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Fast wave heating and current drive in

tokamaks

Martin Laxåback

Doctoral Thesis

Alfvén Laboratory

Division of Fusion Plasma Physics

Royal Institute of Technology

Stockholm 2005

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v

Laxåback, Martin

Fast wave heating and current drive in tokamaks (in English).

Alfvén Laboratory, Division of Fusion Plasma Physics,

Royal Institute of Technology, Stockholm 2005.

Abstract

This thesis concerns heating and current drive in tokamak plasmas using the fast magnetosonic

wave in the ion cyclotron range of frequencies. Fast wave heating is a versatile heating method for

thermonuclear fusion plasmas and can provide both ion and electron heating and non-inductive

current drive. Predicting and interpreting realistic heating scenarios is however difficult due to

the coupled evolution of the cyclotron resonant ion velocity distributions and the wave field. The

SELFO code, which solves the coupled wave equation and Fokker-Planck equation for cyclotron

resonant ion species in a self-consistent manner, has been upgraded to allow the study of more

advanced fast wave heating and current drive scenarios in present day experiments and in prepa-

ration for the ITER tokamak.

Theoretical and experimental studies related to fast wave heating and current drive with em-

phasis on fast ion effects are presented. Analysis of minority ion cyclotron current drive in ITER

indicates that the use of a hydrogen minority rather than the proposed helium-3 minority results

in substantially more efficient current drive. The parasitic losses of power to fusion born alpha

particles and beam injected ions are concluded to be acceptably low. Experiments performed at

the JET tokamak on polychromatic ion cyclotron resonance heating and on fast wave electron cur-

rent drive are presented and analysed. Polychromatic heating is demonstrated to increase the

bulk plasma ion to electron heating ratio, in line with theoretical expectations, but the fast wave

electron current drive is found to be severely degraded by parasitic power losses outside of the

plasma. A theoretical analysis of parasitic power losses at radio frequency antennas indicates that

the losses can be significantly increased in scenarios with low wave damping and with narrow

antenna spectra, such as in electron current drive scenarios.

Descriptors

Tokamak, JET, ITER, thermonuclear fusion, fast wave, heating, current drive, ion cyclotron res-

onance, polychromatic, finite orbit widths, RF-induced transport, neutral beam injection, fusion

born alpha particles, magnetosonic eigenmodes, parasitic absorption, modelling, weighted Monte

Carlo scheme

vii

Acknowledgement

First of all I would like to extend my sincerest gratitude to my supervisor Prof. Torbjörn Hellsten

for his contagious enthusiasm and for many hours of stimulating discussion. I would also like to

thank all the past and present Ph.D. students at the Lab who have been my colleagues and friends

these years, particularly Johan Hedin for introducing me to the many idiosyncrasies of life as a

Ph.D. student, Thomas Johnson for being such a great post conference travelling companion - from

Death Valley and Yellowstone to the Vatican and Pompeii, Tommy Bergkvist for always pushing for

more movie pubs according to the “quantity over quality” principle and Tomas Hurtig for dragging

me along in his pursuit of life, limb and sanity threatening extra-curricular activities. I am deeply

indebted to Jerzy Brzozowski for his unquestionable hospitality and entertaining company during

my visits to JET. At JET, I have had the pleasure of working and interacting with far to many people

to name in this limited space, but Mervi Mantsinen, Philippe Lamalle and Antti Salmi deserve special

mention and thanks.

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List of publications

Included in the thesis

This thesis is based on the work presented in the following included papers:

I. M. Laxåback, T. Hellsten and T. Johnson,

“Self-consistent simulations of ICRH in ITB plasmas”,

Proceedings of the 14:th Topical Conference on Radio Frequency Power In Plasmas, Oxnard,

California 2001.

II. M. Laxåback, T. Hellsten and T. Johnson,

“Self-consistent RF modelling of beam and ICRF heated plasmas”,

Proceedings of the Joint Varenna-Lausanne International Workshop, Varenna, Italy 2002.

III. T. Hellsten, T. Johnson, J. Carlsson, L.-G. Eriksson, J. Hedin, M. Laxåback and M. Mantsinen,

“Effects of finite drift orbit width and RF-induced spatial transport on plasma heated by ICRH”,

Nuclear Fusion 44, 892-908, 2004.

IV. M. Laxåback and T. Hellsten,

“Modelling of minority ion cyclotron current drive during the activated phase of ITER”,

Submitted to Nuclear Fusion, 2004.

V. M. Laxåback, T. Johnson, T. Hellsten and M. Mantsinen,

“Self-consistent modelling of polychromatic ICRH in tokamaks”,

Proceedings of the 15:th Topical Conference on Radio Frequency Power In Plasmas, Moran,

Wyoming 2003.

VI. M. Mantsinen, V. Kiptily, M. Laxåback, A. Salmi, Yu. Baranov, R. Barnsley, P. Beaumont, S.

Conroy, P. de Vries, C. Giroud, C. Gowers, T. Hellsten, L.C. Ingesson, T. Johnson, H. Leggate,

M.-L. Mayoral, I. Monakhov, J.-M. Noterdaeme, S. Podda, S. Sharapov, A.A. Tucillo, D. Van

Eester and EFDA JET contributors,

“Fast ion distributions driven by polychromatic ICRF waves on JET”,

Draft of manuscript to be submitted to Plasma Physics and Controlled Fusion, 2005.

VII. T. Hellsten, M. Laxåback, T. Bergkvist, T Johnson, F. Meo, F. Nguyen, C.C. Petty, M. Mantsinen,

G. Matthews, J.-M. Noterdaeme, T. Tala, D. Van Eester, P. Andrew, P. Beaumont, M. Brix, J.

Brzozowski, L.-G. Eriksson, C. Giroud, E. Joffrin, V. Kiptily, J. Mailloux, M.-L. Mayoral, I. Mon-

akhov, R. Sartori, A. Staebler, E. Rachlev, E. Tennfors, A.A. Tucillo, V. Bobkov, K.-D. Zastrow

and contributors to the EFDA-JET Workprogramme,

“On the parasitic absorption in FWCD experiments in JET ITB plasmas”,


VIII. T. Hellsten and M. Laxåback,

“Influence of coupling to spectra of weakly damped eigenmodes in the ion cyclotron range of

frequencies on parasitic absorption in rectified radio frequency sheaths”,

Accepted for publication in Physics of Plasmas, 2004.

IX. T. Hellsten and M. Laxåback,

“Edge localized magnetosonic eigenmodes in the ion cyclotron frequency range”,

Physics of Plasmas 10, 4371-4377, 2003.

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Not included in the thesis

The following papers are not included in the thesis since since they represent intermediate steps

or since the contributions from the thesis author are minor:

X. T. Hellsten, J. Hedin, T. Johnson, M. Laxåback and E. Tennfors,

“Self-consistent modelling of ICRH”,

Proceedings of the 18:th IAEA Fusion Energy Conference, Sorrento, Italy 2000.

XI. A.A. Tucillo, Y. Baranov, E. Barbato, Ph. Bibet, C. Castaldo, R. Cesario, V. Cocilovo, F. Crisanti,

R. De Angelis, A.C. Ekedahl, A. Figueiredo, M. Graham, G. Granucci, D. Hartmann, J. Heikki-

nen, T. Hellsten, F. Imbeaux, T.T.H. Jones, T. Johnson, K.V. Kirov, P. Lamalle, M. Laxåback, F.

Leuterer, X. Litaudon, P. Maget, J. Mailloux, M.J. Mantsinen, M.L. Mayoral, F. Meo, I. Monakhov,

F. Nguyen, J-M. Noterdaeme, V. Pericoli-Ridolfini, S. Podda, L. Panaccione, E. Righi, F. Rimini,

Y. Sarazin, A. Sibley, A. Staebler, T. Tala, D. Van Eester and EFDA-JET Work-Programme Con-

tributors,

“Recent heating and current drive results on JET”,


California 2001.

XII. T. Hellsten, J. Hedin, J. Carlsson, L.-G. Eriksson, T. Johnson, M. Laxåback and M. Mantsinen,

“Effects of finite drift orbit width and RF-induced spatial transport on plasmas heated by ICRH”,


California 2001.

XIII. R.C. Wolf, Y. Baranov, C. Giroud, M. Mantsinen, D. Mazon, K.-D. Zastrow, N. Hawkes, T. Hell-

sten, J. Hobirk, M. Laxåback, F. Rimini, A. Stäbler, F. Ryter, J. Stober, H. Zohm, the ASDEX

Upgrade Team and the Contributors to the EFDA-JET Workprogramme,

“Influence of electron heating on confinement in JET and ASDEX Upgrade internal transport

barrier plasmas”,

Proceedings of the 28:th EPS Conference on Controlled Fusion and Plasma Physics, Funchal,

Portugal 2001.

XIV. T. Hellsten and M. Laxåback,

“Edge localized magnetosonic eigenmodes in the ion cyclotron frequency range”,


XV. T. Johnson, T. Hellsten, L.-G. Eriksson and M. Laxåback,

“Numerical modelling of ICRH induced torques”,


XVI. T. Hellsten, T. Johnson, L.-G. Eriksson and M. Laxåback,

“Effects of finite drift orbit width and RF-induced spatial transport on plasma rotation by ICRH”,

Proceedings of the 29:th EPS Conference on Controlled Fusion and Plasma Physics, Montreux,

Switzerland 2002.

XVII. M. Mantsinen, M. Laxåback, A. Salmi, V. Kiptily, D. Testa, Yu. Baranov, R. Barnsley, P. Beau-

mont, S. Conroy, P. de Vries, C. Giroud, C. Gowers, T. Hellsten, L.C. Ingesson, T. Johnson, H.

Leggate, M.-L. Mayoral, I. Monakhov, J.-M. Noterdaeme, S. Podda, S. Sharapov, A.A. Tucillo, D.

Van Eester and EFDA-JET contributors,

“Comparison of monochromatic and polychromatic ICRH on JET”,


Wyoming 2003.

XVIII. T. Hellsten and M. Laxåback,

“The effect of weak single pass damping on the coupled ICRH power spectrum”,


Wyoming 2003.

XIX. T. Hellsten, T. Johnson, M. Laxåback, M. Mantsinen, G. Matthews, P. Beaumont, V. Bobkov,

C. Challis, D. Van Eester, E. Rachlev, T. Bergkvist, C. Giroud, E. Joffrin, A. Huber, V. Kiptily,

xi

F. Nguyen, J.-M. Noterdaeme, J. Mailloux, M.-L. Mayoral, F. Meo, I. Monakhov, F. Sartori, A.

Staebler, E. Tennfors, A.A. Tucillo and contributors to the EFDA-JET Workprogramme,

“Fast wave current drive in JET ITB-plasmas”,


Wyoming 2003.

XX. T. Johnson, T. Hellsten, L.-G. Eriksson and M. Laxåback,

“Modelling of ICRH induced current and rotation”,


Wyoming 2003.

XXI. T. Bergkvist, T. Hellsten, T. Johnson and M. Laxåback,

“Nonlinear interaction between RF-heated high energy ions and MHD-modes”,


Wyoming 2003.

XXII. P.U. Lamalle, M.J. Mantsinen, B. Alper, P. Beaumont, L. Bertalot, Vl.V. Bobkov, G. Bonheure, J.

Brzozowski, S. Conroy, M. de Baar, P. de Vries, G. Ericsson, V. Kiptily, M. Laxåback, K. Lawson,

M. Mironov, J.-M. Noterdaeme, S. Popovichev, M. Santala, M. Tardocchi, D. Van Eester and JET

EFDA contributors,

“Investigation of low concentration tritium ICRF heating on JET”,

Proceedings of the 31:st EPS Conference on Controlled Fusion and Plasma Physics, London,

England 2004.

XXIII. A. Salmi, P. Beaumont, P. de Vries, L.-G. Eriksson, C. Gowers, P. Helander, M. Laxåback, M.J.

Mantsinen, J.-M. Noterdaeme, D. Testa and EFDA JET contributors,

“JET experiments to assess finite Larmor radius effects on resonant ion energy distribution

during ICRF heating”,

Proceedings of the 31:st EPS Conference on Controlled Fusion and Plasma Physics, London,

England 2004.

XXIV. T. Hellsten, T. Bergkvist, T. Johnson and M. Laxåback,

“Non-linear study of fast particle excitation of global Alfvén eigenmodes during ICRH”,

Proceedings of the 20:th IAEA Fusion Energy Conference, Vilamoura, Portugal 2004.

XXV. P.U. Lamalle, M.J. Mantsinen, J.-M. Noterdaeme, B. Alper, P. Beaumont, L. Bertalot, T. Blackman,

Vl.V. Bobkov, G. Bonheure, J. Brzozowski, C. Castaldo, S. Conroy, M. de Baar, E. de la Luna, P.

de Vries, F. Durodié, G. Ericsson, L.-G. Eriksson, C. Gowers, R. Felton, J. Heikkinen, T. Hellsten,

V. Kiptily, K. Lawson, M. Laxåback, E. Lerche, P. Lomas, A. Lyssoivan, M.-L. Mayoral, F. Meo, M.

Mironov, I. Monakhov, I. Nunes, S. Popovichev, A. Salmi, M.I.K. Santala, S. Sharapov, T. Tala,

M. Tardocci, D. Van Eester, B. Weyssow and JET EFDA contributors,

“Expanding the operating space of ICRF on JET with a view to ITER”,

Proceedings of the 20:th IAEA Fusion Energy Conference, Vilamoura, Portugal 2004.

XXVI. T. Bergkvist, T. Hellsten, T. Johnson and M. Laxåback,

“Non-linear study of fast particle excitation of global Alfvén eigenmodes during ICRH”,


XXVII. P.U. Lamalle, M.J. Mantsinen, J.-M. Noterdaeme, B. Alper, P. Beaumont, L. Bertalot, T. Blackman,

Vl.V. Bobkov, G. Bonheure, J. Brzozowski, C. Castaldo, S. Conroy, M. de Baar, E. de la Luna, P.

de Vries, F. Durodié, G. Ericsson, L.-G. Eriksson, C. Gowers, R. Felton, J. Heikkinen, T. Hellsten,

V. Kiptily, K. Lawson, M. Laxåback, E. Lerche, P. Lomas, A. Lyssoivan, M.-L. Mayoral, F. Meo, M.

Mironov, I. Monakhov, I. Nunes, G. Piazza, S. Popovichev, A. Salmi, M.I.K. Santala, S. Sharapov,

T. Tala, M. Tardocci, D. Van Eester, B. Weyssow and JET EFDA contributors,

“Expanding the operating space of ICRF on JET with a view to ITER”,


XXVIII. A. Salmi, P. Beaumont, P. de Vries, L.-G. Eriksson, C. Gowers, P. Helander, M. Laxåback, M.J.

Mantsinen, J.-M. Noterdaeme, D. Testa and EFDA JET contributors,

“JET experiments to assess the clamping of the fast ion energy distribution during ICRF heating

due to finite Larmor radius effects”,

To be submitted to Plasma Physics and Controlled Fusion, 2005.

xiii

Contents

1 Introduction 1

1.1 Thermonuclear fusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Magnetic confinement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.3 Plasma heating . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.3.1 Neutral beam heating . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.3.2 Radio frequency heating . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.4 The tokamak . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.4.1 Experimental devices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.4.2 JET - the Joint European Torus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.4.3 ITER - “the way” towards fusion energy . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2 The fast magnetosonic wave 11

2.1 Wave equation and dielectric tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.2 Dispersion relation and polarisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.3 Ion cyclotron resonance interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.4 Evolution of ion drift orbits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.5 Minority ion cyclotron current drive . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.6 TTMP/ELD and electron current drive . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

3 Antenna coupling 21

3.1 Coupled toroidal mode spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

3.2 Impurities and parasitic power losses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

4 Numerics and modelling 25

4.1 The LION code . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

4.2 The FIDO code . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

4.3 The SELFO code . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

4.3.1 Weighted Monte Carlo scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

4.3.2 Neutral beam injection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

4.3.3 Fusion born α particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

5 Summary of publications 29

5.1 Modelling of ICRH and fast particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

5.1.1 Paper I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

5.1.2 Paper II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

5.1.3 Paper III . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

5.1.4 Paper IV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

5.2 Polychromatic ICRH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

5.2.1 Paper V . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

5.2.2 Paper VI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

5.3 FWCD, parasitic absorption and magnetosonic eigenmodes . . . . . . . . . . . . . . . . . 31

5.3.1 Paper VII . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

5.3.2 Paper VIII . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

5.3.3 Paper IX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

5.4 Contributions from the thesis author . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

6 Conclusions 35

1

1 Introduction

Between the years 1971 and 2001 the total primary energy supply of the world increased from

5 012 to 10 029 Mtoe1 and the electricity generation increased from 5 224 to 15 476 TWh [1, 2]. In

the period until 2020 these quantities are projected to continue to increase, annually by about 2%

and 2.7% respectively [1]. While the increased energy demand is expected to be manageable in the

short term through increased production of crude oil, alternative energy sources must inevitably

be found, out of economical necessity if not for ecological reasons. Cost-effective production of

conventional oil is expected to peak, followed by a sharp drop, some time between 2026 and 2047

depending on the ultimately recoverable reserves of oil [3].

In spite of their many apparent virtues the renewable energy sources popularly suggested to re-

place oil, e.g. wind and biomass, can be proved rather unsuitable for large scale energy production

through back-of-an-envelope calculations. Their inefficient conversion of solar energy to electricity

would require that unrealistically large areas be used for energy production in densely populated

areas. Alternatives more suitable for efficient energy production are nuclear fission, including ad-

vanced reactors of the breed- and spallation type to meet the limited uranium resources and the

nuclear waste problem [4,5], and thermonuclear fusion.

1.1 Thermonuclear fusion

Thermonuclear fusion is the process that powers the stars. Light atomic nuclei are fused together

in the immense temperatures and pressures of stellar cores, forming ever heavier elements and

releasing energy as the fusion products descend the binding energy potential. On Earth, ther-

monuclear fusion has the potential to provide humanity with an abundance of clean and safe

energy from hydrogen and lithium isotopes present in ordinary sea water.

In a future fusion reactor power plant the nuclear reaction

2H+ 3H→ 4He+ n (∆E = 17.6 MeV) (1.1)

is planned to be used as the source of energy production, where the hydrogen isotopes 2H and 3H

are also known as deuterium, D, and tritium, T. In the reaction 17.6 MeV of energy is released in the

form of kinetic energy, 14.1 MeV of which goes to the neutron and 3.5 MeV to the helium nucleus,

or α particle. Since the neutron irradiation will activate the surrounding structure tritium is rarely

used in present day experiments.

Deuterium is a stable isotope and present in ordinary sea water at a deuterium to protium ratio

of nD/nH ≈ 1.5× 10−4 [6]. Tritium on the other hand is radioactive and β decays with a half-life

of 12.3 years. It is therefore not naturally occurring in any significant amount. Fortunately, tritium

can be produced on-site, using the neutrons emitted from the reactor core to induce fission of the

common lithium isotopes 6Li and 7Li [6] to tritium and helium in a blanket around the reactor.

There are sufficient reserves of lithium in the oceans to supply electricity generated by DT fusion

at the 2001 level of electricity generation for approximately 100 million years. The deuterium

reserves are expected to last approximately 300 billion years [7].

For two nuclei to fuse they must overcome the mutual Coulomb repulsion between their positive

charges in order to get close enough for the nuclear forces to act. This implies that the nuclei must

have very high kinetic energies, as is illustrated in figure 1.1 which shows how the DT fusion cross

section peaks at over 100 keV [6]. Figure 1.1 also indicates that other fusion reactions are possible,

such as D+D (∆E = 3.7 MeV) and D+3He (∆E = 18.3 MeV).

1Million tonnes oil equivalent.

2 1. Introduction

101

102

103

104

10−33

10−32

10−31

10−30

10−29

10−28

10−27

D+D

D+TD+3He

E [keV]

σ [m

−2 ]

Figure 1.1 Fusion cross sections for some reactions proposed for use in reactors.

Integrating the fusion cross sections over the Maxwellian velocity distributions of thermalised

gaseous media gives the fusion reaction rates as function of isotropic temperature, illustrated in

figure 1.2 [6]. Note that the temperature is given in units of electron volts2, eV, as is customary

in fusion research. Fusion reactors are projected to operate in the 10–20 keV temperature range,

corresponding to roughly 100–200 million K. Since the magnitude of these temperatures is of-

ten brought forward as an argument against the feasibility of thermonuclear fusion it should be

mentioned that temperatures of up to 40 keV have already been reached in fusion experiments [8].

1.2 Magnetic confinement

At the temperatures required for thermonuclear fusion matter exists in the form of plasma3,

ionised, quasi-neutral, gas where the electrically charged particles display a collective behaviour

influenced by electric and magnetic fields in the plasma. From including the Lorentz force in

Newton’s equation of motion:

mdv

dt= Ze(E+ v× B) (1.2)

it is evident that a particle with massm and charge Ze will not only experience an acceleration par-

allel to any electric field E present, but also perpendicular to it’s own velocity v and any magnetic

field B. Thus charged particles will spiral magnetic field lines with a frequency called the gyro- or

cyclotron frequency fc =ωc/2π :

ωc =Ze|B|

m(1.3)

Since there is no velocity dependence on ωc all particles with common charge to mass ratios will

share the same cyclotron frequency at the same magnetic field. The radius of gyration is denoted

21 eV corresponds to 11 600 K.3The word “plasma” is of Greek origin, meaning “to mold” or “shape”.

1.2. Magnetic confinement 3

100

101

102

103

10−26

10−25

10−24

10−23

10−22

10−21

D+D

D+T D+3He

T [keV]

<σ

v>

[m

3 s−1 ]

Figure 1.2 Fusion reaction rates averaged over Maxwellian velocity distributions.

the gyro- or Larmor radius ρ:

ρ =v⊥ωc

=mv⊥Ze|B|

(1.4)

where v⊥ is the component of the velocity that is perpendicular to the magnetic field. Parallel to

the magnetic field particles move freely and the ratio of parallel to total velocity is called the pitch

angle, ξ = v‖/|v|. The centre of the projection of the gyro motion on a plane perpendicular to the

magnetic field is called the gyro centre.

The influence of magnetic fields on the motion of charged particles can be used to confine the

plasma. Ideally, the magnetic field lines should either close upon themselves, be confined to a

surface or at least remain confined within a finite volume in order to prevent losses of particles

following field lines out of the confinement device. This can be achieved by making the plasma

toroidal, figure 1.3, with the dominant magnetic fields either in the poloidal direction, like in the

RFP4 [9], or in the toroidal direction, like in the tokamak [10–12] and the stellarator [13]. Coordinate

systems for toroidal devices are illustrated in figure 1.4.

However, evaluating equation (1.2) for the gyro-averaged motion of a charged particle reveals

that inhomogeneities in the magnetic field causes the gyro centres to drift across magnetic field

lines [15]. A field line curvature radius Rc or a gradient in the magnetic field strength B introduces

the respective drift velocities vc and v∇B :

vc = −mv2

‖

Ze|B|2|Rc|2B×Rc (1.5)

v∇B =mv2

⊥

2Ze|B|3B×∇B (1.6)

Devices with purely toroidal magnetic fields are therefore subject to severe plasma instabilities,

as are devices with purely poloidal magnetic fields induced by toroidal plasma currents [16]. By

4Reversed Field Pinch.

4 1. Introduction

Figure 1.3 A charged particle confined in a toroidal device with a helical magnetic field made up of both

toroidal and poloidal components. From reference [14].

combining both toroidal and poloidal field components the gyro centre drifts can however be made

quasi-periodic and the plasma can be made stable. The projection in the poloidal plane of the path

taken by a gyro centre during one such periodic orbit is called a drift orbit.

Another effect of the toroidal form of the plasma is that the toroidal magnetic field strength will

be higher on the inside of the plasma than on the outside:

B ≈B0R0

R(1.7)

The gyro centre of a charged particle will therefore experience a magnetic mirror force, F‖ =

−µ∇‖B, from the conservation of magnetic moment, µ = mv2⊥/2B [15], as it follows the helical

field lines around the plasma torus towards the high field side. Particles with sufficiently low

magnetic moment, µ < E/Bmax where E is the particle energy and Bmax is the highest magnetic

field along the orbit, can circulate around the torus and are said to be passing. Particles with higher

µ will be reflected back towards the low field side at the point along the orbit where B = E/µ. These

particles are said to be trapped.

1.3 Plasma heating

A fusion reactor operating in steady-state is planned to sustain a temperature in the optimum

10–20 keV range through the bulk plasma heating by the slowing down of fusion born 3.5 MeV

α particles. In order to reach these temperatures and initiate the burning plasma5 a method of

heating the plasma to thermonuclear temperatures must be devised.

Early experiments relied on the ohmic heating from the plasma current to heat the plasma to

temperatures of around 1 keV. Higher temperatures are increasingly more difficult to reach with

only ohmic heating due to the low electrical resistivity of high temperature plasmas, η∝ T−3/2e [18],

and due to limitations on the plasma current from stability considerations [12,16]. Heating systems

other than ohmic- or α-particle heating are referred to as auxiliary heating systems, despite the

fact that they usually represent the primary source of plasma heating in fusion experiments today.

Since the fusion reaction rates increase with plasma temperature, applying auxiliary heating to

a fusion plasma will in general increase the fusion power. The quotient of the fusion power and

the auxiliary heating power is denoted the energy multiplication factor, or Q = Pfus/Paux . For a

burning plasma the energy multiplication factor will be infinite, but an energy producing reactor

can also operate at finite Q.

An efficient auxiliary heating method should deliver energy to the central part of the plasma and

reliably deposit it there. The two major classes of auxiliary heating commonly used are neutral

beam injection (NBI) heating, and radio frequency (RF) heating.

5A fusion plasma is considered burning if it can sustain fusion without auxiliary heating.

1.3. Plasma heating 5

p

Bφ

θBθB

IpBφ

I

Bθ

a

θ

Side view of cross section

rZ

R

R

R

φ

0

Top view

Figure 1.4 Coordinate systems for toroidal devices. (R,Z,φ), where R is the major radius, Z the height

above the midplane and φ the toroidal angle in the direction of the toroidal magnetic field, is often used

when referring to quantities strongly affected by toroidicity, such as local magnetic field, cyclotron resonance

positions etc. (r ,φ,θ), where r is the minor radius and θ the poloidal angle, is more common for quantities

with weak or no dependence on toroidicity, such as temperature, density and current profiles. Note that the

plasma current is negative in this picture. From reference [17].

6 1. Introduction

1.3.1 Neutral beam heating

Injecting charged particles into the plasma core is difficult due to the strong magnetic fields used

to confine the plasma. Neutral atoms, unaffected by the fields, can however reach deep into the

core if they are injected at high enough velocities to delay ionisation. Efficient central heating can

be achieved by adjusting the injection energy to the plasma density and to the size of the device.

This implies injection energies around 50–150 keV in most current experiments and up to 1 MeV

in the next generation of experiments.

The beams are produced by letting electrostatically accelerated ions pass through a neutralisa-

tion cell before they enter the plasma. Once in the plasma the beam atoms are ionised by charge

exchange, particularly with high-Z impurity ions, or by collisions. Since their energy is consider-

ably higher than the bulk plasma temperature they will collisionally heat the plasma as they slow

down to thermal energies.

Beams injected with a velocity component parallel to the toroidal direction can also drive plasma

current, provided that the charge of the beam ions differs sufficiently from the effective charge

Zeff =∑

iniZ2i /∑

iniZi of the plasma ions so that the mean parallel velocity of the scattered

electrons differs from the mean parallel ion velocity [19], c.f. equation (2.19).

1.3.2 Radio frequency heating

Power can be electromagnetically coupled to the plasma over a wide range of frequencies [20]

where it can be absorbed when a local wave-plasma resonance condition is fulfilled, e.g. through

ion or electron cyclotron resonance damping [21] or Landau damping [22]. Usually a small fraction

of the resonant plasma species absorbs the RF power and transfers it to the bulk plasma through

Coulomb collisions. Radio frequency current drive is possible if the properties of the wave are

chosen such that asymmetries in velocity space are created in the distribution functions of the

resonant plasma species.

The most commonly used RF heating schemes are ion cyclotron resonance heating (ICRH), elec-

tron cyclotron resonance heating (ECRH), and lower hybrid current drive (LHCD). The possibility

to control the spatial position of the resonance, and thereby the position of localised heating and

possible current drive, makes RF heating an effective control tool for both present experiments

and future reactors.

1.4 The tokamak

The tokamak6 [10–12], originally conceived in the 1960:s in the Soviet Union, is currently the most

promising concept for the realisation of controlled thermonuclear fusion. Characteristic for the

tokamak equilibrium is a strong toroidal magnetic field, created by external magnetic field coils,

and a weaker poloidal magnetic field, created by a current flowing in the toroidal direction in the

plasma [12]. The plasma current is usually driven inductively by using the plasma torus as the

secondary winding of a transformer, figure 1.5.

The combined toroidal and poloidal magnetic fields create helical magnetic field lines spiralling

the magnetic axis of the device, the centre of the plasma current distribution where the poloidal

magnetic field goes to zero. In an axisymmetric equilibrium these field lines form nested flux

surfaces, i.e. surfaces of constant ψ, where the flux function ψ is proportional to the amount of

poloidal magnetic flux contained inside the surface. Charged particles are in general confined to

the vicinity of individual flux surfaces, but for high-energy ions the excursion across flux surfaces

during a quasi-periodic drift orbit can be significant. The number of toroidal turns a field line on a

flux surface must complete in order to return to the same poloidal angle is called the safety factor,

or q value. The q profile is typically around unity at the magnetic axis and increases to about 3–6

at the plasma edge. If the central current density increases sufficiently for q to drop below unity

an instability occurs and a change in the magnetic topology transfers energy, particles and current

across the q = 1 surface. After this crash the central temperature and current density slowly

recovers until q drops below unity again and the instability is repeated, causing characteristic

6TOroidalnaya KAmera i MAgnitnaya Katushka.

1.4. The tokamak 7

magnetic fieldResultant helical

(twist exaggerated)

field coilsToroidal

magnetic fieldPoloidal

(primary circuit)Transformer winding

(secondary circuit)Plasma current

coreIron transformer

Toroidalmagnetic field

Figure 1.5 Schematic drawing of the tokamak configuration. Courtesy of JET (JG91.256/4).

oscillations in the time trace of the central temperature which have led to the name “sawtooth

instability”.

Several distinct modes of tokamak operation have been identified with respect to thermal and

particle transport. At low levels of auxiliary heating the plasma is said to be in L-mode7, charac-

terised by low edge temperature and density and strong Dα radiation. Increasing the power so

that the energy flux across the last closed flux surface exceeds a certain threshold spontaneously

triggers a transport barrier at the plasma edge, resulting in high edge temperature and density and

low Dα radiation. In this H-mode8 the energy confinement time, which is defined as the plasma

thermal energy divided by the total heating power and is a measure of the insulating properties of

the plasma, is increased by a factor of around 2 compared to L-mode [23]. Under specific condi-

tions an internal transport barrier (ITB) can form in the plasma, particularly at a rational q surface

in a region of the plasma where the magnetic shear s = (r/q)dq/dr is low or negative [24]. The

confinement improvement with an ITB over the H-mode varies and depends on the radial location

and the strength of the barrier.

1.4.1 Experimental devices

Tokamak experiments range from small, “table-top”, devices with major radii of a few decimetres

to flagship class machines such as JET and JT-60U with major radii up to 3 m. Toroidal magnetic

fields typically range from 0.5–5 T and plasma currents up to several MA are used. The shell

surrounding the plasma, the vacuum vessel, either has a circular poloidal cross section (TFTR, Tore-

Supra, Textor, Alcator C-Mod etc) or a more complex triangular D-shaped cross section (JET, JT-60U,

Asdex Upgrade, D-IIID etc). Most devices with non-circular cross section utilise the possibility of

easily including a divertor in the design, a pumped dump for energy and particles escaping from

7Low confinement mode.8High confinement mode.

8 1. Introduction

Figure 1.6 The interior of JET. Two of the four A2 ICRH antennas are visible. Note the tilted Faraday screens

in front of the current straps [28].

the plasma. The presence of a divertor, or at least a magnetic X-point, is in general a prerequisite

for H-modes to form.

Most tokamak experiments today are run in pulsed operation, with pulse lengths ranging from

milliseconds to tens of seconds. Pulsed experiments can use conventional electromagnets to create

the toroidal magnetic field and rely on the transformer action to drive most of the plasma current.

Steady-state physics can be explored provided that the plasma pulse is significantly longer than

the typical time scales in the plasma, which is usually the case. True steady-state operation can

be achieved by using superconducting toroidal field coils and by driving the plasma current non-

inductively using a combination of the auxiliary heating systems described in section 1.3 and a

self-organising pressure gradient driven current aptly named the bootstrap current [25]. Steady-

state operation using superconducting magnetic field coils and fully non-inductive current drive

by bootstrap- and lower hybrid range radio frequency driven currents has been demonstrated on

the TRIAM-1M and Tore-Supra tokamaks [26,27].

1.4.2 JET - the Joint European Torus

JET [28, 29] is the largest magnetic confinement fusion experiment in the world and after the

decommissioning of TFTR9 the only machine with the capability of operating with tritium. JET cur-

rently holds the records for maximum fusion power and energy multiplication factor Q, 16.1 MW

and 0.62 respectively [30,31] (corresponding to a steady-state Q of 0.95 [32]).

The plasma has a roughly D-shaped cross section with a major radius of 3 m, a minor radius in

the midplane of around 1 m and an elongation around 1.6, yielding a total plasma volume around

80 m3, depending on the equilibrium configuration [29]. The toroidal magnetic field is nominally

3.45 T but can be varied from around 1 T up to 4 T. Typical plasma currents range from 2–4 MA.

Auxiliary heating and current drive is handled by up to 23 MW of NBI (2 sets of injectors, 80 keV

9Tokamak Fusion Test Reactor.

1.4. The tokamak 9

Figure 1.7 Cutaway diagram of JET [28].

and 140 keV injection energies), 10 MW of ICRH (4 antennas, 23–57 MHz) and 5 MW of LHCD. The

planned addition of a new ICRH antenna is expected to add another 7 MW.

Experiments at JET are co-ordinated through EFDA, the European Fusion Development Agree-

ment [33].

1.4.3 ITER - “the way” towards fusion energy

While the major fusion experiments today can reach the conditions required for thermonuclear

fusion they are still too small to properly confine the fusion born α particles and to provide long

enough energy confinement times for the demonstration of a burning plasma. ITER10 [34, 35] is

designed to demonstrate the energy producing capability of magnetically confined thermonuclear

fusion and the reactor potential of the tokamak.

In the present iteration of the design the machine has major and minor radii of 6.2 and 2 m re-

spectively, an elongation of 1.8 and a plasma volume of 840 m3. During nominal H-mode operation

with a toroidal magnetic field of 5.3 T, an inductively driven plasma current of 15 MA and 40 MW of

auxiliary heating the fusion power is expected to be 400 MW for an energy multiplication factor Q

10Originally an acronym for International Thermonuclear Experimental Reactor.

10 1. Introduction

Figure 1.8 Cutaway diagram of ITER [35].

of 10. Steady-state operation relying on the bootstrap current and currents driven by the auxiliary

heating systems is envisaged at Q ≈ 2 in H-mode and at Q ≈ 5 with an ITB [36]. Auxiliary heating

is planned to initially consist of 33 MW of NBI (1 MeV injection energy), 20 MW of ICRH (40–55 MHz)

and 20 MW of ECRH (170 GHz) [37].

11

2 The fast magnetosonic wave

The fast magnetosonic wave1 propagates in the ion cyclotron range of frequencies in typical fusion

plasmas [38]. The wave can be absorbed in the plasma by cyclotron resonant ions [39], so called

ion cyclotron resonance heating (ICRH) or by electrons through transit time magnetic pumping

(TTMP) [21], and electron Landau damping (ELD) [22]. Ion cyclotron resonance occurs when the

wave frequency matches a harmonic of the cyclotron frequency of an ion species in the plasma.

The variation of the toroidal magnetic field, and thereby also of the cyclotron frequencies, with

major radius according to equations (1.7) and (1.3) allows the resonance layers to be positioned at

specific major radii by combinations of magnetic field and wave frequency. By launching the wave

from the midplane of the machine the refractive index and the convex shape of the plasma will

focus the wave towards the centre and, for scenarios with strong damping, yield efficient central

absorption, c.f. figures 2.2 and 2.3. For scenarios with weaker damping the wave will spread over

a larger plasma volume and the absorption will be broader. Magnetosonic eigenmode structures

may also form in plasmas with weak damping.

2.1 Wave equation and dielectric tensor

Combining Maxwell’s equations with the equation of motion and electric charge conservation gives

the wave equation:

∇×∇× E−ω2

c2ε · E = iµ0ωJext (2.1)

which describes the electric field E driven by the external current Jext and oscillating with the

frequency ω in a quasi-homogeneous medium described by the dielectric response tensor ε. The

local dielectric tensor of a plasma is the sum of the identity tensor 1 and the susceptibility χs of

each species s:

ε(ω,k) = 1+∑

s

χs(ω,k) (2.2)

The susceptibility of species s in a magnetic field can be calculated from the linear response of the

equilibrium distribution function f0,s , described by the linearised Vlasov equation, to a spatially

and temporally varying electric field E(x, t) = E0 exp[i(k · x−ωt)]:

χs =ω2p0,s

ωωc0,s

∞∫

0

2πp⊥dp⊥

∞∫

−∞

dp‖

[

e‖e‖ωc,s

ω

(

1

p‖

∂f0

∂p‖−

1

p⊥

∂f0

∂p⊥

)

p2‖

+

∞∑

n=−∞

ωc,sp⊥U

ω− k‖v‖ −nωc,sTn

]

s

(2.3)

Here ωp0,s is the plasma frequency, ωc0,s the non-relativistic cyclotron frequency and ωc,s the

relativistic cyclotron frequency of species s and p is the particle momentum.

U =∂f0

∂p⊥+k‖

ω

(

v⊥∂f0

∂p‖− v‖

∂f0

∂p⊥

)

(2.4)

1Also known as the compressional Alfvén wave.

12 2. The fast magnetosonic wave

and

Tn =

n2J2n

z2inJnJ

′n

znJ2

np‖zp⊥

−inJnJ

′n

z (J′n)2 −

iJnJ′np‖p⊥

nJ2np‖zp⊥

iJnJ′np‖p⊥

J2np

2‖

p2⊥

(2.5)

where Jn = Jn(z) is the n:th order Bessel function of argument z = k⊥v⊥/ωc and J′n = ∂Jn(z)/∂z

[21]. Note the resonant denominator ω − k‖v‖ − nωc,s in equation (2.3). It appears due to the

linearised treatment because the particle at the Doppler shifted resonance, where nωc,s =ω−k‖v‖,

experiences a constant electric field and is uniformly accelerated.

For species with non-relativistic Maxwellian velocity distributions the susceptibilities reduce to:

χs =ω2p

ω

∞∑

n=−∞

e−λYn,s(λ) (2.6)

where

Yn,s(λ) =

n2InλAn −in(In − I

′n)An

k⊥ωcnInλBn

in(In−I′n)An

(

n2

λIn+2λIn−2λI′n

)

Anik⊥ωc(In−I

′n)Bn

k⊥ωcnInλBn −

ik⊥ωc (In − I

′n)Bn

2(ω−nωc)k‖w

2⊥

InBn

(2.7)

In = In(λ) is the n:th order modified Bessel function of argument λ =k2⊥T⊥mω2

cand

An =1

ω

T⊥ − T‖

T‖+

1

k‖w‖

(ω−nωc)T⊥ +nωcT‖

ωT‖Z0 (2.8a)

Bn =1

k‖


ωT‖(2.8b)

+1

k‖

ω−nωck‖w‖


ωT‖Z0

Z0 =Z0(ζn), ζn =ω−nωck‖w‖

(2.8c)

where asymmetries from parallel plasma rotation have been neglected. T‖ and T⊥ are here respec-

tively the parallel and perpendicular temperatures, w‖ and w⊥ the thermal velocities and Z0 is the

plasma dispersion function [21].

It can be demonstrated that non-dissipative plasma processes are described by the Hermitian

parts of the dielectric tensor, εHij =12

(

εij + ε∗ji

)

, and dissipative processes by the anti-Hermitian

parts, εAij =12

(

εij − ε∗ji

)

[21,40].

2.2 Dispersion relation and polarisation

With an expression for the dielectric tensor it is possible to solve the dispersion relation for the

fast wave. After Fourier transformation the homogeneous wave equation can be written on matrix

form:

εxx −c2

ω2k2z εxy

c4

ω4kxkz

εyx εyy −c2

ω2k2x −

c2

ω2k2z 0

c4

ω4kxkz 0 εzz −c2

ω2k2x

ExEyEz

= 0 (2.9)

where the magnetic field has been placed in the z direction and the perpendicular part of the wave

vector k in the x direction. Noting that the frequency range of interest is much lower than the

electron plasma frequency we assume that any parallel electric fields are efficiently short-circuited

and set Ez = 0. The dispersion relation then becomes:∣

∣

∣

∣

∣

∣

εxx −c2

ω2k2z εxy

εyx εyy −c2

ω2k2x −

c2

ω2k2z

∣

∣

∣

∣

∣

∣

= 0 (2.10)

2.3. Ion cyclotron resonance interactions 13

2 2.5 3 3.5 4−200

0

200

400

600

800

1000

R [m]

k⊥2

[m

−2 ]

2 2.5 3 3.5 40

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

R [m]

|E

+/

E−|

Figure 2.1 Perpendicular wave number squared and wave field polarisation for a JET-like plasma; R0 = 3 m,

a = 1 m, B0 = 3.45 T, f = 51 MHz, nφ = 25, nH/nD = 5%, ne = 3 × 1019[1 − (r/a)2]0.5m−3, Zeff = 2.5 and

TH = TD = Te = 10 [1− 0.2 (r/a)2]10keV. The dashed line represents the H cyclotron resonance.

A solution to the dispersion relation in the midplane of a JET plasma, using the dielectric tensor

presented in equations (2.2) and (2.6), is illustrated in figure 2.1. k‖ is assumed to vary as nφ/R,

where nφ is the toroidal Fourier mode number of the wave. The electric field components are here

defined as E± =12(Ex ± iEy) and represent the co- and counter ion rotating electric fields respec-

tively. Note that at the hydrogen cyclotron resonance the E+ component is partially screened by

the minority hydrogen ions. Since the screening increases with the concentration of the resonant

species the E+ component will nearly vanish with the fundamental cyclotron resonance of a major-

ity ion species in the plasma. Also note that in the vacuum between the plasma and the antenna

the wave is evanescent in the radial direction (k2⊥ < 0), i.e. spatially decaying without absorption.

The wave field excited by the antenna has to tunnel through this layer in order to couple to the

plasma wave.

A full wave solution in a JET equilibrium with the same parameters as used in figure 2.1 is

illustrated in figures 2.2 and 2.3, calculated with the LION code which is described in section 4.1.

Figure 2.2 clearly demonstrates the focusing of the wave field towards the centre of the plasma,

the E+ screening at the cyclotron resonance and the stronger E+ field just on the high field side of

it. Although cyclotron absorption takes place all along the resonance, figure 2.3 demonstrates the

strong central absorption achieved with the focused wave field.

2.3 Ion cyclotron resonance interactions

The evolution of the distribution function fi = fi(x,v) of the resonant ion species i during ion

cyclotron resonance heating can be described by the Fokker-Planck equation:

∂fi∂t

+ v · ∇fi = Q(fi,E)+ C(fi)+ Si − L(fi) (2.11)

where Q is the quasi-linear RF operator, C is the Coulomb collision operator, Si is a general source

term, for example representing beam injected ions or fusion born α particles, and L describes

losses to the walls.


2 2.5 3 3.5

−1

−0.5

0

0.5

1

1.5

R [m]

Z [

m]

Figure 2.2 E+ wave field component for the scenario

in figure 2.1.

2 2.5 3 3.5

−1

−0.5

0

0.5

1

1.5

R [m]

Z [

m]

Figure 2.3 Absorbed power for the scenario in figure

2.1.

The quasi-linear RF operator gives the first order perturbation to the distribution function from

the interactions with the electric field of the wave. For a single ion the rate of change in energy

from the interaction with the E± wave field components can be written

∂E

∂t= Zev · E = Zev⊥

∞∑

n=−∞

(E+Jn−1 + E−Jn+1) eiν(t,n) (2.12)

where the perpendicular components of the electric field have been expanded around the gyro

centre in the quasi-homogeneous plasma approximation. Jn = Jn(k⊥ρ) is the n:th order Bessel

function and ν(t,n) =∫ tdτ(ω − nωc − k‖v‖) is the phase difference between wave oscillation

and ion gyro motion. When the condition nωc = ω − k‖v‖ for the Doppler shifted resonance is

met the phase difference is constant in time, leading to a constant electric field in the reference

frame of the ion and a constant acceleration or deceleration. If the phase difference is decorrelated

by collisions and non-linear effects between successive cyclotron interactions with the same wave

mode then the quasi-linear RF operator can be modelled by a diffusion process in velocity space

with a diffusion coefficient of the form

D ∼ |E+Jn−1(k⊥ρ)+ E−Jn+1(k⊥ρ)|2 (2.13)

where n is the harmonic of the cyclotron frequency at the local resonance. With the typical distri-

butions which decrease with increasing perpendicular velocity the diffusion leads to a net increase

in perpendicular velocity and absorption of wave energy.

Thermal ions, for which k⊥ρ << 1, are effectively heated only at the fundamental cyclotron fre-

quency, D ∼ |E+J0|2, and then only if the E+ screening at the resonance is minimised by resonating

only with a minority species. Heating at higher harmonic resonances is a finite Larmor radius effect

and requires that the resonating ion has a sufficiently large Larmor radius so that variations in the

wave field over a gyration results in a net acceleration or deceleration.

The modifications to the distribution function by the RF interactions work against the Coulomb

collisions with the approximately Maxwellian background plasma. The collisions both transfer en-

ergy from the accelerated ions to the bulk plasma and influence the distribution functions of the

2.4. Evolution of ion drift orbits 15

0 200 400 600 800 1000 1200 1400 1600 1800 2000

101

102

f

TEFF

f

E [keV]0 200 400 600 800 1000 1200 1400 1600 1800 2000

0

200

400

600

800

1000

1200

TE

FF [

keV

]

Figure 2.4 Energy spectrum and effective temperature of hydrogen after heating the scenario in figure 2.1

with 5 MW for 2 s using the symmetric spectrum nφ = ±25.

resonant ions. Since Coulomb interactions are strongest between particles with similar velocities

the RF-accelerated high-energy ions will preferentially interact with background electrons, where

the low momentum exchange predominantly leads to a slowing down of the ions. Ions with ener-

gies below the critical energy, Ec = 14.8TeA(∑

j njZ2j /(Ajne))

2/3, will interact more strongly with

background ions and are also strongly pitch-angle scattered [39, 41]. The resulting distribution

function of a resonant ion species during high-power ICRH therefore consists of a more or less

thermalised, isotropic, bulk distribution and a strongly anisotropic high-energy tail.

The energy spectrum and effective temperature, Teff = −[∂(ln f)/∂E]−1 [39], of the “steady-

state” distribution function of hydrogen in the JET-like plasma described in figure 2.1 is illustrated

in figure 2.4, calculated with the FIDO code which is described in section 4.2. Note that the high-

energy tail extends to several MeV and that the effective tail temperature varies significantly with

energy.

2.4 Evolution of ion drift orbits

In magnetically confined fusion plasmas the characteristic time scales for changes in ion energy

or pitch angle by wave-particle interactions and collisions are much longer than the characteristic

orbit times. This allows the distribution functions to be expressed in terms of invariants of motion

of unperturbed drift orbits and it is sufficient to evaluate only the orbit-averaged changes to the

invariants when solving the Fokker-Planck equation.

The unperturbed particle orbits in an axisymmetric tokamak can be described by three invariants

of motion and three angles [42]. Here we choose the three invariants to be energy E, normalised

magnetic moment Λ and canonical toroidal angular momentum Pφ:

E =mv2

2(2.14)

Λ =B0v

2⊥

Bv2(2.15)

Pφ = Rmvφ − Zeψ (2.16)


Figure 2.5 Regions in the (Λ, Pφ) invariant space of a 100 keV proton. Ip = −2 MA, R0 = 3 m and

a = 1 m. Dotted lines indicate where the orbit intersects the wall. Pφ = Pφ/m. From reference [17].

From the invariance of Λ follows that the ratio of perpendicular to parallel velocity for ions travel-

ling around the torus must vary to reflect the local magnetic field strength. For Pφ to be invariant

as R and vφ ≈ v‖ changes along the orbit the ion must also drift across the flux surfaces ψ. For

high-energy trapped ions these changes lead to broad drift orbits and for high-energy passing ions

the drift orbits get dislocated towards the low field side for co-current passing ions and towards

the high field side for counter-current passing ions.

The invariant space (E,Λ, Pφ) can be divided into nine regions characterised by the topology

of the drift orbits in the region [43]. Regions I–IV contain co- and counter-current passing orbits,

regions V and VII contain trapped orbits and regions VI and VIII contain co-current passing orbits.

There are no orbits in region IX. Since Pφ is degenerate with respect to co- and counter passing

orbits in regions I–IV [44] an extra label σ is needed to distinguish between these. The regions in

the invariant space of a 100 keV proton in a JET-like plasma are illustrated in figure 2.5 and the

corresponding drift orbit topologies in figure 2.6.

Although the dominant effect of cyclotron interactions is to change the perpendicular velocity of

the resonant particles there exists also a finite change in the parallel velocity due to the absorption

of parallel wave momentum via the Lorentz force from the perpendicular magnetic wave field

component [39, 42, 45–47]. Changes ∆E in energy from wave-particle interactions are therefore

accompanied by changes in the invariants Λ and Pφ according to:

∆Λ =

(

nωc0ω

−Λ

)

∆E

E(2.17)

∆Pφ =nφ

ω∆E (2.18)

2.4. Evolution of ion drift orbits 17

Figure 2.6 Projections in the poloidal plane of the guiding centre orbits in the invariant space

regions illustrated in Fig. 2.5. Trapped orbits are solid, counter-current orbits are dotted and

co-current orbits are dashed. From reference [17].

where ωc0 is the cyclotron frequency at the magnetic axis. Interactions with a single wave mode

thus give rise to one-dimensional diffusion in phase space along characteristics that, with increas-

ing ion energies, asymptotically approach Λ = Λres = nωc0/ω at continuously increasing or de-

creasing Pφ depending on the sign of nφ. For a spectrum of toroidal mode numbers the diffusion

becomes two-dimensional as the ions may follow different characteristics towards Λres .

While pitch-angle scattering tends to restore isotropic distribution functions at low energies the

exchange of toroidal momentum with the wave has important consequences for the properties of

the high-energy ion population. For trapped orbits the Λ drift has the physical effect of driving

the turning points towards the unshifted cyclotron resonance. Also, since the flux surface of

the turning points (where vφ ≈ 0) is given by ψ = −Pφ/Ze, the Pφ drift will cause the turning

points to drift inwards or outwards along the resonance depending on the relative directions of

the wave propagation and the plasma current [48]. This RF-induced spatial transport affects the

radial fast ion density and thereby the pressure and bulk plasma heating profiles. If the Pφ drift

from interactions with co-current propagating waves (Ip > 0 and nφ > 0 or Ip < 0 and nφ < 0)

is sufficiently strong for the turning points of a trapped orbit to meet in the plasma midplane the

orbit may detrap into a passing orbit. Effects of RF-induced spatial transport and detrapping have

been observed experimentally with γ-emission tomography [49,50]

During high-power ICRH the RF interactions may produce distribution functions that are in-

verted, i.e. not monotonically decreasing, along characteristics in phase space other than those

defined by equations (2.17) and (2.18). If the cyclotron resonance condition is met this can give


rise to ion cyclotron emission. While relatively rare, ion cyclotron emission has been observed

experimentally [51].

2.5 Minority ion cyclotron current drive

Minority current drive was first proposed by Fisch [52] for non-inductive plasma current drive.

The scheme is based on heating minority ions with a specific direction of toroidal propagation and

thereby decreasing their collisionality with the background plasma. This establishes a net drift

of minority ions in the direction of the propagation of the heated ions and, due to conservation

of toroidal momentum, an opposite drift of the background plasma. Taking into account the

background plasma response the total driven current Jtot from a minority current Jmin becomes

[53]:

Jtot = Jmin

(

1−ZminZeff

− C1 + fTC2

)

(2.19)

where

C1 =

λmmin

∑

i Zini

(

1−ZiZeff

)

ZminZeff∑

inimi(2.20)

and

C2 =

(

ZminZeff

−λmmin

∑

iniZ2i

ZminZeff∑

inimi

)

(2.21)

The sums over i include all ion species except the heated minority and λ = 1 for undamped

background plasma rotation. fT ≈ 1.46A(Zeff )√

r/R0 is the effective fraction of trapped electrons,

where A(Zeff ) is tabulated in references [54] and [55] and varies from A(1) ≈ 1.68 to A(4) ≈ 1.18

in typical fusion plasmas. The unity term in equation (2.19) represents the minority current, the

second term represents the electrons dragged along by the heated minority ions and C1 represents

the drift of the background plasma opposite to the minority drift. These terms are formulated

assuming no trapped electrons, the effects of which are corrected for by the term fTC2. For neutral

beam current drive equation (2.19) is slightly modified with C1 = 0 and C2 = Zbeam/Zeff [55].

The physics of minority ion cyclotron current drive is strongly modified when taking into account

the 1/R dependence of the ion cyclotron frequency. Since cyclotron interactions take place at the

Doppler shifted resonance, nωc = ω − k‖v‖, a wave with a given frequency and parallel wave

number resonates with ions having opposite parallel velocities on opposite sides of the unshifted

resonance. The driven current density to be expected from the discussion above would therefore

take a bipolar shape, co-current on one side of the resonance and counter-current on the other. In

general, the total driven current would be insignificant. Taking ion trapping, finite ion drift orbit

widths and the RF-induced spatial transport into account further modifies the current drive and

introduces new current drive mechanisms [56,57]. Finite orbit widths introduce a bipolar current

which is always co-current on the outside and counter-current on the inside of a flux surface

that lies outside the flux surface tangent to the unshifted cyclotron resonance. The RF-induced

transport creates a net co-current due to asymmetric detrapping and modifications to the ion drift

orbits, particularly for co-current propagating waves.

2.6 TTMP/ELD and electron current drive

In section 2.3 was discussed how the resonant wave-particle interactions when the phase differ-

ence between ion gyration and the rotating electric field is stationary lead to a net acceleration or

deceleration of the ion and, provided the phase is decorrelated between interactions, a velocity

diffusion. These resonances correspond to the resonance conditions nωc =ω− k‖v‖ with n ≥ 1.

In the linearised theory a resonance also appears for n = 0 ⇒ v‖ =ω/k‖ = c‖, i.e. when the parallel

velocity of the particle matches the parallel part of the phase velocity of the wave. For the fast wave

in fusion plasmas this resonance condition is most likely to be fulfilled by electrons and there are

2.6. TTMP/ELD and electron current drive 19

two mechanisms of wave-particle interactions associated with it; electron Landau damping (ELD)

and transit time magnetic pumping (TTMP).

The linear electron Landau damping is, akin to cyclotron damping, an electrostatic acceleration

or deceleration when the wave-particle phase difference is stationary, combined with collisional

or non-linear phase decorrelation. Transit time magnetic pumping accelerates the gyro centres

of resonant particles with the magnetic mirror force, F‖ = −µ∇‖B, which arises from the parallel

gradients in the equilibrium magnetic field formed through the modulation of the field by the

propagating wave. However, the mirror force acceleration gives rise to a parallel electric field and

thereby leads to a counter-directed Landau-type interaction. This reduces the total damping to

half that of the original TTMP [39]. Direct electron Landau damping of the fast wave is in general

negligible due to the insignificant parallel electric wave field.

TTMP/ELD is, like cyclotron damping, a diffusive process. With the typically decreasing v‖ dis-

tribution at the resonant v‖ = ω/k‖ = c‖ the interactions will therefore lead to net parallel ac-

celeration of the electron population and, with an asymmetric k‖ spectrum, provide non-inductive

plasma current drive. The current drive efficiency for power damped by TTMP/ELD is relatively

high, γCD = ICD <ne >R/Pe ≈ 0.3 × 1020 AW−1m−2, but the total current drive efficiency is often

significantly degraded by parasitic absorption of RF power on cyclotron resonant ions and, due

to the low damping in many TTMP/ELD current drive scenarios, by power losses outside of the

plasma.

21

3 Antenna coupling

In the preceding chapter was discussed how the fast magnetosonic wave can propagate and be

absorbed in a quasi-homogeneous plasma. This chapter comments on the coupling of electromag-

netic power to the plasma wave and some associated problems.

Since the vacuum wave lengths at ion cyclotron frequencies in typical tokamaks are comparable

to the machine dimensions, large antennas are required for high coupling resistance and efficient

power coupling, c.f. figure 3.1. The coupling resistance is here defined as the proportionality factor

between the coupled power and the square of the currents in the antenna straps, PRF =12RCI

2A.

The large size of the antennas, and the proximity to the plasma edge that is necessitated by the

evanescence of the wave in the vacuum region, makes them susceptible to detrimental interactions

with the plasma.

3.1 Coupled toroidal mode spectrum

ICRH antennas typically consist of several vertically directed parallel current straps covered by a

Faraday screen that is designed to minimise the electric field component parallel to the equilibrium

magnetic field, c.f. figures 1.6, 3.1 and 3.2. The excited k‖ spectrum is chiefly determined by the

toroidal separation between the current straps, their widths and the phase between the currents in

them. In the near vacuum between the antenna and the plasma the associated k⊥ spectrum is re-

lated to k‖ through k2⊥ = k

20−k

2‖, where k0 is the vacuum wave numberω/c. For typical experimen-

tal parameters and antenna designs k2‖ > k

20 except at the very longest toroidal wave lengths and

most of the spectrum is thus evanescent, i.e. k⊥ is imaginary. Since the decay length of the evanes-

cent wave decreases with increasing parallel wave number, E(x, t) = Eant exp i(√

k20 − k

2‖x −ωt),

the k‖ spectrum coupled to the plasma will differ from that excited by the antenna.

The coupled toroidal Fourier mode power spectrum as function of current strap phase difference

for a simplified antenna with toroidal dimensions similar to the JET A2 antenna [58] is illustrated in

figure 3.3, assuming complete single pass damping. The evanescent decay is modelled for 51 MHz

and assuming that the current straps are located 10 cm from a plasma density step where the

whole spectrum can propagate. Neglecting poloidal magnetic field effects the k‖ spectrum in the

plasma can be approximated from the toroidal mode spectrum by k‖ ≈ kφ = nφ/R.

Operation with no phase difference between the currents in the straps is usually referred to as

“Monopole phasing” and results in a symmetric spectrum with the dominant toroidal mode num-

ber 0. The coupling resistance with the monopole phasing is usually high since a significant part

of the spectrum can either propagate in vacuum or has a small evanescent decay. Introducing an

increasing phase difference between the strap currents shifts the dominant mode number and pro-

duces an asymmetric spectrum. The most commonly used asymmetric spectra are the ones with

±90◦ phase differences, which in JET have dominant mode numbers nφ ≈ ±10 and subdominant

peaks at nφ ≈ ∓20 and nφ ≈ ∓35. For all asymmetric spectra the sign of the phase differences

will determine the toroidal direction of wave propagation and allow for RF current drive co- and

counter the plasma current etc. With a phase difference of 180◦, the so called “dipole phasing”,

the spectrum is once again symmetric, but now consists of two maxima at higher absolute toroidal

mode numbers, nφ ≈ ±25 for JET, and no modes with nφ=0.

3.2 Impurities and parasitic power losses

The plasma performance of early machines using ion cyclotron resonance heating was found to be

severely degraded by a massive influx of high-Z metallic impurities during the application of RF

22 3. Antenna coupling

Figure 3.1 JET cross section, including an A2 antenna.

From reference [58].

Figure 3.2 One half of a JET A2 antenna. From

reference [59].

power [60]. The impurities were identified to be caused by sputtering of the wall and the antenna

by ions accelerated in rectified RF sheaths, i.e. potentials between the plasma and the wall or the

antenna caused by electrons accelerated by parallel electric fields into the walls or other conducting

structures. The problem was largely mitigated by the introduction of antenna Faraday screens that

short-circuited most of the parallel electric fields.

Impurity influx during RF heating is however still observed in certain scenarios with finite mis-

alignment angles between the Faraday screens and the equilibrium magnetic field [61–64], often

in connection with parasitic losses of RF power [61, 62, 65–68]. These effects have been found to

depend on the antenna phasing and on the damping in the plasma. The dependence on phasing

is usually attributed to the different total potential induced along field lines passing the current

straps if the strap currents are in or out of phase. However, different antenna designs also display

different dependencies on the phasing, e.g. the four-strap JET A2 antennas display a greater sensi-

tivity to the phasing than the old two-strap A1 antennas did [66, 69]. Also, using only two of the

four current straps in the A2 antennas, and thereby launching a broader toroidal mode spectrum,

results in lower losses of RF power [68].

Although the average coupling resistance of individual magnetosonic eigenmodes, and thereby

also the average total coupling resistance, is independent of the damping [70], the average antenna

voltage at constant coupled power is not [71, 72]. At low damping, and particularly with narrow

antenna spectra, the infrequent coupling to resonant eigenmodes induces strong variations in the

coupling resistance as equilibrium quantities vary and results in an increased average antenna

voltage. Since the rectified sheath potentials are proportional to the antenna voltage [64, 73], the

parasitic losses of RF power through the dissipation of the sheath potentials are increased.

23

−50−25

025

50

0

45

90

135

1800

0.05

0.1

0.15

nφPhasing

Pn

φ/ P

RF

Figure 3.3 Toroidal power spectrum for the JET A2 antenna. The amplitude is normalised for constant coupled

power.

25

4 Numerics and modelling

The theoretical basis for ICRH and TTMP/ELD has been known for a long time [39,74,75], but only

relatively recently has development in experimental diagnostics made direct verification of theory

and numerical models realistic. These refined diagnostics also benefit from accurate and robust

theoretical models and numerical codes for benchmarking and for interpretation of experimental

results.

Due to the complexities introduced by the non-linear coupling between wave fields and distribu-

tion functions during ICRH, most modelling efforts to date have focused either on solving discrete

toroidal Fourier modes of the wave field by assuming Maxwellian distribution functions [76–80]

or on calculating the evolution of a distribution function from interactions with a model wave

field [81–83]. Much fundamental physics can unarguably be explored using this decoupled ap-

proach, but for comparison with experimental measurements the non-linear evolution of wave

fields and distribution functions is often critical. For high-power heating scenarios the anisotropic

and strongly non-Maxwellian resonant ion distribution functions and broad ion drift orbits will

have significant consequences for the heating performance, e.g. affecting the partition of RF power

on resonant species and the collisional bulk plasma heating.

Starting with separate codes developed to study the decoupled problems of wave propagation

and distribution function evolution it is possible to solve the coupled problem in a self-consistent

manner by taking advantage of the different time scales for the wave oscillation and the evolution

of the distribution functions.

4.1 The LION code

The global wave code LION1 [76,84] solves the wave equation for discrete toroidal Fourier modes of

the compressible fast magnetosonic wave and the shear Alfvén wave in an axisymmetric torus with

arbitrary cross section using a finite hybrid element method [85]. Since the wave equation is solved

only to second order in ∂/∂ψ the Bernstein wave and the kinetic Alvén wave are not included in

the solution. The plasma is described by a local dielectric tensor wherein k‖ is approximated by

nφ/R, i.e. poloidal magnetic field effects are neglected. Higher harmonic cyclotron absorption is

included with the full Bessel functions in the dielectric tensor and by obtaining k⊥ from the local

fast wave dispersion relation, assuming the plasma to be quasi-homogeneous. Electron damping

and current drive through TTMP/ELD is calculated by modelling the effect of a finite parallel wave

field perturbatively. The current drive calculations include the reduction in current drive efficiency

by trapped electrons [86,87].

LION accepts the externally calculated susceptibilities of an arbitrary number of ion species,

and can thus be used to calculate the wave field in a plasma with non-Maxellian ion distribution

functions. The susceptibilities of ion species that do not significantly differ from Maxwellian can

be evaluated using equation (2.6).

4.2 The FIDO code

The FIDO2 code [17,82,88,89] solves the Langevin equivalent of the orbit-averaged Fokker-Planck

equation using a Monte Carlo method [45, 90, 91] in an axisymmetric torus with circular cross

section and concentric flux surfaces.

1Lausanne ION-cyclotron-2D-toroidal-global-wave-code.2Finite Ion Drift Orbits.

26 4. Numerics and modelling

The Monte Carlo method describes the distribution function with a statistical ensemble of test

particles. Each test particle represents a point in the invariant space (E,Λ, Pφ, σ) and the distribu-

tion function is given by the test particle density function. The test particle ensemble is advanced

in time by calculating the orbit-averaged drift- and diffusive increments to the invariants of each

test particle according to the quasi-linear RF interactions and Coulomb collisions, including the ef-

fects of finite drift orbit widths, absorption of parallel momentum from the wave and RF-induced

spatial transport discussed in section 2.4.

FIDO has the ability to import an arbitrarily sized spectrum of individual electric wave fields

calculated by LION for different toroidal mode numbers and/or RF generator frequencies [92]. For

a self-consistent coupling of wave field and distribution function evolution it can also calculate the

susceptibility of the simulated ion species according to equation (2.3) [93].

4.3 The SELFO code

The SELFO code [92–98] solves the coupled Fokker-Planck and wave equations iteratively using the

FIDO and LION codes according to the scheme outlined in figure 4.1. Each iteration FIDO advances

the distribution functions fi of the resonant ion species in time and calculates the susceptibilities

χi,ω,nφ from the heated distribution functions. LION then updates the electric wave fields Eω,nφand the partition of RF power Pi,ω,nφ on the different plasma species for the next iteration. Due

to the modular design of the code an arbitrary number of resonant ion species, RF generator

frequencies and toroidal mode numbers can be treated simultaneously. The number of FIDO time

steps each iteration can also be varied during the simulation to reflect the rate of change of the

distribution functions.

Modelling experimental RF heating scenarios using SELFO requires that equilibrium quantities

are somehow transformed to the circular equilibrium used in FIDO. Following reference [89] it is

suggested that the following experimental quantities should as far as possible be preserved:

• The major radius of the magnetic axis and the minor radius on the low field side of the

magnetic axis.

• The major radii of the intersections of the cyclotron resonances with the midplane.

• Density and temperature profiles in the midplane on the low field side of the magnetic axis.

• The frequency of the launched wave.

• The average power absorbed per particle.

• The poloidal flux.

Since the SELFO code, and particularly FIDO, is constantly evolving, e.g. with interactions between

fast ions and MHD modes [99] and an updated quasi-linear RF operator [100] recently implemented,

an overview of some of the upgrades of importance for the included papers is presented below.

4.3.1 Weighted Monte Carlo scheme

Detailed modelling of ICRH requires an accurate description of the small high-energy tails of the

distribution functions of the resonant ion species. This is critical for higher harmonic heating,

which is a finite Larmor radius effect, but it is also important for heating at the fundamental

cyclotron resonance. When representing the distribution function with Monte Carlo test particles

this implies that a large number of test particles is needed for the tail description. With a uniform

sampling probability, i.e. letting each test particle represent a fixed fraction of the distribution,

most test particles will only contribute to the description of the thermal part of the distribution

function and the total number of test particles required to ensure a good statistical description of

the tail will be very large. Together with the longer computation times of thermal test particles

due to their high collisionalities this leads to an inefficient and slow scheme. A more efficient

description of the distribution function and of derived quantities, as well as faster computations,

can be achieved by attributing numerical weights to the test particles and introducing a weighted

Monte Carlo scheme.

4.3. The SELFO code 27

Initialise fi, Eω,nφ , Pi,ω,nφ

�

Iterate in time

�

Loop over fi�

Advance fi using Eω,nφ , Pi,ω,nφ

�

Loop over ω

�

Loop over nφ

�

Calculate χi,ω,nφ from fi

�

�

�

�

Loop over ω

�

Loop over nφ

�

Construct εω,nφ from χi,ω,nφ

�

Calculate Eω,nφ , Pi,ω,nφ from εω,nφ

�

�

�

Diagnostics

�

�

Done

“FIDO”

“LION”

Figure 4.1 Schematic flow chart of the SELFO code.

A weighted Monte Carlo scheme has been implemented in the FIDO code in order to optimise the

use of the test particles during self-consistent simulations and thereby increase both the accuracy

and the performance of the calculations [98]. Identifying the anti-Hermitian parts of the dielec-

tric susceptibilities as the critical quantities the scheme is defined according to the principles of

importance sampling [101]. Regions are defined in ascending energy space such that the average

contribution per test particle weight to the anti-Hermitian parts of the susceptibilities doubles in

each higher region, assuming that the contribution per test particle is strictly increasing with en-

ergy. Efficient sampling is then achieved by splitting each test particle that is moved into a higher

energy region by RF interactions or collisions in two parts. The two new particles retain the proper-

ties of the original, but carry only half the numerical weight each. Correspondingly, as particles are

slowed down into lower energy regions their number should decrease with preserved total weight.

Since the probability of finding two identical test particles to merge is inappreciable, each particle

is instead subjected to a Russian roulette which preserves the average weight by either removing

the particle from the distribution or, with equal probability, doubling the numerical weight of the

particle.

A dynamically adapting scheme is required to ensure optimum sampling as the distribution

28 4. Numerics and modelling

function evolves. This is achieved by re-evaluating the boundaries of the energy regions each

SELFO iteration as the susceptibilities are updated. Also, a minimum number of test particles must

always be allocated for the bulk plasma description. In the FIDO scheme the boundary between

bulk and tail is set at two thermal ion energies and the particles are divided evenly between the

bulk and tail description.

4.3.2 Neutral beam injection

Neutral beam injection represents a source of high-energy ions which, due to their strong higher

harmonic RF absorption and significantly Doppler broadened cyclotron resonances, are important

contributors to the distribution function when considering ion cyclotron heating. Models for the

JET and ITER neutral beam injection systems have therefore been implemented in FIDO [97,102].

The model for the JET NBI system accepts beam deposition data for specific JET pulses as calcu-

lated with the JET NBP2 PENCIL [103] code for energy components and injection angles consistent

with the installed beams. The beam ions are assumed to be deposited on the low field side of

the magnetic axis, which the simulated beam trajectories pass through. These approximations are

largely unavoidable due to the folded radial profiles provided by PENCIL, but of limited concern

for the accuracy of the modelling since the real trajectories pass close to the magnetic axis and

the beam injection energies (30–140 keV) are of the order of, or lower than, the critical energy

where pitch-angle scattering becomes strong and tends to quickly make the beam ion population

isotropic.

The NBI heating system that has been suggested for ITER consists of two monoenergetic 1 MeV

beams directed tangentially to the torus with a tangency radius of 5.3 m [37]. These beam trajecto-

ries have been implemented in FIDO. A constant ionisation probability along the path is assumed,

adjustable by specifying the estimated shine-through fraction.

4.3.3 Fusion born α particles

Similarly to beam ions, the high-energy fusion born α particles in future high-Q experiments and

reactors have been considered a possible competitor for RF power. As such they could affect

the partition of RF power on resonant species, the RF power deposition profiles and represent a

possible source of RF heating degradation. A model for introducing 3.5 MeV α particles according

to a prescribed radial fusion yield profile has therefore been implemented in FIDO.

29

5 Summary of publications

This thesis is based on the results presented in nine included papers previously presented at

international conferences or either published in, or submitted to, refereed journals. The treated

topics are related to the heating and current drive in tokamak plasmas using the fast magnetosonic

wave. A short summary of the papers is given below, followed by an account of the thesis author’s

contributions.

5.1 Modelling of ICRH and fast particles

5.1.1 Paper I

The standard JET scenario is deuterium plasmas heated by deuterium NBI combined with hydrogen

minority ICRH. In this scenario the fundamental cyclotron resonance of hydrogen coincides with

the second harmonic cyclotron resonance of deuterium. Second harmonic resonance absorption is

a finite Larmor radius effect and therefore weak at thermal energies, but appreciable at the beam

injection energies. Paper I reports on the upgrade of SELFO to allow simulating multiple ion species

self-consistently, and on the implementation of neutral beam injection in FIDO.

The new capabilities are tested by modelling heating of an ITB plasma with a small hydrogen

minority and directed antenna spectra. Due to the high temperatures in ITB plasmas, and thereby

long slowing down times for fast ions, large high-energy tails on the distribution functions of the

absorbing ion species will form at high RF power. Combined with the low central plasma current

this leads to very broad orbits. The simulations show that beam ions will absorb RF power in this

scenario, and the overall power partition turns out to be sensitive to the antenna phasing. With an

antenna phasing producing counter-current propagating waves the hydrogen ions are driven out

from the plasma centre, leading to stronger wave fields as the absorption by hydrogen decreases

and the formation of a deuterium tail even without beams. With the opposite direction of wave

propagation hydrogen is accumulated in the plasma centre and without beams no deuterium tail

is formed.

5.1.2 Paper II

Detailed knowledge of the physics of fusion born 3.5 MeV α particles is essential for predicting the

performance of the next generation of high-Q experiments and future reactors. Experiments on

simulating α particles were performed at JET by accelerating 4He ions to MeV energies using third

harmonic ion cyclotron heating of 4He plasmas with and without 4He neutral beams for providing

high-energy seed ions. Modelling these experiments, and other scenarios with higher harmonic

resonances, with SELFO required the development of a weighted Monte Carlo scheme for acquiring

sufficient statistics in the descriptions of the high-energy tails of the distribution functions. These

upgrades are reported on in paper II together with simulations of the JET experiments.

The weighted Monte Carlo scheme was defined according to the principles of importance sam-

pling when regarding the anti-Hermitian elements of the dielectric tensor as as the critical quantity

and assuming that the dominating variations in the contributions from the resonant ions were

with energy. A dynamically updated scheme was required since the contributions to the tensor el-

ements change as the distribution functions evolve. The dynamically weighted Monte Carlo scheme

significantly improved the convergence properties of the calculations; tests at the third harmonic

cyclotron resonance indicated that 12 500 weighted test particles provided similar statistics in the

calculations of the anti-Hermitian tensor elements as 400 000 unweighted particles.

30 5. Summary of publications

The simulations using the weighted scheme recovered the noteworthy experimental result of

forming a 4He tail only in combination with 4He neutral beam injection. Without 4He beams the

absorption was instead dominated by residual deuterium due to the larger Larmor radii of the

thermal deuterium ions and the longer slowing down times of accelerated deuterium ions. The

simulations show that strong damping is possible even at higher harmonic resonances if a high-

energy tail can develop. This result is of importance for fast wave electron current drive, where

absorption by ions decreases the current drive efficiency. It should however be noted that parasitic

absorption of RF power at the antennas or the walls could be important in the thermal plasmas

due to the weak damping. It is conceivable that this could sufficiently decrease the power density

in the plasma to prevent a high-energy tail from forming.

5.1.3 Paper III

In paper III the effects of finite drift orbit width and RF-induced spatial transport on plasmas heated

by ICRH are discussed. Significant differences in the heated ion distribution functions appear when

heating with co-current, counter-current and symmetric antenna spectra. Heating with co-current

propagating waves drives the turning points of trapped ions vertically inwards. If the turning

points meet in the midplane the orbit will detrap, for high field side resonances or high-energy

ions preferentially into a co-current passing orbit. As these ions continue to be accelerated the

orbits are shifted towards the low field side until the Doppler shifted resonance is tangential to

the orbit, introducing a maximum energy the ions can be heated to. Heating with counter-current

propagating waves drives the turning points of trapped orbits vertically outwards, curtailing the

high-energy tails as the orbit moves out into the colder edge plasma. Symmetric wave spectra heats

the resonant ions to high energies, limited only by by cancellation of the RF diffusion over a drift

orbit or by the ions hitting the walls. Since the parallel ion velocity on the outer legs of trapped

orbits is co-current, losses of ions hitting the walls produce a net counter-current torque on the

plasma. For directed spectra the main source of steady-state momentum transfer to the plasma is

from the wave momentum.

5.1.4 Paper IV

The large non-thermal energy content from fusion born α particles in future tokamaks with high

fusion yield is expected to lead to long sawtooth periods. These have experimentally been found to

trigger neoclassical tearing modes which could impose a severe β limit on the plasma and limit the

attainable performance. Localised minority ion cyclotron current drive at the sawtooth inversion

radius has been demonstrated to destabilise sawteeth at JET and to help avoid neoclassical tearing

modes. For ITER, 3He minority current drive at the outboard sawtooth inversion radius has been

proposed. It is also the only minority heating scenario available within the projected frequency

span of the RF system at full magnetic field strength.

In paper IV the 3He minority current drive scenario proposed for ITER is evaluated together with3He current drive at the inboard sawtooth inversion radius and H minority current drive at the

outboard and inboard sawtooth inversion radii. It is found that none of the 3He minority current

drive scenarios provide any appreciable current drive due to the strong collisionality of the 3He and

due to the bulk plasma drag current. The fundamental H cyclotron resonance is degenerate with

the second harmonic resonances of beam injected D ions and fusion born α particles and therefore

feared to be susceptible to parasitic absorption of RF power. The simulations however indicate

that the parasitic absorption is acceptably low and that the H minority current drive efficiency is

significantly higher than that of 3He current drive.

5.2 Polychromatic ICRH

5.2.1 Paper V

Polychromatic ICRH, with the intent of increasing the bulk ion heating and decreasing the fast

particle pressure by decreasing the average fast ion energy, has been successfully used at JET and

other tokamaks. The lower average fast ion energy with polychromatic ICRH is usually attributed

to a lower power density. Taking finite orbit width effects into consideration however provides

5.3. FWCD, parasitic absorption and magnetosonic eigenmodes 31

additional mechanisms which may have part in the phenomenon. In paper V the particle physics

of polychromatic ICRH is discussed. For monochromatic ICRH with an arbitrary number of toroidal

mode numbers the RF interactions describe a two-dimensional diffusion in phase space as is dis-

cussed in paper III; an orbit may approach Λ = Λres along different characteristics associated

with different toroidal mode numbers. For polychromatic ICRH the diffusion process becomes

three-dimensional. Each frequency is associated with one Λres which acts as an attractor for the

RF diffusion characteristics and makes the RF induced drift non-trivial. A characteristic may well

approach different Λres asymptotically at different values of ion energy or canonical toroidal mo-

mentum depending on with which frequencies the ion is in resonance and the strength of the

individual interactions. Due to the asymmetric radial extent of the orbits of trapped particles to-

wards the low field side of the turning points, and the large Doppler shift therefore required for

interactions with resonances on the high field side of the turning points, the cumulative drift will

have a trend towards the low field side.

5.2.2 Paper VI

Paper VI reports on the experimental comparison between monochromatic and polychromatic

ICRH performed at JET in the autumn of 2002. Monochromatic 3He and H minority heating with

the fundamental cyclotron resonance in the plasma centre was compared to polychromatic heat-

ing with the resonances spread over 30–40 cm either on the low- or high field side of the magnetic

axis. The polychromatic pulses displayed lower diamagnetic energy content, shorter period and

lower amplitude sawteeth and higher ion to electron temperature ratios than the corresponding

monochromatic pulses, in line with theoretical expectations from the lower fast ion energy content

with polychromatic operation.

The experiments were analysed with the SELFO and PION codes. The latter code uses a one-

dimensional pitch-angle averaged Fokker-Planck solver combined with a semi-empirical model for

the RF power deposition. Agreement between the codes, as well as between the codes and experi-

mental measurements, was in general good. Information on the spatial distribution of fast particles

was provided by gamma emission tomography and found to be consistent with the fast particle

densities calculated with SELFO. The SELFO simulations indicated that polychromatic heating with

the resonances on the high-field side predominantly creates high-energy passing orbits and with

the resonances on the low-field side predominantly high-energy trapped orbits. This is a result of

the resonance positions and thereby orbit type regions the different Λres intersects, c.f. figure 2.5.

5.3 FWCD, parasitic absorption and magnetosonic eigenmodes

5.3.1 Paper VII

Fast wave electron heating and current drive is potentially a very useful tool for controlling the

central electron temperature and the central plasma current in tokamaks. Since the absorption

through transit time magnetic pumping and electron Landau damping is weak for typical experi-

mental scenarios it is however crucial to avoid parasitic absorption, e.g. by cyclotron resonant ions

or rectified RF sheaths. Paper VII describes and analyses a series of fast wave electron heating and

current drive experiments performed at JET during 2003–2004. The heating and current drive was

found to be significantly degraded, partly because of damping on residual 3He ions. Up to half

of the applied RF power was however also found to be unaccounted for by the divertor thermo-

couplers and the bolometers measuring energy radiated from the plasma. Such discrepancies are

consistent with energy lost to the walls with fast particles or with RF power absorbed outside of

the last closed flux surface. SELFO modelling indicated that the power lost with fast 3He ions was

insufficient to explain the lost power. Strong BeII and CIV line radiation, concluded to come from

impurities emanating from the beryllium Faraday screens and carbon limiters of the antennas, was

seen and indicated that the power was lost in rectified RF sheaths at the antennas. More RF power

was found to be lost with the directed ±90◦ phasings than with the symmetric dipole phasing. A

reference discharge with higher plasma current, and thereby smaller misalignment angle between

the equilibrium magnetic field and the Faraday screens, showed better heating efficiency in line

with expectations from RF sheath theory.

32 5. Summary of publications

The current drive was calculated with the LION code, using the power damped on electrons from

power modulation measurements for input, and fed to the JETTO transport code in order to take

current diffusion and changes to the bootstrap current into account. The calculated response of

the central current densities to the driven currents were significantly smaller than that measured

by a Faraday rotation polarimeter. This indicates that the current diffusion in these plasmas with

reversed central magnetic shear is anomalous rather than neoclassical as is assumed in the JETTO

code.

5.3.2 Paper VIII

Parasitic absorption of RF power in rectified RF sheaths at the antennas is a known phenomenon

appearing in heating scenarios with low single pass damping and/or large misalignment angles

between the equilibrium magnetic field and the Faraday screens of the antennas. Existing theory

suggests that the losses should be proportional to the antenna voltage, which is given by the cou-

pling resistance and the coupled power, and depend on the relative phases between the currents

in the antenna straps. Differences in parasitic losses have however also been found between dif-

ferent antenna designs and when operating the same antenna with a varying number of current

straps. In paper VIII a statistical model of the parasitic losses in rectified RF sheaths is presented.

While the average coupling resistance is independent of single pass damping, at constant coupled

power the infrequent coupling to resonant magnetosonic eigenmodes at low single pass damping

with narrow antenna spectra will increase the parasitic losses due to the resulting higher average

antenna voltage. It is found that the parasitic losses increase rapidly with the misalignment angle

for small angles and that the losses can be significant even with relatively strong damping.

5.3.3 Paper IX

Magnetosonic eigenmode structures may appear in plasmas with low damping. Such eigenmodes,

localised at the plasma edge, are of interest in connection with the study of ion cyclotron emission

by suprathermal ions. A model for edge localised modes, based on an ansatz where the poloidal

variation of the radial extension of the mode is neglected, has been proposed by others and is

frequently used in the literature. For small aspect ratios and for modes with high poloidal mode

numbers this leads to a strong localisation of the mode to the low field side of the torus. However,

except for modes with high toroidal mode numbers no physical mechanism for such a localisation

exists in the absence of a fast wave cut-off in the plasma. In paper IX edge localised modes are

studied using the LION code and an alternative model based on the conservation of Poynting flux is

proposed. This model gives good agreement with LION calculations in regions where convergence

can be achieved. Whether discernible edge localised modes exist at all in realistic tokamak equilib-

ria, or if they are destroyed by resonant mode coupling, is unclear due to limited code resolution.

5.4. Contributions from the thesis author 33

5.4 Contributions from the thesis author

• Papers I and II: The thesis author performed the code upgrades, the simulations, the analysis

of the results and wrote the papers.

• Paper III: The thesis author participated mainly in the development of the numerical codes

used in the preparation of the paper.

• Paper IV: The thesis author initiated and performed this study.

• Paper V: The thesis author performed the simulations, the analysis of the results and wrote

the paper. The code upgrades were performed in collaboration with Dr. T. Johnson.

• Paper VI: The thesis author participated in the execution and analysis of the experiments,

including performing the SELFO simulations and related analysis of fast particle effects, and

participated in writing the paper.

• Paper VII: The thesis author participated actively in the preparation, execution and analysis

of the experiments as well as in writing the paper. Specific tasks included premodelling,

development of analysis tools for evaluating the fraction of RF power absorbed in the plasma

from experimental measurements, analysis of the losses of power to the walls with fast 3He

ions and analysis of the driven currents given the measured electron damping.

• Paper VIII: The thesis author participated in the development of the theoretical model for the

parasitic absorption, wrote the code used for the statistical analysis, performed the numerical

calculations and actively participated in the analysis and in writing the paper.

• Paper IX: The thesis author performed the numerical calculations, including developing the

tools necessary for tracking individual magnetosonic eigenmodes through changing plasma

equilibria, and participated in writing the paper.

35

6 Conclusions

Theoretical and experimental work related to heating and current drive with the fast magnetosonic

wave in tokamak plasmas has been presented. The coupling between the velocity distribution func-

tions of cyclotron resonant ion species and the propagation and damping of the wave necessitates a

self-consistent modelling. This is particularly important when heating at higher cyclotron harmon-

ics, when several ion species are resonant in the plasma or when resonant neutral beam injected

ions or fusion born α particles are present. Finite Larmor radius effects are important in many

ion cyclotron resonance heating scenarios and require an accurate description of the high-energy

tails of the distribution functions. This description can be significantly improved by a dynamically

weighted Monte Carlo scheme.

Finite orbit width effects and the absorption of toroidal momentum from the wave are also criti-

cal in predicting and interpreting many heating and current drive scenarios such as polychromatic

ICRH, minority ion cyclotron current drive and losses of fast ions to the walls. Modelling of mi-

nority ion cyclotron current drive in ITER indicated that although the parasitic losses of RF power

to fusion born α particles and to beam ions should be acceptable, the current drive efficiency with

the proposed 3He minority scheme will be poor. H minority current drive was found to greatly

increase the current drive at the expense of bulk ion heating.

Experiments at JET demonstrated that polychromatic ICRH can increase the ion to electron tem-

perature ratio and decrease the fast ion energy content, leading to smaller sawteeth. Experiments

on fast wave electron current drive in ITB plasmas were found to be degraded by parasitic losses

of RF power. Only part of the losses could be explained by losses of fast residual 3He ions, but evi-

dence was found of strong interactions at the RF antennas during times of low wave damping. This

lead to the conclusion that the losses were probably caused by rectified RF sheaths. A separate

study showed the losses in rectified sheaths to be increased when using narrow antenna spectra in

scenarios with low damping due to the infrequent coupling to resonant magnetosonic eigenmodes

and thereby higher average antenna voltage.

37

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