# Transport and confinement physics: Chapter 2 of the special issue: on the path to tokamak burning plasma operation

M. Yoshida, R.M. Mcdermott, C. Angioni, Yann Camenen, J. Citrin, M.

Jakubowski, J.W. Hughes, Y. Idomura, P. Mantica, A. Mariani, et al.

► To cite this version:

M. Yoshida, R.M. Mcdermott, C. Angioni, Yann Camenen, J. Citrin, et al.. Transport and confinement physics: Chapter 2 of the special issue: on the path to tokamak burning plasma operation. Nuclear Fusion, 2025, 65 (3), pp.033001. 10.1088/1741-4326/ad8ced. hal-05373162

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# Transport and confinement physics Chapter 2 of the special issue: on the path to tokamak burning plasma operation

To cite this article: M. Yoshida (Chair Transport and Confinement) et al 2025 *Nucl. Fusion* **65** 033001

View the [article online](#) for updates and enhancements.

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# Transport and confinement physics

## Chapter 2 of the special issue: on the path to tokamak burning plasma operation

M. Yoshida (Chair Transport and Confinement)<fnref n="1" />,*, C. Angioni<fnref n="2" />, Y. Camenen<fnref n="9" />, J. Citrin<fnref n="3" />, M. Jakubowski<fnref n="4" />,
J.W. Hughes<fnref n="5" />, Y. Idomura<fnref n="14" />, P. Mantica<fnref n="6" />, A. Mariani<fnref n="6" />, S. Mordijck<fnref n="15" />,
E.J. Paul<fnref n="17" />, T. Tala<fnref n="7" />, G. Verdoolaege<fnref n="8" />, A. Zocco<fnref n="4" />, F.J. Casson<fnref n="10" />,
G. Dif-Pradalier<fnref n="11" />, B. Duval<fnref n="12" />, B.A. Grierson<fnref n="20" />, S.M. Kaye<fnref n="13" />, P. Manas<fnref n="11" />,
M. Maslov<fnref n="10" />, T. Odstrcil<fnref n="20" />, J.E. Rice<fnref n="5" />, L. Schmitz<fnref n="18" />, F. Sciortino<fnref n="2" />,
E.R. Solano<fnref n="19" />, G. Staebler<fnref n="16" />, M. Valovič<fnref n="10" />, E. Wolfrum<fnref n="2" />, J.A. Snipes (co-Chair
Transport and Confinement)<fnref n="13" /> and the Transport and Confinement Groupª

<figure><img src="image_2.png" /></figure>

<footnote n="1">¹ National Institutes for Quantum Science and Technology (QST), Naka, Ibaraki 311-0193, Japan</footnote>

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

<footnote n="3">³ DIFFER—Dutch Institute for Fundamental Energy Research, 5612 AJ Eindhoven, Netherlands</footnote>

<footnote n="4">⁴ Max Planck Institute for Plasma Physics, 17491 Greifswald, Germany</footnote>

<footnote n="5">⁵ MIT Plasma Science and Fusion Center, Cambridge, MA 02139, United States of America</footnote>

<footnote n="6">⁶ Institute for Plasma Science and Technology, CNR, Milano, Italy</footnote>

<footnote n="7">⁷ VTT, PO Box 1000, FI-02044 VTT Espoo, Finland</footnote>

<footnote n="8">⁸ Department of Applied Physics, Ghent University, 9000 Ghent, Belgium</footnote>

<footnote n="9">⁹ CNRS, Aix-Marseille University, PIIM UMR7345, Marseille, France</footnote>

<footnote n="10">¹⁰ UKAEA, Culham Science Centre, Abingdon OX14 3DB, United Kingdom of Great Britain and Northern Ireland</footnote>

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

<footnote n="12">¹² Ecole Polytechnique Fédérale de Lausanne, Swiss Plasma Center, CH-1015 Lausanne, Switzerland</footnote>

<footnote n="13">¹³ Princeton Plasma Physics Laboratory, Princeton University, Princeton, NJ 08543, United States of America</footnote>

<footnote n="14">¹⁴ Japan Atomic Energy Agency, Kashiwa, Chiba 277-0871, Japan</footnote>

<footnote n="15">¹⁵ Department of Physics, College of William & Mary, Williamsburg, VA, United States of America</footnote>

<footnote n="16">¹⁶ Oak Ridge National Laboratory, Oak Ridge, TN, United States of America</footnote>

<footnote n="17">¹⁷ Columbia University, New York, NY10027, United States of America</footnote>

<footnote n="18">¹⁸ University of California Los Angeles, Los Angeles, CA 90095, United States of America</footnote>

<footnote n="19">¹⁹ Laboratorio Nacional de Fusión, CIEMAT, Madrid, Spain</footnote>

<footnote n="20">²⁰ General Atomics, PO Box 85608, San Diego, CA 92186-5608, United States of America</footnote>

<footnote>ª https://www.iter.org/scientists/itpa/itpa-transport-and-confinement-topical-group</footnote>

<footnote>* Author to whom any correspondence should be addressed.</footnote>
E-mail: yoshida.maiko@qst.go.jp

Received 12 January 2024, revised 22 September 2024

Accepted for publication 30 October 2024

Published 19 February 2025

<figure><img src="image_3.png" /></figure>

CrossMark

Abstract

Progress in physics understanding and theoretical model development of plasma transport and confinement (TC) in the ITPA TC Topical Group since the publication of the ITER Physics Basis (IPB) document (Doyle *et al* 2007 *Nucl. Fusion* **47** S18) was summarized focusing on the contributions to ITER and burning plasma prediction and control. This paper provides a general and streamlined overview on the advances that were mainly led by the ITPA TC joint experiments and joint activities for the last 15 years (see JEX/IA table in appendix). This paper starts with the scientific strategy and scope of the ITPA TC Topical group and overall picture of the major progress, followed by the progress of each research field: particle transport, impurity transport, ion and electron thermal turbulent transport, momentum transport, impact of 3D magnetic fields on transport, confinement mode transitions, global confinement, and reduced transport modeling. Cross references with other Topical Groups are given in order to highlight overlapped topics, such as the 3D effect on the plasma transport in the edge and L-H transition physics. The increasing overlap between the topical groups is a reflection of the progress on integrating the known physics into comprehensive models that are better and better able to reproduce the plasma transport. In recent years, such integration has become increasingly prevalent when considering transport from the SOL, through the edge pedestal, and into the plasma core. In the near future, increased collaboration also with the magneto-hydrodynamic and energetic particles community will be important as we approach burning plasma conditions in next-step fusion devices. A summary of remaining challenges and next steps for each research field is given in the Summary section.

This paper is part of the Special Issue: *On the Path to Tokamak Burning Plasma Operation: A collection of papers prepared by the ITPA Topical Physics Groups reviewing progress in the development of the physics basis for burning plasma operation.*

Keywords: plasma transport, plasma confinement, ITPA

(Some figures may appear in colour only in the online journal)

Contents

<table><tr><td>1. Introduction</td><td>3</td></tr><tr><td>2. Particle transport</td><td>5</td></tr><tr><td>3. Impurity transport</td><td>17</td></tr><tr><td>4. Ion and electron thermal turbulent transport</td><td>31</td></tr><tr><td>5. Momentum transport</td><td>42</td></tr><tr><td>6. Impact of 3D fields on transport</td><td>61</td></tr><tr><td>7. Confinement mode transitions</td><td>76</td></tr><tr><td>8. Global confinement</td><td>91</td></tr><tr><td>9. Reduced core transport modeling</td><td>96</td></tr><tr><td>10. Summary</td><td>111</td></tr><tr><td>Appendix</td><td>114</td></tr><tr><td>References</td><td>115</td></tr></table>
# 1. Introduction

M. Yoshida, R.M. McDermott and J. Snipes

Most plasma transport in fusion plasmas (particle, thermal, and momentum transport) is governed by turbulent processes. The turbulent transport changes with variations in the plasma parameters, such as collisionality, gyroradius, the ratio of electron and ion temperature, beta, and magnetic shear. In addition, the global plasma confinement depends not only on the core confinement, but also on the boundary conditions, such as the first wall material and recycling ratio. To improve the predictive capability of ITER plasmas, an understanding is needed of the underlying transport physics together with integrated models that can be applied at ITER relevant conditions. Considerable progress has been made on obtaining systematic data sets that cover ITER relevant regimes by means of mutually complementary and comparable joint experiments. Significant progress on model validation and verification has been made, assisted by the expanded experimental databases, newly developed gyrokinetic databases, and an increase in computational power. Within the ITPA Transport and Confinement (TC) group, the direction of theoretical work is to develop reduced models using the latest gyrokinetic and gyrofluid models and introducing neural network (NN) schemes to substantially reduce computational time. That will support ITER scenario- and plasma control scheme development.

## 1.1. Scope and major goals of TC topical group

The overall scope of the TC topical group is to explore and to develop a fundamental understanding of the TC physics governing plasma performance, and to provide a predictive capability for ITER and burning plasmas in general. This scope includes:

* maintaining the confinement and L-H threshold databases, and augmenting them as necessary,

* developing an understanding of the basic processes controlling plasma particle, thermal, and momentum transport,

* supporting the identification of experiments, inter-machine comparisons and analyses to address critical transport issues, and

* facilitating the validation of physics-based transport models in support of developing a fully predictive transport capability that can be used for integrated scenario modeling.

The tasks of the TC topical group are wide-ranging, covering experiment, theory, and modeling. The group works not only on characterizing TC properties, but also towards developing physics-based models with the aim of using these models to predict performance in future devices. In the future, the topics on which the group will be active will depend on both the immediate needs of ITER and the interests of the group.

The major goal of the TC topical group is to predict the plasma profiles, such as ion and electron density, impurity, temperature, rotation and the safety factor, and plasma confinement performance in ITER. A key task is to identify characteristics of each transport channel in terms of the background parameters and sources and use these observations to validate theoretical models. Another task is to find methods to improve confinement and fusion performance. Response and controllability of electron, ion and impurity densities have also been assessed. Key results from the ITPA TC group since the 2007 Update of the ITER physics basis (IPB) are presented in this Nuclear Fusion Special Issue Contribution. These results are divided into 8 separate papers focusing on: main ion and electron particle transport, impurity ion transport, thermal transport, momentum transport, the impact of 3D fields on transport, changes in confinement regimes, global confinement scalings, and the development of reduced models. The major progress in each of these is introduced briefly below, in section 1.

## 1.2. Major progress from the previous special issues

The progress on understanding main ion and electron particle transport is presented in section 2. Particle transport studies have explored particle transport and fueling effects by neutral beam injection (NBI) in the core region as well as pellet fueling in the peripheral region to describe density profiles over all radii. A multi-machine density peaking database revealed the role of collisionality in the density peaking in H-mode plasmas. Both ion and electron particle transport coefficients were obtained with improved diagnostics and perturbation techniques. Particle transport mechanisms in the presence of the ion temperature gradient (ITG) mode and the trapped electron mode (TEM) have been established by both analytical models and gyrokinetic simulations. Deposition of pellets on the High Field Side (HFS) has been generally reproduced by a neutral gas and plasma shielding model combined with a model for redeposition due to grad B drift of plasmons. The isotope mix ratio was successfully controlled by D pellets in both HD and DT plasmas.

Progress in impurity transport studies is presented in section 3. These studies have covered both light and heavy impurities (helium to tungsten) and identified critical elements of transport characteristics for the prediction and control of the impurity concentration in ITER. Systematic analyses of a database of impurity density profiles and theoretical investigations have identified that turbulent transport is often dominant for light impurities in present day machines. In contrast, neoclassical transport can be dominant for heavy impurities. However, this is not necessarily the case in ITER, where turbulence is expected to play a stronger role, also for high-Z impurity transport. A clear dependence of the light impurity diffusion coefficients on the electron to ion heat flux ratio was documented, showing the importance of off-diagonal components.
High-Z impurity accumulation is affected by the tempera-
ure gradient, collisionality and toroidal rotation, and is well
described by neoclassical transport theory. The importance of
poloidal asymmetries was also recognized by the 2D meas-
urements, local and global non-linear gyrokinetic simulations.
Avoidance of impurity accumulation in the core by central
heating has been well established in multiple devices, and the
mechanisms were elaborated on with turbulence and neoclas-
sical effects.

Thermal transport studies, see section 4 made great pro-
gress on understanding the mechanisms of ion and electron
turbulent transport focusing on critical gradients and profile
stiffness. Mechanisms for ITG stabilization have been doc-
umented by utilizing heating scans in multi-machine exper-
iments and high fidelity non-linear gyrokinetic codes. The
reduction of ion stiffness with increasing power has been
explained with the nonlinear electromagnetic stabilization
effects in addition to $E \times B$ shear effects. Quasi-linear model-
ing captures well the ion and electron critical gradients (lin-
ear physics) but do not include in the saturation rule non-
linear electromagnetic effects which are particularly important
in advanced scenarios. Validation of Trapped Gyro Landau
Fluid (TGLF) ion and electron stiffness against gyro-kinetics
has led to improved versions of TGLF. Discrepancies between
flux-driven gyrokinetics and local or quasilinear approaches
have been documented near marginal turbulence stability and
stress the importance of nonlocal phenomena and secondary
flow organization in both core and edge. The role of elec-
tron scale turbulence and multi-scale interactions on elec-
tron heat transport has been examined to explain measured
high electron stiffness in tokamak devices. Multi-scale flux-
tube simulations have enabled the assessment, from a theor-
etical perspective, of the electron heat flux in the presence of
cross-scale interactions between ion and electron scale turbu-
lence. In general, however, predicted electron scale transport
is lower than seen in experiment. Further development is then
needed.

The theory of momentum transport, discussed in section 5,
has developed significantly in the last two decades and now
includes a validated theory of momentum convection and the
identification of residual stress terms likely to be relevant
in future burning plasma machines. These mechanisms will
play an out-sized role in such devices, compared to present
day machines, on account of the low external torque com-
pared to plasma inertia. Characteristics of diffusive, convect-
ive, and residual stress driven momentum fluxes were explored
in torque perturbation experiments in multiple machines and
with theoretical approaches. Great advances have also been
made in gyrokinetic modeling including global modeling,
which has succeeded in reproducing measured intrinsic rota-
tion gradients. These developments have greatly improved our
understanding of turbulent diffusive, convective, and residual
stress momentum fluxes, contributing to an improved predict-
ive capability of toroidal rotation profiles in existing and future
devices.

The impact of non-axisymmetric fields (3D fields) on trans-
port is discussed in section 6. There, the influence of 3D fields
on neoclassical transport, turbulent transport, and the L-H

transition have been investigated in tokamaks with external
magnetic perturbations and in stellarators, featuring intrins-
ically 3D magnetic geometries. Neoclassical physics can
strongly influence toroidal rotation profiles through the neo-
classical toroidal viscosity (NTV) torque and density profiles
(density pump-out) through collisional particle transport for
both main- and impurity- species. The 3D magnetic fields
can impact turbulent transport by, for example, changing the
plasma profiles, the radial electric field, and inducing the form-
ation of magnetic islands. Another key physics observation is
the impact of 3D fields on the L-H transition conditions. The
3D fields can increase the power threshold for the L-H trans-
ition and change the pre-LH-transition plasma conditions. The
mechanisms of those observations can be described by tur-
bulence and zonal flow (ZF) characteristics, and neoclassical
effects, respectively.

The progress on understanding transitions between confine-
ment regimes is presented in section 7. These studies have
advanced the understanding of global threshold conditions
for confinement mode transitions (L to H-mode, and L to I-
mode), identified potential critical requirements on edge para-
meters to the transitions, and dynamic interactions between
edge flows and turbulence. The former scalings for the L-H
power threshold were replaced by a new one using a subset
of discharges most applicable to ITER operations from the
larger ITPA threshold database [1]. Potential critical common
influences on the H-mode threshold, such as isotope mixture
and boundary configurations have been found, which allows us
to improve the scaling. Numerical modeling provided insight
into the threshold conditions, such as the density branches
and the critical pressure gradient to H- mode access. The L-
to I-mode transitions have been identified in many tokamaks,
which allowed the examination of the parameter dependencies
and the role of ion $B \times \nabla B$ drift direction.

Global confinement scalings can be a critical tool for the
design of future machines and one that requires updating as
improved data and information becomes available. The TC
Group's critical role in this task is discussed in section 8.
Global confinement studies have expanded the confinement
database and developed ITER-relevant scalings for the global
energy confinement time. The database consists of data from
nineteen tokamaks, is extended to heating power, plasma cur-
rent, density and inverse aspect ratio. The data from metal-
lic wall conditions were also added. This allowed the exten-
sion of the range of the normalized gyroradius, toroidal beta,
and safety factor. The most ITER-relevant scaling has been
obtained with the ITER like ELMy H-mode restrictions and
robust statistics analysis of the database. The new scaling
shows weaker dependence on density, power loss, and major
radius comparing with IPB98(y,2). The influence of a metallic
wall on the scaling was found. Those modifications of para-
meter dependence result in the lower predicted confinement
time in the ITER baseline scenario.

Finally, significant progress has been made on the
development of reduced transport modeling capability. These
developments are discussed in section 9, which details the sig-
nificant efforts in developing, validating, and applying quasi-
linear turbulent transport models. Fast reduced turbulence
models have been coupled to integrated modeling frame-
works to calculate temporal evolution of core plasma pro-
files in reasonable computational time. Validations of the
reduced models with gyrokinetic turbulence simulations and
experimental data were conducted with respect to multiple
aspects. Nonlinear physics was captured by updating satur-
ation rules, and the predictive capability from the core to
the pedestal- forming-region in L-mode plasmas was signi-
ficantly improved. Benchmarks between reduced transport
models have been made for ITER baseline and PFPO-1 scen-
arios, showing reasonable agreement between the models.
Faster surrogate transport modeling has been constructed with
machine learning techniques to accelerate high-fidelity non-
linear simulations.

The TC chapter reflects the research that has been per-
formed mainly within the TC ITPA joint activities and exper-
iments. Note that this is not a comprehensive review of
all the research carried out in the ITPA TC topical group.
The structure of the sections is generally based on the
experimental observations, the theoretical understanding and
predictive modeling capabilities of the observations, the theory
validations of each research area, and the areas requiring fur-
ther work. There is also one section exclusively dedicated to
the development of theoretical models.

The progress on R&D described in this chapter 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 (see ITER Research Plan for 2016
staged approach baseline ITR-2018-3 (provisional version)
and ITR-2024-5 (final version) for details). The new ITER
baseline 2024 currently under elaboration proposes modific-
ations to several of these features (e.g. first wall material,
heating and current drive mix, etc), as introduced in ‘ITER
Progress on ITER manufacturing, construction, commission-
ing and plans, P. Barabaschi, et al Proc. 29th IAEA Fusion
Energy Conference, London, UK, 2023. Paper OV/1-3’, which
will require additional R&D beyond that described in this
chapter.

## 2. Particle transport

T. Tala, M. Valovič and C. Angioni

### 2.1. Introduction

There has been a lot of progress in the area of electron and
main ion particle transport including the physics of pellets in
the last 15 years, both within the ITPA TC working group
and within the broader fusion community. In this overview,
we emphasize the work performed by the ITPA TC working
group, but also review other key developments within that time
frame. Both the particle transport theory and experiments have
gained improved understanding of various phenomena. Still,
electron and ion particle transport, partly due to its complex-
ity in tokamaks, has received less attention than heat transport.
Although it is very well known and recognized that the shape

of the density profile has a significant influence, for example
on fusion performance (P<sub>fus</sub> ~ n<sup>2</sup>), and on impurity transport,
it is not understood to the same level of detail and accuracy as
heat transport [2]. Moreover, the models tend to predict better
agreement in heat transport channels than for particle trans-
port. Reaching very high density is only possible with peaked
density profiles, indicating that particle transport and density
peaking studies indeed deserve a lot of attention. Peaked dens-
ity profile can, on the other hand, give rise to impurity peaking
and tungsten accumulation. Impurity particle transport is dis-
cussed in section 3 in this special issue.

In the previous IPB [3] the existence of two different kinds
of particle sources (central and edge sources), and the signif-
cant effect of convection on the density profile, in addition
to diffusion, were reported. An improved understanding of
particle transport was emerging by applying experimental and
analytical methods together with theory-based modeling. An
anomalous particle pinch at low collisionality had been seen
in several tokamaks [4, 5] and understood in the framework of
the ITG/TEM transport theory [6, 7]. Single machine density
peaking databases had been reported [4, 6, 8] and the ITER
design assumption of having a flat density had been being ser-
iously questioned. Pellet studies were not reported in the IPB
in the particle transport section, and the transport of the main
particles and the impurities were mixed within a single section.

Since the previous IPB, the theory of turbulent particle
transport has received particular consideration. The progress
obtained in that period has already been reviewed in great
detail in [9]. A detailed description of the transport mechan-
isms for both trapped and passing particles in the presence
of ITG and TEM instabilities can be found in [10]. On the
experimental side, further development from single machine
density peaking databases and single machine datasets [6,
8, 11] led to the creation of multi-machine density peak-
ing databases [12, 13]. Thanks to the improved diagnostics,
such as High Resolution Thomson Scattering (HRTS) and the
profile reflectometry, more accurate measurements of local
electron particle transport coefficients have been performed,
for example in [14, 15]. The ion particle transport was also
experimentally evaluated for the first time [16] and the res-
ults explained in theory and modeling in [17]. Mechanisms of
electron and ion particle transport could be different, but the
radial fluxes are the same due to ambipolarity condition. The
role of the NBI core particle source was found to depend on the
plasma transport and turbulence level [18–20]. The results on
the core transport coefficients and particle source are presen-
ted in this chapter, while the progress in the edge particle
source due to neutral fueling performed by the ITPA PEP
group is described in the pedestal chapter [21]. Major progress
has been achieved in pellet fueling in the last 15 years, and
section 2.4 in this paper has its own short introduction to this
topic.

This section on particle transport in this review paper
subsection 2) is organized as follows. The theory development
is described in section 2.2. It describes the turbulent particle
transport mechanisms including both trapped and passing
particles in a thorough manner. It also describes the influence
of electromagnetic effects on particle transport, the relation
between the ion particle and the electron particle transport, particle transport in plasmas with different hydrogen isotopes and the role of rotational $E \times B$ shear in particle transport. Various experimental particle transport studies are presented in section 2.3, which focuses on particle transport database efforts combined with dedicated experiments to study density peaking for the electron particle transport channel. It also describes experiments on quantifying the core particle source. The local, core electron particle coefficients have been measured on several devices. Furthermore, the faster time scales of the ion particle transport with respect to the electron particle transport observed in a mixed deuterium-hydrogen plasma in ITG dominated plasmas are illustrated in section 2.3. Section 2.4 is devoted to showing the progress in various pellet fueling issues. These include pellet deposition, pellet ablation, plasmoid drift, and many aspects related to post-pellet transport phenomena. Finally, a summary is given in section 2.5 together with a future outlook on open issues.

## 2.2. Particle transport theory development

The theory of turbulent particle transport has received particular consideration during the first decade of this century. The progress obtained in that period is already reviewed in [9]. This reference reviews the physical processes producing electron particle transport in the core of tokamak plasmas starting from the gyrokinetic equation. Furthermore, the experimental observations on particle transport in tokamaks is reviewed and the agreement with the theoretical predictions is discussed. A direct link between macroscopic transport properties and the character of microscopic turbulence is demonstrated. This confirms the paradigm of microinstabilities and turbulence as the main cause of particle transport. A detailed description of the transport mechanisms for both trapped and passing particles in the presence of ITG and TEM instabilities can be specifically found in [10]. This paper reviews the particle transport provided by a balance of outward diffusion and inward convection, driven by either neoclassical or turbulent mechanisms. A detailed quasilinear gyrokinetic analysis of turbulent particle transport including the analytical and numerical analysis show how the normalized density gradient can strongly vary between an ITG dominated and a TEM dominated turbulence regime.

More recently, specific interest has arisen on the impact of electromagnetic effects [22], the relation between ion particle and electron transport [17], particle transport in plasmas with different hydrogen isotopes [23, 24], and the role of rotational $E \times B$ shear [25].

Here below we recall the main aspects of turbulent particle transport. There are two main elements which play critical roles in determining the nature (diffusive and convective) and the properties of particle transport. The first element comes from the general property of gyrokinetic turbulent particle transport of being ambipolar, a condition which is intrinsically fulfilled in whatever magnetic geometry and for both electrostatic and electromagnetic transport [26]. This directly follows from the nature of the turbulent transport

mechanisms and the quasi-neutrality condition. The second element is connected with the properties of the gyro-center drifts in connection with the toroidal magnetic confinement geometry.

From ambipolarity, it derives that in a two species plasma, with electrons and ions, if one of the two particle species has an adiabatic response, that is, density fluctuations which are completely in phase with the electrostatic potential fluctuations, then its particle flux is zero and, therefore, also the particle flux of the other species is zero. This can be seen as a direct consequence of the solution of the dispersion relation, which is determined by quasi-neutrality of the turbulent fluctuations. Historically, the first studies on turbulent transport were dedicated to the ITG instability, starting from the limit of adiabatic electrons, that is, the $\eta_i$ mode. In this limit, the particle flux is zero. A non-zero particle flux is produced in the moment that any effect breaks the adiabatic response of the electrons. This also explains why, historically, turbulent particle transport has been predominantly identified with electron particle transport. The first and most direct mechanism that breaks the adiabatic response of the electrons is trapping. The presence of trapped electrons produces finite particle transport, even when the remaining passing electrons keep an adiabatic response, typical of the electrostatic collisionless limit. However, mechanisms connected with finite collisionality and electromagnetic fluctuations impact the response of the passing electrons, leading to particle transport also from the side of passing electrons. The detailed discussions on these mechanisms can be found in [9, 10, 27].

Considering now specifically trapped electrons in the collisionless limit the characteristics of their transport can be understood through the examination of the properties of the drifts, the $E \times B$ drift and the curvature and $\nabla B$ drifts, which determine their gyro-averaged motion. We consider first the $E \times B$ drift arising from the fluctuating electrostatic potential. The essential aspect of this drift in a toroidal magnetic configuration is that the corresponding $E \times B$ flow is not divergence free, as it would be in the presence of an homogeneous magnetic field with straight field lines. The compression, that is the divergence, of the $E \times B$ velocity, which is caused by the fluctuating electrostatic potential, produces a pure, usually inward directed (with conventional monotonic safety factor profile), convective contribution to the transport, due to both the magnetic field line curvature and the magnetic field strength gradient.

In order to more easily clarify the origin of the various main contributions to the turbulent particle transport, we report here below the expression of the linearized $\delta f$ gyrokinetic equation. The non-adiabatic part $g_k$ of the perturbed electron distribution function $f_k = g_k + F_M J_0 \hat{\phi}$, for a single $k_y$ binormal wave number in Fourier space, is solution of the equation

$$ (\omega - \omega_{Gk} + i\nu_{ei}) \hat{g}_k = \left\{ \omega_{Dk} \left[ \frac{R}{L_n} + \left( \frac{E}{T_e} - \frac{3}{2} \right) \frac{R}{L_{Te}} \right] - \omega \right\} \times F_M J_0(k_{\perp} \rho_s) \hat{\phi}_k \quad (1) $$

where a simple Krook collision operator has been included, to describe electron-ion collisions, with a collision frequency
$\nu_{ei}$. On the left hand side, the motion of the gyrocenters is described by the total frequency $\omega_{Gk} = k_{\parallel}v_{\parallel} + \omega_{dk}$, where the derivative along the field line has been formally replaced by a parallel wave number $k_{\parallel}$. The perpendicular drift frequency arising from the $\nabla B$ and curvature drifts is $\omega_{dk} = \omega_{Dk} [(v_{\parallel}^2 + v_{\perp}^2)/v_{th}^2][\cos\theta + (s\theta - \alpha \sin\theta)\sin\theta]$, with $\omega_{Dk} = k_y \rho_s c_s / R$ and $\theta$ the extended ballooning angle, $s$ the magnetic shear. At the right hand side, the normalized logarithmic gradients of the equilibrium density and temperature $R/L_n = -Rdn/dr/n$ and $R/L_{Te} = -RdT_e/dr/T_e$ stem from the radial derivative of the equilibrium Maxwellian distribution $F_M$. $R$ is the major radius at the center of the magnetic flux surface, $E$ the kinetic energy of the particle, $c_s = \sqrt{Te/m_i}$ the sound speed,

$\rho_s = c_s/\Omega_{ci}$ the ion Larmor radius computed with the sound speed, with $\Omega_{ci}$ the ion cyclotron frequency. Finally, the fluctuating electrostatic potential has been normalized as usual $\hat{\phi}_k = e\phi_k/T_e$.

The flux surface averaged quasi-linear particle flux $\Gamma_{QL}$ produced by the fluctuating $E \times B$ drift $\tilde{v}_{E \times B}$ can be computed as $\langle \tilde{n}_e \tilde{v}_{E \times B} \rangle$, where the brackets provide the flux surface average. The linear phase shift between density and electrostatic potential fluctuations can be computed by formally solving equation (1), with the left hand side of the equation appearing at the denominator of the formal expression of $g_k$. From this, the following expression for the quasi-linear particle flux is obtained

$$ \Gamma_{QL} = \sum_k \frac{k_y c_s^2}{\Omega_{ci}} \left\langle \int d^3 v F_M \times \frac{(\hat{\gamma}_k + \hat{v}_k) [R/L_n + (E/T_e - 3/2)R/L_{Te}] - (\hat{\gamma}_k \hat{\omega}_{Gk} - \hat{\omega}_{rk} \hat{v}_k)}{(\hat{\omega}_{rk} + \hat{\omega}_{Gk})^2 + (\hat{\gamma}_k + \hat{v}_k)^2} J_0(k_{\perp} \rho_s)^2 |\hat{\phi}_k|^2 \right\rangle, \quad (2) $$

where $\gamma_k$ and $\omega_{rk}$ are the growth rate and real frequency of the unstable linear mode and all frequencies have been normalized to the fluid perpendicular drift frequency $\omega_{Dk} = k_y \rho_s c_s / R$. The sign convention is that a positive value for the real frequency $\hat{\omega}_{rk}$ identifies a mode propagating in the ion diamagnetic direction, that is, an ITG mode for instance. The contribution from the curvature and $\nabla B$ drifts, which has been previously introduced as main example of convection for the trapped electrons, can be identified in the last term of the numerator at the right-hand-side (RHS) of equation (2). The effect is reduced by the collision term, producing the well known reduction of the inward convection and of the density peaking with increasing collisionality in ITG turbulence [6, 28].

The radial gradient of the unperturbed distribution function, that is of the background Maxwellian, which appears on the RHS of equation (1), with both the density and temperature gradient terms, provides the additional contributions to the particle transport produced by the fluctuating $E \times B$ velocity. These terms can be identified in the first part of the numerator at the RHS of equation (2).

The density gradient term is responsible for the usual 'diagonal' diffusion, which is regularly outward directed. The temperature gradient term, combined with the energy dependent factor $(E - 3/2)$ coming from the radial gradient of the Maxwellian distribution, produces a more subtle and critically important additional component to the transport, which is usually called 'thermo-diffusion'. It is important to recall at this point that the curvature and $\nabla B$ drifts appear on the left-hand-side of the $\delta f$ gyrokinetic equation (1), and thereby are also present in the resonant denominator of the kinetic integral which provides the electron density perturbation and the particle flux, equation (2). Because these drifts depend on the parallel and perpendicular velocities of the particles,

the kinetic integral connected with the temperature gradient does not average to zero, as it would do in case of an energy independent drift, due to the Maxwellian factor $(E - 3/2)$. Because the curvature and $\nabla B$ drifts are energy dependent, the kinetic integral connected with the temperature gradient does not average to zero and produces a finite contribution to the flux, whose sign is determined by the location of the resonance in the resonant denominator, that is, on the value and sign of the real frequency of the unstable mode. This property directly highlights an important feature of the thermo-diffusive particle transport, namely, that its strength and direction change depending on the frequency of the mode. In particular, it is directed inward for modes propagating in the ion drift direction (ITG modes) and increases in magnitude for mode frequencies which approach the zero or are small in the electron drift direction, whereas it is directed outward for modes with large frequencies in the electron drift direction (typical of TEM). More complete physics descriptions of the physics mechanisms which are described here and produce the off-diagonal components of the particle flux can be found here [10, 27, 29].

The previous arguments also clearly show that different transport levels can be obtained for electrons and ion particles, even though the total particle fluxes remain ambipolar [17]. This can be directly understood considering, for instance, the conditions of ITG modes. With ITG modes, the mode frequency propagates in the ion drift direction, and therefore it is balanced by the ion drift frequency in the resonant denominator of the kinetic integral not only of the ion heat, but also of the ion particle transport, producing large levels of transport. In contrast, in the kinetic integral of the electron particle transport (equation (2)), the ITG mode frequency is opposite in sign with respect to the electron drift frequency and, therefore,
no compensation takes place in the denominator, leading to lower levels of electron heat and, specifically, electron particle transport. Eventually, the electron and ion particle fluxes will be exactly the same, that is, ambipolarity is precisely fulfilled, because mode frequency and growth rates are the eigenvalues which result from the solution of the dispersion relation, which is in fact determined by the quasi-neutrality condition. However, with ITG modes, the ambipolarity condition is obtained with electron diffusive and convective components which are both relatively small as compared to the ion diffusive and convective components, which are both large and reciprocally more strongly balancing [17]. An example of this behavior is presented in figure 1. The ion to electron diffusivity ratio is small for low value of $R/L_{Ti}$ and dominant TEM turbulence and becomes large at $R/L_{Ti} \ge 6$ in correspondence to dominant ITG turbulence. In the bottom plot for figure 1, the ratio of the total convection velocities is shown to follow a similar behavior as the ratio of the diffusivities, presented in the top plot. It is interesting, however, to note that the ratio of the thermodiffusion terms feature a weaker variation than that of the compressibility terms, consistent with the analytical derivation also presented in [17]. Analogous resonance effects also determine the size of the impurity diffusion and convection coefficients with respect to the heat conductivities when moving from TEM to ITG turbulence [31, 32].

The fact that in the $E \times B$ particle transport the main contribution to the electron particle flux is produced by trapped electrons implies that collisions have a large impact. By affecting the trapped particle orbits and reducing the effective fraction of trapped particles, collisions also reduce the particle transport. In particular, since at sufficiently low density gradients in ITG turbulence the trapped electrons are transported inward, the impact of collisions is that of reducing the inward convection, that is the pinch, eventually reducing the density gradient which fulfills the condition of zero particle flux. Moreover, since the collision frequency is scale independent, whereas the characteristic time of the perpendicular drift increases with decreasing binormal wave number, the effect of collisions is stronger at large scales and weaker at small scales. This implies that small scales are less affected by collisions as compared to large scales, and the wave number spectrum can be composed by the balance of positive (at large wave lengths) and negative (at small wave lengths) contributions [28].

This also has some implications for the requirements in terms of resolution in numerical simulations of the turbulent transport, which have to cover both small and large scales sufficiently accurately. Along the same line, the electron collision frequency is practically independent of the mass of the main ions, while the characteristic drift frequency connected with the perpendicular drift motion increases inversely proportional to the square root of the main ion mass, as does the thermal velocity. Therefore, for otherwise the same parameters, the impact of collisions is stronger in a plasma with heavier main ion mass and smaller main ion thermal velocity, and the particle pinch decreases with increasing mass of the hydrogen isotope. This leads to the theoretical result that the density gradient at zero particle flux is larger in hydrogen, and progressively decreases in deuterium and tritium plasmas, with

<figure><img src="image_4.png" /><figcaption>**Figure 1.** (top frame) Ratio of ion to electron particle diffusivities versus $R/L_{Ti}$ (other parameters from the GA standard case, see table 1 in [17]). The red circles are for QuaLiKiz [29] results and the blue stars for gyrokinetic GKW [30] nonlinear results. When the dominant mode moves from TEM (low $R/L_{Ti}$) to ITG ($R/L_{Ti} \ge 6$), the ion particle diffusivity becomes large with respect to the electron particle diffusivity. (bottom frame) Ratio of ion to electron convective velocities, versus $R/L_{Ti}$ as in top frame. Magenta circles: ratio of the total convective velocities. Blue dashed lines: ratio of the compressibility components. Green full lines: ratio of the thermodiffusion components. Reproduced from [17]. © EURATOM 2018. All rights reserved.</figcaption></figure>

a difference among isotopes which increases with increasing collisionality [23]. The experimental results of isotope scaling of particle transport are discussed in section 2.3.3.

Recently, nonlinear gyrokinetic simulations have been performed to also study the impact of the $E \times B$ shearing rate and the parallel velocity gradient (PVG) [25]. The combined effects of PVG and $E \times B$ shearing modify the spectral shapes of the particle flux diffusive and convective components, but have only limited impact on the total particle to heat flux ratio. In contrast, when the $E \times B$ shearing rate is considered alone, this is shown to significantly affect both the spectra and the total particle to heat flux ratio. The analysis shows that in combination with the usual reduction of the amplitude of the large scales in the spectrum produced by the $E \times B$ shear, the inward component of the flux is increased at intermediate scales. The
<figure><img src="image_5.png" /><figcaption>**Figure 2.** Comparison between ASDEX Upgrade predictions and ITER/DEMO-like predictions. The density peaking is plotted as a function of a parameter which is proportional to the plasma collisionality, cyan stars are ASDEX Upgrade experimental values, black open circles are TGLF-SAT1 results including the effect of the particle source from neutral beam injection, the black open diamonds with dashed line are the corresponding modeling results in which the particle source, produced by NBI, is removed ('no source' in the figure), and finally red stars and blue dotted circles are the ITER and DEMO predictions with and without the inclusion of electromagnetic effects. Reproduced from [19]. © 2019 EURATOM All rights reserved.</figcaption></figure>

analysis shows that this can be ascribed to an increased effect of inward thermo-diffusion in the ITG turbulence simulations [25]. The impact of $E \times B$ shearing also reduces the overall level of the turbulent transport, also reducing the diffusion, which can make the impact of the particle source relatively more important.

As initially mentioned, also passing electrons can contrib-
ute to particle transport when physical effects prevent their
response from being adiabatic. A particularly important case
is that of the electromagnetic effects induced by finite plasma
pressure [22]. In the presence of perpendicular magnetic field
fluctuations, the particle flux produced by passing electrons
is still dominated by the $E \times B$ transport and is consistently
shown to be directed outwards by both an analytical model
and gyrokinetic simulations. The consequent reduction of the
predicted density peaking with increasing plasma pressure is
only partly mitigated by the consistent inclusion of the total
pressure term in the expression of the curvature drift [22]. This
effect has non-negligible implications on the predictions of the
density peaking in a reactor when electromagnetic effects are
included [19]. In figure 2 (reproduced from figure 11 of [19]),
the peaking factor of the ITER and DEMO density profiles
predicted by the ASTRA transport code with the TGLF-SAT1
[33] transport model are shown as a function of the collision-
ality parameter $Rn_e/T_e^2$. This simple ratio, with $R$ in meters,
$n_e$ in $10^{19} \text{ m}^{-3}$ and $T_e$ in (keV) is proportional to the effect-
ive collisionality, which was introduced in [6] to describe the
density peaking. This effective collisionality $\nu_{\text{eff}}$ is defined
as the ratio between the electron-ion collision frequency $\nu_{\text{ei}}$

and a characteristic frequency for ion Larmor radius scale
micro-instabilities, which can be approximately parameter-
ized as $k_y \rho_i c_s / (RL_T)^{0.5}$, with $k_y \rho_i$ the normalized binormal
wave number, $c_s$ the sound speed and $L_T$ the scale length of
the temperature profile. A simple manipulation, adopting typ-
ical values for $k_y \rho_i$ and $L_T$, shows that this parameter can be
expressed in the simple form $\nu_{\text{eff}} = 0.1 Z_{\text{eff}} R n_e / T_e^2$. In figure 2,
the reduction of the peaking factor for ITER and DEMO pro-
duced by the inclusion of electromagnetic effects is illustrated.
The figure also shows a comparison of ASTRA/TGLF-SAT1
predictions with ASDEX Upgrade experimental results, which
were obtained in discharges that reproduce a set of reactor rel-
evant parameters [19]. In particular, these experiments aimed
at reproducing the low core particle source impact in com-
parison to the intrinsic role of transport, the low momentum
input, the ratio of electron to ion heat flux and the heat flux
in gyro-Bohm units. The resulting plasma conditions had
however lower $\beta_N$ and higher collisionality in comparison to
the expected situation of a reactor [19]. The good perform-
ance of the TGLF-SAT1 transport model as well as the almost
negligible impact of the particle source in these reactor rel-
evant conditions are demonstrated, together with the favor-
able scaling of increasing density peaking with decreasing
collisionality, confirming the predicted peaked density pro-
files in low collisionality reactor plasmas. The increase of
density peaking with decreasing collisionality saturates at low
collisionality, when the collisionless limit is reached. This
property, in combination with the reduced electron temperat-
ure logarithmic gradients of the ITER and DEMO predicted
plasmas as compared to the ASDEX Upgrade low collision-
ality low $\beta$ plasmas in figure 2, explains why the DEMO and
ITER predictions do not feature a stronger peaking than the
low collisionality ASDEX Upgrade plasmas, despite the lower
collisionality.

2.3. *Experimental core particle transport studies*

Section 2.3 reviews the recent progress in experimental
particle transport work tightly coupled with theory and model
validation, concentrating on the work within the ITPA TC
working group. It focuses only on the core electron and ion
particle transport aspects, leaving the edge particle transport
in the [21] chapter and impurity transport in section 3 in this
paper. Furthermore, the 3D effects on the particle transport,
such as the density pump-out due to RMP coils, is discussed
in section 6.

2.3.1. *Empirical scaling of electron density peaking and the role particle sources.*
Two approaches have been employed to obtain further understanding of density peaking and its dependencies on various plasma parameters: regression over large databases and analysis of selected discharges close to ITER parameters. Special attention has been devoted firstly to confirm the role of collisionality in density peaking and secondly, to quantifying the role of the core NBI fueling source in contributing to density peaking in various conditions.
Several studies using the regression method over a large database using data from individual machines have been reported [4, 6–8]. These single machine density peaking scaling studies indicated that collisionality (often in these papers estimated as $\nu_{\text{eff}}$) has statistically the strongest correlation with density peaking while the NBI particle source, temperature ratio, magnetic shear and Greenwald fraction $n_G$ have somewhat smaller, but still significant correlation with the density peaking. The need to eliminate or reduce some of the parameters in the density peaking scaling led to creating multi-machine density peaking databases within the ITPA TC working group [12, 13]. In the JET-AUG combined density peaking scaling database [12], $\nu_{\text{eff}}$ was found to be statistically the most relevant and important parameter in multi-variable regressions to describe density peaking. The contribution of the beam fueling alone was found insufficient to describe the observed variation of density peaking. The multi-machine scaling showed that the observed density peaking can only be explained by the existence of an anomalous inward pinch in low collisionality H-mode plasmas. This experimental multi-machine scaling on the role of collisionality in contributing to density peaking is consistent with the particle transport theory described in section 2.2. The derived scaling for density peaking, which includes $\nu_{\text{eff}}$ in the regression variables, predicts a peaked density profile for ITER. The estimated peaking factor is of the order of $n_e(\rho_{\text{pol}} = 0.2)/\langle n_e \rangle = 1.45$ where $\langle n_e \rangle$ is the volume averaged density. In JET-AUG density peaking database, density peaking was also strongly correlated with $n_e/n_G$, the ratio of the plasma density to the Greenwald density limit. Based only on two machines, it was not possible to eliminate the latter parameter as one of the most important scaling variables. Without knowing the relative importance between $\nu_{\text{eff}}$ and $n_e/n_G$, it is not possible to predict unambiguously density peaking for ITER. The addition of C-Mod data allowed the correlation to be broken, strengthening the case for $\nu_{\text{eff}}$ as the controlling parameter. Data from the three devices strongly suggest that density peaking in ELMy H-mode plasmas in ITER will be in the range $n_e(\rho_{\text{pol}} = 0.2)/\langle n_e \rangle = 1.4–1.6$, as illustrated in figure 3. The relative importance between $\nu_{\text{eff}}$ and $n_e/n_G$ as the principal scaling parameter for density peaking is illustrated as the difference between figures 3(a) and (b).

Further studies using similar approaches were performed as independent complements to the previous studies on density peaking in the JET tokamak. These focused on low collisionality data as the most representative of the reactor conditions [34]. The study confirms that $\nu^*$ is the most important parameter governing density peaking in H-mode, followed by the NBI particle flux and the $T_i/T_e$ temperature ratio. Moreover, similar studies on electron density peaking were performed in JT-60U tokamak and LHD helical plasmas at low collisionality [35]. The electron density peaking increased with decreasing collisionality in ELMy H-mode plasmas of JT-60U when the collisionality at half the minor radius was in the collisionless regime. The collisionality dependence of density profiles in LHD plasmas was similar to that in JT-60U plasmas in the same collisionality

<figure><img src="image_6.png" /><figcaption>**Figure 3.** (a) The density peaking factor $n_e(\rho_{\text{pol}} = 0.2)/\langle n_e \rangle$ is plotted versus $\nu_{\text{eff}}$ for C-Mod, JET and AUG data. (b) As in (a), but scaled against $n_e/n_G$ for the same data. As in seen in (b), the C-Mod data break the covariance and demonstrates that collisionality is the better fitting parameter. Reproduced from [13]. © 2007 IAEA, Vienna. All rights reserved.</figcaption></figure>

regime when neoclassical transport was reduced by geometrical optimization. JT-60U data with respect to density peaking versus $\nu^*$ fits well in a multi-machine database as shown in [9].

There are other approaches to study density peaking in particular with respect to the possible role of the NBI particle source. In JET, an analysis of a restricted number of JET plasmas with sufficiently ITER-similar parameters (low NBI, ITER like low $\nu^*$ and $q_{95}$) [20] was performed. It showed a clear correlation between density profile and particle flux. As a consequence, particle sources must be taken into account when extrapolating the density profiles to ITER or other next-step devices. Transport simulations of similar JET plasmas by adding the particle source due to NBI and an anomalous pinch proportional to $D\nabla q/q$ have been performed [36]. These simulations gave results that are compatible with the experimental measurements, however, the effects of the two terms are comparable and difficult to separate and compare in magnitude. The presence of an anomalous pinch can only be clearly identified in shots heated mainly by ion cyclotron resonance heating (ICRH) where the NBI particle source is negligible and, therefore, further experiment to quantify the magnitude of NBI fueling was needed.

Consequently, the role of NBI fueling was studied on JET with a dedicated experiment. This is needed as the analysis is complicated by the fact that often $T_i/T_e$, $\nu^*$ and the NBI particle source are correlated, and simultaneously, they have an opposite influence on the density peaking [34]. As a consequence, this correlation between density peaking and the source may cause such a bias in the database type of analysis in such a way that the role of NBI cannot be resolved precisely. An H-mode experiment with a comparison of 8 MW of NBI heating/fueling with an identity discharge but with 8 MW of ICRH heating was performed in JET [18]. In the plasma with NBI fueling, density peaking was increased by a factor of
two. This experimental result where the NBI fueling increases the density peaking by a factor of two was confirmed both with gyro-kinetic simulations and with full-radius transport modeling. On the other hand, on AUG in an experiment with plasma density and auxiliary power levels scanned to cover a wide parameter space indicated that the density peaking is mainly sustained by turbulence, with only a minor role for the NBI fueling source [19]. These at first sight contradicting results from AUG and JET with respect to the role of NBI fueling can be explained by the different turbulence properties and partly by the machine size. When the turbulence intensity and particle diffusion coefficients are large, NBI fueling has a minor role while in the opposite case it may even dominate. Both on JET and AUG, the NBI fueling is quite centrally located which will not be the case for example in ITER. However, there is not yet a full characterization on the plasma regimes in which the NBI fueling has a significant role in density peaking and in which it does not have that.

Furthermore, electron density peaking has been found to increase in electron cyclotron resonance heating (ECRH) heated plasmas on AUG with respect to the corresponding NBI ones [37]. The higher density peaking with increased central electron heating is explained by a concurrent reduction of the outward convection and a simultaneous increase in the inward thermo-diffusion. Further for AUG, density peaking was found to correlate with ITG-TEM turbulence, the highest peaking observed in the vicinity of ITG-TEM threshold [27, 38]. Similar conclusion was reached on DIII-D with NBI versus ECRH heating, i.e. the maximum density peaking takes place at ITG-TEM threshold, and then rapidly decreases when moving deeper into the TEM regime [39, 40]. Concerning the effect of rotation on density peaking on DIII-D, a strong increase in the inward particle pinch and consequent increase in density peaking in the counter-torque injected plasma is reported [40] In ITER, due to large volume with respect to NBI fueling, the role of NBI fueling in contributing to density peaking is not believed to be dominant and in most operational scenarios it will not be even particularly significant.

### 2.3.2. Local particle transport coefficients.
Major progress has been achieved in measuring local particle transport coefficients in the last decade, thanks to the improvements in profile reflectometry and High-Resolution Thomson Scattering (HRTS) diagnostics, such as those ones upgraded in JET [41, 42]. Local particle transport has been analyzed to quantify the magnitudes of particle diffusion, particle convection and thereby the NBI fueling in contributing to density peaking. The particle flux can be expressed

$$ \Gamma = -D\nabla n + nV, \quad (3) $$

where D is the particle diffusivity and V is the convection velocity. They are defined separately for electrons $D_e$ and ions $D_i$. A typical notation is that the negative value of $V_e$ or $V_i$ indicates an inward particle pinch. The local particle transport experiments and analyses are described separately for electron particle transport and ion particle transport below.

### 2.3.2.1. Electron particle transport.
Plasma core electron particle transport has been studied by executing numerous dimensionless collisionality $\nu^*$ scans while simultaneously keeping otherdimensionless profiles fixed in many different plasma scenarios on JET and DIII-D [14, 15]. These experiments are the first ones where electron particle transport coefficients (diffusion and convection) in H-mode were measured on JET with high resolution diagnostics. A low-frequency gas puff modulation technique has been developed and optimized in JET to study core particle transport coefficients as well as core and edge particle sources [43]. The local electron density response to the gas puff modulation was measured with a multi-band density profile reflectometry which iscapable of both good spatial and temporal resolution, and complemented with HRTS diagnostics.

The diffusive and convective electron particle transport coefficients have been measured in JET H-mode plasmas where $\nu^*$ has been scanned over 0.1–0.5 by keeping other dimensionless plasma parameters $\rho^*, \beta_N, q$ and $T_e/T_i$ constant. Similar three-point dimensionless collisionality scan was performed in L-mode in JET. The electron particle transport coefficients from both the H-mode and L-mode dimensionless $\nu^*$ scans are presented in figure 4. This also demonstrates the existence of the inward particle pinch from the experimental side in both L- and H-mode. The gyro-kinetic code GENE and TGLF and EDWM transport models have also been used to calculate diffusion and pinch for the same $\nu^*$ scans [44–46]. The simulation results showed good match with the experimentally measured electron particle transport coefficients. The particle transport coefficients are significantly larger in L-mode plasma than in the H-mode plasma, giving rise to the fact that the NBI fueling plays a larger role in H-mode plasmas in contributing to density peaking, and a smaller role in L-mode plasmas. The significant role of NBI fueling in JET plasmas was already discussed in section 2.3.1. Another density modulation method, high frequency (up to 20 Hz) strike-point-sweeping, was also developed to study particle transport and edge fueling properties in JET [43]. This paper includes also the newly developed workflow to estimate the particle transport coefficients from a modulated density data beyond the previously used Takenaga’s formulas [47].

A similar three-point $\nu^*$ scan in H-mode was also performed in DIII-D [15]. Similarly to JET, $\nu^*$ was varied by a factor 5, while keeping $\rho^*, \beta_N, q$ and $T_e/T_i$ fixed. Moreover, a 3 Hz gas puff modulation was applied to modulate the density profile and extract the particle transport coefficients. The perturbative particle transport analysis indicated that the increase in density peaking at small $\nu^*$ is associated with an increase in the inward particle pinch and not an increase in core fueling. Furthermore, the experimental dimensionless scan confirmed that there is an increase in density peaking towards lower $\nu^*$, linked to a shift in the turbulence regime from ITG towards TEM. The existence of the electron particle pinch with high-quality diagnostics was also reported earlier on DIII-D [39, 40].

The electron particle diffusion coefficient and electron convection velocity in ECRH heated plasmas have been evaluated
<figure><img src="image_7.png" /><figcaption>**Figure 4.** The experimentally determined $D_e$ (a) and $V_e$ (b) in the JET low $\beta$ H-mode $\nu^*$ scan (upper row) and in lower row for the corresponding JET L-mode $\nu^*$ scan of $D_e$ (c) and $V_e$ (d). Negative value of $V_e$ refers to an inward convection (pinch). The particle transport coefficients are radially averaged from $\rho_{tor} = 0.5$ to $\rho_{tor} = 0.8$. The colors refer to the magnitude of the collisionality in the $\nu_*$ scans. Reproduced from [14]. © EURATOM 2019. All rights reserved.</figcaption></figure>

using a density modulation experiment in LHD [48]. The values of the perturbative electron particle diffusion coefficient, $D_{e,mod}$, were significantly higher than those obtained by neoclassical prediction. On the other hand, the perturbative electron particle convection coefficients, $V_{e,mod}$, were comparable with the neoclassical estimations. Turbulence measurement at ion-scale was compared with the measured $D_{e,mod}$ and $V_{e,mod}$, suggesting a clear link between the two. This low anomalous pinch of the same order as the neo-classical pinch seems like a specific feature in stellarator particle transport not observed in tokamaks.

### 2.3.2.2. Ion particle transport.
There are few diagnostics that measure ion density similarly to the electron density and, therefore, the methods to evaluate the ion particle transport coefficient are different from the ones for electrons. A novel method exploiting the beam-target fusion neutrons in a plasma composed of two hydrogenic isotopes, protium (H) and deuterium (D), was developed in JET to estimate the ion particle diffusion and convection coefficients [16]. The core particle source was deuterium, while the background thermal plasma was H majority. With two different isotopes mixed in the plasma, it is possible to separate the beam fueling effect from transport by observing how the D/H isotope radial profile varied from the edge to the core. The ion particle diffusivity $D_i$ estimated in this way was significantly larger than the values of electron particle transport coefficients reported in typical JET plasma [14]. Furthermore, a significant inward ion particle convection $V_i$ was evaluated. It was concluded that using the isotope density profiles enables the unambiguous evaluation of the ion particle transport coefficients, which can be significantly different from those of the electrons without breaking the quasi-neutrality constraint. As a consequence, even if the electron and ion fluxes equal to each, keeping quasi-neutrality,

the diffusion and convection coefficients can be significantly different between the electron and ion channels. As a result, ion density peaking can be relatively insensitive to the source, i.e. determined by the transport and $D_i/V_i$ ratio. This is potentially an important result, for example in getting the fusion born Helium out of the plasma center. These experimental results are also consistent with the theory observation reported in section 2.2.

The previous experimental result in ITG driven turbulence was explained both analytically and in nonlinear gyro-kinetic simulations [17]. The simulations demonstrated that the resonance condition leads to ion particle transport coefficients significantly larger than electron particle transport coefficients. This simulation result was further confirmed by JETTO-QuaLiKiz integrated modeling. The consequence of the large ion particle transport coefficients implies that the ion density profiles are uncorrelated to the corresponding ion source. Therefore, it is possible to obtain peaked isotope density profiles even in the absence of core source. Consequently, the simulations suggest no strong core accumulation of He ash. While we expect fast isotope mixing and fast impurity transport in ITG regimes, in TEM regime the situation is the opposite, i.e. ion particle turbulent transport coefficients are smaller than their electron counterparts. This simulation result was presented in figure 1.

### 2.3.3. Isotope dependence of particle transport.
The theoretical prediction for the isotope scaling of particle transport was discussed in section 2.2, concluding that for otherwise similar parameters except the isotope, the particle pinch decreases with increasing mass of the hydrogen isotope. This leads to the theoretical result that the density gradient at zero particle flux is largest in hydrogen, and progressively decreases in deuterium and tritium plasmas, with a difference among the isotopes which increases with increasing collisionality [23].

A dimensionless ion mass scaling experiment between the hydrogen and deuterium plasma in JET was performed to study confinement as well as heat and particle transport [49]. Experimentally, no isotope scaling in particle transport was found between hydrogen and deuterium at matched $\rho_i^*, \nu^*, \beta_N$, $q$ and $T_e/T_i$. This is confirmed in TGLF simulations with JINTRAC transport code where an agreement between the experimental and simulated density profiles was achieved between the dimensionlessly matched hydrogen and deuterium plasmas. The collisionality $\nu^*$ was small in these experiments and therefore, we do not expect to see a collisionality dependence in the particle transport in this regime, in accordance with theory [23] in section 2.2. At larger collisionality regime, theory predicts an isotope dependence on collisionality [23].

In the recent JET DT campaign, a similar, dimensionless ion isotope mass scaling experiment in L-mode between pure deuterium and pure tritium plasmas with matched $\rho^*, \nu^*, \beta_n$, $q$ and $T_e/T_i$ was also performed in JET [50]. Moreover in recent JET DT campaign, a dimensional isotope scaling experiment between hydrogen, deuterium and tritium was performed [51].
In the dimensionless isotope mass scaling experiment, the isotope mass dependence in particle transport was found to be negligible, supported also by the perturbative particle transport analysis with gas puff modulation [50]. The same conclusion, i.e. negligible isotope scaling in particle transport and density peaking was found in H-mode [51]. Local GENE simulations in the L-mode pulses indicated, in agreement with the experimental results, a negligible effect on particle transport. On the other hand, in TGLF-SAT2 simulations, the effective turbulent particle diffusivity shows a very large increase in the tritium discharge outside of mid-radius, which is due to the change of the isotope mass. Similar observations, i.e. too flat density profiles were predicted for the H-mode discharges outside the mid-radius. This unfavorable isotope scaling of particle transport from D to T seen in the TGLF-SAT2 simulations on the outer radii at $0.5 \le \rho \le 0.8$ is not observed in the experiment. Further work is needed to validate the particle transport model in TGLF-SAT2 to obtain the experimentally observed isotope scaling, when moving from deuterium plasma to tritium plasma, and this work is already ongoing [52].

Isotope scaling of particle transport was also studied in LHD and CHS stellarators [53–55]. A reduction, favoring the deuterium plasma, in particle transport and turbulence characteristics between the hydrogen and deuterium plasmas was found in the core region at $\rho \le 0.8$ in high collisionality regime on LHD. Gas puff modulation experiments in the high collisionality regime also showed correspondingly smaller particle diffusion coefficients in the deuterium plasma with respect to the hydrogen plasma. On the other hand, no such reduction in particle transport and turbulence was observed in low collisionality regime. On CHS, the particle confinement was improved in the deuterium dominant plasma with respect to the hydrogen plasma. The gas puff modulation experiments showed the quantitative difference in the particle transport coefficients, i.e. a lower particle diffusivity and a larger inward pinch velocity were observed in the deuterium plasma with respect to the hydrogen one.

## 2.4. Pellet fueling

In tokamak fusion reactors the rate of burning DT fuel is rather small and consists of only a few percent of fueling throughput in the vacuum vessel. The latter is dictated by the need to keep the plasma density at the required value and helium removal. Therefore, the term fueling is perhaps somewhat misleading and density control might be more appropriate. Nevertheless, in this review we will keep the traditional term fueling for historical reasons.

Traditionally, tokamaks are fuelled by puffing neutral gas into the vacuum vessel. This method, however, becomes inefficient when plasma parameters approach reactor grade values, i.e. become hot, dense and large. Under these conditions gas puffing transforms itself from a plasma density control tool into an ELM control actuator. More specifically, increasing gas puffing increases the ELM frequency [56], which in turn could even reduce the plasma density.

To overcome this problem a number of tokamaks are now equipped with injectors of cryogenic pellets which deliver the

fuel deeper into the plasma e.g. [57, 58]. Such pellets are several mm in size, have repetition rate of tens of Hz, and velocity hundreds of m s⁻¹. By this method plasma density can be significantly increased compared to gas puffing.

As an example, pellet injection on ASDEX Upgrade increased the plasma line-averaged density by a factor of 1.5 above the Greenwald density [59]. In these plasmas density profiles become more peaked inside the pedestal and hence, the Greenwald limit can be regarded as an edge limit which can easily be overcome by pellet injection. In a more recent work [60] the plasma is refined so that it is stationary and well controlled. In addition, application of pellet fueling in this experiment does not change the energy confinement while ELMs change from type-I to essentially type-III ELMs with a few type-I ELMs of low magnitude.

Pellet fueling is now part of the ITER and DEMO designs. It is, therefore, of interest to update its physics basis with results as found in recent experiments and their implication for extrapolation towards reactors.

When a pellet is injected into a plasma it ablates and deposits its material along its trajectory. This process is dominated by pellet specific 3D physics, including pellet triggered magneto-hydrodynamic (MHD). After the pellet is evaporated plasma profiles are expected to become 2D so that the post-pellet particle transport can be studied in a conventional transport framework. These two phases, pellet deposition and post-pellet transport, are discussed separately in the following sections.

### 2.4.1. Pellet deposition

#### 2.4.1.1. Pellet ablation.
The main technologically limiting factor of pellet fueling is the maximum velocity at which pellets can be delivered to HFS of the plasma [61] without breakage. For repetitive pellets and shaped flight tube, as planned in ITER, the pellet velocity is limited to about 300 m s⁻¹. When such a pellet is injected into a reactor grade pedestal with a temperature of ~5 keV, the neutral gas and plasma shielding model (NGPS) [62–64] predicts that pellet fully ablates within the pedestal, i.e. inside the normalized minor radius $r/a = 0.95–1.0$ [65, 66]. The NGPS ablation model was originally calibrated against the experimental dataset for ohmic [64] and auxiliary heated plasmas [63]. Comparison of these ablation datasets with the situation expected in ITER shows that in the present data, the temperature at the pedestal is too low and the relative pellet size is too large. The combination of these two effects results in deep pellet penetration in present day tokamaks. This size gap in datasets is difficult to bridge in present medium size machines as the ITER-equivalent pellet would be sub-mm in size which is difficult to produce and deliver.

The aforementioned gap was recently reduced during the experiments in JET. This new dataset [67] contains data on pellets launched from vertical HFS into the plasma with high current and high heating power, up to 3.5 MA and 30 MW resp. Pellet size was changed between 2 mm and 4 mm. All technically available isotope combinations were used with H, D and T
plasma and H and D pellets. For individual pellets, high temporal resolution core interferometry is used to determine the duration of the pellet ablation process and the number of atoms deposited in the plasma. The ablation depth $r_{abl}/a$ is calculated from the duration of ablation light and nominal pellet velocity. The dataset spans the interval of $r_{abl}/a = 0.82–0.98$, which covers the expected ITER point $r_{abl}/a = 0.95$ from both sides. The relative line averaged perturbation of the plasma density by pellet spans the interval from 0 to 16%, which again covers the ITER design value. Pellet deposition efficiency is calculated as the ratio of number of pellet particles deposited in the plasma divided by the number of pellet particles measured in the flight line. For ablation depth $r_{abl}/a < 0.95$ the deposition efficiency is 50%–100%. However, for pellets with shallow ablation $r_{abl}/a > 0.95$ the deposition efficiency sharply decreases to zero. The reason for such low deposition efficiency is that pellet ablated within the edge transport barrier (ETB) typically triggers ELM-like event that are synchronous with the ablation process (oscillations on fast interferometer signal) and as a result majority of pellet material is promptly removed from the plasma.

Regarding the NGPS model several comparisons with data were reported. Broadly speaking no serious discrepancy was found in the overall duration of the ablation process. Interestingly, at the end of the ablation interval high temporal resolution interferometer signal often shows a sharp inflection point (change in slope of line averaged density signal) [67, 68]. This effect could be used for calibration of NGPS model [63].

### 2.4.1.2. Plasmoid drift.
As discussed above, in ITER, pellet fueling that relies only on ablation alone cannot fulfill its fueling function. In order to keep pellet fueling capability, pellets have to be injected from the HFS with the aim of maximizing redeposition of the pellet material due to $\nabla B$ drift. This drift is facilitated by plasmoids [69–71] which emanate from the neutral cloud around the pellets and move in the direction of decreasing magnetic field (deeper into the plasma). Experimentally, the length of the redeposition can be deduced from comparison of the measured post-pellet density profile with modeling. This was done on several machines and typically good agreement is found (e.g. [68, 72] for MAST). Despite that, there is still a large gap between present data and the situation expected in ITER. In particular, in experiments the ablation length is typically larger compared to the drift length, i.e. opposite to that in ITER [73–76]. This will need attention because the physics of plasmoid drift is quite delicate [70, 77] and, for example, the effect of the H-mode pedestal on plasmoids is not documented yet.

### 2.4.2. Pellet deposition modeling.
As described above the pellet deposition is a result of two simultaneous processes: pellet ablation and redeposition of pellet material. Such description defines the structure of any pellet deposition code. There are several leading codes that simulate pellet deposition process: PELDEPD2 [78], HPI2 [79] and PAM [80–82]. These codes differ in detail of calculations of such parameters as

<figure><img src="image_8.png" /><figcaption>**Figure 5.** Pellet deposition profile predicted in ITER 15 MA/5.3 T/Q = 10 plasma using 90 mm³ pellet. Solid line before, and dashed line after pellet deposition. Reproduced from [83]. © 2018 ITER Organization. All rights reserved.</figcaption></figure>

pellet lifetime, the size of plasmoids emanating from pellet and their trajectory in ambient plasma. In this respect recent PAM code can predict the pellet ablation and deposition very quickly while maintaining good accuracy. Finally pellet deposition codes also differ into which transport code the pellet model is implemented. It has to be noted that conventional transport codes describe the particle transport after pellet is ablated and homogenized on magnetic surfaces. The time of changeover from pellet deposition to conventional particle transport approximately coincides with the end of pellet evaporation processes and thus is conveniently handled by plug in of pellet deposition module into transport codes.

### 2.4.3. Post-pellet transport.
After a pellet is deposited, the plasma density quickly homogenizes along the flux surfaces. Such a post-pellet density profile takes the form of bump located at the plasma periphery. In ITER the maximum of the perturbation is located at $r_{abl}/a < 0.82$ and its amplitude is about 30% [83] as seen in figure 5. Such a density perturbation creates two distinct zones: (1) inner zone with inverted density gradient and (2) outer zone with increased density gradient. These two zones broadly coincide with zones of inward and outward particle flux, respectively. The overall pellet fueling efficiency then depends on the balance between inward and outward particle fluxes in the pellet deposition zone. For example, when the inward flux dominates and the outward flux is small, then the fueling efficiency will be high. In the opposite case, when the inward flux is small and the pellet material is transported outwards, the fueling efficiency of the plasma will be low. The real situation is more complicated as inward and outward particle fluxes act simultaneously and also change in time during the pellet cycle.
2.4.3.1. **Inward flux (inner pellet deposition zone).** The most distinct feature of pellet injection is the formation of a zone with inverted density gradient as shown in figure 5. It is true that the size of the inversion of density gradient depends on pellet size and local intensity of particle transport. Nevertheless this density profile feature (bump) is found in all simulations of ITER baseline scenario fuelled by discrete pellets including the most recent work [84]. Regarding the electron temperature the profile remains conventional but the gradient could be steeper due to adiabatic pellet deposition.

If the plasma turbulence were unaffected by changes in density profiles then such inversion would create inward plasma diffusion, which would be a favorable effect. Linear gyrokinetic simulations of MAST data, however, shows that the situation in this zone could be more complicated. In particular, inversion of the density gradient can result in stabilization of micro-turbulence for both ITG and ETG modes [85]. This is understood as a result of the reduction of the parameter $\eta_{e,i}$ due to steep density gradient. Such stabilization of turbulence would be unfavorable as it will reduce the efficiency of pellet fueling.

The aforementioned picture is modified when one includes the change of the temperature profile. Because the pellet deposition is close to adiabatic (plasma pressure is approximately constant) the temperature gradient in inner deposition zone will increase due to the pellet. This will have a destabilizing effect on micro-turbulence and can counteract the stabilising effect of the strong density gradient mentioned above. This effect was analyzed in detail using local gyrokinetic integral equations [86]. From comprehensive parametric scans, the authors obtained stability diagrams for ITG modes and TEMs in terms of density and temperature gradient scale lengths. The results show that, for inverted density profile, there exists a normalized threshold temperature gradient above which the ITG mode and the TEM are either separately or simultaneously unstable.

This problem was also extensively studied using the data from AUG [87]. Both linear and nonlinear simulations were used and similar conclusions to those above were drawn. In addition, a new mode driven by a positive density gradient (hollow density profile) was identified for the first time. This instability is located in the proximity of the HFS of the poloidal cross section. It is produced by the non-adiabatic response of passing particles and it becomes more unstable with increasing collision frequency, producing a strong inward flux. This, however, is unlikely to be destabilized at the low collisionality of a reactor plasma.

2.4.3.2. **Inward propagation of isotope mix.** In fusion plasmas pellet fueling is expected to control not only electron density, but also the deuterium tritium mixture in a plasma core. Note that in ITER there are no tritium neutral beams. As a result, the tritium fraction in the core is controlled from the periphery by pellets. It has to be noted that the requirements for isotope mix control are rather modest: the timescale

on which the plasma drifts from a 50:50 mix is expected to be long and also the accuracy of mix ratio is not very demanding. The main issue with isotope mix control is the separation of hydrogen isotopes in real time. This process is very expensive and its efficiency determines the amount of tritium in the fuel loop, which is an important parameter for the nuclear regulation process [88]. For this reason it is considered that only a fraction of the deuterium-tritium (DT) stream will undergo a isotope separation process and the rest will be re-injected in the form of 50:50 mix. The fraction of necessary tritium separation depends on the efficiency of propagation of the isotope mix from the edge to the core. Unfortunately, there are no tritium pellets available in present tokamaks, including JET and, therefore, it is impossible to validate isotope mix control exactly in a reactor like configuration. Instead of that, only pellets with H/D mixture were used in experiments so far. In ITER it is planned that pellets can be made of pre-mixed D/T fuel [89]. In DEMO the use of un-separated D/T fuel goes one step further by introducing a direct internal recycling loop [90].

In a first experiment, pre-mixed 50:50 HD pellets were injected in AUG H-mode plasma, initially fuelled by gas with 50:50 HD mixture [91, 92]. It was shown that at low particle flux there are cases in which the isotope mixture of the plasma and applied flux deviate significantly. This is attributed to a not yet fully equilibrated wall reservoir. However, once a sufficiently high flux is applied—either by gas, pellets or combining both—the plasma H/(H+D) fraction approaches the set point value within about 1 s.

In a second experiment, deuterium pellets were injected into an initially pure hydrogen plasma in JET (H gas, H beams and RF) [93]. Application of pellets changed the H/(H+D) fraction in the core from 1 to ~0.5 as measured by core charge-exchange recombination spectroscopy (CXRS) spectroscopy and the (thermal-thermal) neutron rate. Transport of both isotopes was simulated by gyro-kinetic code QuaLiKiz [94]. Simulations show that in the ITG driven turbulence regime the diffusivities for both ion species are higher than the electron diffusivity $D_H \sim D_D > D_e$. This could explain the fast propagation of the isotope mix from the edge to the core even with similar diffusivities for hydrogen isotopes. Generally, fast propagation of isotope mix would be a favorable observation, provided the mix at the periphery can be adequately controlled.

Regarding the simulation of isotope mix control in ITER baseline scenario, more accurate results were obtained recently [95, 96]. In simulations the D and T ions are evolving independently according to EDWM transport model. In this situation D and T ion diffusivities are anomalous. Tritium and deuterium is supplied by pellets while D ions are supplied by gas. Pellet deposition is calculated by HPI2 model. Simulations show that within this model it is possible to keep the D/T mix in the core at 50:50 ratio. For Greenwald fraction 0.85 the required pellet composition should be 45:55 DT. Such asymmetry is combined effect of higher T diffusivity, neoclassical pinch and additional deuterium gas puffing used.
Note that these are whole plasma simulations encompassing core/pedestal/SOL/divertor and hence able to study fuel mixing.'

### 2.4.3.3. Outward flux (outer pellet deposition zone).

Deposition of pellets in the plasma periphery creates a zone with increased (negative) density gradient ($\partial n/\partial r < 0$). On ITER this zone spans from normalized minor radius $r/a \sim 0.8$ of up to the separatrix. Outward particle flux in this zone determines the pellet fueling throughput needed to keep the plasma density at the required value. In other words, it is the particle transport in this zone which controls the pellet fueling efficiency and ultimately the reactor burn-up ratio [88]. The later is the most critical parameter of fusion power plant as it determines the amount of tritium on site.

Particle transport in the outer pellet deposition zone is complex. Firstly, it has to be remembered that the outward flux is dynamic and changes during the pellet cycle. Secondly the outer pellet deposition zone (i.e. zone with negative density gradient) covers both the pedestal and the pedestal top regions so that the particle transport is not expected to be described by single physics model. For practical reasons, one can treat the outer flux as a sum of two fluxes: (1) flux due to micro-turbulence and (2) flux resulting from ELM mitigation or suppression. These are now discussed in next paragraphs.

The modes due to micro-turbulence can be studied by a conventional gyro-kinetic model as discussed above for inner pellet deposition zone. In a simulation for the AUG tokamak [87] it is found that the dominant mode is driven by an increased negative density gradient due to the pellets, and corresponds to the usual TEM instability. This mode produces large diffusive particle flux directed outwards, and becomes more unstable with increasing trapped particle fraction and with decreasing collisionality. This means that this mode is expected to be relevant at reactor conditions.

Concerning the flux due to ELM suppression/mitigation, there is now a consensus that, in fusion power plants, ELMs should be mitigated or completely suppressed in order to protect the divertor from high power bursts. Several ELM control methods are considered to achieve this, such as application of RMPs, pellet ELM pacing, and utilizing natural MHD modes. Among these methods only one regime has been subject to pellet fueling so far, namely the application of RMPs. The second method, testing pellet fueling on a plasma with pellet ELM pacing, would require two pellet injectors and such a system is not yet available in present tokamaks. Nevertheless a pellet injection system, including two pellet injectors, has been installed on EAST for exploring ELM pacing performance on EAST simultaneously with pellet fueling. Finally, pellet fueling of plasmas with ELM suppression/mitigation by natural MHD modes has not yet been tested.

Pellet fueling of plasma with RMP ELM mitigation was tested on MAST [97], EAST [98, 99], AUG [100] and DIII-D [101, 102]. A common feature of ELM mitigation/suppression is the so called density pump-out i.e. increased outward particle flux caused by RMP fields. These experiments showed that pump-out can be refueled by pellets. The pellet particle

flux $\Phi_{pel}$ required for refueling of density pump-out on AUG is [103]

$$ \Phi_{pel} \sim 0.07 P_{aux} / T_{i,ped} \quad (4) $$

where $P_{aux}$ is the heating power and $T_{i,ped}$ is the pedestal ion temperature. When the density pump-out is refueled by pellets, the plasma transitions from ELM suppression to ELM regime, but a significant degree of ELM mitigation is preserved. The pellet particle flux required will, therefore, depend on which level of mitigation/suppression plasma has to be controlled.

Phenomenology of the transition from ELM suppression to an ELM regime was studied in detail in AUG [100] using long trains of shallow pellets. It is found that each pellet triggers an ELM-like event which is followed by a transient (20 ms) ELM free H-mode, and then the plasma returns to the pre-pellet phase with ELM suppression. When the plasma density gradually increases (spontaneously or by increasing pellet rate) then ELMs start to occur during the post-pellet H-mode phase with increasing frequency until ELM suppression is completely lost in the pellet cycle. This behavior is in line with existence of density limit for ELM suppression and is also observed with gas fueling.

The physics basis for particle transport in the outer pellet deposition zone under condition of ELM suppression/mitigation is under development. Based on experimental observations the main elements of such model should be (1) the particle transport in presence of RMPs and (2) role of the steep density gradient due to pellets [104]. To identify the character of overall particle transport it would be useful to distinguish between convective or conductive losses. In MAST [97] post-pellet ELM losses seem to be convective as the pellet material is removed by ELM outwards even from inner pellet deposition zone.

## 2.5. Summary and outlook

Very good progress in understanding particle transport and pellet physics has been made in the last decade and most of this work was performed within the auspices of the TC ITPA working group. Theory of turbulent particle transport has received particular consideration since the last ITER physics update. A detailed description of the particle transport mechanisms for both trapped and passing particles in the presence of ITG and TEM instabilities has been established. Moreover, the theoretical framework to understand the impact of electromagnetic effects, the relation between ion and electron particle transport, particle transport in plasmas with different hydrogen isotopes, and the role of rotational $E \times B$ shear has been achieved.

On the experimental side, further development starting from single machine density peaking databases have progressed towards creating multi-machine density peaking databases, which have unambiguously revealed the unique role of collisionality to influence the peaking of the density profile. Further experiments and modeling have shown that the role NBI fueling has in affecting density peaking depends on the turbulence type or magnitude of the particle diffusion,
and varies from very significant to almost negligible. Thanks to the improved HRTS and profile reflectometry diagnostics, more accurate measurements of local electron particle transport coefficients have been performed on many tokamaks and stellarators. The measurements of local particle transport coefficients have confirmed the increase in particle pinch with decreasing collisionality, and the significantly higher ion particle transport coefficients with respect to the electron particle transport coefficients in ITG dominated plasma. The local measurements of the particle transport coefficients have also shown that the isotope scaling of particle transport, for example between deuterium and tritium, can be different from that one of heat transport, i.e. while strong favorable scaling for tritium with respect to deuterium is seen on the heat transport channel, no or very small isotope scaling is seen on the particle transport channel.

The particle sources, with the exception of the role of the NBI particle source, are not discussed in this review. The experimental quantification of the edge neutral particle source remains challenging, and this is reviewed in the pedestal chapter of this Nuclear Fusion Special Issue [21]. In addition, 3D effects may have a significant influence on particle sources and also transport in the edge part of the plasma, such as the density pump-out effect due to RMP coils or magnetic ripple effects, and these are reviewed in section 6.

According to present consensus, reactor grade plasmas have to be fuelled by pellets and simultaneously ELMs should be suppressed. The corresponding actuators, pellet fueling and ELM control, are both acting on the same plasma region i.e. on plasma periphery. This coupling causes competing control issues that have not yet been fully resolved. In order to move forward one can cut the problem into smaller elements, test them locally against experimental data, and then build an integrated physics basis which is used for prediction. During prediction, particular attention should be paid to the validity of a particular physics element at the prediction point. This usually points toward the weakest part in the integrated physics basis.

In this review we listed all physics elements which we believe contribute to the physics basis of pellet fueling. The quality of their comparison with experimental data and prediction toward ITER is varying. In our opinion, one of the weakest parts of the model is the pellet deposition when the pellet is within the ETB. Another uncertainty is the value of the outward particle flux in the outer deposition zone under conditions of ELM suppression and steep density gradients due to shallow pellets. The main difficulty is that successful ELM suppression technique results in a replacement of ELMy regime by particle transport mechanism which is not yet analyzed. For example ELM control by RMPs could lead to a particle flux which is a mixture of micro-turbulence and flow due to 3D magnetic field.

The predictive capability of particle transport in the plasma core has greatly improved in the last decade thanks to the theory development. Further work is needed in some specific conditions to predict the core density profile, such as plasmas with tritium and ICRH dominated plasmas. The role of the

plasma edge, including the magnitude of the separatrix density in influencing the density profile, has become an active area of research. Full radius predictive capability for the core density profile will require good understanding on the connection of SOL and edge with the core, and the relevant particle sources and sinks involved, and further work is required on this front.

## Acknowledgments

The authors are very grateful to P.T. Lang, L. Baylor, L. Garzotti, H. Jhang, M. Jia, F. Köchl and M. Marin for their contributions to this manuscript.

## 3. Impurity transport

C. Angioni, F.J. Casson, B.A. Grierson, P. Manas, R.M. McDermott, T. Odstrcil and F. Sciortino

### 3.1. Introduction

The need for collaborative research on impurity transport within the ITPA was already recognized 15 years ago, and a joint activity was established in the programme of the TC topical group, starting from the end of 2008. This joint activity was initially specifically motivated by the need to make progress in the description of helium transport, with the aim of obtaining a consistent set of experimental data, possibly from different tokamaks, in order to validate theoretical predictions. This specific scope remains an important goal still today, since only a relatively limited amount of helium density profiles and helium transport coefficients have been made available in the meanwhile, since the first pioneering works at TFTR [105–107] and at DIII-D [108]. The limited availability of helium impurity data, also caused by the complexity of helium CXRS measurements [109–111] as compared to other light impurities, has, since the start, suggested to also extend the study to other light impurities. These can be expected to share some of the properties of helium impurity transport, but are more regularly measured in experiments. For this reason, intrinsic impurities like boron, carbon or other externally injected light impurities, have received increased consideration, and their behavior is now largely documented by several recent experimental works. In the meanwhile, and in particular from 2012, motivated by the studies required for the decision on the start of ITER operation with a tungsten divertor [112], also heavy, highly charged impurities, such as tungsten, have been included in the research activities of the ITPA group. These have covered a wide range of experimental investigations and diagnostic techniques, which have been also improved, in order to answer to the increasing requirements of more comprehensive and conclusive comparisons with the theoretical predictions. From the theoretical and modeling standpoints, both turbulent transport, regularly more dominant for light impurities, and collisional transport, which can become dominant for heavy, highly charged
impurities like tungsten, have been considered. The dominance of collisional and turbulent transport also varies along the minor radius. Collisional transport usually has stronger effect close to the magnetic axis and at the edge, in the presence of an ETB characterizing the H-mode, whereas in the confinement region turbulent transport can play a more dominant role.

In this way, the ITPA joint activity progressively answered the research needs which are determined by the fact that in a burning plasma, impurities with extremely large variations of mass and charge are present. A central source of helium is produced by fusion reactions, while, from the materials of the plasma facing components, a variety of impurities can enter the plasma together with the impurities which are actively seeded to obtain appropriate mitigation of the heat fluxes on the walls. These impurities, spanning a wide range of masses and charges, are radially transported by both collisional and turbulent processes, which can be both simultaneously important, or can relatively dominate one over the other, depending on the plasma conditions and on the impurity charge and mass. These transport processes can also directly interact, and be reciprocally modified. The theoretical study of the impurity transport, therefore, requires a variety of numerical tools which can accurately compute these separate transport mechanisms as well as their direct interaction, with increasing level of complexity and computational costs. At the same time, the experimental behaviors of impurities with largely different charges and masses requires different diagnostics methods, based on CXRS for light impurities, allowing local measurements, or based on measurements of radiation emitted along lines of sight for heavy, highly charged impurities. The latter method requires tomographic inversions in order to obtain a local reconstruction of the impurity emissivity and, ultimately, of the impurity density.

As a consequence of the large spectrum of different experimental and theoretical investigations which are required for a comprehensive study of impurity transport, the joint research of the ITPA has been structured according to the different characteristics of the activities, with two main blocks, one dedicated to light impurities, usually measured by CXRS and for which turbulent transport dominates in the majority of plasma conditions, and the other dedicated to heavy impurities, which are usually measured with soft X-ray spectroscopy or bolometry and for which neoclassical transport can provide the dominant component of the convective flux.

This review is therefore organized following the same structure of these research activities. A review of impurity transport studies, from the theoretical and modeling standpoints, has very recently been published [113], which also summarizes the comparisons between experimental and modeling results.

The present chapter uses this recent review as a reference and limits the overlap by focusing on the research which has been performed under the ITPA coordination, providing the appropriate links to the more general framework of that reference. Following the historical development of the research within the ITPA group, in section 3.2 the activities connected with light impurity transport are reviewed. This section

also allows us to briefly present the recent progress on the theoretical aspects of turbulent impurity transport, which is usually dominant for light impurities. The main properties of the turbulent diffusion and of the turbulent convective terms are presented in this section, together with the progress in the comparisons between theoretical predictions and experimental observations. In section 3.3 the activities connected with heavy, high charge impurity transport are described. Here recent advances in the description of neoclassical transport will be briefly reviewed. The unfavorable consequences of collisional transport on plasma operation in the presence of highly charged impurities are also discussed. In particular, the problems connected with central impurity accumulation and the experimental solutions which have been identified to avoid accumulation in tokamak operation are presented, together with the related theoretical understanding and predictions.

The connections between the research activities on turbulent and neoclassical impurity transport will be highlighted. These should not be considered separately, particularly in view of the fact that in future reactors the low collisionality and the expected lower plasma toroidal rotation will lead to a more dominant role of turbulent transport also for heavy impurities. Section 3.4 is dedicated to the direct interactions between the two transport processes, collisional and turbulent, which are regularly computed as independent mechanisms in the modeling, but which recent theoretical research has shown to have various ways of direct interaction, which can mutually affect their properties. Finally, main conclusions and an outlook are provided in section 3.5, with specific consideration of the expected plasma conditions in a fusion reactor.

## 3.2. Turbulent impurity transport, main theoretical aspects and comparison with experimental results, with emphasis on light impurities

First observations of impurity density profiles revealed that several trends were consistent with the neoclassical transport expectation that the peaking of the impurity density is mainly determined by the peaking of the main ion density profile [114, 115]. However, the presence of an anomalous component in impurity transport was also recognized, and identified as a critical element preventing strong accumulation in the plasma center. The magnitude of helium transport was acknowledged to be of critical importance in determining the peaking of the helium density profile and, therefore, the central concentration in a burning plasma. An increased peaking of the helium density profile provides stronger constraints at the edge in terms of helium pumping efficiency, where the peripheral concentration has to be decreased with increasing helium profile peaking to ensure acceptable fuel dilution in the center [107, 116–118]. Back in the early years of this century the collisional theory of impurity transport was largely established and regularly applied in the modeling of the impurity behavior, at least in the limit of non-rotating plasmas with poloidally homogeneous impurity densities [119]. At that time, turbulent transport theory was already largely established for the prediction of heat transport [120] and was gaining reliability for the prediction of at least some features of the electron particle transport [121].
Then the critical question arose of whether the same theory could also be applied to describe impurity transport, and this was directly considered by the ITPA joint activity.

### 3.2.1. First studies on helium transport.
Pioneering results on helium density profiles and transport were obtained in TFTR [105–107] and DIII-D [108]. First measurements of helium transport coefficients [105] revealed that diffusion was at least one order of magnitude higher than neoclassical at the edge and decreased towards the plasma center. Similar radial variation was observed for the inward directed convection, with centrally peaked helium density profiles. The inward convection was measured to be stronger than the neoclassical predictions in TFTR supershots and the helium diffusivity was observed to increase with increasing ion heat transport [106]. The confinement time of helium ash produced by D–T fusion reactions in TFTR was observed to be dominated by the edge pumping rates and up to one order of magnitude longer than the energy confinement time [107], still compatible with ignition in a reactor [116]. At DIII–D, an extensive analysis of helium density profile shapes in deuterium plasmas in various regimes showed that these have similar shapes to the profiles of the main plasma electron density [108]. Moreover, helium diffusivities were also measured which were of the same size as the effective heat conductivity $D_{He}/\chi_{eff} \sim 1$ [108] and, therefore, higher than the neoclassical expectations. Research in JT-60U [122, 123] also found helium diffusivities which were significantly larger than the neoclassical predictions, with the additional result that no observation of central accumulation of light impurities like helium and carbon was obtained even in internal transport barriers which featured ion heat transport at the neoclassical level. In contrast, an impurity with higher charge, such as argon, was observed to accumulate. The unfavorable behavior of impurities with increasing charge in the presence of internal transport barriers was also observed at JET [124], consistent with the predictions of neoclassical transport theory.

### 3.2.2. First observations of avoidance of impurity accumulation by central wave heating.
A reduction of the argon accumulation, corresponding to a flattening of the argon density profile, was found to occur as a consequence of the application of central ECRH [123]. The leading mechanism producing the flattening was identified to be connected with a reduction of the peaking of the main plasma density profile, also produced by the central electron heating in the high $\beta_p$ transport barriers of JT-60U. At the same time, experiments in ASDEX Upgrade in normal H-mode plasmas also demonstrated the beneficial role of central electron heating in reducing the peaking of the impurity density profiles [125]. Silicon laser ablation experiments revealed that central ECRH significantly increased the Si central diffusivity to values which were clearly above the neoclassical level, while, concomitantly, reducing the inward convection or even reversing it to an outward convection [125]. This body of results provided the basis for one of the main control tools to avoid central accumulation of heavy impurities in the operation of present tokamaks with tungsten plasma

facing components [126], namely, the application of central wave heating. As it will be described below in more detail, in these conditions the electron heating fraction plays the dominant role.

### 3.2.3. Turbulent impurity transport theory.
At the same time, these experimental evidences clearly showed that predictions of impurity transport could not be based on collisional transport only, and this motivated the investigation and development of a more comprehensive theoretical description of turbulent impurity transport [29, 127–131]. The main mechanisms of turbulent impurity transport have direct analogues to those which produce the transport of the main plasma species. In addition to the diagonal diffusive component, connected to the density gradient of the impurity itself, off-diagonal contributions are also present. Pure convective terms are produced by the compression of the $E \times B$ drift in toroidal geometry and by the impurity parallel dynamics [128]. Moreover, the presence of the temperature gradient of the impurity causes an additional off-diagonal contribution, usually dubbed thermo-diffusion [128–130]. A more complete description also includes an effect which was identified only later [132] and which provides an additional convective component, proportional to the gradient of the toroidal rotation, dubbed roto-diffusion. A comprehensive review of these mechanisms can be found in section 3.3 of [113] and is not reported here. It is, however, important for what follows to recall some of the general properties of turbulent impurity transport, which are also connected with several aspects of the comparisons with the experimental results. The ratio of the turbulent impurity diffusion coefficient to the turbulent heat conductivities is theoretically found not to be constant, but in fact to feature a strong non-monotonic dependence on the electron to ion heat flux ratio. This is produced by the different types of turbulence, from ITG to TEM, and the corresponding impact of the relevant resonance, determined by the mode frequency, on the different transport channels [31]. This effect is particularly strong for highly charged impurities, which are predicted to reach values of diffusivities that are significantly larger than the effective turbulent heat conductivity, in conditions when electron and ion heat fluxes become comparable. In contrast, for large dominant ion or electron heat fluxes, the ratio of the impurity diffusivity to the heat conductivity becomes significantly smaller than one, resulting in a variation of this ratio by more than one order of magnitude depending on the value of the electron to ion heat flux ratio. Off-diagonal components also exhibit the property that their direction can change sign depending on the type of turbulence [27]. In particular both thermo-diffusion and roto-diffusion are directed outward in the general conditions of dominant ITG turbulence, and oppose the usually inward directed pure convective term. In contrast, they can be directed inward in strong TEM turbulence. Moreover, in conditions of comparable electron and ion heat fluxes, also subdominant modes can play an important role in determining the impurity flux, as confirmed by comparisons with nonlinear results [32].
### 3.2.4. First comparisons with theoretical predictions.

Concomitant with the first theoretical works on the properties of turbulent impurity transport, a growing number of experimental observations [133, 134] allowed the first dedicated comparisons between theoretical predictions and experimental results. These first applied still limited theoretical models [135–137], but already revealed some elements of general, qualitative agreement, particularly in the overall consistency that no evidence of impurity accumulation in conditions of dominant turbulent transport was obtained, in agreement with general expectations from theory. However, also elements of disagreement were identified, which motivated further research.

An increase of the central diffusivity of nickel with increasing electron temperature logarithmic gradient was reported in Tore Supra [138, 139]. This increase was demonstrated to be of turbulent origin, and to be consistent with theoretical quasi-linear calculations of impurity diffusivity. Two complementary works for the validation of turbulent impurity transport theory were performed in ASDEX Upgrade and Alcator C-Mod. The first [37] was dedicated to the comparison of quasi-linear gyrokinetic predictions with a set of intrinsic boron density profiles measured in H-mode plasmas [140] with NBI heating and different levels of central ECRH. Here an experimental trend, which is largely confirmed by future works, is identified. The boron density profiles are increasingly peaked with increasing central ECRH, while still exhibiting a peaking that is lower than that of the electron density profile. The electron density profile also peaks in the confinement region with increasing ECRH power in these conditions, consistent with theoretical predictions of electron particle transport in ITG turbulence, see discussion in section 2.2. In contrast, with NBI heating only, the boron density profile becomes flat or even hollow around mid-radius. These experimental trends are at least qualitatively consistent with the combined outward roles of thermo-diffusion and roto-diffusion in ITG turbulence, which increase with decreasing ECRH heating. However, as described later, the pursuit of quantitative agreement with the increasingly accurate measurements of the observed hollowness of the impurity density profiles has been further investigated in the ITPA activity. In fact, this aspect still challenges the theoretical predictions, despite several attempts and recent additional improvements in the completeness of the theoretical description. The second work [141] compares, for the first time, global non-linear gyrokinetic simulations with the gyrokinetic code GYRO [142], matching the heat fluxes, with the separate measurements of the diffusion and convection coefficients of a calcium impurity, injected in Alcator C-Mod L-mode ICRH plasmas with dedicated laser ablation experiments [143]. The results provide a demonstration, for the first time, that, within the experimental uncertainties, gyrokinetic simulations can reproduce experimentally estimated values of the impurity transport coefficients, with large diffusion and large inward convection.

These promising results motivated more specific and extended research, aiming at improving the quality of the measurements, extending the investigated domain of parameters

and thereby more strongly constraining the theoretical validation and informing on the reliability of our current prediction capabilities of the impurity density profiles predominantly determined by turbulent transport. The impact of experimental uncertainties were comprehensively considered [144], and linear and non-linear gyrokinetic simulations were found to reproduce the trend of hollow boron density profiles with strong NBI heating and strong plasma rotation, but an underprediction of the level of hollowness was regularly reported. Impurity transport modeling activity at JET consistently documented the inability of the gyrokinetic modeling to reproduce the observed hollowness of the intrinsic carbon density profiles [145, 146]. In these studies also the dependence of the impurity peaking factor with increasing impurity charge was investigated, and found to be qualitatively reproduced by the theoretical predictions, with an increase of the peaking factor with increasing impurity charge for $Z \le 20$ and a saturation to constant values for larger values of Z, much below the neoclassical predictions. This is consistent with the theoretical expectation of turbulent impurity transport which becomes independent of the impurity charge at high charge.

### 3.2.5. Recent studies on the validation of impurity transport theory.
These considerations in connection with the disagreement on the prediction of hollow impurity density profiles has further motivated the joint research under the ITPA. Analyses and modeling of both JET C-Wall [147] intrinsic carbon profiles and JET-ILW [148] profiles of extrinsic helium, berillium, carbon, nitrogen and neon confirmed that the gyrokinetic simulations were predicting weakly peaked impurity density profiles in conditions where measured profiles were flat or hollow. Moreover, while the experimental results exhibit a significant variation in impurity density peaking, with helium as peaked as the electron density profile, and flat to slightly hollow carbon and nitrogen around mid-radius, the theoretical results are characterized by a more limited variation of the density peaking.

This has also motivated new experimental studies on intrinsic impurities at ASDEX Upgrade [149], with an extensive comparison of boron and helium density profiles, as well as on injected impurities with transient transport experiments, separately determining diffusion and convection in DIII-D [150] and ASDEX Upgrade [151, 152]. Consistent with the specific profile measurements of different impurities in JET-ILW [148], in [149] a large database of helium density profiles is observed to have a density peaking which mainly follows the peaking of the electron density, whereas the corresponding profiles of boron are generally flatter than the electron density. Cases with hollow density profiles were observed for both helium and boron. The comparison with quasi-linear gyrokinetic calculations of turbulent impurity transport at mid-radius is shown in figure 6. The main parametric trends are correctly reproduced by the gyrokinetic results. However, the helium normalized logarithmic density gradient $R/L_{nHe}$ is systematically underpredicted in the low NBI power cases, whereas, in contrast, the boron normalized logarithmic density gradient
<figure><img src="image_9.png" /><figcaption>**Figure 6.** The local normalized logarithmic gradient of the helium density $R/L_{nHe}$ (a)–(c) and of the boron density $R/L_{nB}$ (d)–(f) versus $R/L_{ne}$, $R/LT_i$ and $u'$ (for D) at $\rho_{tor} = 0.5$. The database points correspond to various levels of centrally applied $P_{ECRH}$, from 0.5 to 1.4 MW, on top of 2.5 MW (red and blue circles) and 7.5 MW of $P_{NBI}$ (orange and magenta squares). The open symbols correspond to the experimental gradients and the filled symbols correspond to the predictions obtained with quasi-linear gyrokinetic calculations with the GKW code [30]. Reproduced from [149]. © EURATOM 2019. All rights reserved.</figcaption></figure>

$R/L_{nB}$ is usually underpredicted for the low NBI power cases, and is largely overpredicted for the high NBI power cases, in particular at low ECRH power, where the observed boron holowness is not reproduced.

These extensive datasets of boron and helium profiles were also made possible by upgrades and improvements in the CXRS diagnostics, for light impurities [153] and helium in particular [111]. In [153], the importance of the additional emission from the beam halo population for the interpretation of the CXRS spectra has been highlighted. In [111], a kinetic forward model for the helium plume [109] emission has been developed.

In [150, 151], where the experimental analysis of the injected impurities transients allows the separate determination of diffusion and convection, measurements are compared with both quasi-linear gyrofluid and quasi-linear and nonlinear gyrokinetic simulations respectively. In [150], dedicated to high confinement regimes at DIII-D without edge localized modes (ELMs), agreement with quasi-linear gyrofluid predictions is mainly identified in the convective term, not in the diffusion. In [151], a 0.5 MW ECRH and 5 MW NBI heated H-mode plasma at ASDEX Upgrade, with additional time modulation of 1 MW ICRH power to produce a modulated edge boron source from the antenna limiters, reports experimentally determined boron diffusion and inward convection coefficients both quantitatively consistent with quasi-linear and heat flux matched nonlinear gyrokinetic predictions within experimental uncertainties. It is worth observing that this case falls in the domain of parameters where an agreement between

gyrokinetic results and observations can be expected from the results in [149].

### 3.2.6. The theoretical challenge of explaining observations of outward impurity convection.
The systematic analyses of databases of intrinsic carbon and boron density profiles, respectively from JET C-Wall [147] and ASDEX Upgrade [149] H-mode plasmas, allowed the identification of the main parameter domains in which the disagreement between theory and carbon and boron experimental profiles was large. In JET the disagreement is found to be particularly large at high collisionality, while in ASDEX Upgrade the disagreement is found to be particularly large at high ion temperature logarithmic gradient and rotation gradients [149], correlating with large NBI powers. The comparison between the data of the two devices, performed within the ITPA joint activity, showed that the ASDEX Upgrade cases with higher ion temperature and rotation gradients were close to the JET parameter space investigated in [147] and confirmed the overprediction of impurity density peaking found in that previous work. These results motivated two lines of theoretical investigation, which were performed within the ITPA group. The first, related to the high collisionality, lead to the study of the direct impact of a neoclassical equilibrium on the turbulent impurity transport [154]. The second, connected with the correlation of the disagreement with high NBI power, focused on the impact of fast ions [155]. The first study found only minor effects, which become negligible for light impurities like boron or carbon at
<figure><img src="image_10.png" /><figcaption>**Figure 7.** Comparison between experimentally inferred calcium impurity transport coefficients (solid lines and shaded areas) and corresponding theoretical predictions (dashed lines and stars for the gyrokinetic results) in L-mode (left), I-mode (center) and EDA H-mode (right). Top plots show the diffusion coefficients *D*, middle plots the convective velocity *v*, bottom plots the normalized convection to diffusion ratio $Rv/D = -R/L_n$ (from [157]).</figcaption></figure>

core plasma conditions. The second study identified an important additional ingredient, the beam ion density, which was missing in the modeling activity performed so far, and which leads to a strong reduction or even reversal of the convection at high values of beam ion density and gradients, due to a combination of effects involving both turbulent and neoclassical transport.

In Alcator C-Mod separate diffusive and convective transport coefficients of calcium have been experimentally inferred with a fully Bayesian approach and the corresponding measured density profiles [156]. The results have been compared with theoretical predictions obtained from both quasi-linear gyro-fluid and non-linear gyrokinetic simulations [157], and a good agreement has been found in general, particularly for the diffusion coefficient of all examined cases and for conditions with inward impurity convection, as shown in figure 7. At the same time, clear experimental evidence of outward impurity convection and hollow calcium density profile is obtained in an ICRH EDA H-mode plasma (figure 2, bottom-right), which cannot be reproduced by the theoretical predictions. As compared to the other examined cases, in L-mode and I-mode, this EDA H-mode case features lower current, higher collisionality, lower logarithmic density gradient, higher logarithmic electron temperature gradient, and comparable logarithmic ITG. The results provide a strong indication that, while the impact of beam ions, as identified in [155], is certainly an important physical ingredient to be included in the theoretical modeling, at the same time it cannot be considered the only missing piece leading to the solution of the disagreement between observations and predictions in conditions of outward impurity convection.

From the impurity diagnostic standpoint, clear evidence has mounted that impurity transport studies approaching the outer part of the plasma, also well inside of the pedestal, cannot ignore the impact of charge exchange between impurities and background (deuterium) neutrals [157–159].

This has important ramifications for spectroscopic measurements. As a result of these reactions, the ionization equilibrium of impurities can be radically modified, with higher charge states recombining to lower ones. This has been shown to be a fundamental ingredient for experimental analysis and interpretation of spectroscopic studies on ASDEX Upgrade [158], DIII-D [159], and even C-Mod [157], where plasma densities were much higher and neutral penetration comparably lower. Charge exchange with NBI neutrals, often used for diagnostic purposes, has also been quantified and observed to play a more limited role in modifying the ionization balance, while possibly still leading to misinterpretation of spectroscopic data in the core, especially for partially stripped ions. The consideration of the role of edge neutrals is challenging since their densities are poorly known. In the C-Mod [157] and DIII-D [159] works, neutral densities where estimated based on SOLPS-ITER runs using the EIRENE Monte Carlo neutral code, demonstrating a new element of integration between core and edge models. As the community expands its activities towards validation of pedestal and scrape-off layer (SOL) impurity transport, these efforts gain clear importance.

At ASDEX Upgrade, further developing the experimental technique which allows the separate identification of the diffusive and convective components of boron transport [151], a large database of boron transport coefficients has been assembled, allowing a practically unprecedented analysis of the experimental properties of light impurity transport in terms of separate diffusion and convection coefficients as a function of a variety of plasma parameters [160], in particular including a very large variation in NBI and ECRH powers, and a limited variation in edge safety factor. The normalized ion temperature logarithmic gradient $R/L_{Ti}$ is identified as the strongest ordering parameter for both boron diffusion and convection. It is also shown that a large value of $R/L_{Ti}$ is a necessary condition to obtain outward convection and hollow density
<figure><img src="image_11.png" /><figcaption>**Figure 8.** (Left) The measured ratio of the aluminium diffusion coefficient to the power balance effective heat conductivity in DIII-D experiments (circles) is compared to the results from nonlinear gyrokinetics simulation with the CGYRO code (stars) as a function of the corresponding ratio of the electron to ion heat flux $Q_e/Q_i$. Reprinted from [161], with the permission of AIP Publishing. (Right) The measured ratio of the boron diffusivity to the power balance ion heat conductivity in ASDEX Upgrade experiments (full circles) is compared to the quasi-linear gyrokinetic results of the GKW code (open squares) as a function of the corresponding experimental ratio of the electron to ion heat flux $Q_e/Q_i$ from power balance. Reproduced from [160]. 2021 The Author(s). Published on behalf of IAEA by IOP Publishing Ltd CC BY 4.0.</figcaption></figure>

profiles of boron. A critically important result, which confirms and extends previous knowledge [37, 125, 140], is that, while even large variations of the NBI power have relatively limited impact on the impurity transport, the inclusion of low levels of ECRH can dramatically increase both the diffusive and the convective coefficients, also leading to a moderate increase of the boron density peaking. Similar to the results on Alcator C-Mod, the comparison of these experimental results to a combination of neoclassical and quasi-linear gyrokinetic results shows a relatively good agreement on the diffusion coefficient and on the convection coefficient when this is inward directed. In particular, a non-monotonic dependence of the impurity diffusivity to heat conductivity ratio on the electron to ion heat flux ratio is demonstrated by the experimental analysis, qualitatively consistent with the theoretical predictions [31]. In contrast, even with the inclusion of the impact of beam ions, the outward convection measured in NBI dominated plasmas, featuring strong rotation and ITGs, is not reproduced by the theoretical predictions. A strong variation of the impurity diffusivity to heat conductivity ratio as a function of the electron to ion heat flux has been also observed in DIII-D experiments of aluminium transport [161], demonstrating a commonality in this generic property of turbulent impurity transport. A comparison of the experimental results, as well as of the related gyrokinetic calculations, obtained in these recent works dedicated to the separate experimental determination of the impurity diffusion and convection coefficients is presented in figure 8. The ratios of the impurity diffusion coefficients to heat conductivities for aluminium and boron in DIII-D and ASDEX Upgrade, respectively, are plotted as a function of the electron to ion heat flux ratio. In the left plot, from [161], the DIII-D experimental results are compared with nonlinear electrostatic gyrokinetic simulations. In these simulations the ion heat flux was matched by modifying the ITG in input. The experimental and theoretical aluminium diffusivity to heat conductivity ratios are plotted as a function of the electron

to ion heat flux ratio from power balance and from simulations respectively. In the right plot, from [160], the ASDEX Upgrade experimental results are compared to quasi-linear gyrokinetic results. These have been performed with nominal input parameters from profile measurements only, without attempting any matching of the electron to ion heat flux ratio. In this case, both the experimental and the theoretical results of the boron diffusivity to heat conductivity ratio are plotted as a function of the experimental electron to ion heat flux ratio from power balance. Current work, still within the ITPA, is now dedicated to nonlinear electromagnetic gyrokinetic simulations of a subset of these data.

An interesting result has recently been obtained within a statistical test-particle approach by the explicit inclusion of the polarization drift in the gyro-center test-particle motion [162]. Within this approach, the direction of the impact of the polarization drift on the impurity transport is determined by the mode propagation frequency and it is directed outward for ITG modes and inward for TEM modes. Therefore, it provides an additional contribution that, in particular, allows the ASDEX Upgrade experimental observation of hollow impurity density profiles with large NBI heating to be reproduced. In the conventional gyrokinetic formulation, the polarization effect is treated as a shielding term in the gyrokinetic Poisson equation, provided by the so-called polarization density [163]. Alternative formulations of the gyrokinetic system of equations which explicitly include the polarization drift have been derived [164, 165]. However, so far the effect of the polarization drift on the impurity transport has not been computed within a consistent gyrokinetic formalism. Moreover, an additional term in the impurity particle flux, which is produced by an externally applied toroidal momentum torque, typically by NBI, and has been largely neglected in the last decades, has been rederived very recently [166], confirming past results [167–169]. This term is directed outward with co-current injected torque and, therefore, also contributes to
the explanation of the observation of hollow light impurity density profiles with high co-current NBI. It is caused by the interaction between parallel impurity inertia, poloidal flow damping and friction, and by the link between parallel motion and perpendicular particle flux caused by the poloidal stress tensor. It arises from the impact of the turbulent toroidal momentum flow on the radial neoclassical particle flux and, as such, is not included in separate descriptions of turbulent and collisional impurity transport, and its effect has been ignored so far in recent comparisons between theoretical predictions and experimental impurity density profiles. Its effect is estimated to be large enough to provide an observable modification of the predicted profile shape of light impurities, potentially leading to the observed level of hollowness in conditions of large NBI heating power [160]. In contrast it should be expected to become relatively small in comparison to the neoclassical convection of highly charged and heavy impurities, particularly in the concomitant presence of strong toroidal rotation, typical of high NBI heating powers.

### 3.2.7. Impurity transport research in spherical tokamaks.

Investigations of impurity transport were also performed in spherical tokamaks. The dominance of neoclassical transport for externally injected neon was first reported in NSTX H-mode plasmas with NBI heating [170]. The transport of intrinsic carbon and lithium impurities was analyzed at NSTX [171], consistently revealing a dominance of neoclassical transport effects. In particular, the analysis shows that carbon tends to accumulate during the ELM-free phases (induced by lithium coating). The observed accumulation is explained by neoclassical transport in connection with modifications of the ion temperature and density profiles, whereas the high carbon concentration enhances the neoclassical lithium diffusivities, limiting the lithium concentration to very low values. One of the few recent studies providing experimentally inferred helium transport coefficients has been performed on MAST [110], by helium gas puff transient transport analysis. In the outer part of the L-mode plasmas, helium transport is dominated by turbulence, providing large diffusion and inward convection. In contrast, in H-mode the diffusivity is found at the predicted neoclassical levels, without evidence of a pinch. The turbulent pinch in L-mode is consistent with the gyrokinetic predictions for collisionless TEM, in connection with steep density gradients, while ITG are stabilized by the strong $E \times B$ shear. Building on these helium transport results, also the charge dependencies of neoclassical and turbulent light impurity transport have been experimentally investigated in MAST [172]. While carbon and nitrogen are observed to be weakly screened in the core, with an outward convection at mid-radius, helium is peaked over the entire plasma radius. Over a large central portion of the plasma, the transport is at the neoclassical level, whereas it is turbulent in the outer region, outside mid-radius, where gyrokinetic predictions confirmed the dominant role of TEM. Here the modeling, combining neoclassical with quasi-linear turbulent transport, provides reasonable estimates of the impurity transport coefficients and the impurity charge dependence. The reversal from inward to

outward convection observed around mid-radius when moving from the periphery to the center of the plasma for both carbon and nitrogen is found to be consistent with the neoclassical predictions of banana-plateau transport. Neoclassical transport can play a dominant role in this extended radial region for light impurities in a spherical tokamak, characterized by strong rotation, stabilizing ITG turbulence, and with TEM only unstable outside mid-radius, due to flat density profiles in the core. These conditions with dominance of neoclassical transport over a large portion of the radial domain appear to be specific to spherical tokamaks like MAST and NSTX, whereas in conventional aspect ratio tokamaks turbulent transport dominates over neoclassical transport for light impurities over a broader radial region approaching the plasma center. This implies that an explanation of the observed hollow impurity density profiles in the confinement region of conventional aspect ratio tokamaks has to be obtained from the turbulent transport predictions.

### 3.3. Collisional impurity transport and highly charged impurities, experimental results and operational aspects related to accumulation avoidance

Complementary to turbulent transport, which in present devices dominates the transport of light impurities in the absence of transport barriers, neoclassical transport often produces the dominant contribution to the convection of heavy, highly charged impurities. There are two basic effects which make neoclassical convection large for heavy and highly charged impurities. The first comes from a general property of collisional transport. The collision frequency is proportional to the square of the charge of each colliding particle species, and the collisional convective velocity is larger than the diffusion coefficient by an impurity charge factor. This implies that, for highly charged impurities, the collisional convective component can become dominant, or at least comparable, with respect to the turbulent convection, which, in contrast, becomes constant with increasing charge at high charge. The second effect is connected with the impact of plasma rotation or any other mechanism which produces a poloidal inhomogeneity of the impurity density. Low field side localization of the impurity density, as produced by plasma toroidal rotation, strongly increases the neoclassical transport, which, in highly rotating plasmas, can cause convective terms of heavy impurities that are largely dominant over the turbulent component.

For general considerations on the collisional transport theory we refer to the essential review by Hinton and Hazeltine [173], and, more specifically for collisional impurity transport, to the equally famous paper by Hirshman and Sigmar [119]. This review comprehensively describes the properties of neoclassical impurity transport in the non-rotating limit. The recent review on impurity transport [113] also relies on these two references for all the more general aspects, and more specifically describes the progress in parallel and radial transport obtained over the last years. We refer to this recent review for more details on all of these recent general theoretical aspects.
**3.3.1. The impact of poloidal asymmetries on neoclassical impurity transport.** One theoretical element, that is not included in that very recent review (as it became clear even more recently), is related to the impact of plasma rotation on neoclassical impurity transport in the low collisionality limit. This aspect has been identified in very recent calculations of neoclassical tungsten transport with the drift-kinetic code NEO [174-176] applied to plasma parameters achieved in the KSTAR tokamak [177], which are characterized by very high rotation at low collisionality. It is shown that in this limit of the parameters, the relative effect of the screening produced by the ITG [119, 178] is increased, leading to more favorable conditions for the avoidance of central impurity accumulation. This behavior, which is characteristic of the impurity transport in the banana regime [179], was in fact already present in the results of the first study on the impact of rotation on impurity transport [180], although this specific favorable property was not highlighted in that pioneering paper. Recent experimental results at JET-ILW [181, 182] have entered this regime, where the collisionality is low enough to reach conditions in which the impact of the plasma toroidal rotation increases the relative strength of the temperature screening.

This behavior is also consistent with the modeling results of the transient phase of an ASDEX Upgrade plasma with very high energy confinement, low collisionality and very high rotation [183]. This regime can be expected to be achieved at the low collisionalities of a burning plasma, in case of sufficiently large rotation, and could be particularly interesting for low aspect ratio devices, which are often characterized by strong plasma rotation. In contrast, at high collisionality, with highly charged impurities which typically are in the Pfirsch-Schlüter regime, toroidal rotation not only strongly increases the neoclassical transport [184-188], but can also reduce the relative strength of the temperature screening [186, 189, 190] more easily leading to central accumulation of the impurities. This effect is clearly exacerbated in the operation with NBI heating, as already documented in first observations [191, 192], since NBI produces the highly unfavorable combination of a central particle source and an external torque, which both have a negative impact on the behavior of heavy, highly charged impurities like tungsten. This unfavorable effect has been clearly demonstrated on JET-ILW and ASDEX Upgrade [183, 193-195], and has been very recently confirmed also on EAST [196]. The role of plasma rotation in inducing poloidal asymmetries of the tungsten density distribution and in modifying the tungsten impurity flux has been experimentally verified in ASDEX Upgrade, consistent with the theoretical predictions [197]. The mitigation of this effect, which unfavorably impacts the operation of present tokamak plasmas in the most common high collisionality regime of heavy impurities, critically relies on the modification of the plasma profiles in order to reduce or reverse the inward convection, typically by reducing the ion density gradient and by increasing the ITG, and/or on the relative increase of the turbulent transport, which has to offset the neoclassical convection, by

central wave heating and/or by reducing the plasma rotation and/or the poloidal asymmetry. These elements clearly explain the interest in understanding and realistically predicting the physics and the magnitude of the turbulent diffusion, already presented in section 3.2, given the critical role that it can have to avoid heavy impurity accumulation. At the same time, the strong impact of the inhomogeneity of the poloidal distribution of heavy impurities on the neoclassical perpendicular transport has renewed the interest in the study of the parallel transport of heavy impurities.

**3.3.2. Causes of poloidal asymmetries of the impurity density.** Here below, we briefly review the main aspects of these physical mechanisms, particularly those that have been considered within the ITPA.

There are three main effects which can break the homogeneous distribution of the impurity density on the magnetic flux surface. The first is the plasma toroidal rotation by the centrifugal force [180, 184-188, 198], the second is the impurity- ion friction [185, 188, 199-202], the third is the presence of particle species featuring significant temperature anisotropies, like resonant particle species in the presence of ICRH, which also develop non homogeneous poloidal distributions [203-209] and impact the impurity distribution by quasi-neutrality. Also the usually limited temperature anisotropy developed by NBI ions can have a non-negligible impact on the poloidal distribution of heavy impurities [197, 206]. Depending on the poloidal asymmetry which is generated, the neoclassical transport of the impurities, and in particular the Pfirsch-Schlüter transport, can be strongly increased, for instance in the case of strong low field side localization as produced by the centrifugal force, or decreased, for low HFS asymmetry which can offset the effect of the poloidal asymmetry of the magnetic field [186].

While the impact of toroidal rotation on the neoclassical transport has been known for a long time [180, 184, 185], the realization of its strong impact and the requirement of its inclusion in the modeling of experiments has been highlighted only recently, particularly within the ITPA [183, 189, 193], thanks to the modeling possibilities offered by the NEO drift-kinetic code. Figure 4 shows the impact of rotation on the NEO predictions for the diffusion and the convection terms of W for three time slices of a JET-ILW discharge, from [189]. Not only the diffusion and the convection are strongly increased, figure 9(a), but also the convection to diffusion ratio is modified, figure 9(b), and the convection becomes more inward, due to a reduction of the strength of the temperature screening component produced by the centrifugal effects. Considering the edge region of the plasma, including the H-mode pedestal, parallel friction [185, 199, 202] and centrifugal effects from poloidal rotation [210] can become important. These effects have been also shown to produce poloidal asymmetries in B2SOLPS5.2 simulations [211], with increased impurity density on the HFS in the region of the ETB, consistent with observations in AUG and MAST.
<figure><img src="image_12.png" /><figcaption>**Figure 9.** Profiles of the neoclassical W diffusion to ion heat conductivity ratio $D_{WNEO}/\chi_{iNEO}$ (a) and the neoclassical W convection to diffusion ratio $RV_{WNEO}/D_{WNEO}$ (b) computed by NEO, with (full symbols) and without (open symbols) the inclusion of rotational (CF) effects, at the three time slices 9.32 s (triangles pointing down) and 10.42 s (circles) and 12.37 s (triangles pointing up) of the JET-ILW pulse #83351. The inclusion of rotational effects strongly increases both neoclassical diffusion and convection, and can produce less outward, or more inward, convection, due to a reduction of the temperature screening effect. Reprinted from [189], with the permission of AIP Publishing.</figcaption></figure>

### 3.3.3. Impact of the unfavorable properties of neoclassical transport on tokamak operation to avoid impurity accumulation.
The unfavorable properties of collisional transport imply that operation with highly charged metallic elements, such as tungsten or molybdenum, in the plasma facing components requires some precautions in order to avoid accumulation [212–214]. Two concomitant main lines have to be pursued for safe operation. The first element is related to the impurity concentration at the edge, which has to be kept small. This can be obtained by limiting the impurity sources from the wall, as well as by sufficient transport in the pedestal region. The latter can be produced by regular ELMs [215, 216] or by favorable pedestal conditions, in particular with strong ITGs [217]. ELMs are observed and predicted to produce a net tungsten outward transport in present conditions with inward neoclassical convection and peaked impurity density profiles in the pedestal. At the same time, in the case of weak main ion density gradients in the pedestal, and consequent outward neoclassical convection and hollow tungsten density profiles, ELMs are predicted to produce an opposite effect and increase the tungsten density in the core [216]. The impurity transport in the H-mode pedestal region is experimentally found

to be mainly caused by neoclassical convection [215], which strongly increases the impurity confinement time in H-mode plasmas when the main ion density gradient inward component dominates the neoclassical convection.

In these conditions, a sufficiently high ELM frequency is required to limit the edge tungsten concentration, which typically imposes operational constraints to the minimal level of allowed gas puff. High gas puff levels can have detrimental effects on the energy confinement [218–223]. Non-stationary ELM-free conditions are prone to an increase of the tungsten concentration at the edge and eventually to a radiation collapse. The same property of relatively long impurity confinement times characterizes stationary high confinement regimes with small or no ELMs that maintain a H-mode density pedestal, featuring an ETB with reduced particle turbulent transport, such as the EDA H-mode regime [224, 225], as well as the quiescent H-mode regime and regimes characterized by active ELM-suppression [150]. However, while particle transport, and therefore also impurity transport, is in general reduced in the radial region of the density pedestal, specific properties of the different conditions which characterize the various high confinement regimes of stationary operation, with small or no ELMs, allow certain regimes to be less prone to, or even free from, problems connected to W accumulation. This is particularly the case of the conditions of the quasi-continuous exhaust regime, with small type II ELMs, [226], due to the high density in the SOL and consequent cold far SOL and reduced tungsten sources, in combination with possible increased transport close to the separatrix. This is also the case of conditions of ELM suppression with externally applied magnetic perturbations [227, 228], because of the usually lower edge density and reduced density pedestal and the possible increased transport in connection with the peripherally modified magnetic topology. Stationary ELM-free regimes, like the EDA H-mode [224, 225, 229, 230] are often characterized by the presence of quasi-coherent fluctuations at the edge. These keep the operational conditions below those which would lead to an ELM trigger and can also have favorable consequences on the impurity behavior, leading to a reduction of the impurity content in the plasma [224, 229, 230] with respect to similar plasma conditions featuring ELMs. A stronger reduction of the impurity confinement time with increasing heating power as compared to the reduction of the energy confinement time has been observed in the EDA H-mode regime in ASDEX Upgrade. The general property of relatively long impurity confinement times which characterizes the regimes with a strong density pedestal consistently also implies that, in contrast, a reduced impurity confinement time is present in regimes which are not characterized by a strong density gradient at the edge. This is the case of the L-mode or the I-mode, as demonstrated by a systematic comparison in Alcator C-Mod [225]. Here the increased particle transport as well as the reduced neoclassical inward convection, due to a reduced main ion density gradient at the edge, lead to significantly shorter impurity confinement times, where in particular the I-mode regime breaks the otherwise usual correlation between impurity confinement and energy confinement times.
A second element which allows the avoidance of heavy impurity accumulation is related to the impurity profile shape in the core. The impurity density profile has to feature a limited peaking in the core, particularly without developing strong central gradients close to the magnetic axis. A central particle source, as produced by operation with NBI heating, has an unfavorable impact by increasing the peaking of the main ion density profile close to the magnetic axis. In contrast, operation with wave heating only is clearly much less prone to accumulation, because of the combined beneficial effects of absence of a central particle source and of an external torque, as shown in both C-Mod [231] and more recently at WEST [232].

Central wave heating also has very beneficial effects in the case of operation with NBI heating, due to a combination of effects, which can favorably act concomitantly. A first element is provided by the increase of the impurity turbulent diffusion [125, 160, 161, 233]. The increase of central turbulent transport can also reduce the peaking of the density profile close to the magnetic axis, more strongly balancing the impact of the NBI particle source. An additional and related element, also provided by the increase of the central turbulent transport due to the central wave heating, is connected with the increase of central toroidal viscosity and the reduction of the plasma toroidal rotation [234], which can also be produced by a modification of the turbulent residual stress. These effects, mainly connected with an increase of central turbulent transport, can be considered to be common to both ECRH and ICRH. There are, however, also some differences between the effects of ECRH and ICRH. First, it has to be mentioned that ICRH with W-coated antennas has been observed to increase the tungsten sputtering, due to radio-frequency (RF) sheaths driven by parallel RF electric fields, an effect which has been strongly reduced on ASDEX Upgrade by offsetting the effect of the electric fields with a three-strap antenna [235]. Second, ICRH produces highly energetic ions with strong temperature anisotropy, providing an additional source of tungsten poloidal asymmetry [203-209].

Fast ions can also contribute to the total temperature screening [183], an effect which strongly depends on the actual peaking of the effective temperature profile of the fast ions. The predicted strength of this effect is observed to depend on the impact of finite orbit width effects in the modeling of the fast ion distribution [195]. Central ICRH and ECRH have the tendency to increase the overall turbulent transport level. This can reduce the impact of the central particle source, particularly in case NBI heating is applied, or of the Ware pinch, in conditions in which this is significant, close to the magnetic axis. Therefore, central ECRH and ICRH can reduce the central peaking of the main plasma density [191, 236]. Finally, ICRH produces both electron and ion heating, whereas ECRH only heats the electrons. While ion heating can be efficient in increasing the ITG and increasing the neoclassical screening, depending on the plasma heating conditions, the electron heating can have a stronger effect in increasing the turbulent diffusion [31]. A comparison on the efficiency of ICRH and ECRH in preventing tungsten accumulation in

ASDEX Upgrade has shown that it is the electron heating fraction which orders the peaking of the tungsten concentration, providing experimental indication that, at least in the conditions of those experiments, it is the electron heating fraction that has to be maximized for a more efficient avoidance of tungsten accumulation [194]. It has to be underlined that while on-axis ECRH and ICRH are widely documented to have these beneficial effects in avoiding central impurity accumulation, the application of too off-axis ECRH or off-axis ICRH usually does not allow one to reach the same type of effects and thereby does not lead to the avoidance of central impurity accumulation [125, 207, 237]. Recently it was reported that in EAST the application of LHW also helps to avoid the central tungsten accumulation in dominant NBI heating plasmas [214, 238]. Integrated modeling is performed like in [233], and the results suggest an increase in the turbulent diffusion by increasing the electron temperature and a reduced inward neoclassical convection by decreasing the toroidal rotation.

### 3.3.4. Integrated modeling including impurity transport.

The increased understanding of both the neoclassical and the turbulent transport processes in impurity transport, and particularly the clear identification of the critical ingredients leading to neoclassical central accumulation of heavy impurities, allows the integrated modeling of the impurity behavior to be performed, making use of a combination of theory-based models. Recent state-of-the-art examples of this activity, which apply completely theoretical models for the description of neoclassical and turbulent transport, are [195, 239]. The latter [239] is based on a fully integrated core-pedestal modeling framework, including a model for the pedestal structure and the application of neural-network accelerated models, also for light impurity turbulent transport. It makes use of the TGLF-NN turbulent transport model, also trained for impurity transport, coupling the transport solver TGYRO [240] with the STRAHL impurity transport code [241]. It compares predictions of this highly integrated modeling workflow with DIII-D experimental results and also includes predictions of the impact of increasing concentration of neon on the ITER baseline scenario. The former integrated modeling approach [195] adopts the JINTRAC [242] integrated transport simulator and also models the transport of heavy impurities like tungsten and nickel for comparison with JET experimental results and to optimize tungsten control with ICRH. It further develops a previous multi-channel integrated modeling effort [243] and applies the new modeling framework to the prediction of the time evolution of the tungsten behavior in JET. It combines the comprehensive description of neoclassical impurity transport provided by the NEO code [176], also applied in [239], with the turbulent transport, including impurities, of the QuaLiKiz [244] quasi-linear transport model. The turbulent transport model includes the impact of poloidal asymmetries [245, 246]. These transport simulations allow comprehensive descriptions in an integrated modeling approach of the physical effects which have been separately
<figure><img src="image_13.png" /><figcaption>**Figure 10.** (Left) Evolution of 1D profiles of soft X-ray tomography for JET pulse #92398 compared to forward modeled SXR emission from tungsten, nickel and beryllium in predictive simulation. The magnetic axis is at $R = 3.05$ m. Before central accumulation, tungsten and nickel are localized on the LFS by rotation. The solid curves are at the same times in sim. and experiment, the dashed curves indicate later times in expt. (Right) Poloidal cross-sections of predicted 2D density distributions of tungsten before and after the start of central accumulation. nickel evolves similarly. Reproduced from [195]. © EURATOM 2020. All rights reserved.</figcaption></figure>

identified in previous dedicated investigations with the drift-kinetic and gyro-kinetic modeling of specific experimental observations. An example of the results obtained with this comprehensive integrated modeling approach is presented in figure 10. Following the approach developed in [239], more recent works have been dedicated to the prediction of tungsten impurity and radiated power density profiles in EAST plasmas [196, 233, 247] respectively considering an improved high-performance fully non-inductive discharge, a hybrid scenario discharge and a long-pulse steady-state H-mode discharge.

At the same time, the computational requirements of the drift-kinetic code NEO as a coupled module in the framework of transport modeling are heavy and have motivated the development of reduced analytical models, which can closely reproduce the NEO results. An example is the FACIT model [179, 188], which has been also extended to include the impact of rotation at low collisionality [179].

A even more recent application [248] has been dedicated to the modeling of a pair of JET plasmas in which an increase of confinement is observed with neon seeding in comparison to the unseeded case. This includes the modeling of the neon density profiles by combining the QuaLiKiz and NCLASS for turbulent and neoclassical transport and also includes a comparison between QuaLiKiz neon impurity transport predictions with the results of the GENE gyrokinetic code.

### 3.3.5. Impact of core MHD modes on the impurity behavior.
Although not directly connected with actual transport mechanisms produced by collisions or turbulence, additional important effects on the impurity behavior are caused by the occurrence of MHD instabilities. These can be mainly classified in two groups. The first group is given by the impact of modes connected with the presence of the $q=1$ surface, in particular sawtooth relaxations as well as ideal or resistive deformations of the magnetic equilibrium in the innermost region of the plasma, affecting the impurity behavior in the proximity of the magnetic axis. The second group is connected with the occurrence of neoclassical tearing modes (NTMs). Therefore, these are effects which impact the impurity behavior in a

radial domain which is more usually located around mid-radius. Sawteeth, as well as fishbones, have been historically recognized to have a beneficial effect against central impurity accumulation, producing a flattening of the central impurity density profile and consequently a more off-axis redistribution of the profile of the radiated power density [192, 249, 250]. Consistent observations of this phenomenology have been also reported and analyzed more recently [251]. Moreover, deformations of the magnetic equilibrium in the proximity of the magnetic axis, as produced by the presence of long-living, saturated internal kink modes at the $q=1$ surface, can significantly modify the impurity behavior, particularly through an interplay with central ECRH, leading to experimentally measured centrally hollow impurity density profiles [252, 253]. These central impurity holes cannot be explained by the usual combination of neoclassical and turbulent transport effects in the presence of an axisymmetric magnetic equilibrium [194, 253] and still await a physical explanation, likely connected with 3D effects of this helical magnetic core on the transport. The occurrence of NTMs is observed to impact the impurity behavior [193, 254] and experimental evidence is obtained that the tungsten accumulation can be initiated or accelerated by the appearance of the modes [254].

### 3.4. Direct interaction between collisional and turbulent transport mechanisms

Within the ITPA on impurity transport, regular emphasis has been given also to theoretical investigations of mechanisms of synergy between collisional and turbulent transport. Usual modeling computes the collisional and the turbulent transport components separately. The disagreement which is obtained between theoretical predictions and experimental results in some conditions, particularly in cases of measured hollow impurity density profiles, have motivated the investigation of effects which can produce direct mutual impact between collisional and turbulent transport. As already mentioned, specific work has been dedicated to the role of a background distribution function provided by the neoclassical local equilibrium on the turbulent transport predictions [154]. Different effects are
identified which modify the roto-diffusion component of the turbulent transport, but none of these has been found to be large enough to significantly impact the predictions of light impurity turbulent transport in the core. In contrast, these effects can become non-negligible and experimentally measurable only at high collisionality for highly charged impurities. Another line of research, which has been also regularly included in the ITPA, is dedicated to the identification and quantification of the impact of turbulence on the collisional transport. These effects can arise when turbulence can modify the macroscopic properties of the plasma directly on the turbulence time scales, in particular through the development and sustainment of mesoscale structures [255-257] and of large scale poloidal asymmetries [258, 259]. These effects usually require global gyrokinetic codes for their computation. Through these structures, a direct interaction between turbulent and collisional transport can take place, which affects the transport properties so that the transport cannot be considered to be provided by the sum of the collisional and turbulent processes computed separately. This becomes particularly important for high-charge impurities like tungsten [258, 260], with the collisional transport being modified by turbulence driven poloidal asymmetries [260]. An additional synergy effect is due to the transient development of non-ambipolar turbulent fluxes of electrons and main ions by bursty ITG turbulence, leading to a fast growth of the radial electric field and the consequent development of up-down asymmetric density perturbations and enhancement of the neoclassical transport [261].

All of the mechanisms connected with the modification of the collisional transport component by direct effects provided by turbulence, or, more in general, whatever type of synergistic effect between the two transport components, are predicted to be larger at high charge. Any experimental demonstration of the presence of these effects therefore is expected to be more easily obtained in the analysis of the transport of heavy, highly charged impurities, possibly in the absence of a strong toroidal rotation or other strong external sources of poloidal asymmetries.

### 3.5. Conclusions and outlook

Over the last decades, significant progress has been made in the experimental characterization and the theoretical prediction of impurity transport in tokamak plasmas. It is widely recognized that the impurities will play a critical role in the operation of a tokamak nuclear fusion reactor, particularly for achieving practical solutions for the combination of the requirements of good core confinement and of tolerable edge particle and heat exhaust. At the same time, the acceptable levels of impurity contamination remain stringent with respect to central dilution and central radiation [118]. The capability of predicting and controlling the concentration levels and the density profile shapes of both intrinsic and extrinsic seeded impurities plays therefore an important role. Light impurities are regularly observed to not produce any significant accumulation, with concentration profiles which are flat or even hollow. The diffusion coefficients are

large when turbulent transport dominates, and are observed to strongly depend on the electron to ion heat flux ratio, in at least qualitative consistency with the theoretical expectations. However, a more complete characterization of the transport coefficients of helium from the experimental standpoint and a more stringent validation of the present predictive capabilities remain important research goals, given the importance that core helium diffusivity plays in the determination of the central helium concentration in a burning plasma. This is also directly related to the requirements in the pumping efficiency, for the control of the concentration levels at the plasma edge. The validation of the turbulent transport predictions against observations of light impurity transport has received increased consideration in recent years. While several experimental trends are qualitatively reproduced and cases of quantitative agreement are reported, there are still effects, particularly connected with experimentally measured outward impurity convection and the observation of hollow impurity density profiles, which are not quantitatively predicted by the theory, although some related trends are reproduced. The possibility of predicting a reversal of the impurity convection can be considered an important and challenging test for theory validation, particularly with respect to the completeness of the theoretical model and the adequacy of the modeling approach. At the same time, theoretical models are in agreement with the general experimental observation of absence of turbulent transport mechanisms leading to strong central accumulation. This overall consistency gives a fundamental level of reliability to the present modeling capabilities which should be pursued also within integrated modeling approaches in transport codes, enabling the quantification of profile effects. In this respect, the under-prediction of outward convection can be considered to provide conservative results regarding the impurity behavior. The goal of achieving quantitatively validated theoretical predictions of the impurity diffusion coefficients should be regarded with equal priority, in particular in connection with the need of predicting the helium density profile in a burning plasma. Here qualitative agreement in some trends is obtained and specific evidences of quantitative agreement are also reported, particularly when physics comprehensive nonlinear gyrokinetic simulations are performed. However, a complete identification of the conditions under which quantitative agreement is not obtained and of the causes are still outstanding elements.

The use of highly charged heavy impurities as plasma facing material, such as tungsten, particularly on the divertor plates, has given strong priority to the complete understanding of the behavior of these impurities, from both the experimental standpoint, also for the practical purpose of safe tokamak operation, and the theoretical standpoint. Heavy, highly charged impurities are also the preferred seeded radiators in a reactor, since they have significantly higher cooling factors and thereby produce less fuel dilution for the same level of required radiated power. The importance of poloidal asymmetries developed by heavy impurities in determining the observed transport levels, particularly for the usually large neoclassical convection, is now widely recognized, and well
described by the neoclassical transport theory. The role of neoclassical transport in the peripheral region is particularly critical when the main plasma density gradient in the pedestal is strong, leading to strong inward convection. Reduced strength of the density gradient in the pedestal region has clear beneficial effects in reducing the concentration of impurities at the pedestal top, as observed in confinement regimes which do not feature strong density pedestals, like the L-mode and the I-mode. In general these confinement regimes do not lead to problems of impurity accumulation and do not require the adoption of specific precautions in the operation. In contrast, the strong density gradients of the main plasma developed in the pedestal of the H-mode confinement regime in present devices, particularly at high currents and low edge safety factors, are reason of concern and lead to specific requirements in the operation to avoid central accumulation. The operational solutions, which have been identified and are regularly adopted, are also well understood with respect to the edge behavior, including the role of gas puffing and of ELMs, and the behavior at the plasma center, particularly in connection with the role of central wave heating, and particularly electron heating. Stationary ELM-free H-mode regimes or H-modes regimes characterized by small (type II) ELMs, which also feature relatively long impurity confinement times as compared to L-mode and I-mode, usually exhibit more favorable impurity behaviors with respect to regular H-modes with type I ELMs. In contrast, non-stationary ELM-free phases are highly prone to impurity accumulation.

The deleterious role of plasma rotation on neoclassical heavy impurity transport, as typically produced by NBI heating in the usual intermediate to high collisionality regime of operation, is robustly documented in present devices. This deleterious effect vanishes, and is even reversed into a favorable effect, characterized by increased temperature screening, at very low collisionality and high rotation. A more complete experimental documentation of this low collisionality, high rotation regime in present devices would be of interest and would enable a more reliable validation of the theoretical predictions. This would be of particular interest for the development of high confinement and high rotation advanced tokamak scenarios, which could be naturally free from heavy impurity accumulation. At the same time, the expected levels of rotation in a reactor from both external NBI torque and intrinsic turbulent residual stress are relatively limited. Therefore, while the possibility of accurately predicting the rotation level in various reactor scenarios would allow a more precise determination of the impurity transport coefficients of heavy, highly charged impurities, at the same time it can be at least concluded that these relatively limited rotation levels should not be expected to produce operational complications from the standpoint of the heavy impurity behavior. Indeed, H-mode operation in present devices in the absence of strong toroidal rotation is regularly observed to be free from central accumulation, consistent with theoretical expectations. This gives confidence on a more favorable behavior of heavy impurities like tungsten in a reactor with respect to present devices operated with strong

NBI heating and featuring strong rotation. Moreover, if in a reactor the density profiles at the edge will also be characterized by relatively low density gradients, in connection with reduced neutral penetration and high separatrix densities, this will also be favorable in limiting the inward neoclassical convection in the pedestal region, or even reversing it, leading to hollow impurity density profiles. However, in case of a hollow impurity density profile in the pedestal, ELMs are predicted to produce a net tungsten transport directed inward. Strong ELMs should in any case be avoided in a reactor plasma. A more complete characterization of the impurity transport behavior in these potentially relevant conditions of weak pedestal density gradients in reactor H-mode plasmas would certainly be of interest, although these conditions are very difficult to obtain in present devices.

Finally, the low collisionality regime which will be achieved in a reactor burning plasma will naturally reduce the role of neoclassical transport in comparison to the role of turbulent transport. Given that turbulent transport in general leads to relatively flat or even hollow impurity concentration profiles, this is an additional beneficial element in the comparison between operational conditions in present devices with those expected in a reactor burning plasma.

An extremely important research area is dedicated to the experimental investigation and the theoretical prediction of transport mechanisms of impurities in the plasma region across the last closed flux surface (LCFS). This relatively thin layer of plasma determines the impact of the peripheral sources of impurities from the wall on the impurity levels at the foot of the pedestal region. It provides the boundary conditions for the confined plasma and connects it to the regions with open field lines. In this peripheral plasma region also the behaviors related to the impact of impurity seeding and divertor plasma detachment are determined. All of these elements require theoretical understanding and consistent integration from the modeling standpoint in order to more robustly tackle the general problem of the integration of core and edge requirements in tokamak operation. This critical research area has not been considered among the topics of the ITPA TC topical group, which has always been more dedicated to the plasma core, and is not reviewed in this chapter. Work related to this topic has been done in both the pedestal and the divertor-SOL ITPA groups and is presented in those contributions to this Nuclear Fusion Special Issue [21, 262]. We limit ourselves in providing a short list of recent publications, in which various aspects related to the impurity behavior are considered [263–269].

## Acknowledgments

The Authors would like to warmly thank all of the colleagues who have contributed to the ITPA on impurity transport, for the many interesting and valuable discussions during the ITPA meetings, and for the very fruitful collaborations that the ITPA has fostered.
# 4. Ion and electron thermal turbulent transport

P. Mantica, J. Citrin and A. Mariani

## 4.1. Introduction

The understanding of core heat transport (where by core we mean the region inside the separatrix in L-mode and inside the pedestal top in H-mode) is of key importance for the optimization of tokamak global performance, particularly in the presence of metallic walls posing constraints on edge parameters. The theoretical description and experimental validation of turbulent transport has been a topic of intense investigation in the past two decades (see e.g. [163, 270–273] as reviews). This has led to enormous progress both in the understanding and in the modeling capability of heat transport processes, thanks also to the availability of very sophisticated diagnostics and the execution of refined dedicated heat transport experiments.

Core heat transport has been a topic of collaborative research within the ITPA TC topical group since early times and was, therefore, a focus of the two previous IPB papers: in [274] only the phenomenological aspects were addressed, whilst in [3] the fundamental turbulent transport processes were reviewed, together with the modeling capabilities by means of local gyrokinetic codes and first quasilinear transport models, and their validation against experiments. Since [3], research has progressed in several directions. Gyrokinetic theory has been extended to include fully electromagnetic, nonlinear, gyrokinetic equations for tokamak edge turbulence [275], improved collisional operators [276] and $E \times B$ treatment [277], global simulations, flux driven simulations within a full-f approach [278–280], interplay with neoclassical transport [281, 282] and with energetic particles [283–286], and multi-scale interactions at real mass ratios (see section 4.3), with a general increase in numerical resolution routinely used as available computer power has increased. The development of quasilinear models has seen major breakthroughs, as summarized in section 9, leading to two mainly used reduced models, TGLF [33, 287, 288] and QuaLiKiz [29, 246], that provide very good capabilities to reproduce experimental results with multi-channel integrated modeling. New experiments have been made with improved diagnostic capabilities, significantly expanding the validation efforts for various devices, providing the response of ion and electron temperature profile gradients to ion and electron heat flux scans and/or power modulation, to compare with theory predictions, as well as turbulence measurements coupled with synthetic diagnostics. All this work has allowed many open issues in the transport field to be solved, an example, among many, being the well known mystery of the improvement of confinement with isotope mass, at variance with simple turbulence scalings (see [289, 290] for a review of main results on isotope effects on JET and AUG).

Out of this enormous mass of results, this paper will focus specifically on those produced within the ITPA Transport & Confinement Joint Activity dedicated to the study of the ion

and electron critical gradient and profile stiffness and their parametric dependencies. We recall here that drift wave turbulence is driven by the logarithmic temperature gradients $R/L_T = -R\nabla T/T$, with R a normalizing factor taken here as the tokamak major radius. We define as the ‘critical gradient’ the $R/L_{T(i,e)}$ value above which linear instabilities are excited (with ion temperature for ion driven instabilities, and electron temperature for electron driven instabilities), and as ‘stiffness’ the rate of change of the gyroBohm normalized heat flux with increasing logarithmic driving gradient. This joint activity was established in the programme of the ITPA TC group starting from the end of 2008, motivated by the need for deeper experimental investigations of electron and ion heat transport, to provide more stringent tests for model validation than just profile modeling or a single point flux match. We recall that the transport mechanisms described in this section are not restricted to specific operational scenarios but hold in general in the plasma core, with the exception of reverse magnetic shear scenarios which have not been included in the investigation due to their limited extrapolability to a reactor. The work has been structured in two main blocks, respectively on ion critical gradient and stiffness, covered in section 4.2, and on electron-scale heat transport and multi-scale interactions, covered in section 4.3. Main conclusions and an outlook are provided in section 4.4.

## 4.2. Ion critical gradient and stiffness

Experiments to characterize ion heat transport via determination of the ion critical gradient and stiffness started later with respect to those on electron heat transport (see e.g. two reviews [291, 292]) due to the lack of spatially and temporally resolved ion temperature ($T_i$) diagnostics and to the difficulty of obtaining a local ion heat source using ICRH or NBI, as opposed to the strong electron heat source localization possible with ECRH.

Improved Active Charge Exchange (CX) diagnostics made these studies possible on JET, ASDEX-Upgrade (AUG) and DIII-D, which have been the 3 devices mainly contributing to this topic. These studies are therefore completely new since the 2007 IPB review [3]. The ($^3$He)–D ICRH absorption scheme was successfully used on JET [293–295] to provide localized ion heating both on and off axis, for scans of the ion heat flux at near constant temperature, to probe the flux vs logarithmic gradient dependence and identify ion temperature critical gradients and stiffness. On AUG [296] and DIII-D [297, 298], NBI was used as the ion heating source instead, providing valuable results although requiring a more involved interpretation.

These new results have provided very constraining experimental evidence for theory validation, in particular for the striking and unprecedented observation that in some regimes, ion stiffness decreases with increasing power. Understanding this effect involved the incorporation of electromagnetic (e.m.) and energetic (fast) ion effects within nonlinear gyrokinetic simulations, requiring a decade of theoretical studies, and brought up a number of modeling challenges, especially for
advanced scenarios, which are still not solved. A review of such effects has been recently published [299], to which the present paper refers for a broader and more detailed overview of both experiments and theory.

### 4.2.1. Parametric dependencies of ion critical gradient.
The threshold value of $R/L_T$ for turbulence onset is identified experimentally by reaching the condition of heat flux close to 0 inside a given radius, i.e. by localizing external power off-axis, or alternatively by performing a scan of heat flux by changing the proportion of on- vs off-axis heating and calculating the $R/L_T$ intercept for 0 flux, whilst the slope yields the stiffness. This implies that neoclassical ion heat transport is negligible, which is a good assumption for tokamaks outside the magnetic axis region [240]. To aid with the interpretation of the measurements against local gyrokinetics, it is useful to normalize to the natural dimensionless units of the system, i.e. gyroBohm units, e.g. $q_{i,e}^{gB} = q_{i,e}(eBL^2)/(nT^2\rho_{ref})$, with n the reference density, T the reference temperature, L a reference length, and B the reference magnetic field. $\rho_{ref}$ is the reference Larmor radius $\rho_{ref} = (mT)^{1/2}/eB$, with m the reference mass, most often taken as the main ion mass, and e the electron charge. The specific reference values (e.g. $T_i$ or $T_e$ for reference temperature, major or minor radius for reference length, etc) may vary for different authors or codes, and care must be taken when comparing. If the heat flux scan is experimentally repeated with different values of one dimensionless plasma parameter (to be kept constant during each heat flux scan) then the threshold dimensionless parametric dependencies can be measured. This requires very careful dedicated transport experiments, which were performed on JET [295] and AUG [296]. An example of expected parametric trends can be seen in an analytical formula from [300], which in the flat density limit predicts dependencies of ion threshold on $T_e/T_i$ and $\hat{s}/q$ as

$$R/L_{T_i}^{\text{crit}} = 4/3(1+T_i/T_e)(1+2\hat{s}/q) \quad (5)$$

where $q$ is the safety factor and $\hat{s}$ the magnetic shear. More exact absolute predictions and multivariate parametric trends are found through carrying out gyrokinetic calculations, e.g. as in [301]. Figure 10 in [295] shows the ion threshold measurements at different $T_e/T_i$, in good agreement with linear predictions by the gyrokinetic code GKW [30], whilst figure 11 in [295] shows the dependence on $\hat{s}/q$, also found in good agreement with the gyrokinetic code GS2 [302]. Also figure 7 in [296] supports the $T_e/T_i$ dependence of the ion threshold predictions.

Particular attention has been given both in experiments and modeling to the stabilizing effect of the $E \times B$ flow shear. This introduces a shift in the effective ion threshold, as discussed e.g. in [270]. Early simplified treatment of this effect [303, 304] was based on the replacement in a transport model of the maximum linear growth rate $\gamma_{max}$ by an effective growth rate $\gamma_{eff} = \gamma_{max} - \alpha_E \gamma_E$ where

$$\gamma_E = \left| \frac{RB_p}{B_T^2} \frac{d}{dr} \frac{E_f}{RB_p} \right| \quad (6)$$

with $B_p$ and $B_T$ being the toroidal and poloidal fields, $E_r$ the radial electric field and $\alpha_E$ a constant. This ‘quench rule’ often provides a significant degree of $E \times B$ stabilization, although this can be reduced by kinetic electrons, TEM drive, and destabilizing PVG which depends on the magnetic geometry [305]. A general quench rule model would demand a parameterization of $\alpha_E$ taking this into account. A more nuanced approach revolves around the finding that the $E \times B$ flow shear induces a shift in the peak of the radial wavenumber spectrum of electric potential fluctuations, and a reduction in their amplitudes. The occurrence of this shift has been recently measured directly in a dedicated experiment [306]. Such a spectral shift can be modeled analytically [307] and is now the paradigm for the $E \times B$ implementation in the TGLF model, with a generally reduced effect of stabilization compared to the previous implementation, in better agreement with experiments. In QuaLiKiz both parallel and perpendicular flow shears are taken into account through a sheared Doppler shift of the mode frequency, a modification to the diamagnetic frequency, and through inducing a shift in the ansatz Gaussian in the eigenfunction calculation, which leads to symmetry breaking and momentum transport [308].

In higher fidelity gyrokinetic codes such as CGYRO [277, 309] and GENE [310, 311], the implementation of the $E \times B$ shear can be taken into account through radial wavevector evolution methods. GENE uses a time-dependent wavevector shift scheme [312], which has been recently corrected to avoid spurious nonlinear couplings [313], whereas CGYRO uses a continuous wavevector advection scheme [277]. The impact of $E \times B$ shear has been shown to be negligible in nonlinear gyrokinetic simulations, for low-flux regimes with strong ITG stabilization due to e.m. effects [314]. For the same settings but taking the electrostatic (ES) limit, $E \times B$ stabilization is predicted. While conjectured to be related to predicted decreases in auto-correlation time and increased parallel mode structure in the e.m. case, the reason for this observation is still under investigation.

### 4.2.2. Ion stiffness measurements.
Measurements of ion or electron stiffness are obtained by performing a scan of the ion or electron heat flux by changing the proportion of on- vs off-axis heating and calculating the slope of the $q_{i,e}^{gB}$ vs $R/L_{T_i,e}$ curve. Alternatively, power modulation experiments allow an independent determination of stiffness, extracting from the heat pulse propagation the local derivative of the $q_{i,e}^{gB}$ vs $R/L_{T_i,e}$ curve, as described in detail in [292]. The first measurements of ion stiffness were performed in JET L-mode plasmas, thanks to well time and space resolved CX diagnostics for $T_i$ and to the availability of ICRH power in the ($^3$He)–D scheme, providing localized ion heating with short slowing down time [293–295]. This allowed both the heat flux scan and the modulation techniques to be used. On AUG, the heat flux scan was made by using on- and off-axis NBI [296], without modulation. In addition, ICRH in ($^3$He)–D was applied, but only centrally to study the effects on local transport without heat flux scans or modulation [315].
On DIII-D, different levels of centrally localized NBI power with co- or balanced injection were applied in H-mode plasmas to study the effects on local transport with and without rotation, but no heat flux scans or modulation were performed [297, 298].

Figure 11(a) shows the results of the JET ion heat flux scan. A significant reduction of core ion stiffness at high NBI power was observed, leading to very peaked $T_i$ profiles in the inner half of the plasma (figure 11(b)). The local slope of the $q_i^{gB}$ curve from modulation (indicated by the two segments on the encircled points) confirms the stiffness reduction. This effect was found to be enhanced by low magnetic shear ($\hat{s}$). Although in JET, with NBI heating, power and rotation are inherently coupled, this effect cannot be ascribed to increased rotation, which is mainly producing a threshold up-shift, partly compensated by the destabilization due to the PVG, as discussed in [316, 317]. Instead, an explanation was found in terms of non-linear electromagnetic (e.m.) stabilization linked to thermal and suprathermal pressure [316, 317]; the associated complex physics is discussed in detail in [299] and will be summarized in section 4.2.3. These effects have been the object of intense theoretical studies in the last decade. Additional interesting JET results were the repetition of the ion stiffness measurements in Hydrogen, finding a much lower reduction of ion stiffness [318]. This was found due to differences in the fast ion content between H and D plasmas, inducing lower nonlinear e.m. stabilization in H, thus providing an explanation for the observed anti-gyroBohm dependence of the core transport in these discharges. A similar effect was found in H versus D H-mode plasmas in AUG [319]. Instead, in AUG L-mode plasmas with pure electron heating and no fast ions, ion stiffness was found independent of mass, with mass entering only in the energy exchange term between electrons and ions [320].

The experiments on AUG and DIII-D support the same type of phenomenology. In AUG [296], moving NBI power from off-axis to on-axis leads to a non-monotonic $q_i^{gB}$ vs $R/L_{T_i}$ trend, interpreted as an increased ion stabilization for central NBI power. This seems less effective for higher $T_c/T_i$. Also on AUG, central ICRH in ($^3$He)–D minority scheme leads to very peaked $T_i$ profiles in the inner region [315], very similar to those observed on JET (figure 11(b)). On DIII-D the role of e.m. stabilization and of rotation were clearly identified by a NBI power scan with co- and balanced injection, the results of which are shown in figure 12 [297]. Here the shift between co- and balanced NBI points is indicative of the rotation effect, but in both sets we see the trend of decreasing $q_i^{gB}$ with increasing $R/L_{T_i}$ seen in JET and AUG. We remind that in figure 12, being a power scan for central heating and not a heat flux scan, we only see the tips of the curves shown in figure 11. These observations have been explained by gyrokinetic modeling as due to nonlinear e.m. stabilization associated with increasing thermal pressure, with little contribution from fast ions [298], which were instead a dominant contribution in the cases of JET and AUG experiments.

Besides these specific transport experiments, a general experimental observation is that in the core of Hybrid plasmas, where $\hat{s}$ is low and pressure is high, very high $T_i$ peaking is observed for very low $q_i^{gB}$, which is suggestive of very low

<figure><img src="image_14.png" /><figcaption>**Figure 11.** (a) Experimental $q_i^{gB}$ vs $R/L_{T_i}$ at different NBI power levels in JET L-mode plasmas. The two segments on the encircled points indicate the slope inferred from $T_i$ modulation; (b) $T_i$ experimental profiles of the two encircled points in (a) [293]. Reproduced from [272]. © IOP Publishing Ltd All rights reserved.</figcaption></figure>

<figure><img src="image_15.png" /><figcaption>**Figure 12.** Experimental $q_i^{gB}$ vs $R/L_{T_i}$ at different NBI power levels in DIII-D H-mode plasmas. Diamonds are discharges with balanced NBI and circles with co-NBI. Reproduced from [297]. © 2018 IAEA, Vienna. All rights reserved.</figcaption></figure>

ion stiffness, e.g. in JET [314, 321, 322] and AUG [323]. This has also been ascribed by gyrokinetic simulations to the same physics of nonlinear e.m. stabilization, with a small effect of $E \times B$ shear, which is instead more dominant in the external part of the plasma, where the nonlinear e.m. stabilization is reduced due to increased $\hat{s}$ and reduced thermal and suprathermal pressure [317, 321]. However, e.m. stabilization can still play a (diminished) stabilizing role also at outer radii, according to gyrokinetic simulations [299].

Finally, all the above experimental results on ion stiffness reduction concerned plasmas with moderate energy ($\sim 100$ keV) fast ions, as provided by present-day NBI and ICRH systems. An obvious question arises whether such stabilization effects would be present also in a reactor characterized by fast ions in the MeV range. Recent JET experiments using ICRH in a three-ion scheme (see [324]) revealed that stabilization could take place also in the presence of fast ions in the MeV range [325]. In spite of a variety of Toroidal Alfvèn Eigenmodes (TAEs) with different toroidal mode numbers excited in the central region, the ion heat transport was found improved with respect to a discharge with similar power and low energy NBI fast ions only. This encouraging observation, nevertheless, requires further verification and will, hopefully, be corroborated by new results from the JET 2021 D-T campaign, which will enable the investigation, in detail, of the