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Damping of Alfvén Eigenmodes in Complicated Tokamak and Stellarator Geometries

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Damping of Alfvén Eigenmodes in Complicated Tokamak and Stellarator Geometries Damping of Alfvén eigenmodes in complicated tokamak and stellarator geometries George Bowden A thesis submitted for the degree of Doctor of Philosophy of The Australian National University June 2017 © George Bowden 2017 Except where otherwise indicated, this thesis is my own original work. George Bowden 19 June 2017 To my parents, Jeff and Janine Acknowledgements I would like to thank the chair of my supervisory panel, Matthew Hole, for intro- ducing me to the field of plasma physics and overseeing my research. His expertise and guidance have been invaluable to the development of work presented here. I am grateful also to my other supervisors, Axel Könies, Robert Dewar, and Michael Fitzgerald, for many discussions which have enriched my understanding of this area of physics. I am particularly indebted to Axel for his hospitality during my visits to IPP Greifswald and for providing access to CKA and its associated codes. Other members of the Plasma Theory and Modelling group have provided in- valuable expertise, which I want to acknowledge here. These include Graham Den- nis, Brett Layden, and Greg von Nessi. I also thank Bernhard Siewald and Henry Leyh for their patience and generosity in assisting me with many computing issues. I have enjoyed working alongside and friendship with fellow Plasma Theory and Modelling postgraduate students Ley Chen, Sebastian Cox, Hooman Hezaveh, Zhisong Qu, Jake Ross, and Alexis Tuen. I thank Sebastian particularly for the many entertaining and discursive discussions we had when we shared an office, some even relating to physics. I greatly appreciate the assistance of Boyd Blackwell and Shaun Haskey from the H-1NF experimental group in sharing their expertise regarding wave activity on that machine. Thanks also to Christoph Slaby for his input regarding kinetic extensions to MHD. I gratefully acknowledge the collaboration of Nikolai Gorelenkov who pro- vided access to NOVA and Samuel Lazerson who provided GIST output for H-1NF. Thank you also to Gopal Gopalsamy for assistance in editing previous drafts of this thesis. I thank my friends in Adelaide for staying in touch while I have been studying in Canberra, I look forward to catching up whenever I return. Thank you also to the new friends who have helped welcome me to Canberra. Finally, and above all, I wish to thank my parents, Jeff and Janine, and my sisters, Phoebe and Jane, for their love, encouragement and support. vii Abstract A variety of Alfvén wave phenomena are found in toroidal magnetically confined fusion plasmas. Shear Alfvén eigenmodes may exist, which can be driven unstable by interaction with energetic particles. The linear stability of such modes depends on damping through several mechanisms. Continuum resonances cause damping of the modes, which occurs even in non-dissipative ideal magnetohydrodynamic (MHD) theory given appropriate treatment of resulting poles. Additional damping of the modes occurs due to conversion to kinetic Alfvén waves and finite parallel electric fields when kinetic extensions to MHD are considered. In this thesis, methods for calculating the damping of Alfvén eigenmodes are developed, with particular focus on the continuum damping component. Damping of modes in complicated two- and three-dimensional magnetic geometries characteristic of tokamak and stellarator plasmas is considered. In this work, shear Alfvén eigenmodes are analysed based on reduced MHD models. A background is provided, covering relevant theoretical aspects of plasma equilibrium, coordinate systems and linearised MHD waves. A coordinate inde- pendent reduced MHD wave equation is derived for Alfvén eigenmodes in low b tokamaks and stellarators. Coupled wave equations in terms of Fourier harmonics of the eigenmode are then derived for large aspect-ratio plasmas. Expressions for continuum damping are derived perturbatively from the coor- dinate independent and coupled harmonic wave equations. Application of the ex- pressions using Galerkin and shooting methods is described. Damping computed in this manner is compared with values from an accepted method for the benchmark case of a TAE in a large aspect-ratio circular cross-section tokamak. The perturbative technique is shown to produce significant errors, even where continuum damping is small. A novel singular finite element method is developed to compute continuum damping. The Galerkin method adopted employs special basis functions reflecting the asymptotic form of the solution near continuum resonance poles. For particu- lar eigenmodes, the unknown complex eigenvalue and pole location are computed iteratively. The procedure is verified by application to a TAE in a large aspect-ratio circular cross-section tokamak, where well converged and accurate complex eigen- value and mode structure are obtained. Continuum damping can be computed numerically by solving the ideal MHD eigenvalue problem over a complex contour which circumvents continuum reso- nance poles according to the causality condition. This calculation is implemented in the ideal MHD eigenvalue code CKA, using analytic continuation of equilibrium quantities. The method is verified through application to a TAE in a tokamak, where the complex eigenvalue computed agrees closely with that found using the accepted ix x resistive method, but converges faster with increasing radial mesh resolution. Con- tinuum damping of shear Alfvén eigenmodes is computed for three-dimensional configurations in torsatron, helias and heliac stellarators. Extensions to the ideal MHD wave equations allow non-ideal kinetic effects to be modelled. The damping of a TAE in a tokamak case through these effects is computed using different models for magnetic geometry and kinetic effects. Choice of the former strongly influences results, while choice of the latter does not. Damping from kinetic effects is also computed for an NGAE in a heliac. Contents Acknowledgements vii Abstract ix 1 Introduction 1 1.1 Nuclear fusion . .1 1.1.1 Controlled fusion . .1 1.1.2 Conditions for fusion . .2 1.1.3 Magnetic confinement . .3 1.1.4 Tokamaks . .5 1.1.5 Stellarators . .5 1.2 The physics of plasmas . .5 1.2.1 Plasma definition . .5 1.2.2 Magnetised plasmas . .6 1.2.3 Magnetohydrodynamics . .7 1.2.4 Alfvén waves . .8 1.2.5 Fast particles . 13 1.2.6 Continuum damping . 13 1.2.7 Radiative damping . 15 1.3 Motivation and aims . 16 1.4 Outline . 16 2 Ideal MHD Equilibrium 19 2.1 Equilibrium . 19 2.2 Coordinate systems . 21 2.2.1 Toroidal coordinates . 21 2.2.2 Flux coordinates . 21 2.2.3 Straight field line coordinates . 22 2.3 Computing plasma equilibria . 23 2.3.1 VMEC . 23 2.3.2 Mapping to Boozer coordinates . 24 2.3.3 Recalculation of MHD equilibria . 24 3 Ideal MHD Waves 27 3.1 Shear Alfvén wave equation . 27 3.1.1 General geometry . 28 3.1.2 Large aspect-ratio . 34 xi xii Contents 3.2 Variational formulation . 39 3.3 MHD Spectra . 40 3.3.1 Continuum modes . 40 3.3.2 Discrete modes . 43 3.4 Continuum damping of discrete modes . 44 3.5 Non-ideal effects . 47 3.6 Finite element method . 50 3.7 CKA . 52 4 Perturbative Calculation of Continuum Damping 55 4.1 Perturbative formalism . 55 4.2 Evaluation of quadratic form perturbation . 58 4.3 Quadratic form variation with frequency . 62 4.4 Implementation . 64 4.4.1 Finite element method . 64 4.4.2 Shooting method . 66 5 Complex Integration Contour Calculation of Continuum Damping 69 5.1 Specification of the contour . 69 5.2 Analytic calculation of discontinuities . 71 5.3 Numerical implementation . 73 5.3.1 Finite element method . 73 5.3.2 Shooting method . 75 6 Singular Finite Element Calculation of Continuum Damping 77 6.1 Variational formulation with continuum resonances . 77 6.2 Singular basis functions . 78 6.3 Iterative solution procedure . 82 6.4 Verification . 83 7 Damping of Alfvén Eigenmodes due to Kinetic Effects 89 7.1 Calculation of collision frequency . 90 7.2 ITPA benchmark TAE case . 90 7.3 Effect of higher order geometric terms . 92 7.4 Effect of E∥ model . 94 7.5 Temperature dependence . 94 7.6 Heliac NGAE case . 97 8 Continuum Damping of Alfvén Eigenmodes in Large Aspect-Ratio Toka- maks 101 8.1 Large aspect-ratio circular cross-section tokamak TAE case . 101 8.2 Variation of equilibrium quantities . 102 8.3 Wave function perturbation . 107 Contents xiii 9 Continuum Damping of Alfvén Eigenmodes in General 2D and 3D Geom- etry 111 9.1 Circular cross-section tokamak TAE case . 112 9.2 Torsatron TAE and EAE cases . 116 9.3 Helias TAE case . 123 9.4 Heliac NGAE cases . 124 10 Conclusion 133 A Continuum Resonance Singularities 137 A.1 General geometry . 137 A.2 Large aspect-ratio . 143 A.3 Self-adjointness of continuum resonance condition operator . 145 B Finite Element Method Implementation Using MATLAB 147 References 149 xiv Contents List of Figures 1.1 Coordinates and fluxes in toroidal geometry. .4 1.2 Continuum resonance frequency for a periodic cylinder and a tokamak. 12 2.1 Simple toroidal coordinate system. 22 2.2 Magnetic field lines on a flux surface. 23 3.1 Continuum resonance poles along with causal integration contour. 45 3.2 Logarithmic functions with branch cuts. 46 5.1 Causal integration contour used to calculate discontinuity due to branch cut............................................ 72 6.1 Logarithmic basis functions. 80 6.2 Constant basis function. 81 6.3 Linear basis function. 82 6.4 Continuum resonance frequency for singular finite element method verification case. 84 6.5 Convergence of TAE continuum damping with respect to excluded region over five iterations. 85 6.6 Convergence of TAE continuum damping with respect to inner region width. 86 6.7 Convergence of TAE continuum damping with respect to radial reso-.
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