INOM EXAMENSARBETE TEKNISK FYSIK,AVANCERAD NIVÅ, 30 HP

, STOCKHOLM SVERIGE 2019

3D Finite Element Modelling of ICRH in JET

BJÖRN LJUNGBERG

KTHSKOLAN FÖR ELEKTROTEKNIK OCH DATAVETENSKAP

Abstract

This master’s thesis assesses the possibility of using the finite element method to solve the electromagneticwave equation in a fusion plasma in 3D. In particular, the frequency is chosen to match that of ioncyclotron resonance heating in the fusion experiment JET. In this work, a brief introduction on fusion isgiven, followed by an explanation of the damping process in a plasma. A projection of the 3D wave fieldonto a poloidal plane is compared to the 2D wave field produced by the code FEMIC for validation ofthe developed 3D code. The comparison was done with good results.

The power spectrum and coupling resistance per toroidal mode obtained from the 3D model arealso compared to the corresponding quantities obtained from an analytical slab model. Though somediscrepancies can be seen near the toroidal mode number n = 0 and for higher mode numbers (|n| > 70),the appearance of the power spectra are similar. The difference near n = 0 is attributed to inducedcurrents in the reactor wall, whereas for higher mode numbers, the difference is likely due to bad resolution.The induced currents in the wall causes singularities in the chosen model of the coupling resistance. Thisproduces unreliable predictions of the coupling resistance.

Sammanfattning

Denna masteruppsats utvarderar mojligheten att anvanda finita elementmetoden till att losa den elektro-magnetiska vagekvationen i ett fusionsplasma i 3D. Speciellt valjs frekvensen for att matcha frekvensenfor uppvarmning genom joncyklotronresonans i fusionsexperimentet JET. I detta arbete ges en oversiktligintroduktion till fusion, atfoljd av en forklaring av dampningsprocessen i ett plasma. En projektion av3D-vagfaltet pa ett poloidalt plan jamfors med 2D-vagfaltet producerat av 2D-koden FEMIC for attvalidera den utvecklade 3D-koden. Jamforelsen gjordes med gott resultat.

Effektspektrumet och kopplingsresistansen per toroidal mod fran 3D-modellen jamfors ocksa medmotsvarande storheter fran en analytisk 1D-modell. Trots att vissa skillnader kan ses nara det toroidalamodtalet n = 0 och for hogre modtal (|n| > 70), ar utseendet pa effektspektrumen lika. Skillnaden naran = 0 tillskrivs de inducerade strommarna i reaktorvaggen, medan for hogre modtal beror skillnadentroligen pa dalig upplosning. De inducerade strommarna i vaggen ger upphov till singulariteter i denvalda modellen for kopplingsresistansen. Det resulterar i otillforlitliga varden pa kopplingsresistansen.

1

Contents

1 Introduction 3

2 The Tokamak 52.1 Toroidal Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.2 Confinement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.3 The Flux Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.4 The Antenna . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

3 Wave and Plasma Physics 93.1 The Wave Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93.2 The Dielectric Tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93.3 The Scrape-off Layer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103.4 Absorption . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113.5 The Poynting Vector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

4 Model 144.1 Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144.2 Antenna Current . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154.3 Mesh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154.4 Coordinate Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

5 Results 185.1 The 3D wave field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185.2 Comparison of Wave Fields with FEMIC . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205.3 Comparison of the Power Spectrum with the Analytical Solution . . . . . . . . . . . . . . 205.4 Comparison of the Coupling Resistance with the Analytical Solution . . . . . . . . . . . . 21

6 Discussion 22

7 Conclusion 23

Appendices 25

A Analytical Solution to the Wave Equation 25A.1 Deriving the Wave Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25A.2 Geometry and Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26A.3 Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

B Fast Wave Dispersion Relation 32

2

1 Introduction

Society faces challenges regarding CO2 emissions and it is vital that a transition from fossil to sustainableenergy is made. One alternative to fossil energy that has been proposed by the scientific community isfusion energy. Fusion plasma physics has been researched for half a century, and researchers have persisteddue to the many attractive features of fusion power. For example, the process does not release any CO2,and the induced radioactivity in the reactor decays significantly faster than fission fuel. There is also avirtually endless supply of fuel if we consider today’s energy consumption [1]. However, due to the manycomplexities of fusion plasma physics, progress has been slower than anticipated. There is currentlya large fusion experiment called ITER being built in France, which will hopefully show the way to aresearch reactor DEMO (DEMOnstration Power Station), which in turn is supposed to show that it ispossible to commercialize fusion energy. The DEMO project is still in its planning stage, but ITER iswell under way and is expected to have its first plasma by 2025.

In a fusion process, two lighter nuclei combine into a heavier nucleus, releasing energy. The amountof energy can be calculated from E = ∆mc2, where ∆m is the mass difference between the reactantsand products. The energy is released in the form of a kinetic energy of the fusion products. The mostpromising fusion reaction is deuterium (2H or D) and tritium (3H or T) fusion [1], given by

D + T→ 4He + n + 17.6 MeV. (1)

Atomic nuclei are positively charged, and will therefore repel one another at a long range. To overcomethe Coulomb barrier, the two nuclei must be close enough for the short ranged strong force attraction todominate. This is achieved by heating the fuel to high enough temperatures, of the order of ∼ 10 keV(∼ 1× 108 K). At these temperatures the particles move at very high velocities (∼ 1× 106 m s−1), andcan tunnel through the Coulomb barrier and fuse [1].

The temperatures in a fusion reactor are high enough to fully ionize the fuel. An ionized gas is called aplasma. A fully ionized plasma can be confined by magnetic fields. There are several different confinementschemes, and one of the most promising candidates is the tokamak, which is a toroidal reactor with astrong toroidal magnetic field and a weak poloidal field [1]. There are many different active tokamakexperiments, such as ASDEX upgrade, WEST, JET, JT-60SA, EAST etc. The subject of study in thisthesis will be JET (Joint European Torus), located in the UK. The antenna in the model will be the JETITER like antenna (ILA). The tokamak design will be described more in detail in section two.

A plasma is conductive, which means that it can be heated through ohmic (resistive) heating. Ohmicheating is effective up to a certain temperature; however, the efficiency decays as the resistivity of theplasma decreases as T−3/2. Therefore, auxiliary heating is required [2]. One method of heating is NeutralBeam Injection (NBI), where high energy neutrals are injected into the plasma. As they travel throughthe plasma, the neutrals give away their electrons to ions, which have lower energy. This process isknow as charge exchange. The high energy particles can now be confined, and the newly formed neutralsexit the plasma. The high energy particles will deposit their kinetic energy in the bulk plasma throughcollisions, and the temperature of the plasma will increase [2].

Another method is radio frequency (RF) heating, where electromagnetic waves are emitted into theplasma. There is generally not much damping in plasmas, but there may be certain resonances, relatedto the gyration of charged particles around magnetic field lines. Here, the wave energy is convertedinto particle kinetic energy. This gyration frequency is called the cyclotron frequency, and is given byΩ = qB/m, where q is the charge of the particle, B is the magnetic flux density and m is the mass of theparticle. Electromagnetic waves emitted into the plasma at the cyclotron frequency or a multiple of thecyclotron frequency can interact with these gyrating particles at wave-particle resonances. The electroncyclotron frequency is typically in the order of ∼ 100 GHz, while the ion cyclotron frequency is noticeablylower, usually around ∼ 40 MHz [1]. Ion cyclotron resonance heating (ICRH) has proven successful inseveral experiments [3], and will also be the subject of study in this work. ICRH will be discussed ingreater detail in section three.

Due to the complex geometry of a fusion reactor, it is necessary to use numerical tools to computethe wave field. Several codes have been developed for this purpose, such as TORIC, EVE, CYRANO,LEMan, LION, AORSA and FEMIC . These codes use Fourier methods and/or the finite element method(FEM) to compute the wave field from an ICRH antenna [4]. Usually, these codes use a simple geometryoutside the plasma. There are few 3D codes, particularly fast 3D codes. One 3D code is HIS-TORIC,which uses Fourier methods in the core plasma and FEM in the edge plasma. It then uses mode matchingto combine the solution inside the plasma with the solution outside the plasma. The computation is verytime consuming [5]. Other 3D codes, such as LEMan [6] and AORSA [7], also use Fourier methods.

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However, it is difficult to accurately describe the antenna using Fourier methods, especially the coaxialfeeders. Other things that also cause the geometry to be asymmetric, such as delimiters, are also difficultto describe using Fourier methods.

The feasibility of a pure 3D FEM code has to the best of my knowledge not yet been investigated.Therefore, the goal with this thesis is to:

• Assess the possibility to model the wave fields for ICRH applications in a tokamak in 3D usingFEM.

• Validate the electric fields and absorption with the FEMIC code by comparing the correspondingquantities generated by FEMIC

• Study the coupled power from the antenna to the plasma as well as corresponding coupling resistancefor each toroidal mode.

• Validate the coupled power and coupling resistance using a 1D analytical slab model.

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2 The Tokamak

In this section, the tokamak design is described in further detail. First, toroidal coordinates are intro-duced. Then, the plasma confinement mechanism of the tokamak will be briefly explained. The fluxfunction is also introduced as a method of describing the temperature and density of the plasma.

2.1 Toroidal Coordinates

In a tokamak, the geometry of the plasma is toroidal. Suitable coordinate are therefore either cylindricalor toroidal. A toroidal coordinate system (r, φ, θ) can be defined in several ways, but perhaps most simplyas

R = R0 + r cos θ

Z = r sin θ

where R and Z refer to the cylindrical coordinates. R is called the major radius, and r is the minorradius. R0 is the distance from the Z-axis to the centre of the cross section of the torus. This coordinatesystem is illustrated in Figure 1. The r-direction is called the radial direction, the φ-direction the toroidaldirection, and the θ-direction the poloidal direction [8]. In general, the cross section of a fusion plasma isnot circular, and a more complex set of coordinates is needed. The poloidal angle θ can, instead of beingthe geometrical angle, be a generalized angle. The coordinate r can also be replaced with a generalizedradius ρpol. In this work, both cylindrical and generalized toroidal coordinates will be used.

Figure 1: Illustration of the toroidal coordinate system. R is the major radius, φ is the toroidal direction,and θ is the poloidal direction

2.2 Confinement

The tokamak has a strong toroidal magnetic field. From Ampere’s law, it is possible to show that thetoroidal magnetic field decays as 1/R, where R is the radial distance from the centre of the tokamak.Therefore, the tokamak will have a high field side and a low field side. Most instruments and diagnostics,including the ICRH antenna, are located on the low field side.

A current is induced in the plasma to heat the plasma and to produce a poloidal component of themagnetic field, which will twist the magnetic field lines into helices. A poloidal field is required to confinethe plasma. Particles will drift across the field lines due to the curvature and the decay of the magneticfield. However, as the particles are drifting, the poloidal magnetic field will bring them around to theother side of the plasma, compensating for the drift.

The surfaces spanned by the field lines are called flux surfaces, and the innermost surface, whichessentially is a line, is called the magnetic axis. A visualization of the flux surfaces can be found inFigures 2 and 3. In a tokamak with a circular cross section, the minor radius r is constant over theflux surfaces. It is also assumed that densities and temperature are constant over each flux surface. Inmodern tokamaks, with non-circular cross sections, the generalized radius ρpol is constant over the fluxsurfaces. ρpol assumes the value 0 at the magnetic axis and the value 1 at the last closed flux surface,which is called the separatrix. The space outside the separatrix is called the scrape-off layer (SOL).

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Figure 2: Illustration of the flux surfaces in the tokamak [9]. The flux surfaces are the surfaces spannedby the magnetic field lines, which circle the magnetic axis helically.

Figure 3: Several flux surfaces (black) between ψ = 0.035 (the smallest contour) and ψ = 1. ψ = 1corresponds to the separatrix (orange). The reactor wall (blue) is plotted outside the separatrix.

2.3 The Flux Function

The flux surfaces can be seen as the contours of the flux function ψ. The flux function is scaled in sucha way that the contour ψ = 1 corresponds to the separatrix. In this work, an analytic flux functionhas been used. It was created such that the contours are sufficiently similar to a real plasma. The fluxfunction is given by

ψ = ψ+(R0, Z0) + ψ−(R0,−Z0 + 2X), (2)

where

ψ±(R0, Z0) = ψ0

[(r cos θ)2 +

(r sin θ

1 + αr

)2

+ βr3 cos 3θ

]eγr cos θ×

×(

1± tanh (b(r sin θ −X))

2

),

(3)

with

6

r =√

(R−R0)2 + (Z − Z0)2

θ = arctan

(Z − Z0

R−R0

).

(4)

The different parameters are explained in Table 1. The flux surfaces are illustrated in Figure 3. The fluxfunction can be used to define a generalized radius ρpol, given by

ρpol =

(

ψ − ψaxis

ψsep − ψaxis

)1/2

, for R,Z inside separatrix(1 +

∣∣∣∣ ψ − ψsep

ψsep − ψaxis

∣∣∣∣)1/2

, for R,Z outside separatrix

(5)

Here, ψaxis and ψsep are the values of the flux function on the magnetic axis and separatrix respectively.

Table 1: Explanation and values of the flux function parameters.

Parameter Value Unit Description

ψ0 1.15 T Scale factorα 0.4 m−1 Elongation paramterβ −0.04 m−1 Triangularity parameterγ 0.4 m−1 Shafranov shift paramaterR0 3.05 m Radial coordinate of magnetic axisZ0 0.3 m Z-coordiante of magnetic axisX −1.3918 m X-point parameterb 2 m−1 Steepness of step function

2.4 The Antenna

The antenna that will be studied is the JET ITER like antenna (ILA). It consists of two antenna strapsin the toroidal direction and four straps in the poloidal direction. It is possible to control the directionof the radiated power by modifying the phase difference between the currents in the different straps.This is called phasing. Having different phases in the toroidal direction is called toroidal phasing. If thephase is different in the poloidal direction, it is called poloidal phasing. Phasing can be used to changethe direction of the waves emitted into the plasma. For example, having a toroidal phase difference of180 will result in two lobes extending into the plasma at an angle to the major radius axis, due to thenegative interference which causes the electric field to vanish on the axis. A CAD model of the JET ILAcan be seen in Figure 4

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Figure 4: The JET ITER like antenna.

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3 Wave and Plasma Physics

A plasma is a complex medium, here described by the dielectric tensor, which is a complex valuedanisotropic nondiagonal tensor. If the dielectric tensor is dependent on the frequency, it is said to betemporally dispersive. Similarly, if the tensor is dependent on the wave number, it is said to be spatiallydispersive. The expressions for the different tensor components are generally very complicated anddependent on both frequency and wave number, which makes the general dielectric tensor both spatiallyand temporally dispersive. In the simplest of cases, the dielectric tensor is given by the scalar relativepermittivity, εr.

In this section we will briefly discuss the electromagnetic wave equation, the dielectric tensor of a hotplasma and how the wave field is absorbed in the plasma.

3.1 The Wave Equation

The wave fields in the tokamak are governed by the electromagnetic wave equation, which can be derivedfrom Maxwell’s equations (see appendix A). It is given by

∇×∇×E − ω2

c2K ·E = iωµ0Jant (6)

where K is the dielectric tensor, ω is the frequency of the wave, c is the speed of light in vacuum, E isthe electric field and Jant is the antenna current. In a tokamak geometry, the parallel and perpendicularwave numbers can be approximated by

k‖ =nφR

(7)

k2⊥ =ω2

c2

(K1 +K0 − n2‖ +

K22

K1 − n2‖

)(8)

where n‖ = ck‖/ω is the refractive index in the direction parallel to the magnetic field, and K0, K1 andK2 are components of the dielectric tensor, which will be introduced in more detail shortly. Equation (8)is the dispersion relation for the fast magnetosonic wave, derived in appendix B. However, k⊥ dependson k‖, so we assume that k‖ is given by equation (7). Spatial dispersive effects tend to generate integro-differential operators. However, by using equations (7-8), we get rid of these integro-differential operatorsand FEM is applicable.

3.2 The Dielectric Tensor

A magnetized plasma is an anisotropic medium [10], which means that the response is described by atensor (as opposed to a scalar in an isotropic material). The plasma is also gyrotropic. The gyrotropyarises from the fact that particles 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. Thisimplies that an electric field in the x-direction gives rise to a current in the y-direction and vice versa, ifwe assume that B ‖ z. If we also assume that the wave vector is in the xz-plane, the dielectric tensorfor a hot plasma is given by [11]

K0 =

K1 K2 K4

−K2 K1 +K0 −K5

K4 K5 K3

(9)

where

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K0 = 2∑j

ω2pje−λj

ωk‖v‖j

∞∑n=−∞

λj(In − I ′n)

[(1−

k‖v0j

ω

)Z(ζnj)+

+k‖v‖j

ω

(1− T⊥j

T‖j

)Z ′(ζnj)

2

] (10)

K1 = 1 +∑j

ω2pje−λj

ωk‖v‖j

∞∑n=−∞

n2Inλj

[(1−

k‖v0j

ω

)Z(ζnj)+

+k‖v‖j

ω

(1− T⊥j

T‖j

)Z ′(ζnj)

2

] (11)

K2 = i∑j

εjω2pje−λj

ωk‖v‖j

∞∑n=−∞

n(In − I ′n)

[(1−

k‖v0j

ω

)Z(ζnj)+

+k‖v‖j

ω

(1− T⊥j

T‖j

)Z ′(ζnj)

2

] (12)

K3 = 1−∑j

ω2pje−λj

ωk‖v‖j

∞∑n=−∞

In

(ω + nΩjk‖v‖j

)×

×[

1 +nΩjω

(1−

T‖j

T⊥j

)]Z ′(ζnj) +

2nΩjT‖jv0j

ωT⊥jv‖j

[Z(ζnj) +

k‖v‖j

ω + nΩj

] (13)

K4 =∑j

k⊥ω2pje−λj

k‖ωΩj

∞∑n=−∞

nInλj

nΩjv0jωv‖j

Z(ζnj)+

+

[T⊥jT‖j− nΩj

ω

(1− T⊥j

T‖j

)]Z ′(ζnj)

2

(14)

K5 = i∑j

k⊥εjω2pje−λj

k‖ωΩj

∞∑n=−∞

(In − I ′n)

nΩjv0jωv‖j

Z(ζnj)+

+

[T⊥jT‖j− nΩj

ω

(1− T⊥j

T‖j

)]Z ′(ζnj)

2

(15)

Here j indicates the species, ωpj and Ωj are the plasma and cyclotron frequencies respectively, k‖ is thewave number parallel to the magnetic field, k⊥ is the wave number perpendicular to the magnetic field,v‖j is the parallel thermal velocity, v⊥j is the perpendicular velocity, v0j is the net parallel flow velocity,T‖j is the parallel temperature, T⊥j is the perpendicular temperature, εj is the sign of the charge. Finally,In = In(λj) is the nth order modified Bessel function evaluated at λj = k2⊥v

2⊥j/(2Ω2

j ), and Z(ζnj) is theplasma dispersion function evaluated at ζnj = (ω+nΩj − k‖v0j)/(k‖v‖j). Prime denotes derivative. Theplasma dispersion function is defined by [11]

Z(ζ) ≡ 1√π

ˆ ∞−∞

e−ξ2

ξ − ζdξ. (16)

K2 describes the gyrotropy of the plasma, while the other nondiagonal terms K4 and K5 come from thefact that the plasma is hot.

3.3 The Scrape-off Layer

In an isotropic medium, the index of refraction is given by

n2 =c2

ω2

(k2x + k2y + k2z

), (17)

where c is the speed of light in vacuum, ω is the frequency of the wave and ki is the wave number in thei-direction. If we assume that the antenna is located in the yz-plane, that y is the poloidal direction andthat z is the toroidal direction, we can estimate ky and kz just by the size of the antenna. We assumethat the dimensions of the antenna are in the order of 0.1 m by 1 m. The wavelength of the emitted wavesare determined by the size of the antenna. This implies that k2y ∼ 100 m−2 and that k2z ∼ 1 m−2, sincek = 2π/λ. Solving for k2x yields

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k2x =n2ω2

c2− k2y − k2z . (18)

Assuming that we can approximate the SOL as a perfect vacuum and using the antenna frequency50 MHz, k2x must be negative, implying evanescent waves in the vacuum region. On the other hand, theindex of refraction is significantly higher inside the plasma, yielding propagating waves. This means thatthe waves have to tunnel through the SOL into the plasma. They will therefore be subject to couplingresistance. Consequently, in order to reduce the fraction of reflected energy and increase the coupledpower P , the antennas have to be placed as close as possible to the plasma. This, however, shortens theirlifespan, as they will be exposed to high energy particles. The coupling resistance can be defined througha generalization of Ohm’s law

R =P

I2, (19)

where I is the total current in the antenna. R can be seen as an effective resistance of the system,hence the name coupling resistance. There are different versions of equation (19) that can be studied.For example, the coupling resistance of each Fourier mode is given by Pn/I

2, where Pn is Poyntingvector from the nth Fourier components of the E- and B-fields integrated over the separatrix. Thiswill be described in greater detail later. Another variant worth investigating is Pn/J

2n, where Jn is

the nth Fourier component of the current density in the antenna. The Fourier decomposition of theelectromagnetic fields is of interest due to the fact that both the absorption and the coupling resistanceis mode dependent. Typically, the coupling resistance Pn/J

2n increases with the mode number, since the

evanescent waves decay faster for higher mode numbers. The absorption on the other hand, is low forlow mode numbers.

3.4 Absorption

To obtain ion cyclotron resonance heating, the wave field first needs to have a component which is lefthand polarized, i.e. same as the direction of rotation of the ions. The resonance condition is given by

ω = nΩ + v‖k‖ (20)

where n = 0, 1, 2... is the harmonic number, Ω is the cyclotron frequency and v‖k‖ is the Doppler shift. v‖is the particle velocity parallel to the magnetic field. Here n = 0 corresponds to Landau damping, n = 1corresponds to the fundamental cyclotron resonance, and n = 2 corresponds to the second harmoniccyclotron resonance [12]. If this criterion for the frequency is fulfilled, the wave energy is transferred tothe ions in the form of kinetic energy.

The first naıve choice of frequency would be ω = Ω + v‖k‖, but at this frequency, the polarization ofthe wave is zero. To show this, we can use a cold plasma model (particles have no thermal motion) andcalculate the ratio of the left hand polarized wave (E+) and the right hand polarized wave (E−). Thisratio turns out to be

E+

E−=ω − Ω

ω + Ω. (21)

In a cold plasma model, the Doppler shift vanishes, so a wave with frequency corresponding to the thefundamental cyclotron resonance frequency would therefore be completely right hand polarized at theresonance [12]. There are two ways around this. According to equation 21, a wave at the second harmonicresonance frequency would have a finite left handed polarized component, since

E+

E−=

1

3. (22)

when ω = 2Ω. Therefore, E+ can be absorbed by the ions at the resonance.It would also be possible to introduce a minority species with a different cyclotron frequency than the

bulk ions. If the minority species has the same cyclotron frequency as the second harmonic frequency ofthe bulk ions, both species may be heated at the same point in space with the same antenna. This isvalid in for example a deuterium plasma with minority hydrogen (i.e. protons).

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3.5 The Poynting Vector

Assuming that the plasma is cylindrically symmetric, it is valid to assume that all normal energy flux atthe separatrix will heat the plasma. The total power absorbed in the plasma will be given by the normalcomponent of the Poynting vector integrated over the separatrix. The Poynting vector is given by

S =1

µ0E ×B. (23)

Usually, the quantity of interest is the time averaged Poynting vector, given by

〈S〉 =1

2µ0ReE ×B∗. (24)

To understand how each Fourier mode couples into the plasma, the Poynting vector has to be Fourierdecomposed. However, if the Fourier decomposition was to be done directly on the Poynting vector, everycomponent except S0 would vanish when integrated over the separatrix. This representation would notgive us any information, except for the total flux S0. Instead, the electric and magnetic fields can bedecomposed one at a time. Starting with only toroidal decomposition we obtain

E = eα∑n

Eαneinφ (25)

B∗ = eβ∑n′

Bβ∗n′ e−in′φ. (26)

It is worth noting that the Fourier spectrum is coordinate dependent, so care has to be taken to use anappropriate coordinate system.

E ×B∗ =∑n

∑n′

EαnBβ∗n′ eα × eβe

i(n−n′)φ (27)

To obtain the power absorbed in the plasma, the time averaged Poynting vector is integrated over theseparatrix, yielding

P =1

2µ0

‹ ∑n

∑n′

ReEαnB

β∗n′ e

i(n−n′)φn · (eα × eβ) dA. (28)

Explicitly writing out the integration limits, the coupled power can be expressed as

P =1

2µ0

ˆ 2π

0

ˆ 2π

0

∑n

∑n′

ReEαnB

β∗n′ e

i(n−n′)φn · (eα × eβ) Jdθdφ, (29)

where J is the Jacobian. To simplify the integral over φ,

ReEαnB

β∗n′

n · (eα × eβ) (30)

must be independent of φ. Neither Eαn nor Bβ∗n′ depends on φ, since the φ dependency is containedin the exponential function due to the Fourier decomposition. The direction of n (and sometimes ei,depending on coordinate system) however, depends on φ. Since the separatrix is cylindrically symmetric,it is convenient to change into cylindrical coordinates. The normal vector is then given by

n = nr(r, θ)er + nz(r, θ)ez (31)

The coordinate transformation is done before Fourier decomposing the electric and magnetic field, sointegrating over φ yields

P =π

µ0

ˆ 2π

0

∑n

∑n′

ReEαnB

β∗n′

n · (eα × eβ) δnn′J

θRdθ = (32)

=∑n

π

µ0

ˆ 2π

0

ReEαnB

β∗n′

n · (eα × eβ) Jdθ ≡

∑n

Pn, (33)

12

where we have defined

Pn ≡π

µ0

ˆ 2π

0

ReEαnB

β∗n′

n · (eα × eβ) Jdθ. (34)

If we instead Fourier decompose in both the toroidal and the poloidal directions, we similarly obtain

P =1

2µ0

‹ ∑mn

∑m′n′

ReEαmnB

β∗m′n′e

i(m−m′)θei(n−n′)φn · (eα × eβ) dA (35)

or in toroidal coordinates

P =1

2µ0

ˆ 2π

0

ˆ 2π

0

∑mn

∑m′n′

ReEαmnB

β∗m′n′e

i(m−m′)θei(n−n′)φn · (eα × eβ) Jdθdφ (36)

The φ dependence of the normal component of the Poynting vector can again be circumvented by intro-ducing cylindrical coordinates, but this time we face a larger problem. First, there is a θ dependencein n. A coordinate transformation moved the φ dependence from the vector components of n to theunit vectors. This can be repeated for θ by introducing a coordinate system (n, t, φ), where n is the(inward) normal coordinate, t is the (counter-clockwise) tangential component and φ again is the toroidalcomponent. The normal vector is then simply given by

n = en (37)

Secondly, the Jacobian J is θ dependent. Therefore, a change of variables is required. The differentialJdθdφ can be rewritten as

Jdθdφ = 2πJdθdφ

2π= dAθ

dφ

2π(38)

The integral over φ is unproblematic, and the integral over θ simply yields the area of the separatrix. Asuitable change of variables is then

dt = kdAθ, (39)

where k can be determined by integrating the differential

˛dt = k

˛dAθ ⇒ 2π = kA⇒ k =

2π

A(40)

where A is the area of the separatrix. The absorbed power is then given by

P =A

8π2µ0

ˆ 2π

0

ˆ 2π

0

∑mn

∑m′n′

ReEαmnB

β∗m′n′e

i(m−m′)θei(n−n′)φn · (eα × eβ) dtdφ (41)

Now that the θ dependence has been eliminated, it is possible to integrate and make use of theorthogonality of the complex exponential functions to obtain

P =∑mn

A

2µ0ReEαmnB

β∗m′n′

n · (eα × eβ) ≡

∑mn

Pmn, (42)

where we have defined

Pmn ≡A

2µ0ReEαmnB

β∗m′n′

n · (eα × eβ) (43)

13

4 Model

The simulations were made in COMSOL Multiphysics using the RF module to model the wave field. Thegeometry was designed in the 3D CAD software SolidWorks and imported into COMSOL. Post-processingwas done in COMSOL’s graphical user interface and in MATLAB.

4.1 Geometry

Only 70 of the tokamak was modelled in order to reduce the memory requirements. The antenna boxwas placed on the middle of this section, at φ = 35 (see Figure 5a). In an attempt to minimize thenumber of mesh elements, the JET cross section was trimmed in areas which would require very smallmesh elements to resolve. The difference between the actual cross section and the trimmed cross sectioncan be seen in Figure 5b.

The modelled antenna was a simplified version of JET ILA. Instead of eight separate straps, the modelonly used two straps at two different toroidal angles. Therefore, it is not possible to simulate poloidalphasing in this model. The geometry of the antenna can be seen in Figure 5c.

(a) (b)

(c)

Figure 5: The geometry of the tokamak (a), the trimmed cross section (grey) compared to the originalcross section (blue) (b) and a more detailed view of the antenna box (c).

14

4.2 Antenna Current

The current in the antenna is modelled as a prescribed current of 1 A/m in both straps with a 180

phase difference. The Fourier decomposition of with current distribution can be seen in Figure 6. Thisspectrum is important, since it will determine the spectrum of the power radiated into the plasma.

Figure 6: Fourier decomposition of the antenna current in the toroidal direction. The 180 phase differ-ences results in two different lobes due to the destructive interference.

4.3 Mesh

Normally, the size of the mesh elements should be approximately one fifth of the wavelength, using secondorder elements. Therefore, it is useful to investigate the wave numbers of the propagating wave. As statedin section three, the wave numbers are given by equations (7-8). The parallel wavelength is then givenby

λ‖ =2πR

nφ≈ 0.7 m. (44)

Figure 7: The real and imaginary parts of k2⊥ plotted against R

To obtain the perpendicular wave number, we need to choose some realistic values of the plasmaparameters. The on axis temperature can be chosen to be 8 keV for all species, the on axis electrondensity to be 8× 1019 m−3 and the on axis magnetic flux density to be 3.2 T. The hydrogen minoritydensity is 4% of the electron density and the remaining ions are deuterium ions. Inserting these parameter

15

values into the expression for k2⊥ yields the dispersion relation for the perpendicular wave number. Thedispersion relation can be seen in Figure 7. The minimum wavelength to resolve is therefore approximatelygiven by

λ⊥ =2π√

max(Re(k2⊥))(45)

which turns out to be approximately 0.1 m. This implies that the mesh needs to be much finer perpen-dicular to the magnetic field than parallel to it.

(a) (b)

Figure 8: The full mesh of the model (a) and a cross section of the swept mesh (b).

COMSOL has a feature called swept mesh, which takes a 2D mesh and sweeps it along the geometry.By using swept mesh, it is possible to reduce the total number of mesh elements in the model. However,the boundaries need to be cylindrically symmetric, which means that it is not possible to use swept meshin the scrape-off layer, since the antenna is on this boundary. Therefore, a free tetrahedral mesh wasused in the SOL and the antenna box. Finally, on the plasma cross section and on the antennas, a freetriangular mesh was used. The plasma cross section was split into three regions. The middle region,located around the plasma centre, was given a much finer mesh, since most of the absorption take placethere. The mesh above and below the plasma centre was given a coarser mesh, in order to reduce the totalnumber of mesh elements. Finally, in order to improve the resolution at the plasma edge, a boundarylayer mesh was introduced. In this model, the boundary layer mesh was only one element thick. The meshcan be seen in Figure 8, and the mesh sizes in Table 2. The mesh parameters were chosen consideringmemory constraints.

Table 2: Size of mesh elements in different domains of the model. All values are given in metres.

Size Domain

0.026 m Plasma cross section, fine0.07 m Plasma cross section, coarse0.03 m Antennas0.06 m Antenna box0.14 m SOL

37 Number of elements in toroidal sweep

4.4 Coordinate Transformations

When the dielectric tensor in section 3.2 was defined, a different coordinate system than COMSOLMultiphysics’ is used [11](see Figure 9), so the dielectric tensor has to be transformed. The transformationmatrix R1 from the coordinate system in [11] to COMSOL Multiphysics’ Cartesian coordinate system isgiven by

16

R1 =

1 0 00 0 −10 1 0

(46)

Due to the (mostly) cylindrical symmetry of the tokamak, it is convenient to use cylindrical coordi-nates. Therefore, the dielectric tensor has to be transformed again. The transformation matrix R2 isgiven by

R2 =

cosφ sinφ 0− sinφ cosφ 0

0 0 1

(47)

where φ again is the toroidal angle. In the end, the transformed dielectric tensor is given by

K = R2 ·(R1 ·K0 ·R−11

)·R−12 . (48)

x

z

y

(a)

x

y

z

(b)

Figure 9: COMSOL Multiphysics’ coordinate system (a) and Swanson’s coordinate system (b).

17

5 Results

Before studying the results from the 3D model, it is worthwhile to briefly investigate the convergenceof the simulation. The model was solved with two different meshes. The first mesh had the parametersgiven in Table 2 in the previous section, while the second mesh was approximately 10% coarser (i.e. themesh parameters were 10% larger, except the number of elements in the toroidal sweep, which of coursewas lower). The convergence can be visualized by comparing for example the E+ wave from the twosolutions. The E+ waves can be seen plotted against a distance x along one of the lobes of the electricfield in Figure 10. It seems like the solution has not fully converged, and that the amplitude is slightlyhigher in the finer mesh.

Figure 10: E+ wave from the simulation with finer mesh (solid blue) and coarser mesh (dashed orange).x is the distance from the edge of the plasma along one of the lobes.

5.1 The 3D wave field

The solution of the 3D model can be visualized in different ways, but a simple way of getting an overviewof the solution is to use different cut planes. The amplitude of the wave field can be seen together withE+, E− and the absorption in Figure 11. As expected, the wave in the plasma is separated into two largelobes, due to the toroidal phasing. The field magnitude between these lobes is zero due to destructiveinterference. We can also see that the wave propagates to the centre of the plasma, where it is absorbed.The ion cyclotron resonance occurs at the centre of the plasma, which can be seen in Figure 11d. Theabsorption between the centre and the plasma boundary is mostly electron absorption through variousmechanisms.

(a) (b)

(c) (d)

Figure 11: Electric field and absorption from the 3D model. (a) shows the norm of electric field, (b)shows the real part of E+, (c) shows the real part of E− and (d) shows the total absorption.

18

(a) (b)

(c) (d)

(e) (f)

(g) (h)

Figure 12: Wave fields and absorption from FEMIC (a, c, e, g) and from the 3D solution projected onto2D (b, d, f, h). (a) and (b) show norm of electric field, (c) and (d) show the real part of E+, (e) and (f)show the real part of E− and (g) and (h) show the total absorption.

19

5.2 Comparison of Wave Fields with FEMIC

To compare solution from the 3D model to the 2D solution from FEMIC, it may be helpful to projectthe wave field from 3D onto a poloidal plane. The projection was done with the COMSOL MultiphysicsGeneral Projection tool, which integrates the solution along a specified path. Therefore, there is adifference in units. However, it should be possible to compare the qualitative behaviour between the twomodels. The comparison between |E|, |E+|, |E−| and the absorption can be seen in Figure 12. Only theabsolute value of the wave fields can be compared this way, since any negative contributions would cancelout the positive contributions, and the projections would essentially be zero.

By making a cumulative integral of the absorbed power in the plasma, it is possible to see how muchis absorbed within a certain value of ρpol. It is then possible to study the amount of cyclotron resonancedamping and the amount of electron damping in the plasma. If the volume is computed through asimilar cumulative integral, it is possible to find the derivative of the absorbed power with respect to thevolume. This is nothing but the density of absorbed power. Both the normalized cumulative power andthe normalized power density can be seen in Figure 13.

Figure 13: Normalized cumulative power (left) and normalized power density (right) from both theFEMIC code (black) and the 3D model (blue).

We expect ion cyclotron resonance close to the magnetic axis, in the centre of the plasma. It can beseen in Figures 12g and 12h that this is indeed the case. We can also see that there is absorption betweenthe centre and the edge of the plasma. This is the electron absorption. Again, studying Figure 13, itseems like there is more electron damping in the 3D model than in the FEMIC code.

5.3 Comparison of the Power Spectrum with the Analytical Solution

The toroidal power spectrum from the analytical model and the 3D model can both be seen in Figure 14.Both spectra have been normalized to simplify comparison. Near n = 0, the amplitude of the numericalpower spectrum is lower compared to the analytical mode spectrum.

It can also be seen that the amplitudes in the second lobes differ between the two models. This maybe due to the limited resolution. At high mode numbers (|n| > 70), the fields may not be properlyresolved, yielding inaccurate results for the wave fields in these regions.

A contour plot of the two dimensional Fourier decomposition of the power spectra can be seen inFigure 15. The general shape of the contour plots is very similar, but there are certain differences. Forexample, there is one poloidal lobe and two toroidal lobes with maxima at the same poloidal and toroidalmode numbers. However, the shape of each individual contour differs.

20

Figure 14: Comparison of analytic (black) and 3D (blue) power spectrum.

(a) (b)

Figure 15: Power spectra of the analytic solution (a) and 3D model (b).

5.4 Comparison of the Coupling Resistance with the Analytical Solution

Clearly, the coupling resistance in the 1D and 3D cases are very different. While the coupling resistancein the analytical solution looks Gaussian, there are singularities present in the 3D model. However,both models have in common that the coupling resistance decreases with higher mode numbers, and isrelatively high for intermediate mode numbers (n ≈ 30).

(a) (b)

Figure 16: Coupling resistance for the analytical solution (a) and the 3D simulation (b). There are severalsingularities in the 3D coupling resistance, arising from the induced wall current.

21

6 Discussion

The dielectric tensor used in this model was the dielectric tensor for n = 27. This is near the maximumof the power spectrum, which means that the tensor is a good approximation for the nearby modes (andin this case, the most important modes). Due to the plasma dispersion function (equation 16) in thedielectric tensor, the resonances will be narrower for lower mode numbers. This increases the risk ofmode conversion. For higher mode numbers, the opposite happens.

Structurally, the wave fields from FEMIC and from the 3D solution projected onto the poloidal planeare very similar. This is promising for the 3D model, which indicates that with limited available memory,it is possible to reproduce results of lower dimensionality. The projected wave fields however, seem tohave a grainy resolution. This likely comes from the COMSOL general projection feature, which mayneed a finer mesh to produce a result with high resolution. Even when projecting a constant quantity,the resulting plot was found to be grainy.

The main difference between the FEMIC model and the 3D model seems to be the absorption. The3D model has a larger fraction of electron absorption. Without more available RAM, it is difficult tosay where this comes from. Something that became apparent during the creation of the 3D model wasthat lower resolution leads to higher absorption outside the plasma centre. When the resolution waslow enough, or when linear elements were used, there was no absorption at the centre of the plasma.Therefore, it is important that these simulations are done with a fine mesh. With sufficient RAM, itwould be possible to investigate whether the difference in absorption between FEMIC and this 3D modelvanishes with an even finer grid.

The difference in absorption may also come from the difference in dimensionality. In the FEMIC code,we only have the mode corresponding to n = 27, and therefore only the k‖ corresponding to this modenumber. In the 3D model, we have included all k‖. The parallel wave vector is involved in the absorptionprocess, so this may partly explain the discrepancy.

This discrepancy around n = 0 in the toroidal power spectrum likely comes from the current inducedin the walls in the 3D model. These currents give rise to their own electromagnetic fields, which mayinterfere destructively with the fields from the antenna. This effect is predominant for the lower modenumbers, as they decay slower, or may even be propagating in the SOL.

In the 2D power spectrum, there is only one lobe in the poloidal direction, which is explained bythe design of the antenna. The antenna is not divided into different sections in the poloidal direction,meaning there can be no poloidal phase shift. The phase shift is, in this simple model, the most importantfactor in the shape of the spectrum. Similarly, the 180 phase shift in the toroidal direction gives rise tothe two toroidal lobes.

As for differences, the shape of the contours differs slightly, but this is to be expected consideringdifferences in geometry as well as the wall currents previously discussed.

The coupling resistance for each mode is given by Rn = Pn/|In|2, where In is the Fourier decomposedantenna current. The coupling resistance behaves very differently in the two cases. In the analytical case,the zeros in the (squared) current spectrum exactly match the zeros in the power spectrum, giving riseto apparent singularities. The result is a well behaved and smooth coupling resistance.

In the 3D simulation however, this is not the case. The electromagnetic fields radiated from theantenna induce a current in the reactor walls. This current has a different spectrum than the antennacurrent, and the true source of the radiated power is the superimposed current in the antenna and thewall. Near the zeros in the current spectrum, the assumption that the radiated power only comes fromthe antenna is no longer valid, so when we try to obtain the coupling resistance in the same manner as inthe analytical case, the apparent singularities turn into real singularities, yielding the behaviour we seein Figure 16b. This means that the coupling resistance far away from the zeros of the current spectrummay be more reliable.

Once minor issues such as the computation of the coupling resistance and and the (possibly) excessiveelectron absorption have been resolved, the goal is to integrate this code into the FEMIC code. FEMICalready has support for some features that are helpful when modelling a fusion plasma, such as adding apedestal at the plasma boundary, and simulating a low density plasma in the SOL.

In the future, it would be interesting to add a poloidal field component. This component is todayneglected. It would also be beneficial to implement an iterative solver that can take the spatially dispersiveeffects of the dielectric tensor into account. This would make the results more reliable.

Many of these improvements would require more RAM. To further optimize the mesh, and thereforeuse the available RAM more efficiently, it would be helpful to use adaptive meshing, which automaticallyimproves the mesh there there are large gradients.

22

7 Conclusion

In this thesis we have assessed the possibility of making 3D FEM models with limited access to RAM(128 GB). The results were compared with the 2D code FEMIC by projecting the 3D wave field onto thepoloidal plane with good structural agreement of the wave fields.

The 3D model was also compared to an analytical slab model. For the power spectrum, the agreementis good, with some differences near the mode number n = 0 due to the induced current in the conductingwalls. Due to these currents, the radiated power and the current in the antenna don’t match, sincethe radiated power spectrum comes from the superposition of these two current spectra. This gives riseto singularities in the coupling resistance. Therefore, this method of computing the coupling resistancecannot be used to make reliable predictions of the coupling resistance.

With more RAM and an iterative solver that can solve the electromagnetic wave equation with aspatially dispersive dielectric tensor, the code developed would be promising.

23

References

[1] J. Scheffel and P. Brunsell. Fusion Physics - Introduction to the Physics Behind Fusion Energy, 6thEdition. Fusion Plasma Physics, Alfven Laboratory, KTH, 2016.

[2] J. Wesson and D.J. Campbell. Tokamaks. Clarendon Press, Oxford, 2004.

[3] Heating ITER Physics Expert Group on Energetic Particles, Current Drive, and ITER Physics BasisEditors. Chapter 6: Plasma auxiliary heating and current drive. Nuclear Fusion, 39(12):2495, 1999.

[4] P. Vallejos et al. Effect of poloidal phasing on ICRH power absorption. Unpublished, 2018.

[5] S. Shiraiwa et al. HIS-TORIC: Extending core ICRF wave simulation to include realistic SOLplasmas. Nuclear Fusion, 57, 2017.

[6] P. Popovich et al. A full-wave solver of the maxwell’s equations in 3d cold plasmas. ComputerPhysics Communications, 175:250–263, 2006.

[7] E. F. Jaeger et al. Advances in full-wave modeling of radio frequency heated, multidimensionalplasmas. Physics of Plasmas, 9:1873, 2002.

[8] R.D. Hazeltine and J.D. Meiss. Plasma Confinement. Dover Publications, 2003.

[9] J.P. Freidberg. Ideal MHD. Cambridge University Press, 2014.

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

[11] D.G. Swanson. Plasma Waves Second Edition. IOP Publishing Ltd, 2003.

[12] J.P. Freidberg. Plasma Physics and Fusion Energy. Cambridge University Press, 2008.

24

Appendices

A Analytical Solution to the Wave Equation

A.1 Deriving the Wave Equation

Starting with Maxwell’s equations, we want to derive the electromagnetic wave equation.

∇×B = µ0J +1

c2∂E

∂t(1)

∇×E = −∂B∂t

(2)

∇ ·B = 0 (3)

∇ ·E =ρ

ε0(4)

By taking the time derivative of Ampere’s law (1) and inserting Faraday’s law (2) into the result, weobtain

∇×∇×E = −µ0∂J

∂t− 1

c2∂2E

∂t2. (5)

Rearranging this gives us the electromagnetic wave equation.

∇×∇×E +1

c2∂2E

∂t2= −µ0

∂J

∂t(6)

We can take the temporal Fourier transform of this and obtain

∇×∇×E − ω2

c2E = iωµ0J . (7)

The current density can be divided into two components. One source current corresponding to theantenna current, call it Jant, and one current induced into the medium, call it J ind. The induced currentcan be described by the conductivity σ of the medium (note that σ is zero in vacuum). This is known asthe generalized Ohms law. Note that σ is a tensor

J ind = σ ·E (8)

Inserting this into (7) yields

∇×∇×E − ω2

c2

(I +

iσ

ωε0

)·E = iωµ0Jant. (9)

We can now identify the dielectric tensor

K = I +iσ

ωε0(10)

and obtain

∇×∇×E − ω2

c2K ·E = iωµ0Jant, (11)

which is the final form of the electromagnetic wave equation. For simplicity, we will here use the unmag-netized cold plasma description. The conductivity is then given by

σ = iε0ω2p

ω, (12)

where ωp is the plasma frequency, which is given by

ω2p =

nee2

mε0(13)

The electromagnetic wave equation can now be written as

25

∇×∇×E − ω2

c2E + µ0ε0ω

2pE = iωµ0Jant (14)

or, rearranging

∇×∇×E − ω2

c2

(1−

ω2p

ω2

)E = iωµ0Jant. (15)

We can now introduce the speed of light in the medium

c2m = c2

(1−

ω2p

ω2

)−1(16)

and obtain

∇×∇×E − ω2

c2mE = iωµ0Jant, (17)

which will be the basis of our analysis. In order to simplify the analysis, we can rewrite the curl curloperator as

∇×∇×E = ∇(∇ ·E)−∇2E, (18)

where ∇2 is the vector Laplacian, which can be expressed in Cartesian coordinates as

∇2E = (∇2Ex,∇2Ey,∇2Ez). (19)

A.2 Geometry and Boundary Conditions

To simplify the calculations, we can assume a slab geometry with periodic boundary conditions in the”poloidal” and the ”toroidal” directions y and z. At the wall, the toroidal circumference of the tokamakis given by 2π(R0 + r) where R0 and r are the major and minor radii respectively. In the poloidaldirection, the circumference is 2πr, if we for simplicity assume that the tokamak is a torus. Therefore, atz = ±π(R0+r) and at y = ±πr we impose periodic boundary conditions. The antenna will be representedby a two rectangular surfaces with the dimensions 2a× 2b located at the distance x0 from the wall andseparated by a distance 2d. The two will have uniform surface currents K0 = ±K0y respectively. Thisis visualized in figure 17. The mathematical expression for the current density Jant will then be

Jant = K0δ(x− x0)[θ(y + a)− θ(y − a)][θ(z + d+ b)− θ(z + d− b)− θ(z − d+ b) + θ(z − d− b)]y (20)

where θ(x) is the Heaviside step function.Between the wall and the plasma, we will have a scrape-off layer (SOL), in which we will assume that

there is vacuum. The tangential components of the electric field will be continuous at all boundaries.We will also assume that the derivative of the tangential components of the electric field are continuousat x = xp, the boundary between the SOL and the plasma. At the antenna however, there will be adiscontinuity in the derivative, since there is a delta function. The magnitude of this discontinuity is tobe determined later. Finally, we will assume that the entire wave is absorbed in the plasma, so that wehave no reflected wave from inside the plasma.

The plasma density is assumed to be a step function, given by

ne(x) =

0 x < xp

ne x ≥ xp(21)

26

x

z

y

Reactor Wall

x = x0

Antenna

2a

2b2d

Plasma

x = xp

Figure 17: Slab geometry

A.3 Solution

Due to the periodicity in the y- and z-directions, we start by expanding the different components of thesolution in Fourier series.

Ei =

∞∑mn

eikm,yyeikn,zzEimn(x), (22)

where i = x, y, z. km,y and kmzcan be determined from the periodic boundary conditions. In order to

have periodicity we require

2πrkm,y = 2πm⇒ km,y =m

r(23)

and

2π(R0 + r)kn,z = 2πn⇒ kn,z =n

R0 + r(24)

Then we can write out the curl curl operator in terms of partial derivatives. In the x-component we have

(∇(∇ ·E))x −∇2Ex =∂2Ey∂x∂y

+∂2Ez∂x∂z

− ∂2Ex∂y2

− ∂2Ex∂z2

(25)

In the y- and z-components we get

(∇(∇ ·E))y −∇2Ey =∂2Ex∂x∂y

+∂2Ez∂y∂z

− ∂2Ey∂x2

− ∂2Ey∂z2

(26)

and

(∇(∇ ·E))z −∇2Ez =∂2Ex∂x∂z

+∂2Ey∂y∂z

− ∂2Ez∂x2

− ∂2Ez∂y2

(27)

respectively. We can now solve the vector valued partial differential equation component by component.This means that we have to solve three coupled scalar valued partial differential equations. Namely

27

∂2Ey∂x∂y

+∂2Ez∂x∂z

− ∂2Ex∂y2

− ∂2Ex∂z2

− ω2

c2mEx = 0 (28)

∂2Ex∂x∂y

+∂2Ez∂y∂z

− ∂2Ey∂x2

− ∂2Ey∂z2

− ω2

c2mEy = iωµ0Jant (29)

∂2Ex∂x∂z

+∂2Ey∂y∂z

− ∂2Ez∂x2

− ∂2Ez∂y2

− ω2

c2mEz = 0 (30)

Inserting the solution ansatz into these equations gives us

∑mn

(ikm,y

∂Eymn∂x

+ ikn,z∂Ezmn∂x

+

(k2m,y + k2n,z −

ω2

c2m

)Exmn

)eikm,yyeikn,zz = 0 (31)

∑mn

(ikm,y

∂Exmn∂x

− km,ykn,zEzmn −∂2Eymn∂x2

+

(k2n,z −

ω2

c2m

)Eymn

)eikm,yyeikn,zz = iωµ0Jant (32)

∑mn

(ikn,z

∂Exmn∂x

− km,ykn,zEymn −∂2Ezmn∂x2

+

(k2m,y −

ω2

c2m

)Ezmn

)eikm,yyeikn,zz = 0 (33)

In order to solve this, we first need to also expand the source term in a Fourier series.

Jant =

∞∑mn

eikm,yyeikn,zzJmn(x), (34)

where the coefficient Jmn(x) needs to be determined. We will do this by making use of the orthogonalityof the Fourier series, and multiply both sides with e−ikm,yye−ikn,zz and integrate. We obtain

ˆ πr

−πr

ˆ π(R0+r)

−π(R0+r)

Jante−ikm,yye−ikn,zzdydz = 4π2r(R0 + r)Jmn(x), (35)

or

ˆ a

−ae−ikm,yydy

(ˆ −d+b−d−b

e−ikn,zzdz +

ˆ d−b

d+b

e−ikn,zzdz

)K0δ(x− x0) = 4π2r(R0 + r)Jmn(x), (36)

which can be evaluated to be

Jmn(x) =iK0 sin (km,ya)δ(x− x0)

π2km,ykn,zr(R0 + r)[cos (kn,z(d− b))− cos (kn,z(d+ b))] (37)

The equations (31)-(33) must hold for each term separately, so we arrive at the simplified system ofequations

ikm,y∂Eymn∂x

+ ikn,z∂Ezmn∂x

− k2xExmn = 0 (38)

ikm,y∂Exmn∂x

− km,ykn,zEzmn −∂2Eymn∂x2

−(k2x + k2m,y

)Eymn = iωµ0Jmn(x) (39)

ikn,z∂Exmn∂x

− km,ykn,zEymn −∂2Ezmn∂x2

−(k2x + k2n,z

)Ezmn = 0 (40)

where we have defined

k2x =ω2

c2m− k2m,y − k2n,z. (41)

We can now use equations (39) and (40) to eliminate the x-component and obtain

km,ykn,z

(∂2Ezmn∂x2

+ k2xEzmn

)−(∂2Eymn∂x2

+ k2xEymn

)= iωµ0Jmn(x) (42)

28

In the SOL, cm is the speed of light in vacuum, so k2x is typically negative. This means that the waveswill be evanescent in this region. Inside the plasma however, the refractive index may be in the order of100, causing the speed of light in the medium to be much lower than in vacuum. Therefore, we expectk2x to be positive, implying sinusoidal wave solutions in the plasma. The electromagnetic waves will haveto tunnel from the antenna through the SOL into the plasma.

To obtain the boundary condition at the antenna, we integrate equations (38)-(40) over a very smallinterval around x = x0 and obtain

ˆ x0+ε

x0−εikm,y

∂Eymn∂x

+ ikn,z∂Ezmn∂x

− k2xExmndx = 0 (43)

ˆ x0+ε

x0−εikm,y

∂Exmn∂x

− km,ykn,zEzmn −∂2Eymn∂x2

−(k2x + k2m,y

)Eymndx =

= iωµ0

ˆ x0+ε

x0−εJmn(x)dx

(44)

ˆ x0+ε

x0−εikn,z

∂Exmn∂x

− km,ykn,zEymn −∂2Ezmn∂x2

−(k2x + k2n,z

)Ezmndx = 0 (45)

Using the continuity of Eimn, we see that (43) reduces to zero. Equation (45) reduces to

∂Ezmn∂x

∣∣∣∣x−0

− ∂Ezmn∂x

∣∣∣∣x+0

= 0 (46)

implying continuity in the x-derivative of Ezmn. In the y-component however, we don’t have continuity.Carrying out the integration on both sides of equation (44) gives us

∂Eymn∂x

∣∣∣∣x−0

− ∂Eymn∂x

∣∣∣∣x+0

= iωµ0Kmn (47)

where we define Kmn as

Kmn =iK0

π2km,ykn,zr(R0 + r)sin (km,ya)[cos (kn,z(d− b))− cos (kn,z(d+ b))] (48)

This determines the discontinuity of the x-derivative of the y-component of the electric field.We need solve the differential equation in three different regions: inside the antenna, between the

antenna and the plasma, and finally in the plasma. We then need to match the solutions with theappropriate boundary condition. Starting with the region between the antenna and the conducting wall,the solution to equation (42) is

Eimn = Aimneikx(0)x +Bimne

−ikx(0)x (49)

for i = y, z. At the conducting surface, the electric field strength is zero, so we must have

Aimn = −Bimn. (50)

This is not surprising, since a wave has to be fully reflected at a perfect electrical conductor. In theregion between the plasma and the antenna, the general solution is identical to the first region, namely

Eimn = Cimneikx(0)x +Di

mne−ikx(0)x (51)

In the plasma, the solution is the same as in the first two regions, but now we have to recall that kx(x) isimaginary, so the solution is a sinusoidal propagating wave, and not evanescent. Recalling the assumptionthat there is no reflection from the middle of the plasma, the solution is given by

Eimn = F imneikx(xp)x (52)

Imposing the continuity in the electric field at the antenna yields

Aimneikx(0)x0 +Bimne

−ikx(0)x0 = Cimneikx(0)x0 +Di

mne−ikx(0)x0 , (53)

and the discontinuity of the derivative at x = x0 yields

29

Aimneikx(0)x0 −Bimne−ikx(0)x0 − Cimneikx(0)x0 +Di

mne−ikx(0)x0 =

ωµ0Kmn

kx(0)δiy, (54)

where δiy is the Kronecker delta. For the last boundary, we obtain

Cimneikx(0)xp +Di

mne−ikx(0)xp = F imne

ikx(xp)xp (55)

Cimnkx(0)eikx(0)xp −Dimnkx(0)e−ikx(0)xp = F imnkx(xp)e

ikx(xp)xp (56)

We can now use equations (50) and (53)-(56) to determine the electric field everywhere. Inserting equa-tions (50) and (55) into the others gives us the three equations

2iAimn sin (kx(0)x0) = Cimneikx(0)x0 +Di

mne−ikx(0)x0 (57)

2Aimn cos (kx(0)x0) = Cimneikx(0)x0 −Di

mne−ikx(0)x0 +

ωµ0Kmn

kx(0)δiy (58)

Cimnkx(0)e2ikx(0)xp −Dimnkx(0) = Cimnkx(xp)e

2ikx(0)xp +Dimnkx(xp) (59)

It is possible to rearrange equation (59) to obtain

Dimn = Cimn

kx(0)− kx(xp)

kx(0) + kx(xp)e2ikx(0)xp , (60)

which we can insert back into equations (57) and (58) and obtain

Aimni sin (kx(0)x0)

kx(0) cos (kx(0)(xp − x0))− ikx(xp) sin (kx(0)(xp − x0))= Cimn

eikx(0)xp

kx(0) + kx(xp)(61)

Aimn cos (kx(0)x0)

−ikx(0) sin (kx(0)(xp − x0)) + kx(xp) cos (kx(0)(xp − x0))= Cimn

eikx(0)xp

kx(0) + kx(xp)+ωµ0Kmn

kx(0)δiy (62)

For the z-component, equations (61) and (62) is a homogeneous two dimensional system of equations,which only has the trivial solution, giving us Azmn = Czmn = 0. This implies that also Bzmn = Dz

mn =F zmn = 0. We can therefore conclude that we have no electric field in the toroidal direction. For they-component however, we can combine (61) and (62) and obtain

Aymn =ωµ0Kmn

kx(0)

kx(0) cos (kx(0)(xp − x0))− ikx(xp) sin (kx(0)(xp − x0))

kx(0) cos (kx(0)xp)− ikx(xp) sin (kx(0)xp)(63)

Bymn = −Aymn (64)

Cymn =ωµ0Kmn

kx(0)

i sin (kx(0)x0) (kx(0) + kx(xp)) e−ikx(0)xp

kx(0) cos (kx(0)xp)− ikx(xp) sin (kx(0)xp)(65)

Dymn =

ωµ0Kmn

kx(0)

i sin (kx(0)x0) (kx(0)− kx(xp)) eikx(0)xp

kx(0) cos (kx(0)xp)− ikx(xp) sin (kx(0)xp)(66)

F ymn = 2ωµ0Kmni sin (kx(0)x0)e−ikx(xp)xp

kx(0) cos (kx(0)xp)− ikx(xp) sin (kx(0)xp)(67)

The x-component can then be obtained from (38)

Exmn =ikm,yk2x

∂Eymn∂x

(68)

From this relation, we obtain

30

Axmn = − km,ykx(0)

Aymn (69)

Bxmn =km,ykx(0)

Bymn (70)

Cxmn = − km,ykx(0)

Cymn (71)

Dxmn =

km,ykx(0)

Dymn (72)

F xmn = − km,ykx(xp)

F ymn (73)

31

B Fast Wave Dispersion Relation

To derive the dispersion relation of the fast magnetosonic wave, wave start with the homogeneous Fouriertransformed wave equation (see appendix A for derivation)

k × k ×E +ω2

cK ·E = 0, (1)

or in index notation,

kikjEj − k2δijEj +ω2

c2KijEj = 0. (2)

We can rewrite this as

ΛijEj = 0 (3)

by defining the wave operator Λij as

Λij =c2

ω2(kikj − k2δij) +Kij . (4)

Equation (3) only has non-trivial solutions if the determinant of the wave operator is 0. By assuming thatthe magnetic field is in the z-direction, and that the wave propagates in the xz-plane, it is possible towrite the wave operator in matrix form. We also assume that the dielectric tensor is a hot bi-Maxwellianplasma tensor (given in section 3.2). The wave operator is then given by

Λ =

n2⊥ 0 n⊥n‖0 0 0

n⊥n‖ 0 n2‖

−n2 0 0

0 n2 00 0 n2

+

K1 K2 K4

−K2 K1 +K0 −K5

K4 K5 K3

, (5)

where we have defined n = ck/ω, n⊥ = ck⊥/ω and n‖ = ck‖/ω. We get

Λ =

K1 − n2‖ K2 K4 + n⊥n‖−K2 K1 +K0 − n2 −K5

K4 + n⊥n‖ K5 K3 − n2⊥

. (6)

Assuming that K3 is so large that the parallel electric field component can be ignored, it is sufficient tocompute ∣∣∣∣K1 − n2‖ K2

−K2 K1 +K0 − n2

∣∣∣∣ = 0. (7)

We obtain

(K1 − n2‖)(K1 +K0 − n2‖ − n2⊥) +K2

2 = 0. (8)

Solving for n2⊥ gives us

n2⊥ = K1 +K0 − n2‖ +K2

2

K1 − n2‖, (9)

or

k2⊥ =ω2

c2

(K1 +K0 − n2‖ +

K22

K1 − n2‖

). (10)

This is known as the fast wave dispersion relation.

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