that the strength of the noise increases with the strengthof the poloidal magnetic field. This explanation would quadrate with the fact that no numerical noiseis observed when RF-heating simulations are performed without a poloidal magnetic field.

By comparing Figure 20 and Figure 21 together with Figure 22, it can be deduced that the numericalnoise decreases with the mesh size of the plasma. However, the exponential dependence of the numericalnoise on mesh size of the plasma, as seen in Figure 22, needs further verification. The origin of thedependence is unknown.

It can be noted that all the simulations in this project were solved in a Linux/Ubuntu workstationwith two Intel Xeon E5-2687W v4 @3.00 GHz processors.

5.4 Benchmark

The obtained results for the cyclotron resonances (Figure 23) were found to be in agreement with thebenchmark data in Bilato et al [1].

The acquired absorbed power density (Figure 24) was found to be in good agreement with thecorresponding plots by the LION [11] and the EVE [9] code, with a slightly closer resemblance to theEVE code for ⇢tor > 0.15 and a slightly closer resemblance to the LION code for ⇢tor < 0.15. Thisgood agreement provides validity to the assumption that the implementation completed in this projectis correct.

The benchmarked data for the absorbed power partition as a function of helium-3 concentrationcan be seen in Figure 25. Comparing with the data in Bilato et al [1], the obtained project resultsgive an absorbed power dependence between that of the EVE code [9] and the LION code [11], witha slightly closer resemblance to the EVE code. The resemblance has increased with the addition ofa poloidal magnetic field, compared to the non-poloidal case in the old version of FEMIC [16]. Thisfurther strengthens the validity of the assumption that the implementation is correct. This is becausethe agreement of FEMIC with the EVE code should increase with the addition of a poloidal magneticfield. Furthermore, at zero helium-3 concentration the obtained absorbed power is in agreement withthe LION code [11] and not the EVE code [9]. This may be due to the fact that the EVE codehas implemented measures to account for the downshift of the parallel wave vector number [9] while

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the LION code [11] and FEMIC do not [16]. This could explain why the absorbed power partitionobtained with the new version of the FEMIC code is in agreement with the LION code at zero helium-3concentration and not the EVE code. In the old version of FEMIC, it was found that the obtainedabsorbed power partition was in close agreement with the LION code [16]. In the new version of FEMICwith a poloidal field, the agreement with the EVE code [9] is strengthened. Lastly, the fact that theabsorbed power obtained by FEMIC is in agreement with both the LION code and the EVE code fora helium-3 concentration of 1 % and above indicates that the FLR-e↵ects implemented in FEMIC arein line with the FLR-e↵ects implemented by the LION and EVE codes respectively.

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6 Conclusion

In this master’s thesis, a poloidal magnetic field has been added to the FEMIC code. The electric wavefield experienced a bending of the wavefront, most apparent on the high-field side of the tokamak. Theshape of the electric wave field on the low-field side was also a↵ected. The total absorbed power by theplasma and the coupling to the plasma decreased slightly. This could be due to the fact that with apoloidal field the antenna straps are no longer aligned with the magnetic field lines, causing a smallerportion of the electric field to successfully couple to the plasma.

The power partition between particle species is shifted in favor of the ions at the expense of theelectron power absorption. This is explained by lower magnitude in k|| compared to the case withouta poloidal magnetic field. The change is small compared to the power partition in the purely toroidalcase.

The benchmark of the project results against Bilato et al [1] proved successful. The cyclotronresonances were found to be in good agreement, as well the absorbed power density and the powerpartition versus helium-3 concentration. With the addition of a poloidal field, FEMIC has movedtowards a closer agreement with the EVE code [9]. However, a lack of implementation of the up- anddownshift of the parallel wave number still causes discrepancy.

Furthermore, it was found that simulating RF-heating with a poloidal magnetic field with FEMICrequires a subcentimeter mesh grid to acquire satisfactory results. If the mesh grid is too coarse, thegreat di↵erence in absolute magnitude between the di↵erent dieletric tensor elements inevitably causesthe FEM solver in COMSOL Multiphysics® to produce noisy results. This noise seems to decreaseexponentially with plasma mesh size, however the origin of this particular dependence remains unknown.Future work could verify the exponential dependence and attempt to explain its origin. In addition,a side e↵ect of the need for a finer mesh compared to the purely toroidal case is that RF-heatingsimulations with a poloidal magnetic field with the new version of FEMIC requires longer computationtimes.

With a poloidal magnetic field, the simulations had to be performed with a vacuum model in thescrape-o↵ layer (SOL). This was due to a great increase in complexity when modelling the SOL withthe dielectric tensor. Future work could be to enable RF-heating simulations to be performed withFEMIC without having to use a vacuum model in the SOL, as well as reducing the overall computationtime while maintaining the possibility of using a helical magnetic field. Future work should also focuson implementing an up- and downshift of the parallel wave number and benchmark the results againstBilato et al [1] once again.

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7 References

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[15] V. Lancellotti et al. TOPICA: an accurate and e�cient numerical tool for analysis and design ofICRF antennas. Nuclear Fusion, 46(S476), 2006.

[16] P Vallejos et al. E↵ect of poloidal phasing on ICRH power absorption. Nuclear Fusion, 59(076022),2019.

[17] D.B. Melrose and R.C. McPhedran. Electromagnetic Processes in Dispersive Media. CambridgeUniversity Press, 2005.

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[22] Assoc. Prof. Thomas Jonsson. Conversations throughout the course of this project, 2019. Schoolof Electric Engineering, KTH, Sweden.

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