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 component, 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 optimisation 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 optimisation 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.

The main reseach topics are:

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 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 (Magneto-Hydro-Dynamics) 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, ELM occur in the edge plasma and their modeling requires to take in account not only the intricate magnetic topology of this region where co-exist both open and closed field lines 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-UCA and Inria. A comprehensive description of JOREK is given in 21

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.

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.

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 optimisation 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.

In the short paper 18, we present the first application of the IMAS compatible code NICE to equilibrium reconstruction for ITER geometry. The inverse problem is formulated as a least square problem and the numerical methods implemented in NICE in order to solve it are presented. The results of a numerical experiment are shown: a reference equilibrium is computed from which a set of synthetic magnetic measurements are extracted. Then these measurements are used successfully to reconstruct the equilibrium of the plasma.

Equilibrium reconstruction codes are the corner stone of many workflows for analysis of tokamak discharges. Code benchmarking can be facilitated when the codes share the same data ontology, which also allows for easier porting to new devices. Such an approach has been adopted by the Work Package for Code Development (WPCD) using the ITER Integrated Modelling and Analysis Suite (IMAS), building upon the work of the European Integrated Modelling framework. Following up on some initial work, we report here on a benchmarking exercise using the EQRECONSTRUCT workflow, developed within WPCD and sporting the EQUAL and NICE equilibrium reconstruction codes, using data from the TCV, AUG and JET EUROfusion tokamaks. Most reconstructions presented here will use magnetics data only but the workflow is already equipped to also handle data from interferometry, polarimetry or MSE diagnostics. The plasma pressure in the outer radius can also be constrained by input profiles. Results from the EQUAL and NICE codes will be compared to the local codes used for reconstruction on each machine

The first plasmas have been achieved in the HL-2M tokamak by fulfilling the usual criteria for plasma breakdown,
i.e. optimal working gas pressure, field null configuration of poloidal magnetic field, reasonable toroidal electrical field and clean wall conditions. The characterization of plasma initiation, such as identification of initiation time and position, is performed by diagnostic data analysis and numerical simulation. A model-based method, which enables dedicated pick-up sensors to estimate plasma radial position in real time, is applied for the limiter plasma shape in the first plasma commissioning phase.
For a circular cross-section plasma, feedback control on plasma current and radial position allowed us to extend discharge durations to

The brand-new RAPTOR suite of codes is here presented. The suite has been developed for JET to combine real-time model-based predictions of the plasma state with the available diagnostic measurements. The suite embeds: the upgraded equilibrium reconstruction EQUINOX code, the FLUXMAP algorithm, which maps the diagnostic measurements from geometric to normalized magnetic flux coordinates; the RABBIT code for the NBI reconstruction and eventually RAPTOR state observer, which combines the output from all these codes with the predictions of 1D control-oriented transport code. The suite is both implemented in MATLAB/Simulink and it is being integrated in the C++ real-time MARTe2 framework. Thanks to its user-friendly interfaces, which are based on the MDSplus I/O and visualization tools, the RAPTOR suite can be used both offline, for a fast reconstruction of the plasma state, and in integrated control algorithms once it will be deployed in the JET real-time data network. This work is detailed in the paper

11.

We wish to compute numerically the equilibrium for a hot plasma in a tokamak. For such a problem in an axisymmetric configuration, we present a non-overlapping mortar element approach, that couples piece-wise linear finite elements in a region that does not contain the plasma and reduced Hsieh-Clough-Tocher finite elements elsewhere, to approximate the magnetic flux field on a triangular mesh of the poloidal tokamak section. This approach has the flexibility to achieve easily and at low cost higher order regularity for the approximation of the flux function in the domain covered by the plasma, while preserving accurate meshing of the geometric details in the rest of the computational domain and simplifying the inclusion of ferromagnetic parts. Details can be found in 19.

The numerical simulation of the equilibrium of the plasma in a tokamak as well as its self-consistent coupling with resistive diffusion should benefit from higher regularity of the approximation of the magnetic flux map. In this work 4, we propose a finite element approach on a triangular mesh of the poloidal section, that couples piece-wise linear finite elements in a region that does not contain the plasma and reduced Hsieh-Clough-Tocher finite elements elsewhere. This approach gives the flexibility to achieve easily and at low cost higher order regularity for the approximation of the flux function in the domain covered by the plasma, while preserving accurate meshing of the geometric details in the rest of the computational domain. The continuity of the numerical solution at the coupling interface is weakly enforced by mortar projection. A new technique for the computation of the geometrical coefficients is also presented.

The Jorek code 7 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. We have begun to study using the Mortar technique the coupling of the Hermite-Bezier quadrangular elements used in Jorek and of Clough Tocher triangular elements that will allow a easier representation of the boundary and solves the problem of mesh singularities.

We recall the classical tree-cotree technique in magnetostatics. We extend it in the frame of high-order finite elements in general domains. We focus on its connection with the question of the invertibility of the final algebraic system arising from a high-order edge finite element discretization of the magnetostatic problem formulated in terms of the magnetic vector potential. With the same purpose of invertibility, we analyse another classically used condition, the Coulomb gauge. We conclude by underlying that the two gauges can be naturally considered in a high order framework without any restriction on the topology of the domain. (hal-03426096 = [15] )

The well-known tree-cotree gauging method for low-order edge finite elements is extended to high-order approximations within the first family of Nédélec finite element spaces. The starting point of the algorithm is a spanning tree of the graph given by vertices and edges of the mesh (the so-called global spanning tree, that is the one used in the low-order case). This global step, interpreted in the high-order sense, is enriched locally, with a loop over the elements of the mesh, with arcs corresponding to edge, face and volume degrees of freedom required in the high-order case. (See 12 )

In 20, a new model describing the evolution of the diamagnetic function

Neural networks have been used to compute the plasma boundary using experimental magnetic measurements on a database of the Tokamak West. As the plasma boundary is not directly avalaible on the machine, we used plasma boundary computed by NICE to train the model. A neural network taking as input the magnetic measurement and providing the plasma boundary has been implemented. The first results are encouraging. Other neural networks has been implemented to compute the 2D magnetic flux and the function

Non linear MHD simulations for tokamaks are now mature enough to be used for control, optimization or data assimilation purposes. The use of these tool can largely benefit from the possibility to compute gradients of the results of the simulations with respect to initial data or parameters. In collaboration with the ECUADOR team, we have used the automatic differentiation tool TAPENADE to obtain the adjoint code of a reduced MHD model. Several problems related with FORTRAN arrays have been solved and an adjoint code for a reduced MHD model coded in the CTfem framework have been obtained.

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. We have begun a numerical investigation of acoustic streaming motion (of the Rayleigh type) in a compressible gas inside two-dimensional rectangular enclosures. To numerically study the effects of the sound field intensity on the formation process of streaming structures, the full compressible two-dimensional Navier-Stokes equations have been discretized using a high-order compact scheme. An article describing the numerical method and the results of this investigation has been submitted.

In the framework of an intership with Master students of the engineering school Polytech'Nice Sophia, we have begun the development of a numerical code based on the diffusive wave approximation of the shallow water equations. Collecting topographic data related to the Vesubie river watershed have been done and some preliminary computations corresponding to the Alex tempest of October 2020 have been realized.

This year's objectives included field surveys and observations to calibrate the dissipative effects and friction coefficients in the target study basin in the city of Douala. Due to the health crisis, this could not take place and delays the realistic applications of our SSW model. Nevertheless, we have continued the development of the simplified and the more accurate modelling. The results obtained on these themes have being submitted for publication in an international journal 9. A two-week training course was held in Douala on the derivation of "shallow-water" (SW) equations, attended by a dozen students (M2 and PhD) and four senior researchers. We have anticipated our simulation program by performing the first regularization and refinement of the mesh associated with the Togo-Bassa target basin.

Our goal for next year is to make a significant effort to gather the field data needed to perform the first realistic simulations on a portion of the Tongo-Bassa Basin.This is where there are strong interactions between rainfall run-off and oceanic tidal effects. The SSW model seems to be adapted for these complex interactions that generate turbulence. For simulation purposes, we need a fairly accurate topography, including the internal profile of the rivers. These data will be obtained during measurement campaigns using remote-controlled miniature boats equipped with sonar, as well as flying drones. We also wish to organize a week of sustained work at CIRM on modelling and numerical methods.

During the scientific stay of D. Balsara in nice in November 2021, we continued our investigation on the derivation of a genuinely multi-dimensional Riemann solver for non-conservative hyperbolic systems. In our first attempts, no difference was made between possible conservative and non-conservative contributions. It turns out that it is desirable, in a high-order MultiD Riemann solver, to preserve the contributions that are expressed in terms of flows. One can thus use Simpson's rules for numerical integrations on the edges of the cells. The theoretical analyses performed for different numerical strategies are very encouraging. This trend will have to be confirmed by future numerical simulations.

Dynamo instabilities power long-lived magnetic fields in the electrically conducting layers of astrophysical flows. The effect of various flow properties, such as the spatial variability of the electrical conductivity, or the intensity of stratification, are studied numerically in simplified flow models motivated by astrophysical, and in particular stellar applications. Particular attention is paid to the case where the flow is linearly stable with respect to the dynamo instability, but unstable to finite-amplitude perturbations. An important part of the work here consists in the development of a flexible adjoint-based optimisation code for fully nonlinear, 3D, unsteady MHD flows, and its application to the identification of subcritical dynamo instabilities. This workpackage involves collaborations with researchers from various institutes, in particular Y. Ponty (Observatoire de la Cote d’Azur), L. Petitdemange (LERMA / Observatoire de Paris), C. Gissinger (ENS), F. Petrelis (ENS), B. Gallet (CEA Saclay).

Cédric Boulbe and Blaise Faugeras participate to the Eurofusion Workpackage WPSA

Boniface Nkonga and Hervé Guillard participate to the EuroFusion
workpackage 01 (Tokamak exploitation)TSVV

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

"Graines minimales de dynamos célestes" May 2020-2022, PI : Florence Marcotte.

F. Marcotte has joined the Conseil Scientifique of Académie « Systèmes Complexes », Université Cote d’Azur.