# Energetic particle physics: Chapter 7 of the special issue: on the path to tokamak burning plasma operation

M. Salewski, D.A. Spong, P. Aleynikov, R. Bilato, B.N. Breizman, S.

Briguglio, H. Cai, L. Chen, W. Chen, V.N. Duarte, et al.

► To cite this version:

M. Salewski, D.A. Spong, P. Aleynikov, R. Bilato, B.N. Breizman, et al. Energetic particle physics: Chapter 7 of the special issue: on the path to tokamak burning plasma operation. Nuclear Fusion, 2025, 65 (4), pp.043002. 10.1088/1741-4326/adb763. hal-05468163

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# Energetic Particle Physics

## Chapter 7 of the Special Issue: On the Path to Tokamak Burning Plasma Operation

M. Salewski<fnref n="1" />, D.A. Spong<fnref n="2" />, P. Aleynikov<fnref n="3" />, R. Bilato<fnref n="4" />,
B.N. Breizman<fnref n="5" />, S. Briguglio<fnref n="6" />, H. Cai<fnref n="7" />, L. Chen<fnref n="8" />, W. Chen<fnref n="9" />,
V.N. Duarte<fnref n="10" />, R.J. Dumont<fnref n="11" />, M.V. Falessi<fnref n="6" />,<fnref n="12" />,<fnref n="13" />,
M. Fitzgerald<fnref n="14" />, E.D. Fredrickson<fnref n="10" />, M. García-Muñoz<fnref n="17" />,
N.N. Gorelenkov<fnref n="10" />, T. Hayward-Schneider<fnref n="4" />,
W.W. Heidbrink<fnref n="8" />, M.J. Hole<fnref n="16" />, Ye.O. Kazakov<fnref n="17" />,
V.G. Kiptily<fnref n="14" />, A. Könies<fnref n="3" />, T. Kurki-Suonio<fnref n="18" />, Ph. Lauber<fnref n="4" />,
S.A. Lazerson<fnref n="3" />,¹⁹, Z. Lin<fnref n="8" />, A. Mishchenko<fnref n="3" />, D. Moseev<fnref n="3" />,
C.M. Muscatello²⁰, M. Nocente²¹, M. Podestà<fnref n="10" />,²²,
A. Polevoi²³, M. Schneider²³, S.E. Sharapov<fnref n="14" />,
A. Snicker<fnref n="18" />,²⁴, Y. Todo²⁵, Z. Qiu²⁶, G. Vlad<fnref n="6" />, X. Wang<fnref n="4" />,
D. Zarzoso²⁷, M.A. Van Zeeland²⁰, F. Zonca<fnref n="6" />,<fnref n="12" />,²⁶, and
S.D. Pinches²³

<footnote n="1">¹ Department of Physics, Technical University of Denmark, Kgs. Lyngby, Denmark</footnote>

<footnote n="2">² Oak Ridge National Laboratory, 1 Bethel Valley Rd, Oak Ridge, TN 37830, United States of America</footnote>

<footnote n="3">³ Max-Planck Institute for Plasma Physics, Wendelsteinstrasse 1, 17491 Greifswald, Germany</footnote>

<footnote n="4">⁴ Max-Planck Institute for Plasma Physics, Boltzmannstrasse 2, 85748 Garching, Germany</footnote>

<footnote n="5">⁵ Institute for Fusion Studies, The University of Texas, Austin, TX 78712, United States of America</footnote>

<footnote n="6">⁶ ENEA, Nuclear Department, C.R. Frascati, Via E. Fermi 45, 00044 Frascati, Italy</footnote>

<footnote n="7">⁷ School of Nuclear Science and Technology, University of Science and Technology of China, Hefei 230026, People's Republic of China</footnote>

<footnote n="8">⁸ Department of Physics and Astronomy, University of California, Irvine, California 92697-4574, USA</footnote>

<footnote n="9">⁹ Southwestern Institute of Physics, PO Box 432, Chengdu 610041, People's Republic of China</footnote>

<footnote n="10">¹⁰ Princeton Plasma Physics Laboratory, Princeton University, Princeton, NJ 08543-0451, USA</footnote>

<footnote n="11">¹¹ CEA, IRFM, F-13108 Saint-Paul-lez-Durance, France</footnote>

<footnote n="12">¹² Center for Nonlinear Plasma Science and C.R. ENEA Frascati, C.P. 65, 00044 Frascati, Italy</footnote>

<footnote n="13">¹³ Istituto Nazionale di Fisica Nucleare (INFN), Sezione di Roma, Piazzale Aldo Moro 2, Roma, Italy</footnote>

<footnote n="14">¹⁴ United Kingdom Atomic Energy Authority, Culham Campus, Abingdon, Oxon OX14 3DB, UK</footnote>

<footnote n="15">¹⁵ University of Seville, 41012 Seville, Spain</footnote>

<footnote n="16">¹⁶ Australian National University, Canberra, ACT 0200, Australia</footnote>

<footnote n="17">¹⁷ Laboratory for Plasma Physics, LPP-ERM/KMS, Royal Military Academy, Avenue de la Renaissance 30, B-1000 Brussels, Belgium</footnote>

<footnote n="18">¹⁸ Department of Applied Physics, Aalto University, FI-00076 AALTO, Finland</footnote>
¹⁹ Gauss Fusion GmbH, 85748 Garching, Germany ²⁰ General Atomics, P.O. Box 85608 San Diego, California 92186-5608, USA

²¹ Department of Physics 'G. Occhialini', University of Milano-Bicocca, Piazza della Scienza 3, 20126 Milano, Italy

²² Ecole Polytechnique Fédérale de Lausanne, Swiss Plasma Center, CH-1015 Lausanne, Switzerland

²³ ITER Organization, Route de Vinon-sur-Verdon, 13115, St. Paul-lez-Durance, France

²⁴ VTT, PO Box 1000, FI-02044 VTT, Espoo, Finland

²⁵ National Institute for Fusion Science, National Institutes of Natural Sciences, Toki, 509-5292, Japan

²⁶ Institute for Fusion Theory and Simulation, School of Physics, Zhejiang University, Hangzhou 310027, People's Republic of China

²⁷ Aix Marseille Univ, CNRS, Centrale Med, M2P2, Marseille, France

E-mail: msal@dtu.dk

**Abstract.** We review the physics of energetic particles (EPs) in magnetically confined burning fusion plasmas with focus on advances since the last update of the ITER Physics Basis [A. Fasoli et al. 2007 Nucl. Fusion 47 S264]. Topics include basic EP physics, EP generation, diagnostics of EPs and instabilities, the interaction of EPs and thermal plasma instabilities, EP-driven instabilities, energetic particle modes (EPMs), and turbulence, linear and nonlinear stability and simulation of EP-driven instabilities and EPMs, 3D effects, scenario optimization strategies based on EP phase-space control, EPs in reduced field scenarios in ITER before DT, and the physics of runaway electrons. We describe the simulation and modelling of EPs in fusion plasmas, including instability drive and damping as well as EP transport, with a range of approaches from first-principles to reduced models, including gyrokinetic simulations, kinetic-MHD models, gyrofluid models, reduced models, and semi-analytical approaches.
# Contents

1 Introduction

2 Basic physics of EPs: sources, plasma heating, confinement and losses

2.1 Equilibrium on the collisionless timescale

2.2 Drift orbit approximations

2.3 Equilibrium on the collisional timescale

2.4 EP sources

2.5 Collisional energy transfer

2.6 Non-dimensional EP parameters

3 EP generation and plasma heating by ICRF waves

3.1 Basic physics of ICRF heating

3.1.1 Wave propagation and cyclotron damping.

3.1.2 ICRF heating scenarios for fast-ion generation.

3.1.3 ICRF heating and collisional relaxation.

3.2 Application of ICRF heating for EP physics studies

3.2.1 Sawtooth dynamics.

3.2.2 AE destabilization.

3.3 ICRF heating scenarios for EP and AE studies before DT operation in ITER.

3.4 Bulk-ion ICRF heating in DT plasmas of ITER and future fusion reactors

4 Diagnostics of EPs and instabilities

4.1 Diagnostics of confined EPs

4.1.1 Gamma-ray spectroscopy.

4.1.2 Collective Thomson scattering.

4.1.3 Neutron emission spectroscopy and cameras.

4.1.4 Neutral particle analyzers.

4.1.5 Ion cyclotron emission.

4.1.6 Phase-space sensitivity of EP diagnostics.

4.1.7 Measurements of velocity-space and phase-space distribution functions.

4.2 Diagnostic of lost EPs

4.3 Runaway electron diagnostic

4.4 Diagnostics of instabilities in fusion plasmas

4.4.1 Toroidal interferometer polarimeter.

4.4.2 ECE radiometer.

4.4.3 Microwave reflectometry.

4.4.4 Magnetic probes.

4.4.5 Other potential diagnostics of EP-driven instabilities at ITER.

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4.4.6 AE spectroscopy.

5 Interaction of EPs with thermal-plasma-driven instabilities

5.1 Neoclassical tearing modes

5.1.1 Non-resonant EP-drive of NTMs.

5.1.2 Numerical simulations of EP effects on NTM stability.

5.1.3 Resonant interaction between EPs and NTMs.

5.1.4 EP transport by tearing modes.

5.2 Sawteeth and kink modes

5.2.1 EP redistribution by sawteeth and kink modes.

5.2.2 Pacing of sawtooth instabilities with EPs.

5.3 Ballooning instabilities

5.4 Resistive wall modes

5.5 Edge-localized modes.

6 Linear stability of EP-driven modes

6.1 Drive and damping of AEs

6.2 Overview of AEs

6.3 AE eigenfunction

6.4 Alfvén frequency range: TAEs, EAEs, and RSAEs.

6.4.1 Perturbative and non-perturbative simulations.

6.4.2 AE antenna measurements and benchmarks.

6.4.3 Code benchmarks.

6.4.4 ITER predictions.

6.5 High-frequency modes: CAEs and GAEs.

6.5.1 CAEs.

6.5.2 GAEs.

6.5.3 ICE-based diagnostics for burning plasma devices.

6.6 Low frequency modes: BAEs, BAAEs, LFAMs, EGAMs

6.6.1 BAEs.

6.6.2 BAAEs and LFAMs.

6.6.3 EGAMs.

6.7 Linear stability of EPMs.

6.8 Impact of anisotropy and toroidal flows

7 Nonlinear mode evolution, theory and simulation

7.1 Fishbones

7.2 Nonlinear gyrokinetic theory

7.2.1 Nonlinear wave-wave interactions.

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scenario. These mission goals in terms of Q directly imply necessary values for the α-particle heating fraction $η_α$ which indicates how much plasma heating originates from fusion reactions, leading towards a self-sustained system, and how much heating originates from auxiliary heating systems directly controlled from the outside. The α-particles from the deuterium-tritium (DT)‡ fusion reaction, $D + T \rightarrow \alpha + n$, receive 1/5 of the released fusion energy due to energy and momentum conservation, so the α-particle heating power is $P_α = P_{fus}/5 = QP_{aux}/5$. Hence, $η_α$ is related to Q by

$$η_α = \frac{P_α}{P_α + P_{aux}} = \frac{Q}{Q + 5}, \qquad (1)$$

as illustrated in figure 1. At ITER's mission goal $Q = 5$, we have $η_α = 0.5$, so self-heating by energetic α-particles equals auxiliary heating ($P_α = P_{aux}$), and at ITER's mission goal $Q = 10$, we have $η_α = 0.67$, so the plasma is predominantly self-heated by energetic α-particles.

<figure><img src="image_2.png" /><figcaption>Figure 1. Plasma self-heating fraction as a function of the power amplification $Q$. Plasmas are predominantly heated by MeV-range alphas for $Q > 5$. Reproduced from [3].</figcaption></figure>

In this review, we will focus on energetic particle (EP) physics in burning plasmas in magnetic confinement devices.§ We consider a plasma to be a burning plasma if the self-heating by alphas is larger than the auxiliary heating, leading to a high degree

‡ We will introduce acronyms when they first appear but have also appended a complete list of acronyms.

§ Note by the ITER Organization: The progress on R&D described in this volume includes topics of relevance to ITER, as well as to burning plasmas more generally. In some cases, the underlying physics R&D activities were motivated by specific features of the ITER 2016 staged approach baseline [4] and ITR-2024-5 (final version) [5]. The new ITER baseline 2024 currently under elaboration proposes modifications to several of these features (e.g. first wall material, heating and current drive mix, etc.), as introduced in [6], which will require additional R&D beyond that described in this chapter.

of self-sustainment and self-organization. For burning plasmas, $P_α > P_{aux}$, so $η_α > 0.5$ and $Q > 5$. Such self-organized plasmas constitute a new regime of EP physics. The highest achieved $Q$ in magnetic confinement fusion to date is $Q = 0.64$, corresponding to $η_α = 0.11$, which was achieved at JET in 1997 [7]. We denote such plasmas with significant fusion power but below the burn condition as weakly burning plasmas. Scientific break-even is defined by $Q = 1$, corresponding to $η_α = 0.17$. Reactor-grade burning plasmas in the first power plants will require a much higher $Q$ than our burn condition, $Q > 5$, to have a substantial net electricity output [8]. For example, various DEMO designs have $Q \sim 20-50$ and $η_α \sim 0.8-0.91$. The ideal burning plasma is completely self-heated and has $Q \rightarrow \infty$ and $η_α \rightarrow 1$. This is also called ignition. In recent years, several devices have been designed or are already being constructed, which currently aim at operating burning plasmas, among them SPARC [9], STEP [10, 11], CFETR [12], and BEST. The various burning plasma regimes and relevant magnetic confinement devices are summarized in table 1. Characteristics of burning fusion plasma devices are reviewed in chapter 1 of this volume [13].

ITER is currently scheduled to start low-power plasma operation in a few years and burning fusion plasma operation a few years later. Some of the privately and publicly funded fusion efforts may access the burning-plasma regime even earlier. The next phase of fusion power generation beyond ITER, hopefully early in the second half of this century, is the operation of reactor-grade burning plasmas in fusion power plants, delivering fusion power to the electricity grid for humankind to use. These long timescales make fusion research necessarily a multi-generational effort. The next generation of fusion scientists will base the operation of reactor-grade burning plasmas on the choices our generation is making now. This generation changeover will be well on its way even for the burning plasma operation in ITER. This aspect of fusion research is a primary motivation for the ITPA Topical Groups to periodically condense the knowledge and progress towards burning plasma operation achieved in recent years into a single volume to explain the rationale for the choices we have made [13-19]. We will refer the reader to these other 7 chapters of this volume where the referred topics are discussed from the perspective of the other ITPA Topical Groups.

The essential defining characteristic of a burning fusion plasma is the self-sustained heating by the energetic α-particles, which we will call alphas in the following. To ensure self-sustained heating, the alphas must be well-confined. The enhanced transport of alphas caused by 3D effects and instabilities must be minimized. This central role of alphas in any burning
<table><caption>**Table 1.** Burning plasma regimes in terms of power amplification, alpha heating fraction, and heating, and a selection of devices aiming to achieve these.</caption><thead><tr><th></th><th>Q</th><th>η<sub>α</sub></th><th>Heating power</th><th>Devices</th></tr></thead><tbody><tr><td>weakly burning plasma below break-even</td><td>0 &lt; Q &lt; 1</td><td>0 &lt; η<sub>α</sub> &lt; 17%</td><td>0 &lt; P<sub>fus</sub> &lt; P<sub>aux</sub></td><td>JET, TFTR</td></tr><tr><td>weakly burning plasma above break-even</td><td>1 &lt; Q &lt; 5</td><td>17% &lt; η<sub>α</sub> &lt; 50%</td><td>P<sub>α</sub> &lt; P<sub>aux</sub> &lt; P<sub>fus</sub></td><td>ITER, STEP, BEST, SPARC, CFETR</td></tr><tr><td>burning plasma</td><td>Q &gt; 5</td><td>η<sub>α</sub> &gt; 50%</td><td>P<sub>α</sub> &gt; P<sub>aux</sub></td><td>ITER, STEP, BEST, SPARC, CFETR</td></tr><tr><td>reactor-grade burning plasma</td><td>Q &gt; 5</td><td>η<sub>α</sub> &gt; 50%</td><td>P<sub>α</sub> &gt; P<sub>aux</sub></td><td>DEMO, CFETR, STEP</td></tr><tr><td>ideal ignited plasma</td><td>Q &gt; &infin;</td><td>η<sub>α</sub> = 100%</td><td>P<sub>α</sub> &gt; P<sub>aux</sub> = 0</td><td></td></tr></tbody></table>

plasma system makes EP physics a core part of this volume.

Here, we will review the most essential EP physics in burning plasmas in magnetic confinement devices. We consider a particle in a plasma to be energetic, if its energy $E_f$ is significantly larger than the plasma temperature $T$ (in eV), $E_f \gg T$. EP populations consist of fusion-born alphas, fast ions from neutral beam injection (NBI), fast ions accelerated by electromagnetic wave heating in the ion cyclotron range of frequencies (ICRF), and finally the undesired runaway electrons that can get accelerated to MeV-range energies by parallel electric fields. While we will attempt to cover EP physics thoroughly and emphasize current results, the starting point and depth we have in mind will be appropriate to the non-specialist reader. We will focus on the progress achieved in the field of EP physics for burning plasmas since the last update of the ITER Physics Basis [20]. Most work of the ITPA Topical Group for Energetic Particle Physics has been done for ITER which will provide early experimental tests of our predictions on EP physics in a burning plasma. Furthermore, we will discuss implications for the fusion power plants currently being designed around the world. For further and in some areas more comprehensive information, we also point to the EP physics chapters in the earlier ITER Physics Bases [20, 21] as well as several reviews on EP physics [22–38]. We also highlight the recent JET DT experiments which allowed EP studies in mildly burning plasmas [39–54].

First, let us start by briefly reviewing the historical context of EP physics in magnetically confined fusion plasmas. Experimental investigations of EP physics began in the 1980s, when powerful NBI and ICRF heating technologies were developed and applied to toroidal plasmas. Initially favorable heating results were obtained. However, as scaling information accumulated over a range of devices and regimes, the negative effects of auxiliary heating power on

confinement became recognized and were incorporated into empirical scaling laws.

On the theoretical front, new forms of global Alfvén waves susceptible to resonant destabilization by EPs were identified. These modes exist in frequency gaps of the shear Alfvén spectrum created by couplings across poloidal mode numbers [27]. Such couplings arise from the periodic variation of magnetic field strength in the poloidal direction and are analogous to the bandgaps between the Brillouin zones of lattices with periodically varying potential, as described in solid state physics. Also, modes such as the fishbone instability and the kinetic ballooning mode that involve interaction of EPs with low-frequency magnetohydrodynamic (MHD) activity were observed in experiments and analyzed. The DT operation of TFTR and JET in the 1990s motivated an intense search for alpha-driven instabilities. However, the significant EP population from NBI heating and the relatively dilute alpha population made it difficult to identify instabilities specifically driven by alphas, except for in brief intervals following the turn-off of the beams.

Fast forwarding to current times, a large zoology of Alfvén eigenmodes (AEs) has been identified, and carefully diagnosed in many toroidal devices, including imaging of two-dimensional mode structures and diagnosis of rapid frequency variations and nonlinear dynamics. Among these are the toroidicity-induced AE (TAE), reversed-shear AE (RSAE), ellipticity-induced (EAE), non-circular-triangularity-induced (NAE), global AE (GAE), compressional AE (CAE), beta-induced AE (BAE), the beta-induced Alfvén-acoustic eigenmode (BAAE), and the low-frequency Alfvén mode (LFAM), see figure 26 in section 6. It is also known that these AEs can lead to substantial EP transport, removing up to 40 to 60% of the beam power that would normally be available for core plasma heating. Measurements and simulations of AEs can be very precise, leading to an often remarkable
agreement between theory and simulation, which is one
of the major quantitative successes of MHD in toroidal
devices. We will discuss AEs from section 6.

In addition to energetic ions, tokamaks can
also spontaneously create in-situ beams of relativistic
electrons, known as runaway electrons. The runaway
phenomenon is caused by the acceleration in toroidal
electric fields present in tokamak plasmas, coupled
with the decreasing collisional drag on electrons with
increasing velocity. Such runaway beams can attain
a sufficient intensity to cause damage to the vacuum
vessel walls. Runaway electrons were first a concern
in the early days of tokamak research due to their
generation by the Ohmic electric field in the early
breakdown phase of the discharge. More recently, it
has been recognized that intense runaway generation
can be possible due to avalanche phenomena and the
large induced electric fields that will be driven by the
current collapse phase of disruptive instabilities which
sometimes terminate tokamak discharges. We will
treat runaway electrons in section 12, and disruptions
and runaway electrons are also treated in chapter 4 of
this volume [16].

Understanding and predicting EP-driven instabil-
ities requires both linear and nonlinear modeling. Lin-
ear modeling is important because EP-driven modes
experience damping, resulting in thresholds that have
to be surpassed before a mode can exist. Nonlin-
ear modeling is important because observed EP-driven
modes typically survive for many e-folding times, based
on linear growth rates. Thus, the instabilities dynami-
cally reach some balance between sources and sinks, i.e.
drive strength vs. mitigating effects such as flattened
regions in phase space, zonal flows and currents, and
turbulent cascades. It has become recognized that EP-
driven modes set limits on the profiles and parameters
that can be achieved through critical-gradient behav-
ior and enhanced transport. In current devices, en-
hanced EP transport can conveniently be character-
ized through the deuterium-deuterium (DD) neutron
rate which is almost linearly related to the EP den-
sity since DD neutrons in existing devices are predom-
inantly generated through beam-target reactions, i.e.
reactions between slow ions and EPs. An observed neu-
tron deficit indicates an EP density deficit and hence
enhanced transport. Nonlinear EP physics remains an
active area of experimental and theoretical research.
The nonlinear effects of EP-driven instabilities can de-
pend strongly on the q-profile, the mode classification,
the mode frequency range, and the number of modes
that are active concurrently.

A significant challenge arising in burning plasmas
with high power amplification *Q* is that the tempera-
ture and density profiles are self-organized, i.e. self-
consistently determined by self-heating from alphas

rather than directly controlled from the outside by aux-
iliary heating. The physics and stability properties of
plasma with self-sustained heating by alpha popula-
tions lead to a new regime of operation in ITER and
future burning plasma devices. Several plasma char-
acteristics are substantially different, such as the high
degree of isotropy of energetic alpha populations com-
pared to the highly anisotropic ions generated by aux-
iliary heating, the shape of the EP profile, the EP Lar-
mor radius normalized by the minor radius, which is
around ρ<sub>Lf</sub><sup>*</sup> ~ 0.02 in present devices and 0.04 in ITER,
and the Alfvén Mach number (the ratio of EP speed
and Alfvén speed), which often is 0.3 to 0.5 in present
devices but will be 1 or above in ITER. EP-driven in-
stabilities will have different characteristics compared
to those seen in present experiments. They are most
active around k<sub>⊥</sub>δ<sub>f</sub> ~ 1, where k<sub>⊥</sub> is the perpendicular
wave number and δ<sub>f</sub> is the drift orbit width, and will
therefore predominantly exist at higher mode numbers
and be more radially localized than in present experi-
ments. This can lead to increased local transport and
the potential for radial avalanches as compared to ex-
isting devices where radial wavelengths are typically 20
to 50% of the minor radius. This new regime of oper-
ation cannot be directly tested on non-burning, exter-
nally heated plasmas in present machines and forms
a major part of the rationale for building ITER and
other experimental burning plasma devices.

One might have hoped to tackle this change in
parameters by a “wind-tunnel” approach, where we
carefully diagnose the EP populations and instabilities
in experiments on a sequence of smaller devices and
develop scaling laws describing EP-related phenomena.
However, since the resonant nature of EP-driven
instabilities causes strong dependencies on the plasma
profiles, such a “wind-tunnel” approach by itself
cannot be relied on to predict the performance
of reactor-grade fusion plasmas from present high-
performance plasmas or even from weakly burning
plasmas. Instead, our approach is to make predictions
of experimental outcomes in present devices by
modeling and simulation and test these predictions
against the experimental data. The application of
codes validated this way constitutes our best prediction
of EP physics in burning fusion plasmas. To gain a
full understanding of alpha heating and instabilities
in burning plasmas, it will be crucial to test these
predictions against experimental data at ITER and
other burning plasma devices, both in the operation
phases with and without burning fusion plasmas. Good
diagnostic coverage at ITER and the other devices
will therefore be essential. However, many existing
diagnostic methods will not be available in the high
neutron flux environment of burning plasmas.

Thus, in order to prepare for future plasma
operation with significant alpha populations and self-
sustained heating, EP physics research must support
three main research areas, which comprise the main
topics of this paper:

(i) Experiments on existing devices,

(ii) Diagnostic development, and

(iii) Theory, modeling, and simulation.

Studies of EPs further need to consider the
helium ash, i.e., alphas that have slowed down and
delivered a major part of their energy to the bulk
plasma. Whereas the high-energy alphas, which heat
the plasma, are a cherished population for sustaining
fusion power, the helium ash is an undesired impurity
population, diluting the DT fuel mixture. If these
thermalized alphas are not transported out of the
plasma at a sufficient rate, the fuel dilution can lead
to an extinction of the burning plasma state. The
transport of helium ash is discussed comprehensively
in chapter 2 of this volume [14]. The understanding
of alpha physics must extend over a broad range of
energies from the 3.5 MeV alpha birth energy to a
few times the thermal plasma energy. Ideally, control
mechanisms should be identified that rapidly transport
alphas out of the plasma once they have deposited
a large fraction of their energy in the thermal bulk
plasma. These mechanisms could include EP-driven
instabilities, externally driven radiofrequency (RF)
waves, 3D fields, or MHD instabilities such as sawteeth
or edge-localized modes (ELMs).

The physics of the helium ash regime has not
attracted nearly as much attention as the effects related
to the higher energy alphas. ITER will face this
issue in an especially direct way since the pulse length
significantly exceeds the slowing-down time of alphas.
Dilution of the DT fuel by thermalized helium ash
can threaten this mission goal for the pulse length
of burning plasmas. The need to understand alpha
transport over the whole energy range leads to an
unusual bimodal challenge. At high energies, the
alphas must have good confinement and instabilities
suppressed; at low energies comparable to the thermal
energy, alpha confinement should be degraded and
instabilities excited if possible. ITER will offer a
crucial experimental platform to consistently address
both extremes of alpha physics. This motivates careful
diagnostic and modeling over the full energy range of
the alpha distribution function.

This paper is organized as follows. Section 2
reviews elementary EP physics in idealized plasmas,
including drift orbit theory and associated frequencies,
EP sources, plasma heating, confinement, and losses,
which constitutes a basis for more complex physics
discussed in later sections. Section 3 highlights
recent developments in the area of ICRF heating,

such as the efficient three-ion scheme, which opens
new possibilities not known when ITER was designed.
Furthermore, in the early operational phases of ITER
before DT, ICRF heating will be the only significant
source of EPs. Section 4 describes the main diagnostics
used to detect EPs and instabilities, in particular the
diagnostics capable of working in the high neutron-
and gamma-ray flux environment of burning plasmas.
We focus on what physics can be diagnosed and
refer to chapter 8 of this volume for the technical
details of the diagnostics [19]. Section 5 describes
the mutual interaction of EPs and thermal-plasma
instabilities, i.e. instabilities typically not driven by
EPs. In particular, we describe the interaction of EPs
and neoclassical tearing modes, sawteeth, ballooning
instabilities, resistive wall modes, and edge-localized
modes, which are thought to have the strongest
interaction with EPs. Section 6 discusses linear
stability theory of EP-driven modes, which remains a
crucial theoretical framework to predict their existence.
While section 6 can describe the existence of EP-
driven modes, section 7 deals with their evolution,
requiring nonlinear theory and simulation. This field
has grown substantially since the 2007 ITER Physics
Basis [20], partially owing to the ever increasing
computer power. Taking the fishbone instability as
our starting point requiring nonlinear theory and
simulation, we review various approaches to modeling.
Gyrokinetic theory and simulation constitutes the
most fundamental approach to EP simulations,
requiring the fewest modeling assumptions. We then
review various approaches relying on progressively
more modeling throughout the section: kinetic-MHD
models, gyrofluid models, reduced models, and semi-
analytic approaches. Simulations using such reduced
models are often more tracktable since they allow
us to focus on the most essential nonlinear physics.
Section 8 deals with 3D effects on EP confinement.
3D effects arise since plasmas in real magnetic
fusion devices are never exactly axisymmetric, which
would be required for a 2D treatment, due to
the magnetic coil systems generating magnetic field
ripples and due to the 3D structures of modes
in the plasma. Section 9 considers multi-scale
synergistic interactions between EP-driven modes, low-
frequency perturbations, especially EP-driven geodesic
acoustic modes (EGAMs), and turbulence, which are
interactions where several phenomena are concurrently
active. In section 10 we explore options to
optimize plasma scenarios and to enhance the plasma
performance through EPs by considering the EP phase
space and possible actuators we have to control the EP
phase space. section 11 deals with the EP physics in
ITER plasmas before fusion power operation, where
plasmas are heated by NBI and ICRF heating, but
are not burning. Here we study EP physics in plasma scenarios with reduced magnetic fields, in which it is easier to access the H-mode. Finally, while runaway electrons are considered throughout this review as a group of EPs, section 12 focuses on the physics of runaway electrons in particular. Section 13 concludes this review by highlighting the state-of-the-art of EP physics research and by pointing to possible future avenues of EP physics research in burning plasmas.

## 2. Basic physics of EPs: sources, plasma heating, confinement and losses

Most of the plasma in ITER will be in a state near local thermodynamic equilibrium. The thermal ions and electrons that comprise the bulk gyrate with Larmor orbits with radii $\rho_{Li}$ and $\rho_{Le}$ which are small when compared with the scales of the vessel, and they drift along magnetic field lines with small deviations $\delta_i$ and $\delta_e$ from the concentric flux surfaces on which the magnetic field lines lie. Typical characteristic speeds of phenomena in the bulk plasma include the thermal speeds of the ions $v_{th,i} = \sqrt{T_i/m_i}$ and electrons $v_{th,e} = \sqrt{T_e/m_e}$, the propagation speeds of Alfvén waves $v_A = B/\sqrt{\mu_0 m_i n_i}$ and sound waves $c_s = \sqrt{(T_e + \Gamma_i T_i)/m_i}$ with $\Gamma_i$ the adiabatic index, and the diamagnetic drift velocities of electrons $v_{*e} = T_e/(eB)\partial_r \log(p_e)$ and ions $v_{*i} = T_i/(Z_i e B)\partial_r \log(p_i)$. Several thermal ion species will be present in burning plasmas, each with their own temperatures and velocities.

In order to sustain the thermal plasma at constant temperature, a steady source of bulk plasma heating must be provided by more energetic ions or electrons that are constantly generated or introduced into the plasma. They slow down due to collisional processes and gradually become thermalized. In addition to the plasma self-heating from 3.5 MeV alphas produced in the DT fusion reactions, also the auxiliary heating methods ICRF heating and NBI on ITER introduce MeV-range ions into the plasma. Electron cyclotron radio-frequency (ECRF) heating is another main auxiliary heating scheme which may generate energetic electrons.

Energetic ions reach speeds exceeding some of the important characteristic speeds of the bulk plasma processes, and the balance between ion creation and slowing-down produces energy distributions which deviate strongly from local thermal equilibrium distributions. The motion of fast ions is more complicated than the motion of thermal ions; in addition to the large free-streaming velocity $v_{\parallel L}$ and perpendicular gyration $v_{\perp L}$, the drift motion away from the magnetic field lines $v_D$ can become significant as well. Importantly, the drift velocity depends not only on the particle energy, but also on the

magnetic field strength and geometry. Confined fast ions will traverse ITER with these velocities, with the dimensions of ITER implying a corresponding set of timescales. For phenomena much slower than the ion gyration and lengthscales larger than the gyroradius, it is sufficient to ignore the gyration and focus on the timescales associated with the gyro-averaged quantities $v_{\parallel}$ and $v_D$.

### 2.1. Equilibrium on the collisionless timescale

For magnetic fields with slow variations in space and time, and for sufficiently slow and weak electric fields, particles possess nearly invariant properties. The existence of these invariants guarantees that particles are confined to imaginary surfaces that can be labelled by the invariants [55]. When considering distributions of particles which are steady in time, confined particles belong to equilibrium distributions. The timescale of the equilibrium governs the form that the distribution must take [56]. For equilibrium on a longer timescale than a cyclotron period $2\pi/\omega_c$, where $\omega_c = ZeB/m$ is the cyclotron frequency, the particle distribution must be independent of gyroangle and expressible as

$$F = F(x, \mathcal{E}, \mu, \sigma), \qquad (2)$$

where $\sigma = \text{sign}(v_{\parallel})$ and $x$ is the position. The particle energy $\mathcal{E}$ is never changed by the magnetic field. The invariant $\mu$ associated with the gyration of the particle is given by the magnetic moment $\mu \approx mv_{\perp}^2/(2B)$, provided the magnetic field is nearly uniform on the scale of the gyration. The next fastest timescale is the time taken to follow a magnetic field line $L/v_{\parallel}$ for some typical distance $L$, meaning that equilibrium must be independent of distance along a magnetic field line, $B \cdot \nabla F = 0$. If magnetic field lines are labeled with Clebsch-type coordinates $\alpha_{Cl}$ and $\beta_{Cl}$ [57], so that $B = \nabla\alpha_{Cl} \times \nabla\beta_{Cl}$, and a length coordinate along the field line is represented by $s$, then we will have the positional dependence on $x$ replaced by

$$F = F(\alpha_{Cl}, \beta_{Cl}, \mathcal{E}, \mu, \sigma), \qquad (3)$$

independent of $s$. For axisymmetric idealised tokamaks, the magnetic field lines lie within poloidal flux surfaces (constant poloidal flux $\psi$), and the magnetic field can be written in terms of $\psi$ and toroidal angle $\phi$ as $B_{\theta} = \nabla\psi \times \nabla\phi$. The poloidal flux is related to the magnetic vector potential according to $A_{\phi} = \psi/R$. For timescales shorter than $L/v_{\parallel}$, fast ions are approximately confined to those flux surfaces

$$F = F(\mathcal{E}, \mu, \psi, \sigma). \qquad (4)$$

Eventually, the drift away from magnetic field lines becomes important for equilibrium on a timescale $L/v_D$, meaning that the surface traced by an invariant such as $J^{**} = \oint (v_{\parallel} + \frac{Ze}{m}A_{\parallel})ds$, where $A_{\parallel}$ is the
parallel magnetic vector potential, (for almost closed field lines) defines the longer equilibrium

$$F = F(\mathcal{E}, \mu, J^{**}, \sigma) \qquad (5)$$

with purely spatial position coordinates no longer appearing. If the field is exactly axisymmetric, then the canonical toroidal angular momentum $P_\phi = mRv_\phi + Ze\psi$ is an exact invariant providing perfect confinement with the equilibrium

$$F = F(\mathcal{E}, \mu, P_\phi, \sigma), \qquad (6)$$

until collisions or other processes violate the symmetries underlying the constants of motion (COMs) [37]. Most notably, departures from axisymmetric magnetic fields violate the conditions for equilibrium because of the loss of the exact invariant $P_\phi$.

Static or slowly time-varying perturbations to the magnetic field can each be decomposed into toroidal modes of the form

$$\delta \mathbf{B} (R, z, \phi, t) = \tilde{\mathbf{B}} (R, z, t; n, \omega) \exp (i (n\phi - \omega t)), \quad (7)$$

where R is the major radius coordinate, z the vertical coordinate, $\omega$ is the angular frequency, and n is the toroidal mode number. For a particle interacting with this perturbation, the toroidal component of the Lorentz force changes the toroidal momentum in $P_\phi$, whilst the perturbed vector potential changes the poloidal flux, so the canonical toroidal angular momentum changes at the rate

$$\begin{aligned} \frac{dP_{\phi}}{dt} &= ZeR\hat{\phi} \cdot \left( \delta \mathbf{E} + \mathbf{v} \times \delta \mathbf{B} + \frac{d\delta \mathbf{A}}{dt} \right) \\ &= inZe (\delta \mathbf{A} \cdot \mathbf{v} - \delta \Phi), \end{aligned} \qquad (8)$$

where we have used Faraday's law to obtain the electric field $\delta \mathbf{E}$ from the vector and scalar potentials $\delta \mathbf{A}$ and $\delta \Phi$.

Important characteristic frequencies of the orbital motion of the ions are the transit frequencies in the toroidal and poloidal directions $\omega_\phi$ and $\omega_\theta$, based on the gyro-averaged quantities $v_\|$ and $v_D$. If the time it takes a particle to complete a full revolution in the tokamak in the poloidal direction is denoted by $\tau_\theta$, these orbital frequencies are given by

$$\omega_{\theta} = 2\pi/\tau_{\theta}, \qquad (9)$$

$$\omega_{\phi} = \frac{1}{\tau_{\theta}} \int_{0}^{\tau_{\theta}} \frac{d\phi}{dt} dt \qquad (10)$$

Any violation of equilibrium is most efficient when there is coherent synchronization between the orbital phase and the phase of the perturbation with frequency $\omega$, leading to the resonance condition

$$\omega - n\omega_{\phi} - l\omega_{\theta} = 0, \qquad (11)$$

where the integer l is arbitrary. The details of the orbit interaction with the 2D perturbation structure $\tilde{\mathbf{B}}(R, z, t)$ governs the magnitude of the effect on the orbit for different values of l. Even when there is

synchronization, the variation in the $P_\phi$ can be periodic and average to zero for sufficiently small $\delta B/B$. However, for orbits closest to being in resonance, small deviations due to collisions or due to overlapping resonances can lead to chaotic motion. The number of orbits sensitive to this effect scales with the magnitude of the perturbation.

The slow time variation of the magnetic field of a single toroidal mode implies an induced electric field. This electric field does work on the particles and hence exchanges energy with the particle with a rate depending on the frequency according to

$$\frac{d\mathcal{E}}{dt} = Ze \left( \mathbf{v} \cdot \delta \mathbf{E} + \frac{d\delta \Phi}{dt} \right) = i\omega Ze (\delta \mathbf{A} \cdot \mathbf{v} - \delta \Phi). \quad (12)$$

The rates of change of the energy and the canonical toroidal angular momentum from each toroidal mode can be compared, giving immediately

$$n\Delta\mathcal{E} = \omega\Delta P_{\phi}. \qquad (13)$$

This important equation can also be deduced from a quantum mechanical perspective if we imagined the perturbation of particles from toroidally propagating modes as the resonant absorption (or emission) of photons; for an incoming photon, the energy absorbed is $\Delta\mathcal{E} = \hbar\omega$ and the absorbed toroidal momentum is $\Delta P_\phi = R\hat{\phi} \cdot \hbar\mathbf{k} = \hbar n$. An important consequence of equation (13) is that a particle undergoing radial excursion due to a perturbation will also experience a change in energy. This change in energy is the result of the induced electric field associated with the perturbation. For shear Alfvénic perturbations, this energy change is small, and vanishes entirely for magnetic perturbations approaching zero frequency and growth rate. For low-frequency large length-scale perturbations, the gyro-invariant is preserved, $\Delta\mu \approx 0$, giving together with equation (13) a complete description of how orbits are affected.

So far, we have only considered perturbations where the gyration of the particles plays no role in the coherent synchronization between the orbit and the waves. If the perturbations become narrow compared with the size of the gyration, or if the frequency of the perturbation becomes comparable to the gyrofrequency, considering only the resonance of guiding centres in equation (11) will not suffice and a general resonance condition is required [58]. Care must be taken to consider the coherent synchronization between the perturbation and the orbit-averaged poloidal, toroidal, and cyclotron frequencies. When these frequencies are well separated, such as on ITER, the resonant interaction can be taken as a local phenomenon giving

$$\omega = N\omega_c + k_{\|}v_{\|}. \qquad (14)$$

At high frequency, the changes in magnetic moment and the energy are no longer small, and a change
in energy leads to a change in normalized magnetic moment $\Lambda = \frac{\mu B_0}{\epsilon}$, where $B_0$ is the toroidal magnetic field on axis, according to [59]

$$ \Delta\Lambda = \left( \frac{N\omega_c}{\omega} - \Lambda \right) \frac{\Delta\mathcal{E}}{\mathcal{E}}, \qquad (15) $$

or equivalently

$$ \Delta\mu = \frac{N\omega_c}{\omega B_0} \Delta\mathcal{E}. \qquad (16) $$

Note that the resonance condition in equation (14) applies when the particle interacts with the wave on a portion of its drift orbit, whereas the resonance condition in equation (11) applies when the particle experiences the same wave phase on multiple drift orbits. In addition to the resonances, efficient wave-particle energy transfer can occur when an EP encounters a constant phase during a single pass through an instability [60]. This phenomenon can cause AE-induced losses of fast ions that are born near the plasma edge [61].

## 2.2. Drift orbit approximations

For a perfectly confined particle, the drift velocity across magnetic field lines $v_D$ is a periodic function, and the particles return to flux surfaces after a poloidal transit time $\tau_\theta = 2\pi/\omega_\theta$. The width of a drift orbit $\delta$ is then the distance traversed by $v_D$ during half a poloidal transit time, which is the distance between the innermost and outermost flux surface the ion reaches. For realistic tokamak geometries, these characteristic frequency- and lengthscales can be computed using orbit-following codes, e.g. [62]. If $L/v_D \gg \tau_\theta$, we may neglect the orbit width in the equilibrium and take $\psi$ to be invariant instead of $P_\phi$. Under these conditions and for circular, large-aspect-ratio flux surfaces, we can estimate the orbital frequencies and widths. The toroidal and poloidal transit frequencies of strongly passing particles can be estimated as

$$ \omega_{\phi,\text{pa}} \approx \frac{v_{\parallel}}{R_0}, \qquad (17) $$

$$ \omega_{\theta,\text{pa}} \approx \frac{v_{\parallel}}{qR_0}, \qquad (18) $$

where $R_0$ is the major radius of the magnetic axis and

$$ q \approx \frac{2\pi r^2 B_0}{\mu_0 I(r) R_0} \qquad (19) $$

is the safety factor where $r$ is the minor radius coordinate and $I(r)$ is the plasma current contained within the minor radius $r$. The safety factor $q$ is the number of toroidal turns per poloidal turn of a magnetic field line around the torus. For trapped particles, orbits do not complete a circular transit, and $\omega_\theta$ depends on a more complicated expression in terms of elliptic functions in analogy with the period for a

pendulum [63]. As for a small amplitude pendulum, deeply trapped particles approach a constant bounce period $\tau_{\theta,\text{tr}} \approx 2\pi(\frac{2}{\epsilon})^{1/2} \frac{qR_0}{v}$ with the corresponding bounce frequency

$$ \omega_{\theta,\text{tr}} \approx \left(\frac{\epsilon}{2}\right)^{1/2} \frac{v}{qR_0}, \qquad (20) $$

where $\epsilon = r/R_0$ is the inverse aspect ratio.

The parallel velocity of trapped particles changes sign, with inexact cancellation of toroidal motion over an orbit and a resulting precession with frequency $\omega_{\phi,\text{tr}}$, which scales with the drift velocity according to $\omega_{\phi,\text{tr}} = qv_D/r$. The drift velocity $v_D$ is the sum of grad-B and curvature drifts. The grad-B drift depends on $v_\perp$ while the curvature drift depends on $v_\parallel$. A first approximation to the drift is again a constant value taken at the magnetic axis $v_D \approx \mu B_0/(mR_0\omega_c) + v_\parallel^2/(R_0\omega_c)$. Introducing the particle energy into the expression cancels the mass dependence giving

$$ v_D \approx \frac{\mathcal{E}}{ZeB_0 R_0}. \qquad (21) $$

Hence, the drift velocity is proportional to the energy of the particle, where now the mass of the particle only enters through the energy. This cancellation of the mass dependence in the drift velocity implies the important result that electrons and ions at the same energy drift at the same speed. For deeply trapped particles, the toroidal precession frequency is thus in terms of energy

$$ \omega_{\phi,\text{tr}} \approx \frac{q\mathcal{E}}{ZeB_0 R_0 r}. \qquad (22) $$

The precession frequency $\omega_{\phi,\text{tr}}$ for trapped electrons and ions with a single charge is the same, but they drift in opposite directions. The maximum excursion of a strongly passing particle from a flux surface and the orbit width of deeply trapped particle $\delta$ can both also be estimated immediately using the drift velocity and bounce times, but a more accurate calculation based on $P_\phi$ conservation gives

$$ \delta_{\text{pa}} \approx 2q\rho_{\text{L}}, \qquad (23) $$

$$ \delta_{\text{tr}} \approx 2q\rho_{\text{L}}\epsilon^{-1/2}, \qquad (24) $$

showing that trapped particles drift further from flux surfaces than passing particles. The dependence on $q\rho_L$ makes clear that it is the poloidal Larmor radius which matters to the orbit width, which is not dependent on the toroidal field, but rather on the plasma current.

## 2.3. Equilibrium on the collisional timescale

The kinetic picture of fast-ion equilibrium presented above derives from ignoring collisional processes. Connection can be made with the neoclassical theory of transport (e.g: [64, 65]) by comparing the
collision frequencies with the orbital frequencies of collisionless motion. If collision frequencies are small when compared with orbital frequencies, as will be the case on ITER, equilibrium established on collisional timescales will be obtained in the ‘banana limit’ of the neoclassical theory. These equilibrium distribution functions, established over timescales exceeding the collision time, will not resemble Maxwellian distributions. The effect of Coulomb collisions on the equilibrium is modelled with a Fokker-Planck equation

$$ \frac{\partial f_s}{\partial t} + \mathbf{v} \cdot \frac{\partial f_s}{\partial \mathbf{x}} + \frac{Z_s c}{m_s} (\mathbf{E} + \mathbf{v} \times \mathbf{B}) \cdot \frac{\partial f_s}{\partial \mathbf{v}} = Q[f_s] + \sum_{s'} \{C[f_s, f_{s'}] + S[f_s, f_{s'}]\} \quad (25) $$

where we have also included terms for the creation and loss of particles s and for the quasi-linear heating Q from rapidly varying electric fields that are not included in the left-hand side of the equation. The left-hand side of the equation represents the collisionless motion of particles including fast gyration and can be approximated by transformation to guiding-centre variables [66]. Solving the Fokker-Planck equation for timescales where $\frac{\partial}{\partial t} = 0$ gives the equilibrium distribution function expressible as $f_s = F(\mathcal{E}, \mu, P_\phi, \sigma)$.

Source and sink terms include the production and loss of fast ions in fusion reactions, the production and loss of charges in charge exchange reactions, or simply the external introduction of fast ion species.

## 2.4. EP sources

The main sources of energetic ions in fusion plasmas are fusion reactions, NBI and ICRF heating. We will discuss ICRF heating in section 3 and will here highlight certain aspects of alphas and other charged fusion products, EPs from NBI heating and energetic electrons generated by parallel electric fields. The heating systems at ITER are further described in chapter 6 of this volume [18].

The fusion reaction between a deuteron and a triton results in an alpha confined by the magnetic field and a neutron which escapes the plasma. The energy of the alpha in the center-of-mass frame of the fusion reaction is 3.52 MeV with no preferred velocity direction. The large energy and near isotropic distribution makes alpha physics difficult to imitate using ions produced by NBI or ICRF heating, which generate highly anisotropic distributions. Record fusion energy production, and equivalently the largest alpha population, was achieved in the recent JET DT campaign by injection a deuterium beam in tritium-rich plasma [39], but Q was lower than in the earlier experiments at JET and TFTR. A high $Q \approx 2.2$ at

$P_\alpha = 1.8$ MW was achieved in an afterglow experiment, i.e. after the NBIs were switched off [47].

In presently operating tokamaks, NBI is the most reliable and commonly used method to heat the plasma to high temperatures relevant for fusion reactions to occur. ITER will have two 16.5 MW NBIs with energies up to 870 keV for hydrogen operation and 1 MeV for deuterium operation. In many present NBI sources, positive ions are brought up to energies in the range $\sim 20-180$ keV in a particle accelerator and then neutralized by recombination reactions with electrons. In these positive-ion NBI sources, a mixture of ions and molecular ions, e.g. $D^+, D_2^+$, and $D_3^+$, is typically present when hydrogenic gas is ionized and accelerated in the voltage drop. The fast neutral molecules then dissociate into fast neutral atoms, dividing the energy of the molecule equally between the constituent atoms. The fast atoms entering the tokamak therefore have energies matching the full injection energy, half the injection energy, or one-third of the injection energy.

The resulting high-energy atoms are guided to the plasma through an NBI port, i.e. an opening in the vessel wall. The neutral atoms are not deflected by the magnetic field and move on straight paths, until they are ionized in the plasma. The beam of fast neutral atoms is highly directional and has a maximum energy chosen for optimal penetration into the core of the plasma before ionization and capture by the magnetic field. This ionization process corresponds to the source term S in the Fokker-Planck equation (25). After ionization, the ions proceed on a drift orbit dictated by the magnetic field. Subsequently, the energetic ions will heat the bulk plasma via Coulomb collisions and eventually become part of the thermal bulk plasma.

However, NBIs with typical energies of 100 kV cannot heat the plasma core in large machines with dense plasma such as ITER as most ionization already occurs in the plasma periphery. The penetration depth can be increased by increasing the particle energy, i.e., the acceleration voltage. But when the acceleration voltage is increased to significantly above 100 kV, the neutralization efficiency rapidly drops and is unacceptably low at energies required for ITER plasmas.

This problem can be circumvented by accelerating negative ions instead of positive ions. These two methods are referred to as negative-ion neutral beam injection (NNBI) and positive-ion neutral beam injection (PNBI). In NNBI, the additional electron is only loosely attached to the accelerated particle, so the neutralization efficiency is high. Since the ITER neutral beams will operate with 1 MeV energy, it is clear that NNBI is vastly superior to PNBI. NNBI has been experimentally demonstrated at the JT-60U tokamak and the LHD stellarator. NNBI do not inject
neutrals at half- and one-third injection energy.

Also the orientation of the beams has to be carefully considered, both to fully harness the potential of the beams and for safety reasons. Tangential injection, i.e. the beam path is tangent to some flux surface, produces a large fraction of ions on passing orbits, which drive current in the plasma. Therefore, neutral beam current drive (NBCD) is considered as a potential means for allowing non-inductive operation of tokamak plasmas [67]. In contrast, perpendicular injection produces energetic ions on trapped orbits that drive negligible current. We will discuss EP current drive in section 10.

An injected neutral that does not ionize in the plasma strikes the vacuum vessel wall of the inner column. When the plasma density is lower than the nominal design value, a large fraction of injected neutrals may traverse the plasma without ionizing, heating the wall materials instead of the plasma. This undesired wall heating is called “shinethrough”. For ITER, plasma scenarios requiring extra care in this respect are the ramp-up and ramp-down phases in the beginning and end of the discharges, and in particular in scenarios with lower magnetic field strengths, such as in scenarios at 1/2 and 1/3 of the nominal magnetic field and plasma current foreseen in the early phases of ITER operations. Low magnetic fields lead to low plasma densities that promote shinethrough. The needed adjustments of the ITER beams in these plasma scenarios will be discussed in section 11.

Electrons in future burning plasmas are also anticipated to have energetic non-thermal populations. Heating and current drive of electrons can be achieved through exploiting electron cyclotron or lower hybrid wave resonances. Additionally, very large loop voltages can be induced in the plasma during a disruption, resulting in a population of runaway electrons whose drag due to collisions vanishes as they accelerate. These channels of relativistic electrons are highly directional and contain a large energy density since the energy gains of the runaway electrons are not balanced by energy losses due to collisions. These runaway electrons will be discussed in section 12.

## 2.5. Collisional energy transfer

Once EPs are introduced by the various sources, they slow down due to collisions, transferring energy to the thermal particles. Collisions between charged particles in a fully ionized plasma are dominated by the sum of many small angle collisions occurring within a Debye length [78]. The collisions are represented as an outflow of particles in velocity space

$$C[f_s, f_{s'}] = -\nabla_v \cdot \mathbf{j}_{s,s'} \quad (26)$$

The flow in velocity space depends in turn on the velocity gradients of the distribution:

$$\dot{\mathbf{j}}_{s,s'} \equiv -\mathbf{A}_{s'} f_s - \nabla_v \cdot \underline{\underline{\mathbf{D}}}_{s'} f_s. \quad (27)$$

The vector $\mathbf{A}_{s'}$ represents a drag force slowing particles down, but not altering their directions. The tensor $\underline{\underline{\mathbf{D}}}_{s'}$ represents the diffusion of particles. When considering any given fast ion, collisions with both electrons and ions must be accounted for. Due to the large mass difference between ions and electrons, diffusion is negligible for ions colliding with electrons. For fusion products and neutral beam heating, the equilibrium distribution is approximately solved by balancing the source rate $S$ with the Coulomb collisions $C$.

In the range of energies typical for energetic ions in tokamaks, $v_f$ always satisfies $v_{th,i} \ll v_f \ll v_{th,e}$. In this case, analytic solutions to the Fokker-Planck equation for uniform plasmas are called slowing-down distribution functions [79, 80]. Since the Coulomb cross-section is determined by the relative velocity of the scattering particles, the scattering rate with electrons is determined by $v_{th,e}$, while the scattering rate with thermal ions is determined by $v_f$, nearly independent of $v_{th,i}$. The drag caused by thermal electrons (alone) leads deceleration of the particles with the associated Spitzer slowing-down time

$$\tau_s = 6.27 \times 10^{14} \frac{A_f T_e^{3/2}}{Z_f^2 n_e \ln \Lambda} \frac{m^{-3}}{\text{eV}^{3/2}} \text{s}, \quad (28)$$

where $\ln \Lambda$ is the Coulomb logarithm, $T_e$ is the background electron temperature in eV, $n_e$ is the density in m$^{-3}$, and $A_f$ and $Z_f$ the EP atomic mass and charge numbers. The relative importance of electron and ion friction depends upon both $v_{th,e}$ and $v_f$. The fast ion energy at which the electron friction just balances the bulk ion friction is known as the critical energy $\mathcal{E}_{crit}$ and is given by

$$\mathcal{E}_{crit} \equiv 14.8 T_e A_f \left( \sum_i \frac{n_i Z_i^2}{n_e A_i} \right)^{2/3}, \quad (29)$$

where the sum is over the thermal-ion species. Above this critical energy, collisions with electrons dominate, leading to slowing down of the fast ions with little pitch angle scattering. Below the critical energy, collision with ions dominate, leading to slowing down of fast ions with significant pitch angle scattering. On ITER, the birth energies for fusion products and NNBI will exceed the critical energy and will, at first, be dominated by electron-ion drag collisions. Electron heating by alphas was directly observed in the recent JET DT campaign [45]. The Spitzer time describes the rate of change of the test particle velocity caused by the friction on background electrons, i.e. $dv_f/dt = -v_f/\tau_{se}$ (assuming only electrons are involved in the slowing-down process). The thermalization time of a fast
<table><caption>**Table 2.** Non-dimensional parameters characterizing various EP effects. ◦ Orbit-width estimate modified to include ellipticity $\delta_f \approx 2\rho_{Lq}/(\kappa\epsilon^{1/2})$. † This value is large because $q_{95} = 10$ is not representative of the average $q$ over the orbit. Numerical results [77] indicate a value closer to $\delta_f/a \sim 1.2$. • Field ripple assumed as largest non-axisymmetric field.</caption><thead><tr><th></th><th>ITER</th><th>JET</th><th>NSTX</th><th>SPARC</th><th>STEP</th><th>DEMO</th><th>CFETR</th></tr><tr><th></th><th>f<sub>α</sub></th><th>f<sub>α</sub></th><th>f<sub>NBI</sub></th><th>f<sub>α</sub></th><th>f<sub>α</sub></th><th>f<sub>α</sub></th><th>f<sub>α</sub></th></tr></thead><tbody><tr><td>References</td><td>[31, 68]</td><td>[7, 68, 69]</td><td>[33, 70, 71]</td><td>[72, 73]</td><td>[11]</td><td>[8, 74]</td><td>[12, 75, 76]</td></tr><tr><td>a/R<sub>0</sub></td><td>0.32</td><td>0.33</td><td>0.76</td><td>0.31</td><td>0.56</td><td>0.3</td><td>0.31</td></tr><tr><td>v<sub>f</sub>/v<sub>A</sub></td><td>1.91</td><td>1.68</td><td>3.34</td><td>1.55</td><td>4.17</td><td>1.46</td><td>1.66</td></tr><tr><td>v<sub>A</sub>/v<sub>th,i</sub></td><td>4.70</td><td>4.74</td><td>2.19</td><td>6.08</td><td>2.26</td><td>5.34</td><td>4.48</td></tr><tr><td>ρ<sub>L,f</sub>/a</td><td>0.03</td><td>0.08</td><td>0.2</td><td>0.04</td><td>0.04</td><td>0.01</td><td>0.02</td></tr><tr><td>δ<sub>f</sub>/a °</td><td>0.16</td><td>0.53</td><td>1.88†</td><td>0.21</td><td>0.37</td><td>0.12</td><td>0.19</td></tr><tr><td>β<sub>f</sub>(0)/β(0)</td><td>0.15</td><td>0.13</td><td>0.15</td><td>0.13</td><td>0.58</td><td>0.4</td><td>0.16</td></tr><tr><td>Max δB/B •</td><td>0.5%</td><td>0.08%</td><td>0.6%</td><td>0.15%</td><td>0.65%</td><td>0.3%</td><td>0.1%</td></tr></tbody></table>

ion is related to the Spitzer slowing down time by $\tau_{th} \equiv (\tau_{se}/3) \ln(1 + (\mathcal{E}_f/\mathcal{E}_{crit})^{3/2})$.

For the fast ions to deposit their heat into the thermal electrons, the invariants of the motion must not vary much within a slowing down time. As we have already mentioned for low-frequency phenomena, the dominant loss of invariance occurs due to loss of axisymmetry. We therefore require that the variation in canonical momentum over an orbit $\Delta P_\phi/Ze(\psi_{edge} - \psi_{core})$ be much less than the number of orbits required to slow down $\tau_s/\tau_\theta$.

## 2.6. Non-dimensional EP parameters

Several essential non-dimensional parameters characteristic for the size of various EP effects are given in Table 2. The larger the parameter, the stronger the EP effects will be. $v_f/v_A$ measures the EP ability to resonate with Alfvénic modes in the plasma, and $v_A/v_{th,i}$ measures if the Alfvénic modes will be dominated by EPs or by a combination of EP and thermal effects [81]. The normalized Larmor radius $\rho_{L,f}/a$ is a fundamental measure of how strongly magnetized the plasma is, with ramifications for both stability and confinement of the EPs. The normalised orbit-width $\delta_f/a$ measures the deviation from flux surfaces, which relates to how well the asymmetric poloidal magnetic field is able to confine the EPs. The ratio $\beta_f(0)/\beta(0)$ on axis with $\beta = 2\mu_0p/B^2$ measures the relative sizes of the EP pressure and the plasma pressure. The EP pressure drives instabilities, and the plasma pressure results in damping. The perturbed magnetic field ripple $\delta B/B$ is responsible for the rate of momentum change $[1/Ze(\psi_{edge} - \psi_{core})]dP_\phi/dt$ which if sufficiently large will create chaotic orbits and result in radial transport.

# 3. EP generation and plasma heating by ICRF waves

## 3.1. Basic physics of ICRF heating

ICRF waves have been used early on in magnetic fusion research to generate EPs (see section 2.3 of [22] for a review). The thermal plasma is heated by two effects: direct wave damping and collisional heating due to EPs accelerated by the ICRF waves. In the frequency range of ICRF waves ($f \approx 20 - 100$ MHz), part of the wave power is usually absorbed by the thermal particles. A physics description of RF heating therefore involves elements related to waves (propagation and damping) and to particles (e.g. wave-induced diffusion and collisional relaxation).

RF heating is described by the Fokker-Planck equation

$$ \frac{df_s}{dt} = \hat{C}(f_s) + \hat{Q}(f_s) + S_s - L_s. \quad (30) $$

Here, $f_s$ is the (quasilinearly time-space averaged) distribution function of ion species s, $\hat{C}$ is the collision operator, $S_s$ is a particle source term, and $L_s$ is a loss term. $\hat{Q}$ is the quasilinear operator representing the effect of waves interacting with species 's'. In quasilinear theory, this term takes the form of a diffusion or friction term and involves a quasilinear diffusion tensor, $D_{ql}$:

$$ \hat{Q}(f_s) \equiv \nabla_v \cdot (D_{ql} \cdot \nabla_v f_s). \quad (31) $$

Depending on the level of sophistication of the description used, $\hat{Q}$ can have a rather complicated form. Important features of the quasilinear operator can nevertheless be illustrated by writing its components in the symbolic form

$$ D_{ql} \equiv A \sum_{n} |d^{(n)}(\mathbf{E})|^2 \delta(\omega - N\omega_{cs} - k_{\parallel}v_{\parallel}). \quad (32) $$

Here, $A$ is a constant, $d^{(n)}(\mathbf{E})$ is a differential operator acting on the electromagnetic wave field, $k_{\parallel}$ is the component of the wave vector $\mathbf{k}$ along the local
magnetic field, $\omega_{cs}$ is the cyclotron frequency of species 's'. The harmonic number $N$ is any integer (including zero). The delta function $\delta(\omega - N\omega_{cs} - k_{\parallel}v_{\parallel})$ picks out the particles resonating with the wave, i.e. satisfying the wave-particle resonance condition equation (14).

A simplified representation of the process of plasma heating with ICRF waves appears in figure 2. The cold plasma dispersion relation in the ICRF frequency range allows two plasma waves, the slow and the fast magnetosonic waves (SW and FW) [82] which are both excited by the ICRF antenna. Whereas the slow wave is evanescent, the fast wave can pass through a thin evanescence region to heat the core plasma through different branches illustrated in figure 2. In large-scale tokamaks such as ITER, the dominant branch is the generation of energetic ions rather than the mode-conversion branch.

However, other processes than those illustrated in figure 2 may play a significant role, too. For example, mode-converted waves (ion Bernstein waves and ion cyclotron waves) can also be absorbed by ions, and part of the fast-wave power can be directly absorbed by the thermal ions. Furthermore, it is possible to drive a small central electron current if part of the fast-wave power is directly absorbed by thermal electrons for an adequate toroidal antenna phasing [83, 84].

<figure><img src="image_3.png" /><figcaption>Figure 2. A simplified overview of fast-ion generation and plasma heating with ICRF waves.</figcaption></figure>

### 3.1.1. Wave propagation and cyclotron damping.
ICRF antennas typically consist of sets of metallic straps enclosed in boxes near the plasma edge. The currents in the straps can excite waves in the nearby plasma. In the ICRF frequency range, it is convenient to view the plasma and the antenna together as a global electrical circuit. In practical terms, the antenna needs to be adapted to the plasma by ensuring that the plasma and antenna impedances match [85, 86].

Indeed, the issue of ICRF power coupling to the plasma is a challenging problem requiring considerable effort.

The fast and slow magnetosonic waves are characterized by their refractive indeces $n = kc/\omega$. The parallel component of the refractive index with respect to the magnetic field, $n_{\parallel}$, is essentially determined by the antenna geometry, whereas the perpendicular component is determined by the dispersion relation. For conditions typical for the plasma edge in tokamaks, the slow wave is characterized by $n_{\perp,SW}^2 \simeq -\omega_{pe}^2/\omega_{ci}^2$, where $\omega_{pe}$ is the electron plasma frequency and $\omega_{ci} \equiv Z_i eB/m_i$ is the cyclotron frequency of the main ion species. Since the slow wave is evanescent ($n_{\perp,SW}^2 < 0$) with a typical decay length of less then 1 mm, it does not propagate to the regions far from the antenna and thus cannot heat the plasma core. However, the slow wave plays an important role in the problem of wave coupling and in the possible development of performance limiting electrical sheaths.

The propagation of the fast magnetosonic wave for a cold plasma is described by the dispersion relation [87]

$$n_{\perp,FW}^2 \approx \frac{(L - n_{\parallel}^2)(R - n_{\parallel}^2)}{S - n_{\parallel}^2} \quad (33)$$

Here, $S$, $L$, and $R$ are the components of the plasma dielectric tensor in the notation of Stix, with $S = 1 - \sum_i \frac{\omega_{pi}^2}{\omega^2 - \omega_{ci}^2}$, $L = 1 - \sum_i \frac{\omega_{pi}^2}{\omega_{ci}(\omega - \omega_{ci})}$, and $R = 1 + \sum_i \frac{\omega_{pi}^2}{\omega_{ci}(\omega + \omega_{ci})}$. The fast-wave dispersion relation (equation 33) suggests quite complex behaviour, especially in plasmas with several ion species.

The presence of the right-hand cutoff, given by the condition $R = n_{\parallel}^2$, implies that the fast wave does not propagate below a certain density, $n_e \lesssim 1 \times 10^{18} \text{ m}^{-3}$, which typically corresponds to a location in the plasma scrape-off layer. Therefore, the fast wave is evanescent near the plasma edge, too, just as the slow wave. However, since its decay length is typically on the order of 10 cm, a significant fraction of the fast-wave power can readily tunnel through the evanescence layer to reach the right-hand cutoff and propagate onward towards the plasma core. Part of the power reflected at the right-hand cutoff couples back to the antenna circuit, which leads to the requirement of adequate antenna-plasma matching.

In a single-ion plasma, the left-hand cutoff, $L = n_{\parallel}^2$, and the fast-wave resonance, $S = n_{\parallel}^2$, are reached at very low plasma densities, below the right-hand cutoff density. At the resonance layer, warm plasma effects (non-zero temperature) need to be taken into account, and the fast-wave resonance is bent into a confluence with a kinetic plasma mode. In plasmas with two or more ion species, both the left-hand cutoff and the fast-wave resonance can be located near the
plasma core. In this case, the region bounded by this cutoff-resonance pair is often referred to as the ion-ion hybrid layer. For example, in plasmas with two ion species and for low $k_\|$, the ion-ion hybrid frequencies are given by

$$ \omega_S \approx \sqrt{\frac{\omega_{p1}^2 \omega_{c2}^2 + \omega_{p2}^2 \omega_{c1}^2}{\omega_{p1}^2 + \omega_{p2}^2}}, \quad (34) $$

$$ \omega_L \approx \frac{\omega_{p1}^2 \omega_{c2}^2 + \omega_{p2}^2 \omega_{c1}^2}{\omega_{p1}^2 \omega_{c2}^2 + \omega_{p2}^2 \omega_{c1}^2}. \quad (35) $$

We always have $\omega_S < \omega_L$, and both ion-ion hybrid frequencies are in between the cyclotron frequencies $\omega_{c1}$ and $\omega_{c2}$.

The dominant damping mechanism of the fast magnetosonic wave is cyclotron damping due to ions fulfilling the wave-particle resonance condition equation 14 [88]. $N=1$ to $N=3$ correspond to fundamental, 2nd harmonic, and 3rd harmonic ICRF heating at $\omega \approx \omega_{ci}$, $2\omega_{ci}$ and $3\omega_{ci}$. Equation (14) is usually fulfilled in the vicinity of the ion cyclotron resonance layers, $R_{IC}$, where $\omega = N\omega_{ci}$. The radial position of these layers in the plasma can be found considering on the $1/R$ radial dependence of the magnetic field in tokamaks by the handy formula,

$$ R_{IC} \approx R_0 \frac{N Z_i 15.25 B_0 \text{ MHz}}{A_i f T} \quad (36) $$

where $R_0$ is the major radius, $B_0$ is the magnetic field on-axis in Tesla, and $f$ is the RF frequency in MHz.

A simplified form of the quasilinear diffusion coefficient (equation 32) is given by

$$ D_{ql} = A \sum_{N} |E_{+}J_{N-1}(k_{\perp}\rho_{Li}) + E_{-}J_{N+1}(k_{\perp}\rho_{Li})|^2 \times \delta(\omega - N\omega_{ci} - k_{\|}v_{\|}), \quad (37) $$

where $J_N$ is the Bessel function of order $N$, $k_\perp$ is the perpendicular wavenumber, and $\rho_{Li} \equiv v_\perp/\omega_{ci}$ is the heated ion Larmor radius. In equation (37), $E_+$ and $E_-$ represent the left-hand and right-hand components of the RF electric field. For thermal and moderately energetic ions with energies of a few hundred keV, the argument of the Bessel functions is small, so we can approximate $J_0 \approx 1$ and $J_N(k_\perp\rho_{Li}) \approx (k_\perp\rho_{Li}/2)^N/N!$. As $J_0$ is the only Bessel function with an appreciable value at low $k_\perp\rho_{Li}$, cyclotron damping at the fundamental frequency ($N=1$) has a strong wave-particle interaction for thermal or fast ions, leading to potentially very effective heating (but not for single-ion plasmas as we will discuss below). For the harmonic ICRF heating ($N \ge 2$), the efficiency of the wave-particle interaction relies on the presence of higher-energy ions with a large Larmor radius, which is usually called the finite Larmor radius (FLR) effect.

Equation (37) also shows that the ion cyclotron damping is to lowest order determined by the left-hand

component of the RF electric field, which rotates in the ion direction. However, the two components of the RF electric field are not independent. In fact, the local RF polarization (the ratio $E_+/E_-$) and its spatial distribution in the plasma volume is to a large extent determined by the plasma composition, rather than by the operational parameters of the RF system. For the fast wave, the RF polarization is given by

$$ \frac{E_+}{E_-} \approx - \frac{R - n_\|^2}{L - n_\|^2}. \quad (38) $$

The combination of equations (33), (37) and (38) determines the characteristics of fast-wave propagation and ion cyclotron interaction in the plasma. The efficiency of the ICRF heating depends crucially on the plasma composition, i.e. the number of ion species with a different charge-to-mass ratio and their concentrations.

### 3.1.2. ICRF heating scenarios for fast-ion generation.
It follows from equation (38) that, in a single-ion plasma, $E_+/E_- \approx (\omega - \omega_{ci})/(\omega + \omega_{ci})$. As the left-hand component of the RF electric field nearly vanishes for $\omega = \omega_{ci}$, the fundamental cyclotron damping in single-ion species plasmas is weak. Since ICRF heating in metallic-wall machines is more dependent on adequate single-pass wave damping, this constraint limits the applicability of the fundamental ICRF heating in single-ion plasmas in both present-day and future tokamaks with metallic plasma-facing components.

To overcome this limitation, several methods have been developed [43, 87, 90–93], summarized in table 3. First, using harmonic resonances ($N=2, 3, ...$) is an option, both for single-ion and multi-ion species plasmas. As outlined above, the harmonic damping is an FLR effect, requiring $k_\perp\rho_{Li}$ to be large enough for efficient heating to occur. For this reason, harmonic ICRF heating is often applied in combination with NBI. Then the NBI system provides a seed of resonant fast ions with energies of about 50-100 keV that are accelerated to higher energies by ICRF heating. As fast ions reach MeV-range energies, the term for the right-hand component of the electric field $E_-J_{N+1}(k_\perp\rho_{Li})$ in equation (37) becomes comparable or even larger than the term for the left-hand component $E_+J_{N-1}(k_\perp\rho_{Li})$. Eventually, a tail of fast ions is generated, with a maximum energy determined by the condition $D_{ql} \approx 0$ [94]. The energy barrier scales approximately as $B^2/n_e$ and usually reaches a few MeV. Being an efficient technique for generating MeV-range ions, the 3rd harmonic ICRF heating of D-beam ions is often used for fast-ion studies in D-D and D-$^3$He plasmas, see e.g. [95]. The form of the EP tail generated by 3rd harmonic ICRF heating simulated by the ASCOT-RFOF and the SPOT-RFOF
<table><caption>**Table 3.** Selected ICRF heating scenarios for EP generation in fusion plasmas. The first three rows are minority ICRF heating schemes, followed by the 2nd and the 3rd harmonic ICRF heating scenarios, and the three-ion ICRF heating schemes.</caption><thead><tr><th>ICRF scenario</th><th>Resonant ion</th><th>Typical EP energies</th><th>Background plasma</th><th>Comments</th></tr></thead><tbody><tr><td>ω = ω<sub>ci</sub>(H)</td><td>H</td><td>A few hundred keV</td><td>D, T, DT mix, incl. 50%-50%</td><td>Efficient in the range n<sub>H</sub>/n<sub>e</sub> ≈ 2–10%</td></tr><tr><td>ω = ω<sub>ci</sub>(<sup>3</sup>He)</td><td><sup>3</sup>He</td><td>A few hundred keV</td><td>D, T, DT mix, incl. 50%-50%</td><td>Efficient in the range n<sub><sup>3</sup>He</sub>/n<sub>e</sub> ≈ 2–5%</td></tr><tr><td>ω = ω<sub>ci</sub>(D)</td><td>D</td><td>A few hundred keV</td><td>T-rich plasma</td><td>Efficient in the range n<sub>D</sub>/n<sub>e</sub> ≈ 5–15%</td></tr><tr><td>ω = 2ω<sub>ci</sub>(H)</td><td>H</td><td>A few MeV</td><td>H, H-<sup>4</sup>He mix, H-D mix</td><td>Often applied at low magnetic field</td></tr><tr><td>ω = 2ω<sub>ci</sub>(D)</td><td>D</td><td>A few MeV</td><td>D, DT mix, incl. 50%-50%</td><td>Requires low n<sub>H</sub>, typically n<sub>H</sub>/n<sub>e</sub> ≲ 2%, efficient in combination with NBI</td></tr><tr><td>ω = 2ω<sub>ci</sub>(T)</td><td>T</td><td>A few MeV</td><td>T, DT mix, incl. 50%-50%</td><td>Efficient at T<sub>i</sub> ≳ 10 keV; main scenario for the flat-top plasma in ITER</td></tr><tr><td>ω = 3ω<sub>ci</sub>(D)</td><td>D</td><td>A few MeV</td><td>D, D-<sup>3</sup>He mix, DT mix, incl. 50%-50%</td><td>Efficient in combination with NBI</td></tr><tr><td><sup>4</sup>He-(<sup>3</sup>He)-H, D-(<sup>3</sup>He)-H</td><td><sup>3</sup>He</td><td>A few MeV</td><td>Mix H+~5-15% <sup>4</sup>He, mix H+~10-30% D</td><td>Three-ion scheme to generate MeV-range EPs in hydrogen majority plasmas</td></tr><tr><td>D-(D<sub>NBI</sub>)-<sup>3</sup>He</td><td>D</td><td>≥ 500 keV</td><td>D-<sup>3</sup>He mix</td><td>Three-ion scheme to generate EPs (D and alphas) in D-<sup>3</sup>He plasma</td></tr></tbody></table>

<figure><img src="image_4.png" /><figcaption>**Figure 3.** Measurement (left) and simulation (right) of a fast-ion velocity distribution function [a.u.] in the center of a plasma at JET heated by NBI and 3rd harmonic ICRF heating. The inversion was done for energies larger than 120 keV. Reproduced from [89].</figcaption></figure>

codes was corroborated experimentally by velocity-space tomography (see section 4) based on NES and GRS measurements at JET as illustrated in figure 3 [89]. For this 3rd harmonic ICRF heating scheme, the coupling between the electromagnetic waves and the fast ions becomes weak at an energy of about 2 MeV which is corroborated by the measurement. Also the widths of the measured and simulated functions in $v_{\perp}$ direction agree well. A 3rd harmonic ICRF heating scheme was also used to generate an alpha tail by accelerating helium injected by NBI [96], which demonstrated the detection of alphas by gamma-ray spectroscopy (see section 4).

Nevertheless, the harmonic cyclotron interaction

does not necessarily need a seed population of beam ions and can be an effective technique for plasma heating and fast-ion generation with ICRF heating alone [97]. For example, the 2nd harmonic heating of hydrogen ions, $\omega = 2\omega_{ci}(H)$ is routinely applied for heating JET hydrogen plasmas at $B_0 \approx 1.7$ T. As discussed in [87], for harmonic absorption, the damping rate scales with the ion beta as $\beta_i^{(N-1)}$, and therefore the wave absorptivity increases at lower magnetic fields. In general, in modern plasmas containing species with temperatures sufficiently large prior to the ICRF power application, harmonic heating has proven quite efficient, including during DT operation in TFTR [98,99] and JET [100,101]. The 2nd harmonic
heating of tritium ions, $\omega = 2\omega_{ci}(T)$ is currently considered as the main scenario for ICRF heating of high-temperature DT plasmas in ITER [102].

A second possibility, referred to as minority heating [103], is to resort to fuel mixes containing only a small fraction of the targeted ion species. In this case, the RF frequency is tuned to the cyclotron frequency of the minority ions, $\omega = \omega_{ci}^{\text{min}}$, which usually have a concentration of a few percent ($n_{\text{min}}/n_e \approx 5\%$) [104]. In this case, the proximity of the left-hand cutoff and of the hybrid resonance results in the creation of a particular wave structure called the ion-ion resonance-cutoff pair. The practical consequence is that the wave polarization near the minority ion cyclotron layer is now dictated by the proximity of this structure, so that $E_+$ can remain significant close to the cyclotron resonance of minority ions. The minority heating scheme is efficient and robust with respect to changes in the features of the heated ion distribution function, since the corresponding leading term in equation (37) is independent of $k_{\perp}\rho_{Li}$. The minority heating scheme has, therefore, been the main ICRF heating scenario in many fusion devices in operation until now. Minority heating of thermal and NBI-generated deuterons has been applied in recent JET DT experiments in T-rich plasmas, aiming to maximize beam-target fusion, and high performance DT hybrid plasmas [39, 43].

<figure><img src="image_5.png" /><figcaption>**Figure 4.** Energy spectra of neutral tritium fluxes measured by a neutral particle analyzer in JET high performance DT hybrid plasmas with different ICRF heating schemes. The fluxes have been averaged over a one-second period (8-9 s) in each discharge. Reproduced from [43].</figcaption></figure>

Figure 4 shows neutral particle analyzer measurements of tritium resolved in energy for high-performance DT hybrid discharges with 2^nd^ harmonic tritium ICRF heating and hydrogen and ^3He^ minority ICRF heating [43]. The neutral particle fluxes are largest for 2^nd^ harmonic tritium ICRF heating without ^3He^ gas injection. When ^3He^ is injected, the neutral particle fluxes of tritons decrease since less power is absorbed by triton ions when ^3He^ is present, leading to a less energetic triton population and lower neutral

particle fluxes.

Finally, the recently proposed three-ion scheme has been successfully applied for plasma heating in several devices [106]. Whereas standard minority heating consists of introducing an ion species in a quantity suitable for the creation of the hybrid resonance-cutoff pair close to its own cyclotron layer, the underlying principle of the three-ion scheme is to have two main ion species, characterized by mass and charge numbers $A_1, Z_1$ and $A_2, Z_2$, with a density ratio chosen to adequately locate the ion-ion resonance, typically near the magnetic axis, and introduce a third ion species ($A_3, Z_3$) in a small fraction with its cyclotron layer located close to the ion-ion resonance of the main ion species. This can be achieved by ensuring that the third species satisfies $Z_1/A_1 < Z_3/A_3 < Z_2/A_2$, which introduces a new ion-ion resonance layer in the plasma which benefits from the large $|E_+|$ component obtained at the ion cyclotron resonance of the minority ion. By separating the creation of a wave structure adequate for minority ion heating from the ion acceleration process at the cyclotron layer, the three-ion scheme achieves very good per-pass damping rate and effectively relaxes the stringent constraints on the minority ion concentration in the classical scheme. Alternatively, it is possible to choose a minority ion identical to one of the ion species already present in the plasma, but with a distribution function such that a substantial number of particles have a parallel velocity component $v_\parallel$ such that they are able to satisfy the resonance condition (equation 14) owing to their Doppler shift. In practice, this is achieved by targeting the NBI ions injected in the plasma, which has been demonstrated to yield efficient heating [38, 107]. The ICRF tail distribution function has been measured by velocity-space tomography at JET [108]. Figure 5 illustrates that the power can be deposited in a small region close to the plasma core, as experimentally confirmed in the measured 2D neutron emission profile.

The ICRF schemes designed for generating energetic ions in plasma are subject to several limitations. These include constraints related to plasma composition and the range of minority concentrations where ICRF wave damping remains effective, as summarized in Table 3. Technical factors also play a crucial role, such as the cost of operational gases like ^3He^ and the challenges in controlling minority concentrations, particularly in the plasma core, where natural uncertainties are quite significant [109]. Furthermore, the efficient coupling of ICRF power to the plasma can be challenging and may depend on the specific ICRF scheme and target plasma conditions. This challenge might be, however, mitigated by employing localized gas puffing near the
<figure><img src="image_6.png" /><figcaption>**Figure 5.** Illustration of the strong core localization of the RF power deposition and generation of energetic deuterium ions in JET pulse #91256 heated with the three-ion scheme. (a) Spatial distribution of $|E_+|^2$ (computed by the 2D full-wave code TORIC) exhibits the strong enhancement of $|E_+|^2$ at the ion-ion hybrid layer in the plasma core. (b) Tomographic reconstruction of neutron emission the neutron camera measurements. Reproduced from [105].</figcaption></figure>

antennas [110, 111].

### 3.1.3. ICRF heating and collisional relaxation.
Collisional relaxation during ICRF heating is more complex than for fusion alphas and NBI ions (section 2), since RF power continuously accelerates the targeted ions, counteracting the slowing-down process. Nevertheless, the RF power absorbed by superthermal ions is eventually transferred to the bulk plasma by collisional energy transfer. Accurately describing the process of ICRF heating requires solving the wave equation and the Fokker-Planck equation (30) self-consistently (see, e.g. Ref. [112]), now with a power source. The Fokker-Planck equation can be solved efficiently using, e.g., Monte Carlo methods, finite differences or finite elements.

Nevertheless, much insight into the relaxation process can be gained by examining the relaxation of a single test ion on the background plasma species outlined in section 2. The critical energy $\mathcal{E}_{crit}$ suggest some flexibility of ICRF heating schemes to selectively heat predominantly electrons or ions. For example, in minority heating, targeting light ions at low concentrations (typically hydrogen) results in ions with energies $\mathcal{E} \gg \mathcal{E}_{crit}$ and hence in dominant bulk electron heating. Targeting heavier ions (such as $^3$He) or larger concentrations of the minority species, or both, results in less energetic ions and hence promotes bulk ion heating. Finally, the three-ion scheme allows heavy intrinsic impurities to be targeted, and exhibits only weak dependence on the their concentration [113].

## 3.2. Application of ICRF heating for EP physics studies

### 3.2.1. Sawtooth dynamics.
Controlling MHD events by external means has been an early priority in fusion research. Energetic ions have a stabilizing effect on various MHD instabilities, e.g. sawteeth [114–116]. By applying NBI or RF power, the sawtooth period can be increased, which appears to be an appealing idea at first sight, since a very hot core is formed. However, this leads to massive sawtooth crashes nicknamed “monster sawteeth” or “giant sawteeth”, which can have deleterious outcomes. Monster sawtooth crashes can destabilize neoclassical tearing modes (NTMs), which are much more detrimental to the global confinement than the sawteeth themselves [117]. We will describe the physics of sawteeth and NTMs in detail in section 5. In addition to the possible modification of the sawtooth period, it has been recognized that sawteeth may sometimes be desirable as they eject heavy impurities from the plasma center to the periphery. Therefore, flexible tools capable of modulating the sawtooth activity are desirable. Such tools include ICRF and NBI heating to tailor the fast-ion population [118].

Initially, sawtooth destabilization has been attributed to shear variations in the vicinity of the $q=1$ surface. This has triggered the development of advanced schemes consisting of driving localized current by means of EC waves [119, 120] or IC waves, using the rather subtle ion cyclotron current drive (ICCD) effect [121]. ICCD requires asymmetric antenna phasing, which are more prone to operational difficulties
than the more classical dipole phasing. More recently, however, it has been established that energetic ions themselves could have either a stabilizing or a destabilizing effect, depending on their distribution function [122,123]. This has relaxed some of the constraints on the asymmetric antenna phasing and allowed more classical ICRF heating conditions to be used to successfully control sawteeth [124,125].

### 3.2.2. AE destabilization.
Fusion-born alphas in ITER and future fusion reactors are expected to destabilize a range of AEs. These instabilities could play a crucial role in a nonlinear impact of alphas on plasma heating and confinement in ITER. AE physics is described in sections 6 to 10. ICRF heating has been used to accelerate ions to velocities large enough to destabilize AEs and to study the impact of AEs on the plasma at different tokamaks. In particular, hydrogen minority heating was applied to destabilize TAEs and study mechanisms of their control on several devices [126-129]. 2^nd^ and 3^rd^ harmonic ICRF heating schemes were also efficient in destabilizing AEs in tokamak plasmas [130-132]. In JET experiments with 2^nd^ harmonic deuterium ICRF heating, TAEs were shown to transport energetic deuterons with specific energies away from the plasma core, thereby resulting in the formation of a local bump-on-tail distribution of fast ions in the plasma [133,134].

TAEs and EAEs were regularly observed in fast-ion experiments with the three-ion ICRF heating scenarios on Alcator C-Mod, ASDEX Upgrade (AUG) and JET [113, 135-137]. Three-ion scheme ICRF heating scenarios efficiently generate large populations of passing fast ions at JET and can sustain plasmas with inverted q-profiles with $q_{min} < 1$ [38, 138]. Consequently, this scenario provides necessary conditions for the destabilization of RSAEs, including high-frequency RSAEs [138]. Surprisingly, in most cases, the observed complex AE activity in fast-ion experiments with the three-ion ICRF heating scheme was not detrimental for plasma confinement. The improved thermal plasma confinement in the presence of a large population of MeV-range fast ions and AEs was observed at JET [139].

### 3.3. ICRF heating scenarios for EP and AE studies before DT operation in ITER

In the original ITER baseline, the plasma-facing components were planned to be beryllium for the first wall and tungsten for the divertor. Currently, the plan is to switch the material of the first wall to tungsten, too. As a result, a new ITER baseline plan is under development, accompanied by revisiting the ITER heating mix [140]. The ICRF system in ITER will operate in the frequency range 40-55 MHz [141].

A phased approach for increasing the ICRF power from 10 MW up to 20 MW during the later phases of ITER operations has been proposed.

A large variety of efficient ICRF heating scenarios can be applied at ITER, depending on the background plasma mix, the targeted species, and the magnetic field [142, 143]. The most promising ICRF heating scenarios for testing plasma heating and initial fast-ion studies in the plasma scenarios before DT operation in ITER include minority heating of hydrogen ions in deuterium plasma ($B_0 = 2.65$ T, $f \approx 40$ MHz), minority heating of $^3$He ions in hydrogen plasma ($n_{3He}/n_e \approx 2-3\%$, $B_0 = 5.3$ T, $f \approx 53$ MHz) [144] and the three-ion $^4$He-($^3$He)-H scenario with a small amount of $^3$He resonant ions in H-$^4$He plasmas ($n_{3He}/n_e < 0.5\%$, $B_0 = 5.3$ T, $f \approx 53$ MHz. (or 10-30% of deuterium ions)). This three-ion $^4$He-($^3$He)-H scenario was validated at JET and demonstrated robustness for both plasma heating and AE destabilization across a range of $^4$He concentrations $n_{4He}/n_e \approx 5-15\%$ (which could be replaced by 10-30% of deuterium ions).

Because of a strongly increased RF power absorbed per resonant ion, which maximizes the generation of MeV-range fast ions, the three-ion $^4$He-($^3$He)-H ICRF heating scheme is particularly suited for the destabilization of AEs in the plasma scenarios before the DT operation phase and initial studies of the impact of AEs on the dynamics and confinement in ITER plasmas. The high efficiency of this ICRF heating scheme for AE studies in non-active H-$^4$He plasmas was experimentally confirmed at JET [145]. In line with modeling results [146], AEs were destabilized in H-$^4$He plasmas for a wide range of $^4$He concentrations, varying from ~5% to ~15%. The efficiency of fast-ion generation in these JET experiments was enhanced by utilizing the asymmetric ICRF antenna phasing, predominantly launching waves in the co-current direction.

### 3.4. Bulk-ion ICRF heating in DT plasmas of ITER and future fusion reactors

High ion temperatures $T_i \approx 15-20$ keV are essential for economical energy production in magnetic confinement fusion devices. Under these conditions, fusion-born alphas provide the dominant source of plasma heating and maintain the high rate of fusion reactions. However, reaching such ion temperatures during the ramp-up phase and ultimately the high-Q operational point cannot be done without auxiliary heating systems.

In future fusion devices, ICRF heating can provide a significant fraction of bulk-ion heating. In ITER, radiofrequency heating of a few percent of $^3$He ions (~3-5%) is currently considered as the main option for increasing $T_i$ with ICRF heating during the ramp-
up [88, 147]. This technique was experimentally
demonstrated in DT experiments on TFTR and JET
[98, 100, 101], including recent JET DT experiments
[43, 102]. This scenario can provide ~50-60% of bulk-
ion heating in approximately 50/50 DT plasmas in
ITER [84].

The efficiency of bulk-ion ICRF heating can be
increased further by channeling RF power to a small
amount of selected impurities with $1/3 < (Z/A)_{\text{imp}} <
1/2$ [148]. As a result of their higher atomic mass,
these impurity ions provide an even larger fraction
of bulk-ion heating, as compared to hydrogen and
$^3$He minority ions. The efficiency of the three-ion
ICRF heating scenario for heating DT plasmas was
demonstrated in recent experiments at JET, with
intrinsic $^9$Be impurities as resonant absorbers [145].
For ITER, a similar technique can be applied using
neon, argon, or boron impurities. More recently, an
extension of this technique with $^7$Li impurities was
proposed for bulk-ion heating in CFETR [149].

4. Diagnostics of EPs and instabilities

In this section, we will discuss the diagnostic of
confined and lost energetic ions, runaway electrons,
and instabilities. It is essential to diagnose EPs
and instabilities in ITER and other burning plasma
machines in order to optimize plasma scenarios and
fusion performance and possibly for plasma control.
EP and instability diagnostic in ITER will allow us to
test theory against experimental data, strengthening
the predictive power of theory for devices beyond ITER
and the first generation of burning plasma experiments.
Measurements of alphas and alpha-driven instabilities
are of particular interest, since alphas are the key
ingredient of burning, predominantly self-heated and
self-organized plasmas. We will focus on diagnostics
that can survive the neutron and radiation fluxes
emitted by burning plasmas. The design of the EP
diagnostics in ITER is described in detail in chapter
8 of this volume [19]. Here we will focus on the EP
physics and especially the alpha physics we can learn
due to recent progress in the detector technology and
integrated data analysis procedures.

4.1. Diagnostics of confined EPs

The great majority of alphas and EPs from auxiliary
heating will be confined in the fusion plasma. The
distribution functions of the EPs in the plasma
are complex functions in a 6D phase space, which
can be reduced to 3D assuming symmetries and
near symmetries (see section 2). However, toroidal
symmetry of the tokamak is not always a good
assumption (see section 8). There is seldom enough
measurement data to determine even the reduced 3D

phase-space distribution functions fully, even though a
promising first demonstration has been accomplished
[150]. At ITER, integrated data analysis of all available
experimental data is possible either by synthetic
diagnostics or by tomographic inversion [3].

For measurements of confined alphas, ITER will
be equipped with γ-ray spectroscopy (GRS) [151–153]
and with collective Thomson scattering (CTS) [154–
158]. Fast-ion charge-exchange recombination spec-
troscopy (CXRS) might detect low-energy alphas in
the few 100 keV range. The high-energy neutral parti-
cle analyzer (NPA) [159] is quite insensitive to alphas
and cannot distinguish them from deuterium. Note
that a neutralization by charge-exchange reactions is
far more likely for deuterium (Z = 1) than for an alpha
(Z = 2). Neutron emission spectroscopy (NES) [160]
is not directly sensitive to alphas, but it is sensitive
to velocities of the fuel-ion populations. Alphas might
produce a knock-on tail in the fuel-ion population, that
has been detected by NES in the 1997 DT campaign
at JET [161]. However, this specific measurement, be-
sides being indirect, is possible only by using a mag-
netic proton recoil neutron spectrometer. Ion cyclotron
emission (ICE) is hoped to give further diagnostic in-
formation.

For measurements of energetic hydrogen, deu-
terium and tritium, ITER will be equipped with a
neutral particle analyzer (NPA) [159,162] and an ar-
ray of neutron diagnostics [160]. CTS and GRS are
also sensitive to energetic deuterium and tritium. For
measurement of lost alphas and other EPs, ITER will
be equipped with a fast-ion loss detector (FILD). ICE
might be another option to measure lost alphas and
other EPs. While these diagnostics are described in de-
tail in various sections of chapter 8 of this volume [19],
we will summarize EP diagnostics briefly here in the
following.

4.1.1. Gamma-ray spectroscopy. GRS detects γ-ray emission from nuclear reactions in the plasma, either between an EP and an impurity or, in some cases, by the fusion reactions themselves [163, 164]. The impurity (especially carbon, beryllium or boron) is generally naturally present due to the composition of the machine first wall, but it can also be injected on purpose to enhance the signal. The instruments are placed in shielded areas at the end of collimated sightlines. Simultaneous measurements along distinct sightlines make it possible to infer spatial information on the emission and hence the EPs by tomographic inversion. Two GRS instruments are planned for ITER, both with a radial sightline: one is a spectrometer integrated in the ITER NPA [162]; the other is a set of GRS diagnostics integrated in the ITER radial neutron camera [153].
First GRS measurements have been made in the early '90s, and routine GRS measurements have been established at JET since then. A major advancement was the deployment of high-resolution detectors with MHz counting rate capabilities in 2010. This allowed expanding the range of experimental conditions where measurements are possible, for example towards high-performance deuterium [105, 165, 166] and DT plasmas [44]. It also allowed measuring the spectral shapes of individual peaks [167, 168]. The spectral shape is determined by the Doppler shift caused by the EPs, which hence gives experimental access to the EP velocities, either via velocity-space tomography [89], or by comparison between data and a synthetic signal starting from a model of the EP distribution function [169, 170]. Different detector options, notably lanthanum bromide (LaBr) [171] or high-purity germanium (HpGe) [167] are used, depending on whether the scope is to make measurements in scenarios with a significant residual neutron background or to enable measurements of the spectral shapes of the individual lines. When they are used in combination with silicon photo-multipliers for light readout [172], the detectors can be made very compact, allowing their installation on multiple sightlines to obtain information on the spatial profile of the emission via tomographic inversion. An example of a high-resolution GRS measurement of the 2868 keV peak of the ⁹Be(D,nγ)¹⁰B reaction from JET appears in figure 6.

Measurements of confined alphas by GRS in the recent JET DT experiments were demonstrated in [44,47]. Figure 7 shows a gamma-ray spectrum from a JET DT plasma [44]. The gamma-ray spectrum shows three main features. First and foremost, the dominant 4.44 MeV peak from the ⁹Be(α,nγ)¹²C* reaction is clearly detected, along with its single escape peak at 3.93 MeV (an instrumental feature). Second, two high-energy, low-intensity peaks at $E_γ = 8.53$ MeV and $E_γ = 9.0$ MeV are detected, originating from the capture of thermalized neutrons on the impurity nickel. Third, a broad feature at low intensity extending up to about $E_γ = 20$ MeV is detected which can give information about the fusion power in this discharge. The inset shows a synthetic GRS spectrum computed for ITER for a beryllium wall [153]. While GRS measurements on ITER will need a different reaction following the new 2024 ITER baseline, the overall similarity between JET DT measurements and the simulation nevertheless demonstrates that expected peaks in GRS spectra can be detected in the recent DT experiments at JET.

#### 4.1.2. Collective Thomson scattering.

ITER will be equipped with a CTS diagnostic, which detects alphas and other EPs spatially resolved in seven measurement

<figure><img src="image_7.png" /><figcaption>**Figure 6.** High-resolution GRS measurement at JET. The spectrum resolves the 2868 keV peak of the ⁹Be(D,nγ)¹⁰B reaction. Reproduced from [169].</figcaption></figure>

<figure><img src="image_8.png" /><figcaption>**Figure 7.** Gamma-ray spectrum measured by the vertical gamma-ray detector at JET in a typical DT discharge. The inset shows a calculation of the gamma-ray spectrum expected from a ITER DT plasma (from [153]). Reproduced from [44].</figcaption></figure>

volumes where a probe beam from a 60 GHz gyrotron overlaps receiver beams [154, 156]. A synthetic spectrum for ITER and the corresponding inferred 1D projected velocity alpha- and NBI distributions appear in figure 8. The radiation is scattered due to EPs in the measurement volume. Alphas and other EPs cause large Doppler shifts in this scattered radiation which, as for GRS, gives experimental access to the velocities of the EPs [173,174]. The spectral resolution of CTS measurements has substantially increased in the last decades. The first-generation CTS receivers split the signal into tens of channels via bandpass filters [175–178]. Modern CTS receivers digitize the signal at high rate [179–182], and the spectrum is found by Fourier transformation, leading to a substantially higher spectral resolution and hence better diagnosis of the EP velocities. Unlike GRS, CTS cannot distinguish alphas and other EPs since the Doppler shift of any ion moving at the same velocity is the same [157, 158].
<figure><img src="image_9.png" /><figcaption>**Figure 8.** (a) Synthetic spectrum (black) and a corresponding randomized, resampled spectrum accounting for noise (red). The cyan curve shows the best-fit spectrum and associated uncertainties. (b) 1D projected distribution functions for alphas and fast deuterium NBI ions leading to the synthetic spectrum and the average fitted distributions with uncertainties. The bulk ion feature leads to large spectral power densities at small Doppler shifts, masking the fast-ion feature. Reproduced from [156].</figcaption></figure>

<figure><img src="image_10.png" /><figcaption>**Figure 9.** Spectrum of 14.1 MeV neutrons measured by the vertical line-of-sight diamond detector in the recent DT campaign in JET, compared with synthetic spectra for different D/T plasma compositions. The neutron energies are 5.702 MeV larger than deposited energies measured by the diamond detector, which are shown on the abscissa. Reproduced from [39].</figcaption></figure>

### 4.1.3. Neutron emission spectroscopy and cameras.
The array of neutron diagnostics consists of neutron flux monitors, a neutron activation system, a neutron calibration, vertical and radial neutron cameras and neutron emission spectrometry (NES) [160]. The neutron diagnostics measure neutrons moving towards the detector along collimated sightlines, similar to

GRS. The neutron cameras yield 2D profiles of the neutron emission in the poloidal plane by tomography. The NES diagnostics are based on three detection principles: a thin proton-recoil spectrometer, two time-of-flight spectrometers, one using forward scattered neutrons and the other backward scattered neutrons, and a single-crystal diamond detector. The thin proton-recoil and time-of-flight spectrometers are optimized for fuel-ion-ratio measurements but might also give information on fast ions. An advantage of diamond detectors is that they are much more compact than time-of-flight detectors. The NES diagnostics can provide information about the velocities of the fuels, i.e. energetic deuterium and energetic tritium [170, 183, 184]. Additionally, since equally many neutrons and alphas are generated in the DT reaction, the alpha birth rate can be inferred, too, and it may be possible to infer the alpha birth velocity distribution, which is expected to be slightly anisotropic [3]. High-resolution NES measurements of DT neutrons with diamond detectors have recently been demonstrated at JET as shown in figure 9 [39, 44].

### 4.1.4. Neutral particle analyzers.
Energetic hydrogen, deuterium and tritium can be detected by a neutral particle analyzer (NPA) [159]. The NPA will measure energies of the neutral particles leaving the plasma, which have been generated from the corresponding ions in the plasma in charge-exchange reactions. NPAs hence provide measurements of the confined hydrogen, deuterium and tritium in the plasma. The detected neutrals can be born along the sightline but signals from the plasma edge typically dominate in passive NPAs since a large population of neutral particles must be in the sightline in order to have many charge-exchange reactions. The NPA measures a narrow pitch range of the detected species, since the charge-exchange reactions leaves the momentum of the detected species almost unchanged and the neutral must move along the sightline towards the detector. The energy, on the other hand, is well resolved over a broad energy range, so that energy spectra for a given pitch are obtained. NPAs are quite insensitive to alphas, since the neutralization of an alpha requires that two electrons are transferred in charge-exchange reactions to generate a neutral helium atom, which is more unlikely than for deuterium, which only requires the transfer of one electron. NPA measurements of tritons during high-performance JET DT discharges for various ICRF heating schemes are shown in figure 4 in section 3 [43].

### 4.1.5. Ion cyclotron emission.
Due to the crucial role played by alphas in sustaining the thermonuclear burn in burning plasmas, the alphas must be studied
<figure><img src="image_11.png" /><figcaption>**Figure 10.** (a) An H-mode shot at DIII-D showing both core and edge-localized ICE, where horizontal lines indicate $f_{ci}$ evaluated at the magnetic axis in L-mode (left, yellow) and the LCFS in H-mode (right, white). In L-mode, core ICE harmonics 1–4 $f_{ci}$ are excited. In H-mode, edge ICE harmonics 1–9 $f_{ci}$ are excited. (b) NBI power (red) and electron density (black). Reproduced from [185].</figcaption></figure>

using any methods compatible with the challenging environment of reactor-grade plasmas. ICE is likely to provide one such method and could in the best case provide crucial information on the EP phase-space distribution function. However, it remains a challenge to interpret ICE signals quantitatively, since it requires understanding of the drive, damping and saturation levels of the instability. We will review the possibility to use ICE as diagnostic when we discuss instabilities in section 6. ICE radiation is frequently observed in current tokamaks [185–198] and stellarators [199–201] and can usually be associated clearly with EP-driven instabilities, even though ICE has sometimes been observed in Ohmic discharges. ICE signals can be related to either the core or the edge plasma by the frequency separation of peaks in measured spectra as illustrated in figure 10. Edge ICE may be related to lost ions. The ICE linear theory outlined in section 6 indicates that the EP distribution function is related to the ICE signal, but it is not yet clear how that information can be extracted from the ICE signals. Further efforts in theory and in experiments on various conventional tokamaks, spherical tokamaks and stellarators are still needed. At ITER, ICE could be measured with the ICRF antennas foreseen if the ICRF transmission line was equipped with a spectrometer. The fast-wave reflectometer could also be used. Lastly, ICE is also worth detailed studies because it may offer the possibility of phase-space engineering of the EP distribution function.

4.1.6. *Phase-space sensitivity of EP diagnostics.*
The development of diagnostics with high spectral resolution allowed better experimental access to the EP

<figure><img src="image_12.png" /><figcaption>**Figure 11.** Total alpha velocity-space sensitivity of all CTS and GRS measurements at ITER combined, i.e. the weight function is computed for the entire spectral range of all CTS and GRS measurements. The color scale shows the base ten logarithm such that the sensitivity is highest in black regions and lowest in white regions. Reproduced from [3].</figcaption></figure>

velocity space or even phase-space which has triggered new methods exploiting this frequency resolution in recent years. The velocity-space sensitivity of several EP measurements has now been understood and can be visualised using so-called weight functions [202]. Velocity-space weight functions show the signal generated per EP in velocity space for each small bin in a measured spectrum and thus reveal the velocity-space sensitivities of a given measurement. Such weight functions have been computed for the several fast-ion diagnostics, and their velocity-space space sensitivity has been understood: FIDA [202, 203], NPAs [202], CTS [174], NES [204, 205], GRS [206, 207] and FILD [208, 209], 3 MeV proton diagnostics [210], ICE spectroscopy [211, 212], and imaging neutral particle analyzers (INPAs) [213]. Recently, the phase-space sensitivity of FIDA, CTS, NES and GRS diagnostics to the possible orbits in a tokamak have been numerically computed, too, [49, 62, 150, 214–216], as well as using a semi-analytically tractable model in COM space [217].

An overall summary of the velocity-space sensitivity of alpha measurements at ITER appears in figure 11 [3]. Typical weight functions for the individual measurements can be found in [3]. Alphas with energies higher than about 1.7 MeV are diagnosed most accurately in ITER due to detection by CTS and GRS. This study was done assuming a reaction between beryllium and alphas, but other GRS reactions will typically lead to a similar overall pattern. For reactions between alphas and beryllium, the GRS diagnostic has the peak sensitivity near the nuclear resonances at 2 MeV and 4 MeV [218, 219]. Alphas with energies between 300 keV and 1.7 MeV are diagnosed by CTS only. The yellow region below 300 keV and extending up to 1.7 MeV for pitches $p \sim \pm 1$ is not observable by either CTS or GRS installed at ITER. This is a consequence of the nearly perpendicular viewing geometry
of CTS. A second consequence of the perpendicular viewing geometry is that the sign of the pitch cannot be measured, i.e. co- and counter-going particles cannot be told apart.

4.1.7. *Measurements of velocity-space and phase-space distribution functions.* The new high-resolution measurements and the understanding of their velocity-space sensitivity have allowed the formulation of tomography problems in velocity space [220, 221] and phase space [150], which can be solved by standard methods of tomography [222]. It has been shown that velocity-space and phase-space tomography allow integrated data analysis of the different diagnostics [223], e.g. FIDA and CTS [224], GRS and NES [89] or FIDA and NES [225]. The formalism is described in detail in section 10 in Chapter 8 of this volume [19].

<figure><img src="image_13.png" /><figcaption>**Figure 12.** Measurement (left) and TRANSP/NUBEAM simulation (right) of an EP velocity distribution function [a.u.] in the center of a plasma at AUG heated by NBI. The measurement is a tomographic inversion of five simultaneously acquired FIDA spectra. Reproduced from [222].</figcaption></figure>

An example of a measured 2D fast-ion velocity distribution function by velocity-space tomography and a corresponding numerical simulation appears in figure 12 [222]. This inversion is based on five simultaneously acquired FIDA spectra with lines-of-sight crossing the NBI path in the plasma center. The overall shape of the NBI velocity distribution function is captured well by the tomographic inversion. The NBI distribution is biased towards positive pitch due to the co-current NBI injection. The NBI injection energy is 60 keV, and the half-energy is 30 keV leading to local bumps-on-tail in the distribution which are found in the TRANSP/NUBEAM simulation and in the tomographic inversion of FIDA measurements. An inversion of GRS and NES measurements at JET was illustrated in figure 3 in section 3.

Until now velocity-space tomography has been applied at AUG [221, 222, 224, 226-229], JET [54, 89, 108, 230], MAST [231], DIII-D [232], EAST [225, 233, 234] and TCV [235]. Two to five simultaneous detectors have been used at these tokamaks, measuring velocity distribution functions in plasma with NBI or

ICRF heating. Based on the orbit sensitivities, first inversions of a 3D phase-space distribution describing the possible orbits have been computed [150, 236] (see also section 5). Recently, possibilities for phase-space tomography on stellarators have been explored [237-239].

At ITER, velocity-space tomography of the alpha distribution function based on GRS and CTS has been shown to be feasible for energies from about 1.7 MeV upwards [3]. However, since all currently foreseen diagnostics observe in a perpendicular direction with respect to the magnetic field, the sign of the pitch *p* of the alphas cannot be determined. But the absolute value |*p*| can be determined, so that the velocity distribution function *f*(*E*, |*p*|) can be measured. If an oblique γ-ray detector were to be installed, the sign of the pitch could be found, too [3]. Since below 1.7 MeV CTS is the only diagnostic for confined alpha-particles, a 2D inference seems difficult if not impossible, unless cogent prior information can be supplied. Energy spectra can be determined by 1D inversion of CTS spectra, for example by assuming isotropy. Reviews of velocity-space tomography are available in references [35, 240] as well as in chapter 8 of this volume [19].

Looking into the future, we need to deal with the heavily underdetermined nature of the velocity-space and phase-space tomography problems based on sparse measurement data. Promising approaches incorporate physics-based prior information, such as nonnegativity [222], the unlikely phase-space based on null-measurements [222, 231], the geometry of heating NBIs [222], numerical simulations [222], near-isotropy and anisotropy [3, 241, 242], monotonicity [232], the physics of collisions and slowing down of EPs [225, 238] or the physics of wave-particle interactions [243]. Recently, neural networks have been used to infer phase-space distribution functions from FILD and INPA measurements as well as from synthetic ICE measurements [211, 244]. This new approach to tomographic phase-space inversion problem should certainly be developed further, which might allow very rapid tomographic inversion.

## 4.2. Diagnostic of lost EPs

Despite the good confinement in ITER, some alphas and other EP are inevitably lost from the plasma and hit the first wall. EP losses are to be minimized in fusion devices since EPs can generate hot spots leading to localized melting of plasma-facing components. To understand the transport and loss of alphas and other EPs, both the confined and the lost EPs must be diagnosed.

Diagnosis of lost ions might be possible by ICE as described in the previous subsection. Additionally, ITER is planned to be equipped with one FILD
detector [245]. In FILDs, the EPs enter the FILD through an entrance slit, are dispersed in the magnetic field, and hit a scintillator plate emitting light due to the impact. The time resolution of FILD measurements has greatly improved in recent years. The spectral content of the light can reveal what mode has ejected the ions (figure 13) [246, 247].

<figure><img src="image_14.png" /><figcaption>**Figure 13.** FILD spectrogram measured at AUG showing the spectral content of the fluctuation that has ejected the ions. Reproduced from [247].</figcaption></figure>

<figure><img src="image_15.png" /><figcaption>**Figure 14.** FILD measurement from AUG showing the formation of a high-energy tail above the NBI injection energy. Tomographic inversion shows that the most likely velocity distribution at the entrance slit of the FILD detector is localized in velocity space. Reproduced from [248].</figcaption></figure>

The light can also be imaged to study the velocity-space distribution (figure 14) [246]. Such a measurement appears in figure 14 which shows the formation of a tail above the injection energy of the ions during ELMs. The fast-ion distribution in the entrance slit can be found from the image on the scintillator plate by tomographic inversion [208, 242, 249]. Here the most likely solution is that an ion population entering the slit with a distinct Larmor radius leads to the formation of a tail due to instrumental broadening. This points to a wave-particle resonant interaction between the ELM electromagnetic perturbation and a certain edge fast-ion population leading to an acceleration of this population [250]. A slice through the tomographic inversion is shown in figure 14b. Recent full orbit ASCOT simulations with time-dependent ELM electromagnetic fields reveal that, indeed, the measured fast-ion acceleration arises from a resonant interaction between the fast-ions and both the parallel and perpendicular electric fields observed during the ELM crash [251]. The acceleration parallel to the magnetic field is caused by the parallel electric field emerging during the reconnection event that appears at the ELM crash. The acceleration in the direction perpendicular to the magnetic field arises from a gyrotron acceleration process induced by the vertical polarization of outward-propagating ELM filaments with sizes smaller than the fast ion gyroradius [252].

Measurements of lost alphas from the recent JET DT experiments using a FILD have been demonstrated in [41, 46, 47], as figure 15 illustrates. The detected Larmor radii are consistent with alphas at the magnetic field of the scintillator plate. At the time of the measurement, modes in the TAE frequency range were observed [41].

### 4.3. Runaway electron diagnostic

Runaway electron (RE) diagnostics at ITER should measure the maximum energy of the runaway electrons up to 100 MeV with a time resolution of 10 ms, and the RE current after the thermal quench and for failed breakdown from 1–15 MA with a time resolution of 10 ms. This measurement capability has been demonstrated in current tokamaks using both GRS$\|$ [164, 253, 254] and synchrotron emission in the infrared band [255]. A measured spectrum of HXR emission at DIII-D and the corresponding inferred RE energy spectrum appear in figure 16.

RE diagnosis by GRS is based on measuring the spectrum of the bremsstrahlung hard x-ray emission

$\|$ As customary, we will use the term GRS to indicate measurements of hard x-ray (HXR) emission in the MeV energy range, even though the origin of this radiation is not a nuclear decay.
<figure><img src="image_16.png" /><figcaption>**Figure 15.** Measurements of lost alphas during JET DT shot #99500 from the scintillator probe showing counts resolved in pitch angles and Larmor radii (cm) during the time interval 6.5–7.5 s. The major radius coordinates the alphas most likely originated from are indicated. Reproduced from [41].</figcaption></figure>

<figure><img src="image_17.png" /><figcaption>**Figure 16.** (a) Measured hard x-ray (HXR) emission at DIII-D and (b) inversion to runaway electron energy spectra. Reproduced from [256].</figcaption></figure>

produced when (predominately MeV) runaways collide with ions. The detectors must be able to cope with signals at MHz counting rates to have sufficient statistics in the measured spectrum (say, at least $10^4$ counts) within the required time resolution (10 ms or less). Measurements under these conditions have been possible only in the past few years, mostly thanks to the rapid development of fast digitizers [171], and have been established in a number of major tokamaks, such as JET [257], DIII-D [256], AUG [258] and EAST [259]. The maximum energy of the REs is obtained by the inversion of the measured gamma-ray spectrum, starting from the known response function of the detector and the cross sections of hard x-ray emission

[260]. This gives the one-dimensional runaway electron distribution function that is most compatible with the measurements and from which the maximum runaway energy can be determined. The runaway electron current is calculated from the integral of the runaway electron distribution function obtained by inversion.

RE diagnostics by measurements of their synchrotron emission in the infrared band have been established at TEXTOR [255,261], and applied at other major tokamaks [262–264]. Modeling tools to relate the properties of the infrared emission to the runaway electron distribution function have also been developed [265,266] and applied to interpret the emission in present experiments, instead of the former analytic models based on several additional approximations. Interpretation of data from the infrared cameras provides the location and size of the runaway electron beam, while the analysis of its wavelength spectrum gives information on the RE energies.

At ITER, both gamma-ray and infrared diagnostics will be used to measure REs. Two gamma-ray systems are currently planned: a hard x-ray monitor [267] and the radial gamma-ray spectrometers [153]. The hard x-ray monitor is installed right behind the machine first wall and, being sensitive to runaway current as low as few kA, will provide a first signal as runaway electrons develop in the plasma. However, due to the harsh measurement conditions behind the first wall, the hard x-ray monitor may not be able to provide reliable information on the RE energies.

The REs are detected with the already described radial gamma-ray spectrometers which will also provide clean measurements of the gamma-ray spectrum from RE bremsstrahlung and are expected to be the main instrument providing information on the RE energies and their evolution in ITER.

As far as infrared diagnostics are concerned, no dedicated synchrotron camera for runaway electrons is currently planned for ITER, but standard infrared (as well as visible) cameras will be used. As a consequence, infrared cameras will remain the main tool to determine the spatial extent of the runaway electron beam but, unlike present experiments, they will not be able to provide information on the runaway electron energies, which will be measured by the radial GRS only.

### 4.4. Diagnostics of instabilities in fusion plasmas

Key to interpreting the EP measurements discussed in the previous section and fully leveraging ITER’s unique access to alpha physics will be measurements of any instabilities driven by or transporting EPs. Ensuring appropriate measurements of AEs in ITER will allow the validation of predictions of alpha-driven instabilities in plasmas dominated by self-heating. Alpha-driven
TAEs have recently been detected in a DT discharge at JET with magnetics, soft-xray, interferometry and reflectometry diagnostics [40]. Ensuring appropriate measurements will further provide the control capability for optimizing performance, reducing the alpha losses, and minimizing potential first wall damage. In addition, such measurements may provide key information on the bulk plasma through Alfvén spectroscopy [268]. The discussion here will focus on recent advances in measurements of AEs with a particular emphasis on those techniques and systems that will be possible on ITER; this need is called out in ITER measurement specifications as “TAE mode-induced perturbations in B, T, n with a bandwidth of 30-300 kHz” [269]. None of the ITER diagnostics we will describe will be capable of resolving toroidal mode number or core EP-driven instabilities. This is a significant deficit and will impact the ability to rigorously validate models for EP transport by instabilities in ITER.

<figure><img src="image_18.png" /><figcaption>**Figure 17.** (a) TIP chord layout on ITER, (b) TIP prototype interferometer spectrogram from DIII-D discharge 169632. Reproduced from [270]</figcaption></figure>

### 4.4.1. Toroidal interferometer polarimeter.
One of the primary diagnostics expected to contribute to the detection of EP-driven instabilities on ITER is the toroidal interferometer and polarimeter (TIP) [270–273]. A detailed description of key design elements appears in chapter 8 of this volume [19]. TIP is essentially a two-color CO₂ laser based mid-

infrared (4.6 and 10.59 µm) interferometer system [274–279], probing line-integrated electron density with a combined polarimetry measurement to probe line-integrated Faraday rotation (product of $n_e B_\|$). As shown in figure 17a, TIP will feature five independent tangentially viewing chords spaced across the device midplane. Two-color interferometers like these are extremely valuable for the detection of core instabilities and are capable of producing high quality fluctuation data for both coherent and incoherent fluctuations with wavenumbers $k < 2/a_o$, where $a_o$ is the Gaussian beam-waist, and with frequencies up to several MHz [280]. They can operate in almost all tokamak plasma conditions, rarely suffer from refraction, are based on well developed CO₂ laser and detector technology and are able to take advantage of modern digital phase demodulation techniques for very low noise phase measurement. The TIP interferometric measurement provides the mode induced line-integrated density fluctuation along each sightline while the polarimetry Faraday effect measurement, in principle, will provide a combination of density and magnetic fluctuations, although the minimum resolvable fluctuation levels are significantly higher due to the relatively small effect being exploited [272, 280–282]. An example interferometric density fluctuation spectrum from the ITER TIP prototype tested on DIII-D is shown in figure 17b, where different types of beam-driven AEs as well as other coherent fluctuations are present [270]. In the figure, BAEs, RSAEs and NTMs are labeled. These modes will be discussed in detail in the following sections. Resolvable fluctuation levels in the AE frequency range are predicted to be on the order of $\delta n_e/n_e \approx 10^{-5}$ [272]. Additionally, because of its role in plasma control, ITER TIP data will be available in “real-time,” allowing its use as a monitor of core AEs and other fluctuations for instability control purposes.

### 4.4.2. ECE radiometer.
Another diagnostic technique that will be available on ITER and has proven to be extremely valuable for EP physics studies is electron cyclotron emission (ECE) radiometer measurements of electron temperature fluctuations [283–286]. Figure 18a is an example radial profile of ECE power spectra from a reversed magnetic shear DIII-D discharge in which various RSAEs and TAEs are clearly observed with ECE [283, 287]. Each type of mode is labeled along with the detailed radial structure of an individual RSAE and TAE in figure 18b and figure 18c respectively [283]. The RSAEs are localized near the minimum in the safety factor profile and the TAEs are seen for radii outside of $q_{min}$ extending to the plasma edge. As an example of the type of validation enabled by ECE measurements, figures 18b and 18c show a comparison of measurements and calculations of the
<figure><img src="image_19.png" /><figcaption>**Figure 18.** ECE radiometer data from a DIII-D beam-heated discharge with AEs. (a) Radial profile of ECE power spectra vs. normalized minor radius. (b) Radial profile of ECE radiometer measured temperature perturbation vs. major radius for an *n* = 3 RSAE and for (c) an *n* = 3 TAE. Reproduced from [283].</figcaption></figure>

ideal MHD eigenmode electron temperature perturbation from the NOVA code. The ECE radiometer in ITER will span approximately 70% of the device mid-plane from inside the magnetic axis to the last closed flux surface on the low-field side ($R \approx 5.2 - 8.2$ m) with about 64-128 channels giving a rough spacing of $\Delta R=2.3-4.6$ cm [287]. The poloidal and toroidal resolution is determined by the beam spot size, which is expected to be < 10 cm in ITER [287]. To avoid averaging effects, the poloidal wavelength should be less than the beam size which, at mid-radius, would correspond to poloidal mode numbers of *m* < 60. The radial resolution is determined by the instrumental RF frequency channel widths and relativistic broadening, which depends on electron temperature and is expected to dominate in ITER high temperature plasmas. For 1st harmonic O-mode measurements in ITER scenario 2 plasmas, the expected radial resolution is < 10 cm across the device midplane and significantly better at lower temperatures [287]. The resolvable mode amplitudes will depend on several factors including electron temperature, analysis approach (Fourier interval, spatial averaging, correlation measurements, etc.) and detailed hardware choices. The modes shown in figure 18 have amplitudes $\delta T_e/T_e \approx 0.5 \times 10^{-3}$.

<figure><img src="image_20.png" /><figcaption>**Figure 19.** AUG discharge #25528. (a) ECEI spectrogram, (b) Reflectometer spectrogram, (c) model for RSAE frequencies. Calculated RSAE minimum frequency overlaid as solid white line. Yellow labels and dashed vertical lines show $q_{min}$ values used as input to RSAE model. Reproduced from [288].</figcaption></figure>

4.4.3. *Microwave reflectometry.* ITER will also be equipped with multiple microwave reflectometry systems that will likely be capable of resolving density fluctuations due to coherent EP-driven instabilities. Reflectometry is an established approach with a long history of research in its use for measurement of both coherent and incoherent core fluctuations [289–294]. Density fluctuations along the path of the beam can modulate the phase or amplitude of the reflectometer signals or both. Under certain conditions, phase modulation of the probe beam is localized at the cutoff [295–297] (the point of reflection of the microwaves), so that localized measurements of density fluctuations are possible. A more sophisticated analysis approach using multiple channels has also been developed that takes the nonlocal response of reflectometers into account, and estimates of both the density and magnetic perturbations can be derived [293].

The two main reflectometer systems on ITER,
each with planned capabilities for measuring EP-
driven instabilities, are the low-field side reflectometer
(LFSR) [298, 299] and the high-field side reflectome-
ter (HFSR). Each will supply simultaneously measure-
ments of the n<sub>e</sub>-profile and n<sub>e</sub> fluctuations. The broad
measurement capability of LFSR is enabled by an ar-
ray of six monostatic antennas which inject from an
equatorial port on the outboard side of the ITER ves-
sel. A low-loss transmission line transmits the 30 – 165
GHz, O- and X-mode signals to and from the ITER
plasma. The design of LFSR has been optimized to
probe the edge region (r/a > 0.85) for a broad range of
ITER conditions. However, penetration into the core
with X-mode is possible with either reduced toroidal
field or full field and n<sub>e</sub> not exceeding <approx> 7 × 10<sup>19</sup>m<sup>-3</sup>.
Broadband transmission signals are realized by full-
band microwave transceivers combined with quasi-
optical multiplexing. Both frequency-modulated con-
tinuous wave (FMCW) and fixed-frequency continuous
wave (FFCW) operations are incorporated. LFSR is
equipped with conventional and Doppler reflectome-
try systems, enabling detection of low-to-intermediate
wavenumber fluctuations. Full-band FMCW opera-
tions with repetition rate of 1 MHz are planned for
LFSR, providing a means to measure n<sub>e</sub> fluctuation
profiles with high time resolution. Ignoring the effects
of the profile gradient and turbulence, the radial res-
olution depends explicitly on the signal-to-noise ratio
and the RF bandwidth. LFSR is expected to achieve
better than 26 dB signal-to-noise ratio over most of its
operating space, and with a RF bandwidth of 250 MHz,
a radial resolution of 5 mm is achievable. Flat profile
gradients and turbulence tend to broaden the radial
resolution, and the cumulative effect is strongly depen-
dent on the plasma conditions. Full-wave reflectometry
simulations indicate that LFSR can access and should
be sensitive to core-localized weak density fluctuations
(<span style="text-decoration: overline">δn</span><sub>e</sub>/<span style="text-decoration: overline">n</span><sub>e</sub> <span style="text-decoration: overline">≈</span> 10<sup>-6</sup> – 10<sup>-4</sup>) similar to TAEs. The ability
to make this measurement unambiguous can be com-
plicated by strong fluctuations (<span style="text-decoration: overline">δn</span><sub>e</sub>/<span style="text-decoration: overline">n</span><sub>e</sub> > 10<sup>-3</sup>) in the
path of the beam that can also modulate the signal,
scrambling the phase. However, reflectometer mea-
surements combined with synthetic diagnostic model-
ing provide a powerful tool for interpreting the signals
and resolving possible ambiguities. An example reflec-
tometry measurement of RSAEs in a reversed mag-
netic shear AUG plasma is shown in figure 19a where
a broad spectrum of RSAEs is clearly visible [288,300].
These measurements are to be compared to ECE imag-
ing measurements shown in figure 19b, where it can be
seen that the different localizations result in a slightly
different spectrum of observed modes [301]. This data
allows the inference of q<sub>min</sub> by MHD spectroscopy, see
also [302]. The q<sub>min</sub> resolution based on the data
shown in figure 19 is very good since every observed

mode has a different (m/n) value and gives a separate
constraint on q<sub>min</sub>.

4.4.4. Magnetic probes. Like the majority of present machines, ITER will be equipped with a large array of magnetic sensors with varying sensitivities (bandwidth, polarization, amplitude, etc.) Several “High-frequency” coils have been designated to meet the measurement requirement 062 for the measurement of TAEs. These probes are a standard inductive pickup coil design and will have a minimum bandwidth of 0.5 MHz and sensitivities in the range 1 × 10<sup>-6</sup> – 3.5 × 10<sup>-4</sup> T over 30 kHz to 0.5 MHz [303]. The system will consist of an array of coils spaced toroidally and poloidally for resolution of toroidal mode number and poloidal structure. While edge modes will likely be clearly resolved by these sensors, it is questionable whether core modes that do not reach the edge can be measured.

4.4.5. Other potential diagnostics of EP-driven insta-
bilities at ITER. Several other diagnostics on ITER
have the potential to measure EP-driven instabilities.
However, due to hardware or approach choices or other
issues, they may not have the required bandwidth
or sensitivity. These diagnostics include: poloidal
polarimeter (PoPola), Mirnov/magnetic pickup loops,
dispersion interferometer polarimeter (DIP) and soft
x-ray (SXR) spectroscopy. The PoPola is a fan of
119 µm poloidal polarimeter channels with potential
to measure core density and magnetic fluctuations. As
discussed in chapter 8 of this volume [19], this diag-
nostic will have 100 Hz bandwidth but a modified ap-
proach could extend its bandwidth and utility to the
measurement of instabilities. The ITER DIP, similar
to TIP, is a multi-channel CO₂ laser based interfer-
ometer system that will use a photoelastic modulator
approach limiting its fluctuation measurement band-
width to ≈10 kHz [304]. Soft x-ray systems, like those
planned for ITER, have been used successfully to mea-
sure AEs in the past [305]. Unfortunately, the specified
bandwidth of the ITER system appears to be 100 kHz
which is marginal for measurements of AEs in ITER
[306]. Recently, a promising diagnostic technique has
been demonstrated using launched fast Alfvén waves in
an interferometer configuration - the fast wave interfer-
ometer (FWI) [307,308]. Because the fast wave velocity
depends on ion mass density and magnetic field, the
diagnostic is capable of probing line-integrated mass
density along with probing core density and magnetic
field fluctuations. Recent experiments clearly show the
diagnostic ability to resolve core fluctuations for the
first time. The FWI is not currently an assigned ITER
diagnostic but its use is under investigation.
4.4.6. AE spectroscopy. Beyond measurements of the modes themselves, work since the last ITER physics basis has clearly demonstrated the potential of EP-driven instability measurements for providing information about the equilibrium plasma, so-called “AE spectroscopy” [268]. A particularly compelling application of this technique is the determination of the minimum safety factor $q_{min}$ through measurements of RSAEs. As mentioned, figure 19a and figure 19b show measurements of RSAEs in AUG reversed shear plasmas during the current ramp phase. During this time period, $q_{min}$ and the mode spectrum are evolving rapidly. The RSAE spectral evolution is well described using a simple ad hoc model [273,280,309], the results of which are shown in figure 19c; the dominant feature of the RSAE frequency sweep is the $q_{min}$ dependence which roughly scales as $(m-nq_{min})/q_{min}$ [268,310,311]. In figure 19b and 19c, the overlaid $q_{min}$ evolution has been determined by adjusting the values until the predicted RSAE timing matches that measured experimentally. Discharges on DIII-D with motional Stark effect (MSE) polarimetry have confirmed the validity of this approach and show the precision with which $q_{min}$ can be determined [280]. Further, when RSAE localization information is present through measurements of the AE mode structure with ECE or even coarsely with interferometry, this information can be used to place constraints on not only the value of $q_{min}$, but its location. Also, the RSAE minimum frequency depends on $T_e$ and $T_i$ and also provide a check on those values [311].

## 5. Interaction of EPs with thermal-plasma-driven instabilities

In this section, we review the interaction between EPs and thermal-plasma-driven instabilities, which are instabilities also appearing in the absence of EPs. The EP-driven Alfvénic modes are discussed in the following sections. Important thermal-plasma-driven instabilities interacting significantly with EPs are neoclassical tearing modes (NTMs), kink modes, sawteeth, kinetic ballooning modes (KBMs), resistive wall modes (RWMs), and edge-localized modes (ELMs), which we will review in the following. The recent JET DT experiments highlight the importance of the interaction between alphas and thermal-plasma-driven instabilities. Alpha losses due to NTMs, sawteeth, and ELMs were observed in these experiments [46,47].

### 5.1. Neoclassical tearing modes

NTMs are a type of helical MHD instability in tokamak plasmas where magnetic islands with low poloidal and

toroidal mode numbers $(m/n)$ form on rational $q$-surfaces. Inside the island, the pressure profile is flat, which leads to a low bootstrap current. NTMs are discussed comprehensively in chapter 4 of this volume [16]. EP transport and losses can be strongly affected by NTMs, and in turn EPs can play an important role in the stability and growth of NTMs [312]. This has been documented in numerous experimental and theoretical studies of the stability of NTMs and their effect on EP confinement [313–338]. Analytic and numerical modeling predicts that the EP distribution function and equilibrium parameters influence the NTM stability in complex ways, e.g. using the M3D-K [339], NIMROD [319,340] and GTC [330] codes.

In turn, the influence of NTMs on EP losses has been studied analytically as well as numerically with orbit following codes. Recently the reduced “kick” model in TRANSP has been extended to include EP transport by NTMs [341]. The “kick” model simulations suggest that (N)TMs, as well as fishbones (see section 7), can interact through modification of the EP distribution in phase space, which influences the drive.

Analytic and numerical modeling [322, 324, 325, 339] indicates that EPs can have a stabilizing or a destabilizing effect on tearing modes, depending on the relative balance of trapped EPs compared to co- and counter-passing EPs, on the presence of direct resonances between EP drift orbit frequencies and the tearing mode rotation frequency, and on the equilibrium parameters, notably the EP orbit size and the magnetic shear. Analytic models find that the island frequency in the local plasma frame, an important but not well understood or measured parameter, can have an impact on the stability (leading to stabilizing or destabilizing EP terms in the extended Rutherford equation) [326]. Additionally, the interaction between EPs and NTMs is qualitatively different if orbital frequencies resonate with the NTM frequency. We will deal with both non-resonant and resonant interaction in the following.

5.1.1. Non-resonant EP-drive of NTMs. Analytic modeling of the interaction of EPs with tearing modes, done in the context of the extended Rutherford equation [327], can provide useful physical insight into the interaction mechanisms. The stability of resistive instabilities such as tearing modes is described by a 4^th^-order differential equation [342] which is difficult to deal with analytically. Rutherford had the insight that this 4^th^-order differential equation could be reduced to a 2^nd^-order differential equation in the region outside the island, except for in a narrow boundary (tearing) layer. Furthermore, the tearing mode stability and growth rate can be determined
by matching the solutions of the 2nd-order equation outside the island across the tearing layer, with no need to solve the 4th-order equation.

The original Rutherford equation not modified to include EP effects is of the form

$$ \frac{\tau_s}{r_s} \frac{dW}{dt} = \Delta_c, \qquad (39) $$

where W is the island width, $\tau_s = \frac{\mu_0 r_s^2}{1.22\eta}$ is the resistive time, $r_s$ is the radius of the rational surface, $\eta$ is the resistivity, $\mu_0$ is the vacuum permeability, and $\Delta_c = r_s\Delta'$ is the classical stability parameter. $\Delta'$ is calculated from the equilibrium q-profile and current-profile in the regions outside the tearing layer [343]. It is proportional to the perturbed sheet current at the rational surface required to match the external solutions across the tearing layer. The perturbation decays in time for a negative sheet current, whereas it grows for a positive sheet current. The Rutherford equation can be extended by adding other sources of current perturbations. For example, a magnetic island flattens the pressure profile in the vicinity of the island, resulting in a reduction in the perturbation current (in normal shear plasmas), which leads to the "bootstrap drive" term in the modified Rutherford equation [344].

The extended Rutherford equation including non-resonant EP drive terms for NTMs $\Delta_{EP}$ is of the form

$$ \frac{\tau_s}{r_s} \frac{dW}{dt} = \Delta_c + \Delta_{GGJ} + \Delta_{bs} + \Delta_{pol} + \Delta_{EP}. \qquad (40) $$

The classical stability parameter $\Delta_c$ from the original Rutherford equation may be affected by EPs through the EP-driven currents changing the equilibrium current profile. The additional terms are the Glasser-Greene-Johnson curvature term $\Delta_{GGJ}$ [345], the bootstrap current drive term $\Delta_{bs}$ [344, 346], the thermal-plasma polarization current term $\Delta_{pol}$ [347, 348] arising from island motion through the equilibrium plasma, and the EP term $\Delta_{EP}$, which results from the perturbed currents on the rational surface arising from the effect of the island magnetic perturbations on the EP population. It consists of several contributions due to various sub-populations of the EPs, such as co-passing, counter-passing and trapped populations, usually derived for magnetic island widths either much larger or much smaller than the EP orbit width. To determine if the plasma is linearly stable or unstable to NTM formation, one determines if an initially small island grows, so the island width is generally assumed to be much smaller than the EP orbit width. For currently operating tokamaks and especially spherical tokamaks, which are smaller than ITER and have lower magnetic fields, even most saturated islands are smaller than the EP orbit width. However, the extended Rutherford equation typically predicts islands to be metastable

with a threshold island size. An ideal linearly unstable "classical" tearing mode grows from small a amplitude. Additionally, many linearly stable tearing modes can also become unstable when an MHD event (like an ELM or sawtooth) creates an initial perturbation that is greater than the threshold island size.

**Influence of EPs in the outer region** The first term on the right-hand-side of the extended Rutherford equation captures the effect of the current drive from EPs on the current profile shape in the region outside the island. In practice, the effects of the EPs on the current profile are independently calculated with codes such as TRANSP. However, it is useful to have a qualitative understanding of how EP current drive affects the tearing mode stability through this term. Co-passing EPs tend to decrease the value of $\Delta'$ and have a stabilizing effect. Conversely, counter-passing EPs tend to increase the value of $\Delta'$, a destabilizing effect decreasing the onset threshold of the NTMs [321, 339]. The effect of trapped EPs on the instability criterion is primarily through the asymmetric pressure of trapped EPs because the parallel current they produce is small [321]. Analytic studies have found that trapped EPs can be destabilizing in weak magnetic shear, but stabilizing in strong magnetic shear [322, 325].

Many experiments have investigated the effect of EPs on NTM stability. Counter-NBIs have been found to have different effects for various plasma scenarios. In DIII-D experiments [313] it was found that increased co-NBI power stabilized NTMs, but there was no corresponding decrease in stability for increasing counter-NBI power. In beam-heated DIII-D hybrid plasmas, the most unstable mode switched from AEs to chirping NTMs when the perpendicular beam pressure decreased [335]. In AUG experiments [314], both increasing co-NBI power and increasing counter-NBI power stabilized NTMs. The analytic and simulation results discussed above do not explain these experimental discrepancies for the counter-NBI. A possible reason is that the toroidal rotation could not be controlled in these experiments and may have been partly responsible for the observed changes in stability [328, 349-352]. However, the effects of toroidal rotation should be independent of the rotation direction. Unfortunately, appropriate codes have not yet been developed to model both the changes in the EP populations and the changes in equilibrium parameters. It is also important to remember that experiments, by their nature, will include all types of EP effects on NTM stability, including potential effects of EP resonances and EP-driven polarization currents discussed below.
Influence of EPs in the island region In addition to the neoclassical perturbed polarization current generated by the thermal plasma, EPs can also generate a perturbed cross-field E × B current which induces a parallel electron current to satisfy charge neutrality [326]. The contribution of this polarization current can be expressed as

$$ \Delta'_{u} = -G_3 \frac{r_s^2}{s^2 L_{n_i}^2} \frac{\beta_i}{W} \frac{\omega}{\omega_{*i}} \frac{L_{n_i}}{L_{n_f}} \frac{n_f}{n_i}, \quad (41) $$

where $G_3 \approx 1.58$ is a numerical coefficient, s is the magnetic shear, and $L_{n_i}$ and $L_{n_f}$ are the lengthscales of the thermal-ion and EP densities, $\beta_i$ is the ion beta, and $\omega$ is the island frequency relative to the background plasma. $\Delta'_{u}$ depends on the magnetic shear, island propagation frequency, and EP density gradient at the rational surface. $\Delta'_{u} > 0$ if the mode propagates in the electron diamagnetic drift direction, $\omega < 0$ and the density gradients of thermal ions and EP on the resonance surface have the same sign, i.e., the onset threshold of NTMs is decreased. In weak shear plasmas with large EP fractions, the EP polarization term can become comparable to the bootstrap term, and $|\Delta'_{u}/\Delta'_{bs}|$ increases as $n_f/n_i$ increases or s decreases. Thus, this term can affect the island growth when the EP orbit size is greater than the island width.

Some operational regimes in ITER steady-state and hybrid scenarios are expected to have weak magnetic shear [20], which hence deserve particular attention. In a low-shear, high-performance regime on JT-60U, no NTMs were observed in several discharges during NBI with high bootstrap fraction [353]. In these experiments, the effects of EPs may be important. The relative importance of the EP polarization current term has also been investigated for tearing modes in the spherical tokamak NSTX [327, 354]. For these experiments, it was found that the destabilizing EP polarization term overcame the stabilizing effect of the thermal polarization term, allowing growth of the island to widths where the other terms in the extended Rutherford equation dominate the island dynamics.

### 5.1.2. Numerical simulations of EP effects on NTM stability.
Simulations find that EPs interact with tearing modes mainly in the outer region and the growth rate of the tearing modes is reduced dramatically by EPs [319]. Furthermore, the net effect of co-passing EPs is weakly stabilizing whereas the net effect of counter-passing EPs is destabilizing [321]. The simulation results are in qualitative agreement with the analytic results for the passing particles discussed above.

Simulations using similar parameters to the DIII-D experiment [317] were performed using the global gyrokinetic toroidal code GTC. The perturbed parallel current induced by EPs was added to Ampère's

law [330], resulting in a small stabilizing effect on the excitation of NTMs.

Simulations of DIII-D equilibria with the NIMROD code have also found that EPs were generally stabilizing, although this effect was from EPs in the plasma bulk rather than a tearing layer effect. However, NIMROD simulations also found that a precession drift resonance with the core mode destabilizes the 2/1 mode [340]. The M3D-K code [355] has been used to study the effect of EPs on tearing modes [321]. The results agreed in large parts with previous analytic results for the effects of passing EPs. It was also found that trapped EPs are much more destabilizing than counter-passing EPs at the same beta. In M3D-K simulations of HL-2A equilibria, co-passing EPs were responsible for destabilizing the bursting/chirping 2/1 modes [334, 337]. Additionally, it was found that $q(0) = 1.5$ was most unstable for a scan of the safety factor $1.2 \le q(0) \le 1.9$.

Plasmas with hollow current profiles can have two rational surfaces with magnetic islands of the same helicity. For such double tearing modes, simulations find that passing EPs are destabilizing, although this result is limited by the assumption of a “top-hat” model for the double tearing mode [356]. The effects of EPs on double tearing modes were studied by M3D-K [357], where co-passing EPs are destabilizing but trapped EPs are stabilizing. Double tearing modes can undergo a transition to a fishbone-like mode if the EP beta $\beta_f$ is larger than a threshold.

We note that self-consistent codes for describing NTMs that include the physics of the EPs and the effects of toroidal rotation are still lacking, but would be essential to explain the experimental results and to explore methods for controlling NTMs.

### 5.1.3. Resonant interaction between EPs and NTMs.
At first sight, resonances of NTMs with EPs were thought to be unlikely because the typical EP bounce, transit and precessional frequencies are much larger than typical NTM frequencies. However, indications of resonant interaction between EPs and NTMs have been found in several tokamaks [331–335] where the NTM frequency jumps up on a timescale of about 0.1 ms and chirps down over a longer time period of one to two ms (figure 20 [335]). During each chirp the neutron rate is reduced by about 1%, as was also observed at TFTR. This implies that there is a redistribution or loss of EPs during each chirp. Experiments at AUG suggest that this is caused by a resonance between EPs and NTMs [332]. However, it is not clear from presently available experimental data whether the resonances of passing or trapped EPs cause chirping NTMs.

In [336], it was shown using a particle model that an additional toroidal torque is generated by the
<figure><img src="image_21.png" /><figcaption>**Figure 20.** Cross-power spectrograms of electron temperature fluctuations measured by the ECE diagnostic for two hybrid NBI- and ECRF-heated discharges in DIII-D. It is often observed that steady NTMs and chirping $n = 1$ fishbones coexist for more than 100 ms [see (a)] and then trigger chirping NTMs [see (c)]. It is occasionally observed that NTMs can be fully stabilized and that fishbones and harmonics become dominant [see (d)]. The central toroidal rotation frequency is plotted in (b) and (d) as the dashed white lines. The rotation frequency at the $q = 4/3$ rational surface is shown by the blue line in (b). Reproduced from [335].</figcaption></figure>

resonance of NTMs and trapped EPs which increases the propagation frequency of NTMs. However, the simulated duration of the frequency chirping up is up to a factor of 50 times longer than in the experiments. A model based on drift kinetic theory reduces the discrepancy in comparison with the DIII-D experimental results, although this model doesn't address the chirp-down phase [320]. An alternative interpretation is that the frequency chirping may be caused by a fishbone-like mode resonance with an EP precession frequency, which is supported by analytic and simulation studies [337, 338]. Analytic results predict that the fishbone-like mode is driven by trapped EPs above an EP beta threshold [338] but simulations have also found that a fishbone-like mode can be driven by co-passing EPs [337].

Related phenomena have been found in HL-2A where a fishbone-like mode rather than a chirping tearing mode or NTM were observed. Similarly, it is believed that the fishbone-like mode is excited by resonance of the tearing mode with trapped EPs [337, 338]. In simulations for a reversed $q$-profile, a fishbone-like mode and a double tearing mode were found to co-exist [357, 358]. This fishbone-like mode is excited by resonance between the tearing mode and co-passing EPs. These disparate results suggest that the physical

mechanism of frequency chirping during NTMs still remains unresolved. They have also not yet revealed whether trapped or co-passing EPs are most important in this process. The physical mechanism is still under debate and will likely require further experiments and investigations.

5.1.4. EP transport by tearing modes. Since magnetic islands break the tokamak axisymmetry in a way similar to the toroidal field ripple, enhanced particle diffusion can be expected due to NTMs. However, the mechanisms affecting the EP confinement have proven a lot richer. In DIII-D, large coherent MHD modes were observed to reduce the neutral beam current drive efficiency [359] and 2.5 MeV neutron emission [360] by as much as 80% and 65%, respectively. Guiding-center simulations with the ORBIT code suggest that the intrinsic orbit stochasticity leads to transport of co-passing EPs, similarly to the finding of earlier work for low-$n$ magnetic perturbations [361, 362]. In AUG experiments, time-resolved pitch and energy measurements of the EP losses found that the losses were modulated at the frequency of a (2,1) NTM [363, 364]. Even trapped EPs experience increased transport when the NTM resonates with the bounce and toroidal precession frequencies [365], as confirmed
by recent drift-kinetic calculations [366]. Also within
a drift kinetic framework, the passing EPs were
found to experience resonances with the rotating
islands, providing an explanation to the observed phase
locking between the enhanced transport and the NTM
perturbation [356]. More recently, the resonances for
trapped, co- and counter-passing MeV alpha particles
with a (2,1) tearing mode were found using the GPU-
accelerated TAPAS code, showing the impact of the
rotation frequency of the magnetic island on the losses
of EPs [367].

NTM-induced EP losses can be caused by
stochastization of drift orbits (see e.g. [360, 363, 367]).
Simulations of an AUG discharge with a (2,1) NTM
revealed that the NTM-induced losses of passing beam
ions were caused by the formation of a (4,1) NTM
[368]. In a purely RF-heated discharge in AUG,
resonant interaction between the EPs in the high-
energy tail of the ICRF distribution and a (5,4) NTM
was found to lead to enhanced EP losses [208]. In
DIII-D experiments, beam modulation was applied to
study the interaction between EPs and a (2,1) NTM in
different parts of the phase space [317]. The mode was
observed to affect the confinement of both the counter-
passing and trapped beam ions. The kick model in
TRANSP, extended to NTMs [341], allowed analyzing
the effect of NTMs on EPs in a variety of DIII-D
plasmas [369]. Above a threshold width for the island,
EP losses were found to increase, leading to reduction
in beam-driven current and torque across the entire
plasma. Kick modeling in conjunction with velocity-
space tomography show that NTMs alter the positive-
pitch distribution [232].

Note that the experimental studies described
above have used plasmas with very large island sizes
to help identify the loss mechanisms. Such plasmas
cannot be tolerated in reactor-level plasmas. For
ITER, studies on the effect of MHD modes on EP
confinement have mainly been done for the foreseen
main operating phases with fusion alphas using the
ASCOT code following full gyro-orbits, which can be
important when the magnetic field has strong toroidal
inhomogeneities [370,371]. Both (2,1) and (3,2) NTMs
were simulated, together with realistic toroidal-field
(TF) ripples, using a theory-based model for the
NTM island built into the magnetic background. The
simulations indicate that the total power density on
the wall strongly depends on the NTM perturbation
amplitude. However, even for an excessive amplitude,
the wall power densities remain below the ITER design
limits. Furthermore, the NTMs were not found to
cause additional hot spots on the wall, but only
increase the wall power density in the places also found
for simulated MHD-quiescent plasmas.

EP transport by NTMs was observed in the recent

JET DT experiment [52]. Discharge #99886 was an MHD-active DT discharge with an NTM, fishbones, RSAEs, and TAEs as detected with the Mirnov coil measurements shown in figure 21. Despite this MHD activity, the energy confinement in this discharge was very good. FILD measurements sensitive to lost alphas detected the NTMs and the fishbones which suggests transport and loss of alphas due to the NTMs and fishbones as shown in figure 22.

<figure><img src="image_22.png" /><figcaption>**Figure 21.** Mirnov coil spectrogram during JET DT discharge #99886 showing an NTM, fishbones, RSAEs, and TAEs. Alpha losses at the fishbone and NTM frequencies were detected with the FILD (figure 22). Reproduced from [52].</figcaption></figure>

<figure><img src="image_23.png" /><figcaption>**Figure 22.** FILD measurements showing alpha losses during NTMs and fishbones in JET DT discharge #99886. Mirnov coil measurements detected the fishbones and NTM at the same frequency (figure 21). Reproduced from [52].</figcaption></figure>

5.2. Sawteeth and kink modes

The redistribution of EPs by sawteeth and kink modes
and the effect of trapped and passing EPs on sawtooth
stability have long been studied theoretically and experimentally. The sawtooth instability causes a periodic cycle of the core plasma, where the core density and temperature slowly increase and then suddenly drop, an event referred to as the sawtooth crash, giving their timetraces the form of sawteeth. A kink mode has often been observed to become unstable right before the sawtooth crash. The sawtooth instability is triggered at an on-axis safety factor of $q(0) < 1$. During the sawtooth crash, the on-axis safety factor jumps up to $q(0) \approx 1$ and the plasma profiles $T_e$ and $n_e$ change abruptly. This is thought to result from reconnection of the helical magnetic flux, leading to rapid redistribution of particles from within the $q=1$ surface to outside. Figure 23 shows this redistribution for the EP population at AUG [150]. The EP density before and after the crash has been inferred by orbit tomography based on 27 simultaneously acquired FIDA spectra (see section 4). After the sawtooth crash, the current density profile peaks and $q(0)$ decreases below unity on the resistive timescale of the plasma. This pattern repeats with the characteristic time called the sawtooth period. Sawteeth are discussed comprehensively in chapter 4 of this volume [16].

<figure><img src="image_24.png" /><figcaption>**Figure 23.** Ejection of EPs from the inside to outside the $q=1$ surface during a sawtooth crash at AUG, as shown by the difference in EP densities before and after the crash. The image has been computed by orbit tomography based on FIDA measurements. Reproduced from [150].</figcaption></figure>

It is known that EPs within the $q=1$ surface can stabilize the internal $n=1$ kink mode and increase the sawtooth period significantly [114,372]. However, long sawtooth periods lead to “monster” sawtooth crashes with particularly strong pressure drops [373], which can trigger NTMs leading to significant reduction of plasma confinement [374]. Both trapped and passing EPs can stabilize sawteeth. Experiments on JET

[116, 375], MAST [376], TEXTOR [377] and DIII-D [378] have shown that off-axis passing EPs stabilize sawteeth if they move in opposite direction to the plasma current, whereas on-axis EPs stabilize sawteeth if they move in the same direction as the plasma current.

Sawteeth are a concern for burning plasmas since the alphas and EPs from auxiliary heating will stabilize sawteeth [379], so we might expect monster sawteeth in burning plasmas, and the deleterious effects that go with that. Modeling has predicted that the fusion alpha population will increase the sawtooth period on ITER [380]. It would be advantageous to develop methods to pace sawteeth, resulting in a short sawtooth period with weaker sawteeth. Sawteeth also have positive applications since they eject impurities, and so sawteeth could be used to remove helium ash from the plasma core in burning plasmas.

### 5.2.1. EP redistribution by sawteeth and kink modes.
The redistribution of EPs by sawteeth is relatively well understood theoretically. EPs with moderate energies are thought to remain attached to reconnecting field lines and behave like thermal particles. They suffer appreciable transport, since the finite $E \times B$ drift tends to attach the particles to the evolving flux surface [381]. However, for high-energy EPs, their large orbits can decouple them from the evolving flux surfaces. Three timescales are important [382]: the crash duration $\tau_{cr}$, the bounce-averaged toroidal precession time $\tau_{\phi}$ and, for passing particles, the timescale of longitudinal motion (period around a perturbed flux surface) $\tau_{\psi}$. Particles experience significant transport when their energy $\mathcal{E}$ is smaller than a critical energy $\mathcal{E}_{crit,st}$; however, because $\tau_{\phi}$, $\tau_{\psi}$, and the orbit size differ for passing and trapped particles, $\mathcal{E}_{crit,st}$ has different values for different orbit types. For trapped particles, $\mathcal{E}_{crit,st}$ is sufficiently low that trapped EPs usually decouple from the flux surfaces and suffer little transport. In contrast, for passing particles, $\mathcal{E}_{crit,st}$ is higher, so passing EPs with $\mathcal{E} \lesssim \mathcal{E}_{crit,st}$ often experience significant radial transport. This selective transport pattern is corroborated in figure 24 which shows a comparison of fast-ion densities calculated using TRANSP/NUBEAM with a sawtooth model and calculated from velocity distribution functions inferred by velocity-space tomography based on FIDA measurements at AUG [222]. Whereas the sawtooth model in TRANSP/NUBEAM ejects EPs at all pitches, the measurements show that the EPs with pitches close to zero are not affected by the sawteeth, as predicted by theory.

In addition to these mechanisms associated with reconnection at the sawtooth crash, resonances with the internal kink that triggers the sawtooth are also
<figure><img src="image_25.png" /><figcaption>**Figure 24.** Comparison of simulated and measured EP densities in the plasma center for various pitch ranges in a sawtoothing plasma at AUG. The full lines are measured EP densities found by velocity-space tomography based on FIDA measurements. EPs with pitches $p > 0.25$ are ejected by the sawtooth, whereas EPs with pitches $|p| < 0.25$ are not. The dotted lines are simulations with TRANSP/NUBEAM with a sawtooth model ejecting particles at all pitches. Reproduced from [222].</figcaption></figure>

possible. However, the roles of trapped and passing particles are reversed: the passing-particle transit frequency is too high to resonate with the internal kink but the trapped-particle precession frequency may resonate, so only trapped particles experience significant transport by this mechanism.

In DT plasmas in TFTR, the core density of alphas with energies between 0.15 and 0.6 MeV significantly decreased during the sawtooth crash [24, 383]. A similar redistribution of energetic ions was also observed on JET [384]. Additionally, neutron spectroscopy showed that the EPs in the 100 keV range generated by ICRF heating in JET were redistributed by sawteeth [385], whereas 1 MeV tritons only experienced minor redistribution at sawtooth crashes [100].

Recent experimental work has focused largely on validating the detailed predictions of our theoretical understanding regarding redistribution of EPs in different regions of EP phase-space as in the AUG example discussed above. The theoretical model has been found to be consistent with the recent measurements of EP redistribution on TEXTOR, DIII-D and AUG [150, 222, 224, 229, 382, 386-389].

The study of EP interactions with sawteeth has also been extended to the low field, tight aspect ratio tokamak regime on NSTX (including NSTX-U) and MAST, testing the models in this new regime. NSTX comparisons between tangentially and radially viewing FIDA and NPA diagnostics show that passing particles experience greater transport [390], as theoretically expected. Modeling of sawtooth crashes on NSTX found that it was necessary to introduce energy selectivity to improve the agreement

between experimentally measured and simulated neutron rates and the EP profiles measured with FIDA. At lower energy, trapped EPs showed the strongest redistribution [391, 392]. The redistribution is consistent with theoretical models which predict that redistribution will be stronger for those trapped and barely trapped ions whose precession periods are longer than the sawtooth crash time [381]. For the NSTX crash time of 40-50 µs, the estimated critical energy was ≈65 keV for nearly co-parallel EPs and down to 25-30 keV for trapped EPs. In contrast to the NSTX results, trapped and passing EPs are found to be equally affected by sawteeth on MAST, based on neutron camera data [393] [394] and FIDA data [395]. These results might be reconciled with theory predictions as the beam injection energy is lower in MAST, so that the EP distribution is at lower energy [341]. Thus, the experimental data support the predictions of redistribution dependence on energy, pitch and sawtooth crash time [381, 396]. Models of EP redistribution have been used to predict strong alpha redistribution by sawteeth in ITER [397]. Sawteeth do not usually occur in stellarators, but they can be induced by NBCD and ECCD. Sawteeth induced this way at the stellarator LHD showed no measurable effect on the EPs [398].

Kink modes are often observed just before sawtooth crashes. It has long been known that kink modes or other low frequency magnetic perturbations can cause losses of EPs. The losses have been explained with a model of random “kicks” to the EP orbits making EP orbits stochastic at magnetic perturbation levels lower than those needed to make the magnetic fields stochastic [361, 362]. There has been substantial work on modeling of the redistribution of EPs during the sub-millisecond reconnection period of the sawtooth crash where the (1/1) kink-tearing mode is growing rapidly. However, many sawteeth are preceded for a long period by a saturated (1/1) internal kink instability. Recent work has focused on modeling EP redistribution by ideal kink modes. A reduced “kick” model for EP transport has been developed [399] and used to study the EP redistribution associated with sawteeth [400, 401] in JET plasmas. In a comparison of the EP redistribution from both a saturated internal kink mode and a sawtooth in a JET discharge, the losses from the saturated internal kink mode were found to be larger or comparable to the losses from the sawteeth [402].

### 5.2.2. Pacing of sawtooth instabilities with EPs.
Sawteeth could affect fusion performance on ITER by redistributing the fusion alphas, helping control impurities in the plasma core, and they are implicated in the triggering of tearing modes which can lead to
performance degradation or disruptions. For these reasons, it is desirable to have a method to control sawteeth in ITER to minimize their impact on plasma performance. ICRF heating has been proposed and used for active feedback sawtooth control on JET [403-406], see section 3. Sawtooth pacing is expected to be an important capability needed for ITER to avoid potential giant sawteeth.

The EP population created by minority heating schemes can either suppress sawteeth (core heating), or trigger sawteeth by heating near the $q=1$ surface. Both on-axis ICRF heating with notches and modulated ICRF heating outside the $q=1$ surface have been used on JET for sawtooth control. As already mentioned, long-term suppression of sawteeth lead to large, typically more deleterious, sawteeth. Sawtooth pacing, keeping the sawtooth period short with smaller sawteeth can minimize the deleterious effects of sawteeth. Such pacing has recently been demonstrated on JET [406]. With on-axis ICRF heating, the sawtooth period is lengthened, but notches in the ICRF power, which reduce the core stabilizing EP population, can trigger sawteeth, reducing the sawtooth period to close to its natural period. This may not be sufficient for ITER where the fast-alpha population may result in an excessively long sawtooth period. Conversely, heating near the $q=1$ radius can trigger sawteeth, but requires dynamic steering of the ICRF heating deposition radius. ICRF power is considered among the actuators for sawtooth control in ITER [31]. Owing to the superconducting nature of the magnet system at ITER, however, dynamic control can be achieved through variations of the ICRH frequency only [407].

### 5.3. Ballooning instabilities

The effects of EP populations on core plasma ballooning instabilities were initially analyzed in the mid 1980s to early 1990s time period prior to the discovery of EP destabilized Alfvén gap instabilities. High-$n$ ballooning instability analysis was a convenient paradigm for the study of EP effects at that time since the ballooning transformation reduced the usual 3D MHD mode equations to a 1D ordinary differential equation, which was well within the computational capabilities of that period. The earliest models either examined FLR effects [408], EP diamagnetic drifts [409], or assumed that the EP drifts were large compared to the frequency of the ballooning mode [410–412]. The general conclusion of these studies was that the EP population had a stabilizing effect on ballooning instabilities; the presence of EPs could increase the beta threshold and open up access between first and second stability regimes. The reactor profile model assumed at that time was that

both plasma and alpha pressure profiles would be peaked near the magnetic axis with the expectation that this region would be the most susceptible to ballooning instabilities. The next step in model development was removal of the assumption that EP drifts and transit frequencies were large relative to the ballooning mode frequency [413–420] and inclusion of thermal ion diamagnetic drifts (resulting in a real frequency $\approx \omega_{*i}/2$ for the ballooning mode). The trapped alpha particle precessional drift frequency could be comparable to the ballooning mode real frequency, resulting in the introduction of wave-particle resonances. These generally had a destabilizing effect and could degrade ballooning mode stability boundaries.

<figure><img src="image_26.png" /><figcaption>**Figure 25.** Marginal stability boundaries for $\alpha_H/\alpha_c = 0.3$, $T_c = 10-40$ keV, $\lambda_p = 0.25$, $b_i = 0.04$, $q = 1.7$. A case with no hot component is also shown ($\alpha_H/\alpha_c = 0$, solid line) Reproduced from [414].</figcaption></figure>

Figure 25 shows an example of this in the form of an $s-\alpha$ ballooning stability plot (normalized shear vs. normalized pressure gradient, $s = rq'/q$, $\alpha = -8\pi q^2 B^{-2} R_0 dp/dr$) for the case of a slowing-down distribution of deeply trapped alpha particles. Here $\alpha_c$ refers to the normalized pressure gradient of the core (thermal) plasma, and $\alpha_H$ refers to the normalized pressure gradient of the fast ion component; $b_i = k_\perp \rho_i^2 / 2$, and $\lambda_p$ is an equilibrium profile factor with $\lambda_p = 0.25$ indicating a parabolic pressure profile. The region to the upper right of these curves is ballooning unstable, while the region to the left is stable. As the electron temperature is increased, the mean energy of the alpha distribution increases, resulting in stronger coupling/destabilization and a blocking off access between first and second stability regimes. While alpha-destabilized ballooning instabilities were of concern during this period, interest rapidly shifted
in the early 1990's to the Alfvén gap modes and fishbone instabilities since the gap modes were more global and could exist independently of plasma-driven instabilities; also there were early observations of the fishbone instabilities and their ability to eject fast ions. Furthermore, since ballooning instabilities required higher β's than experiments were achieving at those times, they evaded direct observation. The ballooning analysis technique was also applied toward the study of high-n Alfvén instabilities [421, 422].

Currently the role of thermal plasma-driven ballooning instabilities in tokamak operation is seen from a somewhat different perspective than at the time EP effects on ballooning modes were studied. Access to interesting high performance advanced tokamak regimes relies on a delicate balance between pressure and current gradient driven instability boundaries [423]. The most likely region for encountering ballooning limits is at the pedestal region near the plasma edge. While this would seem well-removed from the core region where the alpha pressure is dominant, there will be a transport flux of energetic alphas from the core through the edge pedestal region, likely enhanced over classical levels by EP instabilities. Since trapped alpha precessional drifts may resonate with and destabilize ballooning modes in this region, potentially degrading the pedestal region β limit, this area of EP physics deserves reexamination taking into account the newer scenario/profile modeling.

### 5.4. Resistive wall modes

In burning plasmas operated above the no-wall β-limit, e.g. in the ITER steady-state scenario, resistive wall modes (RWMs) are prone to become unstable [424]. The q-profile in this scenario is reversed, and there are no internal transport barriers. Without a surrounding wall, the plasma would be stable against kink modes until $β > β^∞$ where $β^∞$ is the no-wall β-limit. If the plasma was surrounded by an ideally conducting wall, it would remain stable up to a higher critical value $β^b$ throughout the so-called wall-stabilised region. However, real vessel walls have finite resistivity, which leads to loss of wall-stabilisation. Resistive wall modes are discussed comprehensively in chapter 4 of this volume [16].

Several tokamaks have operated in the wall-stabilized region where the interaction between EPs and RWM must be understood to make predictions for ITER and burning plasmas [425–429]. Numerical simulations have shown that damping resulting from resonance with the precession frequencies of thermal ions or EPs could play a role, in addition to sound-wave damping and ion Landau damping [31, 430–434]. Thermal kinetic effects on RWM stability in generalized toroidal geometry were assessed, both

with perturbative and self-consistent treatments of the interaction EPs and the RWM displacement. Both thermal particles [435] and alphas [31] stabilize the RWM above the ideal no-wall limit. Finite orbit width effects of the EPs have been shown to stabilize the RWM, too [436, 437]. Trapped EPs are stabilizing for the RWM [438] but, as discussed in section 7.1, the off-axis fishbone, a type of EPM related to the RWM can be driven unstable when the EP pressure is sufficiently large.

NSTX experiments investigated the effect of rotation on RWM stability. For increasing rotation, RWM stability first improves, then degrades when the mode is in resonance with the precession frequency of the bulk ions, then improves again at higher rotation [439]. Experiments at DIII-D varying the EP distribution have shown a change in the damping of the RWM as measured by the resonant field amplification [440]. The RWMs were observed to become more stable at lower rotation. The stabilisation due to rotation counteracted the resonance between EP precession frequency and the Doppler-shifted mode frequency.

### 5.5. Edge-localized modes

H-modes in tokamaks have steep edge pressure gradients that typically drive ELMs unstable, as described comprehensively in chapters 3 and 5 of this volume [15, 17]. Recent experiments at several tokamaks have shown that ELMs eject the edge EP population to the wall which can form on localized hot spots on the wall [441]. Velocity-space resolved FILD measurements indicate that the EPs are significantly accelerated by the ELM perturbation that cause their losses (figure 14), highlighting the strong interplay between the EP population at the plasma edge and the electromagnetic perturbation developed during an ELM crash [248, 250, 251]. However, it should be noted that a recent work questions the role of ELMs in producing the observed fast-ion acceleration. Analysis of the measured lost ions using neural networks for tomographic reconstructions of the ion velocity-space suggests that the fast-ion acceleration might be observed over a wide range of events and not only correlated with ELMs [244].

Nonlinear 3D kinetic-MHD simulations with the MEGA code indicate that EPs may play a key role in the non-linear spatio-temporal structure of ELMs [442]. The interaction between the EP drift orbits at the edge and the electromagnetic perturbations of ELMs results in wave-particle energy and momentum exchange, influencing the overall structure of ELMs. Although ELMs are initially triggered by steep thermal plasma pressure gradients at the edge, the effects of EP kinetics significantly alter various characteristics of ELMs, such as the
growth rate, crash timing, amplitude, frequency spectrum, and ballooning structure. In the presence of EP effects, a broad radial structure leads to a considerable redistribution of the EPs. MEGA simulations replicate key ELM observations seen in low-collisionality plasmas with high levels of EPs in AUG. The simulations show abrupt and sizable type-I ELM crashes with frequencies up to 250 kHz. Notably, filamentary EP losses observed during ELM crashes exhibit a comparable frequency pattern with dominant frequencies spanning a wide range up to 250 kHz, gradually reducing in amplitude and frequency in the post-ELM crash phase. Preliminary estimates predict a strong interaction between alphas and other EPs with ELMs in ITER (if they are allowed to grow unabated).

## 6. Linear stability of EP-driven modes

In all tokamaks with significant EP populations, electromagnetic fluctuations in a wide frequency regime can be observed, from acoustic up to Alfvénic and ion cyclotron frequencies. In some cases they can be excited by externally applied electromagnetic waves, but typically they are destabilized by resonant interaction with the EP population. Two types of EP-driven modes can be distinguished. The first type are the Alfvén eigenmodes (AEs), which are the normal modes of the MHD equations in toroidal geometry. They reside in gaps in the Alfvén continuum or just outside, and their properties are determined by the bulk plasma. They additionally have kinetically modified branches. The second type are the energetic particle modes (EPMs) which can reside in the Alfvén continuum if the drive can overcome the damping. Their properties are mainly determined by the characteristics of the EPs, e.g. their characteristic orbital frequencies. Often, these two types are connected to each other, i.e. the fluctuations start as AEs and then evolve non-linearly into EPMs.

A comprehensive and predictive description of a burning fusion plasma needs global, nonlinear kinetic models and codes, which we will discuss in section 7. Here, we will focus on linear stability and on the underlying stable or unstable linear mode spectrum which is of crucial importance for the analysis of AE modes for several reasons. First, weakly damped modes are a prerequisite for the excitation by EPs in many experimental situations. Second, measured mode structures of AEs close to marginal stability often agree remarkably well with linear predictions and thus can be used for Alfvén spectroscopy. Third, reduced EP transport models rely on linear mode information, in particular on the damping rate that balances the drive in the saturated state. Here, we review basic concepts of linear theory and summarize recent efforts

applying linear theory to various plasma scenarios of interest. For further discussion, please refer to the recent review papers [27–30, 32, 34] and references therein.

### 6.1. Drive and damping of AEs

The excitation of AEs requires a positive effective growth rate, so the drive must exceed the damping,

$$ \gamma_{\text{eff}} = \gamma_{\text{drive}} - \gamma_{\text{damp}}. \quad (42) $$

The drive can be large when the mode frequency resonates with the gyro-, transit-, bounce- or precession frequencies of EPs (equations 11 and 14), which leads to an efficient energy exchange between the EPs and the mode. Thermal ions in fusion plasma are usually not fast enough to resonate with Alfvén waves, but the birth speed of alphas exceeds the Alfvén speed, and hence wave-particle interaction between the alphas and AEs is possible. Free energy to drive the mode via inverse Landau damping is available when the phase-space distribution has spatial gradients or deviates significantly from a Maxwellian distribution in velocity space, for example by inverted velocity gradients or anisotropy.

The drive of a distribution via inverse Landau damping depends on the EP beta $\beta_f$ and the gradients in the EP distribution function. To assess this drive, we take the energy derivative of the distribution. Since $\mu$ and $\mathcal{E}' = \mathcal{E} - \omega P_\phi/n$ of a resonating particle are constant during the wave-particle interaction for modes in Alfvén frequency range and below (section 2), we take the energy derivative of the distribution at fixed $\mu$ and $\mathcal{E}'$ [34]:

$$ \gamma_{\text{drive}} \propto \beta_f \omega \frac{\partial f}{\partial \mathcal{E}} \Big|_{\mu, \mathcal{E}'} \quad (43) $$

This gives $\frac{\partial f}{\partial \mathcal{E}}|_{\mu, \mathcal{E}'} = \frac{\partial f}{\partial \mathcal{E}}|_{\mu, P_\phi} + \frac{\partial P_\phi}{\partial \mathcal{E}} \frac{\partial f}{\partial P_\phi}|_{\mu, \mathcal{E}}$, so with $\frac{\partial P_\phi}{\partial \mathcal{E}} = \frac{n}{\omega}$, we get the drive

$$ \gamma_{\text{drive}} \propto \beta_f \left( \omega \frac{\partial f}{\partial \mathcal{E}} \Big|_{\mu, P_\phi} + n \frac{\partial f}{\partial P_\phi} \Big|_{\mu, \mathcal{E}} \right). \quad (44) $$

At high frequencies on the order of the cyclotron frequency, $\mu$ is no longer constant but varies with energy, so that $\frac{\partial \mu}{\partial \mathcal{E}} = \frac{N\omega_c}{B_0\omega}$ (see section 2). If the distribution is anisotropic in velocity space, we express this anisotropy as a dependence on the normalized magnetic moment $\Lambda = \mu B_0/\mathcal{E}$ [34]. Then the energy derivative at fixed $P_\phi$ (but no longer fixed $\mu$) becomes $\frac{\partial f}{\partial \mathcal{E}}|_{P_\phi} = \frac{\partial f}{\partial \mathcal{E}}|_{\Lambda, P_\phi} + \frac{\partial \Lambda}{\partial \mathcal{E}} \frac{\partial f}{\partial \Lambda}|_{\mathcal{E}, P_\phi}$ with $\frac{\partial \Lambda}{\partial \mathcal{E}} = -\Lambda/\mathcal{E}$. After insertion into equation (44), we find three derivatives of the distribution function that can potentially drive a mode:

$$ \gamma_{\text{drive}} \propto \beta_f \left( \omega \frac{\partial f}{\partial \mathcal{E}} \Big|_{\Lambda, P_\phi} + n \frac{\partial f}{\partial P_\phi} \Big|_{\Lambda, \mathcal{E}} - \omega \frac{\Lambda}{\mathcal{E}} \frac{\partial f}{\partial \Lambda} \Big|_{\mathcal{E}, P_\phi} \right). \quad (45) $$
The first term shows that drive is obtained from a positive gradient in energy, which appears for velocity distribution function with a bump-on-tail, as was recently studied in JET DT plasmas [41]. Since the drive is proportional to the mode frequency, velocity-space gradients can significantly contribute to driving high-frequency modes. The second term approximately reflects drive from spatial gradients, which is often the dominant drive for intermediate- to low-frequency AEs in present experiments. If we approximate $P_φ ≈ Zeψ ≈ ZerR₀B_θ$, the second term becomes [34]

$$n \frac{\partial f}{\partial P_{\phi}}|_{\Lambda, \varepsilon} \approx \frac{n}{ZeR_{0}B_{\theta}} \frac{\partial f}{\partial r}|_{\Lambda, \varepsilon'} \quad (46)$$

giving an approximation to the drive due to spatial gradients. The third term reflects drive due to anisotropy of the distribution. In ITER, about 10% more co-going alphas then counter-going alphas are expected, an only slightly anisotropic distribution [3]. On the contrary, EP populations from ICRF and NBI heating are highly anisotropic. This last term is, as the first term, proportional to the mode frequency and therefore its contribution to the excitation of high-frequency modes could be significant. Note also that the first and third term do not depend on the toroidal mode number and are hence the only terms that can excite low-frequency axisymmetric ($n = 0$) modes, such as the EP-driven geodesic acoustic mode (EGAM) [443] discussed in sections 6.6.3 and 9.2.

Several damping mechanisms need to be considered. The so-called continuum damping originates from the non-uniform ion densities and magnetic field, leading to a non-uniform Alfvén speed. In the cylindrical limit, radial variations lead typically to a 'closed' shear Alfvén wave continuum (SAWC): For any shear Alfvén wave (SAW) propagating at a certain frequency, there is at least one intersection with the continuum, where the wave resonates. This causes phase mixing of a wave packet with finite $Δω$ at the resonant layer, leading to dispersion and thus strong continuum damping of the mode, of order

$$\gamma_{\text{cont}} \propto \frac{\partial}{\partial r}(k_{\parallel}(r)v_{\text{A}}(r)). \quad (47)$$

However, local extrema or coupling of two poloidal harmonics (breaking various symmetries), lead to radially local or global gaps of the Alfvén continuum where the gradient in equation (47) vanishes and continuum damping is absent. AEs can easily be excited in these gaps, which we will discuss in the next subsection. Other damping mechanisms are thermal ion and thermal electron Landau damping, electron collisional damping and radiative damping. These damping mechanisms require a kinetic description beyond the MHD model since they rely on the resonant interaction of the thermal ions and electrons with

the mode. Whereas for ion Landau damping only thermal ions in the Maxwellian tail contribute to the damping, electron Landau damping has contributions from passing and trapped electrons, depending on the parallel mode structure of the perturbation [444]. Radiative damping, which is often found to be one of the dominant damping mechanisms in the core of a tokamak, requires a global description, since the mode-converted kinetic Alfvén wave [445] carries wave energy away from the gap location via radial propagation and subsequent electron Landau damping (see also section 6.4.1).

Depending on the effective linear growth rate on the collisionality and on various nonlinear saturation mechanisms to be discussed in the following, the waves can reach amplitudes up to $δB/B \sim 10^{-3}$ in present day experiments. The related radial transport, here represented by the change of the canonical toroidal angular momentum $ΔP_φ$ can be understood by using the Hamilton formalism [446] of the wave-particle system leading to $ΔP_φ = \frac{n}{\omega}Δε$ (see section 2). Note that toroidally symmetric modes with $n = 0$ that tap energy only from velocity-space gradients cannot cause radial transport larger than a poloidal gyroradius [447] unless a topological boundary (e.g. passing-trapped) is crossed. (Even for $ΔP_φ = 0$, a change in $v_‖$ implies a change in $Ψ$.)

## 6.2. Overview of AEs

We will start with a brief overview of the zoology of AEs and their basic characteristics, ordering them by their characteristic frequency bands. A typical Alfvén continuum for an ITER scenario and mode number $n = 12$ is illustrated in figure 26. At intermediate frequencies on the order of the Alfvén frequency much less than the ion cyclotron frequency, $ω \sim ω_A = v_A/R \ll ω_{ci}$, the toroidicity-induced, ellipticity-induced and non-circular triangularity induced AEs (TAEs, EAEs, and NAEs) are found in gaps of the shear Alfvén continuum. RSAEs have similar frequencies but sit close to a local maximum of the continuum. A typical ECE spectrogram with TAEs and RSAEs is shown in figure 18 in section 4. At high frequencies close to the ion cyclotron frequency $ω_{ci}$, the global AEs (GAEs) and the compressional AEs (CAEs) are found, which we refer to as high-frequency AE modes. At frequencies much lower than the Alfvén frequency, the β-induced AEs (BAEs) and the β-induced Alfvén acoustic eigenmodes (BAAEs) are found, which we refer to as low-frequency AE modes. These modes depend on pressure effects and can be described within the MHD model. A spectrogram from the TIP prototype installed at DIII-D showing BAEs and RSAEs appears in figure 17 in section 4. Finally, the low-frequency Alfvénic modes (LFAMs)
have frequencies down to the diamagnetic frequencies (not illustrated in figure 26) and require a kinetic description (see discussion below).

Next, we will estimate the location and frequencies of the TAEs, EAEs and NAEs. For low frequencies compared with the ion cyclotron frequency and neglecting kinetic effects, the dispersion relation of shear Alfvén waves in a uniform magnetic field is

$$ \omega = k_{\parallel} v_A, \qquad (48) $$

where $v_A = B/\sqrt{\mu_0\rho_m}$ is the Alfvén velocity. In a toroidal plasma with a safety factor profile $q$, discrete modes with a toroidal mode number $n$ and a poloidal mode number $m$ form. Periodicity demands that the parallel wave number be given by

$$ k_{\parallel} = (n - m/q)/R. \qquad (49) $$

Gaps in the shear Alfvén continuum arise when two counter-propagating waves with the same toroidal mode number and close-by poloidal mode numbers couple. The mode surface locations, frequencies and resonance conditions with EPs can then be found by solving

$$ k_{\parallel,m} + k_{\parallel,m+\Delta_m} = 0 \qquad (50) $$

where $\Delta_m = \{1, 2, 3, ..\}$ and using equations (48) and (49). The first three values of $\Delta_m = \{1, 2, 3, ..\}$ correspond to the TAEs, EAEs, and NAEs. For example, for TAEs we set $\Delta_m = 1$ giving $(n - m/q_{TAE})/R + (n - (m + 1)/q_{TAE})/R = 0$ and hence

$$ q_{TAE} = (m + 1/2)/n, \qquad (51) $$

$$ k_{\parallel,TAE} = 1/(2q_{TAE}R), \qquad (52) $$

$$ \omega_{TAE} = 1/(2q_{TAE}R)v_A. \qquad (53) $$

In terms of the Alfvén frequency $\omega_A = v_A/R$, we get $\omega_{TAE} = \omega_A/(2q_{TAE})$. Inserting $\omega = \omega_{TAE}$ and the drift orbital frequencies $\omega_\phi$ and $\omega_\theta$ (section 2) for the mode surface location $q_{TAE}$ into the resonance condition,

$$ n\omega_\phi + (m + l)\omega_\theta - \omega = 0, \qquad (54) $$

with arbitrary poloidal harmonic integers $l$ yields the resonances $v_\parallel = v_A/(2(m + l) + 1)$. So EPs and TAEs can resonate for $v_\parallel = v_A, v_A/3, v_A/5$ and so on. The mode locations, parallel wavenumbers, frequencies and resonances of EAEs and NAEs can be found in the same way for $\Delta m = 2$ and $\Delta m = 3$. The mode locations and frequencies of TAEs, EAEs and NAEs as well as other AEs discussed in the following are summarized in table 4. The mode width in the minor radius coordinate can be estimated as $\Delta_{AE} = r_{AE}/m$ where $r_{AE}$ is the minor radius coordinate where $q(r_{AE}) = q_{AE}$. If the mode width is comparable to the drift orbit width of a resonant EP, $\Delta_{AE} \simeq \delta$, the power transfer between EPs and AEs is typically largest. Alphas in ITER can fulfill the resonance condition $v_\parallel = v_A$. In recent JET DT experiments, TAEs outside

the core region were found to be driven by alphas [40], as anticipated in DD afterglow experiments [448]. TAEs in the core region were predicted to be strongly damped by the thermal plasma.

Reversed shear AEs (RSAEs) are located at the shear reversal point for $q$-profiles with a local minimum, $q_{min}$, appearing due to non-monotonic current profiles. This leads to a local maximum in the SAWC where the radial derivatives and hence the continuum damping vanish, $\gamma_{cont} \sim 0$ (equation 47). The parallel wave number of the RSAE is $k_\parallel = |n - m/q_{min}|/R$, giving the frequency $\omega_{RSAE} = |n - m/q_{min}|\omega_A$. Alfvén spectroscopy exploits this relation between $q_{min}$ and the RSAE frequency to infer the evolution of the $q$-profile from the measured mode frequencies. Often $q_{min}$ decreases during plasma discharges, leading to an increasing RSAE frequency. When several RSAEs appear at rational $q_{min}$, they are also called “Alfvén cascades” (ACs) [449].

Global AEs (GAEs) and compressional AEs (CAEs) are high-frequency AEs. GAEs are related to extrema, typically minima, of the global SAWC for the minimum Alfvén frequency $\omega \sim \min(\omega_A)$. They are located at $q = m/n$. The GAE frequency is often higher than the TAE frequency, but it can also be in the same frequency range. Formally this branch arises from the combined effects of finite $\omega/\omega_{ci}$ and the equilibrium current density gradient, but the ion cyclotron effects may be dropped in typical fusion plasma regimes [450].

The compressional AEs (CAEs) have frequencies in the range of the ion cyclotron frequency, often higher. Whereas the AEs discussed so far are typically excited by spatial gradients, the CAEs are typically excited by velocity-space gradients, as $\omega$ in the drive term is large. CAEs are thought to be connected with ion cyclotron emission (ICE) and are hence of strong interest for diagnostic purposes in burning plasmas.

Beta-induced AEs (BAEs), which are Alfvénic modes in the continuum gap at $q = m/n$ caused by finite plasma $\beta$, are related to the geodesic curvature [451–455]. The BAE frequency is $\omega_{BAE} = \sqrt{\beta_i(7/4 + T_e/T_i)\omega_A}$. The electrostatic geodesic acoustic mode (GAM) [456] has the same frequency and $\beta$ dependence. The beta-induced Alfvén-acoustic eigenmode (BAAE) gaps [457–459] are induced by the coupling of the SAW and the $m + 1$ sound wave branches. This coupling is reflected by the intrinsically mixed polarisation of BAAEs, neither purely Alfvénic nor purely compressional, which has important consequences for the damping of the modes [460–462]. Higher-order geodesic Alfvén-acoustic couplings and gaps can be found when taking plasma shaping into account, in particular elongation [463]. The low frequency BAAE gap can be computed