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Simulation Magnéto-Hydro-Dynamiques Des Edge-Localised-Modes Dans Un Tokamak

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Simulation Magnéto-Hydro-Dynamiques Des Edge-Localised-Modes Dans Un Tokamak Laboratoire de Physique des Interactions Institut de Recherche sur la Fusion par Ioniques et Moléculaires (PIIM) confinement Magnétique (IRFM) Université de Provence CEA Cadarache Thèse de doctorat de l’Université de Provence Simulation Magnéto-Hydro-Dynamiques des Edge-Localised-Modes dans un tokamak Présentée par Stanislas PAMELA Soutenue le 27 Septembre 2010 devant le jury composé de : Pr. Howard WILSON Professeur au Department of Physics Rapporteur University of York, United-Kingdom Dr. Hinrich LUTJENS Docteur d’état, Chargé de Recherche au CNRS Rapporteur CPHT, Ecole Polytechnique, Palaiseau Dr. Xavier LITAUDON Chef-Adjoint du Service de Chauffage et Président du Jury Confinement du Plasma, IRFM, CEA Cadarache Pr. Sadruddin BENKADDA Professeur au Laboratoire PIIM, Directeur de Recherche Directeur de thèse CNRS, Université de Provence, Aix-Marseille I Dr. Guido HUYSMANS Chercheur au Service de Chauffage et Responsable CEA Confinement du Plasma, IRFM, CEA Cadarache Pr. Peter BEYER Professeur au Laboratoire PIIM Invité Université de Provence, Aix-Marseille I Dr. Pierre RAMET Maître de Conférences, LABRI, INRIA Invité Université de Bordeaux I TABLE OF CONTENTS 1. Introduction ----------------------------------------------------------------------------------- 1 1.1 Nuclear Fusion ----------------------------------------------------------------------- 1 1.1.1 Fusion Energy ------------------------------------------------------------- 1 1.1.2 Energy Confinement ----------------------------------------------------- 2 1.1.3 Inertial and Magnetic Confinement ------------------------------------ 3 1.1.4 Brief History of Tokamaks ---------------------------------------------- 4 1.1.5 ITER and the Future of Fusion Energy -------------------------------- 5 1.2 Tokamak Configuration ------------------------------------------------------------- 6 1.2.1 Magnetic Configuration -------------------------------------------------- 6 1.2.2 X-point Geometry --------------------------------------------------------- 8 1.2.3 Plasma-Facing Components --------------------------------------------- 9 1.2.4 Heating Systems, Particle Injection ------------------------------------- 11 1.3 Magneto-Hydro-Dynamics --------------------------------------------------------- 13 1.3.1 Ideal MHD Equations ---------------------------------------------------- 13 1.3.2 Validity of MHD ---------------------------------------------------------- 15 1.3.3 Boundary Conditions ----------------------------------------------------- 17 1.3.4 Conservation Laws ------------------------------------------------------- 18 1.3.5 Non-Ideal MHD Models ------------------------------------------------- 19 1.4 Ideal MHD Equilibrium in Tokamaks -------------------------------------------- 21 1.4.1 The Grad-Shafranov Equation ------------------------------------------ 21 1.4.2 Tokamak Equilibrium ---------------------------------------------------- 23 1.5 Turbulent Transport and MHD Instabilities -------------------------------------- 26 1.5.1 Parallel and Perpendicular Transport ----------------------------------- 26 1.5.2 Classical and Neoclassical Predictions --------------------------------- 27 1.5.3 Turbulence ----------------------------------------------------------------- 28 1.5.4 MHD Instabilities --------------------------------------------------------- 29 1.6 Edge-Localised-Modes -------------------------------------------------------------- 32 1.6.1 The H-mode ---------------------------------------------------------------- 32 1.6.2 Edge-Localised-Modes --------------------------------------------------- 33 1.6.3 Characterization of ELMs ------------------------------------------------ 34 1.6.4 Advantages and Disadvantages of ELMs ------------------------------ 36 1.6.5 Avoiding and Controlling ELMs ---------------------------------------- 36 1.7 Understanding ELM Physics ------------------------------------------------------- 37 1.7.1 Experimental Observation ----------------------------------------------- 37 1.7.2 Theoretical Interpretation ------------------------------------------------ 38 1.7.3 Numerical Approach ----------------------------------------------------- 41 1.7.4 Combining the Three Methods ------------------------------------------ 41 1.8 Thesis Plan ---------------------------------------------------------------------------- 42 2. The Numerical Tool: JOREK -------------------------------------------------------------- 45 2.1 Résumé du chapitre ------------------------------------------------------------------ 45 2.2 Discretization ------------------------------------------------------------------------- 47 2.2.1 Poloidal and Toroidal Discretization ----------------------------------- 47 2.2.2 Finite Elements ------------------------------------------------------------ 49 2.2.3 The X-point Grid ---------------------------------------------------------- 54 2.2.4 Continuity at X-point ----------------------------------------------------- 56 2.3 Time Stepping ------------------------------------------------------------------------ 57 2.3.1 Implicit Scheme ----------------------------------------------------------- 57 2.3.2 PastiX Solver -------------------------------------------------------------- 59 2.3.3 GMRES Convergence and Numerical Limits ------------------------- 59 2.4 The Physical Model ----------------------------------------------------------------- 60 2.4.1 Reduced Resistive MHD ------------------------------------------------ 60 2.4.2 The Weak Form of Equations ------------------------------------------ 62 2.4.3 Boundary Conditions ---------------------------------------------------- 63 2.4.4 Normalization ------------------------------------------------------------- 64 2.4.5 Extensions of the Model ------------------------------------------------- 65 2.5 ELMs Simulations ------------------------------------------------------------------ 70 2.5.1 Running Jorek ------------------------------------------------------------- 70 2.5.2 The Choice of Equilibrium and Harmonics --------------------------- 73 2.5.3 The Main Plasma Parameters ------------------------------------------- 74 2.6 Summary ------------------------------------------------------------------------------ 76 3. The Influence of the Equilibrium Poloidal Flow on ELMs -------------------------- 78 3.1 Résumé du chapitre ----------------------------------------------------------------- 78 3.2 Introduction and Motivation -------------------------------------------------------- 79 3.2.1 Starting From an Unstable Equilibrium -------------------------------- 79 3.2.2 Poloidal Flows at Equilibrium and During ELMs -------------------- 84 3.3 Equilibrium Poloidal Flows without Parallel Velocity ------------------------- 86 3.3.1 Poloidal Flows in Circular Plasmas ------------------------------------ 86 3.3.2 Flows in X-point Plasmas ----------------------------------------------- 89 3.4 Circular Poloidal Rotation of the Pedestal Plasma ------------------------------ 93 3.4.1 Transitions Between Equilibrium Flow States ----------------------- 93 3.4.2 Flows in Simulations with Parallel Velocity -------------------------- 95 3.5 Influence of the Flow on ELMs --------------------------------------------------- 101 3.5.1 Effect of the Flow on the Linear Stability of Ballooning Modes -- 101 3.5.2 Effect of the Flow on The Nonlinear Stability of ELMs ------------ 102 3.5.3 Summary and Conclusion ----------------------------------------------- 104 4. Simulation of ELMs in the JET and MAST Tokamaks ------------------------------ 108 4.1 Résumé du chapitre ------------------------------------------------------------------ 108 4.2 Introduction and Motivations ------------------------------------------------------ 109 4.2.1 Comparing Simulations and Experiments to Improve the Understanding of ELMs Physics ---------------------------------- 109 4.2.2 Obtain Some General Qualitative Agreement ------------------------ 111 4.2.3 Towards a Quantitative Validation of Simulations ------------------ 112 4.3 JET Simulations ---------------------------------------------------------------------- 113 4.3.1 Reconstruction of the Equilibrium with JOREK --------------------- 113 4.3.2 Filamentation of the Pedestal -------------------------------------------- 118 4.3.3 Divertor Heat Fluxes ------------------------------------------------------ 123 4.3.4 The ELM Size ------------------------------------------------------------- 132 4.4 MAST Simulations ------------------------------------------------------------------- 137 4.4.1 Reconstruction of the MAST Equilibrium ----------------------------- 137 4.4.2 Simulations of ELMs in MAST ----------------------------------------- 137 4.4.3 Improving MAST Simulations ------------------------------------------ 140 4.5 Conclusion ---------------------------------------------------------------------------- 141 Aknowlegements ----------------------------------------------------------------------------------- 145 References ------------------------------------------------------------------------------------------- 147 Appendix A ----------------------------------------------------------------------------------------- 150 1 Introduction 1.1 Nuclear Fusion 1.1.1 Fusion Energy The goal of present research on controlled nuclear fusion is to produce energy from the fusion reaction between Deuterium (D) and Tritium (T ) nuclei. These two hydrogen isotopes may combine to form a helium He atom (or alpha particle) and a neutron n. In this process, the mass outcome is smaller than the
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