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
   • Solving methods and parallelism
3. Identification and control
   • Inverse problem: Equilibrium reconstruction
   • Open loop control
   • Dynamo effects in plasmas

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 17

4.2 MHD flows for liquid metal blankets

Participants: Herve Guillard, Boniface Nkonga, Stephane Abide, Praveel Chandrashekara.

Understanding of the physics and control of thermonuclear fusion reactions has progressed in recent decades, with several fusion reactors operated experimentally worldwide. Most explored configurations use a confinement system fueled by a Deuterium-Tricium (DT) plasma mixture. Magnetic confinement is the most advanced strategy for harnessing fusion energy for electrical power production. In this context, a strong magnetic field confined the DT plasma. Plasma activity is subject to instabilities (i.e., edge-localize modes and disruptions) that release significant flows of electrons, neutrons, alpha particles, and heat (thermal and radiative) outwards from the plasma confinement. A nuclear blanket protects the superconducting coils from the adverse effects of plasma activity and interfacing with several other components essential to the machine's operation.

Liquid metal blanket face-to-plasma components offer an alternative to the most demanding protection challenges. They could withstand heat fluxes without permanent damage and open the door to entirely new magnetic fusion operating regimes. Liquid lithium surfaces are an innovation that could fulfill the promise of fusion power in electricity generation.

We are interested in the numerical modeling of liquid metal flowing as part of the blanket protection. This thin layer of metal flow is a promising alternative to protect against possible melting damages that Disruptions and MHD instabilities can cause in fusion devices. The liquid metal blanket will operate according to the principles of magnetohydrodynamics (MHD), which are the same principles that produce the Dynamo effect.

The 2024 work focuses on modeling aspects and numerical challenges associated with the dynamic of a thin layer of metal flow under a strong magnetic field. This work is sufficiently mature to involve a PhD student in collaboration with IRFM/CEA in the coming years.

4.3 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.

4.4 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.

4.5 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 computations 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: Herve Guillard, Boniface Nkonga.

The Bezier approximation is now well-established in CAD (Computer Aided Design). Conversely, the Hermite finite element produces a higher continuity approximation space. By combining these two strategies, we will derive a Hermite-Bezier approximation that will help to capture smooth geometries with few finite elements, accurately represent anisotropies arising in plasma physics.

Furthermore, cubic Hermite and other high-order solution spaces have convergence advantages in finite element simulations compared with linear solution spaces and give rise to continuous properties between elements. A proper mapping between the local and global finite element spaces ensures the continuity of field solutions in these finite element problems. By the 2024 investigations, we provide the main steps of this construction in the context of a given parametric curve. The proposed work also opens the door for fully 3D finite element formulations for tokamaks and stelerator devices.

Participants: Ana Alonso Rodriguez, Francesca Rapetti.

We study Nédélec FEs of the first and second family for high order approximations in H(curl) and H(div). We have developed a geometric approach for constructing physical degrees of freedom for sequences of finite element spaces of Nédélec type (first and second families). In the last works, 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.

4.6 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.

5.1 Impact of research results

On the one hand, the 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.

On the other hand, with the study of astrophysical plasmas and the understanding of instabilities, it could be possible to place additional constraints on the structure of the optimized disturbance that can be exploited for experimental purposes to design a way of kickstarting self-excited dynamos at sustainable energy cost in the laboratory.

6.1 New software

7.1 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 in 14 for an X-point formation scenario in the WEST tokamak are presented as a first illustration of the efficiency of the developed numerical methods.

7.2 Magnetic control of WEST plasmas through deep reinforcement learning

Participants: Samy Kerboua-Benlarbi, Remy Nouailletas, Blaise Faugeras, Eric Nardon [IRFM,CEA], Philippe 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 3. In this work, we 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 11.

7.3 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 $ϵ$ [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 $ϵ$ estimated from two additional transport equations allow to introduce some non-locality in the transport model. This is illustrated in 5 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. New results on more reliable cases are presented in 6 and in 15.

7.4 Anisotropic diffusion

Participants: Blaise Faugeras, Hervé Guillard, Boniface Nkonga, Francesca Rapetti-Gabellini.

Heat transfer in magnetically confined plasmas is characterized by extremely high anisotropic diffusion phenomena. At the core of a magnetized plasma, the heat conductivity coefficients in the parallel and perpendicular directions of the induction field can be very different. Their ratio can exceed 1.e8 and the pollution by purely numerical errors can make very diffcult the simulation of the heat transport in the perpendicular direction. Standard numerical methods, generally used in the discretization of classical diffusion problems, are rather inefficient. The present paper analyzes a finite element approach for the solution of a highly anisotropic diffusion equation. Two families of finite elements of class C1, namely bi-cubic Hermite-Bézier and reduced cubic Hsieh-Clough-Tocher finite elements, are compared. Their performances are tested numerically, for various ratios of the diffusion coefficients, on different mesh configurations, even aligned with the induction field. The time stepping is realized by an implicit high-order Gear finite difference scheme. An example of reduced model is also provided in order to comment on some obtained results (see 13).

7.5 High-order Whitney finite elements in electromagnetics modeling

Participants: Francesca Rapetti-Gabellini, Ana Alonso Rodriguez [Univ. di Trento, Italy].

A method classically used in the lower polynomial degree for the construction of a finite element basis of the space of divergence-free functions is extended in 12 to any polynomial degree for a bounded domain without topological restrictions. The method uses graphs associated with two differential operators: the gradient and the divergence, and selects the basis using a spanning tree of the first graph. It can be applied for the two main families of degrees of freedom, weights and moments, used to express finite element differential forms. See also 1.

7.6 MHD model applied to massive material injections

Participants: Ashish Bhole, Boniface Nkonga, José Costa, Guido Huijsmans, Stanislas 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 7.

A collisional-radiative treatment for impurities using coronal equilibrium assumption was implemented, 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 singularity 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) 2.

7.7 Treatment of grid singularities in the Hermite-Bézier approximations

Participants: Ashish Bhole, Boniface Nkonga, Hervé Guillard, Meng Wu, Bernard Mourain.

JOREK uses high-order isoparametric bi-cubic Hermite-Bézier finite element method (FEM) to numerically 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 particular 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 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 4.

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 interpolations. 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.

7.8 Numerical methods for shear shallow water model

Participants: Boniface Nkonga, Ashish Bhole, Praveen Chandrashekar [TIFR, Bangalore], Arno 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 potential. Although the model is fundamentally non-conservative, the generalized jump conditions are compatible with the conservation of total energy. Consequently, the generalized jump conditions consistent 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 9, but we still need to find an optimal solution to this challenging problem.

7.9 Sediment transport in the shear shallow water framework.

Participants: Boniface Nkonga, Ashish Bhole, Praveen Chandrashekar [TIFR, Bangalore], Arno Roland Ndengna Ngatcha, Raphael Onguéné, Abdou Njifenjou.

gatcha, Ndengna Arno Roland, Nkonga, Boniface., Njifenjou, Abdou., Onguene, Raphael., 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 framework 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 correlation 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 16 improves several existing sediment transport theories recently developed in the literature, and we now have some confidence for future applications.

7.10 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 optimization 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.

7.11 A robust, discrete-gradient procedure for optimisation with time-dependent PDE and norm constraints

Participants: Didier Auroux, Paul Mannix, Florence Marcotte, Calum Skene [University of Leeds, UK].

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 18.

7.12 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 mechanism 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 8, 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, magnetized 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 19.

7.13 On nonlinear transitions, minimal seeds and exact solutions for the geodynamo

Participants: Florence Marcotte, Calum Skene [University of Leeds, UK], Steven Tobias [University of Leeds, UK].

Nearly fifty years ago, it was postulated that Earth's magnetic field, which is generated by turbulent motions of liquid metal in its outer core, likely results from a subcritical dynamo instability characterized by a dominant balance between Coriolis, pressure and Lorentz forces. Here we numerically explore the generation of subcritical geomagnetic fields using techniques from optimal control and dynamical systems theory to uncover the nonlinear dynamical landscape underlying dynamo action. Through nonlinear optimization, via direct-adjoint looping, we identify the minimal seed — the smallest magnetic field that attracts to a nonlinear dynamo solution. Additionally, using the Newton-hookstep algorithm, we converge stable and unstable travelling wave solutions to the governing equations. By combining these two techniques, complex nonlinear pathways between attracting states are revealed, providing insight into a potential subcritical origin of the geodynamo. This paper showcases these methods on the widely studied benchmark of Christensen et al. (2001), laying the foundations for future studies in more extreme and realistic parameter regimes. We show that the minimal seed reaches a nonlinear dynamo solution by first attracting to an unstable travelling wave solution, which acts as an edge state separating a hydrodynamic solution from a magnetohydrodynamic one. Furthermore, by carefully examining the choice of cost functional, we establish a robust optimization procedure that can systematically locate dynamo solutions on short time horizons with no prior knowledge of its structure 20.

8.1 ITER

Participants: Blaise Faugeras, Cédric Boulbe.

An 18 month contract starting on August 2024 has been signed between ITER and a consortium IgnitionComputing-CEA-Université Côte d'Azur for the development of a PDS (Plasma Discharge Simulator) involving 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, 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

9.1 European initiatives

9.2 National initiatives

10.1 Promoting scientific activities

10.2 Research administration

• Didier Auroux

  • Responsible of the MSI at UniCA
• Cedric Boulbe

  • Director of the departement MAM (Mathématiques Appliquées et Modélisation) - Polytech Nice Sophia
• Florence Marcotte

  • Member of the scientific committee of the MSI at UniCA

10.3 Teaching - Supervision - Juries

11.1 Major publications

• 1 miscA.Ana Alonso Rodríguez, J.Jessika Camaño, E.Eduardo de Los Santos and F.Francesca Rapetti. Weights for moments' geometrical localization: a canonical isomorphism.October 2023HALback to text
• 2 articleA.Ashish Bhole, B.Boniface Nkonga, J.José Costa, G.Guido Huijsmans, S.Stanislas Pamela and M.Matthias Hoelzl. Stabilized bi-cubic Hermite Bézier finite element method with application to Gas-plasma interactions occurring during massive material injection in Tokamaks.Computers & Mathematics with Applications142July 2023, 225-256HALDOIback to text
• 3 articleJ.Jacques Blum, C.Cedric Boulbe and B.Blaise Faugeras. Reconstruction of the equilibrium of the plasma in a Tokamak and identification of the current density profile in real time.Journal of Computational Physics2312012, 960-980HALback to text
• 4 inproceedingsM.Matthias Hoelzl, G. T.Guido T. A. Huijsmans, F. J.Francisco Javier Artola, E.Eric Nardon, M.Marina Becoulet, S.Stanislas Pamela, B.Boniface Nkonga, K.K Aleynikova, V.V Bandaru, H.H Bergström, A.A Bhole, T.T Bogaarts, D.D Bonfiglio, A.A Cathey, T.T Driessen, S.S Futatani, G.G Hao, F.F Hindenlang, I.I Holod, D.D Hu, S.S Hu, N.N Isernia, H.H Isliker, S.Sk Kim, M.M Kong, S.S Korving, L.L Kos, I.I Krebs, S.Sj Lee, L.L Meier, V.V Mitterauer, N.N Nikulsin, R.R Ramasamy, J.J Reinking, G.G Rubinacci, K.K Särkimäki, N.N Schwarz, C.C Sommariva, R.R Sparago, W.W Tang, F.F Vannini, S.S Ventre, F.F Villone, L.L Wang, H.-H.H-H Wang, F.F Wieschollek and J.J Zielinski. Non-linear MHD Modelling of Transients in Tokamaks: Recent Advances with the Jorek Code.IAEA Fusion Energy ConferenceFEC 2023 - 29th IAEA Fusion Energy ConferenceLondon, United KingdomOctober 2023HALback to text
• 5 articleL.Louis Lamérand, D.Didier Auroux, P.Philippe Ghendrih, F.Francesca Rapetti and E.Eric Serre. Inverse problem for determining free parameters of a reduced turbulent transport model for tokamak plasma.Advances in Computational Mathematics503May 2024, 39HALDOIback to text
• 6 miscL.Louis Lamérand, D.Didier Auroux and F.Francesca Rapetti. Parameter identification for a reduced transport model in fusion plasma.December 2024HALback to text
• 7 articleS.S. Pamela, A.Ashish Bhole, G.Guido Huijsmans, B.Boniface Nkonga, M.Matthias Hoelzl, I.Isabel Krebs and E.Erika Strumberger. Extended full-MHD simulation of non-linear instabilities in tokamak plasmas.Physics of Plasmas2710October 2020, 102510HALDOIback to text
• 8 articleL.Ludovic Petitdemange, F.Florence Marcotte and C.Christophe Gissinger. Spin-down by dynamo action in simulated radiative stellar layers.Science3796629January 2023, 300-303HALDOIback to text

11.2 Publications of the year

11.3 Cited publications

• 17 miscM.M Hoelzl, G.GTA Huijsmans, S.SJP Pamela, M.M Becoulet, E.E Nardon, F.FJ Artola, B.B Nkonga, C.CV Atanasiu, V.V Bandaru, A.A Bhole, D.D Bonfiglio, A.A Cathey, O.O Czarny, A.A Dvornova, T.T Feher, A.A Fil, E.E Franck, S.S Futatani, M.M Gruca, H.H Guillard, J.JW Haverkort, I.I Holod, D.D Hu, S.SK Kim, S.SQ Korving, L.L Kos, I.I Krebs, L.L Kripner, G.G Latu, F.F Liu, P.P Merkel, D.D Meshcheriakov, V.V Mitterauer, S.S Mochalskyy, J.JA Morales, R.R Nies, N.N Nikulsin, F.F Orain, D.D Penko, J.J Pratt, R.R Ramasamy, P.P Ramet, C.C Reux, N.N Schwarz, P. S.P Singh Verma, S.SF Smith, C.C Sommariva, E.E Strumberger, D.DC vanVugt, M.M Verbeek, E.E Westerhof, F.F Wieschollek and J.J Zielinski. The JOREK non-linear extended MHD code and applications to large-scale instabilities and their control in magnetically confined fusion plasmas.2020back to text
• 18 articleP.P.M. Mannix, C.C.S. Skene, D.D. Auroux and F.F. Marcotte. A robust, discrete-gradient procedure for optimisation with time-dependent PDE and norm constraints.SMAI Journal of Computational Mathematics102024, 1--28back to text
• 19 articleL.L. Petitdemange, F.F. Marcotte, C.C. Gissinger and F.F. Daniel. Tayler-Spruit dynamos in simulated radiative stellar layers.Astronomy & Astrophysics681A752024back to text
• 20 articleC.C.S. Skene, F.F. Marcotte and S.S.M. Tobias. On nonlinear transitions, minimal seeds and exact solutions for the geodynamo.arXiv:2411.054992024, 1--21back to text