3.1 Plasma Physics

The main reseach topics are:

1. Modelling 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
   • Pressure correction scheme
   • Anisotropy
   • Solving methods and parallelism
3. Identification and control
   • Inverse problem: Equilibrium reconstruction
   • Open loop control

4.1 MHD and plasma stability in tokamaks

Participants: Hervé Guillard, Boniface Nkonga, Afeintou Sangam, Ali Elarif, Ashish Bhole.

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

4.2 Long term plasma evolution and optimization of scenarii

Participants: Didier Auroux, Jacques Blum, Cédric Boulbe, Blaise Faugeras, Hervé Guillard.

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.

4.3 Turbulence and models for the edge region of tokamaks

Participants: Didier Auroux, Louis Lamerand, Francesca Rapetti.

4.4 Understanding magnetogenesis in stellar systems

Participants: Didier Auroux, Paul Mannix, 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 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.

5.1 First equilibrium reconstruction for ITER with the code NICE

Participants: Blaise Faugeras, Jacques Blum, Cédric Boulbe.

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.

5.2 Equilibrium reconstruction of discharges from EUROfusion tokamaks using the WPCD scientific workflows

Participants: A. Merle, R. Coelho, F. Carpanese, S. Dixon, M. Dunne, B. Faugeras, L. FLeury, J. Hollocombe, F. Imbeaux, L. Kogan.

5.3 Plasma initiation and preliminary magnetic control in the HL-2M tokamak

Participants: X. Song, B. Li, J. Zhou, E. Nardon, H. Heumann, B. Faugeras, J.X. Li, Sh. Wang, SH.Y. Liang.

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 $\sim 250$ ms associated with a 120 kA plasma current in flat-top. Preliminary plasma equilibrium reconstruction by the NICE code from pick-up sensors and flux loops, as well as interferometry measurements, gives plasma boundaries, plasma profiles of toroidal current density and safety factor, as well as electron density, in an offline way.

5.4 Development of the RAPTOR suite of codes towards real-time reconstruction of JET discharges

Participants: C. Piron, F. Felici, B. Faugeras, N. Ferron, G. Manduchi, N. Marconato, C. Meekes, L. Piron, Z. Stancar, D. Valcarcel, D. Voltolina, M. Weiland, JET contributors.

5.5 High-order finite elements in tokamak free-boundary plasma equilibrium computations

Participants: F. Rapetti, B. Faugeras, C. Boulbe.

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.

5.6 Introduction of C1 Clought-Tocher finite elements in NICE

Participants: A. Elarif, B. Faugeras, F. Rapetti.

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.

5.7 Coupling Hermite-Bezier quadrangular elements and Clough Tocher triangular elements

Participants: H. Guillard, B. Nkonga, F. Rapetti.

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.

5.8 High-order Whitney finite elements

Participants: F. Rapetti, A. Alonso Rodriguez (Univ. de Trento, Italy).

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 )

5.9 Coupling Nice-Metis

Participants: J.F. Artaud, C. Boulbe, B. Faugeras.

In 20, a new model describing the evolution of the diamagnetic function $f$ which appears in the right hand side of the well known Grad-Shafranov equation has proposed. This model has been implemented in the free boundary equilibrium code NICE. This new version of the code has been tested. A first version of the coupling of Nice with the fast transport solver has started in Matlab. This coupling should be more robust than the last version of the coupling NICE-METIS which was using the current diffusion equation for the evolution of the right hand side of the Grad-Shafranov equation. This work is done in the framework of the workpackage WPSA EUROFUSION H-EUROPE.

5.10 Equilibrium reconstruction using neural networks

Participants: C. Boulbe, B. Faugeras, G. Gros.

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 $p^{\text{'}}$ and $f{f}^{\text{'}}$ appearing in the right hand side of the Grad-Shafranov equation.

5.11 Development of an adjoint code for a reduced MHD model

Participants: H. Guillard, L. Hascoet, Ecuador team.

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.

5.12 Numerical simulation of nonlinear acoustic streaming in standing waves

Participants: V. Mandikas, Technical University of Crete, A. Delis, Technical University of Crete, H. Guillard.

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.

5.13 Diffusive wave approximation of the shallow water equations

Participants: H. Guillard, C. Boulbe.

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.

5.14 Shear shallow water model

Participants: B. Nkonga, C. Praveen, J. Iroume, A. Ngatcha.

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.

5.15 MultiD Riemann solver for non-conservative hyperbolic equations

Participants: B. Nkonga, D. Balsara.

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.

5.16 Dynamo instabilities and nonlinear optimal control of MHD flows

Participants: Didier Auroux, Paul Mannix, Florence Marcotte.

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).

6.1 International initiatives

6.2 National initiatives

7.1 Promoting scientific activities

7.2 Teaching - Supervision - Juries

8.1 Major publications

• 1 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, 960-980
  HAL
• 2 articleD. A.Daniele Antonio Di Pietro, J.Jerome Droniou and F.Francesca Rapetti. Fully discrete polynomial de Rham sequences of arbitrary degree on polygons and polyhedra.Mathematical Models and Methods in Applied SciencesAugust 2020
  HALDOI
• 3 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, 102510
  HALDOI

8.2 Publications of the year

8.3 Cited publications

• 20 articleH.Holger Heumann. A Galerkin method for the weak formulation of current diffusion and force balance in tokamak plasmas.J. Comput. Phys.4422021, 110483URL: https://doi.org/10.1016/j.jcp.2021.110483
  DOIback to text
• 21 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.2020
  back to text