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Modelling Eruptions and Edge Stability in Tokamak Plasmas

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Modelling Eruptions and Edge Stability in Tokamak Plasmas Modelling Eruptions and Edge Stability in Tokamak Plasmas Amelia Elizabeth Louise Lunniss Doctor of Philosophy University of York Physics September 2016 Abstract In the high confinement mode (H-mode) of tokamak operation, sharp gradients and the resulting high bootstrap current near the edge of a tokamak plasma (the pedestal) typ- ically trigger eruptions called edge localised modes (ELMs). On the ITER scale, these have the potential to cause unacceptable erosion of materials. However, there exist sce- narios, such as the quiescent H-mode (QH), where there are no ELMs. The ELITE code was originally developed to efficiently calculate the edge ideal MHD stability proper- ties of tokamaks, optimised for the intermediate-high toroidal mode number, n, modes associated with ELMs. In QH-mode the limiting MHD is typically low n. Chapter 3 presents the extension of the ELITE code to arbitrary n. Chapter 4 presents successful benchmarks against the original ELITE code as well as GATO and MARG2D at low n. A first application of the new ELITE code was to study the stability of the QH-mode pedestal in DIII-D. Results from this study are presented in Chapter 5, which show the presence of low n phenomena. Additionally, understanding the pedestal performance losses in JET ITER-like wall (ILW) plasmas is vital to the success of future JET and ITER experiments. Chap- ter 6 presents an inter-ELM pedestal stability study, which compares the pedestal evo- lution to the criteria of the pedestal structure model, EPED. These results suggest that maximising the region of plasma that has second stability access will lead to the highest pedestal heights and, therefore, best confinement - a key result for optimising the fusion performance of JET and future tokamaks, such as ITER. 3 Contents Abstract 3 Contents 5 List of Tables 11 List of Figures 13 Acknowledgements 17 Declaration 19 1 Introduction 21 1.1 Global energy outlook . 21 1.2 Fusion energy . 22 1.3 Ignition . 23 1.4 Definition of a plasma . 24 1.5 Magnetic confinement fusion . 25 1.6 The tokamak . 26 1.7 H-mode and the pedestal . 27 1.8 Edge localised modes (ELMs) . 28 1.9 ELM suppression and mitigation . 29 1.9.1 ELM categorisation . 30 1.9.2 QH-mode . 31 1.10 Thesis motivation and outline . 32 2 Theory and ELITE 35 2.1 Magnetohydrodynamics and ideal MHD . 35 2.2 Tokamak equilibria . 37 2.2.1 Flux surfaces . 38 2.2.2 Grad-Shafranov equation . 39 2.2.3 Safety factor, q . 39 2.2.4 Other important plasma parameters and typical plasma profiles 40 2.2.5 Equilibrium codes . 42 2.3 MHD stability . 43 2.3.1 Linear analysis of stability and the energy principle . 45 2.4 MHD instabilities . 48 5 2.4.1 Ballooning modes . 48 2.4.2 Kink modes . 50 2.4.3 Coupled peeling-ballooning modes . 51 2.4.4 Numerical codes for ideal MHD stability analysis . 54 2.5 The ELITE and Mercier-Luc formalisms . 55 2.5.1 The Euler equation . 55 2.5.2 The δW equation . 57 2.5.3 Inertia contribution to δW ....................... 59 2.5.4 Integration by parts . 60 2.5.5 Fourier decomposition and coupled equations . 61 2.5.6 Radial length scales separation . 64 2.5.7 Mercier-Luc formalism . 65 2.5.8 The eigenvalue condition . 68 2.5.9 Example results from ELITE . 68 2.6 Kinetic ballooning modes & the EPED model . 69 2.6.1 Width scaling proxy for the KBM . 71 2.6.2 n=∞ ballooning proxy for the KBM . 72 3 ELITE extension to arbitrary n 75 3.1 Introduction . 75 3.2 δW extension . 75 3.2.1 Integration by parts . 78 3.3 Extension of inertia terms . 79 3.4 Fourier decomposition . 80 3.4.1 Fourier decomposition . 80 3.4.2 Matrix elements and two sets of coupled Euler equations . 83 3.5 Modifying coefficients . 86 3.5.1 Matrix manipulation . 86 3.5.2 Differentiation of the matrix elements . 89 3.6 Conclusions and further work . 91 4 Benchmarks & δW diagnostic 93 4.1 Introduction . 93 4.2 Formalism implementation . 94 4.2.1 High n inertia approximation . 94 4.2.2 Ideal wall edge boundary condition . 94 4.2.3 ELITE: up-down symmetric and non-up-down symmetric . 94 4.3 Circular benchmark . 95 4.3.1 Benchmark of circular test case with an ideal wall . 95 4.3.2 Benchmark of circular test case . 96 4.4 Up-down-symmetric benchmark . 98 4.5 Non-up-down symmetric benchmark . 98 4.6 EFIT-style benchmark . 100 6 4.7 ELITE - calculating the C matrix . 102 4.8 Parameter convergence discussion . 104 4.9 Development of a δW diagnostic . 105 4.10 δW width study . 106 4.11 Conclusions and further work . 108 5 Study of QH-mode in DIII-D 111 5.1 Introduction . 111 5.2 DIII-D discharge 163520 . 113 5.3 Methods . 116 5.4 Results . 117 5.4.1 EHO QH-mode phase: time 2200ms . 117 5.4.2 Wide-pedestal QH-mode phase: time 2650ms . 119 5.4.3 Wide-pedestal QH-mode phase: time 3000ms . 120 5.4.4 Wide-pedestal QH-mode phase: time 3500ms . 121 5.4.5 Wide-pedestal QH-mode phase: time 3985ms . 123 5.4.6 Kinetic ballooning mode constraint . 125 5.5 Conclusions and further work . 129 6 JET-ILW ELM cycle study 131 6.1 Introduction . 131 6.2 JET analysis framework . 132 6.2.1 Equilibrium reconstruction . 132 6.2.2 Equilibrium reconstruction: the profile tool . 134 6.2.3 Equilibrium reconstruction for full JET equilibrium with HELENA136 6.2.4 JET stability analysis scans . 137 6.3 Analysis techniques . 140 6.3.1 EPED model for JET pedestals and current understanding of JET pedestal dynamics . 140 6.3.2 Discussion of ELM cycle windows . 142 6.3.3 Operational points . 143 6.3.4 Discussion of errors on the operational points . 145 6.3.5 Peeling-ballooning boundary . 146 6.4 JET: KBM proxies & second stability . 148 6.5 Introduction to dataset . 151 6.6 Analysis: low gas fuelling . 153 6.6.1 Pressure pedestal height and width evolution . 153 6.6.2 KBM proxy: n=∞ ballooning stability . 155 6.6.3 Overview of low deuterium gas injection rate pulses . 157 6.7 Analysis: medium gas fuelling . 159 6.7.1 Pressure pedestal height and width evolution . 159 6.7.2 KBM proxy: n=∞ ballooning stability . 161 6.7.3 Overview of medium deuterium gas injection rate pulses . 162 7 6.8 Analysis: high gas fuelling . 163 6.8.1 Pressure pedestal height and width evolution . 163 6.8.2 KBM proxy: n=∞ ballooning stability . 164 6.8.3 Overview of high deuterium gas injection rate pulses . 166 6.9 KBM proxy: s − α diagrams . 166 6.10 ELM trace comparison . 169 6.11 Analysis: high triangularity . 173 6.11.1 Pressure pedestal height and width evolution . 173 6.11.2 KBM proxy: n=∞ ballooning stability . 174 6.11.3 Overview of low deuterium gas injection rate, high triangularity pulses . 176 6.12 Summary . 176 7 Conclusions 181 7.1 Conclusions . 181 7.2 Future Work . 183 A The ELITE shooting algorithm 185 A.1 The ELITE shooting algorithm . 185 B.
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