Ion temperature gradient mode mitigation by energetic particles, mediated by
forced-driven zonal flows
J.N Sama1, A. Biancalani2, A. Bottino3, D. Del Sarto1,R. J. Dumont4, G. Di
Giannatale5, A. Ghizzo1, T. Hayward-Schneider3, Ph. Lauber3, B. McMillan6, A.
Mishchenko7, M. Muruggapan5, B. Rettino3, B. Rofman5, F. Vannini3, L. Villard5
and X. Wang3

1Université de Lorraine, CNRS, IJL, Nancy, France,

2Léonard de Vinci Pôle Universitaire, Research Center, Paris La Défense,
France,

3Max Planck Institute for Plasma Physics, Garching, Germany,

4CEA, IRFM, F-13108 Saint-Paul-lez-Durance, France,

5Swiss Plasma Center, EPFL, Lausanne, Switzerland,

6Center for Fusion, Space and Astrophysics, University of Warwick, Coventry,
UK,

7Max Planck Institute for Plasma Physics, Greifswald, Germany,

## Abstract

In this work, we use the global electromagnetic and electrostatic gyro kinetic
approaches to investigate the effects of zonal flows forced-driven by Alfvén
modes due to their excitation by energetic particles (EPs), on the dynamics of
ITG (Ion temperature gradient) instabilities. The equilibrium of the 92416 JET
tokamak shot is considered. The linear and nonlinear Alfvén modes dynamics, as
well as the zonal flow dynamics, are investigated and their respective radial
structures and saturation levels are reported. ITG dynamics in the presence of
the zonal flows excited by these Alfvén modes are also investigated. We find
that, the zonal flows forced-driven by Alfvén modes can significantly impact
the ITG dynamics. A zonal flow amplitude scan reveals the existence of an
inverse relation between the zonal flow amplitude and the ITG growth rate.
These results show that, forced-driven zonal flows can be an important
indirect part of turbulence mitigation due to the injection of energetic
particles.

##  1 Introduction

Drift wave turbulence and its associated heat and particle transport have been
intensively studied due to the role they play in deteriorating plasma
confinement, which is a concern for future fusion plasma reactors. The ion-
temperature gradient modes (ITG) are drift waves driven unstable by the ion-
temperature gradient. The spontaneous non-linear excitation of zonal flows by
ITG modes [1, 2, 3], is believed to be one of their major saturation
mechanisms. Zonal flows (ZF) i.e. axisymmetric perturbations of tokamak
plasmas, play an important role in the self-consistent saturation of
turbulence. There are two types: the zero frequency zonal flow (ZFZF) and the
finite frequency geodesic acoustic modes (GAMs)[4, 5, 6, 7]. The ZFZF can be
generated both by ITGs and by their lower frequency branch, in which only ions
trapped in the banana orbits intervene[8, 9, 10, 11, 12]. Trapped Ion Modes
(TIM) is the name given to this lower frequency branch of ITGs (an electron
counterpart also exists). The turbulence and ZFZF generated by the high
frequency branch of ITGs are generally dominant over those generated by the
TIMs. The ZFZF, driven by drift waves like ITGs via modulational instability,
is dominated by the Reynolds and Maxwell stresses due to thermal species non-
linearity [13].

It has been observed experimentally in some tokamaks such as the JET [14] and
AUG [15] that under certain conditions, the injection of a small population of
energetic particles, can lead to the transition of the tokamak plasma, from a
low confinement regime, characterized by high transport and turbulence, known
as the L-mode to a high confinement regime characterized by a significant
reduction of turbulence and transport, known as the H-mode [16]. Zonal flows
are believed to play a key role in this transition, however, the exact source
of these zonal flows and the role played by energetic particles still remains
an open question. Several analytical theories and models [17, 18, 19, 20, 21]
have been developed in the last decades and numerical flux-tube simulations
performed [22, 23, 24, 25, 26, 27]. Qiu and co-workers [28], proposed a
theoretical model in which zonal flows can be driven by Alfvén modes (AM)[29,
30] in the presence of energetic particles. According to their model, the
energetic particles non-linearity that enters the vorticity equation through
the curvature coupling term, dominates over the bulk particles non-linearity
associated to the Reynolds and Maxwell stresses responsible for the excitation
of zonal flows by drift waves. Under this condition, the Alfvén modes can
drive a zonal flow with a growth rate that is twice that of the Alfvén modes.
The zonal flows excited by this mechanism are said to be forced-driven.

Recently, global simulations with the gyro-kinetic particle-in-cell code ORB5
[31, 32] have been performed, where the self-consistent interaction of ITG
turbulence, zonal flows, and Alfvén modes has been studied in the presence of
energetic particles [33, 34, 35]. Similar studies have been made with other
codes [36, 37]. As a result, it has been conjectured that zonal flows force-
driven by Alfvén instabilities may be used to reduce the level of ITG
turbulence.This conjecture proposes a possible mechanism, where the indirect
interaction between energetic particles and turbulence is mediated by forced-
driven zonal flows, as illustrated in Fig. 1.

Figure 1: Cartoon summarizing the conjecture we aim to verify with the
numerical simulations discussed herein.

In this work, we design numerical experiments, to test in experimentally
relevant scenarios part of the above conjecture that claims that the forced-
driven zonal flows can significantly mitigate ITG instability. To achieve this
objective, we isolate this mechanism from self-consistent nonlinear
simulations, and we prove that it is indeed effective by measuring the
amplitude and radial structure of the force-driven zonal flows, and by
imposing this radial structure on electrostatic ITG simulations. Imposing this
self-consistent zonal flows leads to a significant mitigation of the ITG
instability, corresponding to a drop of its growth rate. A zonal flow
amplitude scan reveals the existence of inverse relation between the zonal
flow amplitude and the ITG growth rate.

This work is divided into four sections: An introductory section in which the
motivations of the work are given. In section two, we describe the ORB5
numerical scheme, while the dynamics of Alfvén modes and their nonlinear
excitation of zonal flows is presented in section three. The dynamics of ITG
and forced-driven zonal flows is studied in section four.

##  2 Model

The main numerical tool used for the investigations described herein is the
global electromagnetic particle-in-cell code ORB5. The model equations of ORB5
derived using the mixed-variable gyro kinetic formalism are the gyro center
trajectories, equation (1) and (2), quasi-neutrality equation, equation (4),
Ampère’s equations, equation (5) and the Ohm’s law, equation 6.

$\frac{d\textbf{R}}{dt}=v_{||}\textbf{b}_{0}^{*}+\frac{1}{eB_{||}^{*}}\textbf{b}\times\mu\nabla
B+\frac{\textbf{b}}{B_{||}^{*}}\times\nabla\left<\phi-
v_{||}A_{||}^{(s)}-v_{||}A_{||}^{(h)}\right>-\frac{e}{m}\left<A_{||}^{(h)}\right>\textbf{b}_{0}^{*}$
(1) $\frac{d\textbf{v}_{||}}{dt}=-\frac{\mu}{m}\textbf{b}_{0}^{*}\cdot\nabla
B-\frac{e}{m}\left[\textbf{b}^{*}\cdot\nabla\left<\phi-
v_{||}A_{||}^{(h)}\right>+\frac{\partial}{\partial
t}\left<A_{||}^{(s)}\right>\right]-\frac{\mu}{m}\frac{\textbf{b}\times\nabla
B}{B_{||}^{*}}\cdot\nabla\left<A_{||}^{(s)}\right>$ (2)
$\textbf{b}^{*}=\textbf{b}_{0}^{*}+\frac{\nabla\left<A_{||}^{(s)}\right>\times\textbf{b}}{B_{||}^{*}},\quad\textbf{b}_{0}^{*}=\textbf{b}+\frac{mv_{||}}{eB_{||}^{*}}\nabla\times\textbf{b},\quad
B_{||}^{*}=B+\frac{mv_{||}}{e}\textbf{b}\cdot\nabla\times\textbf{b}$ (3)
$-\textbf{$\nabla$}\cdot\left(\frac{n_{0}}{B\Omega_{i}}\textbf{$\nabla$}_{\perp}\phi\right)=\bar{n}_{i}-\bar{n}_{e}$
(4)
$\sum_{s=i,e}\frac{\beta_{s}}{\rho^{2}_{s}}A_{||}^{(h)}+\nabla^{2}_{\perp}A_{||}^{h}=\mu_{0}\sum_{s=i,e}\bar{j}_{s}+\nabla^{2}_{\perp}A_{||}^{s}$
(5) $\frac{\partial}{\partial t}A_{||}^{(s)}+\textbf{b}\cdot\nabla\phi=0$ (6)

The phase-space coordinates are $\textbf{Z}=(\textbf{R},v_{||},\mu)$, i.e.
respectively, the gyro center position, the parallel velocity and the magnetic
moment. The time-dependent fields are the scalar potential, $\phi$ and the
parallel component of the vector potential, which is split into a symplectic
and Hamiltonian part, $A_{||}=A_{||}^{(s)}+A_{||}^{(h)}$. B and b are the
equilibrium magnetic field and magnetic unit vector. $\bar{n}_{s}$ and
$\bar{J}_{s}$ are, respectively, the perturbed density and current, while,
$\Omega_{i}$ is the ion cyclotron frequency. The other quantities are written
in standard notation. The gyro average operator is defined by:

$\left<\phi\right>=\frac{1}{2\pi}\int_{0}^{2\pi}\phi(\textbf{$R+\rho_{L}$})d\alpha$
(7)

where $\alpha$ here is the gyro angle and $\rho_{L}=\rho_{L}(\alpha,\mu)$ is
the Larmor radius. The gyro average operator reduces to the zeroth Bessel
function $J_{0}(k_{\perp}\rho_{L,i})$ if we transform it in Fourier space. The
gyro average is calculated for all ion species. We take into account finite
Larmor radius of ions, and we neglect them for electrons. Hereafter we will
label with ”$s$” the radial coordinate, which is defined in terms of poloidal
flux, $\psi_{p}$ as $s=\sqrt{\psi_{p}/\psi_{p,edge}}$. A nonlinear pullback
scheme is used in all electromagnetic simulations. This scheme includes the
nonlinear flutter terms, whose contribution becomes non-negligible for
strongly unstable MHD-type modes[38].

##  3 Alfvén modes dynamics in JET 92416

The first challenge in this section is to be able to run a global
electromagnetic simulation with experimentally measured magnetic equilibrium,
species temperature and density profiles, in which Alfvén modes excited by
energetic particles drive zonal flows. A good experimental case which we will
consider is the JET shot $92416$ afterglow experiment[39, 40]. In this shot,
$n=4,5,6$ TAEs (Toroidal Alfvén eigen modes) were measured at the time $6.2s$
of the discharge. We shall consider the magnetic equilibrium and profiles at
this particular time. For this study, we focus only on the dynamics of the
$n=5$ mode. A successful benchmark has been performed for this case without
fast particles using the MISHKA code[41] and ORB5.

###  3.1 JET 92416 equilibrium

JET is a tokamak with minor radius, $a_{0}=0.95$m, major radius $R_{0}=2.88$
m, with an on axis magnetic field of $B_{0}=3.4$T. The safety factor on axis
and at the edge are respective $q_{0}=1.74$ and $q_{edge}=4.6$. The magnetic
equilibrium and safety factor profile are generated from experimental data by
the CHEASE code [42], Fig 2. The thermal electrons, energetic species
equilibrium profiles and the ion density were generated by the TRANSP code
[43] from experimental data, Fig. 3. However, in this study, we consider a
scenario in which the electron and main ion species have the same temperature
profile. A reference radial position $s_{0}=0.0$, is chosen from which the
following physical quantities are calculated.
$\rho^{*}=\rho_{s}/a_{0}=3.08\cdot 10^{-3}$ (with $\rho_{s}$ the sound larmor
radius), the Alfvén speed, $V_{a}=7.56\cdot 10^{6}$m/s, the sound speed,
$c_{s}=4.8\cdot 10^{5}$ m/s, the cyclotron frequency, $\Omega_{i}=1.6\cdot
10^{8}$ rads/s and the thermal plasma beta, $\beta=0.9\%$.

(a)  (b)

Figure 2: Magnetic equilibrium and radial profile deduced from experimental
data using the CHEASE code (a) Magnetic field strength (b) Safety factor
profile

(a)  (b)

(c)  (d)

Figure 3: Radial profiles deduced from experimental data thanks to the TRANSP
code, and taken as equilibrium initial conditions for the ORB5 simulation.(a)
Density of thermal species (b)Temperature of thermal species (c) Density of
energetic particles (d) Temperature of energetic particles

Global linear and non-linear electromagnetic simulations were made. We use a
spatial grid of $(ns,nchi,nphi)=(768,192,48)$ points, a time step of
$dt=3\Omega^{-1}_{i}$ and a number of markers of
$(n_{i},n_{e},n_{f})=(32,128,32)\cdot 10^{6}$ respectively for the thermal
ions (Deuterium), electrons and fast particles. A convergence test showing the
validity of these simulation parameters can be found in Fig 11. No collisions
are considered, unicity boundary condition are imposed, and heavy electrons
were used to speed up the simulations ($\frac{m_{e}}{m_{i}}=0.005$). The
radial domain for this simulation is $[0,1.0]$ and a filter is used such that
only the $n=0,5$, toroidal modes are retained. The fraction of energetic
particles for these simulations is set to $10\%$, with the $T_{f}/T_{e}=25$ at
the reference location, $s_{0}$.

###  3.2 Linear Alfvén mode dynamics

An $n=5$, Alfvén mode dominated by the $m=10$, poloidal harmonic is excited by
the energetic particles in the radial domain $[0.2,0.8]$, as shown in Fig 4 a.
In the low-shear large-aspect ratio limit, the eigenmode amplitude is not
strongly localized around half-rational surfaces, and individual poloidal
harmonics are active across an extended radial region. The strong response at
$s=0.5$ results from the coupling between the $m=(9,10)$ poloidal harmonics,
in contrast to the response at $s=0.58$ which is due to the coupling between
the $m=(10,11)$. If we decompose the scalar potential into its poloidal
harmonics and plot the radial structure of these harmonics at a particular
time in the linear growth phase of the modes, we can clearly observe the
dominant poloidal harmonics and deduce the positions of maximum coupling, Fig
4 b. The localization of this mode as well as the dominant poloidal harmonics
are in accordance with theory [44].

(a)  (b)

Figure 4: (a) Structure of TAE in the poloidal plane (b) Radial structure of
individual poloidal harmonics.

The excited mode resulting from the coupling between different poloidal
harmonics at different radial locations, has growth rate and frequency, which
are respectively, $\gamma=0.00153\Omega_{i}$, with a frequency of
$\omega\approx 100$kHz. This frequency fall in the TAE range (which lies
between $60$kHz to $200$kHz). We can check that the excited modes are really
TAEs by comparing their radial localization and frequencies to the Shear
Alfvén continuum spectrum (SAW) corresponding to the equilibrium in question.
The FALCON code [45] was used to calculate the SAW spectrum and its result is
plotted on the frequency spectrum measured from the ORB5 simulation, Fig 5. We
observe from this plot that the frequencies of the excited modes fall in the
TAE gap. So the excited Alfvén modes, are identified as TAEs. This plot also
shows that the strongest TAE results from the coupling between the poloidal
harmonics $m=(9,10)$ at $s=0.5$. It should be noted that due to the high
fraction of energetic particles used in this simulation, the mode growth rate
constitutes a reasonable fraction of the frequency, i.e. the fast particles
are starting to have an effect on mode dispersion. This explains why the mode
touches the continuum.

Figure 5:  Frequency spectrum showing the frequency of the excited TAEs and
their radial localization. The Black curves represent the Alfvén continuum
computed with the FALCON code.

###  3.3 Non-linear Alfvén dynamics

A filter is applied in the non-linear simulation such that only the $n=0,5$,
are retained in the simulation. Fig 6, shows the time evolution of the of
different poloidal harmonics. The linear phase is completely dominated by
$m=10$ harmonic. The excited modes in the linear phase are localizes in the
radial domain $[0.2,0.8]$, Fig 7. In nonlinear phase, however, the dominant
linear harmonic saturates at amplitudes of the same order as the $m=9,11$.
These three modes dominates the nonlinear phase. The nonlinear phase is also
characterized by an increase in the coupling between the dominant poloidal
harmonics and other poloidal harmonics such as the the $m=8,12$, Fig 7 c and
d. The non-linear phase also leads to coupling between different toroidal
harmonics. The couple between the $n=0,5$ will be discussed in the next
subsection. This complex coupling that occurs in the nonlinear phase leads to
a modification of the mode structure of the excited mode and a radial
spreading of the instability. Fig 7, shows how the mode structure of the TAE
changes over time.

Figure 6: Time evolution of poloidal modes at mid-radius.

(a)  (b)

(c)  (d)

Figure 7: Mode structure of excited mode (a) Linear phase (b)Early saturation
phase (c) Early nonlinear phase (d) Deep nonlinear phase.

###  3.4 Non-linear excitation of Zonal flows by TAEs

The nonlinear phase is also characterized by the coupling between the $n=0,5$,
toroidal harmonics. This coupling leads to excitation of the zonal flows
($n=0$) by the $n=5$ TAE in the early nonlinear phase, Fig 8a . According to
Qiu [28], in the presence of energetic particles, Alfvén modes can drive zonal
flows via forced-driven excitation. This excitation mechanism is characterized
by the zonal flows growing at a rate that is approximately twice that of the
Alfvén mode driving it. In Fig 8 b, we can observe that in early nonlinear
phase, the zonal flow driven by the TAE has a growth rate that is
approximately twice that of the driving TAE. These zonal flows are therefore
forced driven. A frequency analysis of this zonal flow shows that it is a zero
frequency zonal flows. The TAE and the zonal flow saturate respectively with
electric field amplitudes of $90$kV/m and $63.5$kV/m. The amplitude of the
zonal electric field is consistent with the magnitude of the same quantity
measured in other JET shots, e.g. Fig. 7 in [46]. The radial structure of the
excited zonal flow and its associated shear is shown in Figs. 8 c and d,
respectively. This zonal flow has a fine radial structure as compared to the
driving TAE radial structure. This fine radial structure of zonal flows
excited by Alfvén modes was also predicted in theory [28]. With these
measurements, we can configure an ”antenna” in ORB5 to emulate this zonal
flow, in a simulation in which we aim to measure its impact, $-$see next
section.

(a)  (b)

(c)  (d)

Figure 8: (a) Time evolution of the electric field showing the excitation of
ZF by TAE (b)Zoom of early nonlinear phase with ZF growing nearly two times
faster than TAE (c) Zonal flow radial structure in the saturation phase. The
fine radial structure of the ZF can be observed (d) The zonal flow ExB
shearing rate in the saturation phase.

##  4 Effects of forced-driven zonal flows on ITG dynamics

###  4.1 Dynamics in the absence of zonal flows

In order to measure the effects of the forced driven zonal flows on ITG
instability, in this experimental scenario, we first have to study the
dynamics of this instability in the absence of any source of zonal flows. We
consider, the same simulation parameters as in the previous case. Electrons in
this case are adiabatic, and we don’t include energetic particles, and we
consider only electrostatic fluctuations. A scan in toroidal mode numbers is
performed and the corresponding linear ITG growth rates are measured. The
fastest growing ITG mode has a toroidal mode number of $n=70$, Fig. 9 a. The
ITG structure in the poloidal plane is shown in Fig. 9 b, with the excited
mode being most unstable in the low field side of the device. These modes are
excited around the mid-radius in the low field side of the device, where there
is a peak in the ion temperature gradient.

(a)  (b)

Figure 9: (a) ITG growth rate versus toroidal mode number (b) $n=20$ ITG
structure in the poloidal plane

###  4.2 The ORB5 antenna module

The antenna module, first implemented in ORB5 by Ohana and co-workers [47,
48], was designed to study the impact of specific plasma modes on the dynamics
of plasma instabilities or turbulence. The antenna acts like an external
source whose field can emulate a specific plasma mode given the frequency
($\omega$), toroidal ($n$) and poloidal ($m$) mode numbers, the radial
structure of the mode($h(s)$) and the phase offset ($\Phi$). However, the
antenna and plasma field are considered separately, so that in linear
simulations, only the plasma fields are linearized keeping the antenna fields
in the particle’s characteristics. This configuration makes it possible to use
the antenna both for linear and non-linear studies. In its simplest form, the
antenna field, $F_{ant}$, can be written as;

$F_{ant}=h(s)Re\left[\sum_{j=1}^{N}A_{j}e^{i\left(m_{j}\theta+n_{j}\varphi+\Phi_{j}\right)}\left(\alpha+\beta
e^{i\omega t}\right)\right]$ (8)

Where $A_{j}$, is the Fourier mode coefficient, $\alpha$ and $\beta$ are
respectively the coefficients of the static and oscillating components. The
quantity $F_{ant}$ can be the electrostatic potential $\phi$ and/or the
parallel vector potential $A_{||}$. In this work, we used the electrostatic
antenna component only. For this present study, the zonal flows described in
the section above can be emulated by the ORB5 antenna module, by setting,
$m_{j}=n_{j}=\omega=\Phi_{j}=0$, while the scalar potentials corresponding to
zonal electric fields structures shown in Fig. 8 c is loaded into $h(s)$.

###  4.3 ITG dynamics in the presence of forced driven zonal flows

In this section, we use the ORB5 antenna module described earlier to model the
zonal flows forced-driven by the $n=5$ TAE. The excited zonal flow is a ZFZF,
with $n=m=0$. The radial structure of the scalar potential from which the
zonal electric field is derived is loaded into the antenna module. We perform
single toroidal mode ITG simulations and scan over different toroidal mode
numbers, and the corresponding ITG growth rates in the presence of the ORB5
antenna emulating the forced-driven zonal flow are measured. This zonal flow
produces a significant mitigation of the ITG instability, measured by the
large reduction of the growth rate of the most unstable mode that dominates
the linear dynamics. All toroidal mode numbers are strongly stabilized in the
presence of forced-driven zonal flows and the range of unstable modes is
correspondingly reduced, Fig. 10 a. This result shows that the mitigation of
ITG instability by forced-driven zonal flows is significant in experimentally
relevant conditions. Therefore, it can serve as an important indirect part of
turbulence mitigation by energetic particle via excitation of forced driven
zonal flows by Alfvén modes. These simulations have shown the effectiveness of
such a process. In different experimental scenarios, we can expect significant
differences in energetic particles population and temperature. These
differences will generally results in a change in the Alfvén modes and zonal
flows saturation levels. It is therefore important to study how the zonal flow
amplitude affects the stabilization of ITG modes. To this end, we focus on the
most unstable linear mode i.e. $n=70$, and perform a zonal flow amplitude
scan, while keeping the radial structure constant. We observe that the
mitigation of the ITG modes doesn’t occur starting from a certain threshold
value of amplitude. The ITG dynamics rather changes with the zonal flow
amplitude, with the mitigation of the instability increasing with the zonal
flow amplitude , Fig 10 b. The zonal flow amplitude is found to be inversely
proportional to the ITG growth rate. This inverse relations have also been
reported in theory[19]. The stabilizing effect of radial electric field on
ITGs has also been reported in [49].

(a)  (b)

Figure 10: (a)ITG growth rate versus toroidal mode number without and with
forced-driven zonal flows, for the parameters and equilibrium configuration
discussed in Sec.3.1. (b) Zonal flow amplitude scan, showing stronger
stabilization for higher amplitudes.

##  5 Conclusion

In this work, we investigated the indirect interaction mechanism of turbulence
and energetic particles, via forced-driven zonal flows. These forced-driven
zonal flows are driven by Alfvén modes excited by energetic particles.
Biancalani and co-workers [34], proposed a mechanism in which the forced
driven zonal flows excited by the Alfvén modes can be used to mitigate
background turbulence. We designed a numerical experiment to test this
conjecture by running linear and non-linear gyro kinetic simulation with the
particle in cell code ORB5. The approach we used consisted in running two
types of simulations. A nonlinear simulation in which zonal flows are forced-
driven by Alfvén modes excited by energetic particles. The excited zonal flows
are measured, and its parameters are used to configure the antenna module of
ORB5, which is then used to emulate this zonal flow in a linear ITGs
simulations in order to study its effects on the ITG dynamics. The study was
carried out using magnetic equilibrium and species profiles generated from
experimental data from the JET shot 92416 Afterglow experiment.

We found in the Alfvén mode simulations that the excited $n=5$ mode was a TAE
as its frequency falls in the TAE gap of continuum spectrum corresponding to
this equilibrium. This TAE nonlinear excited a zero frequency zonal flow that
had a growth rate that was twice that of the TAE, an evidence of forced-driven
excitation [28]. The quantitative features of the excited zonal flow have been
identified, and its parameters were used to configure an antenna in ORB5.
Running linear ITG simulation with the ORB5 antenna emulating the forced-
driven zonal flows, we found that ITG dynamics was significantly affected,
leading to a significant reduction of the growth rate of these modes. These
results show that in experimentally relevant conditions, the zonal flows,
forced driven by Alfvén modes due to their excitation by energetic particles,
can significantly impact the ITG instability dynamics. Hence, the indirect
interaction of energetic particles with turbulence through the excitation of
forced driven zonal flows, can be an important indirect path of turbulence
mitigation. The present ITG study is restricted to the electrostatic,
adiabatic electron model, and further works will examine the effect of forced-
driven ZFs on other types of micro-instabilities, using kinetic electrons.

## Acknowledgement

This work was partially supported by the “Lorraine Université d’Excellence”
Doctorate fundings (project R01PKJUX-PHD21) belonging to the Initiative
”I-SITE LUE”. Part of this work has been carried out within the framework of
the EUROfusion Consortium, funded by the European Union via the Euratom
Research and Training Programme (Grant Agreement No.$101052200$ EUROfusion).
Views and opinions expressed here are however those of the authors only and do
not necessarily reflect those of the European Union or the European
Commission. Neither the European Union nor the European Commission can be held
responsible for them. Numerical calculations for this work have been partially
performed on the cluster Explor of the ”Maison de la simulation Lorraine” (We
are grateful to the partial time allocation under the project No.
2019M4XXX0978), and on the MARCONI FUSION HPC system at CINECA.

##  Appendix A Convergence test of nonlinear Alfvén mode simulation

Figure 11: Convergence test

We performed a convergence test on the simulation parameters. $dr$ and $dt$
are the step heights of the radial and temporal grid, respectively. A test was
also made on the number of markers used for each species. We can conclude from
this test that the simulation results are independent of the parameters used
for the simulation, as illustrated in 11.

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