RESEARCH CENTRE

Inria Centre
at Université Côte d’Azur

IN PARTNERSHIP WITH:

CNRS, Université Côte d’Azur

2023
ACTIVITY REPORT

Project-Team

CASTOR

Control, Analysis and Simulations for
TOkamak Research

IN COLLABORATION WITH: Laboratoire Jean-Alexandre Dieudonné (JAD)

DOMAIN

Digital Health, Biology and Earth

THEME

Earth, Environmental and Energy
Sciences

Contents

Project-Team CASTOR

1 Team members, visitors, external collaborators

2 Overall objectives

3 Research program

3.1 Plasma Physics

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4 Application domains

4.1 MHD and plasma stability in tokamaks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2 Long term plasma evolution and optimization of scenarii
. . . . . . . . . . . . . . . . . . . .
4.3 Turbulence and models for the edge region of tokamaks . . . . . . . . . . . . . . . . . . . . .
4.4 High order accuracy methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.5 Understanding magnetogenesis in stellar systems . . . . . . . . . . . . . . . . . . . . . . . . .

5 Social and environmental responsibility

5.1 Impact of research results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6 Highlights of the year

6.1 Awards . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7 New results

1

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3

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6
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7.1 Tokamak free-boundary plasma equilibrium computations in presence of non-linear ma-
terials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6
7.2 Numerical simulation of Tokamak plasma equilibrium evolution . . . . . . . . . . . . . . . .
6
7.3 Magnetic control of WEST plasmas through deep reinforcement learning . . . . . . . . . . .
6
7.4 Turbulence and models for the edge region of tokamaks . . . . . . . . . . . . . . . . . . . . .
7
7.5 Coupling Hermite-Bezier quadrangular elements and Clough Tocher triangular elements .
7
. . . . . . . . . . . . . .
7.6 High-order Whitney finite elements in electromagnetics modeling
7
7.7 MHD model applied to massive material injections . . . . . . . . . . . . . . . . . . . . . . . .
8
7.8 Treatment of grid singularities in the Hermite-Bézier approximations . . . . . . . . . . . . .
8
. . . . . . . . . . . . . . . . . . . . . . . .
7.9 Numerical methods for shear shallow water model
9
9
. . . . . . . . . . . . . . . . . . .
7.10 Sediment transport in the shear shallow water framework.
7.11 Parameter identification for a MHD model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
7.12 Nonlinear acoustic streaming in standing waves . . . . . . . . . . . . . . . . . . . . . . . . . . 10
7.13 A robust, discrete-gradient procedure for optimisation with time-dependent PDE and norm

constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
. . . . . . . . . . . . . . . . . . . 11

7.14 Tayler-Spruit dynamos in simulated radiative stellar layers

8 Bilateral contracts and grants with industry

11
8.1 UKAEA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
8.2 France-Relance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

9 Partnerships and cooperations

12
9.1 International initiatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
9.1.1 Associate Teams in the framework of an Inria International Lab or in the framework

9.2 International research visitors

of an Inria International Program . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
9.2.1 Visits of international scientists . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
9.3.1 Horizon Europe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
9.4 National initiatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
9.4.1 ANR Sistem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

9.3 European initiatives

10 Dissemination

13
10.1 Promoting scientific activities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
10.1.1 Scientific events: organisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
10.1.2 Invited talks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
10.1.3 Scientific expertise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
10.2 Teaching - Supervision - Juries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
10.2.1 Teaching . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
10.2.2 Supervision . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

11 Scientific production

15
11.1 Major publications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
11.2 Publications of the year
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
11.3 Cited publications

Project CASTOR

1

Project-Team CASTOR

Creation of the Project-Team: 2014 July 01

Keywords

Computer sciences and digital sciences

A6. – Modeling, simulation and control

A6.1. – Methods in mathematical modeling

A6.1.1. – Continuous Modeling (PDE, ODE)

A6.1.4. – Multiscale modeling

A6.1.5. – Multiphysics modeling

A6.2. – Scientific computing, Numerical Analysis & Optimization

A6.2.1. – Numerical analysis of PDE and ODE

A6.2.6. – Optimization

A6.2.7. – High performance computing

A6.2.8. – Computational geometry and meshes

A6.3. – Computation-data interaction

A6.3.1. – Inverse problems

A6.3.2. – Data assimilation

A6.3.4. – Model reduction

A6.4. – Automatic control

A6.4.1. – Deterministic control

A6.4.4. – Stability and Stabilization

A6.5. – Mathematical modeling for physical sciences

Other research topics and application domains

B4. – Energy

B4.2.2. – Fusion

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Inria Annual Report 2023

1 Team members, visitors, external collaborators

Research Scientists

• Hervé Guillard [Team leader, INRIA, Senior Researcher, HDR]

• Blaise Faugeras [CNRS, Researcher, HDR]

• Florence Marcotte [INRIA, Researcher]

Faculty Members

• Didier Auroux [UNIV COTE D’AZUR, Associate Professor, HDR]

• Jacques Blum [UNIV COTE D’AZUR, Emeritus, HDR]

• Cédric Boulbe [UNIV COTE D’AZUR, Associate Professor]

• Boniface Nkonga [UNIV COTE D’AZUR, Professor, HDR]

• Francesca Rapetti-Gabellini [UNIV COTE D’AZUR, Associate Professor, HDR]

• Afeintou Sangam [UNIV COTE D’AZUR, Associate Professor]

Post-Doctoral Fellows

• Ashish Bhole [CNRS, Post-Doctoral Fellow, until Nov 2023]

• Alexandre Vieira [CNRS, Post-Doctoral Fellow]

PhD Students

• Guillaume Gros [UNIV COTE D’AZUR]

• Louis Lamerand [UNIV COTE d’AZUR]

Technical Staff

• Jeaniffer Vides Higueros [LEMMA Company, Engineer]

Interns and Apprentices

• Salma Janati [INRIA, Intern, from May 2023 until Jun 2023]

Administrative Assistant

• Nathalie Nordmann [INRIA, from Apr 2023]

Visiting Scientists

• Praveen Chandrashekarappa [TIFR INDE, from Jun 2023 until Aug 2023]

• Raphael Onguene [Douala IUT , from Sep 2023 until Sep 2023]

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3

2 Overall objectives

In order to fulfill the increasing demand, alternative energy sources have to be developed. Indeed, the
current rate of fossil fuel usage and its serious adverse environmental impacts (pollution, greenhouse gas
emissions, ...) lead to an energy crisis accompanied by potentially disastrous global climate changes.
Controlled fusion power is one of the most promising alternatives to the use of fossil resources, potentially
with a unlimited source of fuel. France with the ITER and Laser Megajoule facilities is strongly involved
in the development of these two parallel approaches to master fusion that are magnetic and inertial
confinement. Although the principles of fusion reaction are well understood from nearly sixty years,
(the design of tokamak dates back from studies done in the ’50 by Igor Tamm and Andreï Sakharov in
the former Soviet Union), the route to an industrial reactor is still long and the application of controlled
fusion for energy production is beyond our present knowledge of related physical processes. In magnetic
confinement, beside technological constraints involving for instance the design of plasma-facing com-
ponent, one of the main difficulties in the building of a controlled fusion reactor is the poor confinement
time reached so far. This confinement time is actually governed by turbulent transport that therefore
determines the performance of fusion plasmas. The prediction of the level of turbulent transport in
large machines such as ITER is therefore of paramount importance for the success of the researches on
controlled magnetic fusion.
The other route for fusion plasma is inertial confinement. In this latter case, large scale hydrodynamical
instabilities prevent a sufficiently large energy deposit and lower the return of the target. Therefore, for
both magnetic and inertial confinement technologies, the success of the projects is deeply linked to the
theoretical understanding of plasma turbulence and flow instabilities as well as to mathematical and
numerical improvements enabling the development of predictive simulation tools.

CASTOR gathers the activities in numerical simulation of fusion plasmas with the activities in control
and optimization done in the laboratory Jean-Alexandre Dieudonné of Université Côte d’Azur. The
main objective of the CASTOR team is to contribute to the development of innovative numerical tools
to improve the computer simulations of complex turbulent or unstable flows in plasma physics and to
develop methods allowing the real-time control of these flows or the optimization of scenarios of plasma
discharges in tokamaks. CASTOR is a common project between Inria, Université Côte d’Azur and CNRS
through the laboratory Jean-Alexandre Dieudonné, UMR UNS-CNRS 7351, LJAD.

3 Research program

3.1 Plasma Physics

The main reseach topics are:

1. Modeling and analysis

• Fluid closure in plasma

• Turbulence

• Plasma anisotropy type instabilities

• Free boundary equilibrium (FBE)

• Coupling FBE – Transport

• MHD instabilities

2. Numerical methods and simulations

• High order methods

• Curvilinear coordinate systems

• Equilibrium simulation

• Anisotropy

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Inria Annual Report 2023

• Solving methods and parallelism

3. Identification and control

• Inverse problem: Equilibrium reconstruction

• Open loop control

• Dynamo effects in plasmas

4 Application domains

4.1 MHD and plasma stability in tokamaks

Participants: Hervé Guillard, Boniface Nkonga, Afeintou Sangam, Ali Elarif,

Ashish Bhole.

The magnetic equilibrium in tokamaks results from a balance between the Lorentz force and the pressure
gradient. Using Ampère law, a convenient description of this equilibrium is provided by the Grad-
Shafranov equation. Of course, the magnetic equilibrium solution of the Grad-Shafranov equation is
required to be stable. Actually any loss of MHD (Magneto-Hydro-Dynamics) stability can lead to the end
of the existence of the plasma, the so-called disruptions that can affect negatively the integrity of the
machine. The primary goal of MHD studies is therefore to determine the stability domain that constraints
the operational range of the machine.
A secondary goal of MHD studies is to evaluate the consequences of possible disruptions in term of heat
loads and stresses on the plasma facing components. In modern machines in the so-called H-mode
some mild instabilities leading to a near oscillatory behavior are also known to exist. In particular, the
so-called ELMs ( Edge Localized Modes) are of particular importance since they can have large effects
on the plasma facing components. The control and understanding of these instabilities is therefore of
crucial importance for the design of future machines as ITER. Unfortunately, ELMs occur in the edge
plasma and their modeling requires to take in account not only the intricate magnetic topology of this
region where both open and closed field lines co-exist but also the existence of molecular and atomic
processes involving neutrals.

At present, the linear theory of MHD stability is relatively well understood. However, the description
of the non-linear behavior is far from being complete. As a consequence and due to the intrinsic difficulty
of the subject, only a few numerical codes worldwide have been developed and validated for non linear
MHD in tokamaks. One of these codes is the JOREK code developped since 2006 from a collaborative work
between CEA-Cadarache (main developer), LABRI Bordeaux, LJAD-UniCA and Inria. A comprehensive
description of JOREK is given in [19]

4.2 Long term plasma evolution and optimization of scenarii

Participants: Didier Auroux, Jacques Blum, Cédric Boulbe, Blaise Faugeras,

Hervé Guillard.

The magnetic equilibrium evolves in time due to diffusion processes on the slow resistive diffusive time
scale and moreover it has to be monitored with active and passive control based on external coils, current
drive, heating system, particle or pellets injections. This set of control mechanism has to be modeled and
this is the goal of real time codes or global evolution codes.
In the same order of ideas, the steering and control of the plasma from the beginning to the end of the
discharge require the research of optimal trajectories through the space of operational parameters. This is
usually performed in an empirical way in present Tokamaks, but the complexity of the problem requires
today the use of optimization techniques for processes governed by MHD and diffusion-type equations.

Project CASTOR

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4.3 Turbulence and models for the edge region of tokamaks

Participants: Didier Auroux, Louis Lamerand, Francesca Rapetti.

The edge region of the plasma is characterized by low temperature and density leading to an increase
of the collision frequency that makes the edge plasma nearly collisional. This combined with the intricate
magnetic topology of this region makes the development of kinetic codes adapted to the edge regions
a real long term adventure. Consequently the fluid approach remains a standard one to study edge
plasma turbulence. The use of optimal control theory to derive simplified models matching data either
experimental or derived from direct numerical simulations is part of the objectives of the team.

4.4 High order accuracy methods

Participants:

Blaise Faugeras, Herve Guillard, Boniface Nkonga, Francesca Rapetti.

We analyze the accuracy and robustness of C1 Finite Element (FE) for plasma equilibrium computa-
tions in presence of strongly anisotropic phenomena. Aligned Hermite Bezier (HB) FEs and non-aligned
reduced Hiesh-Clough-Tocher (rHCT) FEs are coupled by the mortar element method for composite
meshes.

Participants:

Ana Alonso Rodriguez, Francesca Rapetti.

We have shown that high order polynomial interpolation with Nédélec edge elements can suffer from
a Runge phenomenon similar to that well known for high order polynomial interpolation with Lagrange
nodal elements. We have also studied a geometric approach for constructing physical degrees of freedom
for sequences of finite element spaces of Nédélec type (first and second families).

4.5 Understanding magnetogenesis in stellar systems

Participants: Didier Auroux, Florence Marcotte.

The considerable diversity of long-lived magnetic fields observed in the Universe raises fundamental
questions regarding their origin. Although it is now widely accepted that such fields are sustained by a
dynamo instability in the electrically conducting fluid layers of astrophysical bodies, in most cases the
very nature of the flow motions powering the dynamo is essentially unknown, and the conditions required
for amplifying large-scale magnetic fields in non-convective stellar systems are poorly understood. We
claim that optimal control represents a powerful tool to investigate the nonlinear stability of fully 3D,
unsteady magnetohydrodynamic flows with respect to the dynamo instability. Nonlinear optimization
can be also used as a physical diagnostic to gain novel understanding of the mechanisms that are most
favorable to dynamo action in a natural system.

5 Social and environmental responsibility

5.1 Impact of research results

The main objective of the CASTOR team is to contribute to the development of the numerical tools used
for the simulation of fusion plasma. Since the design of the next generation of fusion reactors relies on
numerical simulation, the works done in CASTOR contribute to the search of a clean and decarbonated
energy.

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Inria Annual Report 2023

6 Highlights of the year

6.1 Awards

Florence Marcotte has been awarded a European ERC Starting Grant ’CIRCE’ (2024-2028) to study the
dynamo effect in stellar systems.

7 New results

7.1 Tokamak free-boundary plasma equilibrium computations in presence of non-

linear materials

Participants: Blaise Faugeras, Cédric Boulbe, Guillaume Gros, Francesca Rapetti-

Gabellini.

We consider the axisymmetric formulation of the equilibrium problem for a hot plasma in a tokamak.
We adopt a non-overlapping mortar element approach, that couples piece-wise linear Lagrange finite
elements in a region that does not contain the plasma and piece-wise cubic reduced Hsieh-Clough-Tocher
finite elements elsewhere, to approximate the magnetic flux field on a triangular mesh of the poloidal
tokamak section. The inclusion of ferromagnetic parts is simplified by assuming that they fit within
the axisymmetric modeling and a new formulation of the Newton algorithm for the problem solution is
stated, both in the static and quasi-static evolution cases. [8]

7.2 Numerical simulation of Tokamak plasma equilibrium evolution

Participants: Blaise Faugeras, Cédric Boulbe, Guillaume Gros, Francesca Rapetti-

Gabellini.

This contribution focuses on the numerical methods recently developed in order to simulate the
time evolution of a Tokamak plasma equilibrium at the resistive diffusion time scale. We develop on
the method proposed by Heumann for the coupling of magnetic equilibrium and current diffusion. We
introduce a new space discretization for the poloidal flux using C0 and C1 finite elements. This, together
with the use of spline functions to represent the diamagnetic function in the resistive diffusion equation,
enables to restrain numerical oscillations which can occur with the original method. We add to the model
an evolution equation for electron temperature in the plasma. This enables us to compute consistently
the plasma resistivity and the non-inductive current terms called bootstrap current needed in the resistive
diffusion equation. It also enables us to evolve the pressure term in the simulation. These numerical
methods are implemented in the plasma equilibrium code NICE. The code is coupled with a magnetic
feedback controller through the MUSCLE3 library. This enables to simulate a prescribed plasma scenario.
The results for an X-point formation scenario in the WEST tokamak are presented as a first illustration of
the efficiency of the developed numerical methods.

7.3 Magnetic control of WEST plasmas through deep reinforcement learning

Participants:

S. Kerboua-Benlarbi, R. Nouailletas, Blaise Faugeras, Eric Nar-
don (IRFM,CEA), P. Moreau (IRFM,CEA).

Tokamaks require magnetic control across a wide range of plasma scenarios. The coupled behavior of
plasma dynamics makes deep learning a suitable candidate for efficient control in order to fulfil these
high-dimensional and non-linear situations. For example, on the TCV tokamak, deep reinforcement
learning has already been used for tracking of the plasma’s magnetic equilibrium [1]. In this work, we

Project CASTOR

7

apply such methods to the WEST tokamak, to address control of the plasma’s shape, position, and current,
in several relevant configurations. To this end, we developed a distributed framework to train an actor-
critic agent on a C++ free boundary equilibrium code called NICE, in which resistive diffusion allows
a more representative evolution of current profile throughout the simulation. The interface between
components was done through UDS protocols for fast, asynchronous and reliable communication. The
implemented tool handles feedback control of quantities of interest, with results showing flexibility of the
method regarding the use of different training environments [17].

7.4 Turbulence and models for the edge region of tokamaks

Participants: Didier Auroux, Louis Lamerand, Francesca Rapetti-Gabellini.

The high-dimensional and multiscale nature of fusion plasma flows requires the development of
reduced models to be implemented in numerical codes capable of capturing the main features of
turbulent transport in a sufficiently short time to be useful during tokamak operation. This paper
goes further in the analysis of the dynamics of the k-epsilon model based on the turbulent kinetic energy
k and its dissipation rate epsilon [Baschetti et al., Nuc. Fus 61, 106020 (2021)] to improve the predictability
of the transverse turbulent transport in simulation codes. Present 1D results show further capabilities
with respect to current models (based on constant effective perpendicular diffusion) and on the standard
quasi-linear approach. The nonlinear dependence of D in k and epsilon estimated from two additional
transport equations allow to introduce some non-locality in the transport model. This is illustrated by the
existence of parameter ranges with turbulence spreading. The paper also addresses another issue related
to the uncertainties on the inherent free parameters of such reduced model. The study proposes a new
approach in the fusion community based on a variational data assimilation involving the minimization of
a cost function defined as the distance between the reference data and the calculated values. The results
are good, and show the ability of the data assimilation to reduce uncertainties on the free parameters,
which remains a critical point to ensure the total reliability of such an approach.

7.5 Coupling Hermite-Bezier quadrangular elements and Clough Tocher triangular

elements

Participants:

Ashish Bhole, Boniface Nkonga, Hervé Guillard, Francesca Rapetti-
Gabellini.

The Jorek code for MHD studies uses fourth order C1 quadrangular finite elements. This leads to
results with high spatial accuracy. However with this type of elements, the description of the boundary is
difficult and moreover with polar meshes, geometrical singularities appear on X-points and magnetic
axis. By relying on composite meshes, we have studied the coupling of the Hermite-Bezier quadrangular
elements used in Jorek and of Clough Tocher triangular elements for an easier representation of the
boundary and treatment of the mesh singularities. First results for the Poisson model problem with
Dirichlet boundary conditions can be found in [6].

7.6 High-order Whitney finite elements in electromagnetics modeling

Participants:

Francesca Rapetti-Gabellini, Ana Alonso Rodriguez (Univ. di Trento,
Italy).

It is well known that Lagrange interpolation based on equispaced nodes can yield poor results.
Oscillations may appear when using high degree polynomials. For functions of one variable, the most
celebrated example has been provided by Carl Runge in 1901, who showed that higher degrees do not

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Inria Annual Report 2023

always improve interpolation accuracy. His example was then extended to multivariate calculus and in
this work we show that it is meaningful, in an appropriate sense, also for Whitney edge elements, namely
for differential 1-forms, as explained in [5].

We study a geometric approach for constructing physical degrees of freedom for sequences of finite
element spaces. Within the framework of finite element systems, we propose new degrees of freedom for
the second family of high order Nédélec finite elements in [12]

7.7 MHD model applied to massive material injections

Participants:

Ashish Bhole, Boniface Nkonga, José Costa, Guido Huijsmans, Stan-
islas Pamela, Matthias Hoelzl.

Massive material injection (MMI) experiments in tokamaks aim to inject neutral gases (such as
deuterium, neon, argon, etc.), also called impurities, into the tokamak plasma, giving rise to complex
gas-plasma interactions. The atomic reactions during the interactions produce charged ions at different
ionization levels. Multi-fluid MHD equations are appropriate candidates for gas-plasma interactions,
where each fluid is characterized by its ionization level. In a recent work, under the assumption of coronal
equilibrium, single fluid impurity transport modeling was proposed for the gas-plasma interactions,
which provided satisfactory results for MMI simulations with the reduced MHD models. We have used
this single fluid modeling in the single-temperature full MHD model context to obtain significant results.
To get to this point, we had to face three critical challenges. First, the Galerkin FEMs give central
approximations to the differential operators. Their use in the simulation of the convection-dominated
flows may lead to dispersion errors, yielding entirely wrong numerical solutions. Second, high-order, high-
resolution numerical methods produce high wave-number oscillations near shocks/discontinuities that
adversely affect the numerical stability. Third, the aligned helpful mesh in this context of high anisotropy
had drawbacks at critical points of the magnetic field. Then, we propose a numerical treatment for the
geometric singularity at the polar grid center associated with a SUPG-like numerical stabilization. The
stabilization strategy aims to identify the contributions of the modeling that need smoothing and apply it
locally in space according to fitting criteria. The result is a stabilized bi-cubic Hermite Bézier FEM in the
computational framework of the nonlinear magnetohydrodynamics (MHD) code JOREK.

A collisional-radiative treatment for impurities using coronal equilibrium assumption was imple-
mented, benchmarked, and applied to validate simulations of shattered pellet injection (SPI) in the
JET tokamak. Deuterium and impurity/mixed SPI simulations for the JET tokamak reproduce critical
experimental observations, e.g., regarding radiation, showing that plasmoid drifts play an essential role in
material assimilation, radiation dynamics, and plasma evolution. SPI simulations for the AUG tokamak
are ongoing and successively improved towards entirely realistic plasma parameters; they qualitatively
reproduce experimentally observed double radiation peaks, suggesting that the first peak originates
from the injection location and the second peak from the core. Numerical stabilization, axis singu-
larity treatment, and shock-capturing methods are essential ingredients that allow the carrying out of
highly nonlinear mitigated disruption studies with the full MHD and reduced-MHD models, which were
previously impossible (with Jorek) [7].

7.8 Treatment of grid singularities in the Hermite-Bézier approximations

Participants:

A. Bhole, B. Nkonga, H. Guillard, M. Wu, B. Mourain.

JOREK uses high-order isoparametric bi-cubic Hermite-Bézier finite element method (FEM) to nu-
merically approximate fusion plasma models. One distinguishing feature of JOREK’s numerical method
is the construction of multi-block, flux-aligned grids with curved elements. Such grids may contain
geometrically singular points, such as the polar grid center, where FEM is not well defined. These partic-
ular points may act as a source of numerical error, polluting the numerical solution. We have already
proposed a numerical treatment for the geometric singularity at the polar grid center encountered in the

Project CASTOR

9

application of the isoparametric bi-cubic Hermite Bézier FEM and implemented the treatment in JOREK.
The treatment applies a set of new basis functions at the polar grid center in the numerical algorithm,
where the new basis functions are simply the linear transformations of the original basis functions. The
proposed treatment enforces the C1 regularity in the physical space, preserves the order of the accuracy
of the interpolation, and improves the stability and accuracy of the numerical approximation near the
polar grid center [13].

This year’s studies go beyond the cases investigated in the past years and suggest a way to enforce
regularity when using meshes containing singular points to interpolate smooth functions. The working
context also extends the field of study to higher-order approximations by including bi-quintics inter-
polations. In practice, we use the fact that the meshing vectors differ for each neighbor element of a
singular vertex. Therefore, the meshing vectors will now also contain the element’s index. Consequently,
the degrees of freedom can also differ for each neighbor element. Nevertheless, the physical state and
gradient are shared to enforce the C1-regularity of the interpolations. For the C2-regularity, we also share
the Hermitian matrix. This formal description, mathematically consistent, when included inside the
Jorek platform, will further improve its use in practical and challenging simulations.

7.9 Numerical methods for shear shallow water model

Participants:

Boniface Nkonga, Ashish Bhole, Praveen Chandrashekar (TIFR, Ban-
galore), A. Bhure (TIFR, Bangalore).

A few years ago, we proposed a complete study of the Riemann problem for a genuinely non-
conservative hyperbolic system derived from modeling reasonably turbulent thin flows. This model
coincides with the ten-moment model derived from gas kinetics in the presence of a gravitational po-
tential. Although the model is fundamentally non-conservative, the generalized jump conditions are
compatible with the conservation of total energy. Consequently, the generalized jump conditions con-
sistent with energy conservation are also independent of the path chosen. In this context, we show that
the exact solution of the Riemann problem exists and is unique, except in the case of vacuum. This
Riemann solver can work well within the numerical framework of Glim and Godunov but at a prohibitive
computational cost. We also propose approximate solvers with two, three, or five waves of discontinuities:
HLL, HLLC3, and HLLC5. These different solvers are compatible with the conservation of total energy.
However, the averaging strategy included in each finite volume approach violates this fundamental
conservation law. The main reason for this is that the square of an average quantity is generally not equal
to averaging the square of the same amount. Moreover, some fundamental structures of the model also
need to be preserved. For example, 1D solutions split into two sub-systems with a one-way coupling
between a conservative and a non-conservative sub-systems.

This year’s efforts on this subject have focused on strengthening the total energy conservation and the
structure-preserving properties at the numerical level to ensure the numerical method converges to the
exact solution of the Riemann problem. At least two such strategies have been published in recent years.
One explicitly discretizes the conservation equation and uses it to adjust the numerical dissipation of
non-conservative variables properly. However, this strategy takes advantage of the direction splitting and
does not apply to unstructured meshes. The other approach uses a variable that ensures the symmetry of
the stress tensor at the discrete level. It constructs the numerical dissipation coefficient in such a way as
to verify the conservation of energy at the discrete level. We are developing a third approach applicable
to unstructured meshes that can incorporate multi-wave Riemann solvers. This year, we have some
encouraging preliminary results. But we still need to find an optimal solution to this challenging problem.

7.10 Sediment transport in the shear shallow water framework.

Participants:

Boniface Nkonga, Ashish Bhole, Praveen Chandrashekar (TIFR, Ban-
galore), J. Iroume, A. Ngatcha, R. Onguéné, A. Djifendjou.

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Inria Annual Report 2023

Several shallow water (SW) based sediment transport models have come into the literature over
the last decades. These classical averaged sediment transport models (STM) based on shallow water
equations describe the hydromorphodynamic process, assuming that there is no shear velocity along
the vertical direction. Mainly, these models do not feel the phase shift between the flow and the bed
sediment’s movement. The dynamic properties of the coastal flows for tropical rivers of interest are far
from the SW assumptions. Our short- and medium-term objective is to revise sediment transport models
to include the indispensable shear effects.

The classical models often combined the Exner strategy with the Grass approximations in the frame-
work of the shallow water averaging. This year’s work focuses on deriving suitable formulations for Shear
Shallow Water (SSW) regimes. The derived model includes the distortion (fluctuation with great correl-
ation lengths) that creates the turbulence., We consider the shift between fluid and sediment velocity
(phase-lag) near the bed. The proposed theory significantly reduces the modeling errors observed in
several sediment transport models based on nonhomogeneous shallow water equations. It has excellent
potential to increase the predictive power of sediment transport models in rivers, lakes, coastal flows,
and ocean basins. The proposed theory improves several existing sediment transport theories recently
developed in the literature, and we now have some confidence for future applications.

7.11 Parameter identification for a MHD model

Participants:

Alexandre Viera, Didier Auroux, Hervé Guillard, Florence Marcotte.

The goal of this study is to identify different parameters used in a reduced MHD model. We mainly
focus in two directions. Firstly, the identification of the fluid viscosity and magnetic resistivity. Secondly,
the identification of an initial condition leading to a super-critical unstable equilibrium. In order to
solve these problems, optimal control techniques will be used. This mainly uses two ingredients: the
computation of the gradient of the cost functional, and algorithms to solve optimization problems. In
this regard, we used the Tapenade software in order to differentiate a code implementing a reduced MHD
model using Hsieh-Clough-Tocher finite elements. This approach lets us compute the exact gradient of a
cost functional with respect to the discretization scheme, and not an approximation computed through a
discretization of the continuous adjoint equation. The computed gradient obtained is then used in an
optimisation loop to minimize a cost functional measuring the distance between the computed and a
target solution. Numerical experiments show that the optimization problem converges and allows to
recover the value of the viscosity and resistivity corresponding to the target solution.

7.12 Nonlinear acoustic streaming in standing waves

Participants: Hervé Guillard, Argiris Delis (TUC), V. Mandikas (TUC).

Acoustic streaming is a secondary mean steady flow generated by and superimposed on a primary
oscillatory flow. When a compressible fluid experiences a high-frequency oscillation (e.g. from a sound
source) the nonlinear interactions can often lead to a pattern of time-dependent vortical flows or steady
circulations in the flow field. A numerical investigation of the acoustic streaming motion (of the Rayleigh
type) in a compressible gas inside two-dimensional rectangular enclosure has been realized last year and
published this year [9]

7.13 A robust, discrete-gradient procedure for optimisation with time-dependent

PDE and norm constraints

Participants: Didier Auroux, Florence Marcotte.

Project CASTOR

11

Joint work with Paul Mannix (formerly postdoc in the CASTOR team, now a postdoc at Imperial
College since January 2023) and Calum Skene (postdoc at University of Leeds). Many physical questions
in fluid dynamics can be recast in terms of norm-constrained optimization problems; which in-turn,
can be further recast as unconstrained problems on spherical manifolds. Due to the nonlinearities
of the governing PDEs, and the computational cost of performing optimal control on such systems,
improving the numerical convergence of the optimization procedure is crucial. Borrowing tools from
the optimization on manifolds community we outline a numerically consistent, discrete formulation of
the direct-adjoint looping method accompanied by gradient descent and line-search algorithms with
global convergence guarantees. We numerically demonstrate the robustness of this formulation on three
example problems of relevance in fluid dynamics and provide an accompanying library SphereManOpt.
Paper accepted for publication in SMAI Journal of Computational Mathematics.

7.14 Tayler-Spruit dynamos in simulated radiative stellar layers

Participants:

Florence Marcotte, Ludovic Petitdemange ((Observatoire de
Paris)), Christophe Gissinger
((Ecole Normale Supérieure)),
Florentin Daniel ((Ecole Normale Supérieure)).

The Tayler-Spruit dynamo mechanism has been proposed two decades ago as a plausible mechan-
ism to transport angular momentum in radiative stellar layers. Direct numerical simulations are still
needed to understand its trigger conditions and the saturation mechanisms. The present study follows
up on a previous paper by the same team [11], where we reported the first numerical simulations of a
Tayler-Spruit dynamo cycle. Here we extend the explored parameter space to assess in particular the
influence of stratification on the dynamo solutions. We also present numerical verification of theoretical
assumptions made in (Spruit 2002), which are instrumental in deriving the classical prescription for
angular momentum transport implemented in stellar evolution codes. A simplified radiative layer is
modeled numerically by considering the dynamics of a stably-stratified, differentially rotating, mag-
netized fluid in a spherical shell. Our simulations display a diversity of magnetic field topologies and
amplitudes depending on the flow parameters, including hemispherical solutions. The Tayler-Spruit
dynamos reported here are found to satisfy magnetostrophic equilibrium and achieve efficient turbulent
transport of angular momentum, following Spruit’s heuristic prediction. Paper accepted for publication
in Astronomy & Astrophysics.

8 Bilateral contracts and grants with industry

8.1 UKAEA

Participants:

Blaise Faugeras, Cédric Boulbe.

A contract has been signed with the English Tokamak UKAEA STEP for a license and a training session

on the use of the code NICE.

8.2 France-Relance

Participants:

Boniface Nkonga, Jeaniffer Vides.

As part of the Plan France Relance: financing agreement no. ANR-21-PRRD-0005-01 was signed
between Agence Nationale de la Recherche (ANR) and Inria 15/06/2021. Then follows an associated
contract between Inria, UCA, and CNRS on the one hand and the LEMMA company on the other. (287
K€). The goal was to explore new adaptive sliding methods for rotating machines. At the same time,

12

Inria Annual Report 2023

the Establishments, the Company, and Ms. Jeaniffer VIDES HIGUEROS entered into, on 25/08/2022, an
agreement for the provision of Ms. Jeaniffer VIDES HIGUEROS, an employee of the Company, to the
CASTOR project team for 24 months

9 Partnerships and cooperations

9.1 International initiatives

9.1.1 Associate Teams in the framework of an Inria International Lab or in the framework of an Inria

International Program

AMFoDUC

Title: Advanced Modeling of Flows in Douala City in the strong Urbanization and Climate

Duration: 2021 -> 31/12/2023

Coordinator: Raphael Onguene (ziongra@yahoo.fr)

Partners:

• Université de Douala (Cameroun)

Inria contact: Boniface Nkonga

Summary: The project will bring together students (masters and PhDs) and experts from Cameroun and
France to work towards developing simulation tools for the interaction of fluvial flows and coastal
flows along the Gulf of Guinea and particularly the city of Douala. The project will require the
development of new mathematical tools in the analysis of the Shear Shallow Water (SSW) model,
which will also be beneficial to the team towards developing appropriate understanding of new
methodologies specific to the context of Guinea Gulf. Currently, there is no other known project in
this direction. We aim to further engage neighboring countries along the Gulf of Guinea towards
developing a complete simulation tool and further training in geophysical modeling and simulation.
In collaboration with our colleague of Bangalore (India/University Côte d’Azur collaboration), we
have developed a numerical framework (platform) bringing together the components that are
necessary for the deployment of new applications in the context of the SSW modeling. It takes full
advantage of massively parallel computing and can serve as a practical engineering tool. These
tools are quite flexible, developed in the Finite volume framework and can deal with structured as
well as unstructured meshes. On top of this basis, our objective is to develop additional modules
that will take into account the specificities of coastal flows in the Gulf of Guinea and contribute to
the Douala Sustainable City project. The collected data will help to design the appropriate scaling
to be used in the derivation of simplified modeling to be included in the SSW framework. They will
also give some guidelines for strategies of numerical validations.

9.2 International research visitors

9.2.1 Visits of international scientists

• Praveen Chandrashekar from TIFR, Bangalore India, visited the team to work the shear shallow

water model

• Raphael Onguene from the IUT of Douala, Cameroon, has participated to the Evaluation workshop

of the LIRIMA laboratory.

Project CASTOR

13

9.3 European initiatives

9.3.1 Horizon Europe

Cédric Boulbe and Blaise Faugeras participate to the Eurofusion Workpackage WPSA

Boniface Nkonga and Hervé Guillard participate to the EuroFusion workpackage 01 (Tokamak ex-

ploitation)TSVV

European Starting Grant Project Circe, PI: Florence Marcotte, funded - launching on January 1st, 2024.

9.4 National initiatives

9.4.1 ANR Sistem

Participants: Didier Auroux, Jacques Blum, Cédric Boulbe, Blaise Faugeras,

Francesca Rapetti-Gabellini.

Member of the ANR SISTEM , Oct. 2019 - Sept. 2023 coordinated by the M2P2 Institute of Aix-Marseille
Univ. "SImulations with high-order schemes of tranSport and TurbulencE in tokaMak" programme
Modeles numeriques 2019, Contact : Francesca Rapetti

10 Dissemination

10.1 Promoting scientific activities

Participants:

Boniface Nkonga, Francesca Rapetti, Hervé Guillard.

10.1.1 Scientific events: organisation

Member of the organizing committees B. Nkonga is in the organization board of the ECCOMAS confer-
ence

Member of the editorial boards F. Rapetti is in the editorial board of the Springer Journal ’Advances in
COmputational Mathematics’

10.1.2 Invited talks

• H. Guillard - B. Nkonga, Godunov-type methods for geometrically constrained hyperbolic systems :

MHD and SSW , Colloquium in memory of Serguei Godunov, Marseille, 30/11-01/12 2023

• H. Guillard, Invited talk at IISER Thiruvananthapuram, Kerala state, India

• H. Guillard, Invited talk at TIFR, Bangalore, India.

• L. Lamerand and G. Gros, Invited session 25, M2P 2023 conf., Taormina (IT)

10.1.3 Scientific expertise

Boniface Nkonga has prepared an expertise for TIFR, Mumbai, India for a high performance computer
acquisition project.

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Inria Annual Report 2023

10.2 Teaching - Supervision - Juries

10.2.1 Teaching

• Ecole d’ingénieur, D. Auroux, Mathématiques financières, 66h équivalent TD, niveau M1, Université

Côte d’Azur

• Ecole d’ingénieur, D. Auroux, Probabilités et statistiques, 24h équivalent TD, niveau L3, Université

Côte d’Azur

• Licence : F. Rapetti, Cours Mathématiques 2,60h équivalent TD, L2, Université Côte d’Azur

• Licence : F. Rapetti, TD Mathématiques 2,12h équivalent TD, L2, Université Côte d’Azur

• Licence : F. Rapetti, TD Introduction au calcul scientifique, 24h équivalent TD, L2, Université Côte

d’Azur

• Licence : F. Rapetti, TP Introduction au calcul scientifique, 40h équivalent TD, L2, Université Côte

d’Azur

• Ecole d’ingénieur, C. Boulbe, Analyse Numérique 1, 45.5h équivalent TD, niveau L3, Université

Côte d’Azur

• Ecole d’ingénieur, C. Boulbe, Analyse numérique 2, 45.5h équivalent TD, niveau L3, Université Côte

d’Azur

• Ecole d’ingénieur, C. Boulbe, Algèbre linéaire et Matlab, 26h équivalent TD, Université Côte d’Azur

• Ecole d’ingénieur, C. Boulbe et D. Auroux, Projet 1, 48h équivalent TD, Université Côte d’Azur

• Licence : A. Sangam, Fondements Mathématiques 1, 24 h, Semestre 1 de la Licence, Université Côte

d’Azur, France

• Licence : A. Sangam, Fondements Mathématiques 2, 40 h, Semestre 2 de la Licence, Université Côte

d’Azur, France

• Licence : A. Sangam, Fondements Mathématiques 3, 28 h, Semestre 3 de la Licence, Université Côte

d’Azur, France

• Master : A. Sangam, Analyse de Fourier et Distributions, 36 h, Semestre 2 des Masters Math-
ématiques Fondamentales/Mathématiques Pures et Appliquées, Université Côte d’Azur, France

• Master : A. Sangam, Optimisation et Éléments Finis, 16 h, Semestre 2 des Masters Mathématiques
Fondamentales/Mathématiques Pures et Appliquées, et Ingénierie Mathématique, Université Côte
d’Azur, France

• Master : A. Sangam, Numerical Approximation of Hyperbolic Systems of Conservation Laws, 40 h,

Semestre 3 du Master Mathématiques Pures et Appliquées, Université Côte d’Azur, France

• Ecole d’ingénieur/Master: B. Nkonga, Méthode des éléments finis, 24h, M2, Polytech Nice Sophia,

Université Côte d’Azur

• Ecole d’ingénieur/Master: B. Nkonga, Eléments finis mixtes, 24h, M2, Polytech Nice Sophia, Uni-

versité Côte d’Azur

• Ecole d’ingénieur/Master, H. Guillard, Développement durable et enjeux de gouvernance, 16h, M2,

Polytech Nice Sophia, Université Côte d’Azur

Project CASTOR

10.2.2 Supervision

15

• Internship of Salma JANATI: Development and implementation of a numerical scheme for evolution

PDE of electron temperature in a tokamak plasma

Blaise FAUGERAS and Afeintou SANGAM have supervised the Master 1 Internship of Salma JANATI
from 1st May to 31st July 2023. The internship consisted in designing and implementing of a
numerical scheme of electron temperature evolution PDE in a tokamak plasma taking into account
geometry coefficients. The obtained results give insight for future schemes aiming to couple plasma
profile and electron temperature equations in a tokamak for evolution of the equilibrium at the
diffusion time scale as envisioned in NICE code.

11 Scientific production

11.1 Major publications

[1]

J. Blum, C. Boulbe and B. Faugeras. ‘Reconstruction of the equilibrium of the plasma in a Tokamak
and identification of the current density profile in real time’. In: Journal of Computational Physics
231 (2012), pp. 960–980. URL: https://hal.archives-ouvertes.fr/hal-00419608.

[2] D. A. Di Pietro, J. Droniou and F. Rapetti. ‘Fully discrete polynomial de Rham sequences of arbitrary
degree on polygons and polyhedra’. In: Mathematical Models and Methods in Applied Sciences (Aug.
2020). DOI: 10.1142/S0218202520500372. URL: https://hal.archives-ouvertes.fr/hal-
02356810.

[3] S. Pamela, A. Bhole, G. Huijsmans, B. Nkonga, M. Hoelzl, I. Krebs and E. Strumberger. ‘Extended
full-MHD simulation of non-linear instabilities in tokamak plasmas’. In: Physics of Plasmas 27.10
(Oct. 2020), p. 102510. DOI: 10.1063/5.0018208. URL: https://hal.inria.fr/hal-02974031.

[4] E. Zampa, A. Alonso Rodríguez and F. Rapetti. ‘Using the FES framework to derive new physical
degrees of freedom for finite element spaces of differential forms’. In: Advances in Computational
Mathematics 49.2 (6th Mar. 2023), p. 17. DOI: 10.1007/s10444-022-10001-3. URL: https://ha
l.science/hal-03884149.

11.2 Publications of the year

International journals

[5] A. Alonso Rodríguez, L. Bruni Bruno and F. Rapetti. ‘Whitney edge elements and the Runge phe-
nomenon’. In: Journal of Computational and Applied Mathematics 427 (Aug. 2023), p. 115117. DOI:
10.1016/j.cam.2023.115117. URL: https://hal.science/hal-03893138.

[6] A. Bhole, H. Guillard, B. Nkonga and F. Rapetti. ‘Coupling finite elements of class C1 on composite
curved meshes for second order elliptic problems’. In: International Journal for Numerical Methods
in Fluids (8th Oct. 2023). DOI: 10.1002/fld.5241. URL: https://hal.science/hal-03913143.

[7] A. Bhole, B. Nkonga, J. Costa, G. Huijsmans, S. Pamela and M. Hoelzl. ‘Stabilized bi-cubic Hermite
Bézier finite element method with application to Gas-plasma interactions occurring during massive
material injection in Tokamaks’. In: Computers & Mathematics with Applications 142 (July 2023),
pp. 225–256. DOI: 10.1016/j.camwa.2023.04.034. URL: https://hal.science/hal-038112
24.

[8] C. Boulbe, B. Faugeras, G. Gros and F. Rapetti. ‘Tokamak free-boundary plasma equilibrium com-
putations in presence of non-linear materials’. In: Journal of Scientific Computing 96 (Aug. 2023).
DOI: 10.1007/s10915-023-02265-8. URL: https://hal.science/hal-03423469.

[9] A. Delis, V. Mandikas and H. Guillard. ‘Numerical simulation of acoustic streaming in standing
waves’. In: Computers & Mathematics with Applications 152 (Dec. 2023), pp. 199–220. DOI: 10.101
6/j.camwa.2023.10.027. URL: https://inria.hal.science/hal-04281580.

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Inria Annual Report 2023

[10] A. R. N. Ngatcha and B. Nkonga. ‘Sediment transport models in Generalized shear shallow water
flow equations’. In: Applications in Engineering Science 15 (Sept. 2023), p. 100148. DOI: 10.1016/j
.apples.2023.100148. URL: https://hal.science/hal-04403617.

[11] L. Petitdemange, F. Marcotte and C. Gissinger. ‘Spin-down by dynamo action in simulated radiative
stellar layers’. In: Science 379.6629 (20th Jan. 2023), pp. 300–303. DOI: 10.1126/science.abk2169.
URL: https://hal.science/hal-03948010.

[12] E. Zampa, A. Alonso Rodríguez and F. Rapetti. ‘Using the FES framework to derive new physical
degrees of freedom for finite element spaces of differential forms’. In: Advances in Computational
Mathematics 49.2 (6th Mar. 2023), p. 17. DOI: 10.1007/s10444-022-10001-3. URL: https://ha
l.science/hal-03884149.

International peer-reviewed conferences

[13] M. Hoelzl, G. T. A. Huijsmans, F. J. Artola, E. Nardon, M. Becoulet, S. Pamela, B. Nkonga, K.
Aleynikova, V. Bandaru, H. Bergström, A. Bhole, T. Bogaarts, D. Bonfiglio, A. Cathey, T. Driessen,
S. Futatani, G. Hao, F. Hindenlang, I. Holod, D. Hu, S. Hu, N. Isernia, H. Isliker, S. Kim, M. Kong,
S. Korving, L. Kos, I. Krebs, S. Lee, L. Meier, V. Mitterauer, N. Nikulsin, R. Ramasamy, J. Reinking,
G. Rubinacci, K. Särkimäki, N. Schwarz, C. Sommariva, R. Sparago, W. Tang, F. Vannini, S. Ventre,
F. Villone, L. Wang, H.-H. Wang, F. Wieschollek and J. Zielinski. ‘Non-linear MHD Modelling of
Transients in Tokamaks: Recent Advances with the Jorek Code’. In: IAEA Fusion Energy Conference.
FEC 2023 - 29th IAEA Fusion Energy Conference. London, United Kingdom, 16th Oct. 2023. URL:
https://hal.science/hal-04403692.

Reports & preprints

[14] A. Alonso Rodríguez, J. Camano, E. de Los santos and F. Rapetti. Basis for high order divergence-free

finite element spaces. 31st Jan. 2024. URL: https://hal.science/hal-02429500.

[15] A. Alonso Rodríguez, J. Camaño, E. de Los Santos and F. Rapetti. Weights for moments’ geometrical
localization: a canonical isomorphism. 14th Oct. 2023. URL: https://hal.science/hal-044430
59.

[16] S. Gavrilyuk, B. Nkonga and K.-M. Shyue. The conduit equation : hyperbolic approximation and
generalized Riemann problem. 4th Jan. 2024. URL: https://hal.science/hal-04377682.

[17] S. Kerboua-Benlarbi, R. Nouailletas, B. Faugeras, E. Nardon and P. Moreau. Magnetic control of
WEST plasmas through deep reinforcement learning. Sept. 2023. URL: https://hal.science/hal
-04393963.

[18] S. Tiwari, B. Nkonga, P. Chandrashekar and S. Gavrilyuk. Finite volume approximations of shear
shallow water model on unstructured grids. 18th Jan. 2024. URL: https://hal.science/hal-044
03870.

11.3 Cited publications

[19] M. Hoelzl, G. Huijsmans, S. Pamela, M. Becoulet, E. Nardon, F. Artola, B. Nkonga, C. Atanasiu,
V. Bandaru, A. Bhole et al. The JOREK non-linear extended MHD code and applications to large-scale
instabilities and their control in magnetically confined fusion plasmas. 2020. arXiv: 2011.09120
[physics.plasm-ph].
<scratchpad>
- Main topic: Research Centre CASTOR (Control, Analysis and Simulations for Tokamak Research)
- Key points:
  - Focus on numerical simulation of fusion plasmas and control of turbulent transport
  - Collaboration between Inria, Université Côte d’Azur, and CNRS
  - Research program includes plasma physics, numerical methods, and application domains
- Structure: The document is organized into sections on project-team, activity report, research program, social and environmental responsibility, and dissemination
- Potential artifacts to ignore: Web-related elements such as links, HTML tags, and unnecessary formatting
</scratchpad>

<final_summary>
The CASTOR research centre is a collaboration between Inria, Université Côte d’Azur, and CNRS, focusing on numerical simulation of fusion plasas and control of turbulent transport. The centre's research program includes plasma physics, numerical methods, and application domains such as magnetic confinement and inertial confinement. The team aims to develop innovative numerical tools to improve the computer simulations of complex turbulent or unstable flows in plasma physics and to develop methods allowing the real-time control of these flows or the optimization of scenarios of plasma discharges in tokamaks.
The CASTOR research centre is a collaboration between Inria, Université Côte d’Azur, and CNRS, focusing on numerical simulation of fusion plasas and control of turbulent transport. The centre's research program includes plasma physics, numerical methods, and application domains such as magnetic confinement and inertial confinement. The team aims to develop innovative numerical tools to improve the computer simulations of complex turbulent or unstable flows in plasma physics and to develop methods allowing the real-time control of these flows or the optimization of scenarios of plasma discharges in tokamaks.