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 introducing 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 invaluable 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 provided 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 independent reduced MHD wave equation is derived for Alfvén eigenmodes in low β 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 coordinate independent and coupled harmonic wave equations. Application of the expressions 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 particular 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 eigenvalue 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 resonance 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 resolution...... 86 6.8 Convergence of TAE continuum damping with respect to contour deformation...... 87 6.9 TAE mode structure using singular finite element method...... 87 6.10 TAE mode structure using standard finite element method...... 88

7.1 Continuum resonance frequency for ITPA benchmark case...... 91 7.2 TAE mode structure for ITPA benchmark case, for T 1 keV and T 10keV...... 92 = = 7.3 TAE mode structure for ITPA benchmark case, computed using the large aspect-ratio approximation and CKA...... 93 7.4 Temperature dependence of damping due to kinetic effects in ITPA benchmark case...... 95 7.5 Continuum resonance eigenvalues and global modes in H-1NF case. . 98

xv xvi LIST OF FIGURES

7.6 NGAE mode structure including and excluding kinetic effects in H-1NF case...... 100

8.1 Continuum resonance frequency for large aspect-ratio circular cross- section tokamak...... 102 8.2 Real frequency and damping ratio dependence on qa...... 103 8.3 Real frequency and damping ratio dependence on e...... 104 8.4 Real frequency and damping ratio dependence on ∆1...... 105 8.5 Real frequency and damping ratio dependence on ∆2...... 106 8.6 Comparison of continuum damping calculated using perturbative and complex contour methods...... 107 8.7 TAE mode structure calculated including and excluding continuum resonance discontinuities...... 108

9.1 Continuum resonance eigenvalues in tokamak case...... 113 9.2 Convergence of TAE continuum damping with contour deformation in tokamak case...... 113 9.3 Convergence of TAE continuum damping with conductivity in tokamak case...... 114 9.4 Continuum resonance eigenvalues in torsatron case ...... 117 9.5 Convergence of TAE continuum damping in torsatron case...... 118 9.6 TAE mode structure in torsatron case...... 119 9.7 Convergence of EAE continuum damping in torsatron case...... 121 9.8 EAE mode structure in torsatron case...... 122 9.9 Continuum resonance eigenvalues in W7-X case...... 123 9.10 Convergence of TAE continuum damping in W7-X case...... 125 9.11 TAE mode structure in W7-X case...... 126 9.12 Continuum resonance eigenvalues in first H-1NF case...... 126 9.13 Convergence of NGAE continuum damping in first H-1NF case. . . . . 128 9.14 NGAE mode structure in first H-1NF case...... 129 9.15 Continuum resonance eigenvalues in second H-1NF case...... 129 9.16 Convergence of NGAE continuum damping in second H-1NF case. . . 130 9.17 NGAE mode structure in second H-1NF case...... 131 9.18 NGAE continuum damping density profile dependence in second H-1NF case...... 132 List of Tables

3.1 Geometry induced continuum gaps...... 43

7.1 Parameters for TAE calculation in ITPA benchmark case...... 91 7.2 Parameters for H-1NF NGAE calculation...... 97

xvii xviii LIST OF TABLES Chapter 1

Introduction

1.1 Nuclear fusion

Nuclear fusion is the reaction whereby lighter nuclei combine to form a heavier nucleus. For reactions which produce nuclei with lower masses than that of Fe56, the mass of products is generally less than that of reactants [1]. In accordance with the mass-energy equivalence equation, E mc2, this mass defect corresponds to a net release of energy. Stars, including our sun, are powered by the fusion of light elements. Thus, fusion is the ultimate source= of the vast majority of energy used by life on Earth.

1.1.1 Controlled fusion

Directly harnessing nuclear fusion has been proposed for the generation of electricity. The favoured candidate for controlled fusion is the deuterium-tritium (D-T) reaction,

D T He4 3.5 MeV n 14.1 MeV (1.1)

This reaction has a signiﬁcantly+ → higher( cross-section) + ( at lower) temperatures than the alternatives which are available [2, p. 5]. Moreover, the reaction releases a comparatively large amount of energy via its products. The chargeless n which receive the majority of this energy interact very weakly with the plasma via direct collisions. Thus their energy may be extracted outside the reactor via a moderator. However, the charged He4 nuclei (or alpha particles) can heat the plasma via collisions. In this way it is possible to maintain a self-sustaining reaction. Achieving controlled nuclear fusion would provide an abundant, safe and en- vironmentally benign source of energy. The raw materials consumed by a reactor based on the D-T reaction are deuterium and lithium. Deuterium is obtained from ordinary water and lithium is produced from ore deposits or brines. These resources are widely available, inexpensive, and virtually inexhaustible [3]. Fusion does not produce long-lived radioactive waste products or pose a risk of a runaway process [4]. Base load power can be derived from fusion [3], without the need for inefﬁcient or expensive storage technologies. Fusion does not result in atmospheric pollutants [4] and the resources it consumes can be extracted with low environmental impact.

1 2 Introduction

These attributes are particularly desirable given the rapidly increasing demand for energy from human society. Global energy consumption is projected to grow by 37% above 2014 levels by 2040 [5]. Simultaneously, a reduction in anthropogenic greenhouse gas emissions of between 40% and 70% relative to 2010 levels by 2050 is required for there to be a greater likelihood than not that temperature rise will be limited to no more than 2○C above pre-industrial levels by 2100 [6]. Due to the aforementioned advantages, considerable effort has been made towards harnessing fusion over more than six decades. However, the development of fusion as a viable energy source poses an enormously complex scientific and technical challenge. To derive net energy from fusion, high energy particles must be confined within a reactor with sufficient density and for sufficient time. Under these conditions electrons become dissociated from atoms which form ions and the resulting state is described as a plasma. Great progress has been made, with measures of confinement improving by many orders of magnitude. It is anticipated that a demonstration power plant could be operating by the early 2040s [7].

1.1.2 Conditions for fusion

Fusion reactions require nuclei to cross a large potential barrier due to the long range repulsive electrostatic force between them. They must approach sufficiently closely to experience the short range attractive strong nuclear force in order to combine. Therefore, highly energetic nuclei are required for these reactions to occur. The D-T reaction has a peak cross-section for collisions at 100 keV [2, p. 5]. For a beam of particles undergoing µ interactions per unit length with a target species of number ∼ µ density N, the cross-section of the interaction is defined σ N . While nuclei with such energy can be readily obtained using a particle accelerator, most will rapidly lose energy through scattering interactions with particles in their≡ target preventing a net gain in energy through fusion. A more promising approach is thermonuclear fusion, in which a plasma is heated to a sufficiently high temperature for the reaction to occur. Losses occur from bremsstrahlung (X-ray production due to electron-ion collisions) and heat transport through the plasma. At steady state, these must be balanced by a combination of Ohmic, external, and alpha particle heating. Ignition occurs where alpha particle heating balances losses without the need for external heating and it is envisaged that future reactors will operate close to this condition. Energy production is maximised for constant pressure where particles have characteristic energies of approximately 20 keV, which corresponds to temperatures of 2.3 108 K where the product of particle density and confinement time must be approximately 2 1020 sm−3 for ignition [8]. ×Any material in contact with plasma at such high temperatures will itself be rapidly ionised× and rapidly cool the plasma below the temperatures required for fusion [9, pp. 9-10]. Thus, the necessary confinement of the plasma cannot be provided by a material vessel. The main concepts which have been developed for controlled fusion are inertial and magnetic confinement. In inertial confinement a small pellet of fuel is rapidly compressed and heated by lasers or particle beams. In contrast, in §1.1 Nuclear fusion 3

magnetic conﬁnement hot plasma is held in place by a magnetic ﬁeld.

1.1.3 Magnetic conﬁnement

Plasmas are highly conductive gaseous fluids. In a plasma a significant proportion of atoms are ionised and the resulting ions and electrons can respond to electric and magnetic fields. In a uniform magnetic field the constituent ions and electrons undergo gyrational motion centred on field lines. The Lorentz force F v B, acts perpendicular to the magnetic field and particle velocity to maintain uniform circular motion perpendicular to the field and they can move freely parallel= to the× field. However, particle drifts across magnetic field lines occur where the magnetic field varies in space or time or additional forces are applied to the particles. By performing a Lorentz transformation (in the limit v c) it can be shown that if a uniform electric field E exists in addition to the magnetic field, the charged particle ≪ E×B will execute the previously described motion in a frame moving at v B2 with respect to the original frame [10, pp. 24-26]. Thus, the centre of motion of the particles is observed to drift at this velocity. Replacing the electric field with an= arbitrary and F×B constant force acting on the particles yields the more general expression, v qB2 . In this way, it is possible to obtain expressions for drifts due to field line curvature, mv2 2 = ∥ ˆ ˆ mv⊥ ˆ v qB2 B b b , and non uniform field strength, v 2qB2 b B. It can be shown that the magnetic moment of charged particles in a magnetic = × ∇ × × = × ∇ field is an adiabatic invariant of their motion [11, pp. 41-45]. Kinetic energy is also constant for such motion, and it is found that particles with an insufficient ratio of kinetic energy to magnetic moment cannot pass into a region of higher magnetic field and are instead reflected. Magnetic mirror and cusp confinement schemes have been developed based on this principle [12]. However, such approaches suffer significant leakage of non-trapped particles and are no longer a major focus of fusion research [9, p. 44]. A more promising approach is toroidal magnetic confinement. The equations which describe the static equilibrium of a perfectly conducting magnetised plasma with isotropic pressure are p j B (1.2)

∇ = × B µ0j (1.3)

∇ × = B 0. (1.4)

Here p is pressure, j is current density∇ and ⋅ =B is magnetic field. From equation (1.2) and vector identities it follows that B p 0. This implies that either p 0 or B is tangential to a surface of constant p. If the former is the case for a volume of the plasma, the magnetic field in that region⋅ ∇ does= not contribute to the confinement∇ = of the plasma. The latter requires that surfaces of constant p have toroidal topology [13]. Therefore, to confine the plasma, magnetic field lines must be confined to toroidal 4 Introduction

constant p surfaces which foliate the plasma. These are referred to as flux surfaces and their existence is guaranteed for axially or helically symmetric geometries but not generic three-dimensional geometries [14]. This magnetic field configuration can be obtained by positioning a set of coils with uniform spacing at some distance R0 from the axis of symmetry. However, such a magnetic field must decrease in the radial direction R [15]. This gradient results in opposing paticle drifts for ions and electons, polarising the plasma. The resulting electric field results in an additional drift which rapidly ejects the particles from confinement. An additional poloidal magnetic field component is required to prevent charge separation from occuring. This can be achieved in different ways through the tokamak or stellarator concepts. The helical twist in magnetic field lines on a flux surface can be described by the rotational transform ι or the safety factor q which are defined as 1 dψp ι , (1.5) q dψt where ψp and ψt are the poloidal and≡ toroidal≡ magnetic flux respectively (defined in Figure 1.1). This is equal to the average number of poloidal rotations per toroidal revolution along a field line,

1 N ι lim ∆θi (1.6) 2πN i=1 = Q where N is the number of toroidal rotations.

B ϕ Σ p R

θ Σ t ψp=∫ B⋅d a Σp

ψt =∫ B⋅d a Σt

Figure 1.1: Position on a flux surface is described by the toroidal coordinate φ and poloidal coordinate θ. Surfaces defining toroidal magnetic flux ψt and poloidal magnetic flux ψp are shown. §1.2 The physics of plasmas 5

1.1.4 Tokamaks

In tokamaks a non-zero rotational transform results from toroidal current through the plasma. Induction can be used to produce this current in a pulsed device, though alternative current drive methods must be used for steady-state operation. Additional control over the plasma is provided by plasma positioning and shaping coils carry- ing toroidal currents. Confinement can be improved by pushing more of the plasma toward the high field inboard side, resulting in a D-shaped cross-section. While the finite number of toroidal field coils leads to some toroidal variation in the magnetic field (termed field ripple) tokamak plasmas can generally be treated as axisymmetric and therefore described by two-dimensional coordinates.

1.1.5 Stellarators

In stellarators, the shaping and arrangement of the coils produce a non-zero rotational transform in the absence of plasma currents. This may be achieved either through helical deviation of the magnetic ﬁeld lines in the vicinity of helical currents or through torsion of the magnetic axis [8]. These effects depend on a nonaxisymmetric plasma geometry, in contrast to tokamaks. Magnetic surfaces are helically shaped, resulting in a more complicated three-dimensional plasma geometry. Stellarator design is characterised by a much larger set of parameters than is the case for tokamaks [8]. For these reasons, stellarators are more difﬁcult to design, analyse, and build than tokamaks as they require coils, vacuum vessels, and other components with complicated geometry. However, these devices do not require current drive and therefore may be more suitable for steady state operation and are less prone to disruptions [16].

1.2 The physics of plasmas

1.2.1 Plasma deﬁnition

A formal deﬁnition of a plasma, proposed by Chen, states that: a plasma is a quasineu- tral gas of charged and neutral particles which exhibits collective behaviour [17, p. 3]. Quasineutrality applies over length scales such that L λD, where the Debye length

1 2 ≫ e0kTe λD 2 (1.7) neqe represents the length scale over which= electrons shield charge imbalances due to thermal ﬂuctuations (with Te and ne respectively representing electron temperature and number density). For shielding to occur on this scale, the number of electrons within a Debye sphere (a volume with radius λD)

4 3 ND πλ ne (1.8) 3 D = 6 Introduction

must be sufﬁciently large, ND 1. For collective behaviour to occur, the time scale for electron collective oscillations should be much shorter than that for electron- ≫ neutral collisions, τe τn. The former time scale is based on the plasma frequency

≪ 2π nq2 ω e (1.9) τ ¿e m e Á 0 e ÀÁ While these represent basic conditions= for= a plasma, additional requirements are imposed by the models discussed below.

1.2.2 Magnetised plasmas

Magnetic fields have a significant effect upon plasma behaviour due to its conductivity. The equation of motion for a (non-relativistic) charged particle of mass m and charge q can be found by inserting the expression for the Lorentz force into Newton’s second law dv m q E v B (1.10) dt where v is the velocity of the particle, E=is( the+ electric× ) field and B is the magnetic field. However, the electric and magnetic fields in a plasma are themselves dependent on the position and velocity of many interacting charged particles. A plasma containing a large number of paticles can be decribed statistically through kinetic theory. The plasma is described by distribution functions fα r, v, t , where the probable number of particles of species α within a phase space volume element drdv is fα r, v, t drdv. The time evolution of the system is determined by( the) Boltzmann equation: ( ) ∂ fα qα v fα E v B v fα Cα (1.11) ∂t mα

+ ⋅ ∇ + ∂ ( + × ) ⋅ ∇ = where we deﬁne the operator v ∂v . The term Cα is the collision operator for species α representing the rate of change in its distribution function due to short range interactions between individual∇ = particles. The creation and destruction of ions and electrons is neglected in this case, though these could be represented by additional contributions to these operators. Multiplying equation (1.11) by 1, mvα and 1 2 2 mv and integrating over velocity space, we obtain its zeroth, ﬁrst, and second order moments respectively. These equations respectively correspond to continuity, momentum balance, and energy balance expressed in terms of macroscopic quantities which are stated by Goedbloed and Poedts [9, p. 52]. Closure of this set of equations is obtained by relating these quantities and their gradients using transport theory.

For time scales τ much greater than the collisional relaxation times of ions τi and electrons τe, plasma can be described as consisting of interacting electron and ion fluids. The equations derived from kinetic and transport theory can be combined with those of electrodynamics to model such a plasma. Further simplification can be obtained by neglecting viscosity and heat flow, which usually occur on time-scales §1.2 The physics of plasmas 7

which are much longer than those of macroscopic ﬂuid phenomena in plasmas [9, pp. 67-69]. Thus dissipative effects are limited to resistivity, which is described using Ohm’s law. In many scenarios, including those without external heating, pressure anisotropy can be neglected.

1.2.3 Magnetohydrodynamics

For phenomena with time scales much longer than the electron-ion temperature equilibration time, electrons and ions can be treated as a single ﬂuid. Magneto- hydrodynamics (MHD) is obtained under the additional assumption that the phenomena it describes have length and time scales which are much greater than the −1 radius, ρi, and period, 2πωci , of ion cyclotron motion respectively (here ωci represents ion cyclotron frequency). Further simpliﬁcation can be obtained by assuming non-relativistic speeds (v c). The equations of resistive MHD are [9, p. 69]

≪ ∂ρ ρv 0 continuity (1.12) ∂t ∂v ρ v v j+ ∇B ⋅ ( p) = 0 momentum (1.13) ∂t ∂p v p γ+p ⋅ ∇v −γ ×1 +η ∇j 2 = 0 internal energy (1.14) ∂t + ⋅ ∇ + ∇ ⋅ −E( v− B) SηSj = 0 Ohm’s law (1.15) B 0 no monopoles (1.16) + × − = ∂B E ∇ ⋅ = 0 Faraday’s law (1.17) ∂t ∇ × B + µ0j = 0 Ampère’s law (1.18) where ρ is density, p is pressure and∇j ×is current− density.= The ratio of specific heats 5 is denoted by γ, which is equal to 3 for a monatomic gas. Equation (1.12) and equation (1.13) can be derived from kinetic theory via two-fluid theory under the assumptions indicated above. Ampère’s law (equation (1.18)) is used without the displacement current term as this is found to be negligable for plasma phenomena L with characteristic length scale L and time scale τ such that τ c [18, p.60]. An expression for Ohm’s law (equation (1.15)) which neglects electron inertia and the Hall effect is used. The former term can be neglected as the time≪ scale associated with electron collective motion is much shorter than the time scales typical of MHD, τe τMHD. The latter assumes that the relative speed of the ions and electrons is small compared to their centre of mass velocity ue ui v. ≪The above equations fully determine the evolution of a resistive MHD plasma over time. They are equivalent to a system of 15S scalar− S ≪ differential equations in 14 scalar variables. Equation (1.17) implies that if equation (1.16) is satisfied for some particular value of t, it will be satisfied for all t. Thus, equation (1.16) (corresponding to one of the scalar equations) can be considered a restriction on the initial conditions of a resistive MHD plasma, rather than defining its evolution. A further simplification can be achieved by assuming that the time scale of the 8 Introduction

resistive diffusion of the magnetic ﬁeld is much longer than that of the phenomenon 2 µ0 L being investigated τ τR η [9, pp. 70-71]. Setting η 0 in equations (1.12) - (1.18) on this basis results in the equations of ideal MHD. Using equation (1.17) and ≪ ∼ d ∂ = equation (1.18) to eliminate E and j and deﬁning dt ∂t v , the equations can be expressed in a more compact form [9, p. 71]: ≡ + ⋅ ∇ ∂ρ ρv 0 (1.19) ∂t dv ρ +p ∇ ⋅ (j B) = 0 (1.20) dt d p + ∇ − × = 0 (1.21) dt ργ ∂B = v B 0 (1.22) ∂t − ∇ × ( × B) = 0. (1.23) ∇ ⋅ =

1.2.4 Alfvén waves

The time evolution of small perturbations to an ideal MHD equilibrium can be described by linearising equations (1.19) to (1.23). The variables ρ, v, p and B are expressed in the form f f0 f1, where f0 is the equilibrium value and f1 is the perturbation. Consider the case where the magnetic ﬁeld is uniform and constant = + B0 bˆ 0B0, as are ρ0 and p0, and there is no equilibrium ﬂow, v0. Neglecting small terms due to the products of perturbations results in the equations = ∂ρ 1 ρ v 0 (1.24) ∂t 0 1 ∂v1 1 + ∇ ⋅ ( ) = ρ0 p1 B1 B0 0 (1.25) ∂t µ0 + ∇ − ∂p(1∇ × ) × = γp v 0 (1.26) ∂t 0 1 ∂B 1 + v ∇B ⋅ = 0 (1.27) ∂t 1 0 − ∇ × ( × B)1 = 0. (1.28) Taking the partial derivative of equation (1.25)∇ with ⋅ respect= to t and substituting ∂p1 ∂B1 expressions for ∂t and ∂t obtained from equation (1.26) and equation (1.27) respectively, we obtain the wave equation:

∂2v B 1 p 0 ρ0 2 γ 0 v1 v1 B0 0. (1.29) ∂t µ0 − ∇ (∇ ⋅ ) − (∇ × (∇ × ( × ))) × = §1.2 The physics of plasmas 9

Without loss of generality we can deﬁne Cartesian coordinates such that B0 B0zˆ. Dividing by ρ0, we can rewrite equation (1.29) in terms of the ion sound speed =

γp0 cs (1.30) ¾ ρ0 = and Alfvén speed B0 vA (1.31) µ0ρ0 and thus obtain = √

∂2v 1 c2 v v2 v zˆ zˆ 0. (1.32) ∂t2 s 1 A 1 Using vector calculus identities− ∇ (∇ ⋅ we) ﬁnd− that(∇ × (∇ × ( × ))) × =

v1 zˆ zˆ zˆ v1 zˆ v1 zˆ ∂v1 (∇ × (∇ × ( × ))) × = (∇zˆ × (− ∇ ⋅v1 + ( ⋅ ∇) )) ×zˆ ∂z = × ∇ (∇ ⋅ ) + ∇∂ × × ∂v1 ⊥ v zˆ v zˆ 1 ∂z 1 ∂z 2 = ∇ (∇ ⋅ ) − ∂∇v1⊥ ( ⋅ ) − ( ⋅ ∇) ⊥ v ⊥ . (1.33) 1 ∂z2 Inserting equation (1.33) into equation= ∇ ( (1.32),∇ ⋅ we) + obtain the expression

2 2 ∂ v1 2 2 ∂ v1⊥ c v v ⊥ v ⊥ 0. (1.34) ∂t2 s 1 A 1 ∂z2 − ∇ (∇ ⋅ ) − ∇ (∇ ⋅ ) + = For incompressible solutions, equation (1.12) implies that v1 0. In turn, equation (1.34) implies that for oscillatory motion v1 zˆ and thus becomes ∇ ⋅ = ∂2v ∂2v ⊥ 1⊥ v2 1⊥ 0 (1.35) ∂t2 A ∂z2

i(k⋅x−ωt) − = Let v1 v1,0e , representing a plane wave solution, where v1,0 is a complex vector which is independent of location and time. Thus, the substitiutions ik ∂ = and ∂t iω can be made. Hence, we obtain the shear Alfvén wave dispersion relation ∇ → → − 2 2 2 ωA vAkz. (1.36)

Compressible solutions can be found= by constructing a pair of coupled differential equations, using equation (1.34). Taking the divergence of the perpendicular component of this equation and letting χ v1⊥ and ψ v1 zˆ results in

2 ∂ χ 2 ∂ψ= ∇ ⋅ 2 = ⋅ c ⊥ χ v χ 0. (1.37) ∂t2 s ∂z A − ∇ ⋅ ∇ + − (∇ ⋅ (∇ )) = 10 Introduction

Taking the component of equation (1.34) in thez ˆ direction yields

∂2ψ ∂2ψ ∂χ c2 0. (1.38) ∂t2 s ∂z2 ∂z − + = i(k⋅x−ωt) i(k⋅xωt) Once again considering a plane wave solution, we let χ χ0e and ψ ψ0e . After some algebraic manipulation this allows us to obtain the dispersion relation = = 4 2 2 2 2 2 2 2 2 ω cs vA k ω cs vAk kz 0. (1.39)

This results in two solutions− for+ ω2+, − =

2 2 2 2 cs vA 2 4 kz ω k 1 ¿1 2 (1.40) 2 ⎛ Á cs vA k ⎞ + ⎜ Á vA cs ⎟ = ⎜ ± Á − ⎟ ⎜ ÀÁ ⎟ which are the dispersion relations for⎝ the slow ( )+ and fast ( )⎠ magnetosonic waves (also referred to as magnetoacoustic waves by some sources). In the limit cs vA the dispersion relations for the two waves become − + ≪ 2 2 2 ω kzcs (1.41) and = 2 2 2 2 2 ω vAk cs k⊥. (1.42)

The restoring force in the case of= shear+ Alfvén waves can be thought of as an effective tension on magnetic field lines [9, p. 73]. Potential energy is stored due to the bending of magnetic field lines. According to Alfvén’s theorem, plasma follows magnetic field lines and its inertia is associated with the kinetic energy of the shear Alfvén waves. By contrast, the slow and fast magnetosonic waves have an additional restoring force due to pressure. Thus, for these waves potential energy has a component due to plasma compression and rarefaction. The slow and fast magnetosonic waves respectively correspond to the ion acoustic and compressional Alfvén wave in 2µ0 p the special case where β 1. Here we define β B2 , which is the ratio of plasma pressure to magnetic pressure. The ion acoustic wave is due to the pressure force and occurs even where B≪ 0. It propagates parallel≡ to magnetic field lines with speed cs. The compressional Alfvén wave, like the shear Alfvén wave, results from the effective tension in magnetic= field lines. However, unlike the shear Alfvén wave, it can propagate perpendicular to magnetic field lines. In this thesis we are concerned with shear Alfvén eigenmodes of the plasma. Alfvén waves can have wavelengths comparable to typical fusion plasmas and are therefore affected by their global magnetic geometry [9, p. 73-74]. While the foregoing analysis of MHD waves was limited to a uniform, magnetised plasma, additional phenomena become apparent in the more general case of toroidal magnetised plasmas. Variations in density and magnetic field strength alter the local speed with §1.2 The physics of plasmas 11

which waves propagate. Additional Lorentz forces due to equilibrium currents and magnetic field perturbations can also affect the propagation of the wave. Similarly, forces due to finite pressure gradients can affect wave propagation where magnetic curvature exists. At low frequencies and finite plasma pressures, geometrically induced coupling of shear Alfvén and slow magnetosonic waves can occur due to the curvature of magnetic field lines [19]. The waves which can exist in a toroidal magnetised plasma are constrained by periodicity in the poloidal and toroidal directions. The simplest such case is a periodic cylinder with axial and azimuthal symmetry. Solutions with spatial dependence exp imθ inz can be found. In this expression m and n are the azimuthal and axial mode numbers and 2πR is the axial period of the cylinder. Here the par- ( + ) (n+mι) allel wave number is k∥ R (where ι is defined in terms of azimuthal and axial fluxes, analogous to equation (1.6)). Thus, the relevant dispersion relations derived for these waves above determine= the frequencies at which resonant shear Alfvén and ion acoustic modes exist, which are respectively

2 2 2 B0 n mι ω 2 (1.43) µ0ρ0 R ( + ) = and 2 2 γp0 n mι ω 2 . (1.44) ρ0 R ( + ) If the rotational transform and density= vary radially, the frequency of the modes becomes a function of radial location. A continuous spectrum emerges, consisting of modes which are singular at flux surfaces where the dispersion relation is satisfied. These surfaces correspond to poles of the ideal MHD force operator. An example of the shear Alfvén continuous spectrum for a cylinder is shown in Figure 1.2. The ideal MHD continuum can be considered analogous to the continua found in solutions of the Schrödinger equation of quantum mechanics [9, pp. 451-452]. Alfvén waves can produce global modes in magnetised plasmas. Poloidal or toroidal asymmetry in the plasma equilibrium can induce coupling between Alfvén wave harmonics with different m or n. For example, in the idealised case of a large aspect-ratio circular cross-section tokamak, the variation in magnetic field strength and Shafranov shift produce a coupling between the m and m 1 harmonics [20]. Such couplings remove degeneracies between continuum modes with different m and n, resulting in the formation of frequency gaps in the continuous+ spectrum. Global modes can exist with frequencies in these gaps, localised by the effective potential well due to the avoided crossing of the continuum [21]. Thus, in the large aspect ratio circular cross-section tokamak case toroidicity-induced Alfvén eigenmodes (TAEs) can exist [22]. In more complicated tokamak geometries ellipticity-induced Alfvén eigenmodes (EAEs)[23] and noncircularity-induced Alfvén eigenmodes (NAEs)[24] are also possible due to additional couplings between different n harmonics. In stellarators axial asymmetry leads to coupling of harmonics with different n, producing mirror-induced Alfvén eigenmodes (MAEs)[25] and helicity-induced Alfvén eigen- 12 Introduction

1.0

0.8 m = 2 m = 3 m = 4

0.6 0 Ω 0.4

0.2 m = 1 0.0 0.0 0.2 0.4 0.6 0.8 1.0 r

ωR Figure 1.2: Resonant normalised shear Alfvén frequency (Ω0 ) as a function of vA radial position for a periodic cylinder (green, dashed line) and a circular cross-section tokamak with inverse aspect-ratio e 0.1 (blue, solid line). Resonances= for harmonics with toroidal mode number n 1 and poloidal mode number m 1 to 4 are plotted = 1 for constant ρ and rotational transform proﬁle ι r 2 . 1.0+2.0 r = − = a ( ) = modes (HAEs)[26]. Usually gap modes are dominated by those harmonics which are responsible for the avoided crossing, with these harmonics having comparatively strong coupling in this region. Such modes occur at discrete frequencies and have a radially extended mode structure.

Similar phenomena occur at extrema of the continuum. These modes include global Alfvén eigenmodes (GAEs) [27], reversed shear Alfvén eigenmodes (RSAEs) [28], non-conventional global Alfvén eigenmodes (NGAEs) [29], and beta induced Alfvén eigenmodes (BAEs) [30]. Causes of continuum extrema include magnetic shear, radial density variation, and compressibility. Extremum modes are typically dominated by a single harmonic as they lack the strong couplings associated with avoided crossings. In this work we will examine the stabilisation of shear Alfvén eigenmodes via interactions with continuum modes and kinetic Alfvén waves.

The existence of various Alfvén eigenmodes in fusion experiments is well documented. These modes can be excited through either external antennas [31], electrodes inserted into the plasma [32], or fast particle interaction (discussed further in Subsec- tion 1.2.5). Alfvén eigenmodes may experience signiﬁcant damping due to ion Lan- dau damping, trapped electron collisional damping, continuum resonance damping, and radiative damping from kinetic Alfvén wave coupling [21]. The balance between the aforementioned sources of drive and damping represents the linear growth rate of an Alfvén eigenmode. If a fast particle driven mode has positive growth rate, an initial perturbation will grow to a ﬁnite amplitude imposed by non-linear dissipative effects [21]. §1.2 The physics of plasmas 13

1.2.5 Fast particles The He4 nuclei produced by fusion have significantly greater energy than ions in the thermal background population. Even in the absence of fusion, neutral beam injection, ion-cyclotron resonance heating, and electron-cyclotron resonance heating can lead to fast ion populations. Resonant transfer of energy between MHD waves and fast particles is possible provided that particle orbit and wave phases match after some number of cycles, implying that ω pωθ nωφ, where p is an integer, ωθ is the poloidal precession frequency of the particle and ωφ is its toroidal precession frequency. The sign of the energy exchange between= + a fast particle and resonance is determined by the particle velocity relative to the phase velocity of the wave. Thus, gradients in physical and velocity space ion distribution functions can either drive or damp wave activity in magnetised plasmas [21]. Alfvén eigenmodes are examples of wave phenomena which can be destabilised by resonant fast ions. In the case of shear Alfvén waves, energy transfer is predominantly due to work done due to its transverse electric field component and fast particle velocity drifts [21]. Early research identified the potential for TAEs to be destabilised in a burning plasma [33]. Unstable Alfvén eigenmodes can in turn degrade the confinement of fast particles, which can negatively impact fusion power output and damage plasma facing components. Wave interaction causes increased fast particle transport through several mechanisms [21]. Perturbation of the magnetic equilibrium can modify the orbits of particles so that they collide with a wall. Resonant particles experience a net radial E B drift. Radial transport of energetic particles can destabilise a mode at another position due to increased spatial gradients in their density. A series of these events may× propagate through the plasma, analogous to an avalanche. Unstable TAEs have been shown to cause large losses of fast ions in experiments on the DIII-D [34] and TFTR [35] tokamaks. The frequency of fluctuations in fast ion losses is observed to match that of Alfven eigenmodes in DIII-D [36]. Damage to plasma facing components due to fast particle transport resulting from Alfvén eigenmodes has also been documented for these experiments [34, 35]. The potential for fast particles to destabilise global modes and thereby degrade their confinement is a significant concern for future burning plasma experiments [37]. Indeed evidence for alpha particle driven TAEs has already been obtained for TFTR D-T experiments [21].

1.2.6 Continuum damping Continuum resonances can result in damping of Alfvén eigenmodes. Continuum modes have radial frequency dependence which results in phase mixing of any radially extended perturbation composed of these modes. Large spatial gradients emerge, resulting in heavy damping when dissipative effects are considered [9, p. 515]. Where the frequency of a global mode coincides with the resonant frequency for a particular ﬂux surface, the mode couples to continuum modes. In ideal MHD, this coupling results in a singularity of the global mode occuring at the resonance. Treating this singularity in accordance with the causality condition yields a ﬁnite damping [38, 39]. The counterintuitive result of dissipationless damping is analo- 14 Introduction

gous to the familiar effect of Landau damping [40]. The former results from the inhomogeneity of a plasma in ordinary space in much the same way that the latter results from velocity space inhomogeneity. Formally, the waves which undergo this form of continuum interaction are better described as quasi-modes rather than eigenmodes [9, p. 522]. However, the latter nomenclature is used throughout this thesis, which is expedient recognising the coherent and typically weakly damped nature of the oscillations described. When kinetic effects are considered, the singularity is resolved and a physical basis for the damping emerges. Large spatial gradients in the plasma perturbation in the region around the resonance result in dissipation of the mode due to the combined effects of conversion to kinetic Alfvén waves (KAWs) and charge separation. Alternatively, dissipation due to the continuum resonance can be considered to be due to the resistivity of the plasma from the viewpoint of resistive MHD [41]. In either case, continuum damping represents the damping which occurs in the limit as dissipative effects vanish. The continuum modes are replaced by discrete modes in calculations which include resistive effects [42] or gyro-kinetic effects [43]. Singu- larities associated with continuum resonances are resolved and continuum damping represents the limit of damping as these additional effects vanish [42]. Continuum damping of Alfvén eigenmodes in fusion plasmas has been studied quantitatively using both analytic and numerical techniques. Analytic techniques have typically considered high n TAEs in circular cross-section tokamaks with low β under the large aspect-ratio approximation. In high n cases TAEs can be analysed using a ballooning expansion and continuum resonance interactions represented by small perturbations [38, 44]. Continuum damping of TAEs in cases with high β has been examined using a similar approach [45]. Another method for calculating the continuum damping of high n TAEs in large aspect-ratio tokamaks uses asymptotic matching [39]. Low n TAEs have been analysed in the same limit by considering perturbations to a quadratic form [20]. Commonly, continuum damping is evaluated numerically by determining the damping of an eigenmode in resistive MHD in the limit where resistivity approaches zero [41]. Another numerical technique which has recently been developed is to compute the solution of the ideal MHD wave equations over a complex contour [46]. Such numerical techniques can potentially be applied to compute modes in arbitrary two or three dimensional toroidal geometries, subject to computational limits. The aforementioned analytic treatments of continuum damping indicate its dependence on various parameters in the high n limit. Continuum damping is expected to be most signiﬁcant for Alfvén eigenmodes with low toroidal mode number n [38, 30]. The radial width of eigenmodes is approximately proportional to n−1 and continuum damping is dependent on the amplitude of the mode in the vicinity of a resonance. Continuum damping typically increases with magnetic shear, with continuum resonances occuring closer to the maxima of the modes and their radial extent increasing [39, 44]. Theoretical predictions for continuum damping rates have been compared with decay rate measurements for global modes under conditions where the former is §1.2 The physics of plasmas 15

expected to be the dominant contribution to the latter. Damping measurements for Alfvén eigenmodes excited by an external antenna were performed by Fasoli and collaborators for the JET tokamak[31]. They found that damping rates were highly dependent on magnetic shear, as is theoretically predicted for continuum damping. In the CHS torsatron, a helical device, Matsunaga and colleagues reported damping rate measurements for Alfvén eigenmodes excited using electrodes for a low temperature plasma [32]. They found that the order of magnitude of the damping rate observed was consistent with an analytic model of continuum damping, while radiative damping and neutral collisions were negligible. A study comparing various MHD, gyroﬂuid, and gyrokinetic codes with observations of Alfvén eigenmodes excited by antennas on the JET tokamak found approximate agreement between the damping calculated by these codes and experimental measurements [47]. The computational results indicated continuum damping was an important component at some of the time points examined, though in no case did all the codes agree that it was the largest source of damping.

1.2.7 Radiative damping

Kinetic effects due to the dynamics of thermal electrons and ions can introduce an additional, distinct source of damping of Alfvén eigenmodes. Inclusion of finite ion gyroradius effects contributes to the polarisation drift due to the inhomogenous electric fields associated with the modes [48]. Moreover, the inclusion of collisions and electron Landau damping leads to Alfvén waves having a finite electric field component parallel to the equilibrium magnetic field which is absent from ideal MHD [43]. These effects result in the coupling of shear and kinetic Alfvén waves in a process that can be considered as frequency tunneling [49, 21]. The latter typically oscillate radially on very short length scales and are consequently rapidly dissipated by electron Landau and collisional damping. Kinetic Alfvén waves couple to Alfvén eigenmodes near a continuum gap or extremum and propagate outward for harmonics where frequency is above that of the corresponding continuum branch [49]. Hence energy is redistributed throughout the plasma and the resulting damping is described as radiative. Radiative damping increases with n for Alfvén eigenmodes, and thus produces an upper limit for n of modes that can be destabilised by interaction with fast particles [30]. Theoretical predictions for radiative damping have been found to agree with the order of magnitude of the experimentally observed damping in cases where the former is expected to be dominant [47]. Codes including kinetic effects have been found to more accurately reproduce damping measurements on JET than those limited to continuum damping [50]. Significant correlations have been observed between a non-ideal parameter quantifying kinetic effects and damping measurements for some Alfvén eigenmodes in JET [51]. In future burning plasma experiments and reactors for power generation the ratio of thermal ion gyroradius to plasma radius is expected to be smaller than the devices for which Alfvén instabilities have so far been studied. Theory predicts that this will decrease the stabilisation of Alfvén eigen- 16 Introduction

modes by radiative damping [21].

1.3 Motivation and aims

In this thesis we develop techniques for calculating the continuum and radiative damping in fully two- and three-dimensional magnetic geometries. These methods will be demonstrated though appplication to tokamak and stellarator equilibria. Presently, theoretical predictions of Alfvén eigenmode growth rates contain significant uncertainty, with damping component estimates responsible for a major component of this uncertainty [51]. In conjunction with the development of techniques for calculating fast particle drive in three-dimensional magnetic configurations, improved damping calculations would allow Alvén eigenmode stability to be determined more accurately for such cases. In current fusion experiments this could assist in predicting or identifying observed Alfvén wave activity. Moreover, the nonlinear saturation level of the amplitude of the modes is dependent on the ratio of the linear growth rate and damping rate [52]. The development of such models should assist in designing and operating future fusion reactors to avoid deleterious Alfvén instabilities. Conversely, this research could help limit damping of Alfvén eigenmodes to allow their use in heating [53] or removing low energy alpha particles from the plasma core [54]. Continuum damping is inherently related to the spatial inhomogeneity of magnetised plasmas. Consequently, it is expected that the effect of two or three dimensional geometry can have a significant impact on this phenomenon. Tokamak measurements indicate that the damping of Alfvén eigenmodes can have significant corre- lation with parameters describing the shape of the plasma [51]. To our knowledge, continuum damping has not previously been calculated in fully three-dimensional geometry, accounting for all harmonic couplings. Stellarators such as H-1NF, TJ-II, and Wendelstein 7-X (W7-X) have complicated three-dimensional magnetic geometries which induce many different couplings between different harmonics which will influence the structure and hence continuum damping of Alfvén eigenmodes. Of particular interest in the case of H-1NF is the identification of low frequency wave activity [55, 56, 57]. It is hoped that damping measurements could assist in identifying candidate eigenmodes to explain observations. Three-dimensional geometry can also feature in tokamaks due to field ripple and application of resonant magnetic perturbations and thus potentially influence continuum damping.

1.4 Outline

This thesis is composed of ten chapters. Chapter 2 includes a more detailed discussion of ideal MHD theory, including the calculation of plasma equilibria and magnetic coordinates. This forms the basis for the examination of ideal MHD wave phenomena in Chapter 3, in which the derivation of the shear Alfvén wave equation is detailed. In toroidal magnetic geometry these waves are shown to give rise §1.4 Outline 17

to both continuous and discrete spectra. Extensions to ideal MHD used to analyse the continuum and radiative damping of these modes are also described. In Chap- ter 4 a perturbative technique for calculating the continuum damping is presented. A method for calculating continuum damping using a complex integration contour is detailed in Chapter 5, with a discussion of its implementation in finite element codes. Another method for calculating continuum damping through use of appropriate singular basis functions in a finite element technique is identified in Chapter 6. The calculation of radiative, Landau, and collisional damping based on kinetic extensions to the ideal MHD wave equation is detailed in Chapter 7. For the purpose of comparison, the aforementioned continuum damping calculations are applied to a large aspect-ratio circular cross-section tokamak in Chapter 8. In Chapter 9 we calculate the continuum damping of Alfvén eigenmodes in tokamak and stellarator cases using the finite element MHD code CKA, which we have adapted for use with the complex contour technique. This code allows the computation of this damping in detailed two- and three-dimensional magnetic geometries computed using the equilibrium solver VMEC. Finally, Chapter 10 summarises the outcomes of the research undertaken. 18 Introduction Chapter 2

Ideal MHD Equilibrium

In this chapter we describe the calculation of the equilibrium of a magnetised plasma and coordinate systems which describe it. The equations for plasma equilibrium in ideal MHD are laid out and conditions for which they are valid are identified. Several different toroidal coordinate systems are detailed. Coordinates based on the magnetic geometry of the equilibrium are described, which simplify the description of magnetic field lines and hence their perturbations. Linearised Alfvén waves are represented by small perturbations to the equilibrium, as described in Chapter 3. The plasma equilibrium is required to solve the resulting ideal MHD eigenvalue problem using a finite element code. In this work, the code VMEC is used to numerically compute equilibrium configurations in tokamaks and stellarators. The output from this code is mapped into magnetic coordinates and the equilibrium is recalculated to omit singularities. This equilibrium must be represented by analytic functions to enable extrapolation to complex coordinates as required to solve the problem over a complex contour.

2.1 Equilibrium

Here we repeat the equations for static equilibrium (v 0) in ideal MHD

p j B = (2.1)

∇ = ×

B µ0j (2.2)

∇ × = B 0. (2.3)

These equations follow from the resistive∇ ⋅ = MHD equations (1.13), (1.16), and (1.18) where partial derivatives with respect to time are set to zero. Static equilibrium and perfect conductivity (η 0) cause the solution of the remaining resistive MHD equations to become trivial, yielding E 0 but no information on ρ. Flow can usually be neglected for current and= foreseen fusion experiments [58, p. 261], where the static pressure of the plasma is much greater= than its dynamic pressure. Otherwise,

19 20 Ideal MHD Equilibrium

v v must be added to the left hand side of equation (2.1). In applying the MHD equations to the plasma equilibrium, it is assumed that pressure is isotropic and heat ﬂow⋅ ∇ is negligible.

Equations (2.1) to (2.3) represent a nonlinear system of differential equations. The boundary condition may be specified by the shape of a flux surface at the edge of the plasma, letting n B 0 on that surface where n is a unit vector normal to it. Although the number of equations and unknowns in this system are equal, such a boundary condition⋅ does= not determine a unique solution [13]. The additional information required is equivalent to specifying two additional functions of the flux surface labels, for example p s and ι s . The problem of determining equilibrium can be reformulated as the minimisation of plasma potential energy [13], expressed as ( ) ( ) B2 p W dτ, (2.4) Ω 2µ0 γ 1 = S + where the domain of integration is the plasma− volume, Ω. As shown in Chapter 1, where p 0, magnetic field lines lie on nested toroidal flux surfaces of constant p. Equation 2.1 shows that current is also perpendicular to p, so j also lies on these surfaces.∇ Quantities≠ which are functions of flux surface alone are referred to as flux functions. As enclosed volume decreases, the flux surfaces∇ converge to a closed curve termed the magnetic axis [13].

As stated in Subsection 1.1.3, flux surfaces exist in axially or helically symmetric geometries but not necessarily in general three-dimensional geometries [14]. Where flux surfaces exist, magnetic field lines either connect with themselves after a finite number of toroidal traversals (where rotational transform is rational) or pass arbitrarily close to every point on the surface as the number of toroidal traversals approaches infinity (where rotational transform is irrational) [8]. Alternatively, however, magnetic field lines may pass arbitrarily close to every point in a finite volume of space as the number of toroidal traversals approaches infinity [8]. In this case equations (2.1) and (2.2) imply that constant pressure exists throughout the volume. At rational surfaces, resonant perturbations to a field with nested toroidal flux surfaces produce closed magnetic flux tubes referred to as magnetic islands [8]. When the widths of the islands would become comparable to their separation, field line behaviour changes to ergodically fill a finite volume of space as described above [59, p. 17]. Plasma confinement in toroidal devices is optimised by maximising the volume of the plasma foliated by nested flux surfaces. Consequently, in many cases, it can be assumed that such surfaces are found throughout the plasma.

Equilibria in toroidal magnetic confinement devices typically have certain symmetries. In tokamaks toroidally dependent magnetic components arise due to the breaking of axisymmetry resulting from the finite number of toroidal field coils or use of resonant magnetic perturbation coils. However, these terms are usually negligible in comparison with overall field strength allowing the magnetic equilibrium in tokamaks to be considered axisymmetric. Stellarators are generally designed with §2.2 Coordinate systems 21

coil sets for which coordinates can be deﬁned such that,

f r, θ, φ f r, θ, φ (2.5) for all scalar ﬁelds and ( ) = ( − − )

vr r, θ, φ rˆ vθ r, θ, φ θˆ vφ r, θ, φ φˆ vr r, θ, φ rˆ vθ r, θ, φ θˆ v r, θ, φ φˆ (2.6) ( ) + ( ) + ( ) = − ( − −φ ) + ( − − ) for all vector ﬁelds [60]. This is referred to as stellarator+ symmetry.( − − ) Symmetry breaking terms due to the response of magnetic coils to the applied ﬁeld or errors in the construction of a device can usually be neglected.

2.2 Coordinate systems

2.2.1 Toroidal coordinates A system of toroidal coordinates r, θ, φ can be speciﬁed in terms of cylindrical coordinates as follows, ( )

R R0 r cos θ (2.7) Z r sin θ (2.8) = + ( ) φ φ. (2.9) = ( ) These coordinates are illustrated in Figure= 2.1. Poloidal and toroidal coordinates are represented by multi-valued functions θ and φ respectively, such that r, θ, φ and r, θ 2πm, φ 2πn for integer m and n refer to the same location. The remaining ( ) coordinate r is the distance from the origin. R0 and a are the major and minor radii of( the+ plasma+ respectively.) For plasmas with circular cross section r a at the edge of the plasma, otherwise a may be defined as the mean value at the plasma edge. The −1 R0 = aspect ratio of the plasma is defined as e a . In two-dimensional axisymmetric geometry φ is an ignorable coordinate. Unfortunately, in general two- and three- dimensional cases, flux surfaces have a non-trivial≡ dependence on two and three coordinates respectively.

2.2.2 Flux coordinates Assuming that nested toroidal flux surfaces exist throughout a plasma, these may be used to define a set of coordinates s, ϑ, ϕ . In these coordinates, s is a flux surface label which has its origin at the magnetic axis and monotonically increases with enclosed volume. Thus s satisfies B( s 0) throughout the plasma. Magnetic fluxes Ψt and Ψp increase monotonically with enclosed volume in tokamak and stellarator configurations, and hence can be used⋅ ∇ to= define s in these cases. The multivalued variables ϑ and ϕ are poloidal and toroidal variables respectively. Constant values of these variables on flux surfaces define contours which respectively close poloidally 22 Ideal MHD Equilibrium

axis of symmetry

푎 푟 푅0 휃

Figure 2.1: Simple toroidal coordinates, r, θ, φ , major radius, R0, and minor radius, a, for a tokamak with circular cross section. ( ) and toroidally. Such ﬂux coordinates may not be orthogonal, resulting in a non-trivial metric tensor. Deﬁning coordinates in this way can allow much of the complexity of the magnetic geometry to be ignored.

2.2.3 Straight ﬁeld line coordinates

It is possible to deﬁne ﬂux coordinates such that

B ϑ ιϕ 0. (2.10)

These coordinates are referred to as⋅ ∇ straight( − ) ﬁeld= line, or magnetic, coordinates. In these coordinates, the equation for magnetic ﬁeld lines is

dϑ B ϑ dψp ι (2.11) dϕ B ϕ dψt ⋅ ∇ = = = Hence, the magnetic field lines on a flux⋅ ∇ surface trace straight lines in ϑ, ϕ space, as illustrated in Figure 2.2. The simplicity of this field line representation is advantageous in ideal MHD stability calculations, where plasma follows field line( deforma-) tions [61, p. 253]. The existence of straight field line coordinates follows from the Clebsch representation which exists for solenoidal (divergence free) vector fields [62]. In this representation the magnetic field is expressed as

B ψt α. (2.12)

Hence B α 0, so that α is constant= along∇ × ∇magnetic field lines. Equation (2.10) is satisfied where α ϑ ιϕ. We retain⋅ ∇ significant= freedom in our choice of α and hence ϑ and ϕ. In the Clebsch = − ′ representation of a magnetic field it is possible to replace α with α α f ψt ,

= + ( ) §2.3 Computing plasma equilibria 23

(a)(a) (b)(b)

Δ휑Δ휑 Δ휗Δ휗 휗 휗 휗 휗 Δ휗Δ휗 Δ휗Δ휗 휄 =휄 =limlim 휄 =휄 = Δ휑Δ→휑∞→∞Δ휑Δ휑 Δ휑Δ휑 휑휑 휑휑

(a) (b)

Figure 2.2: Magnetic field lines on a flux surface (a) for arbitrarily chosen flux coordinates s, ϑ, ϕ and (b) in straight field line coordinates s, ϑ, ϕ .

( ) ( ) where f ψt is an arbitrary flux function. Choice of an additional constraint defines different types of straight field line coordinate systems. The magnetic field can be ( ) written B Bj ΦM where Bj is the magnetic field due to current distribution j and ΦM is a magnetic scalar potential due to external coils. In Boozer coordinates = + ∇ the periodic part of the magnetic scalar magnetic potential ΦM vanishes [63]. This results in straight field lines for the quantity ψt B as well as B.

∇ × 2.3 Computing plasma equilibria

In order to compute Alfvén eigenmodes, the ﬁnite element MHD code CKA requires that the equilibrium be deﬁned by

k • covariant and contravariant magnetic ﬁeld components Bk s, ϑ, ϕ and B s, ϑ, ϕ ,

• metric tensor components gi,j s, ϑ, ϕ , ( ) ( )

• Jacobian g s, ϑ, ϕ , and ( ) » • equilibrium pressure( ) p s [64].

The ﬁrst step towards determining( ) these functions is to calculate the magnetic equilibrium by computing a solution to equations (2.1) to (2.3) using VMEC. Another code, called Mapping, is then used to map the resulting equilibrium to Boozer coordinates. Subsequently, unreliable values for equilibrium quantities near the magnetic axis are replaced by extrapolation from functions ﬁtted over the rest of the plasma. Equilibrium quantities are then recalculated to ensure self-consistency.

2.3.1 VMEC VMEC (Variational Moments of Equilibrium Code) is a widely-used code which computes three-dimensional MHD equilibria [65, 66]. It solves the ideal MHD wave equations by iteratively minimising the energy functional in equation (2.4) over a toroidal 24 Ideal MHD Equilibrium

domain, using a steepest descent moment method [65]. This procedure utilises a set of force equations in terms of Fourier components. VMEC uses poloidal coordinates θVMEC that are defined such that they minimise the spectral width of the Fourier representation of cylindrical coordinates R θVMEC, φ , Z θVMEC, φ on each flux surface [67]. The code requires that nested flux surfaces exist throughout the solution domain and is not capable of computing( ( equilibria) in( which magnetic)) islands exist or a field line ergodically fills a finite volume. VMEC may be used to solve for either fixed or free boundary conditions. In the former case, the boundary condition stipulates that n B 0 on a specified flux surface, where n is normal to that surface. The surface is specified by its Fourier decomposition in cylindrical coordinates R, Z, φ . For a free⋅ boundary= calculation, the vacuum magnetic field is specified based on plasma currents and coil configuration. The shape of the boundary flux surface( is calculated) from the force balance at the plasma-vacuum boundary. In the modelling presented in this thesis, the former option is used. The additional information required to calculate equilibria is specified in the form of the flux functions p s and ι s .

( ) ( ) 2.3.2 Mapping to Boozer coordinates

The version of CKA employed in this work uses Boozer coordinates, which are described in Subsection 2.2.3. However, the code does not rely upon any special properties of this coordinate system and other ﬂux coordinate systems may be implemented [64]. A mapping code constructs Boozer coordinates from the magnetic geometry calculated by VMEC. Equilibrium quantities are mapped from VMEC to Boozer coordinates. In the mapping code output, quantities which depend on poloidal or toroidal variables are expressed by their Fourier expansion in poloidal and toroidal coordinates on each of a set of ﬂux surfaces.

2.3.3 Recalculation of MHD equilibria

Improve is a code which extrapolates equilibrium functions to determine values near the magnetic axis and then recalculates the equilibrium self-consistently. A problem arising from the use of VMEC is that solutions show very poor convergence near the magnetic axis. The force equations used by VMEC are singular along this curve [65]. The mapping code also introduces singular behaviour near the magnetic axis. Equilibria are recalculated based on the set of functions in Boozer coordinates ss ′ ′ g s, ϑ, ϕ , g s, ϑ, ϕ , gsϑ s, ϑ, ϕ and Bs s, ϑ, ϕ and flux functions ψt s , ψp s , »Bϑ s , Bϕ s and V s . From these functions it is possible to calculate Bk s, ϑ, ϕ and Bk (s, ϑ, ϕ ), gij (s, ϑ, ϕ )and p( s . Again,) those( quantities) with poloidal and( ) toroidal( ) variation( ) are( ) expressed( ) by decomposition into s dependent Fourier harmonics.( ) (Equilibrium) ( functions) corresponding( ) to these quantities can be found by fitting either polynomial or rational functions to the flux functions and Fourier coeffecients. This is done using a least squares fitting method. Correct asymptotic dependence of functions which are singular near the magnetic axis is assured by dividing by the §2.3 Computing plasma equilibria 25

factor sa corresponding to the asymptotic behaviour and multiplying the resulting fitted function by the same factor. Both polynomial and rational functions can be fitted and evaluated with minimal computational effort. Rational functions can take a wider variety of shapes and are typically smoother than polynomial functions. How- ever, the poles of rational functions limit the domains on which they are analytic and can give rise to spurious asymptotic behaviour. By contrast, polynomial functions are analytic over the domain of all complex numbers. Clearly, the preceeding steps result in an approximation to the equilibrium originally computed using VMEC. Further approximation may be made by fitting splines to values of the equilibrium quantities computed using the aforementioned functions on a grid in s, ϑ, ϕ . In this work, we are interested in the development of numerical techniques for calculating damping in equilibria with complicated geometries and qualitative( conclusions) about the effect of equilibrium parameters on damping. Thus, obtaining self-consistent equilibria is of greater importance than matching experimental equilibrium conditions with high accuracy. 26 Ideal MHD Equilibrium Chapter 3

Ideal MHD Waves

The theory of shear Alfvén waves in ideal MHD is described in this chapter. A coordinate independent reduced MHD wave equation is derived for shear Alfvén waves in devices with low β and large aspect-ratio. Additional terms may be added to the resulting equation to approximate coupling to ion acoustic waves or non-ideal effects. Further approximations allow us to derive coupled equations for these waves in stellarators and tokamaks in terms of spatial Fourier components. These equations form the basis of the damping calculations presented in subsequent chapters. Ideal MHD shear Alfvén wave equations in toroidal geometry result in a continuous spectrum, which is described. Gaps and extrema in this spectrum are identified, which occur due to various features of the magnetic geometry of tokamaks and stellarators and give rise to a discrete spectrum of Alfvén eigenmodes. Continuum and radiative damping of these modes are discussed. Finally, the use of the finite element method to compute Alfvén eigenmodes and its implementation in the code CKA are described. The finite element method can be extended to use of singular basis functions in order to calculate continuum damping as discussed in Chapter 6. CKA is used to implement a complex contour calculation of continuum damping, described in Chapter 5.

3.1 Shear Alfvén wave equation

In Subsection 1.2.4 it was shown that shear Alfvén waves can occur in a magnetised plasma. In Subsection 3.1.1 we derive a coordinate-independent linearised equation for shear Alfvén waves in tokamaks and stellarators with low β. This equation is derived from ideal MHD based upon the assumptions outlined in Subsection 1.2.3. Furthermore, coupling to slow and fast magnetosonic waves is neglected. The former follows from the assumption that the plasma is incompressible ( v1 0), which is approximately correct for negligible β. The latter is implied by the assumption that ∇ ⋅ = k∥ k⊥. This assumption accurately describes shear Alfvén waves, which propagate along magnetic ﬁeld lines. In Subsection 3.1.2 we derive a set of coupled differential equations≪ for the Fourier harmonics of shear Alfvén waves in the large aspect-ratio limit.

27 28 Ideal MHD Waves

3.1.1 General geometry

The derivation begins with the linearised versions of equation (1.12) to (1.18) for incompressible perturbations ( v1 0) to an equilibrium without ﬂow (v0 0) where resistivity is negligible (η 0), ∇ ⋅ = = = ∂ρ 1 ρ v 0 (3.1) ∂t 0 1 ∂v ρ 1 p j +B ∇ ⋅ (j B) = 0 (3.2) 0 ∂t 1 1 0 0 1 ∂p + ∇ − × 1 −v × p = 0 (3.3) ∂t 1 0 ∂B 1+ ⋅ ∇E = 0 (3.4) ∂t 1 1 j1 + ∇ ×B1 = 0 (3.5) µ0 E−1 v∇1 × B0 = 0 (3.6) B 0. (3.7) + × 1 = ∇ ⋅ = Taking the cross product of equation (3.2) with B0 and inserting the expression for j1 given by equation (3.5) results in

∂E1 1 ρ0 p1 B0 B1 B0 B0 j0 B1 B0 0. (3.8) ∂t µ0 − + ∇ × − ((∇ × ) × ) × − ( × ) × = Partial differentiation of this equation with respect to t and substitution of the ex- ∂B1 ∂p1 pressions for ∂t from equation (3.4) and ∂t from equation (3.3) then yields

∂2E 1 1 p ρ0 2 v1 0 B0 E1 B0 B0 ∂t µ0

− − (∇ ( ⋅ ∇ )) × + ((∇ × (∇ × )) × )j0× E1 B0 0. (3.9)

The perpendicular component of a vector can be expressed+ ( × as(f∇⊥ × ))f ×b =b, where b is the unit vector parallel to B0. Taking the cross product of equation (3.6) with B0 E1×B0 = − ( × ) × and rearranging, it is found that v1⊥ 2 . As b p0 0 for static equilibrium B0 E1×B0 in ideal MHD, v1 p0 2 p0. Using these relations and the vector identity B0 = ⋅ ∇ = A B C A C B A B C, equation (3.9) becomes ⋅ ∇ = ⋅ ∇ × ( ∂×2E) = ( ⋅ E) B− ( ⋅ ) 1 1 1 0 p B2 ρ0 2 2 0 B0 E1 ⊥ 0 B0 j0 E1 ∂t B µ0 ×0 − − ∇ ⋅ ∇ × − [∇ × (∇ × )] B+0 ⋅ E∇1 ×j0 0. (3.10)

− ⋅ (∇ × ) = Hereafter we express perturbations to the electric and magnetic ﬁelds in terms of §3.1 Shear Alfvén wave equation 29

the scalar potential Φ and vector potential A, so that

∂A E Φ (3.11) 1 ∂t B1 = −∇ A−. (3.12) Let A 0, which is the Coloumb gauge= ∇ condition. × Combining this condition with the assumption k∥ k⊥, it follows that A∥ A⊥ . Hence the approximation A A∥b ∇ ⋅ = ∂A is valid. From equation (3.6) it is then evident that b Φ ∥ . Consequently, ≪ ≫ S S ∂t ≈ E1 ⊥Φ. Inserting this expression for E1 into equation (3.10) results in, ⋅ ∇ = − = −∇∂2 1 B2 B p 0 ρ0 2 ⊥Φ 0b b 0 Φ bb Φ ⊥ ∂t B0 µ0

∇ − × ∇ B×0b ∇ j0⋅ ∇ bb− Φ[∇ ×B(0∇b × ( ⋅bb ∇ ))]Φ j0 0. (3.13)

The expression above does+ not⋅ explicitly∇ × ( contain⋅ ∇ ) − both⋅ [∇ a ×perturbed( ⋅ ∇ )] and= an unperturbed component for any quantity. Thus, the subscripts for the equilibrium ﬁelds E0, B0, ρ0, p0 and j0 are no longer necessary and are dropped.

The second term in equation (3.13) may be rearranged to explicitly show dependence on the magnetic ﬁeld line curvature and pressure gradient. The curvature, κ, is deﬁned as κ b b b b. (3.14)

First, use vector identities to obtain≡ ( ⋅ ∇) = (∇ × ) ×

1 1 Bb b p Φ B2 bb p Φ B B2 1 1 − × ∇ × ∇ ⋅ ∇ = − ∇ × B×2 ∇ ⋅ ∇ b b p Φ . (3.15) B B

1 + ∇ × × ∇ ⋅ ∇ The term B b in the above equation can be simplified as follows: ∇ × 1 1 1 b B b b (3.16) B B2 B ∇ × = − µ0 ∇ ×1 + (∇1 × ) 1 j⊥ κ b b bb b (3.17) B2 B B ⊥ B µ0 1 1 µ0 = − j⊥ − b× κ + b[∇ ×κ ] +j ⋅ ∇ × (3.18) B2 B B B2 ∥ µ0 2 = − j + j⊥ × b+ κ. × + (3.19) B2 ∥ B The substitution used for= the first term− on+ the× right-hand side of the first line above can be shown as follows

µ0j⊥ µ0 j b b (3.20) B b b (3.21) = − ( × ) × = − ((∇ × ) × ) × 30 Ideal MHD Waves

B b B b b b (3.22) B b Bκ b. (3.23) = − ((∇ × + ∇ × ) × ) × = ∇ × − × µ0 j∥ In deriving equation (3.19) the expression b b B is also used, which is found by rearranging Amperé’s law (equation (1.18)) and taking the component parallel to the ﬁeld. It follows that ⋅ (∇ × ) =

1 1 Bb b p Φ B2 bb p Φ B B2 − × ∇ × ∇ ⋅ ∇ = − ∇ ×2 µ0 × ∇ ⋅ ∇ 2 B j j⊥ b κ b p Φ . (3.24) B3 ∥ B2 + − + × ( × ∇ ⋅ ∇ ) Rearrangement of the third term in equation (3.13) indicates how this term relates to magnetic ﬁeld line bending. This term is perpendicular to magnetic ﬁeld lines and can therefore be written as

2 2 B0 B0 bb Φ ⊥ bb bb Φ µ0 µ0 − [∇ × (∇ × ( ⋅ ∇ ))] = [ ⋅ (∇ × (∇ × ( B2⋅ ∇ )))] 0 bb Φ . (3.25) µ0 − (∇ × (∇ × ( ⋅ ∇ ))) The ﬁrst term on the right-hand side of the above can be expanded as follows

B2 B2 0 bb bb Φ 0 bb b Φ b µ0 µ0 ⋅ (∇ × (∇ × ( ⋅ ∇ ))) = b⋅ (∇Φ × (( b⋅ ∇ ) ∇ × (3.26) B2 +∇0 b( ⋅ ∇ )b× ))Φ b µ0 = (b∇ ⋅ (((Φ ⋅b ∇ )b∇ × b Φ b b Φ b b (3.27) +∇ ( ⋅ ∇ ) × ) × ) + (( ⋅ ∇ ) ∇ × B2 +∇0 b( ⋅ ∇b ) ×Φ ) ⋅κ(∇ ×b ))Φ κ µ0 2 = (∇⊥ (b ⋅ ∇Φ) ⋅ b+ ( Φ⋅ ∇ ) ∇b ⋅ κ b Φ (3.28) −∇ ⋅ ∇ ( ⋅ ∇ ) + ( ⋅ ∇ )S∇ × S 2 B0 − b⋅ ∇ (b ⋅ ∇Φ )) κ ⊥ b Φ µ0 = b (( Φ⋅ ∇ ) ∇b ⋅2 −. ∇ ⋅ ∇ ( ⋅ ∇ ) (3.29)

The ﬁnal two terms in equation (3.13)+ ( can⋅ ∇ be)S rearranged∇ × S into terms with explicit dependence on j∥ and j⊥,

j∥B bb Φ Bb bb Φ j j∥B bb Φ bb bb Φ ∇ × ( ⋅ ∇ ) − ⋅ [∇ × ( ⋅ ∇ )] = (∇ × ( ⋅ ∇ ) − ⋅ [∇ × ( ⋅ ∇ )]) §3.1 Shear Alfvén wave equation 31

Bb bb Φ j j∥b (3.30) j B bb Φ −∥ ⋅ [∇ × ( ⋅ ∇ ⊥)] − Bb bb Φ j . (3.31) = [∇ × ( ⋅ ∇ )] ⊥ The second term on the right-hand side of equation− (3.31)⋅ [∇ × ( can⋅ be ∇ further)] modiﬁed µ0 j∥ using vector identities and b b B ,

Bb bb⋅ (∇ ×Φ) =j⊥ Bb b b Φ b Φ b j (3.32) − ⋅ [∇ × ( ⋅ ∇ )] = − ⋅ ((∇ × ) ⋅ ∇ ⊥ µ j b Φ j . (3.33) + (0∇∥( ⋅ ∇ )) ×⊥ ) Additional simpliﬁcation can be obtained= − by combining( ⋅ ∇ ) the expressions derived µ0 j∥ above for the terms in equation (3.13). From the equation b B b κ b it is µ2 j2 2 2 0 ∥ evident that b κ B2 . Also, note that b p Bj∇⊥. × Thus,= adding− × the last term in equation (3.29) to the part of equation (3.24) proportional to j∥ results in S∇ × S = + × ∇ = 2 2 B 2 µ0j∥ 2 b b Φ b 2 j∥ κ b b Φ µ0 B 2 2 µ(0 ⋅ ∇ )S∇ × S = µ0 + ( ⋅ ∇ ) B j b p Φ j j⊥ Φ (3.34) B3 ∥ B2 ∥ + ( × ∇ ⋅ ∇ ) + ⋅ ∇ µ2 κ2b b Φ 0 j j Φ. (3.35) B2 ∥ Similarly, adding the last term in equation= (3.31)( to⋅ the ∇ component) + ⋅ ∇ of the right hand side of equation (3.24) containing j⊥ yields

Bb bb Φ j⊥ µ0 j∥ Φ j⊥ 2 µ0 B− ⋅ [j∇⊥ × (b ⋅ ∇p )]Φ = −µ0 j⊥ ⋅ ∇Φ j⊥ (3.36) B3 − ( × ∇ ⋅ ∇ ) −µ0 (j ⋅ ∇Φ )j⊥ (3.37)

= − ( ⋅ ∇ ) µ0 Substitute the expressions derived above into equation (3.13). Multiplying by B2 , using the deﬁnition of vA in equation (1.31) and then taking the divergence of this equation yields

1 ∂2Φ 2µ 0 p 2 ⊥ 2 2 b κ b Φ b ⊥ b Φ vA ∂t B ∇ ⋅ ∇ − ∇ ⋅ ( × )( × ∇ ) ⋅ ∇ − ∇2 ⋅ [ ∇ ⋅ ∇ ( ⋅ ∇ )] µ0j∥ µ0 bb Φ j j⊥ j Φ B ⊥ B2 ∥ + ∇ ⋅ [∇ × ( ⋅ ∇ )] + ∇ ⋅ −κ κ2⋅ ∇bb Φ 0. (3.38)

The ﬁrst term on the left-hand side of this equation+ ∇ ⋅ is∇ the ⋅ + polarisation ⋅ ∇ term,= which is associated with the inertial mass of the plasma. The next term results from forces 32 Ideal MHD Waves

due to the combination of magnetic field line curvature and pressure gradient and is responsible for ballooning and interchange instabilities [68]. Forces due to magnetic field line bending are represented by the third term, which is predominantly responsible for shear Alfvén waves. The fourth term represents forces acting due to currents parallel to magnetic field lines and their curvature, which give rise to the kink instability [68]. The final two terms are a correction to the forces due to field line bending arising from their curvature at equilibrium. As is shown below, these terms can be neglected for a low β device.

Equation (3.38) can be further simplified by making ordering assumptions for the quantities that appear based on typical toroidal magnetic confinement devices. Here we follow the ordering used by Fesenyuk et al. [68]. Shear Alfvén eigenmodes are assumed to have frequencies εv ω A , (3.39) a where ε is a small parameter (of order ∼e). It is assumed that most equilibrium and perturbed quantities vary slowly along magnetic field lines, such that ∼ ε , (3.40) ∥ a where a is the transverse length scale of∇ the∼ device. Moreover, it is assumed that field line curvature is small, ε κ . (3.41) a Perpendicular to the magnetic field lines,∼ most equilibrium and perturbed quantities are assumed to vary as 1 ⊥ . (3.42) a The spatial variation in the magnetic field∇ ∼ and curvature are assumed to be weak,

εB B (3.43) a ε2B ∇B ∼ (3.44) ∥ a ε2 ∇ κ ∼ . (3.45) a2 ∇ ∼ Contrary to the work of Fesenyuk et al., it is assumed that β e2 (low β ordering) and hence diamagnetic and Pfirsch-Schlütter current densities are expected to be ∼ ε2B j⊥ j∥ . (3.46) µ0a ∼ ∼ From this ordering we find that the first and third terms in equation (3.38) are dom- e2 inant, being of order a4 . Thus, to first-order, the shear Alfvén wave equation represents the balance between the polarisation and magnetic field line bending terms, ∼ §3.1 Shear Alfvén wave equation 33

1 ∂2Φ 2 ⊥ 2 b ⊥ b Φ 0. (3.47) vA ∂t ∇ ⋅ ∇ − ∇ ⋅ [ ∇ ⋅ε ∇3 ( ⋅ ∇ )] = The second and fourth terms are of order a4 and including these we obtain the equation, ∼ 1 ∂2Φ 2µ 0 p 2 ⊥ 2 2 b κ b Φ b ⊥ b Φ vA ∂t B ∇ ⋅ ∇ − ∇ ⋅ ( × )( × ∇ ) ⋅ ∇ −µ ∇0j ⋅∥[ ∇ ⋅ ∇ ( ⋅ ∇ )] bb Φ 0. (3.48) B ⊥ + ∇ ⋅ [∇ × ( ⋅ ∇ )] = ε4 The final two terms in equation (3.38) are of order a4 and are ignored henceforth. This remains true even if the order of the parallel current in ε is reduced such that ∼ eB j∥ , (3.49) µ0a ∼ as may be the case for an externally driven current. Equation (3.48) is the reduced wave equation which is implemented in CKA for ideal MHD calculations. An alternative derivation of this equation, from gyrokinetic theory, is provided by Fehér [64]. The ideal MHD wave equation derived there includes an additional pressure term resulting from finite β. This term is not incorporated in the work described in this thesis. The reduced ideal MHD wave equation presented here approximates the limit of that derived by Fesenyuk et al. as β 0 [68]. In that work, the magnetic field line bending term in equation (3.48) is replaced by → 1 2 1 b B ⊥ b Φ . (3.50) B B −∇ ⋅ ∇ ⋅ ∇ ⋅ ∇ ε4 The difference between these terms can be shown to be of order a4 . Thus, under our ordering assumptions, either equation is valid. ∼ Assuming a solution of the form Φ x, t Φ x exp iωt in the Shear Alfvén wave equation results in an eigenvalue problem. Consider a Hilbert space composed of analytic functions, Φ Ω C. Here, Ω( R) 3=is a( toroid) ( in− physical) space, bounded by a closed surface S such that Φ x 0 for x S. Let the space be equipped with ∶ → ⊂ inner product Φ, Ψ Ω dτΦ x Ψ x . In this case, the operators acting on Φ in equation (3.48) can be shown to( be) self-adjoint= ∈ ( Φ, TΨ TΦ, Ψ ) [64]. Conse- quently, the eigenvalues⟨ ⟩ = ∫ of equation( ) ( (3.48)) λ ω2 are real [9, p. 249]. This implies that the solutions of the equation are either purely⟨ oscillatory,⟩ = ⟨ exponentially⟩ grow- ing or exponentially decaying. These properties= of the reduced MHD operator and equation are shared with the full ideal MHD operator and equation. Magnetic field-line curvature in a finite β magnetised plasma results in the coupling of shear Alfvén and ion acoustic waves. In a low β plasma this coupling can be incorporated using the slow sound approximation [69]. Considering first-order 34 Ideal MHD Waves

terms in β in the coupled shear Alfvén and acoustic equations of Fesenyuk et al. [68] results in the following correction to the left-hand side of equation (3.48)

2 cs 4 2 b κ b κ Φ . (3.51) vA − ∇ ⋅ ( × )( × ) ⋅ ∇ Integration by parts readily reveals that the operator acting on Φ in this term is self-adjoint. The slow sound approximation term is equal to zero where either β or curvature are zero.

3.1.2 Large aspect-ratio A set of coupled mode equations can be derived which describe shear Alfvén waves in a large aspect-ratio toroidal device, using the method outlined by Kolesnichenko et al. [25]. The coordinate independent shear Alfvén wave equation is rewritten in terms of Boozer coordinates ψ, ϑ, ϕ and then a Fourier decomposition is applied in the poloidal (ϑ) and toroidal (ϕ) coordinates. The resulting equations allow analytical and numerical representations( of shear) Alfvén eigenmodes in simplified geometries. We use as our starting point the wave equation derived by Fesenyuk et al., in the limit that β 0 and j∥ 0 [68]. The resulting shear Alfvén wave equation represents the balance between the polarisation and magnetic field line bending terms, → → 1 ∂2Φ 1 1 B2 2 ⊥ 2 b ⊥ b Φ 0. (3.52) vA ∂t B B ∇ ⋅ ∇ − ∇ ⋅ ∇ ⋅ ∇ ⋅ ∇ = This equation is preferable to the corresponding equation (3.47) derived in Subsec- tion 3.1.1 in that the latter gives rise to additional terms through magnetic field strength dependence which do not arise from the former due to cancellations for the chosen equilibrium. The scalar potential perturbation on each flux surface can be expressed as a Fourier series, Φ Φm,n ψ exp imϑ inϕ iωt (3.53) m,n Further approximations are= Q made( in) describing( + the− equilibrium) configuration of toroidal magnetic confinement devices, simplifying the expression. Consider a stellarator with Nf p toroidal field periods and possesing stellarator symmetry. It is assumed that variation in magnetic field strength throughout the plasma is relatively weak and is therefore expressed in the form B Bh¯ where,

(µ,ν) = h 1 eB ψ cos µϑ νNf p ϕ , (3.54) µ,ν = + Q ( ) + (µ,ν) with eB 1. Under these assumptuions, Kolesnichenko et al. show that in Boozer coordinates the metric tensor elements are approximately ≪ ψψ g 2ψδ0Bh¯ g (3.55)

≈ §3.1 Shear Alfvén wave equation 35

δ B¯ gϑϑ 0 (3.56) 2ψ gψϑ gψϕ≈ 0 (3.57) gϑϕ gϕϕ 0 (3.58) ≈ ≈ where ≈ ≈ (µ,ν) hg 1 eg ψ cos µθ νNf pφ , (3.59) µ,ν

(µ,ν) = + Q ( ) + with eg 1 and δ0 being a positive constant representing the elongation of the plasma [25]. The covariant representation for the unit vector tangential to the magnetic ﬁeld is≪ b bψ ψ bϑ ϑ bϕ ϕ. (3.60)

Assuming the magnetic ﬁeld is dominated= ∇ + by∇ its+ toroidal∇ component, the properties of Boozer coordinates can be used to show that

bψ 0 (3.61)

ϑ hι b = (3.62) R0 h bϕ ≈ (3.63) R0 ≈ where R0 is the mean major radius of the plasma. Under these assumptions, an approximate expression for the Jacobian determinant can also be found,

R g 0 . (3.64) Bh¯ 2 √ ≈ In the following we treat differential operators as acting on terms to their right, regardless of brackets. Using the above expression for magnetic ﬁeld strength, we rewrite the terms on the left-hand side of equation (3.52) in coordinate dependent form. Consequently, the polarisation term becomes,

2 1 ∂ Φ 2 µ0ρ 2 ⊥ 2 ω 2 1 bb Φm,n vA ∂t m,n B ∇ ⋅ ∇ = −exp Qim∇ϑ ⋅ inϕ (iω−t ⋅) ∇ ( (3.65) i 1 ∂ gµ0ρ ∂ b h 2 ( + − ))] gij Lˆ ω i 2 2 j m,n g ∂x √B¯ h ∂x R0

= −Φm,Qn exp√ imϑ inϕ iω t − (3.66) ∂ 1 ∂ ω2 h2 gij ibihk ( ( i +2 4 − ))]j m,n m,n ∂x v¯Ah ∂x = −Φm,Qn exp imϑ inϕ iωt − . (3.67)

( ( ˆ + ∂ − ∂ ))] In the above we have introduced the operator L ι ∂ϑ ∂ϕ , which can be related to = + 36 Ideal MHD Waves

h ˆ the parallel derivative using the covariant expression for b, b R L. The parallel mι+n 0 wave number is given by km,n k . Likewise, the ﬁeld line bending term ∥ R0 becomes, ⋅ ∇ ≈ = = 1 1 1 B2 B B2 b ⊥ b Φ b 2 1 bb B B m,n B −∇ ⋅ ∇ ⋅ ∇ ⋅ ∇ = − Q1 ⋅ ∇ ∇ ⋅ ( − ⋅) b Φ exp imϑ inϕ iωt (3.68) B m,n ∇ h2 ⋅ ∇ ( 1 ∂ ( + −∂ )) bih Lˆ h2 g gij Lˆ 2 i j m,n R0 h g ∂x ∂x R0 √ = − Q1 √ − Lˆ Φm,n exp imϑ inϕ iωt (3.69) R0 2 h( ˆ ∂ ( ij ∂k+m,n − ij )) ∂ i 2 L i ig j ig km,n j b hkm,n m,n R0 ∂x ∂x ∂x

= −ΦQm,n exp imϑ inϕ iωt+ . + (3.70)

( ( + − ))}] Inserting the expressions in equation (3.67) and equation (3.70) into equation (3.52) R0 and multiplying by h2 yields

∂ ω2R − ∂ ∂ ∂k ∂ 0 gij ibihk Lˆ igij m,n igijk bihk2 i 2 4 j m,n i j m,n j m,n m,n ∂x v¯Ah ∂x ∂x ∂x ∂x

Q − + Φm,n exp +imϑ inϕ i+ωt 0.¡ (3.71)

Next we insert expressions for gij and bi into( the above( expression,+ − )) =

2 ∂ 2ψδ0Bh¯ gR0ω ∂ ∂k ∂ iLˆ 2ψδ Bh¯ m,n k 2 4 0 g m,n m,n ∂ψ v¯Ah ∂ψ ∂ψ ∂ψ Q ∂ι ∂ ∂+k ∂ ∂ +ω2R δ B¯ ∂ ih2k ι i 2ψδ Bh¯ m,n k 0 0 m,n 0 g m,n 2 4 ∂ψ ∂ϑ ∂ψ ∂ψ ∂ϑ v¯Ah 2ψ ∂ϑ R0 − ik δ B¯ ∂ h2k2 +ι ∂ +ω2R ih2k h2−k2 Lˆ m,n 0 m,n 0 m,n Lˆ m,n 2 4 2ψ ∂ϑ R0 ∂ϕ v¯Ah R0 R0 + + + −Φm,n exp imϑ+ inϕ iωt ¡ 0. (3.72)

ˆ ∂ f ∂ (ˆ ∂ι ∂ f ( + − )) = The above equation uses the relation L ∂ψ ∂ψ L f ∂ψ ∂ϑ . = − As the amplitudes of the Fourier harmonics in the expressions for h and hg are taken to be small, these can be ignored in coeffecients of Φm,n. However, these terms 2 ∂ Φm,n remain important in coffecients of ∂ψ2 because its zeroth order terms cancel out where ω km,nv¯A. Thus, substituting the deﬁnitions of h and hg into this equation, results in the approximation = §3.1 Shear Alfvén wave equation 37

2 ∂ 2ψδ0BR¯ 0ω 2 1 eg 4eB cos µϑ νϕ m,n ∂ψ ⎡ v¯ µ,ν ⎪⎧ ⎢ A ⎪ ⎢ ⎛ ⎞ Q ⎨ ⎢ + Q − ( − ) ⎪ ⎢ ⎝ ∂ ⎠ ∂ ∂km,n ⎩ 2⎣ψδ0Bk¯ m,n Lˆ 1 eg cos µϑ νϕ 2R0δ0Bk¯ m,n ψ µ,ν ⎤ ∂ψ ∂ψ ∂ψ ⎥ ⎛ ⎞⎥ + 2 + Q ( ¯− 2 ) ⎥ − ω 2⎝ 2 δ0R0Bm ⎠⎥ 2 km,n km,n ⎦Φm,n exp imϑ inϕ iωt 0. (3.73) v¯A 2ψ + − − ¡ ( ( + − )) = 2ψ We define a new radial coordinate r B¯ and from this quantity a new wave function E Φm,n . In these variables, we¼ obtain the set of coupled partial differen- m,n r ≡ tial equations used by Kolesnichenko et al. [25], ≡ 2 2 2 ∂ 3 ω 2 ∂Em,n ω 2 2 2 ∂ ω r 2 km,n r 2 km,n 1 m r 2 Em,n ∂r vA ∂r v¯A ∂r v¯A − + 2 − − + µ,ν ∂ 3 ω µ,ν eg ∂Em+µ,n+νN r eˆ k k + + r 2 m,n m µ,n ν r µ,ν ⎧∂ ⎡ vA 2 ⎤ ∂ ⎪ ⎢ ⎥ + Q ⎨ ⎢ 2− ⎥eµ,ν E ⎪ ⎢∂ 3 ω µ,ν ⎥ g ∂ m−µ,n−νN ⎩ ⎣ r eˆ k k − −⎦ 0. (3.74) r 2 m,n m µ,n ν r ∂ ⎡ vA 2 ⎤ ∂ ⎫ ⎢ ⎥ ⎪ + ⎢ − ⎥ ⎬ = ⎢ ⎥ µ,ν ⎪ ⎣ µ,ν ⎦eg µ,ν ⎭⎪ In the above we introduce the quantity defined as eˆ 2 2eB . The preceeding equation is derived neglecting radial derivatives of km,n. This approximation is valid for gap modes in heliac and helias stellarators, where≡ radial− variation in ι is 2 km,n 1 generally weak. It is also assumed that 2 based on the ordering specified in δ0 a Subsection 3.1.1. From equation (3.74) we observe that µ≪, ν Fourier harmonics of B and gψψ introduce coupling between Em,n and Em±µ,n±νN. Thus, in an Nf p toroidal field period stellarator, coupled families of harmonics exist( ) such that

′ n n Nf pZ, for Z Z. (3.75)

Although equation (3.74) represents− = an infinite set∈ of coupled equations with infinitely many variables, in practice only a finite subset of harmonics are significant in any eigenmode. Thus, the set of equations can be truncated by including only those harmonics with significant coupling to the mode due to gaps or continuum resonances. For a large aspect-ratio circular cross-section tokamak, the above equation can be simplified to that used by Berk et al. [20]. Radial derivatives of km,n are excluded for reasons indicated below. Flux surfaces for such an equilibrium are circular with centres that move inward with increasing r (a phenomenon referred to as Shafranov shift) [70]. In this case the geometry of the device is described by r h 1 e cos ϑ (3.76) a ≈ + 38 Ideal MHD Waves

∂∆ hg 1 2 cos ϑ, (3.77) ∂r where ∆ is the Shafranov shift. Moreover,≈ + coupling between different mode numbers ω is expected to be significant only for km,n km±µ,n±ν so that, the approximation v¯A ω2 km,nkm±µ,n±ν 2 is valid. Inserting these expressions into equation (3.74) results in v¯A = ≈ − ≈ − 2 2 2 ∂ 3 ω 2 ∂Em,n ω 2 2 2 ∂ ω r 2 km,n r 2 km,n 1 m r 2 Em,n ∂r v¯A ∂r v¯A ∂r v¯A 2 − + − ∂ −3 ω + ∂Em+1,n ∂Em−1,n r e˜ 2 0 (3.78) ∂r v¯A ∂r ∂r + + = 1,0 1,0 This equation incorporates the coupling coeffecient defined e˜ eg 2eB . This can 5re be approximated by e˜ 2a . For a constant current profile and negligible β, the 2 ≡ − Shafranov shift is ∆ r e . It is noted that Berk et al. obtain this result under different 8a≈ assumptions to those presented above [20]. The parallel current term in the wave equation is not neglected≈ in their derivation, resulting in a contribution which cancels with that from the radial derivatives of km,n. Thus equation (3.78) is valid even for cases with significant variation in ι. Berk et al. define a different coordinate system in which the radial coordinate is a flux coordinate to e2 .

O It is sometimes convenient to express equation (3.78) in matrix form. Using Ein- stein summation notation

∂ ∂Ej D r A r E 0. (3.79) ∂r i,j ∂r i,j j ( ) + ( ) = where

2 2 3 ω 2 ω Di,j r 2 ki,n δi,j e˜ 2 δi−1,j δi+1,j (3.80) v¯A v¯A 2 2 ≡ ω − 2 + 2 2 ∂ +ω Ai,j r 2 ki,n 1 m r 2 δi,j (3.81) v¯A ∂r v¯A Ej ≡ Ej,n. − − + (3.82)

Equation (3.79) can be expressed≡ as a set of coupled ﬁrst-order differential equations ∂Ej by introducing the new ﬂux-like quantity Ci Di,j ∂r [20], ∂C = i A E (3.83) ∂r i,j j ∂E i = −D−1C . (3.84) ∂r i,j i In the stellarator case, equation (3.74) can= be recast in the form of equation (3.79) or coupled equations (3.83) and (3.84). §3.2 Variational formulation 39

3.2 Variational formulation

An alternative description of linearised shear Alfvén waves is based on variation of quadratic forms. Assume that Φ x, t Φ x exp iωt . Let V be the set of functions Φ R C consistent with the boundary condition Φ x 0 for x S where S is a closed toroidal surface. This boundary( ) = condition( ) ( represents− ) an inﬁnitely conductive wall∶ at→ the edge of the plasma. Additionally, let T V ( V) ′=be the operator∈ acting on Φ in (3.48). Thus, equation (3.48) represents the Euler-Lagrange equation obtained by extremising the functional ∶ →

S Φ Φ, T Φ . (3.85)

This equation can be expressed in a[ symmetric] = ⟨ [ form]⟩ [64],

2 ω ∗ ∗ S Φ dτ ⊥Φ ⊥Φ ⊥ b Φ ⊥ b Φ Ω 2 vA [ ] = ∇ ⋅µ ∇ b − ∇P ( ⋅ ∇ )µ⋅ ∇b ( P⋅ ∇ ) S b κ Φ∗ 0 Φ 0 Φ∗ b κ Φ B2 B2 × ∇ × ∇ + ( × ⋅ ∇ ) j∥ ⋅ ∇ ∗+ ⋅ ∇ ( × ∗ ⋅ ∇ ) ⊥Φ ⊥Φ ⊥Φ ⊥Φ (3.86) 2B + [∇ ⋅ (∇ × ∇ ) + (∇ × ∇ ) ⋅ ∇ ]¡ Surface terms are eliminated by applying the boundary condition at S. Similarly, equation (3.79) is the Euler-Lagrange equation found by extremising the functional

a ∗ ∂Ei ∂Ej ∗ S Ek dr Di,j Ei Ai,jEj . (3.87) 0 ∂r ∂r [ ] = − Likewise surface terms are eliminatedS by assuming boundary conditions such that Ei 0 at r 0 and r a. For solutions to the differential equation T Φ 0, provided that T Φ is deﬁned across the solution domain x Ω, S Φ 0 trivially. = = = [ ] = [ ] ∈ [ ] = The variational formulation leads to a weak formulation when small perturbations to the solution are considered. Consider a perturbation Ψ V to Φ V, such that this quantity becomes Φ εΨ where ε R is small. Given that Φ extremises S, S Φ εΨ S Φ ε2 . Noting that T is a linear operator and∈ considering∈ ε terms we obtain + ∈ [ + ] − [ ] = O Ψ, T Φ 0, Ψ V. (3.88)O ( )

While solutions of T Φ 0 satisfy⟨ [ equation]⟩ = ∀ (3.88),∈ those of equation (3.88) do not necessarily satisfy T Φ 0 pointwise (for weak solutions, T Φ may be undeﬁned at some points x Ω). The[ ] weak= formulation of the problem is used in its discretisation for the purpose of calculating[ ] = a numerical solution. [ ] ∈ 40 Ideal MHD Waves

3.3 MHD Spectra

2 Equation (3.48) is expressed in spectral form, TI ω TW Φ 0, by letting Φ x, t Φ x . Here TI and TW are self-adjoint operators. Explicitly, equation (3.48) becomes − = ( ) = ( ) ω2 ∂2Φ 2µ 0 p 2 ⊥ 2 2 b κ b Φ b ⊥ b Φ vA ∂t B ∇ ⋅ ∇ + ∇ ⋅ ( × )( × ∇ ) ⋅ ∇ +µ ∇0j ⋅∥[ ∇ ⋅ ∇ ( ⋅ ∇ )] bb Φ 0. (3.89) B ⊥ − ∇ ⋅ [∇ × ( ⋅ ∇ )] = This equation and the boundary condition Φ x 0 for x S together represent an eigenvalue problem. For any complex value of ω2 three possibilities( ) = exist [9,∈ p. 252];

2 2 • the inverse of TI ω TW does not exist and hence ω is part of the discrete spectrum, − 2 2 • the inverse of TI ω TW exists but is unbounded and hence ω is part of the continuous spectrum, − 2 2 • the inverse of TI ω TW exists and is bounded and hence ω is part of the resolvent set. − The ﬁrst two possibilities represent non-trivial solutions of the eigenvalue problem and are discussed in further detail below.

3.3.1 Continuum modes

The continuous spectrum corresponds to the set of ω2 for which the ideal MHD oper- 2 2 ator TI ω TW has poles on one or more flux surfaces. Such ω can be considered improper eigenvalues as they are associated with solutions to the eigenvalue problem which − are singular on these flux surfaces. Thus, these solutions do not belong to the Hilbert space introduced above [9, p. 251]. Poles occur where the coeffecient ∂2Φ of ∂ψ2 in equation (3.48) vanishes on a given flux surface. This corresponds to the condition that ω2gψψ gψψ 2 η B0b b η 0 (3.90) vA B0 + ⋅ ∇ ⋅ ∇ = ∂Φ has non-trivial smoothly varying doubly periodic solutions for η ∂ψ on that flux surface. The continuous spectrum is composed of the eigenvalues of the above equation which are calculated on each flux surface. Where only trivial solutions= of equation (3.90) exist for all ψ the solution to the ideal MHD wave equation has smooth flux surface dependence [71]. It is noted that this equation contains contributions from the polarisation and field line bending terms only. Thus the continuous spectrum is determined by the balance of inertial and magnetic field-line bending forces without direct contribution §3.3 MHD Spectra 41

from forces due to parallel current or pressure gradients. If the correction due to coupling between shear Alfvén and ion acoustic waves found using the slow sound approximation is included, the following term is added to the left-hand side of equation (3.90) 2 4cs 2 2 ψ b κ η. (3.91) vA This term is non-zero where there− is finiteS∇ ⋅ β×andS geodesic curvature. In such cases, inclusion of the slow sound approximation results in an upward shift of the continuous spectrum at low frequencies [69]. An inner product on flux surface ψ can be defined as

dS ζ, η ζ∗ ϑ, ϕ η ϑ, ϕ (3.92) ψ ψ ⟨ ⟩ = c dϑdϕ g(ζ∗ ϑ), ϕ( η ϑ), ϕ . (3.93) ψ S∇ S √ = ( ) ( ) This is proportional to the meanc of ζ∗η over the volume enclosed by two flux surfaces as that volume approaches 0. In Appendix A the operator acting on η in equation (3.90) is shown to be self-adjoint with respect to this inner product for real ψ. This remains true when the correction due to the slow sound approximation in equation (3.91) is added. Furthermore, if we express equation (3.90) in the form 2 ω TI η TW η 0, it can be shown that η, TI η 0 (i.e. TI is positive definite) and η, TW η 0 (i.e. TW is positive semi-definite). Consequently, eigenvalues of equation[ ] − (3.90)[ are] = real and positive semi-definite⟨ [ (ω]⟩2> R and ω2 0). For non-zero geodesic⟨ curvature[ ]⟩ ≥ and pressure, the slow sound approximation correction contribu- ∈ 2 ≥ tion to η, TW η is strictly positive and therefore increases ω . Considering the matrix expression for the large aspect-ratio case, equation (3.79), ⟨ [ ]⟩ regular singularities occur for D r 0. The matrix is self-adjoint and consequently the resulting eigenvalues are real. Similarly to the formula for general geometry ω2 can be shown to be positive semi-definite,Y ( )Y = provided that density decreases monoton- 1 ically with r, e˜ 2 , and m 1. As e˜ 1, based on the large aspect-ratio assumption, off-diagonal terms in D are relatively small. Their contribution to roots of the charac- < S S ≥ ≪ 2 2 teristic equation is therefore small, with the exception of regions where km,n km+1,n. Consequently the spectrum approximates that of a periodic cylinder outside such regions. ≈ The parameters in equation (3.90) vary across flux surfaces and thus the continuum resonance frequency ωR is a continuous function of the flux surface label ψ. In the case of a periodic cylinder with uniform magnetic field strength, e˜ 0 in equation (3.79) and hence D r 0 yields the continuum resonance dispersion relation 2 2 2 = ω km,nvA. Each Fourier harmonic corresponds to a different branch of the solution. Y ( )Y = 2 2 The m, n and m µ, n νN branches intersect where km,n km+µ,n+νN, in which case= the solutions to the continuum resonance eigenvalue problem are degenerate. ( ) ( + + ) = This degeneracy is avoided in toroidal geometry where equilibrium quantities vary in ϑ and ϕ. In this case, the Fourier harmonics can couple due to the µ, ν

( ) 42 Ideal MHD Waves

Fourier harmonic in the variation of B and gψψ across the ﬂux surface. The coupling results in destructive interference of the counter-propagating harmonics corresponding to the continuum resonance branches [21]. This interference prevents the two branches of the continuum from crossing, instead forming a gap. Continuum modes in this case are composed of a mixture of harmonics which varies with ﬂux surface label. Where Fourier harmonics are weakly coupled, the coupled m, n and m µ, n νN harmonics can be described approximately by restricting equation (3.74) to two partial differential equations in two variables [25]. In this case, the( continuum) ( resonance+ + ) condition yields the dispersion relation,

k2 k2 4 1 eˆµ,ν 2 2 2 m,n m+µ,n+νN ¿ ω vˆA 1 1 2 . (3.94) µ,ν 2 ⎛ Á k km+µ,n+νN ⎞ 2 1+ eˆ Á m,n − ( ) ⎜ Á km+µ,n+νN km,n ⎟ = ⎜ ± Á − ⎟ ⎜ ÀÁ ⎟ − ( ) + Consequently, the width of the continuum⎝ gap is expected to be ∆⎠ω2 2 eˆµ,ν ω2 centred at vˆ2 µι νN 2 ≈ S S ω2 A . (3.95) 4 1 eˆµ,ν 2 R2 ( + ) 0 = Similarly, in the case of a large aspect-ratio, − ( circular) cross-section tokamak the gap width will be ∆ω2 2 e˜ ω2. It is noted that the continuum gap depends on µ and ν but is independent of the values of m and n chosen for the coupled harmonics. Thus, for each coupling≈ S thereS will be an envelope from which continuum modes are excluded. Even where their coupling coeffecient is zero, the m, n and m µ, n ν , Fourier harmonics can couple indirectly via additional harmonics provided that µ siµi and ( ) µi,(νi + + ) ν si Nf pνi for some choice of si 1 and µi, νi such that eˆ 0. In the limit of µ,ν µi,νi = small eˆ , the strength of this coupling is proportional to i eˆ . The effect of such coupling= is visible in Figure 1.2, where= ± coupling( solely) between m ≠and m 1 gives rise to gaps between branches which represent predominantly∏m and m 2 harmonics. This indirect coupling also results in higher order corrections for the gap+ width for harmonics which couple directly. + Nevertheless, the various continuum gaps that arise can be associated with particular geometrically induced couplings. These are summarised in Table 3.1. In (ideal) tokamaks eˆµ,ν 0 for ν 0 due to axisymmetry so only gaps with ν 0 arise. By contrast, gaps due to any combination of µ and ν may occur in stellarators. = ≠ = An additional important feature of continuum resonance branches is the presence of extrema. Variation in ι and v¯A may result in the occurence such extrema. These extrema may occur even in the case of a one-dimensional periodic cylinder. For example, continuum minima can result from reverse shear, occurring near where a minimum occurs in ι. Moreover, inclusion of coupling to ion acoustic modes using the slow sound approximation in a ﬁnite β plasma in the presence of geodesic curvature results in a non-zero minimum near ψ with rational rotational transform. §3.3 MHD Spectra 43

Name µ ν Toroidicity 1 0 Ellipticity 2 0 Triangularity 3 0 Non-circularity 3 0 Mirror 0 1 Helicity ≥ 0 0

Table 3.1: Geometry induced gaps in the shear≠ Alfvén≠ continuum and associated couplings.

In practice, continuum modes are usually difficult to excite. As the continuum resonance frequency is dependent on the flux surface, any perturbation Φ extending over finite width in ψ which is composed of continuum modes will experience phase mixing [9, p. 515]. At different ψ, Φ which are initially in phase will oscillate at different frequencies and so rapidly become out of phase. Thus large spatial gradients develop, leading to damping when non-ideal effects including charge separation and mode conversion to kinetic Alfvén waves must be considered. The damping rate in ∂ωR this case is proportional to the spatial gradient in the resonance frequency γ ∂r . ∝ 3.3.2 Discrete modes

Where ω2 is part of the discrete spectrum there is an analytic solution to the eigenvalue problem in equation (3.89) which is part of the Hilbert space defined in Subsec- tion 3.1.1 [9, p. 251]. As noted, the ideal MHD operators acting on Φ are self-adjoint and hence the discrete spectrum is composed of real ω2. Such eigenmodes can exist within the coupling induced gaps between branches of the continuous spectrum. Each of the gaps identified in Table 3.1 gives rise to a corresponding class of shear Alfvén eigenmodes. Here the gap plays a role that is analogous to a potential well which leads to discrete solutions of the Schrödinger equation. The global modes display wave-like behaviour in ψ over a region near the avoided crossing and evanescent behaviour elsewhere [39]. Typically the dominant components of the eigenmodes are those Fourier harmonics corresponding to the continuum gap. Where continuum gaps involving common Fourier harmonics coincide, significant coupling of the harmonics responsible for each gap can occur. Thus, modes of mixed identity occur, having significant contribution from the Fourier harmonics responsible for each gap. Moreover, eigenmodes may also be associated with extrema of the continuous spectrum. Such modes are typically dominated by a single Fourier harmonic corresponding to the continuum resonance branch that has the extremum. Under certain conditions, Sturmian sequences of Alfvén eigenmodes labeled GAEs may cluster below minima of the continuum, which are referred to as GAEs [9, p. 462]. Similarly, anti-Sturmian sequences known as NGAEs may cluster above maxima of the continuum. In the important case where the continuum maximum results from a turning 44 Ideal MHD Waves

point of ι, these NGAEs are referred to as RSAEs [28, 72]. Typically, v¯A is monotonically increasing and ι is monotonic, resulting in GAEs in tokamaks and NGAEs in stellarators [29]. This difference results from the different ways in which ι is generated in these devices; in tokamaks j∥ is completely responsible for ι whereas in stellarators external ﬁeld coils contribute to ι. Terms involving radial derivatives of km,n in equation (3.74) must be considered in the NGAE case. When coupling to ion sound waves is considered, clusters of Alfvén eigenmodes may occur below the resulting continuum minima which are referred to as BAEs [30]. Such modes consist of a mixture of shear Alfvén and ion acoustic waves, as is seen in ﬁnite β MHD models.

3.4 Continuum damping of discrete modes

Continuum damping of global modes occurs where they encounter continuum resonances. In equation (3.95) the central frequency of gaps is shown to be dependent on ι and v¯A. Thus, where there is sufficient magnetic shear or variation in the plasma density, gaps at different ψ will not be aligned with one another. Consequently, the frequency of a gap mode may coincide with that of a continuum resonance at some location ψR. Additional continuum extrema may result in a similar resonance occurring for an extremum mode. Continuum resonances correspond to poles of the inverse of the operator acting on Φ in the shear Alfvén wave equation, equation (3.48), and are associated with singularities in the solution to this equation. Where a continuum resonance pole exists for real ω2, the presence of the singularity at real ψR implies that no solution to the wave equation exists for ψ U where U ψ ψ R, 0 ψ ψa [73]. However, deformation of the integration path around the poles yields solutions with complex ω2. This is possible because the operator∈ acting= on{ SΦ in∈ the≤ wave≤ equation} ceases to be self-adjoint when the domain is extended to include non-real x as equilibrium quantities take non-real values. Alternatively, due to the continuum resonance poles, the solutions of the wave equation are necessarily discontinuous on ψ U. Therefore, such solutions to the eigenvalue problem do not belong to the Hilbert space defined in Subsection 3.1.1. The solutions are thus more accurately described∈ as “quasi-modes”, “collective modes” or “virtual eigenmodes” [9, p. 522]. However, the common designation of these solutions as “modes” is adopted in this work. Consider equilibrium quantities B and gψ,ψ which are represented by functions which are analytic for ψ V where U V C. Thus, analytic continuation from U to V allows us to find values for B and gψ,ψ for complex ψ V, which will in almost all cases be complex.∈ For such complex⊂ ⊆ values of ψ the operator acting on η in equation (3.90) consequently ceases to be self-adjoint. Where∈ ω2 is complex continuum resonance poles will occur at complex values of ψ. For complex ω ωr iωi where ωi ωr the location of the continuum resonance pole can be estimated = using a first-order truncated Taylor expansion about ψr0, which is the location of the + ≪ §3.4 Continuum damping of discrete modes 45

pole for ω ωr −1 ∂ωr = ψR ψr iω . (3.96) 0 i ∂ψ ⎛ ψ=ψr0 ⎞ ≈ + W The correct choice of integration contour⎝ is derived⎠ from the causality condition, which requires that a perturbation precedes the response of the plasma. A conse- quence of this condition is that continuum resonances cause damping, rather than growth, of modes so that ωi 0 [73]. A causal path can be found by continuous deformation away from the real axis while circumventing the continuum resonance < pole as ωi goes from being positive to negative [73], as schematically illustrated in Figure 3.1. Equivalently, this condition is satisﬁed by deforming the integration contour such that continuum resonance poles lie between the contour and real axis. This is mathematically analogous to the deformation of the path of the velocity space integral when taking inverse Laplace transforms in Landau’s treatment of collisionless damping of plasma oscillations [40].

ℑ(휓)

ℜ(휓) × ×

Figure 3.1: As ω goes from real ω0 to complex ω1 continuum resonance poles move from real to complex values of ψ. A causal integration contour in the complex ψ plane ensures that these poles lie between itself and the real axis.

The singularities resulting from continuum resonance poles are shown to be logarithmic for devices with stellarator or up-down symmetry in Appendix A. In that ∂ 2 case it is assumed that ∂ψ η ϑ, ϕ , TI ω TW η ϑ, ϕ 0 where ψR is the ψ=ψR flux surface at which the continuum resonance occurs and η is the solution to equa- a ( ) − ( )fU ≠ tion (3.90) on that surface. This is consistent with the findings of Hameiri [74] and Salat and Tataronis [75] regarding continuum resonance singularities. Moreover, continuum resonance singularities are logarithmic for non-symmetric three-dimensional configurations, as shown in Appendix A and in agreement with the aforementioned authors. Physically, asymmetry-induced coupling between counter-propagating harmonics results in a unique (non-degenerate) non-propagating continuum mode for a particular ψR and ωR. Thus, in these cases, near the flux surface on which the 46 Ideal MHD Waves

singularity occurs,

Φ ψ, ϑ, ϕ η1 ϑ, ϕ ln ψ ψR η2 ϑ, ϕ , (3.97) for ψ C. ( ) ≈ ( ) ( − ) + ( ) In equation (3.97) ln x is the complex logarithm which is a multivalued function. ∈ However, physically Φ must be single valued and thus the complex logarithm must be restricted to a single( branch.) Hence, the function depends on the choice of branch cuts; two options are

ln± x ln x i arg ix 2πiΘ R x , (3.98) where a branch cut along( the) = positiveS S + imaginary(∓ ) ± axis( gives( )) ln ln+, taking it along the negative imaginary axis gives ln ln−. These complex functions are plotted in Figure 3.2. Combining equation (3.96) with the causality condition= indicates that = ∂ωR I ∂ωR I where ∂ψ 0, ψR 0, and where ∂ψ 0, ψR 0. Where a con- ψ=ψr0 ψ=ψr0 tinuum resonance exists continuous solutions are not possible, therefore the branch U > ( ) < U < ( ) > cut must intersect with the real axis. This results in a discontinuity in the solution described by equation (3.97) where ψ R ψR . Applying the causality condition, for ∂ωR ∂ωR ∂ψ 0 we take ln ln+, and for ∂ψ 0 we take ln ln−. ψ=ψr0 = (ψ=ψ)r0 InU general,> straight,= axisymmetric or helicallyU < symmetric conﬁgurations= without up-down or stellarator symmetry give rise to solutions of the form

iτ Φ ψ, ϑ, ϕ η1 ϑ, ϕ ψ ψR η2 ϑ, ϕ , (3.99)

( ) ≈ ( )( − ) + ( )

(a) (b)

Figure 3.2: Logarithmic functions with branch cuts (a) along the positive imaginary axis ( sign in equation (3.98)) and (b) along the negative imaginary axis ( sign in equation (3.98)). Brightness indicates modulus of the function and colour indicates + argument (note that the colour map differs between plots). − §3.5 Non-ideal effects 47

near continuum resonances, for some τ R. This is demonstrated in Appendix A. Where τ ln ψ ψR 1 the solution will have approximately logarithmic variation ∈ with ψ ψR . As the conﬁguration approaches stellarator or up-down symmetry, τ 0 andS this( − condition)S ≪ is satisﬁed over an increasing range of ψ. ( − ) → 3.5 Non-ideal effects

Inclusion of kinetic effects introduces important changes to the foregoing description of shear Alfvén waves in plasmas. Finite ion gyroradius and non-zero parallel electric ﬁeld effects are represented by the inclusion of additional terms in the shear Alfvén wave equation, which becomes [64]

1 ∂2Φ 1 3 ∂2Φ 2 i 2 2 ⊥ 2 ⊥ 2 ρi 1 δ ρs ⊥ 2 vA ∂t vA 4 ∂t ∇ ⋅ ∇ + ∇2 ⋅ ∇µ0 + ( − ) ∇ ⋅ ∇ b κ b p Φ b ⊥ b Φ B2 µ0j∥ − ∇ ⋅ ( × )( × ∇ ) ⋅ ∇ − ∇ ⋅ [ ∇ ⋅ ∇ bb( ⋅ ∇Φ)] 0. (3.100) B ⊥ + ∇ ⋅ [∇ × ( ⋅ ∇ )] = The ion gyroradius is given by

mikTi ρi (3.101) ¿ q2B2 Á i = ÀÁ while the sound gyroradius is given by

mikTe ρs , (3.102) ¿ q2B2 Á i = ÀÁ where mi is the ion mass, Ti is the ion temperature, Te is the electron temperature and qi is the ion charge. The correction for non-ideal effects is derived from gyrokinetic theory assuming that ρi a, where a is the average minor radius of the plasma 2 ve (only ﬁrst-order terms in ρ are retained), and that ω vek∥, where √ is the root ≪ 2 mean squared speed of the thermal electrons [48, 76]. The component proportional ≪ to the real positive function δ is not self-adjoint and introduces dissipation and hence damping. This quantity represents the wave dissipation rate due to electron collisional and Landau damping and is a function of ω and k∥ of the mode as well as vA and νei of the plasma. In a large aspect-ratio tokamak with circular cross section, equation (3.100) becomes

2 ∂ ∂ 2 2 2 1 ω 3 2 2 ∂ ∂ 2 2 2 r r m r km,n 2 ρi ρs 1 iδ r rEm,n m Em,n r km,nEm,n ∂r ∂r r v¯A 4 ∂r ∂r − + + ( − ) − + 48 Ideal MHD Waves

2 2 2 ∂ 3 ω 2 ∂Em,n ω 2 2 2 ∂ ω r 2 km,n r 2 km,n 1 m r 2 Em,n ∂r v¯A ∂r v¯A ∂r v¯A 2 − + ∂ − 3 ω ∂−Em+1,+n ∂Em−1,n r e˜ 2 0. (3.103) ∂r v¯A ∂r ∂r + + = In the derivation of equation (3.100) and equation 3.103 the dispersion relation for 2 2 2 shear Alfvén continuum modes in a periodic cylinder, ω km,nv¯A, is used to express non-zero parallel electric ﬁeld effects as a component of the mass-like operator = TI. This approximation assumes that non-ideal effects are important only near the location of the avoided crossing. A more detailed model of non-zero parallel electric ﬁeld effects, which does not apply this dispersion relation and does not assume that νei ω vek∥, results in [76]

≪ ≪ 2 ∂ ∂ 2 2 2 1 3 ω 2 2 1 iν¯m,nZ ξm,n 2 r r m r km,n 2 ρi km,n ρs ∂r ∂r r 4 v¯ 1 ξm,nZ ξm,n A + ( ) − ∂ +∂ + ∂ ω2 ∂E 2 2 2 + ( 3 ) 2 m,n r rEm,n m Em,n r km,nEm,n r 2 km,n ∂r ∂r ∂r v¯A ∂r 2 2 × − ω 2+ 2 +2 ∂ ω − r 2 km,n 1 m r 2 Em,n v¯A ∂r v¯A 2 + − −∂ 3+ ω ∂Em+1,n ∂Em−1,n r e˜ 2 0. (3.104) ∂r v¯A ∂r ∂r + + = Here the non-zero parallel electric field effects are represented as a component of the force-like operator T . We define ξ ω+iνei and ν¯ νei . Here ν is the W m,n Skm,nSve m,n Skm,nSve ei electron-ion collision frequency of the plasma, representing the rate of momentum loss by the former to the latter. The plasma= dispersion function= is defined as

2 1 ∞ e−y Z ξ dy (3.105) π −∞ y ξ ( ) = √ S for I ξ 0, and by analytic continuation for I ξ− 0. While the terms involving 2 Z ξm,n result in a non-linear eigenvalue problem for ω , solutions are found to ( ) > ( ) ≤ converge rapidly using an iterative approach where Z ξm,n is evaluated based on estimates( ) of ω2 [76]. ( ) In deriving the model for parallel electric ﬁeld, the diamagnetic drift of electrons and ions is ignored. This approximation follows from the assumption that the diamagnetic drift frequency, ω⋆, is much smaller than the frequency of the mode [76]. For simplicity trapped particles are neglected in this analysis. Trapped electrons are not subject to collisionless damping and can be modelled by treating passing and trapped electrons as separate species related via collision operators [49]. Thus, the parallel electric ﬁeld becomes dependent on the ratio of trapped and passing electrons, which can be related to e. Moreover, at high temperatures v∥ ve for resonant particles and consequently the proportion which are passing is small, reducing elec- ≪ §3.5 Non-ideal effects 49

tron Landau damping by a large factor [77]. 2 The parameter δ can be estimated by comparing coeffecients of ρs in equation (3.103) and equation (3.104). Where ξ 1, the plasma dispersion function can be approximated by the truncated expansion Z ξ i π 2ξ. Consequently, where ω k∥ve, S S ≪ νei k∥ve, and ω k∥vA, the factor representing√ parallel electric field effects in equation (3.104) becomes ( ) ≈ − ≪ ≪ ≈ 2 2 2 1 iν¯m,nZ ξm,n ω ω km,n 2 i π . (3.106) 1 ξm,nZ ξm,n v¯ vev¯A + ( ) A √ ≈ − In this case, the dissipative component+ ( of parallel) electric field is small relative to non- dissipative component and δ π v¯A . Non-zero δ is found, even in the limit where ν ve ei vanishes, representing Landau√ damping. For ξ 1, the plasma dispersion function ≈ can be approximated by the truncated asymptotic expansion Z ξ 1 1 1 . Thus, S S ≫ ξ ξ2 for ω k∥ve or νei k∥ve, the previous factor representing parallel electric field effects becomes approximately ( ) ≈ + ≫ ≫ 2 1 iν¯m,nZ ξm,n 2ω ω iνei km,n 2 . (3.107) 1 ξm,nZ ξm,n ve + ( ) ( + ) ≈ − Therefore, the dissipative component+ ( of parallel) electric field is dominant and δ 2 νeivA 2 . This limit represents the result obtained using resistive MHD [76]. The quantity ωve ≈ 2 3 2 2 4ω 4 ρi +ρs 2 δ can be related to the resistivity in the limit where 2 2 k⊥R0 1 through ι vA the expression [78] 2 ( ) ≪ 2 ω η µ0ωρs δ ω . (3.108) k∥vA ≈ ( ) The non-ideal terms in the wave equations above contain fourth order derivatives with respect to ψ. The presence of these terms implies that the coefficient of the highest order derivative with respect to ψ no longer disappears when we separate the solution near the resonance into ψ and ϑ, ϕ dependent components, as in Ap- pendix A. Consequently, Φ no longer has singularities due to continuum resonances. Inclusion of non-ideal effects qualitatively changes( ) the spectrum discussed above. As continuum resonance singularities are resolved, the continuous spectrum is replaced by a set of discrete eigenvalues [48, 79]. These results hold even if only dissipative or only non-dissipative terms are included. Where non-ideal effects are included, there ceases to be an absolute distinction between continuum modes and the Alfvén eigenmodes discussed in Subsection 3.3.2 [43]. Nevertheless, the mode components which the non-ideal effects introduce can be identified as kinetic Alfvén waves. These oscillate rapidly in ψ in typical cases where ρi a, ρs a, and δ 1. For small but finite δ, eigenvalues lie along certain curves in the complex plane [61, p. 210]. As δ decreases, eigenvalues move along these curves≪ such≪ that their density≪ increases. However, the curves themselves do not approach arbitrarily close to the real values associated with the continuous spectrum 50 Ideal MHD Waves

[42]. Physically, continuum damping represents the effect of these non-ideal phenomena near continuum resonances. In these regions the large spatial gradients in Φ across flux surfaces give rise to large non-ideal effects. Charge separation due to electron-ion collisions becomes significant and effectively dissipates energy of the mode in a region around the resonance. Mode conversion to kinetic Alfvén waves also occurs, which couple to the mode and propagate away from the resonance. These waves typically have wavelengths which are short relative to shear Alfvén waves and thus experience strong dissipation due to charge separation. Where collisional and non-zero gyroradius effects approach zero, damping due to a continuum resonance becomes independent of the magnitude of these effects. Around the continuum resonance, the region over which the mode loses energy becomes narrower but spatial gradients and hence the loss of energy per unit volume in that region increases. Thus, continuum damping is the damping found approaching ideal MHD. Alternatively, in resistive MHD continuum damping is the limit of damping as resistivity approaches zero. This limit can be found by taking the limit δ 0 in equation (3.100). Coupling of shear and kinetic Alfvén waves is also responsible for radiative damping.→ Near a continuum resonance maximum, Alfvén eigenmodes may have significant coupling between shear Alfvén waves and strongly damped kinetic Alfvén waves. Kinetic Alfvén wave harmonics may propagate where frequency exceeds that of the corresponding shear Alvén wave continuum resonance [49]. Thus, gap modes, RSAEs and NGAEs may experience significant radiative damping while GAEs will not [72]. The strength of the coupling, and hence radiative damping, is dependent on the difference between the frequency of the eigenmode and that of the relevant continuum branch as well as ρi and ρs. If ρi, ρs, and δ are known, radiative damping will form a component of the damping rate computed using equation (3.100). Where dissipative effects are not well modeled, radiative damping can still be estimated numerically for particular ρi and ρs profiles by letting δ be a constant throughout the plasma and varying this parameter [80]. For sufficiently large δ, almost all of the energy transferred to kinetic Alfvén waves will be dissipated and very little transferred back to the shear Alfvén waves. Thus the calculated damping rate becomes nearly independent of δ and approximates the radiative damping, assuming that the actual dissipation satisfies the aforementioned condition. If continuum resonances are present, the continuum damping will form a component of the calculated damping rate. This damping can be distinguished from the additional radiative damping as the component that remains in the limit where ρi 0, ρs 0 and δ 0.

→ → → 3.6 Finite element method

2 Eigenvalue problems of the form TI ω TW Φ 0 can be discretised and solved numerically using a ﬁnite element method. In such methods, a boundary value − = §3.6 Finite element method 51

problem is analysed by dividing the solution domain into a set of smaller, simpler parts on which the solution is assumed to have a particular functional form (typically polynomial). A mesh specifies the division of the solution into subdomains which are referred to as elements. An approximate solution is then obtained by minimising some error functional over the domain. Finite element methods are discussed in detail by Strang and Fix in their textbook [81]. A common finite element approach is the Galerkin method, which is used in this work. In the Galerkin method for solving partial differential equations a problem is expressed in weak form and the solution is approximated by a linear combination of a finite number of basis functions. The weak formulation derived in Section 3.2 can then be used to obtain a discretised eigenvalue problem. Let Vh be the function space spanned by a set of h basis functions in V, which constitutes an h dimensional subset of the function space V. Equation (3.88) is discretised by substituting Φ′ Vh and Ψ′ Vh for Φ and Ψ respectively. This results in a generalised matrix eigenvalue problem, which can be solved numerically. ∈ ∈ Using the Galerkin method, an approximate solution Φ′ to the original continuous problem is obtained such that T Φ′ is orthogonal to Vh. In the limit of decreasing mesh size Φ′ can be shown to converge such that the error e Φ′ Φ is orthogonal to Vh ( Ψ′, e 0 Ψ′ Vh) [82]. In this[ ] limit, Φ′ thus represents an orthogonal pro- jection of Φ onto V′. Numerical convergence is often checked= by ensuring− that the eigenvalue⟨ and⟩ = eigenfunction∀ ∈ obtained remain consistent as the mesh is refined. Approximations for both discrete and continuum modes are found using the Galerkin method with an integration path along the real axis. In these solutions, continuum resonance singularities are regularised due to limited resolution of functions within Vh. Effectively, solutions on either side of the resonance are matched in- accurately, not reflecting the expansion about the continuum resonance determined analytically. The operators and resulting matrices remain self-adjoint, resulting in real ω2 which does not reflect the damping inherent to these modes. As the solution near a continuum resonance is logarithmic, its variation on a mesh interval near the resonance does not change with the scale of the mesh. This is because a change in scale preserves the logarithmic form of the singularity, merely corresponding to the addition of a constant. Ignoring the constant term, the solution over a certain number of finite elements is unchanged when the scale of the elements changes (provided they remain within the region where the solution displays logarithmic behaviour). Hence the solution will not converge to one satisfying a particular matching condition. This results in inconsistent results for discrete modes with continuum interaction as mesh size is varied. However, increasing the resolution of the mesh around the continuum resonance surface results in an increased density of continuum modes modes being computed. Discrete modes with continuum interaction may appear as part of the continuum where this resolution is high. That is, the band of frequencies in which the discrete mode exists will be filled by computed modes with approximately even frequency spacing which are peaked at the continuum resonance. 52 Ideal MHD Waves

3.7 CKA

The finite element solution for MHD eigenmode equations is implemented in CKA (Code for Kinetic Alfvén waves) [64]. CKA can solve equation (3.48), corresponding to the reduced ideal MHD model, or modified versions of this equation ignoring certain terms on its left-hand side. This code can also include the slow sound approximation in expression (3.51). CKA was originally developed by Könies and Fe- hér to determine the mode structure for perturbative calculation of fast particle drive [64]. The code is suitable for investigation of continuum damping as it permits complex values of equilibrium quantities, which are necessary to implement the complex contour method described later in Chapter 5. Moreover, the code can incorporate non-ideal MHD effects discussed in Section 3.5 by solving equation (3.100) and thus include continuum, radiative, electron collisional, and electron Landau damping. CKA uses Boozer coordinates, which are described in Subsection 2.2.3. Volume elements are specified as rectangular cuboids in these coordinates, though these do not necessarily have uniform dimensions. In CKA solutions are represented by basis functions which are the product of B-splines, Bi, that represent variation in each dimension. A spline of order ns is a piecewise function composed of k 1 nsth order polynomials on subintervals delim- ited by k knots and zero elsewhere. This type of function is defined such that it − is continuous to order ns 1 in its derivatives. B-splines have support over a minimal number of knot intervals for a spline of given order (k ns 2). A set of these − functions can be calculated recursively for a given set of knots ti = + n+1 x ti n Bi x Bi x , (3.109) ti+n ti − ( ) = ( ) with the zeroth order B-spline defined by −

1 if x 0, 1 B0 x (3.110) i 0 otherwise ∈ ( ) ( ) = Such a set of B-splines spans a space of all functions composed of ns 1 order ns polynomial pieces joined at ti from which have derivatives that are continuous to + order ns 1. The minimal support of these splines leads to a sparse matrix eigenvalue problem. In this case highly efficient numerical solvers can be applied, such as the Arnoldi− or Kryluv-Schur algorithms [83]. Sets of B-splines in s, ϑ and ϕ are defined by prescribing sets of knots in these coordinates. The B-splines in these coordinates are labelled by indices i, j and k respectively. A unique index l is introduced, corresponding to each combination of i, j and k. We define basis function coeffecients cl ci,j and ci,j,k in the two-dimensional (axisymmetric) and three dimensional (non-axisymmetric) cases = respectively. Define basis functions in these cases as Λl s, ϑ Bi s Bj ϑ and Λl s, ϑ, ϕ Bi s Bj ϑ Bk ϕ respectively. For two-dimensional cases the eigen- ( ) = ( ) ( ) ( ) = ( ) ( ) ( ) §3.7 CKA 53

mode solution for the scalar potential is expressed as

Ns Nϑ ′ Φ s, ϑ clΛl s, ϑ . (3.111) l=1 ( ) = Q ( ) For three-dimensional cases the solution is expressed as

Ns Nϑ Nϕ ′ Φ s, ϑ, ϕ clΛl s, ϑ, ϕ . (3.112) l=1 ( ) = Q ( ) In these expressions, Ns, Nϑ, and Nϕ refer to the number of mesh points used to deﬁne volume elements in the radial, poloidal, and toroidal coordinates respectively. The knots occur at these mesh points. Eigenmodes are required to be periodic in the poloidal and toroidal directions, imposing boundary conditions for Bj ϑ and Bk ϕ . ( ) (Alfvén) eigenmodes typically have a small number of dominant harmonics which have similar m and n. It is often advantageous to use a phase factor extraction method in such cases. In this case the substitution

Φ′ s, ϑ, ϕ Φ˜ ′ s, ϑ, ϕ exp im˜ ϑ in˜ ϕ (3.113) is made. The m, n harmonic( ) in= Φ′( becomes) the( m+ m˜ ,)n n˜ harmonic in Φ˜ ′, which is then represented in terms of B-splines. Thus, the spline basis does not need to represent( the) variation in toroidal and poloidal( − directions− ) associated with the m˜ , n˜ harmonic. This allows for coarser splines in the poloidal and toroidal directions, providing a computational advantage. Quantitatively, the error in the eigenvalue( ) associated with a particular Fourier harmonic is expected to vary with 6 6 2 (m−m˜ ) (n−n˜) poloidal and toroidal parameters as ι 4 4 [84]. This estimate is based on Nϑ Nϕ the convergence of shear Alfvén continuum eigenvalues in a periodic cylinder. As + the period of Φ˜ matches the toroidal ﬁeld period, the value of n˜ determines which mode family the eigenvalue problem solutions obtained belong to. Applying the Galerkin method results in the matrix eigenvalue problem

2 TI,i,jcjω TW,i,jcj 0. (3.114) where the matrix elements are given by − =

∗ TI,i,j dτ exp imϑ inϕ Λi TI exp imϑ inϕ Λj (3.115) Ω f p = (− − ) ∗ ( + ) TW,i,j S dτ exp imϑ inϕ Λi TW exp imϑ inϕ Λj . (3.116) Ω f p = (− − ) ( + ) Although the above integralsS are deﬁned over the entire domain, on any mesh inter- 6 val the integrand will be non-zero for only ns 1 basis functions. CKA evaluates integrals from the values of these integrands at points on a grid using a quadrature ( + ) 54 Ideal MHD Waves

method [64]. The domain Ω f p represents a single toroidal ﬁeld period of the device. Calculation of the MHD equilibrium used by CKA is described in Section 2.3. The equilibrium is assumed to have stellarator symmetry. CKA can represent equilibrium quantities using cubic B-splines. Alternatively, an option has been introduced to represent poloidal and toroidal variation as Fourier series, where coefﬁcients are polynomial functions of ψ. For quantities which are odd functions

f s, ϑ, ϕ pm,n ψ sin mϑ nϕ (3.117) m,n ( ) = Q ( ) ( + ) while for even functions

f s, ϑ, ϕ pm,n ψ cos mϑ nϕ (3.118) m,n ( ) = Q ( ) ( + ) where pm,n ψ are polynomial functions.

( ) Chapter 4

Perturbative Calculation of Continuum Damping

The calculation of continuum damping using a perturbative technique is detailed in this chapter. Use of a perturbative technique offers potential theoretical insights into continuum damping. The resulting equations can be used to infer the qualitative dependence of the damping on plasma parameters. Generalisation to multiple continuum resonances is straightforward and the technique allows the contributions from different resonances to the damping to be distinguished. Moreover, we wish to develop techniques which are applicable to existing MHD codes. The method considered here was developed by Berk et al., who applied it to a large aspect-ratio circular cross section tokamak with low β [20]. Subsequently, a similar method was used in the ﬁnite β case by Zonca and Chen [45]. A perturbative technique of this kind has been implemented in the ideal MHD code NOVA [85]. Here we generalise the perturbative technique for use in three-dimensional geometry. A coordinate independent expression for continuum damping is obtained, based on the model for shear Alfvén waves described in Chapter 3. This would allow implementation of the perturbative damping calculation in CKA with very little additional computational effort. The technique applied by Berk et al. in the large aspect-ratio circular cross-section tokamak is also described, the results of which are compared with those of other techniques in Chapter 8.

4.1 Perturbative formalism

Continuum resonances can be associated with small perturbations to the quadratic forms introduced in Section 3.2 following the approach of Berk et al. [20]. In general, continuum resonances occur at flux surfaces where there are regular singularities in Φ which result in discontinuities in Φ and its derivatives. The reduced ideal MHD expression for shear Alfvén waves in equation (3.48) leads to the quadratic form S Φ presented in equation (3.86). Assume initially that ω2 is real. If there are no continuum resonances, solutions of the former equation for Φ are such that S Φ 0. However,[ ] where such resonances occur the integrand becomes singular and S Φ is no longer well defined. In the case of a single resonance we can redefine the integral[ ] = [ ] 55 56 Perturbative Calculation of Continuum Damping

as the Cauchy principal value,

dτ lim dτ dτ (4.1) δ→0 ψ<ψ− ψ>ψ+

− P+ S = S + S where ψ ψR δ and ψ ψR δ. Applying Gauss’ theorem to the field f Ψ Φ leads to the identity = − = + ∇ ⋅ ∇ dS dτ f Ψ Φ f Ψ ψ Φ dτΨ f Φ , (4.2) Ω S ψ Ω S ∇ ⋅ ∇ = c ∇ ⋅ ∇ − S ∇ ⋅ ( ∇ ) which represents a form of integrationS∇ S by parts. This identity is used to integrate the quadratic form integrand over an infinitesimal volume centred on the continuum resonance and equation (3.48) is substituted into the resulting expression to eliminate the volume integral. Thus a relationship is obtained between the redefined quadratic form and surface integrals in the limit either side of the continuum resonance discontinuity. This is,

dS ω2 S Φ lim Φ∗ ψ Φ b Φ∗ ψ b Φ δ→0 ψ 2 ψ vA [ ] = c2µ0 ∗ − ∇ ⋅ ∇ + ( ⋅ ∇ )µ∇0j∥ ⋅ ∇ ( ⋅ ∇∗ ) S∇Φ S ψ b κ b p Φ ψ Φ ⊥Φ B2 B ψ+ − (∇ ⋅ × )( × ∇ ⋅ ∇ ) − ∇ 1⋅ [ (∇∗ × ∇ ) R ⊥Φ ⊥Φ . (4.3) − 2 ψR − (∇ × ∇ ) V Similarly, if we deﬁne the integral in the large aspect-ratio circular cross section approximation as the Cauchy principal value

a r− a dr lim dr dr (4.4) 0 δ→0 0 r+

− + P S = S + S where r rR δ and r rR δ, we obtain the expression

= − = + r+ ∗ ∂Ej S Ei lim Ei Di,j . (4.5) → δ 0 ∂r r− [ ] = − W The generalisation of either equation (4.3) or equation (4.5) to multiple continuum resonances is straightforward.

The discontinuity associated with the continuum resonance can be considered as a small perturbation to Φ, with the resulting frequency perturbation representing continuum damping. We introduce perturbative expansions for the eigenfunction and eigenfrequency, which are respectively

Φ ψ, ϑ, ϕ Φ(0) ψ, ϑ, ϕ Φ(1) ψ, ϑ, ϕ (4.6)

( ) = ( ) + ( ) §4.1 Perturbative formalism 57

and ω ω(0) ω(1). (4.7) The unperturbed scalar potential, Φ(0), is taken to be continuous across the contin- =+ + (0) ψR (0) uum resonance (limδ→0 Φ ψ, ϑ, ϕ − 0). This function thus satisﬁes S Φ 0 ψR (0) for ω ω . Additionally, if Φ depends logarithmically on ψ ψR the derivatives of ( )T = =+ (0) ψ Φ on either side of the continuum resonance are related by lim ψ ψ ∂Φ R = −δ→0 R ∂ψ − ψR (0) (1) 0. Both Φ and Φ obey the boundary conditions of regularity( at− the) magneticU = axis, and are equal to 0 at the plasma boundary.

Substituting equation (4.6) and equation (4.7) into equation (4.3) and equating terms which are of ﬁrst-order in perturbed quantities, we obtain

∂S ω; Φ(0) S ω(0); Φ(0) Φ(1) ω(1) ∂ω Rω=ω(0) 2 R + dS + ω ( )∗ (R) ( )∗ ( ) Φ 1 ψ Φ R0 b Φ 1 ψ b Φ 0 ψ 2 R ψ vA 2µ0 (=0)c∗ − ∇ ⋅ ∇ (1) + 2µ0⋅ ∇ (1)∗ ∇ ⋅ ∇ ⋅ ∇ (0) Φ S∇ψ Sb κ b p Φ Φ ψ b κ b p Φ B2 B2 − (∇ µ⋅ 0j∥× ) ×( ∇1)∗ ⋅ ∇ −(0) (0)∗(∇ ⋅ × )(1 ) × ∇ ⋅ ∇ ψ Φ ⊥Φ Φ ⊥Φ B + ψR − ∇ ⋅ 1 ∇ × ∇(1)∗ + (0) ∇ ×(0 ∇)∗ (1) ⊥Φ ⊥Φ ⊥Φ ⊥Φ . (4.8) − 2 ψR − ∇ × ∇ + ∇ × ∇ V Terms of second-order in the perturbed quantities are ignored, as these are assumed to be small relative to those of ﬁrst-order. The ﬁrst term on the left-hand side of this equation is evaluated to be

2 (0) (0) (1) ω (1)∗ (0) (0)∗ (1) S ω ; Φ Φ dτ 2 ⊥Φ ⊥Φ ⊥Φ ⊥Φ vA + =(1P)∗ ∇(0) ⋅ ∇ +(0 ∇)∗ ⋅ ∇ (1) ⊥ b Φ S ⊥ b Φ ⊥ b Φ ⊥ b Φ µ b p µ b p b− ∇κ Φ⋅( ∇1)∗ 0 ⋅ ∇ ⋅ ∇Φ(0) + ∇b κ⋅ ∇ Φ(0)∗ ⋅ ∇ 0 ⋅ ∇ Φ(1) B2 B2 µ b p × ∇ µ b p × ∇ + 0× ⋅ ∇ Φ( 1)∗ b κ ⋅ ∇Φ(0) + 0× ⋅ ∇ Φ( 0)∗ b κ ⋅ ∇Φ(1) B2 B2 × ∇ × ∇ + j⋅∥ ∇ (1 )∗ × ⋅ ∇ (0)+ (0)∗ ⋅ ∇ (1)× ⋅ ∇ ⊥Φ ⊥Φ ⊥Φ ⊥Φ 2B (1)∗ (0) (0)∗ (1) + ∇ ⋅ ∇ ×⊥Φ ∇ +⊥Φ ∇ ⋅ ∇ ×⊥Φ ∇ ⊥Φ . (4.9)

Integration by parts leads+ to∇ a partial× ∇ cancellation ⋅ ∇ + with∇ × the ∇ right-hand ⋅ ∇ side of equa- 58 Perturbative Calculation of Continuum Damping

tion (4.8), yielding

∂S ω; Φ(0) dS ω2 ω(1) lim Φ(1) ψ Φ(0)∗ ∂ω δ→0 ψ ψ v2 Rω=ω(0) A R ψ+ R = c ∇(1)⋅ ∇ (0)∗ R R S∇ S b Φ ψ b Φ − . (4.10) R ψR − ⋅ ∇ ∇ ⋅ ∇ ⋅ ∇ U Likewise, in the large aspect-ratio circular cross-section tokamak case we can derive

r+ ∂S ω; E(0) ∂E(0)∗ R (1) i (1) j ω lim E Di,j . (4.11) ∂ω R δ→0 i ∂r R R R − R = (0) Rr Rω ω R R R = R R R If the unperturbed eigenfunction andR eigenfrequency are known,R all that remains is to determine the discontinuity in the perturbed eigenfunction and use this to determine the terms in the above equations.

4.2 Evaluation of quadratic form perturbation

The discontinuity in Φ(1) is determined by how the integration contour is deformed around the continuum resonance pole for ω(0). It is assumed that ω does not coincide with a stationary point of the continuum resonance frequency. The equilibrium is assumed to have either up-down, stellarator or axial symmetry. Therefore, approaching ﬂux surfaces where continuum resonances occur, the asymptotic behaviour of Φ is logarithmic. Considering leading order terms of the two linearly independent solutions, the solution near the continuum resonance pole has the form

Φ ψ, ϑ, ϕ χ1 ϑ, ϕ ln± ψ ψR χ2 ϑ, ϕ (4.12) where the unperturbed( part will) ≈ be( ) ( − ) + ( )

(0) (0) (0) Φ ψ, ϑ, ϕ χ1 ϑ, ϕ ln ψ ψR χ2 ϑ, ϕ . (4.13)

Using equation (4.12)( we can) show≈ ( that) theS third− andS + fourth( ) terms on the right- hand side of equation (4.3) (associated with pressure gradient and parallel current ∂Φ ∂Φ∗ respectively) do not contribute to S Φ . These terms do not contain ∂ψ or ∂ψ and consequently singular components are even about ψR. [ ] The discontinuity in Φ(1) at the continuum resonance is taken to be that in the logarithmic part of Φ(0) when analytically continued around the pole. Here, we ignore (1) higher order contributions due to the change in χ1 and ψR with ω . Differentiating (0) ∂Φ(0) equation (4.13) with respect to ψ and rearranging gives χ1 limψ→ψR ψ ψR ∂ψ . Thus, we obtain the discontinuity in the perturbed part associated with crossing a = ( − ) §4.2 Evaluation of quadratic form perturbation 59

branch cut of the logarithmic singularity

ψ+ (0) (1) R ∂Φ lim Φ ψ, ϑ, ϕ − iπ lim ψ ψR . (4.14) δ→0 ψR ψ→ψR ∂ψ ( )U = ± ( − ) As discussed in Section 3.4, the causality condition implies that the sign in the above ∂ωR equation should be the same as that of ∂ψ . Alternatively, equation (4.14) repre- ψ=ψR sents the result of applying the Sokhotski-Plemelj theorem. U Inserting equation (4.14) into equation (4.10) and rearranging results in

−1 ∂ω ∂S ω; Φ(0) ω(1) iπsgn R ∂ψ ψ=ψ ∂ω R (0) ⎛ R ⎞ Rω=ω = − W 2 ψψ (0) R (0)∗ (0) (0)∗ ⎝ dS ⎠ω g ∂Φ R∂Φ ψψ ∂Φ ∂Φ lim ψ ψR R g b b ψ→ψ ψ 2 R ψ vA ∂ψ ∂ψ ∂ψ ∂ψ × ( − ) c − ⋅ ∇ ⋅ ∇ (4.15)¡ S∇ S The integral in this expression is the quadratic form defined in equation (A.61), which is zero for ψ ψR. The purely imaginary frequency perturbation in equation (4.15) is identified as = (1) ∂ωR the continuum damping ωi iω . The only direct contributions to ∂ψ and ψ=ψR the integral in this equation are from the polarisation and field-line bending terms = − U in the wave equation. Thus, the only direct contribution to continuum damping is from these terms. However, all components of the wave equation will influence the unperturbed eigenfrequency, ω(0), and eigenfunction, Φ(0), and so affect the calculated continuum damping. Now express the damping in terms of the series expansion of the solution about the continuum resonance. In doing so, it is necessary to consider terms which con- ∂Φ(0) tribute to the integral in equation (4.10) through ∂ψ as well as those which contribute through the discontinuity in Φ(0). Therefore, consider a truncated series expansion for the solution near ψR with an additional linear term

(0) (0) (0) (0) Φ ψ, ϑ, ϕ χ1 ϑ, ϕ ln ψ ψR χ2 ϑ, ϕ χ3 ϑ, ϕ ψ ψR . (4.16)

( ) ≈ ( ) S −1 S + ( ) + ( )( − ) Only contributions of O ψ ψR to the integral in equation 4.15 need to be considered when evaluating ω(1). In the generic case, the quadratic form in equa- ( − )(0) (0) tion (A.61) has a single zero for χ1 . Therefore, terms in this integral in which χ1 −1 is repeated are of O ψ ψR . (0) Higher order terms( in− the) expansion which are proportional to χ1 will not contribute to the integral. However, these terms generally contain a component which is (0) orthogonal to χ1 . Integral terms involving coeffecients of both the logarithmic term k k−2 and an O ψ ψR term for k N give an O ψ ψR contribution to the inte-

( − ) ∈ ( − ) 60 Perturbative Calculation of Continuum Damping

−1 gral. Thus, the linear term in the expansion results in an O ψ ψR contribution to the integral and must be included. Integral terms involving coeffecients of two k ( − ) (1) higher order terms are O ψ ψR for k 0 and hence do not contribute to ω . Inserting the expansion in equation (4.16) into equation (4.15), the equation for ( − ) ≥ continuum damping becomes

−1 ∂ω ∂S ω; Φ(0) ω(1) iπsgn R ∂ψ ψ=ψ ∂ω R (0) ⎛ R ⎞ Rω=ω = − W 2 ψψ R d ⎝ dS ω⎠g (0) (0)∗ R ψψ (0) (0)∗ χ χ R g b χ b χ dψ ψ ψ v2 1 1 1 1 A ψ=ψR × c 2 ψψ − ⋅ ∇ ⋅ ∇ ¡W S∇dSS ω g (0) (0)∗ ψψ (0) (0)∗ χ χ3 g b χ b χ3 ψ 2 1 1 ψ vA + c dS ω2gψψ − ⋅ ∇ ⋅ ∇ ¡ S∇ S (0) (0)∗ ψψ (0) (0)∗ χ3 χ g b χ3 b χ (4.17) ψ 2 1 1 ψ vA + c − ⋅ ∇ ⋅ ∇ ¡ Thus damping is dependentS∇ S on the magnitude of the logarithmic component, repre- −1 (0) ∂Sω;Φ(0) sented by χ1 , relative to that of the mode overall, which enters through ∂ω . ω=ω(0) This can be considered to represent the strength of coupling of the global mode to V continuum modes. The movement of the continuum resonance away from the gap or extremum which induces the mode will tend to reduce this coupling and hence continuum damping. The ﬁrst term in the square brackets in equation (4.17) denotes the rate of change in the quadratic form in equation (A.61) at the continuum resonance, where its zero (0) occurs. While damping is explicitly dependent on this term in equation (4.17), χ1 can be shown to be inversely proportional to the term if it is assumed that the term inside the brackets in equation (A.28) is held constant. It is expected that this assumption will be valid for cases where damping is weak and the global mode structure does not change signiﬁcantly. Thus, the overall continuum damping rate is actually inversely proportional to the derivative term. Equation (4.17) indicates that (0) there is also a contribution from the component of χ3 ϑ, ϕ which is orthogonal to χ(0) ϑ, ϕ . 1 ( ) Based on the expression in equation (3.91), the slow sound approximation can be ( ) incorporated into the continuum damping equation by including the term

2 (0) (0)∗ 4cs 2 ∂Φ ∂Φ 2 ψ b κ (4.18) vA ∂ψ ∂ψ − S∇ ⋅ × S in the integrand in equation (4.15). This contribution would reduce ω(1) if Φ(0) and ω(0) were kept constant and the resonance occured for decreasing ρ and increasing km,n at the continuum resonance. However, the introduction of the slow sound approximation also increases ω(0) due to the upshift in the continuum, resulting in an §4.2 Evaluation of quadratic form perturbation 61

approximate contribution of

4c2 ∂Φ(0) ∂Φ(0)∗ s ψ b κ 2 (4.19) v2 ∂ψ ∂ψ A ψ=ψe S∇ ⋅ × S W to the integral in equation (4.15), where ψe is the location of the continuum gap or extremum responsible for the mode. Thus, the overall effect on continuum damping is only expected to be signiﬁcant if the variations in cs and the geodesic curvature term ψ b κ are large.

Returning∇ ⋅ × again to the large-aspect ratio circular cross-section approximation, assume a logarithmic singularity of the form

(0) (0) (0) Em am bm ln r rR . (4.20)

The resulting discontinuity in the= perturbed+ termS − is foundS to be

+ (1) rR (0) Ei − iπ lim Di,jEj . (4.21) rR r→rR U = ± Inserting this expression into equation (4.11) and rearranging yields

−1 ∂S ω; E(0) (0) ∂E(0) (1) ∂ω i ∂Ei j ω iπsgn lim r rR Di,j . (4.22) R r→r ∂r r=rR ∂ω R R ∂r ∂r Rω=ω(0) = − V R ( − ) R R (0) Alternatively, continuum damping can beR expressed in terms of Ci , giving an expression equivalent to that derived by Berk et al. [20]

∂ω iπsgn ∂r (1) r=rR (0) (0)R ω (0) C adjD i,j C R . (4.23) ∂Sω;E i j R i T ∂YDY R ∂ω ∂r R = − ω=ω(0) ( ) Rr=r R R W R (0) R This expression has the advantage that Ci has a well deﬁned limiting value as r rR, as shown by its Fobenius expansion in equation (A.56). Considering the cylindrical limit, where e˜ 0 we ﬁnd that for a resonance with the m, n branch of the→ continuous spectrum → ( )

∂ω iπsgn 2 (1) ∂r r=rR (0) R ω (0) Cm,n R . (4.24) ∂Sω;E 2 R m r3 T∂ ω k2 R ∂ω R ∂r v¯2 m,n R = ω=ω(0) A R Rr=rR W − R R This greatly simpliﬁed expression nevertheless gives some insightR into the dependence of the continuum damping rate on the mode and continuous spectrum. The 62 Perturbative Calculation of Continuum Damping

(0) variable Cm is identified as a flux-like quantity [20]. Based on the Frobenius expansion in Appendix A, this quantity is observed to vary slowly in the region around a continuum resonance. Thus this value will be determined approximately by the value for the mode approaching this region and will not depend strongly on local (0) ωR r . Hence the value of Cm is predominantly determined by the influence of the continuum gap or extremum and regions of high density shear. ( ) (0) ∂Sω;Em Similarly, ∂ω is understood to depend on the global structure of the ω=ω(0) W mode. Up to a constantW factor it is equal to ω , where W is the ideal MHD potential energy associated with the mode. Damping decreases as this term increases because the energy lost due to the continuum resonance each cycle decreases relative to the energy stored in the mode. Unfortunately, the continuum resonance singularity results in an infinite contribution to W for an unperturbed ideal MHD eigenmode. The procedure for redefining the term to remove this unphysical contribution is discussed in Section 4.3. At the continuum resonance, observe that

∂ ω2 1 ∂ω2 k2 R , (4.25) ∂r v¯2 m,n v¯2 ∂r A r=rR A r=rR − W = − W and therefore this term depends entirely on the rate of change in plasma parameters at the continuum resonance. The loss of power at the continuum resonance is inversely proportional to the rate of change in the continuum resonance frequency with the radial coordinate there. This loss is also inversely proportional to the density of the plasma at the continuum resonance. The inverse proportionality of the damping to the rate of change in the continuum resonance frequency is reminiscent of the proportionality of the transition rate from a quantum mechanical eigenstate to the density of ﬁnal states in a continuum, according to Fermi’s Golden Rule [86]. Continuum damping has previously been compared to decay into continuum states in quantum mechanics in the work of Zhang et al. [44].

4.3 Quadratic form variation with frequency

In order to evaluate damping using equation (4.15) it is necessary to determine ∂Sω;Φ(0) ∂ω . However, simply taking the partial derivative of the integrand in ω=ω(0) equation (4.3) results in V (0) ∂S ω; Φ (0) 1 (0)∗ (0) 2ω dτ ⊥Φ ⊥Φ . (4.26) ∂ω v2 Rω=ω(0) A R R = P S ∇ ⋅ ∇ ¡ This equation has a singularR contribution at ψ ψR due to the logarithmic component of Φ(0). If ω is changed, the value of ψ for which the continuum resonance condition is satisﬁed also changes. Thus, this ﬂux surface= no longer corresponds to the singu- §4.3 Quadratic form variation with frequency 63

(0) −2 −1 larity in Φ , and a ψ ψR rather than ψ ψR type singularity occurs in the integrand in the definition of S. ( − ) ( − ) In the method developed by Berk et al. for eliminating the divergent continuum resonance contribution in the large aspect-ratio circular cross-section tokamak case, the partial derivative of the principal value expression is redefined [20]. A new quadratic form is defined such that integration bounds vary with ω. The approach of Berk et al. incorporates higher-order terms in the perturbative expansion to eval- (0) ∂Sω;Φ − (0) uate ∂ω . The principal value is redefined such that ψ ψR δ and ω=ω(0) + (0) (0) (0) ψ ψR δ,V where ψR is the location of the continuum resonance= for ω− .A sesquilinear form F Φ(0), Φ(1) is defined on a flux surface equal to the integral in equation= (4.8).+ The location of the continuum resonance for perturbed frequency ω is −1 then expressed using the approximation ψ ψ(0) χω(1) where χ ∂ωR . R R ∂ψ (0) ψ=ψR Noting that Φ(0) is continuous across the resonance≈ + and using this= expression, V we find that

(0) (0) ψ +δ (1) ∂S ω; Φ (0) (1) R ω lim F Φ , Φ (0) (4.27) → ψ −δ ∂ω R (0) δ 0 R Rω=ω R = ψU(0)+δ R (0) R R lim F Φ , Φ (0) (4.28) δ→0 ψR −δ (1) = UψR−χω +δ lim F Φ(0), Φ (4.29) (1) δ→0 ψR−χω −δ ψ +δ ≈ U ∂Φ R lim F Φ(0), Φ ω(1)χF Φ(0), . (4.30) δ→0 ∂ψ ψR−δ ≈ − W Inserting the expression for the logarithmic singularity used in Section 4.2, the second term in square brackets can be expressed as

(1) (1) (0) ∂Φ 2ω (0) ω χF Φ , χ lim ψ ψR ∂ψ δ (0) ψ→ψR − dS ω(0=)2−gψψ ∂Φ(0) ∂Φ(0)∗ − ∂Φ(0) ∂Φ(0)∗ gψψ b b . (4.31) ψ 2 ψ vA ∂ψ ∂ψ ∂ψ ∂ψ × c − ⋅ ∇ ⋅ ∇ ¡ S∇ S ∂Sω;Φ(0) This term represents the singular contribution to ∂ω . ω=ω(0) Note that in the limit δ 0 [20] V

→ ψ 2 dτ dτ dS ψ<ψ(0)−δ ψ>ψ(0)+δ (0) ψ=ψ(0) R R ψ ψ ψ δ R ⎛ ∇ R ⎞ S + S ∇ ⋅ ⎜ ⎟ = − c 1 ⎝S∇ S − ⎠ dS. (4.32) (0) ψ=ψa ψa ψR + c − 64 Perturbative Calculation of Continuum Damping

We can now deﬁne a new quadratic form where the singular contribution is cancelled out,

˜ (0) (0) ∂S ω; Φ 2ω (0)∗ (0) dτ ⊥Φ ⊥Φ ∂ω v2 Rω=ω(0) A R R = P S ∇ ⋅ ∇ ψ R Q R. (4.33) ψ ψ ψr0 ∇ + ∇ ⋅ ¡ + In this expression we have introduced the constants, S∇ S( − )

1 Q χ ψ ψ(0) dS R ψ=ψr0 ≡ − dS ω−(0)2gψψ ∂Φ(0) ∂Φ(0)∗ ∂Φ(0) ∂Φ(0)∗ ∮ gψψ b b , (4.34) ψ 2 ψ vA ∂ψ ∂ψ ∂ψ ∂ψ × c − ⋅ ∇ ⋅ ∇ ¡ and S∇ S Q R dS. (4.35) (0) ψ=ψa ψa ψR ≡ − c Thus, by replacing S with S˜ in equation− (4.15) we are in principle able to evaluate all of the terms in the resulting expression to obtain an estimate for continuum damping. (0) ∂Sω;Ei The term ∂ω in equation (4.11) represents the rate of change in the ω=ω(0) quadratic form with respectW to frequency as a discontinuity is introduced at the continuum resonance. Berk et al. estimate this quantity by modifying the boundary conditions at the plasma edge [20]. Let Ei ω; r denote a solution to equation (3.79) for ω such that Ei ω; rR δ Ei ω; rR δ . Relax the boundary conditions such that (0) ( ) for ω ω , Em ω; a 0 for exactly one m [20]. Thus, the rate of change in the ( − ) = ( + ) quadratic form can be expressed in terms of that in Ei a ≠ ( ) ≠ (0) ( ) ∂S ω; Ei E ω δω; a C ω ; a lim i 0 i 0 (4.36) ∂ω R δω→0 δω Rω=ω(0) ( + ) ( ) R = − R ∂Em R Cm, (4.37) R ∂ω where there is no summation over=m.− This quantity is readily obtained using the shooting method described in Subsection 4.4.2.

4.4 Implementation

4.4.1 Finite element method

In the conventional ﬁnite element method, logarithmic behaviour close to a continuum resonance is approximated by a linear combination of regular basis functions. §4.4 Implementation 65

Therefore, the finite element solution will not accurately reflect the expected logarithmic behaviour of the exact solution. The finite element method solution will have a finite extremum corresponding to the continuum resonance and therefore in the ∂Φ limit approaching this extremum ∂ψ ψ ψR rather than the analytically predicted behaviour such that ∂Φ ψ ψ −1. Thus, when equation (4.15) and equation (4.33) ∂ψ R ∝ − are used to calculate continuum damping a non-zero value of δ should be used to approximate the principal∝ ( value− integrals) in these expressions. The terms in equation (4.15) can be determined from quantities calculated by (1) ∂ωR the finite element method solver. Provided that ω δ ∂ψ , the correction ψ=ψR term calculated in equation (4.31) will be negligible and equation (4.26) can be used ≪ U without modification. In this case we can use the estimate

∂S ω; Φ(0) 2ω(0)c T′ c . (4.38) ∂ω i I,i,j j Rω=ω(0) R ≈ ′ R The matrix TI,i,j is deﬁned by changingR the bounds of integration for the evaluation of TI,i,j in equation (3.115) such that the volume ψ ψR δ, ψR δ is excluded. This only involves recalculation of the matrix elements i, j where Λi and Λj have mutual support on this volume. Clearly, such elements∈ will( be− zero+ where) their mutual support is completely within the excluded volume. In general, contributions from surface terms must be considered in evaluating these integrals. Substituting the expression for the solution in terms of basis functions into the integral in equation (4.15) yields

2 ψψ (0) (0)∗ (0) (0)∗ dS ω g ∂Φ ∂Φ ψψ ∂Φ ∂Φ lim ψ ψR g b b ψ→ψ ψ 2 R ψ vA ∂ψ ∂ψ ∂ψ ∂ψ ( − ) c − ⋅ ∇ ⋅ ∇δc U c .¡ (4.39) S∇ S i i,j j

In the expression above, the matrix Ui,j is evaluated =

2 ψψ ∂Λ∗ ∂Λ∗ dS ω g ∂Λi j ψψ ∂Λi j Ui,j g b b . (4.40) ψ ψ ⎧ v2 ∂ψ ∂ψ ∂ψ ∂ψ ⎫ ⎪ A ⎛ ⎞⎪ = c ⎨ − ⋅ ∇ ⋅ ∇ ⎬ S∇ S ⎪ ⎝ ⎠⎪ As this expression represents⎩⎪ a surface integral, its evaluation is computationally⎭⎪ inexpensive compared with the volume integrals evaluated using the ﬁnite element solution. Most pairs of basis functions will not have mutual support intersected by the surface in question, resulting in a very sparse matrix. Alternatively, using Gauss’s theorem and the differential expression for shear Alvén waves, we obtain

2 ψψ (0) (0)∗ (0) (0)∗ dS ω g ∂Φ ∂Φ ψψ ∂Φ ∂Φ lim ψ ψR g b b (4.41) ψ→ψ ψ 2 R ψ vA ∂ψ ∂ψ ∂ψ ∂ψ ( − ) c ∂Φ(0) − ⋅ ∇ ⋅ ∇ ¡ Sδ∇ S , T Φ(0) (4.42) ∂ψ Ω′′ ≈ − d i 66 Perturbative Calculation of Continuum Damping

(0)2 ′′ ′′ δdi ω TI,i,j TW,i,j cj. (4.43)

In the ﬁnal expression≈ − above, a spline− ﬁt to the derivative is used

∂Φ(0) Ns Nϑ Nϕ dlΛl s, ϑ, ϕ . (4.44) ∂ψ l=1 ≈ Q ( ) This is necessarily an approximation as the derivatives of Φ(0) Vh, the space spanned by the basis set, do not belong to Vh in general, having continous derivatives ′′ ′′ ∈ to one order lower. The matrices TI,i,j and TW,i,j are defined by changing the region of integration in equation (3.115) and equation (3.116) respectively to the volume Ω′′, such that ψ 0, ψR δ . Again, non-trivial changes to matrix elements correspond to basis functions with mutual support intersecting the boundary surface. In contrast∈ ( to the− assumption) employed in Section 4.2, it is found that finite element solutions are asymmetric about continuum resonances in the surrounding region. The discontinuity is observed to be sensitive to the mesh spacing used. How- ever, as discussed in Section 3.6, the discontinuity does not converge to zero when the resolution of the mesh is increased. This affects the calculated values of the quantities in equation 4.15 which are evaluated near the continuum resonance. In the finite element method, the continuity in the quantities identified in Sec- tion 4.1 across continuum resonances should ideally be refelected in the choice of the set of basis functions. This is complicated by the fact that the locations of continuum resonances are frequency dependent and hence not known a priori. These locations could be found iteratively, updating the relevant basis functions based on the estimated frequency. However, introduction of new basis functions in this manner affords the opportunity to impose the complex jump discontinuities associated with the continuum resonance directly, as outlined in Chapter 6, making the perturbative calculation redundant. Thus, the perturbative technique for calculating continuum damping has generally been implemented using a shooting method, and not a finite element method. Comparison of perturbative continuum damping calculations implemented using the finite element method and shooting method show approximate agreement.

4.4.2 Shooting method

In the shooting method, a boundary value problem is solved by reformulating it in terms of an initial value problem. A subset of boundary values are considered functions of a set of variables, which are computed by solving the differential equation. Values of the variables for which the boundary condition is satisﬁed are determined numerically. Berk et al. describe the use of a shooting method to calculate the continuum damping of a TAE due to the coupling of the 1, 1 , 2, 1 , and 3, 1 harmonics where ι0 1 is the rotational transform at the magnetic axis [20]. This ( − ) ( − ) ( − ) method can readily be generalised to other sets of harmonics, couplings, and ι0. = Initial values for Ei and Ci near the magnetic axis can be obtained by solving the §4.4 Implementation 67

simpliﬁed wave equation in this region analytically. Taking ρ and ι to be constant and neglecting the coupling of different harmonics, equation (3.74) becomes

∂ 3 ∂Em,n 2 r 1 m rEm n 0. (4.45) ∂r ∂r ,

+ − −1±SmS= This equation has solutions of the form Em,n am,nr , with physical (non-singular) solutions taking the positive sign. Consider the m, n harmonic solution and introduce a small toroidicity induced coupling= between this and the m µ, n ν har- µ,ν µ,ν ( ) monic, such that eˆ r and eg r. Under this assumption equation (3.74) becomes ( + + ) ∝ ∝

∂ 3 ∂Em+µ,n+ν r 1 m µ 1 m µ rEm+µ n+ν ∂r ∂r , µ,ν µ,ν +1( − − )( ∂+eˆ + ) =∂eg ∂ ∂E ω2 k k r4 m,n . (4.46) 2 2 2 m,n m+µ,n+ν ω v km+µ,n+ν ∂r ∂r R ∂r ∂r ⎛ A ⎛ ⎞⎞Rr=0 − − R ⎝ − ⎝ ⎠⎠R It is assumed that Em,n is not signiﬁcantly modiﬁed by theR coupling, so that the solution to equation (4.45) can be used to determine the driving term on the right- hand side of equation (4.46). We can assume m 0 without loss of generality, noting that Ei and ω are unchanged by the transformation m, n, µ, ν m, n, µ, ν due to stellarator symmetry. Where µ 1 and µ 2≥m 1 the resulting solution is ( ) → (− − − − ) µ,ν µ,ν 1 ≠∂eˆ ≠ − − ∂eg m 1 m 2 E ω2 k k rm. m+µ,n+ν 2 2 2 m,n m+µ,n+ν ω v km+µ,n+ν ∂r ∂r R 1 µ 2m 1 µ ⎛ A ⎛ ⎞⎞Rr=0 ( − )( + ) = − − R (4.47) ⎝ − ⎝ ⎠⎠R ( − )( + + ) In the exceptional cases where µ 1 or µ 2m 1 the solutionR is

µ,ν µ,ν 1 =∂eˆ = − − ∂eg m 1 m 2 E ω2 k k rm ln r. m+µ,n+ν 2 2 2 m,n m+µ,n+ν ω v km+µ,n+ν ∂r ∂r R 2 m 1 ⎛ A ⎛ ⎞⎞Rr=0 ( − )( + ) = − − R (4.48) ⎝ − ⎝ ⎠⎠R ( + ) R

As equation (3.74) has a regular singular point at the magnetic axis, initial conditions are found for the numerical solution by considering E ζ for some ζ a. Additional boundary conditions for C ζ are obtained based on the derivative of the near magnetic axis solutions obtained above at this point. Considering( ) a set of≪ N harmonics, the near axis solutions above( ) yield N different boundary conditions with a different harmonic considered dominant in each case. Thus, a set of N linearly (i) (i) independent solutions, with the ith solution here denoted Em,n and Cm,n, is obtained by solving the initial value problem for each of these boundary conditions. Conti- + (i) rR nuity across resonance singularities is applied by requiring that Em,n r − 0 and rR r+ (i) R ( )U = Cm,n r − 0 for some ﬁnite δ a. The general solution to the initial value problem rR ( )U = ≪ 68 Perturbative Calculation of Continuum Damping

can be expressed as a linear combination of these solutions

N (0) (i) Em,n r ciEm,n r (4.49) i=1 N (0) ( ) = Q (i) ( ) Cm,n r ciCm,n r . (4.50) i=1 ( ) = Q ( ) (0) (0) The solution for Em,n r and ω is the non-trivial solution satisfying the boundary (0) condition at the edge of the plasma, Em,n a 0. This condition corresponds to (j) ( ) ω E 0, where each i denotes a different harmonic m, n . We search for i ( ) = the solution iteratively using the secant method, where the ith estimate of ω(0) is E ( ) ≡ [ [ = ( )

ωi−1 ωi−2 ωi−2 ωi−1 ωi . (4.51) ωi−2 ωi−1 E ( ) − E ( ) = Equation (4.37) is evaluated based onE ( initial) − valueE ( problem) solutions for sucessive es- (0) (0) timates of ω . Along with the solution for Cm,n r this allows continuum damping to be determined using equation (4.23). The shooting method is advantageous in that( it) guarantees the solution is sym- metrical about r rR in the limit where r rR. Its implementation is straightforward and can take advantage of fast and adaptible differential equation solvers. However, the method outlined= can only compute one→ Alfvén eigenmode at a time. Moreover, the behaviour of ω rapidly changes at accumulation points of the continuous spectrum. This makes it necessary to start the secant method using values within a continuum gap whenE ( calculating) the resultant Alfvén eigenmode. Chapter 5

Complex Integration Contour Calculation of Continuum Damping

The calculation of continuum damping by solving the wave equation along a complex integration path is described in this chapter. A change of variables and analytic continuation of equilibrium quantities allow this problem to be posed in terms of a real contour parameter, rather than complex ﬂux surface label. Thus, the problem can be solved using standard numerical techniques. This approach was applied by Könies and Kleiber to numerically compute continuum damping of shear Alfvén eigenmodes involving two coupled harmonics in large aspect-ratio tokamak and stellarator cases [73]. They found that convergence of continuum damping with respect to the radial mesh resolution in the complex contour method occurs faster than that taking the limit as resistivity vanishes. Here we describe the technique as it applies to more complicated two- and three-dimensional geometries. Alternatively, by considering a complex integration path which traces around branch cuts in the eigenmode solution, the discontinuities due to these branch cuts can be computed. Such approaches are implemented by Berk et al., who used a shooting method [20], and Chu et al., who used a Hamiltonian variational method [73]. The results of these techniques are compared with those of perturbative methods in Chapter 8. The numerical complex contour method is used to compute continuum damping for realistic tokamak and stellarator equilibria in Chapter 9.

5.1 Speciﬁcation of the contour

In Section 3.4 we described how the presence of continuum resonances results in 2 complex ω , leading to poles at complex s sR. The causality condition implies that the shear Alfvén wave equation must be solved along a complex contour in s such that this and the real s-axis enclose a region= of the complex plane containing the continuum resonance poles. A complex integration path is deﬁned in terms of a real

69 70 Complex Integration Contour Calculation of Continuum Damping

parameter x 0, 1 such that

∈ ( ) s x f x ig x (5.1) with real valued functions f and(g)deﬁned= ( ) + such( that) f 0 g 0 g 1 0 and f 1 1. ( ) = ( ) = ( ) = ( ) = The locations of continuum resonance poles in the complex s plane can be estimated using equation (3.96). Furthermore, additional poles may occur which are not associated with continuum resonances of the Alfvén eigenmode. The continuum resonance condition is met at these poles, but they do not form part of a sequence that approaches the real s-axis as I ω2 0. Such complex poles occur close to the location of continuum gaps and extrema along the real s-axis. These poles must be excluded from the region bounded by the→ integration contour and the real axis to avoid spurious contributions to the solution.

In each of our calculations using the complex contour method, we use one of two types of integration path. In the ﬁrst case, we use contours described by

f x x (5.2) g x αx x 1 , (5.3) ( ) = where the parameter α R determines( ) = the scale( − of) the deformation of the contour from the real axis. In the second case we introduce a Gaussian factor to localise this deformation, using contours∈ described by

f x x (5.4) 2 x xγ g (x) = αx x 1 exp , (5.5) xβ ⎛ − ⎞ ( ) = ( − ) − where the parameters α, xβ, and xγ represent⎝ the scale, width,⎠ and location of the deformation of the contour from the real axis. This class of complex contour could be modiﬁed to include multiple continuum resonances by redeﬁning g x as

2 x xγ,i ( ) g x αix x 1 exp , (5.6) x i i ⎛ −β, ⎞ ( ) = Q ( − ) − for an appropriate set of αi, xβ, and xγ in order⎝ to causally circumvent⎠ each pole.

A numerical solution of the eigenvalue problem over a complex contour can be computed by considering Φ to be a function of x, ϑ, ϕ . The partial derivatives with respect to s can be related to those with respect to x through ( ) ∂ 1 ∂ (5.7) ∂s f ′ x ig′ x ∂x = ( ) + ( ) §5.2 Analytic calculation of discontinuities 71

and ∂2 1 ∂2 f ′′ x ig′′ x ∂ . (5.8) ∂s2 f ′ x ig′ x 2 ∂x2 f ′ x ig′ x ∂x ( ) + ( ) = s − x It follows that integrals( with( ) respect+ ( to)) are related( to) those+ ( with) respect to through

ds dx f ′ x ig′ x . (5.9)

′ ′ Let the derivatives of f x Sand g= xS be continuous ( ) + ( to) order k and f x ig x 0 for x 0, 1 . Provided that the contour does not intersect continuum resonance poles, the derivatives of( a) solution( Φ) x, ϑ, ϕ which is analytic on a domain( ) + contain-( ) ≠ ing the∈ integration( ) contour will also be continuous to order k. ( )

5.2 Analytic calculation of discontinuities

Finding the solution of the shear Alfvén wave equation along a complex contour circumventing a continuum resonance pole is equivalent to applying an appropriate jump discontinuity in Φ along the real s-axis. Thus it is possible to determine the mode structure as a function of physical space by including the effect of the continuum resonance interaction. As demonstrated in Section 3.4, such a discontinuity is associated with crossing a branch cut due to the logarithmic singularity. This approach was adopted by Chu et al. to compute continuum damping in a large aspect-ratio circular cross-section tokamak [73]. In their method, jump conditions are derived for continuum resonances and these are incorporated into a Hamiltonian system of differential equations representing the shear Alfvén waves. The complex value of ω2 is found by extremising the corresponding Hamiltonian functional using a ﬁnite element method. The discontinuity can be determined by deﬁning a branch cut in Φ perpendicular to the real axis and terminating in the singularity. This discontinuity can then be evaluated based on an integration contour which traces around the branch cut with a circular loop enclosing the singularity, as shown in Figure 5.1. The contribution due to the loop between W− and W+ can be calculated analytically as the residue of Φ at sR. Let the remainder of the integration path follow the real s-axis. In the case where the singularity is logarithmic,

sW+ ∂ωR ∂Φ lim Φ s, ϑ, ϕ 2πisgn lim s sR , (5.10) sW− s→s δ→0 ∂s s=R(sR) R ∂s ( )S = V ( − ) where sW− and sW+ are the end points of the circular component of the integration − − path, s sR ρ, shown in Figure 5.1. For ωi ωr the contributions from Z W + + and Z W approximately cancel, and the discontinuity can be applied to sZ− and sZ+ S − S = ≪ on the real axis. Moreover, the limit can be approximated by considering s R sR . This results in the following relationship for the quadratic form S ω; Φ , here deﬁned by taking the Cauchy principal value integral to exclude the jump discontinuity→ ( at) [ ] 72 Complex Integration Contour Calculation of Continuum Damping

s R sR ,

= ( ) ∂ω S ω; Φ 2πsgn R ∂s s=R(sR) 2 ss ∗ ∗ [ ] = V dS ω g ∂Φ ∂Φ ss ∂Φ ∂Φ I sR lim g b b . (5.11) s→R(s ) s 2 R s vA ∂s ∂s ∂s ∂s × ( ) c − ⋅ ∇ ⋅ ∇ ¡ Deﬁne a new quadratic formS∇ S ′ ω; Φ by subtracting the right-hand side of this equation from the left. Let V be the set of functions which are continuous over the [ ] R ∂Φ toroidal volume Ω, excluding at the surface s sR , and such that lims→sZ− ∂s ∂Φ lims→s + for Φ V. The Euler-Lagrange equations are obtained by extremising Z ∂s = ( ) = S′ ω; Φ with respect to Φ or Φ∗ (treating these as independent functions), which correspond to the∈ wave equation with the appropriate jump discontinuity. We can then[ derive] a weak formulation of the eigenvalue problem similar to equation (3.88),

∂ωR ′ Ψ, T Φ 2πsgn lim dτI sR δ s s ′ ∂s s=R(sR) s →R(sR) 2 ss ∗ ∗ ⟨ [ ]⟩ − ω gV ∂Φ ∂Ψ ss S ∂Φ( ) ∂−Ψ 2 g b b 0, Ψ V, (5.12) vA ∂s ∂s ∂s ∂s × − ⋅ ∇ ⋅ ∇ ¡ = ∀ ∈ deﬁning the inner product as a Cauchy principal value integral, as was the case for the quadratic form. In this expression we use a function δ s deﬁned such that

dS ( ) dτδ s f s, ϑ, ϕ f s, ϑ, ϕ . (5.13) s s S ( ) ( ) = c ( ) S∇ S ℑ(푠)

푠푅 × 푊− 푊+

푍− 푍+ ℜ(푠)

Figure 5.1: The causal integration contour (blue, solid line) in the complex s plane used to calculate the discontinuity in Φ due to a continuum resonance at sR. The branch cut is also shown (green, dashed line).

Similar expressions can be derived from the wave equation in either the large aspect-ratio stellarator or circular cross-section tokamak case, as represented in equa- §5.3 Numerical implementation 73

tion (3.79). When the continuum resonance discontinuity is excluded by taking the Cauchy principal value, the quadratic form deﬁned in equation (3.87) satisﬁes the equation ∗ ∂ωR ∂Ei ∂Ej S ω; Φ 2πsgn I rR lim Di,j . (5.14) ∂r r=R(rR) r→R(rR) ∂r ∂r

The discontinuity[ can] = also be expressed V in terms( ) of Ci Cm,n, which is continuous at the continuum resonance pole (in the sense that limr→r − Ci limr→r + Ci so that = W W Ci rR can be deﬁned to be equal to these limits), = ( ) Cj rR rW+ ∂ωR lim Ei r 2πisgn adj Di j . (5.15) rW− , ∂YDY δ→0 ∂r r=rR ∂r ( r=)r ( )S = V R The quadratic form is thus expressed as U

a ∗ ∂Ei ∗ ∂Ci S ω; Φ dr Ci Ei , (5.16) 0 ∂r ∂r [ ] = P + and is related to the discontinuity byS

∂ω Ciadj Di,j Cj rR S ω; Φ 2πisgn R . (5.17) ∂YDY ∂r r=rR ∂r r=r ( ) [ ] = V R U 5.3 Numerical implementation

5.3.1 Finite element method

The shear Alfvén wave equation can be solved numerically over a complex contour using the finite element method described in Section 3.6. Basis functions are defined in terms of the parameter x along a chosen complex contour, rather than the real s axis. Radial derivatives are also expressed in terms of x using equations (5.7), (5.8), and (5.9). This allows the use of non-analytic basis functions which are defined solely over the complex contour to represent eigenmodes. Analytic continuation of equilibrium quantities is used to evaluate the elements in the resulting matrix eigenvalue problem. Due to the complex integration path, these matrices are no longer self-adjoint. This approach has been implemented using CKA for the full three-dimensional shear Alfvén wave equation. Complex contour integration has also been incorporated into a MATLAB script which computes a finite element solution for the large aspect-ratio approximation, which is described in Appendix B. Previous versions of CKA use cubic splines defined on real s to represent equilibrium quantities. However, these piecewise functions are clearly not infinitely differentiable and thus do not have analytic continuations that would allow evaluation at complex s. One approach to estimating equilibrium quantities on the complex integration path is use of a truncated Taylor series expansion. Considering a function 74 Complex Integration Contour Calculation of Continuum Damping

h s , expanding about R s gives the following expression

( ) ( ) 1 h s h R s iI R s h′ R s I R s 2 h′′ R s 2 i ( ) = ( ( )) + ( ( )) ( ( )) − ( ( )) I R( s( ))3 h′′′ R s . (5.18) 6 However, it is noted that this expansion is not analytic,− ( ( as)) the complex( ( )) + derivative⋯ is not well defined. Therefore, integrals will have an erroneous dependence on the complex integration path. This corresponds to the error introduced by truncating the series at finite order. Such errors increase moving away from the real s-axis. More- over, inclusion of terms involving derivatives of order greater than or equal to that of the splines (which are not continuous on the real s-axis) results in discontinuities at complex s. These discontinuities also cause errors when integrals are evaluated along a path that crosses them. When this approach is implemented, the aforementioned problems are found to result in poor eigenvalue convergence with respect to changes in the complex integration path. Therefore, CKA evaluates analytic expressions for equilibrium quantities at locations on the complex integration path. The structure of CKA requires that these quantities be evaluated point-wise in order to compute the integrals in the expressions for the matrix elements in equations 3.115 and 3.116. Evaluation of the Fourier series which represent these quantities at a particular s for a given ϑ and ϕ is computationally expensive. To improve computational efficiency, spline functions can be fitted to the values of the equilibrium quantities calculated along the chosen integration path by taking an inverse discrete Fourier transform at each s. It can be verified that the chosen complex integration path satisfies the causality condition by demonstrating that the solution does not vary significantly as the path changes. This may be shown by demonstrating convergence of the solution with respect to the scale of the deformation from the real axis, α. The rate of convergence with respect to α and that with respect to the resolution of the mesh in the radial coordinate are related. If a complex contour passes close to a continuum resonance pole, the mesh must have high resolution with respect to x to accurately represent the continuum resonance interaction. Likewise, high mesh resolution is required to avoid spurious contributions when the complex contour passes close to poles which do not correspond to continuum resonances. Similarly, if s varies rapidly with respect to x, the mesh will also require a high resolution in x. Thus, we wish to choose integration paths which are reasonably smooth and pass as far as possible from poles while satisfying the causality condition. Given that basis functions are defined piecewise along the complex contour in s, straightforward analytic continuation cannot be used to find Φ for real values k I(s) ∂kΦ of s. Nevertheless, in regions where k! ∂sk 1 k 1, Φ s along the complex contour can be taken as a reasonable approximation for Φ R s . Thus, it can be advantageous to choose an integration path that≪ remains∀ ≥ close( ) to the real s-axis, subject to the considerations above. ( ( )) §5.3 Numerical implementation 75

Deformation of the integration path away from the real axis also results in complex ω2 for the calculated continuum modes. These complex values do not reflect the damping of continuum modes, rather, they represent the frequencies at which the continuum resonance condition (equation (3.90)) is satisfied for locations along k k I(s) ∂ ωR I 2 the complex integration path. Where k! ∂sk 1 k 2, ω calculated for the continuum mode varies linearly with I s . This behaviour is also observed for discrete modes with continuum resonances where≪ the deformation∀ ≥ from the real axis is insufficient to satisfy the causality condition.( ) Unfortunately, a straightforward application of the finite element method using equation (5.11) is not possible. Continuum resonance locations are a function of the complex eigenvalue, ω2, that we wish to calculate. Thus, an iterative solution would be necessary, starting from an initial estimate of ω2 and then recalculating 2 sR and ω for each new value the other takes until these values converge. More importantly, if the basis functions are continuous, the finite element solution will represent the transition between branches of the singular solution near the resonance as having finite width in s, rather than as a jump discontinuity. This will give a contribution to the integral, which is ignored by taking the principal value in the treatment of the exact solution in Section 5.2. This problem could be averted by use of a Hamiltonian formulation similar to that of Chu et al., which introduces a new canonical angular momentum which is continuous across continuum resonances [73]. Alternatively, basis functions can be employed which reflect the form of the continuum resonance singularity, including the continuum resonance discontinuity. This approach is described in Chapter 6.

5.3.2 Shooting method In the case of a tokamak with large aspect-ratio and circular cross-section, it is relatively straightforward to use a shooting method to compute the wave equation over the complex contours described above. Using equation (5.7), equations (3.83) and (3.84) are expressed in terms of the contour parameter x. Provided that ρ r and ι r are analytic functions, analytic continuation can be used to compute matrices Di,j r ( ) ( ) and Ai,j r for complex r. Invariance with respect to change in integration path is demonstrated as in the finite element case above. ( ) A shooting( ) method can also include continuum resonance pole contributions calculated analytically as described in Section 5.2. The method described in Sub- section 4.4.2 is modified such that equation (5.15) at x R xR and the coupled differential equations are solved along the complex contour indicated in Figure 5.1. Here limits are simply estimated using a finite ρ a.= Presuming( ) that I ω R ω , the approximations Cm,n rZ− Cm,n rZ+ , adj D rZ− adj D rZ+ , and rW+ rZ+ ≪ ( ) ≪ Em,n r Em,n r can be used. This avoids the need to calculate quantities at rW− rZ− complex( ) locations via analytic continuation.( ) ≈ ( ) ( ( )) ≈ ( ( )) ( )S ≈ ( )S 76 Complex Integration Contour Calculation of Continuum Damping Chapter 6

Singular Finite Element Calculation of Continuum Damping

A technique for calculating continuum damping of shear Alfvén eigenmodes using a singular finite element method is presented in this chapter. A modified weak formulation is obtained for cases where the solution along the real axis has a discontinuity corresponding to a branch cut. The Galerkin method is modified to include basis functions which reflect the asymptotic behaviour of the solution approaching the continuum resonance pole. In this way the appropriate jump condition for the solution at this surface is imposed. This method is used to solve for the complex eigenvalue and continuum resonance pole location iteratively. The complex eigenvalue computed using this method is found to agree closely with that determined using the complex contour method. This procedure obviates the need for analytic continuation of equilibrium quantities, which is advantageous because ideal MHD finite element codes commonly do not represent these using analytic functions. Al- though the discussion below pertains to cases with a single continuum resonance, this method could readily be generalised to compute continuum damping in cases where multiple resonances exist through inclusion of corresponding singular basis functions. In the field of plasma physics, singular finite element methods have previously been put forward by Grimm et al. [87] and Manickam et al. [88] to analyse ideal MHD linear stability in cases where regular singular points exist. Miller and Dewar [89] and Pletzer and Dewar [90] later developed a singular finite element method with improved convergence properties for solving the Newcomb equation, which describes a marginally stable mode of a cylindrical magnetised plasma in accordance with ideal MHD. In the aforementioned treatments, singularities occur at known locations corresponding to rational ι. This differs from the case of Alfvén eigenmodes with continuum resonances, where the pole location is a function of the eigenvalue.

6.1 Variational formulation with continuum resonances

Continuum resonance interactions can be included in the weak formulation of the shear Alfvén wave equation in a large aspect-ratio circular cross-section tokamak

77 78 Singular Finite Element Calculation of Continuum Damping

by permitting discontinuities in Ei corresponding to the intersection of branch cuts with the real axis. This treatment is similar to the derivation of equation (5.12) for a general three-dimensional geometry, though here no assumption is made about behaviour of the solution near the pole. Let Ei V, where V is the set of functions which are continuous on a causal complex contour (but not necessarily along the real r-axis). The complex contour is deﬁned as illustrated∈ in Figure 5.1 and consider only the contribution from integration along the real r-axis. A weak formulation of the eigenvalue problem in this case can then be derived from equation (4.5)

r+ R a ∗ 2 ∗ ∂Ej ∂Fi ∂Ej ∗ ω lim Fi DI,i,j dr DI,i,j Fi AI,i,jEj → ⎧δ 0 ∂r r− 0 ∂r ∂r ⎫ ⎪ R ⎪ r+ ⎨ W + P S R a ∗ − ⎬ = ⎪ ∗ ∂Ej ∂Fi ∂Ej ∗⎪ ⎩ lim Fi DW,i,j dr DW,i,j Fi⎭AW,i,jEj Fi V. (6.1) δ→0 ∂r − 0 ∂r ∂r rR W + P S − ∀ ∈ 2 Here quantities representing the ω dependent and independent parts of Di,j and Ai,j are deﬁned

r3 DI,i,j 2 δi,j e˜ δi−1,j δi+1,j (6.2) v¯A = 3 2 + + DW,i,j r ki,nδi,j (6.3) r ∂ 1 = 2 2 AI,i,j 2 1 m r 2 δi,j (6.4) v¯A ∂r v¯A = 2 − 2 + AW,i,j rki,n 1 m δi,j. (6.5)

The continuum resonance interaction= responsible− for the damping is represented by ∗ ∂Ej the discontinuity in Fi Di,j ∂r . Applying the Galerkin method with discontinuous basis functions, these terms result in a non self-adjoint contribution to the matrices in the resulting discretised eigenvalue problem. The weak formulation above involves integrals over real r, avoiding the need for analytic continuation of equilibrium quantities.

6.2 Singular basis functions

An inner region ra, rb is defined which includes R rR . On this region the solution is assumed to take the form indicated by the lowest order terms in the Frobenius ( ) ( ) expansion about rR, which is calculated in Appendix A. On the outer region, which consists of the remainder of 0, a , the solution is taken to be piecewise polynomial. Solutions of this type are specified in the choice of basis functions detailed below. ( ) Define a set of Nm Nr regular B-spline basis functions in order to represent each of the Nm poloidal harmonics independently on a mesh defined by Nr 1 radial points. For simplicity, these basis functions are chosen to be triangular functions (first-order B-splines) defined on a mesh in r with regular spacing ∆r a . These+ functions are Nr−1 = §6.2 Singular basis functions 79

vectors given by the equation

Sr−riS 1 ∆r δm,j for r ri ∆r, ri ∆r Λm,i,j r . (6.6) ⎧0 for r r ∆r, r ∆r ⎪ − ∈ ( i − i + ) ( ) = ⎨ ⎪ In this expression m is the vector⎩⎪ index representing∈~ ( the− poloidal+ ) harmonic, while i and j are labels denoting radial location and poloidal harmonic. Here, ri represents the centre of the ith basis function, located at i∆r. These basis functions are continuous and have derivatives which are bounded and defined almost everywhere. Therefore, the integral expressions for matrix elements derived from the weak formulation of the problem can be evaluated in a straightforward manner. The boundaries of the inner region are chosen to coincide with mesh points. The regular basis functions defined above which have support on the inner region are replaced with a set of special basis functions which reflect the form of the solution near the pole rR. Terms of first-order in the singular solution and of up to second- order in the non-singular solution are represented in this way. Given rR is dependent on ω2, such a set of basis functions can accurately represent only one ω2. To ensure that they are continuous, but have minimal support, the special basis functions also have linear pieces on mesh intervals on either side of the inner region. First, consider the logarithmic first-order term in the singular solution of the wave equation near rR. Single valued logarithm functions are defined as in equation (3.98), letting x r rR. Thus a branch cut is taken which is perpendicular to and intersects with the real r-axis, starting at rR. This ensures that the function is continuous along a causal complex= − contour. A singular basis function is defined by

ln± r rR δm,j for r ra, rb (ra−r) ⎧ 1 ln± ra rR δm,j for r ra ∆r, ra Λ ⎪ ( −∆r ) ∈ ( ) (6.7) m,ln,j ⎪ (r−rb) ⎪ 1 ln± rb rR δm j for r rb, rb ∆r ⎪ − ∆r ( − ) , ∈ ( − ) = ⎨0 for r r ∆r, r ∆r . ⎪ − ( − ) ∈ ( a + b) ⎪ ⎪ The positive and negative⎩ signs correspond to negative∉ ( and− positive+ I) rR , respectively. Examples of such basis functions for both cases are plotted in Figure 6.1. ( ) These basis functions are differentiable for r 0, a ra ∆r, ra, R rR , rb, rb ∆r . At ra ∆r, ra, rb, and rb ∆r the derivative can be taken to be bounded as this quantity has ﬁnite limits approaching these values∈ ( from) ∖ above{ − and below.( ) The+ weak} − + formulation described in Section 6.1 excludes R rR from the integration domain.

The continuum resonance pole location, rR, is estimated based on the estimated 2 ( ) value of ω , using the continuum resonance condition Di,j 0. If the real component of the error in rR is signiﬁcant compared with the imaginary component of the Z Z = estimate, this can have a large erroneous contribution to integrals involving Λm,ln,j. In such cases both the error and basis functions near the estimated value of rR are very large, leading to underestimation of the coefﬁcient of Λm,ln,j in the solution and thus continuum damping. For this reason, when equation (6.1) is solved numerically 80 Singular Finite Element Calculation of Continuum Damping

(a)

(b)

Figure 6.1: Special basis functions Λln which are logarithmic over the inner region where r ra, rb for cases where (a) I rR 0 and (b) I rR 0. In each case I rR is finite, resulting in a continuous real component (blue, solid line) and discontinuous ∈ ( ) imaginary component( ) > (yellow, dashed( ) line).< ( ) using the Galerkin method, a finite value of δ is used. A region of the real axis − + rR, rR ra, rb is excluded from the path of integration. Effectively, in this case a set of weight functions are used which do not have support on that region, so that no penalty ⊂ ( is applied) to the error there. The remainder of the inner region must be sufficiently large to ensure that the near resonance solution is accurate. The first-order term in the non-singular solution is constant. Thus, the corresponding basis function is defined

δm,j for r ra, rb (ra−r) ⎧ 1 δm,j for r ra ∆r, ra Λ ⎪ ∆r ∈ ( ) (6.8) m,c,j ⎪ (r−rb) ⎪ 1 δm j for r rb, rb ∆r ⎪ − ∆r , ∈ ( − ) = ⎨0 for r r ∆r, r ∆r . ⎪ − ∈ ( a + b) ⎪ ⎪ ⎩⎪ ∉ ( − + ) §6.2 Singular basis functions 81

An example of this type of basis function is shown in Figure 6.2. The next term in the non-singular solution is linear. Thus, the corresponding basis function is deﬁned

r rR δm,j for r ra, rb (ra−r) ⎧ 1 ra rR δm,j for r ra ∆r, ra Λ ⎪( − ∆)r ∈ ( ) (6.9) m,l,j ⎪ (r−rb) ⎪ 1 rb rR δm j for r rb, rb ∆r ⎪ − ∆r ( − ) , ∈ ( − ) = ⎨0 for r r ∆r, r ∆r . ⎪ − ( − ) ∈ ( a + b) ⎪ ⎪ An example of this type⎩⎪ of basis function is shown∉ in( Figure− 6.3.+ Inclusion) of the second-order term ensures that the non-logarithmic component of the inner region solution is a polynomial of the same order as the piecewise polynomials which comprise the outer region solution. Thus, a potential reduction in accuracy as the width of the inner region is increased, which could occur if this component is represented as a constant, is mitigated. Moreover, the contribution to Ci from linear and logarithmic terms is of the same order. Continuum damping can be considered a function of Ci at the continuum resonance, as discussed in Section 4.2. The singular ﬁnite element method incorporating the basis functions described is implemented using a set of MATLAB scripts. Together these scripts deﬁne a mesh, generate basis functions, compute matrix elements in the discretised eigenvalue problem, solve this problem and plot the results. This implementation is described in Appendix B. In principle, this method could be extended to calculate continuum damping using the shear Alfvén wave equation in general geometry. Consider a set of basis functions which span the space of polynomials of order n which are continuous to order n 1 in their derivatives. To ensure that derivatives of the special basis functions are continuous to order n 1 on the outer region, they require piecewise polynomial− components with support over n mesh intervals either side of the inner −

Figure 6.2: Special basis function Λc which is constant over the inner region where r ra, rb .

∈ ( ) 82 Singular Finite Element Calculation of Continuum Damping

Figure 6.3: Special basis function Λl which is linear over the inner region where r ra, rb . In this case, rR R.

∈ ( ) ∈ region. In CKA, if the solution to equation (3.90) is not known a priori, such basis functions would need to be deﬁned for each combination of poloidal and toroidal B-splines used to deﬁne the regular basis functions. If the solution were known, this would determine poloidal and toroidal variation of the logarithmic component of the eigenfunction over the inner region, so that a single logarithmic basis function could be used.

6.3 Iterative solution procedure

Solutions of the continuum resonance condition Di,j 0 are dependent on the unknown mode eigenvalue, ω2. As discussed above, an error in the estimated pole location results in an error in ω2, which is dependentZ onZ = the width of the excluded + − region, rR rR. Therefore, the following iterative procedure is adopted in order to apply the singular ﬁnite element method: − 1. An initial estimate is found for the real component of ω2 using the ﬁnite element method without the special basis functions. This gives a valid approximation in cases where continuum damping is small in relation to the real component of mode frequency (I ω2 R ω2 ).

2 2 2. The location of the pole, rR, is estimated ≪ by solving ωR r ω for r. This equation may be solved either by solving the continuum resonance condition directly or considering the continuum modes calculated by( ) the= ﬁnite element method.

3. The causality condition is satisﬁed by adding a small imaginary component to ∂ωR rR. Based on the discussion in Section 3.4, the sign is opposite that of ∂r . R(rR) This sign can be determined by noting the sign of the change in r estimated R U in the previous step when ω2 is increased by a small amount. §6.4 Veriﬁcation 83

4. The ﬁnite element method is used with the special basis functions described in Section 6.2 to ﬁnd complex ω2 as a function of the width of the excluded region. Where this width is small, an increase in this parameter is expected to decrease the effect of the error in the estimated pole location. However, as the excluded region width approaches that of the inner region, the weight given to errors in the solution over the inner region will approach zero. Hence, this results in a negligible contribution from the logarithmic basis function and therefore an eigenvalue is computed with a negligible imaginary component.

5. An updated estimate for ω2 is found. Its value is taken to be that corresponding to the width of the excluded region for which the sensitivity of ω2 to that parameter is minimised.

6. The estimated pole location rR is recalculated. The real component is approximated similarly to Step 2, using ωR R rR R ω . The imaginary component is approximated based on a truncated Taylor series expansion about ( ( )) ≈ −1 ( ) R I I ∂ωR rR ω , which gives rR ω ∂r . R(rR)

7. The( previous( )) three steps( are) repeated≈ ( ) untilU sufﬁciently accurate values of ω2 and rR are obtained. As these estimates converge on the correct values, the sensitivity of ω2 to the width of the excluded region for small values of this width vanishes.

2 8. It must be demonstrated that ω and rR have converged with respect to the number of radial mesh points Nr and the width of the inner region, rb ra. This can be verified by showing that ω2 does not change significantly when each of these parameters is varied while all other parameters are held fixed. −

6.4 Veriﬁcation

The singular finite element method described above is verified by applying it to a TAE due to the coupling of the 1, 1 and 2, 1 Fourier harmonics. A tokamak with inverse aspect-ratio e 0.1 is considered. In this case, the rotational transform profile is chosen to be ( − ) ( − ) = 1 ι r . (6.10) r 2 q0 qa q0 a ( ) = Here q0 1.0 and qa 3.0, denoting the+ safety( − factor) at the magnetic axis and plasma edge respectively. The prescribed plasma density profile is = = r ρ ∆1 ρ r 0 1 tanh a . (6.11) 2 ∆ − 2 ( ) = − Here ∆1 0.7 and ∆2 0.05, representing the centre and width of the density fall-off respectively in r . These profiles and the resultant continuum resonance frequencies = a = 84 Singular Finite Element Calculation of Continuum Damping

are plotted in Figure 6.4. The TAE exists in the frequency gap due to the avoided crossing of the continuum branches associated with the 1, 1 and 2, 1 harmonics. Near the edge of the plasma the latter branch increases rapidly due to decreasing density, which results in a continuum resonance of the( TAE.− ) ( − )

1. 1.

0.75 0.75 0 Ω 0.5 0.5 ρ,ι

0.25 0.25

0. 0. 0.0 0.2 0.4 0.6 0.8 1.0 r

Figure 6.4: Normalised continuum resonance frequency (blue) as a function of the radial coordinate due to coupled 1, 1 and 2, 1 harmonics in a circular cross- section tokamak with inverse aspect-ratio e 0.1. The density profile (purple, solid line) and rotational transform( − (purple,) ( dashed− ) line) are also plotted. = The iterative procedure described above is performed for a set of basis functions defined by Nr 401 radial mesh points. An inner region of width rb ra 0.0125 2 (corresponding to 5 mesh intervals) is defined. On this narrow region ωR is approx- = ∂ZDi,jZ − = imately linear, indicating that ∂r is approximately constant. Thus, the truncated Frobenius expansion about rR is expected to be a good approximation to the solution there. The damping ratio ωi is plotted as a function of the width of the excluded re- ωr gion over 5 iterations in Figure 6.5. It is found that the damping ratio converges to a value independent of the excluded region width. The normalised complex frequency 1 + − is estimated to be Ω 0.326 0.00572i (for a rR rR 0.005), which corresponds to a damping ratio of ωi 0.0176. ωr= − − = Convergence with respect to inner region width and radial mesh resolution are = − checked for the calculated values of Ω and rR. The width of the excluded region 1 + − is kept constant, a rR rR 0.005. In the former case, the inner region width is 1 varied between rb ra 0.0075 (3 mesh intervals) and 0.0275 (11 mesh intervals) a − = while NR 401 with results plotted in Figure 6.6. The relative difference between the ( − ) = 1 computed damping ratio and that for a rb ra 0.0125 remains between 0.64% = 1 + − and 1.44%. This difference varies linearly with rR rR , with the most accurate ( − ) a= − estimate corresponding to the inner region width used to calculate rR. This variation results from higher order terms in the solution near the− continuum resonance, which are excluded based on truncation of the Frobenius expansion and the assumption that ωR r is linear over the inner region. The resulting error should disappear as the inner region width used in the iterative procedure approaches zero. Convergence with ( ) §6.4 Verification 85

0.010 ○ ○ ○ ○ ▯ ▯ ▯ ) ▯

r 0.001 △ △ /ω i △ △ 10 -4 ▽ Δ(ω ◇ ◇▽ ◇ ▼▼▽ ▽ -5 10 ◆◆◇ 0.000 0.002 0.004 0.006 0.008 0.010 1 + - (r R -r R ) a

Figure 6.5: Convergence of the damping ratio found using the singular finite element method over five iterations. The difference between the damping ratio computed via this method and a reference value determined using the complex contour method is 1 + − plotted against the normalised width of the excluded region, a rR rR . Following each iteration, the eigenvalue estimated using 1 r+ r− 0.01 is used to update the a R R − estimate for rR. The first, second, third, fourth, and fifth iterations are represented by the , , , , and markers respectively. Positive− = and negative values are 2 indicated by hollow and solid markers respectively. [ △ ▽ ◇ respect to the radial mesh resolution is checked by varying this parameter between 1 Nr 81 and 721 while a rb ra 0.0125. Results are plotted in Figure 6.7, with the relative difference between the computed damping ratio and that for Nr 401 mesh points= remaining between( −0.07%) = and 0.36% The complex contour method is used to verify the result of the singular= finite element method. Complex− contours are defined as

r x iαx x 1 (6.12) for complex parameter x 0, 1 and= + deformation( − ) parameter α. In Figure 6.8 it is shown that the computed damping ratio converges with Nr and α. Thus, the complex contour method yields∈ ( ) a complex frequency of Ω 0.326 0.00571i and hence damping ratio of ωi 0.0175. ωr The mode structures computed for the TAE using the singular= and− standard finite element methods are shown= − in Figure 6.9 and Figure 6.10 respectively. Far from the continuum resonance in the core region, the wave function components, Em,n , calculated using the two methods approximately agree. However, there is significant disagreement regarding this quantity near the continuum resonance and nearS theS edge of the plasma. The standard finite element method indicates the presence of a zero in E2,−1 near the continuum resonance which is absent in the mode calculated using the singular finite element method. 86 Singular Finite Element Calculation of Continuum Damping

● ●

-4

) 1.× 10 ○○ ● r -5 /ω 5.× 10 i Δ(ω

1.× 10 -5 ● 0.010 0.015 0.020 0.025 1 (r b -r a ) a

Figure 6.6: Convergence of the damping ratio computed by the singular ﬁnite ele- 1 ment method with normalised inner region width, a rb ra . The excluded region 1 + − is a rR rR 0.005 and the number of radial mesh points is Nr 401. Positive and negative values are indicated by hollow and solid( markers− ) respectively. − = = ● -5 5.× 10 ○● ) r -5

/ω 2.× 10 i

-5 ●

Δ(ω 1.× 10 ●

5.× 10 -6 ○ 0 100 200 300 400 500 600 700

N r

Figure 6.7: Convergence of the damping ratio computed by the singular ﬁnite element method with the number of radial mesh points, Nr. The excluded region width 1 + − 1 is a rR rR 0.005 and the inner region width is a rb ra 0.0125. Positive and negative values are indicated by hollow and solid markers respectively. − = ( − ) =

The singular ﬁnite element method indicates the variation in the phase of Em,n r with r. The phases of E1,−1 and E2,−1 are approximately constant for a 0.0, 0.5 and r r a 0.8, 1.0 . However, they change gradually for a 0.5, 0.7 , corresponding to the region between the toroidicity-induced gap and the continuum resonance,∈ ( and) more ∈ ( ) r ∈ ( ) rapidly close to the continuum resonance where a 0.7, 0.8 . The phase velocity is away from the continuum resonance for the E2,−1 harmonic and towards it for the ∈ ( ) E1,−1 harmonic. §6.4 Veriﬁcation 87

- 0.012 ○ - 0.014

r - 0.016

/ω ▯ ○ i △ △▯ △▯ △▯ ω - 0.018 ○

- 0.020 ○

- 0.022 100 200 300 400 500 600

N r

Figure 6.8: Convergence of the damping ratio as the number of radial points which comprise the mesh increases, Nr. The , , and markers correspond to integration paths deﬁned by α 0.01, 0.02, and 0.052 respectively. The damping ratio calculated using the singular ﬁnite element[ method△ is indicated by the black line. =

0.5

m,n 0 E

−0.5 m = 1, n = −1 m = 2, n = −1 −1 0 0.2 0.4 0.6 0.8 1 r a

Figure 6.9: Mode structure for a TAE due to coupling of the 1, 1 and 2, 1 harmonics computed using the singular ﬁnite element method. In this case, Nr 721 ( − 1) ( − ) radial mesh points were used with normalised inner region width a rb ra and 1 + − = normalised excluded region width a rR rR 0.005. This mode has normalised complex frequency Ω 0.3259 0.005731i and estimated continuum resonance( − ) pole rR − = location a 0.767504 0.000511i. Solid and dashed lines indicate real and imaginary components= respectively.− Units of Em,n are arbitrary. = −

It is noted that limr→0 E1,−1 0 based on the solution to equation (4.45) in Subsec- tion 4.4.2. This contradicts the assumptions made in the derivation of equation (6.1). ≠ 88 Singular Finite Element Calculation of Continuum Damping

0.2

0.1

−0.1 m,n

E −0.2

−0.3

−0.4 m = 1, n = −1 m = 2, n = −1

−0.5 0 0.2 0.4 0.6 0.8 1 r a

Figure 6.10: Mode structure for a TAE due to coupling of the 1, 1 and 2, 1 harmonics computed using a standard ﬁnite element method, with Nr 721 radial mesh points. The mode has a normalised frequency of Ω 0.3264.( − Units) of E( m −are) arbitrary. = =

However, the surface term at r 0 still vanishes as required because the leading order term is r. The mode structure obtained using the ﬁnite element method has the correct dependence on r for=r a. An exception occurs very close to the magnetic axis, over∝ the adjacent mesh interval, where the only non-zero basis function approaches zero as r. This can be shown≪ to have a negligible effect on the computed damping by applying the shooting method implementation of the complex contour method, which applies the initial conditions found in Subsection 4.4.2 at r 0. Using this method, the damping ratio is found to be ωi 0.0175. ωr = = − Chapter 7

Damping of Alfvén Eigenmodes due to Kinetic Effects

Damping of shear Alfvén eigenmodes due to kinetic effects resulting from thermal electron and ion populations is studied in this chapter. Radiative, electron Landau, and collisional damping are considered, as described in Subsection 3.5. These effects can be computed numerically, using an MHD code with kinetic corrections, such as CKA. A previous example of this approach, employing a different code, CASTOR-K, is that of Borba and Kerner [91]. In that example kinetic effects due to thermal electrons and ions are treated as a complex resistivity. Analytic models have been developed for radiative [49, 92], electron Landau [33, 93], and electron-ion collisional damping [77, 94] of shear Alfvén eigenmodes.

A TAE in a large aspect-ratio circular cross-section tokamak case is considered at low and high temperatures. In these cases electron Landau damping is much larger than, and comparable to, radiative damping respectively. The large aspect- ratio model is compared with the reduced MHD model for general geometry, considering kinetic effects to be localised at the avoided crossing. The former model is also compared with the large aspect-ratio model which includes variation in the kinetic effects throughout the plasma. It is found that accurately modelling the kinetic contribution to the shear Alfvén wave equation throughout the plasma is unnecessary to accurately compute damping due to kinetic effects. This damping is predominantly determined by the eigenfunction near its maximum, though this function is sensitive to the global geometry of the eigenvalue problem. Therefore, we conclude that the operator representing kinetic effects in CKA accurately represents damping due to kinetic effects. The temperature dependences of radiative and electron Landau damping are also studied. The results obtained from the numerical model are found to be consistent with the functional form in simple localised models of damping. Finally, damping of an NGAE in a realistic H-1NF equilibrium is considered, which is found to be large and predominantly due to electron-ion collisions.

89 90 Damping of Alfvén Eigenmodes due to Kinetic Effects

7.1 Calculation of collision frequency

While continuum and radiative damping are not sensitive to the value of νei, electron- ion damping can separately be a major source of damping of shear Alfvén eigenmodes. This can occur where νei is not negligible compared with k∥ve and thus the length scale of the mode parallel to the magnetic ﬁeld is not negligible compared with the electron mean free path. These conditions are most common for low temperatures near the edge of the plasma. Assuming Maxwellian distributions of electron and ion velocities, this parameter is given by [95, p.159]

2 4 qiqe 2π νei 3 ni ln Λe. (7.1) 3 4πe0 ¿m kT Á e e = ÀÁ In this equation ln Λe is the Coloumb logarithm,( equal) to

2 e0kTe 4πe0meve Λe (7.2) ¿ nq2 q q Á e i e = ÀÁ for electron-ion collisions. Electron-neutral collisionsS S should play the same role in the parallel electric ﬁeld model as electron-ion collisions. However, these can usually be ignored in fusion plasmas as the ionisation fraction is high in such plasmas and the cross-section for electron-neutral collisions is small compared with that of electron- ion collisions (σen σei).

≪ 7.2 ITPA benchmark TAE case

We study damping due to kinetic effects by considering a TAE in a circular cross- section tokamak with low magnetic shear. This mode is based on that considered in an ITPA (International Tokamak Physics Activity) benchmark study [96], with the plasma parameters summarised in Table 7.1. The mode arises due to the avoided crossing of continuum branches associated with the 10, 6 and 11, 6 harmonics, shown in Figure 7.1. Rotational transform is specified as ( − ) ( − ) 1 ι s . (7.3) 1.71 0.16s ( ) = The ion species is considered to be hydrogen.+ Unlike the case in the reference provided, a uniform density is specified here, avoiding resonant interaction with the continuum. We also consider the modified case where Ti Te 10 keV, where increased coupling to kinetic Alfvén waves is expected. = = An electron-ion collision frequency of νei 230 Hz is assumed, which essentially limits dissipation to that resulting from Landau damping due to the electron velocity space distribution. This is much smaller= than the value computed using equation (7.1), νei 44.2 kHz. However, the substitution does not significantly impact kinetic effects as νei k∥ve. From equation (3.106) it is observed that in this limit the = ≪ §7.2 ITPA benchmark TAE case 91

R0 10.0 m a 1.0 m B0 3.0 T Ti 1.0 keV Te 1.0 keV −27 mi 1.67 10 kg 19 −3 ni0 2.0 10 m × 19 −3 ne0 2.0 10 m × νei 230 Hz × Table 7.1: Parameters for TAE calculation in ITPA benchmark case.

0.8 0.8

0.6 0.6 0 0.4 0.4 ι Ω

0.2 0.2

0. 0. 0.0 0.2 0.4 0.6 0.8 1.0

Figure 7.1: Normalised continuum resonance frequency (blue) as a function of s due to coupled 10, 6 and 11, 6 harmonics in the ITPA benchmark case. The√ rotational transform is also plotted (purple). ( − ) ( − )

3 2 2 −6 2 real and imaginary parts of 4 ρi 1 iδ ρs 1.27 1.42i 10 m are independent of νei. The mode structure of the+ TAE( − computed) = ( using− the) × implementation of equation 7.2 in CKA for each choice of temperature is shown in Figure 7.2. For the lower temperature case Ω 0.282 2.47 10−4i and hence ωi 8.77 10−4. The higher tem- ωr perature case features a small upward frequency shift and increased damping with Ω 0.284 1.27 10−=3i and− hence×ωi 4.45 10−3. In= this− case× oscillations on short ωr radial length scales indicate coupling to kinetic Alfvén waves. These waves are found to= propagate− away× from the peak with= amplitude− × that decays rapidly with radial position, so that negligible reﬂection is expected at the plasma edge. Such oscillations are not apparent in the lower temperature case. In this case there is weaker coupling to kinetic Alfvén waves, which have a shorter radial length scale and are subject to a larger wave dissipation parameter, δ. For the above calculations, the wave dissipation parameter, δ, is computed based on the value of Ω calculated using the large aspect-ratio approximation without the kinetic correction. Using the values of Ω obtained from CKA to compute δ results in changes of just 1.9% and 2.3% to its value respectively. Therefore, additional 92 Damping of Alfvén Eigenmodes due to Kinetic Effects

1.2 − m = 10, n = 6 1.2 m = 11, n = −6 m = 10, n = −6 1 − − m = 9, n = 6 1 m = 11, n = 6 m = 12, n = −6 m = 9, n = −6 0.8 − 0.8 m = 12, n = 6

0.6 0.6 m,n m,n Φ 0.4 Φ 0.4

0.2 0.2

0 0

−0.2 −0.2 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 √s √s (a) (b)

Figure 7.2: The largest Fourier harmonics of the ideal MHD TAE eigenfunction computed using the reduced MHD model of CKA for (a) Ti Te 1 keV and (b) Ti Te 10 keV. Real and imaginary components are represented by solid and dashed lines respectively. = = = = iterations recomputing Ω and δ are deemed unnecessary. Ignoring the real part of the kinetic term in equation (3.100) results in a decrease of 8.33% in damping ratio for the low temperature case and a decrease of 38.9% for the high temperature case. Thus, it is concluded that radiative damping is a minor effect compared with Landau damping of the ideal MHD shear Alfvén eigenmode in the low temperature case. However, in the high temperature case, the two sources of damping are comparable.

7.3 Effect of higher order geometric terms

The inﬂuence of higher order geometric and pressure derivative terms on damping due to kinetic effects is explored through comparison of results of the large aspect- ratio expansion model implemented in the MATLAB ﬁnite element script with those of the reduced MHD model of CKA. As ρi and ρs are typically much smaller than the length scales on which equilibrium quantities vary, kinetic terms in the shear Alfvén wave equation can usually be approximated based on their form in slab geometry. However, the resulting damping is highly dependent on the structure of the underlying ideal MHD mode, which is sensitive to magnetic geometry. The large aspect-ratio tokamak model in equation (3.103) in the low temperature case yields Ω 0.276 4.43 10−4i, with ωi 1.61 10−3. In the high temperature ωr case the model yields Ω 0.279 2.76 10−3i, with ωi 9.88 10−3. The damping = − × = − ×ωr ratios increase by factors of 1.8 and 2.2 respectively compared with the reduced MHD model in full geometry.= − × = − × The mode structures which∼ are computed∼ in the ideal MHD limit using the two methods are shown in Figure 7.3. Higher order terms excluded from the large aspect- ratio expansion increase the frequency of the TAE relative to that of the continuum, §7.3 Effect of higher order geometric terms 93

0.2 0.6 m = 10, n = −6 m = 10, n = −6 m = 11, n = −6 m = 11, n = −6 0.5 − 0.15 m = 9, n = 6 m = 12, n = −6 0.4

0.1 0.3 m,n m,n

Φ Φ 0.2 0.05

0.1 0 0

−0.05 −0.1 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 r a √s (a) (b)

Figure 7.3: The largest Fourier harmonics of the ideal MHD TAE eigenfunction computed (a) using the large aspect-ratio approximation and (b) the reduced MHD model of CKA. broadening the mode. In the absence of kinetic effects, the large aspect-ratio approximation for a circular cross-section tokamak results in a TAE such that ω−ωl 0.153 ωu−ωl where ωl and ωu are respectively the lower and upper bounds of the continuum gap. By contrast, the TAE computed using CKA is such that ω−ωl 0.363 and= the ωu−ωl radial width of the eigenmode is a factor of 1.3 greater. The change in continuum frequencies is small ( 1% for the upper and lower bounds of the gap).= ∼ Kinetic effects can be considered as a perturbation for a narrow mode, provided < that coupling to kinetic Alfvén waves is negligible. In equation (3.100) and equation (3.103) the imaginary non-ideal component of the operator TI contains radial derivatives of up to fourth-order whereas the real ideal component of the operator contains radial derivatives of up to second-order. This difference in order indicates that damping is proportional to σ−2, where σ is the radial length scale of the mode. This accounts for a factor of 1.7 difference in damping due to kinetic effects. Eliminating the pressure gradient and parallel current terms in equation (3.100) ∼ has a negligible effect on the computed frequency of the ideal MHD TAE and continuum modes ( 0.1%). Thus, the frequency shift and associated broadening of the TAE described above must be due to neglected higher order terms in the large aspect- ratio equilibrium< expansion. Terms of order e which are neglected in the derivation of equation (3.78) nevertheless have a signiﬁcant effect on damping due to kinetic effects via the mode structure of the ideal MHD TAE. A simpler estimate of damping can be obtained using the dispersion relation for shear Alfvén waves including kinetic effects on a ﬂux surface [80]

3 ω2 k2 v2 1 ρ2 1 iδ ρ2 k2 , (7.4) ∥ A 4 i s ⊥ = + + ( − ) where k⊥ is the perpendicular wave number. This relation is applied where the ideal 2 MHD eigenfunction has its maximum. Here the estimate k2 1 ∂ Em,n is used, ⊥ Em,n ∂r2 = − 94 Damping of Alfvén Eigenmodes due to Kinetic Effects

based on the radial component at the gap and neglecting the poloidal component k m . In the low temperature case, this yields Ω 0.276 3.84 10−4i, with ωi pol r ωr 1.39 10−3. In the high temperature case the estimate obtained is Ω 0.280 1.23 ≈ ω = − × = 10−3i, with i 4.38 10−3. The difference in accuracy between the two cases can be ωr −ascribed× to the different sources of damping in each case. In the low temperature= − case× damping is highly= − localised,× while in the high temperature case non-local dissipation due to radiative damping becomes signiﬁcant.

7.4 Effect of E model

The kinetic operator∥ used above is simplified by applying the parallel electric field model at the avoided crossing of the continuum branches, as discussed in Section 3.5. The term due to non-zero parallel electric field is effectively obtained assuming that each Fourier harmonic has constant k∥ equal to that at the avoided crossing. In the TAE case examined here, this approximation means that significant variation in k∥ 6 throughout the plasma is ignored, including a resonant surface where ι 11 . The validity of this approximation is investigated by comparing solutions of the eigenvalue = problems in equation (3.103) and equation (3.104), which retains variation in k∥. In the low temperature case, equation (3.104) yields a normalised TAE frequency of Ω 0.276 4.36 10−4i, so that ωi 1.58 10−3. This agrees closely with the ωr estimate from equation (3.103), which is consistent with highly localised dissipation near the= avoided− crossing× of the continuum= − branches.× In the high temperature case, equation (3.104) yields Ω 0.279 2.80 10−3i and thus ωi 1.00 10−2. Again this ωr concurs with the estimate from equation (3.103), and it is concluded that in this case coupling between shear and= kinetic− Alfvén× waves is highly localised= − × near the avoided crossing. Therefore, the approximations underlying equation (3.103) remain valid.

7.5 Temperature dependence

The relationship between the plasma temperature and damping due to kinetic effects is studied using the more general parallel electric ﬁeld model of equation (3.104). It is assumed that ions and electrons continue to be in thermal equilibrium as temperature is varied (T Ti Te). Mett and Mahajan [49] derive expressions for damping of TAEs due to kinetic effects at small T which imply ωi Tδ T and at large T which imply ωr ω 1= = that i T 3 . Expressions for collisional and Landau damping are derived by Candy ωr ∝ ( ) and Rosenbluth, their expression in the small me limit for a TAE due to a single gap ∝ ω ω 1 implies that i depends linearly on Tδ T for small T and i T 2 for large T [94]. ωr ωr By contrast, Berk et al. indicate that for radiative damping ωi exp √α where α ( ) ωr ∝ T is a positive valued constant and for collisional and Landau damping ωi Tδ T ∝ − ωr [92]. The component of damping due to dissipative effects acting on the unperturbed∝ ( ) mode can be estimated by replacing the operator which represents kinetic effects §7.5 Temperature dependence 95

in equation (3.104) with its imaginary component. This model is then equivalent to resistive MHD where resistivity is a real-valued function. Damping consists of both electron Landau and electron-ion collisional components. The latter takes a low arbitrary value based on the choice of νei. The difference in damping between the previous model and this one is attributed to propagation of kinetic Alfvén waves in the former and therefore described as radiative damping. Results of the numerical model are plotted in Figure 7.4. At high temperatures the electron Landau damping is observed to have a power law dependence, with ω 1 i T 2 . At lower temperatures electron Landau damping decreases rapidly and ωr the damping computed becomes a function of the small, arbitrarily chosen νei. The transition∝ between the two regimes occurs at approximately T 0.6 keV, coinciding with k∥ve ω at the avoided crossing. Below this temperature few electrons have = sufﬁcient v∥ to interact with the wave. In the= high temperature limit, radiative damping approaches the relationship ωi T. Thus, this damping grows faster than electron Landau damping and is ωr dominant above T 7 keV. A linear relationship between radiative damping and T, with∝ a small postive intercept on the T axis, extends to lower temperatures. Below T 0.4 keV radiative≈ damping is negligible.

≈ 0.010

0.001 r

/ω -4 i 10 -ω 10 -5

10 -6

0.01 0.10 1 10 T(keV)

Figure 7.4: Damping ratio as a function of T Ti Te. Black solid lines represent total damping, blue short dashed lines represent damping composed of electron collisional and Landau damping, and red long dashed= = lines represent represent radiative damping. Thick lines represent results of the numerical model and thin lines represent results of the analytical models.

As noted in Section 7.3, the local dispersion relation provides a good approximation to electron Landau damping. This implies that ωi Tδ T . In the limit where ωr ω k∥ve and νei k∥ve this equation can be combined with equation 3.106 to give the approximation ∝ ( ) ≪ ≪ 2 ωi k⊥ mekTe π 2 . (7.5) ωr µ0√nivAqi ½ 8 = − 96 Damping of Alfvén Eigenmodes due to Kinetic Effects

Once again kinetic effects are treated as a small pertubation acting on the ideal MHD solution and thus k⊥ is obtained from the eigenfunction in that case. The formula in equation (7.5) is then plotted in Figure 7.4, and is found to accurately compute electron Landau damping in the high temperature limit. A different approach is required in order to estimate the radiative component of the damping analytically, as the strong damping of the kinetic Alfvén waves cannot be treated perturbatively. As pressure is negligible, forces on the plasma are primarily electromagnetic and Poynting’s theorem can be applied. Treating E∥ as a small perturbation in a one-dimensional plasma slab yields the time-averaged Poynting vector S with components ∂Φ 2 k∥ S ∂r (7.6) ∥ 2ωµ T 0T and = ∂Φ∗ k∥Φ ∂r E∥ S⊥ . (7.7) 2ωµ0 Φ The damping rate is = −

ω W˙ i (7.8) ωr 2Wωr S v = − ⊥ A (7.9) 2ωr drS∥ = − where W˙ is the rate at which energy is lost to∫ the kinetic Alfvén wave and W is the energy of the mode. The integral in the denominator is evaluated assuming that the mode has a Gaussian radial profile. In the numerator S⊥ is evaluated based on E∥ Φ from the parallel electric field model used in the derivation of the kinetic terms [76] and polarisation density correction due to finite ion gyroradius effects [64]. It is assumed that coupling between the shear and kinetic Alfvén waves is limited to a narrow region near the avoided crossing. The damping ratio thus obtained is

2 ωi k⊥ 3 2 2 ρi ρs (7.10) ωr π 4 = − m kk 2 +3 i ⊥ T T . (7.11) 2 2 i e πqi B 4 = − + The temperature dependence of this simple model is less accurate than that of the electron Landau damping model. At low temperatures the real part of the factor representing parallel electric ﬁeld effects ceases to be constant (as indicated by equation (3.106) and assumed in the simple model) and instead becomes inversely proportional to T (from equation (3.107)). The sign of the real part of the kinetic operator changes where ω 1.209 (corresponding to T 0.385keV). At lower k∥ve temperatures kinetic Alfvén waves are conﬁned by an effective potential well rather ≈ ≈ than propagating outwards, so radiative damping becomes negligible. Consequently, §7.6 Heliac NGAE case 97

while radiative damping obeys a linear relationship with T, a more accurate model includes the observed small positive intercept on the T axis.

7.6 Heliac NGAE case

Here we consider an NGAE in an H-1NF heliac plasma arising due to a maximum of the 4, 5 branch of the shear Alfvén continuum. The device has Nf p 3 toroidal field periods with a variable magnetic geometry which is determined by the ratio of current( in− its) helical winding to those in other field coils, described by the= parameter κh [97]. A configuration is considered which has previously been modeled using the ideal MHD code CAS3D, with results compared to experimental magnetic probe data [98]. While the plasma is strongly shaped, the mode is dominated by a single Fourier component, namely the 4, 5 harmonic. The temperature in H-1NF is much lower than the previous case while the minor radius is smaller, indicating that the parallel electric field component( resulting− ) from electron-ion collisions is likely to be important. We consider the mode with kinetic effects included using CKA to implement equation (3.100). The plasma conditions are summarised in Table 7.2.

R0 1.02 m a 0.173 m B0 0.478 T κh 0.33 Ti 20 eV Te 20 eV −27 mi 4.99 10 kg 18 −3 ne0 2.2 10 m × Table 7.2: Parameters for× H-1NF NGAE calculation.

Assume that the component of the magnetic field line curvature parallel to the κ⋅b×∇ψ −1 flux surface, κs S∇ψS , is 2 m , representing a typical value for H-1NF obtained from the GIST (Geometry Interface for Stellarators and Tokamaks) code [99]. Hence, =2 −3 −2 the quantity 4βκs 2 10 m , which appears in expression (3.91), is not negli- 2 −3 −2 gible compared with k∥ 6 10 m for the mode. Therefore, the expressions in Section 3.3.2 indicate∼ that× the shear Alfvén continuum could be shifted upwards significantly by interaction with∼ × ion acoustic waves. We therefore consider the effect of compressibility using the slow sound approximation correction in expression (3.51). The vacuum magnetic field of H-1NF has been calculated previously as a function of the current ratio κ and is taken to be an accurate indication of field in H-1NF plasmas due to their very low β [98]. For κ 0.33, the rotational transform profile is

ι s 1.228945 0.000978s 0.048458=s2 0.014175s3 0.008568s4. (7.12)

Based on( measurements) = + of ne over+ a range of− heating powers,+ the electron density 98 Damping of Alfvén Eigenmodes due to Kinetic Effects

x 10−3 6 1.5

5 1.25

NGAE 4 1 ι , 0 e 2 3 NGAE 0.75 Ω /N e

2 0.5 N

1 0.25

0 0 0 0.2 0.4 0.6 0.8 1 √s

(a) (b)

Figure 7.5: (a) Normalised shear Alfvén continuum resonance eigenvalues Ω computed using CKA as a function of the radial coordinate, s ( ). Continuum modes corresponding to the branch shown are dominated by the√ 4, 5 harmonic. The real component of Ω ( ) for modes computed with the kinetic correction[ is also plotted against the value of s where they have their maximum.( NGAE− ) modes computed with and without[ the√ kinetic correction are indicated. Normalised electron density, ne (solid line), and rotational transform, ι (dashed line), are also plotted. (b) The ne0 outermost flux surface for this H-1NF configuration. profile is assumed to be [100]

ne s ne0 1 s . (7.13)

Langmuir probe measurements indicate( ) = that( − typically) Te 20 eV throughout the plasma. Consequently, the uniform temperature profile Ti Te 20 eV is assumed ≈ and the equilibrium pressure profile is calculated via the ideal gas law, p nekTe = = nikTi. The resulting continuous spectrum is plotted in Figure 7.5. = + An effective nuclear charge of Ze f f 1 is assumed based on the relatively low temperature, so that quasi-neutrality implies that the ion density profile matches that of the electrons. The plasma is assumed= to be composed of a 1 2 mix of hydrogen and helium. These quantities are only known approximately for H-1NF discharges, due to impurities in the plasma and incomplete ionisation. Dissipation∶ due to electron-ion collisions for ωI ωR and δ 1 is proportional to the parameter 2 2 δρs . Where νei k∥vA, δρs miZe f f approximately for fixed ne and Te, so that ≪ ≫ inclusion of additional heavy ion» impurities will increase the estimated damping. ≫ ∝ For simplicity, the collision frequency is assumed to be constant with its value calculated based on electron and ion densities at the magnetic axis. This approximation can be made based on the proximity of the maximum of the NGAE eigenfunction to the magnetic axis. Nevertheless, the approximation will increase the computed damping of the mode. If the fluid is taken to include additional neutral particles while maintaining the same number of ions and electrons, its density will increase, reducing vA. The colli- §7.6 Heliac NGAE case 99

sion frequency is not expected to be significantly affected by neutral particles, as the −20 2 cross section of electron-neutral collisions (σen 3 10 m ) is much smaller than −19 2 2 1 that of electron-ion collisions (σei 1.6 10 m ). Hence, it is found that δρs √ ∼ × ρ for fixed ne and Te while density, ρ, changes due to the addition of neutral particles, ∼ × ∝ so that the presence of such particles would be expected to reduce the damping of the mode. Nevertheless, this is not expected to be a large effect, as the neutral particle density in H-1NF is smaller than the ion density. Initial estimates of ω and k∥ are obtained from the ideal MHD equation for shear Alfvén waves in an incompressible periodic cylindrical plasma. The MATLAB finite element script, described in Appendix B, is used to solve equation (3.74) for the 4, 5 harmonic without coupling to other harmonics. An anti-Sturmian sequence of NGAEs is found to be clustered above the maximum in the continuum at (s −0.33.) The mode with the lowest number of nodes is considered, which has normalised√ frequency Ω 0.0870. For the given plasma parameters, it is found ≈ that νei 954 kHz, assuming complete ionisation. By comparison, k∥ve 211 kHz, indicating that electron collisional= damping is the dominant component of the kinetic term≈ in equation (3.100) and thus real components of the term can≈ be neglected. This is confirmed when the factor representing parallel electric field effects in equation (3.104) is computed at the location of the maximum, resulting in 3 2 2 −5 2 4 ρi 1 iδ ρs 0.936 4.21i 10 m . Note that the real component of this quantity is negative, implying that the direction in which kinetic Alfvén waves propagate+ ( is reversed− ) ≈ with(− respect− to that) × described in Section 3.5. Therefore, these waves do not propagate outwards from a mode localised at a continuum maximum. Thus, radiative damping will be negligible and the relatively small real component of the operator representing non-ideal effects is consequently ignored. Equation (3.100) with the slow sound approximation correction in equation (3.51) is solved in three-dimensions using CKA with Ns 80, Nϑ 40, and Nϕ 20. It is found that δ in equation (3.100) changes significantly when recomputed using the resulting complex ω. The estimate of ω changes due= to the inclusion= of both accurate= magnetic geometry and dissipative effects. Therefore, it is necessary to solve the eigenvalue problem iteratively for δ and ω. Values for these quantities converge quickly, δ changes by just 0.0055% based on the change in ω between the third and fourth iterations. The mode is found to− have normalised complex frequency Ω 0.0611 6.51 10−3i, corresponding to a damping ratio of ωi 0.107. This corresponds to a com- ωr plex frequency of 41.4 4.41i kHz. The result is well converged with respect= to− radial,× = − poloidal, and toroidal resolution. Decreases in resolution to Ns 70, Nϑ 30, and N 15 result in changes− in ωi of just 0.0060%, 0.17%, and 0.14% respectively. ϕ ωr The influence that the complicated three-dimensional geometry= and finite= pressure= effects have on damping due to kinetic− effects can be determined through comparison with a simplified model. Inclusion of kinetic effects in the incompressible periodic cylindrical plasma model yields Ω 0.0846 6.77 10−3i and therefore a damping ratio of ωi 0.0801 (following three iterations). It is evident that the com- ωr bined effects cause a pronounced change in= the mode.− To× determine which effect = − 100 Damping of Alfvén Eigenmodes due to Kinetic Effects

1.2 − m = 4, n = −5 m = 4, n = 5 m = 2, n = 1 1.5 m = 2, n = 1 1 m = 2, n = −2 m = 2, n = −2 m = 6, n = −11 m = 6, n = −11 0.8 1 0.6 m,n m,n Φ Φ 0.5 0.4

0.2 0 0

−0.5 −0.2 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 √s √s (a) (b)

Figure 7.6: The largest Fourier harmonics of the NGAE eigenfunction (a) including and (b) excluding the effect of electron-ion collisions. Solid and dashed lines indicate real and imaginary components respectively. is greater, equation (3.100) is again solved in three-dimensions using CKA, this time without the pressure gradient and slow sound approximation terms. This calculation results in Ω 0.0570 6.16 10−3i and hence ωi 0.108. Thus, the pressure gradient ωr and coupling to the ion acoustic wave have a modest effect on frequency, though do not appreciably= affect− damping× ratio. = − The mode structure of the NGAE with and without kinetic effects is shown in Figure 7.6. Inclusion of electron-ion collisions signiﬁcantly broadens the NGAE and shifts its peak towards the edge of the plasma. This effect also reduces the real component of the frequency. Thus NGAEs modelled including kinetic effects are closer to experimental observations of global modes in H-1NF than those modelled using ideal MHD [98]. It is also noted that the predominantly 4, 5 global mode in H-1NF is observed to decay very rapidly when ion cyclotron resonance heating is turned off (with time scale 0.1 ms) [101], consistent with the numerical( − ) result. For non-zero δ, the shear Alfvén continuum is replaced by a small number of discrete non-singular modes,∼ as indicated in Figure 7.5. The real frequency component and location of maxima approximately match ideal MHD continuum resonances for a sequence of modes. While these modes do not have nodes, they resemble an anti-Sturmian sequence in that the change in complex argument increases as Ω decreases. However, as δ is dependent on ω, the value used in the computation is valid only for the mode discussed above. To obtain accurate frequencies, otherS modesS would need to be computed individually using the same iterative approach. Chapter 8

Continuum Damping of Alfvén Eigenmodes in Large Aspect-Ratio Tokamaks

This chapter describes examples of the calculation of continuum damping of a TAE in a large aspect-ratio circular cross-section tokamak. The validity of the perturbative approach described in Chapter 4 is assessed by comparing the resultant values with those of the complex contour approach described in Chapter 5. The accuracy of continuum damping values computed using the latter method have already been well veriﬁed by comparison with resistive MHD in the limit where η 0, with agreement demonstrated to within machine precision [46]. Importantly, prior to this work there had been no direct comparison between results of the perturbative→ technique and accepted numerical approaches. Damping is computed using the two methods for a range of different equilibrium parameters. The dependence of continuum damping on these quantities can be ascribed to variation in the ﬂux-like quantity Cm,n and conditions at the continuum resonance. It is found that the two methods of calculating continuum damping agree qualitatively regarding dependence on equilibrium quantities, however, they differ quantitatively. This discrepancy can be explained through the change in the eigenfunction when the discontinuity due to a continuum resonance is introduced.

8.1 Large aspect-ratio circular cross-section tokamak TAE case

Here we study a TAE due to coupling of the avoided crossing of continuum branches associated with the 2, 2 and 3, 2 harmonics in a tokamak with circular cross- section. For this case the inverse aspect ratio is e 0.1. Density and rotational transform profile have( − the) forms( specified− ) in equation (6.10) and equation (6.11) = respectively. The parameters in these profiles are chosen to be q0 1.05, qa 1.6, ∆1 0.8 and ∆2 0.1. The TAE is analysed by solving the wave equation obtained from the large aspect-ratio approximation, equation (3.78). This case= is analysed= in the= work of Könies= and Kleiber [46]. However, these authors use a somewhat ω2 different model to that used here, avoiding the approximation km,nkm±µ,n±ν 2 . v¯A ≈ − 101 102 Continuum Damping of Alfvén Eigenmodes in Large Aspect-Ratio Tokamaks

Density and rotational transform proﬁles are shown along with the resulting continuum resonance frequencies in Figure 8.1. The TAE is found in the frequency gap due to the avoided crossing of the continuum branches associated with the 2, 2 and 3, 2 harmonics, which has a continuum resonance with the latter near the edge which results from the fall-off in density there. Calculation of the frequency( − ) and( damping− ) of the TAE is performed by solving the coupled ﬁrst-order differential equations (equations (3.83) and (3.84)) using the shooting method as described in Subsection 4.4.2. Applying equation (4.23) results in a complex normalised frequency estimate of Ω 0.387 1.99 10−3i and a continuum damping ratio of ωi 5.15 10−3. Solving the coupled equations over a complex integration path as ωr = − × described in Subsection (5.3.2) yields complex normalised Ω 0.388 1.06 10−3i and = − ω × hence i 2.72 10−3. The complex contour chosen in this case is of the form ωr = − × = − × r x a x iαx x 1 (8.1) with α 0.05 and contour parameter( ) = x ( +0, 1 . ( − ))

= ∈ ( ) 1. 1.

0.75 0.75 0 Ω 0.5 0.5 ρ,ι

0.25 0.25

0. 0. 0.0 0.2 0.4 0.6 0.8 1.0 r

Figure 8.1: Normalised continuum resonance frequency (blue) as a function of the radial coordinate due to coupled 2, 2 and 3, 2 harmonics in a circular tokamak with inverse aspect-ratio e 0.1. The density proﬁle (purple, solid line) and rotational transform (purple,( − dashed) ( line)− are) also plotted. =

8.2 Variation of equilibrium quantities

The continuum damping values calculated using the perturbative and complex contour approaches are compared for a range of different equilibrium parameters. For each of qa, e, ∆1, and ∆2 (viz. edge safety factor, inverse aspect-ratio, density fall- off location, and density fall-off width respectively) a scan is performed, varying that parameter whilst holding the other parameters constant and equal to the values used in Section 8.1. Values of ω and ωi obtained from each method are plotted against r ωr varying qa (Figure 8.2), e (Figure 8.3), ∆1 (Figure 8.4), and ∆2 (Figure 8.5). Dur- ing these scans the parameter α is adjusted so that the deformation of the complex §8.2 Variation of equilibrium quantities 103

0.388

0.387

0.386 r Ω 0.385

0.384

0.383 1.6 1.8 2.0 2.2 2.4 2.6

q a (a)

0.000

- 0.001

- 0.002 r - 0.003 /ω i

ω - 0.004

- 0.005

- 0.006

1.6 1.8 2.0 2.2 2.4 2.6

q a (b)

Figure 8.2: (a) Real normalised frequency component (Ωr) and (b) damping ratio ( ωi ) as functions of q , calculated using the perturbative method (solid line) and ωr a complex contour method (dashed line). Other equilibrium parameters are ﬁxed; e 0.1, ∆1 0.8, and ∆2 0.1.

= = = contour away from the real axis is proportional to the damping rate. This ensures that the continuum resonance pole is circumvented in accordance with the causality condition, whilst any additional poles are excluded from the calculation. The relationship between ωi computed using the perturbative and complex con- ωr tour methods is plotted in Figure 8.6. Clearly, the damping computed using the perturbative method is signiﬁcantly greater than that found using the complex contour method for the range of parameters examined. For scans in qa, ∆1 and ∆2, this discrepancy is greatest where continuum damping is large. Although disagreement between the two damping estimates is reduced as these estimates decrease, a significant discrepancy remains even for very small values of damping. By contrast, the damping values calculated using the perturbative and complex contour techniques for the scan in e differ by an approximately constant factor. There is approximate agreement between the two approaches regarding the values of the parameters for which damping is increasing and decreasing. Moreover, for scans in qa, e, and ∆1 the relationship between the two estimates is approxi- 104 Continuum Damping of Alfvén Eigenmodes in Large Aspect-Ratio Tokamaks

0.395

0.390

0.385 r Ω 0.380

0.375

0.05 0.10 0.15 0.20 0.25 ϵ (a)

0.000

- 0.005 r /ω i

ω - 0.010

- 0.015

0.05 0.10 0.15 0.20 0.25 ϵ (b)

Figure 8.3: (a) Real normalised frequency component (Ω ) and (b) damping ratio ( ωi ) r ωr as functions of e, calculated using the perturbative method (solid line) and complex contour method (dashed line). Other equilibrium parameters are fixed; qa 1.6, ∆1 0.8, and ∆2 0.1. = = = mately linear. Hence, the perturbative formula for continuum damping indicates the qualitative dependence of damping on various factors but does not give accurate quantitative agreement with the proven numerical technique. Terms of higher order in the perturbative expansion give a contribution to the imaginary frequency component which is comparable to the first-order term that constitutes the estimate of the perturbative technique. Additionally, the frequency of the unperturbed eigenmode is found to be consistently smaller than the real frequency component computed using the complex contour method. The difference between the real frequency components calculated in each case is of similar magnitude to that between the imaginary frequency components. Moreover, the difference in the real components is approximately proportional to the imaginary frequency component estimated using the perturbative method for the range of parameters examined. Thus, terms of higher order with respect to the perturbation, which were excluded in the analysis in Chapter 4, are of comparable magnitude to the purely imaginary first-order term which was retained. However, §8.2 Variation of equilibrium quantities 105

0.420 0.415 0.410

r 0.405 Ω 0.400 0.395 0.390

0.70 0.72 0.74 0.76 0.78 0.80

Δ1 (a)

0.000

- 0.001

r - 0.002 /ω i - 0.003 ω

- 0.004

- 0.005

0.70 0.72 0.74 0.76 0.78 0.80

Δ1 (b)

Figure 8.4: (a) Real normalised frequency component (Ω ) and (b) damping ratio ( ωi ) r ωr as functions of ∆1, calculated using the perturbative method (solid line) and complex contour method (dashed line). Other equilibrium parameters are fixed; qa 1.6, e 0.1, and ∆2 0.1. = = = the difference between estimates of the real frequency component is substantially smaller than their variation over the range of qa, e, ∆1, and ∆2 considered here. Using equation (4.23), the dependence of continuum damping on qa can be understood to occur due to the resulting variation in the eigenfunction. As qa increases, the location of the continuum resonance moves towards the magnetic axis. Consequently, at that location, the density increases monotonically while the radial gradient in continuum resonance frequency decreases monotonically. These competing effects act to decrease and increase damping respectively, as discussed in Section 4.2. Also, the flux-like quantity components C2,−2 rr and C3,−2 rr vary with qa. If the wave function component E3,−2 is larger in the region where the density fall-off occurs ( ) ( ) the magnitude of C3,−2 decreases faster. This effect becomes smaller as the avoided crossing moves towards the magnetic axis with increasing qa, tending to increase damping. However, the peak in C2,−2 and C3,−2 at the avoided crossing decreases as the corresponding flux surface moves inwards with increasing qa, tending to decrease damping. These effects combine such that, using either method, damping is found 106 Continuum Damping of Alfvén Eigenmodes in Large Aspect-Ratio Tokamaks

0.400

0.395 r Ω 0.390

0.385

0.06 0.08 0.10 0.12 0.14 0.16

Δ 2 (a)

0.000

- 0.002 r /ω

i - 0.004 ω

- 0.006

0.06 0.08 0.10 0.12 0.14 0.16

Δ 2 (b)

Figure 8.5: (a) Real normalised frequency component (Ω ) and (b) damping ratio ( ωi ) r ωr as functions of ∆2, calculated using the perturbative method (solid line) and complex contour method (dashed line). Other equilibrium parameters are fixed; qa 1.6, e 0.1, and ∆1 0.8. = = = to be greatest where qa 1.65. The case in which e is varied is substantially less complicated. The linear dependence of damping on this≈ parameter can be explained using equation (4.23). It is found that the eigenfunction Em,n does not vary significantly with increasing e except near the avoided crossing, where it becomes more sharply peaked. Furthermore, the continuum resonance frequencies are not significantly changed by increasing e, except near the avoided crossing where the gap between upper and lower branches −1 becomes broader. The change in Em at a due to a small change in ω scales with e for small e while ω scales with e. Hence, equation (4.37) combined with equation (4.23) 2 implies that ωi is proportional to e . Physically, as e decreases, the mode becomes more sharply peaked near the avoided crossing and thus the ratio of the power dissipated by the continuum resonance to the energy of the mode will decrease. The insensitivity of the eigenfunction to e indicates contributions to damping that are of higher order with respect to the perturbation, which are ignored in the perturbative analysis, will have the same scaling with e as the first-order contributions, which are §8.3 Wave function perturbation 107

0.000

- 0.001

Contour - 0.002 ) r

/ω - 0.003 i

(ω - 0.004

- 0.005 - 0.010 - 0.008 - 0.006 - 0.004 - 0.002 0.000

(ω i /ω r ) Perturbative

Figure 8.6: Damping calculated using the perturbative approach plotted against that calculated using the complex contour approach. Results are plotted for varying qa (blue, solid line), e (red, medium dashed line), ∆1 (green, long dashed line), and ∆2 (orange, short dashed line). A thin solid black line indicates where the two quantities are equal. included. Thus, the perturbative and complex contour results differ by a constant factor as e 0. As ∆1 increases over the selected range, the relative increase in density at the continuum→ resonance is faster than the relative decrease in the radial gradient of the continuum resonance frequency there. However, simultaneously, the density fall-off moves further from the peak in the mode, increasing C3,−2 rR . The latter effect dominates for most of the range of ∆1 examined, so that damping increases with ( ) increasing ∆1. The relative rate of increase in C3,−2 rR decreases with ∆1, resulting in greatest damping being found where ∆1 0.79 for either method. ( ) The effect of increasing ∆2 on continuum damping is qualitatively similar to that ≈ of increasing ∆1. The density fall-off becomes more gradual, causing an increase in density at the continuum resonance and a decrease in the radial gradient of the continuum resonance frequency, with the former being dominant. For small ∆2, the absolute value of C3,−2 rr increases rapidly with ∆2 before reaching a maximum. This results in a maximum for damping at ∆2 0.07 using the perturbative method ( ) and ∆2 0.09 using the contour method. ≈ ≈ 8.3 Wave function perturbation

The discrepancy between the perturbative and complex contour methods can be shown to result from the change in the eigenfunction due to the discontinuity at the continuum resonance. The unperturbed eigenfunction is calculated assuming that lim − Em n r lim + Em n r . However, the continuum resonance pole results in r→rR , r→rR , a discontinuity in Em,n along the real r-axis for eigenfunctions calculated along integration paths( satisfying) = the causality( ) condition. The discontinuities can be calculated analytically, as described in Section 5.2, to determine the solution along the real axis. The resulting eigenfunctions are plotted in Figure 8.7. 108 Continuum Damping of Alfvén Eigenmodes in Large Aspect-Ratio Tokamaks

(a)

(b)

Figure 8.7: Variables (a) Em,n and (b) Cm,n as functions of r for the unperturbed eigenfunction (thick line) and the eigenfunction applying resonance pole contributions calculated analytically (solid thin line: real component, dashed thin line: imaginary component). The 2, 2 (blue) and 3, 2 (gold) harmonics are shown. In this case the equilibrium parameters are qa 1.6, e 0.1, ∆1 0.8, and ∆2 0.1. ( − ) ( − ) = = = =

The continuum resonance corresponds to continuum modes where the 3, 2 harmonic is dominant. Consequently, the 2, 2 harmonic of the solution does not ( − ) interact strongly with the resonance and E2,−2 and C2,−2 are found to be approximately the same in each case. In the unperturbed( − ) case, interaction of the 3, 2 harmonic with the continuum resonance results in E3,−2 being logarithmically divergent ( − ) at rR and C3,−2 being approximately constant in that vicinity. When the discontinuity due to the continuum resonance is considered, rR shifts to a complex value and E3,−2 on the real axis remains bounded, though C3,−2 remains approximately constant near rR. The variables E3,−2 and C3,−2 are approximately unchanged between the two cases, except in a small region containing the continuum resonance. It is found that the difference in the real and imaginary components of C3,−2 rR when the discontinuity due to the pole is considered compared with the unperturbed case is of similar ( ) magnitude to the quantity itself. Thus, the assumption that the perturbation to Cm,n §8.3 Wave function perturbation 109

(1) (0) was negligible compared with the unperturbed quantity (Cm,n Cm,n), which was made in deriving the perturbative expression for continuum damping, is not valid. ≪ 110 Continuum Damping of Alfvén Eigenmodes in Large Aspect-Ratio Tokamaks Chapter 9

Continuum Damping of Alfvén Eigenmodes in General 2D and 3D Geometry

In this chapter, continuum damping of Alfvén eigenmodes is computed using the complex contour method in cases with full two- and three-dimensional magnetic geometry. It is anticipated that couplings between numerous harmonics may be important in determining the structure and damping of these modes in strongly shaped tokamak and stellarator plasmas. Continuum damping has previously been calculated for two-dimensional cases by taking the limit of resistive damping in the limit of infinite conductivity [41], though the computational demands associated with resolving the almost singular eigenfunction around the resonance make it difficult to apply this method in three-dimensional cases. The principal finding of this chapter is that the MHD finite element code CKA successfully implements the Alfvén eigenmode calculation over a complex contour described in Chapter 5. Configurations considered here include circular cross-section tokamak, torsatron, heliac, and helias cases. Various TAE, EAE, and NGAE modes with continuum resonances are found in these configurations. In each case, the computed continuum damping is demonstrated to have converged such that it is not sensitive to increases in radial, poloidal, and toroidal mesh resolution as well as deformation of integration path. Moreover, the complex contour calculation in a tokamak is shown to agree very closely with the calculation in the limit where dissipative non-ideal effects vanish, verifying the former method. To our knowledge, the stellarator examples presented here represent the first time that continuum damping has been computed for Alfvén eigenmodes in fully three-dimensional magnetic geometry. Some findings are made based on these continuum damping calculations. Change in the mode structure due to use of a complex contour is discussed with respect to the Alfvén eigenmodes in the torsatron case as an illustrative example. The reduced ideal MHD model in three-dimensional geometry is compared with the large aspect- ratio models for the tokamak and heliac cases. This comparison shows that higher order effects which are ignored in the latter model have a large effect in the heliac

111 112 Continuum Damping of Alfvén Eigenmodes in General 2D and 3D Geometry

case but not in the tokamak case. Continuum damping of the NGAE in the realistic H-1NF equilibria is found to be very small compared to the damping computed in Chapter 7.

9.1 Circular cross-section tokamak TAE case

The implementation of the complex contour calculation of continuum damping for general two- and three-dimensional configurations using CKA is verified by application to a tokamak case. Waves are represented by equation (3.48), the reduced ideal MHD model including shear Alfvén, pressure gradient and parallel current terms. Continuum damping is computed for a TAE in an axisymmetric tokamak equilibrium with a circular cross-section and inverse aspect-ratio e 0.225. The rotational transform is specified by a polynomial function, which approximates that used in Section 8.1, =

ι s 0.95016 0.67944s 0.62286s2 0.41244s3 s4 s5 s6 ( ) = − + − 0.1219 0.0042185 0.0013979 . (9.1) The density is given by the function + + −

ρ s ∆ ρ s 0 1 tanh 1 , (9.2) 2 √ ∆2 − ( ) = − where ∆1 and ∆2 respectively denote the centre location and width of the fall off in density in s. Values of ∆1 0.8 and ∆2 0.1 are selected. The density and rotational transform√ profiles are overlaid on a plot of continuum resonance frequencies in Fig- ure 9.1. A TAE is found with= an eigenvalue= in the gap due to the avoided crossing of the 2, 2 and 3, 2 branches of the continuum. The complex contour is defined by the real and imaginary parts in equation (5.2) and equation( − ) (5.3).( However,− ) the normalised flux, s ψ , is replaced by s as the ψa flux surface label in equation (5.1). Convergence of the TAE eigenvalue with√ respect to the choice of contour is demonstrated by varying α=in Figure 9.2. Here a poloidal mesh resolution of Nϑ 20 is used. A complex normalised TAE frequency of Ω 0.395 0.00174i is obtained, corresponding to a damping ratio of ωi 0.00442. ωr The computed complex= frequency and damping are seen to be well converged= with respect− to radial mesh resolution and α. As expected, it is found= − that the computed continuum damping converges faster with radial mesh resolution for larger values of α, where the continuum resonance pole lies further from the integration contour. The computed complex frequency and damping are found to have ex- tremely weak dependence on α in the limit of high radial mesh resolution. A change from α 0.1 to α 0.08 coincides with a difference of only 0.05% in ωi for N 1000. ωr s This small difference can be ascribed to the approximation of equilibrium quantities on= the complex= contours with cubic splines. The computed complex frequency= and damping can also be shown to be well converged with respect to poloidal mesh §9.1 Circular cross-section tokamak TAE case 113

1.2 1.1 1.1 1 1 0.9 0.9 0.8 0.8 0.7

0.7 ι

0.6 , 2 0

0.6 ρ Ω

0.5 /

0.5 ρ 0.4 0.4 0.3 0.3 0.2 0.2 0.1 0.1 0 0 0 0.2 0.4 0.6 0.8 1 √s

(a) (b)

Figure 9.1: (a) Normalised shear Alfvén continuum resonance eigenvalues Ω2 as a function of the radial coordinate, s, for the tokamak case. The avoided crossing between the 2, 2 ( ) and 3, 2 √( ) branches results in the TAE studied here (not shown). The rotational transform ι (dashed line) and density ρ (solid line) are also ρ0 ( plotted.− ) [ (b) Outermost( − ) [ ﬂux surface of the tokamak case. resolution, with a change from N 20 to 10 resulting in a change in ωi of 0.07%. s ωr As the parameter α is increased, the deformation of the complex contour away = from the real axis grows. This increase causes the gap between the real eigenvalue components of the two branches of continuum modes found along the contour to decrease and ultimately vanish, so that the TAE is no longer found. For the complex eigenvalue, ω2, associated with the TAE, the avoided crossing of continuum branches results in poles which occur at complex locations near the real s-axis. If such poles occur between the chosen complex integration path and the real√ axis they result in a

- 0.0044 ●▲▼ ●▲▼ ● ▲■ ●▲▼▼ ■

- 0.0045 ■ r - 0.0046 /ω i ω - 0.0047

- 0.0048 ■ 0.04 0.05 0.06 0.07 0.08 0.09 0.10

Figure 9.2: Convergence of continuum damping with respect to deformation of the complex contour. Damping ratio, ωi , is plotted against α for N 400 and N 20 ( ), ωr s ϑ Ns 800 and Nϑ 20 ( ), Ns 1200 and Nϑ 20 ( ) and Ns 1200 and Nϑ 10 ( ). = = ∎ = = ● = = ▼ = = ▲ 114 Continuum Damping of Alfvén Eigenmodes in General 2D and 3D Geometry

contribution to the eigenfunction similar to that described for poles associated with continuum resonances. Unlike the latter effect, the former is unphysical, because the poles in that case do not correspond to singularities of the ideal MHD wave equation which occur on real ﬂux surfaces for real ω2. If poles due to the avoided crossing are included in this way, the eigenvalue problem changes such that the TAE no longer represents a solution and no eigenvalue is found for the corresponding ω2. The continuum damping obtained using the complex contour technique can be compared with that found using the accepted technique of taking the limit of damping where non-ideal dissipative effects vanish. This calculation is implemented using CKA to solve equation (3.100), which includes dissipative effects. The real component of the non-ideal term in equation (3.100), responsible for kinetic Alfvén waves, is neglected. The model is then equivalent to the resistive MHD model. From equa- −1 −2 tion (3.108) it is seen that conductivity is δ ρs , excluding a factor which is constant for a particular eigenmode. The convergence of the continuum damping of the TAE with increasing conductivity is shown in Figure 9.3. In these calculations, the density of the mesh in the radial coordinate is increased near the continuum resonance at sR 0.964, where Φ varies most rapidly. Equal numbers of mesh points are deﬁned√ for s 0, sR and s sR, 1 , with spacing approximately proportional to = exp 2 s sR √. √ ∈ ( ) ∈ ( ) √ √ − - 0.00440

- 0.00441 ●■ ▼▲ - 0.00442 ▲▼ ● r

/ω - 0.00443 ●▼▲■ i ω - 0.00444 ■ ●▼▲■ - 0.00445

- 0.00446 0 5.0× 10 7 1.0× 10 8 1.5× 10 8 2.0× 10 8 2 1/( δρ s )

Figure 9.3: Convergence of damping with respect to conductivity. Damping ratio, ωi , ωr 1 is plotted against 2 for Ns 1000 ( ), Ns 1200 ( ), Ns 1400 ( ), and Ns 1600 ( ). δρs = ∎ = ● = ▼ = ▲ In Figure 9.3 it is shown that damping approaches a function of conductivity as radial mesh resolution is increased. To accurately model dissipation due to the continuum resonance, the radial mesh resolution must be sufficiently dense that the basis functions can represent the nearly singular behaviour of the eigenfunction in a narrow region around this flux surface [41]. This region narrows as conductivity increases, resulting in reduced accuracy for lower radial mesh resolution. As conductivity is finite, the TAE will be damped due to dissipation throughout the plasma in addition to that at the continuum resonance. This contribution to the computed §9.1 Circular cross-section tokamak TAE case 115

2 damping is observed for low conductivity in Figure 9.3, where it is δρs . Fitting a 2 function of the form c1 c2δρs to the complex frequency computed for Ns 1600 results in a complex frequency of Ω 0.0395 1.74 10−4i and damping of∝ ωi 0.00442 + ωr = in the limit δ−1ρ−2 . As was the case previously, reduction in the poloidal mesh s = − × = − resolution from Nϑ 20 to 10 results in a change in continuum damping of approximately 0.07%. → ∞ Very close agreement= is found between the continuum damping obtained using the complex contour approach and that found by taking the limit of damping as conductivity approaches infinity (a difference of 0.002%). However, it is found that much faster convergence with respect to radial mesh resolution occurs in the former case than in the latter. For α 0.1, the complex− frequency computed using the complex contour method is found to have converged to 7 significant figures, the = precision of the code output, for Ns 800. This convergence is not matched by the −1 −2 8 −2 calculation for high conductivity (where δ ρs 2 10 m ) where Ns 1600 is used. The accuracy of the computed≥ damping is mainly dependent on obtaining the correct eigenmode behaviour near the continuum= × resonance. Variation≤ of the eigenmode with respect to the radial coordinate is slower where the singularity is avoided using a complex integration path than it is where the singularity is resolved through finite conductivity. The continuum damping computed using the large aspect-ratio model of equation (3.78) can also be compared with that obtained above. Once again the shooting method described in Subsection 5.3.2 is used, considering only the dominant 2, 2 and 3, 2 harmonics. This calculation results in the normalised complex frequency Ω 0.371 0.00174i and damping ratio ωi 0.00470. The latter agrees reasonably( − ) ωr closely( − with) the fully two-dimensional calculation using the reduced ideal MHD model= in equation− (3.48), with a difference of= − just 6.6%. The above difference must result from assumptions made in the derivation of the large aspect-ratio model; namely neglect of the pressure gradient term, coupling to additional harmonics, neglect of higher order terms in the large aspect-ratio expansion and approximation of k∥ in coupling terms by its value for gap regions. If the pressure gradient is excluded from the reduced MHD model, the damping obtained from the fully two-dimensional calculation changes by approximately 0.1%. Thus, the pressure gradient has a minuscule effect on the TAE in this case and the difference between the two models is primarily due to the geometric approxima-− tions identified. Similarly, coupling to additional harmonics can be discounted, as the continuum resonance frequency branches associated with the 2, 2 and 3, 2 harmonics do not have avoided crossings with those of additional harmonics at any flux surface for a plasma with the chosen ι profile. Including the (1, −2) and (4, −2) harmonics in the large aspect-ratio calculation results in damping changing by 0.49% and 1.1% respectively. ( − ) ( − ) The implementation of the complex contour technique for fully three-dimensional configurations is verified by applying this treatment to the above case of a TAE in a tokamak. That is, the discretised eigenvalue problem is constructed performing numerical integration in three-dimensions and basis functions which vary in three 116 Continuum Damping of Alfvén Eigenmodes in General 2D and 3D Geometry

dimensions are chosen. The basis functions are deﬁned on a three-dimensional mesh with of Ns 400, Nϑ 10, and Nϕ 10 radial, poloidal, and toroidal mesh points respectively. It is found that the complex TAE frequency calculated using this method agrees with= the corresponding= two-dimensional= calculation to the precision provided by the output of the code.

9.2 Torsatron TAE and EAE cases

The complex contour method is applied to compute continuum damping of Alfvén eigenmodes in a torsatron case. In these machines, helical coils produce toroidal flux surfaces with helical ripples that generate a non-zero rotational transform [8]. TAEs have been observed previously in torsatrons, with examples reported in the Large Helical Device (LHD) [102, 103] and the Compact Helical System (CHS) [103]. It has been found that continuum damping can be the dominant cause of damping for such modes [32]. Here, a torsatron plasma with Nf p 20 toroidal field periods is considered. For this configuration the rotational transform is defined by the polynomial = ι s 0.4319 0.23407s 0.042125s2 0.008341s3. (9.3)

As is typical for torsatrons,( ) = the+ rotational+ transform has+ a minimum at the magnetic axis and has strong magnetic shear near the edge of the plasma. The same density profile is specified as in the tokamak case in Section 9.1. Based on the average minor radius, this configuration has an inverse aspect ratio of approximately e 0.0487. The magnetic axis is circular while the poloidal plasma cross-section is elliptical. Ellipticity varies toroidally with minimum e 0.426 and maximum e 0.854.= Con- tinuum resonance frequencies are plotted for this configuration in Figure 9.4, along with the density and rotational transform profiles.= Toroidicity and ellipticity= induced gaps are apparent, which indicate small coupling between harmonics which differ by 1, 0 and large coupling between those which differ by 2, 0 . A TAE is found due to the avoided crossing between continuum branches associated( ) with the 3, 2 and 4, 2 harmonics. This TAE( has) continuum resonances at s 0.28 (with the 5, 2 branch) and at s 0.94 (with the 3, 2 branch). In ( − ) ( − ) the√ absence of avoided crossings between the √5, 2 and 4, 2 continuum branches near the≈ TAE frequency,( the− )5, 2 harmonic is not≈ expected to be( a significant− ) component of the mode. As discussed in Subsection( 3.1.2,− ) coupling( − ) terms become important only near avoided crossings,( − where) leading order terms in the large aspect-ratio expansion are small. Therefore, it is assumed that damping at the outer resonance is dominant and that the inner resonance does not significantly affect the mode structure. In this example, the complex integration path is defined as having real and imaginary parts given by equation (5.4) and equation (5.5) respectively, giving a localised deformation away from the real axis. Again, normalised flux s is replaced by s as the flux surface label in equation (5.1). Based on the location of the continuum√ §9.2 Torsatron TAE and EAE cases 117

1.1 0.5 1 0.9 0.4 0.8 0.7

0.3 ι

0.6 , 2 0 ρ Ω

0.5 /

EAE ρ 0.2 0.4 0.3 0.1 TAE 0.2 0.1 0 0 0 0.2 0.4 0.6 0.8 1 √s

(a) (b)

Figure 9.4: (a) Normalised shear Alfvén continuum resonance eigenvalues Ω2 as a function of the radial coordinate s for the torsatron case. The avoided crossing between the 3, 2 ( ) and 4, 2 (√) branches produces a TAE within the resulting spectral gap. Additionally, the avoided crossing between the 3, 2 and the 5, 2 ( ) branches( produces− ) [ an EAE( − within) [ the corresponding spectral gap. The rotational transform ι (dashed line) and density ρ (solid line) are also plotted.( − ) (b) Outermost( − ) ρ0 [ ﬂux surface of the torsatron case.

resonance, the parameters xβ 0.1 and xγ 0.96 are chosen to specify the complex contour. As was the case previously, the contour is varied by changing α to demonstrate convergence. = = The eigenvalue problem is solved numerically using a basis set composed of second-order B-splines and a phase factor of m˜ 3, n˜ 2 . The convergence of the damping ratio of the TAE, ωi , is investigated by varying N , N , and N for different ωr s ϑ ϕ values of α, with results plotted in Figure 9.5.( It= is found= − that) this ratio approaches the same constant value for each choice of α as radial, poloidal, and toroidal mesh resolution are increased, indicating satisfactory convergence of the numerical solution. Thus, a normalised frequency of Ω 0.290 0.00334i and damping ratio of ωi 0.0115 are obtained in this case. ωr = − Continuum damping of the TAE is found to converge rapidly with respect to = − both Nϑ and Nϕ, indicating that coupling of the dominant harmonics to others with a large difference in m and n is not signiﬁcant. While the magnetic geometry of the torsatron indicates substantial coupling coeffecients for harmonics which differ by 0, 20 or 2, 20 , equation (3.95) shows that the corresponding gaps will occur at much higher frequency than that of the toroidicity-induced gap. As signiﬁcant coupling( − of) different( − ) harmonics occurs only due to the continuum gaps shown in Figure 9.4, additional harmonics need not be resolved. The small variation that is found in damping as Nϕ increases is believed to be predominantly due to the resolution of n 2 20 harmonics involved in the aforementioned couplings. The conclusion that couplings of harmonics with different n are small is sup- = − ± ported by the mode structure, shown in Figure 9.6. The 4, 2 harmonic component

( − ) 118 Continuum Damping of Alfvén Eigenmodes in General 2D and 3D Geometry

- 0.0110 ■ - 0.0111

- 0.0112

r - 0.0113 /ω

i - 0.0114 ω - 0.0115 ●▲▼ ●▼▲ ●▼▲■ ●▼▲■ ■ 0.0116 - ●▲▼

- 0.0117 ■ 100 150 200 250 300

N s (a)

■ ■ - 0.01150 ●▲▼■ ●▲▼ ●▲▼ ●▲▼■

- 0.01155 r /ω i

ω - 0.01160

- 0.01165 ●▼▲■ 10 15 20

N ϑ (b)

■ - 0.01150 ●▲▼■ ●▲▼ ●▲▼■ - 0.01152

r - 0.01154 /ω i

ω - 0.01156

- 0.01158

- 0.01160 ●▲▼■ 4 5 6 7 8 9 10

N φ (c)

Figure 9.5: Convergence of continuum damping for the TAE identiﬁed in the torsatron case with respect to (a) radial mesh resolution, Ns, (b) poloidal mesh resolution, Nϑ, and (c) toroidal mesh resolution, Nϕ. For each plot, two of Ns 300, Nϑ 20, and Nϕ 8 are ﬁxed while the other parameter varies. Complex contours have parameters x 0.1 and x 0.96. Damping ratio, ωi , is plotted for α 0.1= ( ), β γ ωr = = α 0.2 ( ), α 0.5 ( ), and α 1.0 ( ). = = = ∎ = ● = ▼ = ▲ §9.2 Torsatron TAE and EAE cases 119

is sharply peaked as the continuum resonance frequency approaches close to that of the TAE at the avoided crossing. By contrast the 3, 2 harmonic component has a much broader peak due to the interaction of the avoided crossing and continuum resonance. This peak is approximately bounded by( these− ) locations, near which its amplitude changes rapidly. In Figure 9.6 the effect of using a complex integration path on the mode structure is shown. The eigenmode computed along the real s-axis has a spike in its 3, 2 harmonic component near x s 0.94 due to the√ continuum resonance pole lo- ( − ) cated there. As a result of the contour√ deformation near s 0.96, the wave function = ≈ along the complex integration path will differ from that along√ the real axis in the surrounding region. A rapid complex change in the 3, 2 harmonic≈ component of the wave function is evident near the continuum resonance pole, in place of the complex discontinuity which would occur if this causal solution( − ) were analytically continued to the real axis. Elsewhere, the mode structure computed along the complex contour can be taken as a reasonable approximation for that at s x. Differences between the computed eigenfunctions extend beyond the region√ in which the complex contour deformation is localised. The differences in the amplitude= and phase of the TAE are most prominent in the region between the avoided crossing and the continuum resonance. The previous assumption that the inner continuum resonance does not signiﬁ- cantly affect continuum damping can be checked by considering the TAE for a mod-

0.8 (3,−2)

0.6

0.4

0.2 (4,−2) m,n Φ 0

−0.2

−0.4

0 0.2 0.4 0.6 0.8 1 x

Figure 9.6: The four largest Fourier harmonics of the TAE identiﬁed in the torsatron case as a function of the contour parameter, x. The dominant 3, 2 (blue) and 4, 2 (red) harmonics are labeled. Modes computed along a complex contour with ( − ) α 0.1, xβ 0.1, and xγ 0.96 (thick lines) and along the real axis (thin lines) are shown.( − ) The real components are indicated by solid lines and imaginary components = = = by dashed lines. 120 Continuum Damping of Alfvén Eigenmodes in General 2D and 3D Geometry

ified density profile. Letting ρ s ρ0, the outer continuum resonance is eliminated and damping is limited to that due to the inner continuum resonance. This damping is calculated using a complex( ) = contour of the same form as previously, choos- ing parameters xβ 0.1 and xγ 0.28. This results in a normalised frequency of Ω 0.262 1.29 10−8i and damping ratio of ωi 4.94 10−8. Although some modi- ωr fication of the mode= structure is expected= due to the change in density near the edge, this= comparatively− × minuscule value for damping= − indicates× that the inner continuum resonance has a negligible effect on the mode structure and continuum damping of the TAE. An EAE is also found, which results from the avoided crossing between the 3, 2 and 5, 2 harmonics. The EAE has continuum resonances at s 0.77 (with the ( − ) 4, 2 branch) and at s 0.96 (with the 3, 2 branch). Strong coupling√ is expected ( − ) ≈ only between the 3, √2 and 5, 2 harmonics due to their avoided crossing, thus (the− outer) resonance is assumed≈ to be the( dominant− ) cause of damping. Therefore, a complex contour is( used− ) which( has− ) the same form as that used above to compute continuum damping of the TAE, again with parameters xβ 0.1 and xγ 0.96. The EAE solution to the eigenvalue problem is computed using the same basis set and phase factor as were used in the TAE case. Convergence= of the damping= ratio of the EAE, ωi , with respect to N , N , and N for various α is shown in Figure 9.7. ωr s ϑ ϕ The computed damping converges more slowly with Ns in the EAE case than it does in the TAE case, particularly for smaller values of α. For a particular α the integration path passes closer to continuum resonance poles in the former case and so the continuum resonance interaction occurs over a shorter radial length scale. Convergence with respect to Nϑ is also slower for the EAE than it was for the TAE, because the difference in m between the phase factor and the 5, 2 harmonic which is important in the EAE case is greater than that with the 4, 2 harmonic which was important in the TAE case. Convergence of continuum( damping− ) of the EAE ( − ) with respect to Nϕ is rapid, as is the case for the TAE. This indicates that coupling to harmonics of different n is negligible for the EAE, which is explained by the much higher frequencies of the relevant continuum gaps. The normalised frequency of the EAE is found to be Ω 0.482 0.00248i, while its damping ratio is ωi 0.00515. From the continuum resonance frequencies dis- ωr played in Figure 9.4 and equation (3.96) it is estimated that= complex− poles lie at s 0.77 5.9 10−4i and= − s 0.96 1.5 10−5i. Thus, the complex contour is found √to circumvent both continuum√ resonance poles as required by the causality condition for≈ the values− × of α considered.≈ Unlike− the× TAE case above, the calculated damping of the EAE contains contributions from both continuum resonances. The mode structure of the EAE is shown in Figure 9.8. Broad 3, 2 and 5, 2 harmonic components arise due to the wide ellipticity-induced gap, which are peaked near the avoided crossing. The effect of computing the EAE over a( complex− ) integra-( − ) tion path rather than the real s-axis is similar to that for the TAE. Solving the eigenvalue problem over the complex√ contour, a rapid complex change in the 3, 2 harmonic component occurs where x 0.96, representing the effect of the continuum resonance. At x 0.77 a much smaller continuum resonance interaction is evident( − ) ≈ ≈ §9.2 Torsatron TAE and EAE cases 121

■ - 0.0046

- 0.0048 r

/ω ■ i

ω - 0.0050 ■ ■ ● ● ■ ▲ ▲▼ ▲▼ ▲▼ ●▼▲ - 0.0052 ▼ ●

● 100 150 200 250 300

N s (a)

- 0.0040 ■ ●▼▲

- 0.0042

- 0.0044 r

/ω - 0.0046 i ω - 0.0048 ■ ●▼▲ - 0.0050 ■ ●▼▲ ■ ●▲▼ ■ - 0.0052 ●▲▼ 10 15 20

N ϑ (b)

■ ■ ■ - 0.00511 ■ - 0.00512 r - 0.00513 /ω i

ω - 0.00514 ● ● ● ● - 0.00515 ▲▼ ▲▼ ▲▼ 0.00516 - ▲▼ 4 5 6 7 8 9 10

N φ (c)

Figure 9.7: Convergence of continuum damping for the EAE identiﬁed in the torsatron case with respect to (a) radial mesh resolution, Ns, (b) poloidal mesh resolution, Nϑ, and (c) toroidal mesh resolution, Nϕ. For each plot, two of Ns 300, Nϑ 20, and Nϕ 8 are ﬁxed while the other parameter varies. Complex contours have parameters x 0.1 and x 0.96. Damping ratio, ωi , is plotted for α 0.1= ( ), β γ ωr = = α 0.2 ( ), α 0.5 ( ), and α 1.0 ( ). = = = ∎ = ● = ▼ = ▲ 122 Continuum Damping of Alfvén Eigenmodes in General 2D and 3D Geometry

for the 4, 2 harmonic. Again, the amplitude and phase of the EAE are found to change substantially on a broader region between the avoided crossing and the continuum( − resonance) when a complex integration path is employed. The relative importance of the two continuum resonances to the damping of the EAE can be illuminated by considering a modified density profile. Density is now specified by the function

ρ ρ ρ ρ s ∆ ρ s 0 a 0 a 1 tanh 1 , (9.4) 2 2 √ ∆2 + − − ( ) = + − which asymptotically tends to ρa as s increases. The role of other parameters is unchanged from equation (9.2). Their√ value is held constant while ρa 0.15 is chosen. The increase in density near the edge eliminates the continuum resonance due to the branch associated with the 3, 2 harmonic. However, this change= has less effect on the density proﬁle near the continuum resonance associated with the 4, 2 harmonic. It is found that the resulting( − ) EAE has a normalised frequency of Ω 0.475 3.03 10−5i and damping ratio ωi 6.37 10−5. Thus, continuum damp- ωr (ing− due) to the resonance with the 4, 2 harmonic branch alone is much smaller than that= found− considering× both resonances. This= implies− × that the continuum resonance associated with the 3, 2 harmonic( − is) the dominant source of continuum damping of the EAE. ( − )

0.8 (3,−2)

0.6

0.4 m,n

Φ 0.2

−0.2 (5,−2) −0.4 0 0.2 0.4 0.6 0.8 1 x

Figure 9.8: The four largest Fourier harmonics of the EAE identiﬁed in the torsatron case as a function of the contour parameter, x. The dominant 3, 2 (blue) and 5, 2 (green) harmonics are labeled. Modes computed along a complex contour ( − ) with α 0.1, xβ 0.1, and xγ 0.96 (thick lines) and along the real axis (thin lines) are shown.( − ) The real components are indicated by solid lines and imaginary components = = = by dashed lines. §9.3 Helias TAE case 123

9.3 Helias TAE case

Next, the complex contour method is used to the calculate continuum damping of a TAE in a helias case. An equilibrium model for a W7-X stellarator conﬁguration with a high mirror-term is used [104]. In this machine, rotational transform is generated partly through torsion of the magnetic axis [105]. Modular coils are used to produce a magnetic geometry which is optimised with respect to current, stability, and transport properties. This results in a complicated magnetic geometry, where equilibrium quantities such as B and gss have broad spectra in m and n. Consequently, a particular Alfvén eigenmode harmonic may experience signiﬁcant coupling to numerous other harmonics. W7-X equilibria have very low magnetic shear [105]. Thus, modes with high m and n can nevertheless have broad radial mode structures.

W7-X plasmas have Nf p 5 toroidal field periods. Its inverse aspect ratio is approximately e 0.0899. A density profile is chosen which has the form stated in = equation 9.2, with parameters ∆1 0.4 and ∆2 0.2. The rotational transform and density profile are= plotted along with the resulting continuum resonance frequencies in Figure 9.9. An avoided crossing between= the 18,= 16 and 19, 16 branches of the continuum results in a TAE. This mode has a continuum resonance with the 18, 16 branch where s 0.67. Additional TAEs with( differing− ) numbers( − of) harmonics can ( − ) be identified within√ the gap. ≈ The density profile and harmonics considered here are chosen to make the problem numerically tractable. Toroidicity induced couplings are reduced for a mode localised closer to the magnetic axis. Hence, the mode will be narrower and the effect of additional couplings will be reduced. Moreover, placing the upshift in continuum resonance frequency introduced by the fall-off in density nearer to the centre

1.4 1.3 0.3 1.2 1.1 1 0.9

0.2 0.8 ι , 2 0.7 0 ρ Ω 0.6 / TAE ρ 0.5 0.1 0.4 0.3 0.2 0.1 0 0 0 0.2 0.4 0.6 0.8 1 √s

(a) (b)

Figure 9.9: (a) Normalised shear Alfvén continuum resonance eigenvalues Ω2 as a function of the radial coordinate s for the W7-X helias case. The avoided crossing between the 18, 16 ( ) and 19,√ 16 ( ) branches produces a TAE in the resulting spectral gap. The rotational transform ι (dashed line) and density ρ (solid line) are ρ0 also plotted.( (b)− Outermost) [ ﬂux( surface− ) [ of the W7-X helias conﬁguration studied. 124 Continuum Damping of Alfvén Eigenmodes in General 2D and 3D Geometry

avoids certain resonances with non-dominant harmonics. Thus, the chosen proﬁles decrease the required poloidal and toroidal mesh resolution, helping to ensure that convergence of computed continuum damping can be obtained within the computational limits imposed by the numerical matrix eigenvalue problem solver. Complex integration paths are deﬁned as in Section 9.2, with parameters xβ 0.1 and xγ 0.65. The solution of the eigenvalue problem is computed using second- order B-spline basis functions and a phase factor of 18, 16 . Convergence of= the TAE damping= ratio, ωi , with respect to N , N , and N for various α is shown in ωr s ϑ ϕ Figure 9.10. For each α, satisfactory convergence of damping( − is) obtained with respect to radial, poloidal and toroidal mesh resolution. Normalised frequency is estimated to be Ω 0.394 0.000477i so that damping ratio is estimated to be ωi 0.00121. ωr Convergence with respect to poloidal and toroidal resolution for the TAE in the W7-X case= is found− to be slower than in the torsatron case. This suggests= − that the former involves coupling to harmonics with a wider range of m and n. The mode structure of the TAE found by solving the eigenvalue problem along the real s axis is shown in Figure 9.11. In addition to the dominant 18, 16 and 19, 16√ harmonics, the mode is found to be composed of 17, 16 and 17, 11 harmonics. The dominant components have 3 nodes inboard of the continuum( − ) resonance( − and) are negligible elsewhere. ( − ) ( − )

9.4 Heliac NGAE cases

Finally, continuum damping is computed for NGAEs in two heliac cases using the complex contour method. Plasma configurations are considered for the H-1NF stellarator, introduced in Section 7.6. Like W7-X, this device has a complicated magnetic geometry resulting in equilibrium quantities having broad spectra in m and n. It has a flat rotational transform partly generated by torsion of the magnetic axis. In the cases investigated here, the dominant harmonics are close to resonant ( n ιm 1) throughout the plasma. In such cases the existence and location of resonant surfaces is very sensitive to changes in the rotational transform profile. S + S ≪ In the first case, an H-1NF equilibrium is considered for which the current ratio 1 is κh 0.33 . The density profile used here is identical to that used in the tokamak case in Section 9.4. Density profile, rotational transform profile, and the resulting continuum= resonance frequencies are plotted in Figure 9.12. This equilibrium is un- usual in that the edge is very nearly a resonant surface for the 10, 13 harmonic. Consequently, while the continuum resonance frequency associated with this harmonic increases where the most rapid density fall-off occurs, it( decreases− ) closer to the edge. The maximum is found to result in an NGAE localised near the edge. It has continuum resonances with the 8, 10 harmonic branch at s 0.81 and with the 15, 19 harmonic branch at s 0.73. √ ( − ) ≈ Complex contours are defined√ which are of same form of as those in Section 9.2, ( − ) ≈ 1 Although this equilibrium nominally corresponds to the same value of κh as that in Section 7.6, it is computed for a somewhat different boundary and ι profile. §9.4 Heliac NGAE cases 125

- 0.00875 ▼■ ● - 0.00880

- 0.00885

r - 0.00890 /ω i - 0.00895 ω

- 0.00900

- 0.00905 ●■ ▼ ●▼■ ●▼■ ●■ - 0.00910 ▼ 40 60 80 100 120 140

N s (a)

- 0.009 ●▼■ ●▼■ ●▼■ ●▼■ - 0.010 ●▼■ r - 0.011 ●▼■ /ω i ω - 0.012

- 0.013 ●▼■ 10 15 20 25 30 35 40

N ϑ (b)

- 0.009 ●▼■ ●▼■ - 0.010 ●▼■

r - 0.011 /ω i

ω - 0.012

- 0.013

●▼■ - 0.014 6 7 8 9 10 11 12

N φ (c)

Figure 9.10: Convergence of continuum damping for the TAE identiﬁed in the W7-X helias case with respect to (a) radial mesh resolution, Ns, (b) poloidal mesh resolution, Nϑ, and (c) toroidal mesh resolution, Nϕ. For each plot, two of Ns 150, Nϑ 40, and Nϕ 12 are ﬁxed while the other parameter varies. Complex contours have parameters x 0.1 and x 0.65. Damping ratio, ωi , is plotted for α 0.2= ( ), α = 0.5 β γ ωr = ( ), and α 1.0 ( ). = = = ∎ = ● = ▼ 126 Continuum Damping of Alfvén Eigenmodes in General 2D and 3D Geometry

(18,−16)

0.5

m,n 0 Φ

(19,−16) −0.5

−1 0 0.2 0.4 0.6 0.8 1 √s

Figure 9.11: The four largest Fourier harmonics of the TAE identiﬁed in the W7-X helias case as a function of the ﬂux surface label, s. The dominant 18, 16 (blue) and 19, 16 (red) harmonics√ are labeled. ( − ) ( − )

1.4 0.09 1.3 0.08 NGAE 1.2 1.1 0.07 1 0.06 0.9

0.8 ι 0.05 , 0 2 0.7 ρ Ω 0.04 0.6 / ρ 0.5 0.03 0.4 0.02 0.3 0.2 0.01 0.1 0 0 0 0.2 0.4 0.6 0.8 1 √s

Figure 9.12: Normalised shear Alfvén continuum resonance eigenvalues Ω2 as a function of the radial coordinate s for the ﬁrst H-1NF heliac case. The maximum of the 10, 13 ( ) branch results in√ an NGAE, which has a resonance with the 8, 10 ( ) branch. The rotational transform ι (dashed line) and density ρ (solid line) are ρ0 ( − ) [ also plotted. ( − ) [ here with parameters xβ 0.1 and xγ 0.8. A phase factor of 10, 13 and basis set composed of second-order B-splines are used. The resulting convergence of the NGAE damping ratio, ωi=, with respect= to N , N , and N for several( − ) values of α ωr s ϑ ϕ is shown in Figure 9.13, and is found to be acceptable. The normalised frequency of the mode is found to be Ω 0.278 0.00162i and hence the damping ratio is

= − §9.4 Heliac NGAE cases 127

ωi 0.00665. ωr Convergence with respect to poloidal and toroidal mesh resolution is slower than = − in the W7-X helias case. Sufficient resolution is required to accurately represent the 8, 10 harmonic component. Solving the eigenvalue problem along the real s-axis results in the mode structure plotted in Figure 9.14. It is found that the ( − ) √dominant 10, 13 harmonic component is highly localised near the continuum resonance frequency maximum. However, the 8, 10 harmonic component has the greatest change( − due) to interaction with the continuum resonance. ( − ) In the second H-1NF NGAE case, the same magnetic geometry is considered as in Section 7.6. It is found that the equilibrium representation near the magnetic axis is particularly poor in this case, which results in rapid changes in j∥ and p. A sharp peak in continuum resonance eigenvalue is found near the magnetic axis, which deviates from the behaviour expected based on the behaviour of the continuum resonance eigenvalue in a periodic cylinder. To avoid spurious continuum resonances due to this feature, the density profile is modified to become

ρ s ρ0 1 s 1 Γ1 exp Γ2 s . (9.5) √ The parameters Γ1 0.2 and( ) =Γ2 (43.8− ) are + chosen based− on the approximate magnitude and radial length scale of the increase in frequency approaching the magnetic axis. Again, an NGAE= due to a= maximum in the 4, 5 branch of the continuum resonance frequency is studied. This branch is plotted in Figure 9.15. ( − ) Acoustic mode coupling, pressure gradient and parallel current terms in the reduced ideal MHD wave equation are ignored and CKA solves equation (3.47). For a stellarator with low β and no external current drive, j∥ should be negligible. Ignoring these terms appears to improve convergence of the continuum damping with respect to poloidal and toroidal mesh resolution, suggesting spurious coupling of different harmonics due to poor representation of the equilibrium near the magnetic axis. Ig- noring the aforementioned terms reduces the continuum resonance eigenvalues of the 4, 5 branch. Consequently, the gap between the real component of ω2 for the NGAE and the continuum maximum increases. ( − ) The eigenvalue calculation is performed using complex contours of the same form as in Section 9.2, with parameters xβ 0.1 and xγ 0.92. A phase factor of 4, 5 and basis set composed of ﬁrst-order B-splines are used. Convergence of the NGAE damping ratio, ωi , with respect to N= , N , and N= for various values of α ωr s ϑ ϕ (is displayed− ) in Figure 9.16. A complex normalised frequency of Ω 0.0637 6.62 10−5i and a damping ratio of ωi 0.00104 are obtained. The continuum damping ωr found here is much smaller than the damping due to kinetic effects= obtained− in× Section 7.6. Convergence with respect= − to poloidal and toroidal resolution is fast in this case, indicating that the role of additional harmonics which couple to the dominant 4, 5 harmonic is not signiﬁcant. Unlike previous cases, convergence with respect to radial resolution is slower for larger contour deformations. This indicates that( − deformation) of the contour increases variation of the solution in x. The poorly converged eigenvalue obtained by including acoustic mode coupling, 128 Continuum Damping of Alfvén Eigenmodes in General 2D and 3D Geometry

■

- 0.0055 ▼

● - 0.0056 r /ω i

ω - 0.0057

- 0.0058 ■ ●▼■ ●▼■ ●▼ 50 60 70 80 90

N s (a)

0.000

- 0.001

- 0.002 ●▼■ r

/ω - 0.003 i ω - 0.004

●▼■ - 0.005 ●▼■ ●▼■ - 0.006 ●▼■ ●▼■ 20 30 40 50

N ϑ (b)

0.000 ■ ● - 0.001 ▼ - 0.002

r ●▼■ - 0.003 /ω i ω - 0.004

- 0.005 ●▼■

- 0.006 ●▼■ ●▼■ 5 10 15 20

N φ (c)

Figure 9.13: Convergence of continuum damping for the NGAE identified in the first H-1NF heliac case with respect to (a) radial mesh resolution, Ns, (b) poloidal mesh resolution, Nϑ, and (c) toroidal mesh resolution, Nϕ. For each plot, two of Ns 96, Nϑ 48, and Nϕ 16 are fixed while the other parameter varies. Complex contours have parameters x 0.1 and x 0.8. Damping ratio, ωi , is plotted for α 0.1= ( ), β γ ωr = = α 0.2 ( ), and α 0.5 ( ). = = = ∎ = ● = ▼ §9.4 Heliac NGAE cases 129

1.2

1 (10,−13)

0.8

0.6 m,n

Φ 0.4

0.2

0 (8,−10) −0.2 0 0.2 0.4 0.6 0.8 1 √s

Figure 9.14: The four largest Fourier harmonics of the NGAE identified in the first H-1NF heliac case as a function of the flux surface label, s. The dominant 10, 13 (blue) harmonic and the 8, 10 (red) harmonic associated√ with the continuum resonance are labeled. ( − ) ( − ) pressure gradient and parallel current terms implies a significantly smaller damping ratio of ωi 3 10−4. ωr The mode structure of the NGAE is plotted in Figure 9.17. It is seen that components other≈ than− × the dominant 4, 5 harmonic are indeed negligible. This mode has

( − )

x 10−3 6 1.5

5 1.25

NGAE 4 1 ι , 2 3 0.75 0 ρ Ω / ρ 2 0.5

1 0.25

0 0 0 0.2 0.4 0.6 0.8 1 √s

Figure 9.15: Normalised shear Alfvén continuum resonance eigenvalues Ω2 as a function of the radial coordinate s for the second H-1NF heliac case. The maximum of the 4, 5 ( ) branch results√ in an NGAE, which has a resonance with the same branch. The rotational transform ι (dashed line) and density ρ (solid line) are also ρ0 ( − ) [ plotted. 130 Continuum Damping of Alfvén Eigenmodes in General 2D and 3D Geometry

- 0.001033 ■

- 0.001034 ▼ - 0.001035 r

/ω - 0.001036 i

ω ▼ - 0.001037 ▼ ● ▼ - 0.001038 ● ■ ■ ■ ● ● 60 70 80 90 100 110 120

N s (a)

▼ ▼ - 0.00104 ●■ ●■ ▼ ●■ - 0.00105

r - 0.00106 /ω

i - 0.00107 ω

- 0.00108

- 0.00109 ●▼■ 10 15 20 25 30 35 40

N ϑ (b)

▼ - 0.001010 ●■

- 0.001015

r - 0.001020 /ω

i - 0.001025 ω - 0.001030

- 0.001035 ▼ ● ▼ ■ ●■ - 0.001040 10 12 14 16 18 20

N φ (c)

Figure 9.16: Convergence of continuum damping for the NGAE identiﬁed in the second H-1NF heliac case with respect to (a) radial mesh resolution, Ns, (b) poloidal mesh resolution, Nϑ, and (c) toroidal mesh resolution, Nϕ. For each plot, two of Ns 80, Nϑ 40, and Nϕ 20 are ﬁxed while the other parameter varies. Complex contours have parameters x 0.1 and x 0.92. Damping ratio, ωi , is plotted for β γ ωr = = α =0.2 ( ), α 0.5 ( ), and α 1.0 ( ). = = = ∎ = ● = ▼ §9.4 Heliac NGAE cases 131

1.2

(4,−5) 1

0.8

0.6 m,n

Φ 0.4

0.2

−0.2 0 0.2 0.4 0.6 0.8 1 √s

Figure 9.17: The four largest Fourier harmonics of the NGAE identified in the second H-1NF heliac case as a function of the flux surface label, s. The dominant 4, 5 (blue) harmonic is labeled. √ ( − ) smooth behaviour near the magnetic axis, confirming that the spurious continuum resonance in this region has been eliminated. The NGAE mode structure here is significantly broader than it is where pressure and parallel current terms are considered (see Figure 7.6). This may account for the significantly greater continuum damping found in the former case. It can be shown that the damping is not sensitive to the density profile modification, provided that the spurious continuum resonance is avoided. NGAE damping estimates for a range of Γ1 are shown in Figure 9.18, which have weak dependence on this parameter for Γ1 0.15. This indicates that the modification has relatively little effect on the overall structure of the mode. Therefore, the calculated damping obtained above is taken to> be an accurate indication of that for the unmodified density profile in equation (7.13). The result obtained using the reduced MHD model in three-dimensions can be compared with that for an incompressible periodic cylindrical plasma. Using Ns 721 and α 0.05, the latter model results in an NGAE with complex normalised frequency of Ω 0.0870 2.71 10−5i and damping of ωi 3.12 10−4. Here= a ωr = 2 density profile of ρ ρ0 1 r is considered. Although coupling between different = − × = − × harmonics is negligible, the approximations in Subsection 3.1.2 introduce significant changes to both the= real frequency− and damping of the NGAE. 132 Continuum Damping of Alfvén Eigenmodes in General 2D and 3D Geometry

■ - 0.00080

- 0.00085 r - 0.00090 /ω i

ω - 0.00095

■ - 0.00100 ■ ■ - 0.00105 ■ ■ 0.05 0.10 0.15 0.20 0.25 0.30

Γ 1

Figure 9.18: NGAE continuum damping ratio, ωi , as a function of density proﬁle ωr parameter Γ1 in the second H-1NF case. Chapter 10

Conclusion

In this thesis, several methods are developed for calculating the damping of shear Alfvén eigenmodes in toroidal magnetically confined plasmas. Using these techniques, the damping of components of these modes due to continuum resonances and kinetic effects are computed in cases with large aspect-ratio, fully two-dimensional, or fully three-dimensional magnetic geometry. Thus, tools are developed to accurately compute damping of shear Alfvén eigenmodes in realistic tokamak and stellarator cases. Reduced ideal MHD equations are derived for general toroidal and large aspect- ratio geomtries, which are used to model shear Alfvén waves in low β plasmas. Ex- tensions to these equations are identified which incorporate coupling to ion acoustic waves and non-ideal kinetic effects. Simplifications to the equations are made based on ordering of terms in a toroidal magnetic confinement device with low beta and low curvature. It is shown that the continuum resonance results in a logarithmic singularity in the generic case for either toroidal or stellarator symmetry. Expressions for continuum damping of shear Alfvén eigenmodes are derived based on perturbation of quadratic forms which are constucted based on the relevant wave equations. The application of these equations using shooting and finite element methods is considered. However, results of the perturbative continuum damping calculation in the large aspect-ratio case are found to differ substantially from those of a benchmark calculation. It is found that the effect of the continuum resonance interaction on mode structure can violate the assumptions of the perturbative treatment, even for small values of damping. Nevertheless, the perturbative technique does indicate the qualitative dependence of continuum damping on equilibrium quantities. Variation in equilibrium quantities may result in competing effects on different terms in the perturbative expression, which can make it difficult to use the form of the expressions to predict changes in damping without computing the modified eigenmode numerically. A singular finite element technique for calculating continuum damping of shear Alfvén eigenmodes in ideal MHD is presented. The damping computed using basis functions of a form that matches the asymptotic expansion about the continuum resonance pole agrees with benchmark values provided by the complex contour method. Such basis functions could readily be incorporated into existing finite element codes. The singular finite element method computes the complex eigenfunction over real

133 134 Conclusion

space, which allows for calculation of interaction with fast particles. However, results of this method must be shown to converge with respect to three numerical parameters (compared with two in the resistive and complex contour methods) and over successive iterations. The MHD eigenvalue code CKA is modified to compute solutions for specified complex contours in the flux-surface label, s. This calculation involves analytic continuation of analytic functions fitted to equilibrium quantities in the code Improve to complex s. Thus, the complex contour technique can be applied to model shear Alfvén waves in fully two- or three-dimensional magnetic geometry. An application of the tool developed here is to accurately compute continuum damping of modes driven by energetic particles in stellarators as part of a linear growth rate calculation, with an example in the H-1NF heliac presented in the work of Hole et al. [100]. It could also be used to provide estimates of dissipative damping and mode structure for non-linear simulations of wave-particle interactions in such cases. An interesting application may be to Alfvénic activity in the recently commissioned W7-X helias. This technique could also be used to model changes in continuum damping of modes in tokamaks due to modification of the shear Alfvén continuum by magnetic perturbations [106]. Experimental validation of the use of the reduced ideal MHD model could be achieved through comparison with observations of antenna or electrode driven Alfvén eigenmodes where dissipation is dominated by continuum resonance interactions, such as in the CHS torsatron [32]. This implementation of the complex contour method is verified by comparison with a benchmark continuum damping calculation provided by the resistive method. It is found that the former method converges faster than the latter with respect to radial mesh resolution, decreasing computational requirements. Continuum damping is calculated for shear Alfvén eigenmodes in fully three-dimensional magnetic geometries in torsatron, helias, and heliac stellarators, with satisfactory convergence with respect to radial, poloidal, and toroidal mesh resolution demonstrated in each case. Convergence with respect to poloidal and toroidal mesh resolution is determined by the range of m and n in the Fourier harmonics of the solution. Even in an NGAE where a single harmonic is dominant, considering full three-dimensional geometry can result in the damping value obtained differing significantly from that found in the corresponding cylindrically symmetric geometry. Damping of shear Alfvén eigenmodes due to kinetic effects beyond ideal MHD is also considered, including radiative, collisional, and electron Landau damping components. For a TAE the computed value for these sources of damping are found to be sensitive to change in the mode structure introduced by the large aspect-ratio approximation for a tokamak. This indicates the importance of considering the full two- or three-dimensional magnetic geometry of a plasma configuration. However, the damping computed is largely determined by kinetic effects at the avoided crossing, and is only weakly depedent on how these are modelled elsewhere. It is found that electron-ion collisional damping is dominant for a GAE modelled for H-1NF parameters, with the calculated order of magnitude matching the experimentally observed rate of decay. 135

Opportunities exist to further develop the damping calculations described here. Currently, for calculations using CKA, the range of m and n that can be resolved is limited by the size of matrices that can be used with the numerical eigenvalue problem solver. The computational efficiency of the finite element method complex contour continuum damping calculation could be improved by using a Fourier basis to represent poloidal and toroidal variation in eigenmodes. Use of such bases is common practice in existing finite element MHD eigenvalue codes, such as NOVA [107] and CAS3D [108]. Use of more complicated complex contours could be explored for use in cases where multiple continuum resonances exist or to determine mode structures for real s. The methods developed for calculating continuum damping in this thesis could also be applied to ideal MHD models which incorporate additional physics. Use of the complex contour method with a compressible ideal MHD model would enable the calculation of damping which occurs due to resonances with the ion acoustic continuum [109] in a plasma with non-zero β. BAEs modelled for H-1NF are believed to experience significant damping due to such continuum resonances [110], calculation of which could help determine whether these modes are observed experimentally. It has been shown that continuous spectra and Alfvén eigenmodes are affected by toroidal flow [111] and pressure anisotropy [112]. A model incorporating these effects could be adopted to investigate their influence on continuum damping. Future work could also extend the analysis of continuum damping to Alfvén eigenmodes in configurations where the plasma is not foliated by nested toroidal flux surfaces. Alfvén eigenmodes can be induced by magnetic islands, occuring due to the resulting modulation of magnetic field strength and metric tensor elements either within these islands [113] or outside of them [114]. Computing modes in such cases would require a change in the way that the equilibrium is calculated and the coordinate system used. Furthermore, methods could be developed to calculate continuum damping in cases where the continuum resonance has a different form from that assumed here. In Appendix A, it is found that the continuum resonance singularity may have a power-law form in configurations with continuous symmetry but without an additional discrete symmetry. Moreover, the behaviour of continuum modes can change abbruptly from filling flux surfaces to exponentially localised along magnetic field lines for sufficiently large non-axisymmetric equilibrium perturbations [115]. The techniques developed for computing continuum damping in this thesis may also be applicable to other physical systems where eigenmodes lose energy due to interactions with resonant layers. For example, consider waves in a dissipationless 2 ω2 2 system described by the Helmholtz equation, Φ v2 Φ 0, where v is a real continuous function of position. Eigenmode solutions have logarithmic singularities on surfaces at boundaries between propagating and∇ evanescent+ = regions, which result in damping that could be analysed using the aforementioned techniques. In inhomogenous magnetised plasmas, such interactions can occur for electromagnetic waves due to ion or electron cyclotron resonances. 136 Conclusion Appendix A

Continuum Resonance Singularities

In this appendix, we show that the generic form of the continuum resonance singularity in Φ is logarithmic for devices with stellarator or both up-down and axial symmetry. This is demonstrated for both the wave equation in general geometry and that for a tokamak with large aspect-ratio and circular cross-section. The operator in the continuum resonance condition is shown to be self-adjoint.

A.1 General geometry

2 Let ωR be an eigenvalue of equation (3.90) for ψ ψR. Consider the corresponding solution of equation (3.48) over a narrow region around ψR. Define orthogonal flux coordinates ψ, ϑ, ϕ such that gψϑ gψϕ gϑϕ 0= and assume that the solution has the form Φ χ ψ η ϑ, ϕ exp iωRt . The separation of the solution into a function dependent on( ψ and) one dependent= on ϑ=, ϕ is= justified due to the small scale in ψ. If we insert= the( expression) ( ) for (Φ− into) equation (3.48) and take the flux surface inner product defined in equation (3.93) with (η we) obtain

η ϑ, ϕ , T χ ψ η ϑ, ϕ 0, (A.1) where T Φ is the differential⟨ ( operator) [ acting( ) on( the)]⟩ left-hand= side of equation (3.48). We will express this equation in coordinate dependent form using certain identi- [ ] ties for ﬂux surface intregrals. Firstly, for a single valued function on a ﬂux surface η, integration by parts and application of Gauss’s theorem and B 0 results in

dS dS ∇ ⋅ = B η Bη (A.2) ψ ψ ψ ψ c ⋅ ∇ = c0. ∇ ⋅ ( ) (A.3) S∇ S S∇ S Moreover, by considering the ﬂux surface= integral of a divergence v can be rewrit-

137 ∇ ⋅ 138 Continuum Resonance Singularities

ten as the derivative of a ﬂux surface integral

dS dV 1 v lim dτ v (A.4) ψ ψ dψ δV→0 δV δV c ∇ ⋅ = dV d S ∇ ⋅ S∇ S dS v (A.5) dψ dV ψ = d dS ⋅ c v ψ. (A.6) dψ ψ ψ = c ⋅ ∇ We will consider the contribution of eachS∇ termS in the operator T Φ to equation (A.1) separately. First, consider the polarisation term applying the identities derived above, [ ]

2 dS ∗ ωR η ϑ, ϕ , TPOL χ ψ η ϑ, ϕ η ⊥ χη (A.7) ψ 2 ψ vA ⟨ ( ) [ ( ) ( )]⟩ = − 2 ∇ ⋅ ∇ ( ) c dS ωR ∗ S∇ S ⊥η ⊥ χη ψ 2 ψ vA = c d dS∇ω2 ⋅ ∇ ( ) S∇ S R η∗ ψ χη (A.8) ψ 2 dψ ψ vA − dSc ω2 ∇ dχ⋅ ∇ ( ) S∇R S η∗ η ψ χ η ψ 2 ψ vA dψ = c ∇dχ ⋅ ∇ + ∇ bS∇ ηS∗b η ψ χ η dψ − d⋅ ∇ dS⋅ ω2 ∇ + ∇ d χ R η∗ ψ η ψ χ η (A.9) ψ 2 dψ ψ vA dψ − d c ω2 gψψ ∇dχ⋅ ∇ + ∇ S∇ RS η, 2 η dψ vA dψ = − d 2 i dS ωR ∗ ⊥η ⊥η χ. (A.10) ψ 2 ψ vA + c ∇ ⋅ ∇ Here we use the orthogonality of ψ and η dueS∇ toS the use of an orthogonal coordinate system. ∇ ∇ Similarly, now consider the ﬁeld line bending term,

dS ∗ η ϑ, ϕ , TFL χ ψ η ϑ, ϕ η b ⊥ b χη (A.11) ψ ψ ⟨ ( ) [ ( ) ( )]⟩ = − c dS ∇ ⋅ [∗ ∇ ⋅ ∇ ( ⋅ ∇ ( ))] S∇ Sb η ⊥ χb η (A.12) ψ ψ = c dS ( ⋅ ∇ ) ∇∗ ⋅ ∇ ( ⋅ ∇ ) S∇ S b η χb η ψ ψ = c dS∇ ⋅ [( ⋅ ∇ ∗ ) ∇ ( ⋅ ∇ )] S∇ S b η 1 bb χb η ψ ψ − c ∇ ( ⋅ ∇ ) ⋅ ( − ⋅) ∇ ( ⋅ ∇ ) S∇ S §A.1 General geometry 139

(A.13) d dS b η∗ ψ dψ ψ ψ = dcχ [( ⋅ ∇ ) ∇ Sψ∇ bS η χ b η dψ ⋅ ∇dS ( ⋅ ∇ ) + ∇ ( ⋅ ∇ ) b η∗ 1 bb ψ ψ − cdχ [∇ ( ⋅ ∇ )( − ⋅ ) S∇ψS b η χ b η (A.14) dψ ⋅d ∇ ( ⋅ ∇ ) + ∇ ( d⋅χ ∇ ) b∗ η, gψψb η dψ dψ = ba∗ ⋅η ∇, ψ ⋅b ∇ fη

∗ ∗ ∗ dχ + [⟨b ⋅ ∇η ,∇ ψ⋅ ∇ ( b⋅ ∇ η)⟩ dψ − ⟨ d ⋅ ∇ ∇ ⋅ ∇ ( ⋅ ∇ )⟩] b∗ η, ψ b η dψ

+ dS⟨ ⋅ ∇ ∇ ⋅ ∇∗ ( ⋅ ∇ )⟩ ⊥ b η ⊥ b η χ. (A.15) ψ ψ − c ∇ ( ⋅ ∇ ) ⋅ ∇ ( ⋅ ∇ ) Here the identitiy in equation (A.3) is usedS∇ to obtainS equation (A.12). Likewise, we are able to replace ⊥ with in the ﬁrst term of equation (A.13). The contribution from the pressure operator is found to be, ∇ ∇ dS ∗ 2µ0 η ϑ, ϕ , TP χ ψ η ϑ, ϕ η b κ b P ψ ψ B2 ⟨ ( ) [ ( ) ( )]⟩ = − c χη ∇ ⋅ ( × )( × ∇ ) (A.16) S∇ S d 2µ0 ⋅∇ ( )]η, ψ b κ b P η χ dψ B2 = − c ∗ ∇ 2⋅µ(0 × )( × ∇ ) ⋅ ∇ h b κ η, b P η χ. (A.17) B2 Finally, consider the contribution of+ c( the× parallel) ⋅ ∇ current( operator,× ∇ ) ⋅ ∇ h

dS ∗ µ0j∥ η ϑ, ϕ , TC χ ψ η ϑ, ϕ η bb χη ⊥ (A.18) ψ ψ B

⟨ ( ) [ ( ) ( )]⟩ = c d dS∇ ⋅µ0j∥ ∗[∇ × ( ⋅ ∇ ( ))] S∇ S η ψ ⊥ χη dψ ψ ψ B

= − cdS µ0j∥ ∗ ∇ ⋅ [∇ × ∇ ( )] S∇ S ⊥η ⊥ χη (A.19) ψ ψ B

+dc dS µ0j∇∥ ∗ ⋅ [∇ × ∇ ( )] S∇ S η χ ψ ⊥η dψ ψ ψ B

= dχc dS µ0j∥ ∇ ⋅ [∗ ∇ × ∇ ] S∇ S ⊥η η ψ dψ ψ ψ B + c ∇ ⋅ (∇ × ∇ ) S∇ S 140 Continuum Resonance Singularities

dS µ0j∥ ∗ χ ⊥η ⊥η (A.20) ψ ψ B

+d c dS µ∇0j∥ ∗⋅ (∇ × ∇ ) χS∇ S η ψ ⊥η dψ ψ ψ B

= dχ c dS µ0j∥ ∇ ⋅ [∇ × ∇ ]∗ S∇ S b ηb ψ η dψ ψ ψ B

+ c dS µ0j∥ ⋅∗ ∇ ⋅ (∇ × ∇ ) χ S∇ S ⊥η ⊥η (A.21) ψ ψ B

+ dc dS ∇∗ ⋅ (∇µ ×0 ∇j∥ ) χS∇ S η b b ψ η dψ ψ ψ B

= − cdS µ0j∥ ⋅∗ ∇ ⋅ (∇ × ∇ ) χ S∇ S ⊥η ⊥η ψ ψ B

+ dc dS µ0j∥∇ ⋅ (∗∇ × ∇ ) S∇ S b η b ψ η dψ ψ ψ B

−dχ c dS µ0j∥ ⋅ ∇ ⋅ (∇ × ∇ ∗) S∇ S b ηb ψ η dψ ψ ψ B

+ cdS µ0j∥ ∗⋅ ∇ ⋅ (∇ × ∇ ) S∇ S b η b ψ η . (A.22) ψ ψ B − c ⋅ ∇ ⋅ (∇ × ∇ ) We can show that S∇ S

j∥ µ B j B b µ 0 j (A.23) 0 B B B2 × ( × ) ⋅ ∇ = µ∇0 ⋅ −B P (A.24) B B2 2µ × ∇ = − 0∇ ⋅b B P. (A.25) B3 Rearranging equation (3.23) and inserting= − into( the× ∇ above) ⋅ ∇ equation yields,

j∥ 2 b µ b κ P. (A.26) 0 B B2 ⋅ ∇ = − ( × ) ⋅ ∇ This identity is then used to express the ﬁrst term in equation (A.22) in terms of pressure gradient and curvature, making use of the fact P ψ,

d dS ∗ 2µ0 ∇ ∥ ∇ η ϑ, ϕ , TC χ ψ η ϑ, ϕ χ η b κ ψb P η dψ ψ ψ B2

⟨ ( ) [ ( ) ( )]⟩ = c dS µ0j∥ ∗( × ) ⋅ ∇ ⋅ (∇ × ∇ ) χ S∇ S ⊥η ⊥η ψ ψ B

+ dc dS µ0j∥∇ ⋅ ∇∗ × ∇ S∇ S b η b ψ η dψ ψ ψ B − c ⋅ ∇ ⋅ (∇ × ∇ ) S∇ S §A.1 General geometry 141

dχ dS µ0j∥ b ηb ψ η∗ dψ ψ ψ B

+ cdS µ0j∥ ∗⋅ ∇ ⋅ (∇ × ∇ ) S∇ S b η b ψ η . (A.27) ψ ψ B − c ⋅ ∇ ⋅ (∇ × ∇ ) Assume that the plasma has stellaratorS∇ symmetry.S This is a design feature of all modern stellarators [60] and therefore not a particularly restrictive assumption. Stel- larator symmetry implies that, if η ψ, θ, φ is a solution of equation (3.90) then so is ζ ψ, θ, φ η ψ, θ, φ . Moreover, because the operator acting on η in this equation is self-adjoint, η∗ is also a solution.( Hence,) where two linearly independent solutions ( ) = ( 2− −∗ ) ∗ exist for some ωR, η aη bζ. As all operators acting on them are linear, η and η may be exchanged in the integrals comprising equation (A.1), leading to cancellation = + dχ of anti-symmetric terms which are found in the coeffecient of dψ . The same result is obtained where only one linearly independent solution to equation (3.90) exists 2 for ωR, which is the generic case for asymmetric fully three-dimensional conﬁgura- tions. This cancellation results in an equation describing the variation in the mode amplitude with ψ near ψR of the form

d dχ D ψ ARχ 0. (A.28) dψ dψ ( ) + = dχ In this equation, the coeffecient of dψ in the brackets is

D ψ η, S η , (A.29) where S η is the operator acting on(η )on= the⟨ left-hand[ ]⟩ side of equation (3.90). Hence, dD D ψR 0 and to ﬁrst-order we can approximate D ψ dψ ψ ψR . [ ] ψ=ψR (Equation) = (A.28) has solutions of the form, ( ) ≈ U ( − )

χ ψ χ1 ln ψ ψR χ2 (A.30) truncating the solution after the( ﬁrst) = term( in− its two) + linearly independent solutions. Clearly, this result can also be applied to a device with both up-down and axial symmetries. Thus, the continuum resonance singularity will usually be logarithmic for tokamak plasmas as well. A possible exception may be for a continuum resonance near the last closed ﬂux surface of a single divertor plasma. In asymmetric cases where there are nevertheless two (or more) linearly independent degenerate solutions to equation (3.90), the integrals which are anti-symmetric in η and η∗ are not expected to cancel. Consequently, these terms provide a purely dχ imaginary contribution to the coeffecient of dψ so that we obtain an equation in the form d dχ dχ D ψ iFR ARχ 0, (A.31) dψ dψ dψ

( ) + + =∞ j+k where FR R. Using the Frobenius method, let χ j=0 Xj ψ ψR . Assuming

∈ = ∑ ( − ) 142 Continuum Resonance Singularities

that D ψ has the same form as previously, and inserting this into equation (A.31) yields the indicial equation ( )

2 dD k ikFR 0. (A.32) dψ ψ=ψR W + = Thus, to ﬁrst-order in each of the independent solutions obtained,

iτ χ ψ χ1 ψ ψR χ2. (A.33)

Here we introduce the parameter( ) = ( − ) +

F τ R . (A.34) dD dψ ψ=ψ = R Using matched asymptotic expansions, itU is found that the expressions in equations (A.30) and (A.33) are approximately equal for τ ln ψ ψR 1. Thus, it can be shown that the discontinuity in the solution in equation (A.33) due to a branch cut from the pole, intersecting the real axis approaches thatS of( equation− )S ≪ (A.30) as τ 0.

The correction due to the slow sound approximation does not change the→ behaviour of the solution near a continuum resonance. Consider the contribution of the slow sound approximation term to the left-hand side of equation (A.1),

2 dS ∗ 4cs η ϑ, ϕ , TSSA χ ψ η ϑ, ϕ η b κ b κ χη ψ 2 ψ vA ⟨ ( ) [ ( ) ( )]⟩ = − c ∇ ⋅ ( × )( × ) ⋅ ∇ ( ) (A.35) S∇ S d dS 4c2 η∗ s ψ b κ b κ ψ 2 dψ ψ vA = − dχc ∇ ⋅ ( × )( × ) η Sψ∇ Sχ η dψ ⋅ dS∇ +4c2∇ s η∗ b κ b κ ψ 2 ψ vA + cdχ ∇ ⋅ ( × )( × ) S∇η ψS χ η (A.36) dψ 2 ⋅ d ∇ 4+cs ∇ 2 dχ η, 2 ψ b κ η dψ vA dψ = − d dS S∇4c2⋅ ( × )S i η∗ s ψ b κ b κ η ψ 2 ψ vA + − cdS 4c2 ∇ ⋅ ( × )( × ) ⋅ ∇ dχ S∇η S s ψ b κ b κ η∗ ψ 2 ψ vA dψ + c d dS∇ 4⋅ (c2 × )( × ) ⋅ ∇ S∇ S η∗ s ψ b κ b κ η ψ 2 dψ ψ vA + − c ∇ ⋅ ( × )( × ) ⋅ ∇ S∇ S §A.2 Large aspect-ratio 143

dS 4c2 s η∗ b κ b κ η χ. (A.37) ψ 2 ψ vA + c ∇ ⋅ ( × )( × ) ⋅ ∇ This contribution has the same form as thatS∇ S from the preceding terms, thus the behaviour of the solution near ψR does not change when the slow sound approximation term is included. Including the slow sound approximation correction to the wave equation, D ψ in equation (A.28) and equation (A.31) is deﬁned as in equation (A.29) letting S η be the slow sound approximation correction in expression (3.91). ( ) [ ]

A.2 Large aspect-ratio

Continuum resonance singularities of Ei in the approximation for the large aspect- ratio circular cross-section tokamak case can also be shown to be logarithmic for generic cases, as expected based on the up-down symmetry of this conﬁguration. Consider ﬂux surfaces where a continuum resonance does not occur, so that Di,j 0, though may be arbitrarily close to such a surface. Letting s r rR, equation (3.79) can be rearranged to give the following expression, Z Z ≠ = − 2 d Ej dDi,j dEj D−1 D−1 A E 0. (A.38) ds2 i,j ds ds i,j i,j j + + = dZDi,jZ Assume that dr 0, corresponding to cases where the continuum resonance r=rR does not occur at aV stationary≠ point of ωR ψ . Thus, near the continuum resonance pole we can use the approximation ( )

−1 adj Di,j Di,j . (A.39) dZDi,jZ s ds ≈ s=0 V Here adj Di,j is the adjugate matrix of Di,j which approaches a ﬁnite constant in the limit ψ ψR. We can rewrite equation (A.38) as → 2 d Ej 1 dEj 1 M N E 0, (A.40) ds2 s i,j ds s i,j j where + + =

adj D i,k dDk,j Mi,j δi,j, (A.41) ⎡ dZDi,jZ ⎤R ⎢ ds ⎥R ⎢ (ds ) ⎥Rs=0 = ⎢ ⎥R = ⎢ ⎥R ⎢ adj D ⎥R ⎣ i,k⎦R Ni,j Ak,j . (A.42) ⎡ dZDi,jZ ⎤R ⎢ ⎥R ⎢ (ds ) ⎥Rs=0 = ⎢ ⎥R ⎢ ⎥R ⎢ ⎥R ⎣ ⎦R 144 Continuum Resonance Singularities

Following the Frobenius method, we assume that the solution near the resonance takes the form ∞ k l Ej s al,js , (A.43) l=0 = Q where a0,i 0 for some i. Inserting equation (A.43) into equation (A.40) yields the indicial equation, ≠ 2 k a0,i 0. (A.44)

As this equation has a double root at k 0,= it provides only one of the two linearly independent solutions. = To obtain a second linearly independent solution, we assume a solution of the form ∞ k l l Ej s al,js bl,js ln s , (A.45) l=0 = Q + ( ) where either a0,i 0 or b0,i 0 for some i. In this case the indicial equations will be

2 ≠ ≠ k a0,i 2kb0,i 0, (A.46) k2b 0. (A.47) + 0,i = As was the case before, k 0 is a solution to= these equations. In this case vector coeffecients a0,i and b0,i can be set arbitrarily and thus represent linearly independent solutions. Hence the solution= near the resonance can be approximated to ﬁrst-order in these two components as

Em am bm ln s . (A.48)

Inclusion of second-order coeffecients= yields+ ( )

Em δm,n sNm,n an 2sNm,n ln s δm,n sNm,n bn. (A.49)

= ( − ) + ( + ( )( − )) If we express the wave equation in terms of Ci rather than Ei near s 0 we obtain

∂2C ∂C 1 = i M i N C 0, (A.50) ∂s2 i,j ∂s s i,j i where + + = ∂ M A A−1 (A.51) i,j i,k ∂r k,j = Ai,k adjD k,j N . (A.52) i,j ∂YDY ( ∂r ) = Assume a solution of the form

∞ k l l Cj s al,js bl,js ln s , (A.53) l=0 = Q + ( ) §A.3 Self-adjointness of continuum resonance condition operator 145

where either a0,i 0 or b0,i 0 for some i. This yields the indicial equations

≠ ≠ k 1 ka0,i 2k 1 b0,i 0 (A.54) k k 1 b 0. (A.55) ( − ) + ( − ) 0,i =

Thus, k 0 and 1 correspond to a0,i 0 for( − some) i and= b0,i 0 for all i. The solution near the singularity to ﬁrst-order in the two linearly independent solutions is = ≠ =

Cm am bms ln s , (A.56) which approaches am as s 0. = + ( )

→ A.3 Self-adjointness of continuum resonance condition operator

It can be shown that the operator acting on the left-hand side of equation (3.90) is self-adjoint with respect to the inner product deﬁned in equation (3.93). Expressing 2 the former in the form TI ω TW applying this to η and then taking the inner product with ζ yields − 2 ψψ ψψ 2 dS ω g ∗ ∗ g ζ, TI ω TW η ζ η B0ζ b b η (A.57) ψ 2 ψ vA B0 2 ψψ ψψ a − f = c dS ω g ∗ + ⋅ ∇g ∗ ⋅ ∇ S∇ S ζ η B0b ζ b η (A.58) ψ 2 ψ vA B0 = c b ζ∗ b η g+ψψ ⋅ ∇ ⋅ ∇ (A.59) S∇ S dS ω2gψψ − ( ⋅ ∇ )( ⋅ ∇ζ∗)η b ζ∗ b η gψψ . (A.60) ψ 2 ψ vA = c − ( ⋅ ∇ )( ⋅ ∇ ) The second term in the integrandS∇ onS the second line is eliminated based on the 2 2 identity in equation (A.3). It can be seen that ζ, TI ω TW η η, TI ω TW ζ , and thus the operator is self-adjoint. a − f = a − f The continuum resonance condition represents the Euler-Lagrange equation obtained by extremising the functional

dS ω2gψψ S η η∗η b η∗ b η gψψ . (A.61) ψ 2 ψ vA [ ] = c − ( ⋅ ∇ )( ⋅ ∇ ) As was the case for the shearS∇ AlfvénS wave equation, S η 0 trivially for solutions of this differential equation. As in Section ??, the variational solution satisﬁes a weak formulation of the continuum resonance condition. For[ ]η= V which satisﬁes the 2 continuum resonance condition on ψ, ζ, TI ω TW η 0 ζ V, where V is the space of smoothly varying doubly periodic functions on ψ. ∈ 2 a − f = ∀ ∈ As the correction to TI ω TW due to the slow sound approximation in equa-

− 146 Continuum Resonance Singularities

tion (3.91) is a real multiplication operator, addition of this term preserves the self- adjointness of the above operator trivially. The corresponding correction to the quadratic form in equation (A.61) is

2 4cs 2 2 ψ b κ η. (A.62) vA − S∇ ⋅ × S Appendix B

Finite Element Method Implementation Using MATLAB

A set of MATLAB scripts have been written to find solutions of the shear Alfvén wave equation in either a large aspect-ratio circular cross-section tokamak (equation (3.78)) or a large aspect-ratio stellarator (equation (3.74)) using a finite element method. Ki- netic effects may be incorporated into the calculation as required, through inclusion of additional terms present in equation (3.103) or equation (3.104). Options for calculating continuum damping using either complex contour integration (described in Chapter 5) or singular basis functions (described in Chapter 6) are incorporated into this code. The purpose of the code is to demonstrate the validity of the latter technique by comparison with the former. As such, it has not been optimised for numerical efficiency. The solution proceeds as described below.

1. The equilibrium and solution parameters are defined. The equilibrium is defined by specifying q r , ρ r and e. The radial mesh, spline order, Fourier harmonics, and matrix eigenvalue solver parameters used in the calculation are also specified. If the( ) complex( ) contour method is to be used, the real and imaginary components of the contour f x and g x are defined using polynomials. An estimate for rr and the dimensions of special basis functions are defined if the singular finite element method( is) to be used.( )

2. The mesh is generated. The mesh can be spaced either uniformly or exponentially about a chosen location. In complex contour method calculations, the mesh is represented in terms of the contour parameter x.

3. Basis functions are generated. B-spline basis functions of order n are calculated using equations (3.109) and (3.110). Knots coincide with mesh points, with knots of multiplicity n 1 occuring at end points to ensure that boundary conditions are met. If the singular ﬁnite element method is used, special basis functions replace these regular+ basis functions over a speciﬁed interval.

4. Elements of matrices in the eigenvalue problem are evaluated. Integral expressions for these terms are approximated by numerical integration using an adap- tive quadrature method provided by MATLAB. To construct the matrix eigen-

147 148 Finite Element Method Implementation Using MATLAB

value problem, each basis function is assigned a unique index (chosen to be l i 1 Nm m mmin 1 for regular basis functions).

5. The= ( matrix− ) eigenvalue+ ( − problem+ ) is solved. The solution is computed using the eigenvalue solver provided by MATLAB, which uses QZ decomposition for non self- adjoint matrices and Cholesky decomposition for self-adjoint matrices. Alter- natively, techniques for sparse matrix eigenvalue problems can be used to ﬁnd a subset of solutions with eigenvalues close to a speciﬁed value.

6. Eigenmode solutions are evaluated and plotted. Both real and imaginary components of a subset of eigenmodes which correspond to eigenvalues near a speci- ﬁed value of interest are evaluated over a grid of r values.

As noted in Section 3.6, approximations for continuum modes appear among com- 2 puted eigenmodes. Thus ωR r can be found from the frequencies and maxima of the computed eigenmodes. ( ) References

1. G. Audi, F.G. Kondev, M. Wang, B. Pfeiffer, X. Sun, J. Blachot, and M. Mac- Cormick. The nubase2012 evaluation of nuclear properties. Chinese Physics C, 36(12):1157, 2012. 1

2. J. Wesson. Tokamaks. Oxford University Press, Oxford, UK, 2004. 1, 2

3. J. Ongena and G. Van Oost. Energy for future centuries: prospects for fusion power as a future energy source. Fusion Science and Technology, 57(2T):3–15, 2010. 1

4. I. Cook. Economic, safety and environmental characteristics of commercial fusion power. In 1st IAEA technical meeting on ﬁrst generation fusion power plants: design and technology, IAEA, Vienna, 2005. 1

5. "World Energy Outlook 2014: Executive Summary". Technical report, Interna- tional Energy Agency, 2014. 2

6. "IPCC, 2014: Summary for Policymakers. In: Climate Change 2014: Mitigation of Climate Change. Contribution of Work- ing Group III to the Fifth Assessment Report of the Intergovernmental Panel on Climate Change". Technical report, Intergovernmental Panel on Climate Change, 2014. 2

7. "A roadmap to the realisation of fusion energy". Technical report, European Fusion Development Agreement, 2012. 2

8. A. H. Boozer. Physics of magnetically conﬁned plasmas. Reviews of Modern Physics, 76(4):1071, 2005. 2, 5, 20, 116

9. H. Goedbloed and S. Poedts. Principles of Magnetohydrodynamics. Cambridge University Press, Cambridge, UK, 2004. 2, 3, 6, 7, 8, 10, 11, 13, 14, 33, 40, 43, 44

10. R. J. Goldston and P. H. Rutherford. Introduction to Plasma Physics. CRC Press, 1995. 3

11. R. J. Goldston and P. H. Rutherford. Introduction to Plasma Physics. Taylor & Francis, New York, USA, 2004. 3

12. R. F. Post. The magnetic mirror approach to fusion. Nuclear Fusion, 27(10):1579, 1987. 3

13. M. D. Kruskal and R. M. Kulsrud. Equilibrium of a magnetically conﬁned plasma in a toroid. Physics of Fluids, 1(4):265–274, 1958. 3, 20

149 150 References

14. R. L. Dewar, D. A. Monticello, and W. N. C. Sy. Magnetic coordinates for equilibria with a continuous symmetry. Physics of Fluids, 27(7):1723, 1984. 4, 20

15. L. Spitzer. The stellarator concept. Physics of Fluids, 1(4):253–264, 1958. 4

16. C. D. Beidler, E. Harmeyer, F. Hernegger, et al. Stellarator fusion reactors-an overview. In Toki Conf. ITC12, 2001. 5

17. F. F. Chen. Introduction to Plasma Physics and Controlled Fusion. Number v. 1 in Introduction to Plasma Physics and Controlled Fusion. Springer, 2006. 5

18. P. M. Bellan. Fundamentals of Plasma Physics. Cambridge University Press, 2006. 7

19. N. N. Gorelenkov, E. Fredrickson, M. A. Van Zeeland, H. L. Berk, N. A. Crocker, D. Darrow, G. Y. Fu, W. W. Heidbrink, S. Kubota, J. Menard, et al. Theory and observations of low frequency eigenmodes due to Alfvén acoustic coupling in toroidal fusion plasmas. In Proc. 22nd Int. Fusion Energy Conf. 2008 (Geneva, Switzerland, 2008), pages 5–2, 2008. 11

20. H. L. Berk, J. W. Van Dam, Z. Guo, and D. M. Lindberg. Continuum damping of low-n toroidicity-induced shear Alfvén eigenmodes. Physics of Fluids B: Plasma Physics, 4(7):1806–1835, 1992. 11, 14, 37, 38, 55, 61, 62, 63, 64, 66, 69

21. W. W. Heidbrink. Basic physics of Alfvén instabilities driven by energetic particles in toroidally conﬁned plasmas. Physics of Plasmas, 15(5):055501, 2008. 11, 12, 13, 15, 16, 42

22. C. Z. Cheng, L. Chen, and M. S. Chance. High-n Ideal and Resistive Shear Alfvén Waves in Tokamaks. Annals of Physics, 161:21–47, 1985. 11

23. R. Betti and J. P. Freidberg. Ellipticity induced Alfvén eigenmodes. Physics of Fluids B, 3(8):1865–1870, 1991. 11

24. R. Betti and J. P. Freidberg. Stability of Alfvén gap modes in burning plasmas. Physics of Fluids B: Plasma Physics, 4(6):1465–1474, 1992. 11

25. Ya. I. Kolesnichenko, V. V. Lutsenko, H. Wobig, Yu. V. Yakovenko, and O. P. Fe- senyuk. Alfvén continuum and high-frequency eigenmodes in optimized stellarators. Physics of Plasmas, 8(2):491–509, 2001. 11, 34, 35, 37, 42

26. N. Nakajima, C. Z. Cheng, and M. Okamoto. High-n helicity-induced shear Alfvén eigenmodes. Physics of Fluids B, 4(5):1115–1121, 1992. 12

27. K. Appert, R. Gruber, F. Troyon, and J. Vaclavik. Excitation of global eigenmodes of the Alfvén wave in Tokamaks. Plasma Physics, 24:1147–1159, 1982. 12

28. S. E. Sharapov, B. Alper, H. L. Berk, D. N. Borba, B. N. Breizman, C. D. Challis, A. Fasoli, N. C. Hawkes, T. C. Hender, J. Mailloux, S. D. Pinches, D. Testa, and References 151

EFDA-JET work programme. Alfvén wave cascades in a tokamak. Physics of Plasmas, 9(5):2027–2036, 2002. 12, 44

29. Y. I. Kolesnichenko, V. V. Lutsenko, A. Weller, A. Werner, Y. V. Yakovenko, J. Geiger, and O. P. Fesenyuk. Conventional and nonconventional global Alfvén eigenmodes in stellarators. Physics of Plasmas, 14:102504, 2007. 12, 44

30. A. D. Turnbull, E. J. Strait, W. W. Heidbrink, M. S. Chu, H. H. Duong, J. M. Greene, L. L. Lao, T. S. Taylor, and S. J. Thompson. Global Alfvén modes: Theory and experiment*. Physics of Fluids B, 5(7):2546–2553, 1993. 12, 14, 15, 44

31. A. Fasoli, D. Borba, G. Bosia, D. J. Campbell, J. A. Dobbing, C. Gormezano, J. Jacquinot, P. Lavanchy, J. B. Lister, P. Marmillod, J. M. Moret, A. Santagiustina, and S. Sharapov. Direct Measurement of the Damping of Toroidicity-Induced Alfvén Eigenmodes. Physical Review Letters, 75(4):645, 1995. 12, 15

32. G. Matsunaga, K. Toi, S. Kawada, J. Kotani, C. Suzuki, K. Matsuoka, and CHS Group. Excitation of Toroidicity-Induced Alfvén Eigenmodes by the Electrodes Inserted in a Heliotron/Torsatron Plasma. Physical Review Letters, 94(22):225005, 2005. 12, 15, 116, 134

33. G. Y. Fu and J. W. Van Dam. Excitation of the toroidicity induced shear Alfvén eigenmode by fusion alpha particles in an ignited tokamak. Physics of Fluids B, 1:1949–1952, October 1989. 13, 89

34. H. H. Duong, W. W. Heidbrink, E. J. Strait, T. W. Petrie, R. Lee, R. A. Moyer, and J. G. Watkins. Loss of energetic beam ions during TAE instabilities. Nuclear Fusion, 33(5):749, 1993. 13

35. R. B. White, E. Fredrickson, D. Darrow, M. Zarnstorff, R. Wilson, S. Zweben, K. Hill, Yang Chen, and Guoyong Fu. Toroidal Alfvén eigenmode-induced ripple trapping. Physics of Plasmas, 2(8):2871–2873, 1995. 13

36. M. A. Van Zeeland, W. W. Heidbrink, R. K. Fisher, M. García Munoz,¯ G. J. Kramer, D. C. Pace, R. B. White, S. Aekaeslompolo, M. E. Austin, J. E. Boom, I. G. J. Classen, S. da Graça, B. Geiger, M. Gorelenkova, N. N. Gorelenkov, A. W. Hyatt, N. Luhmann, M. Maraschek, G. R. McKee, R. A. Moyer, C. M. Muscatello, R. Nazikian, H. Park, S. Sharapov, W. Suttrop, G. Tardini, B. J. Tobias, Y. B. Zhu, DIII-D, and ASDEX Upgrade Teams. Measurements and modeling of Alfvén eigenmode induced fast ion transport and loss in DIII-D and ASDEX Upgrade. Physics of Plasmas, 18(5), 2011. 13

37. G. Vlad, S. Briguglio, G. Fogaccia, F. Zonca, and M. Schneider. Alfvénic instabilities driven by fusion generated alpha particles in ITER scenarios. Nuclear Fusion, 46(1):1, 2006. 13

38. F. Zonca and L. Chen. Resonant Damping of Toroidicity-Induced Shear-Alfvén Eigenmodes in Tokamaks. Physical Review Letters, 68(5):592–595, 1992. 13, 14 152 References

39. M. N. Rosenbluth, H. L. Berk, J. W. Van Dam, and D. M. Lindberg. Continuum damping of high-mode-number toroidal Alfvén waves. Physical review letters, 68(5):596, 1992. 13, 14, 43

40. L. Landau. On the vibrations of the electronic plasma. JETP, 16:574, 1946. 14, 45

41. S. Poedts, W. Kerner, J. P. Goedbloed, B. Keegan, G. T. A. Huysmans, and E. Schwarz. Damping of global Alfvén waves in tokamaks due to resonant absorption. Plasma Physics and Controlled Fusion, 34(8):1397–1422, 1992. 14, 111, 114

42. S. Poedts and W. Kerner. Ideal quasimodes reviewed in resistive magnetohydrodynamics. Phys. Rev. Lett., 66:2871–2874, Jun 1991. 14, 50

43. S. M. Mahajan. Kinetic theory of shear alfvén waves. Physics of Fluids, 27(9):2238–2247, 1984. 14, 15, 49

44. X. D. Zhang, Y. Z. Zhang, and S. M. Mahajan. Continuum damping of ideal toroidal Alfvén eigenmodes. Physics of Plasmas, 1(2):381–389, 1994. 14, 62

45. F. Zonca and L. Chen. Theory of continuum damping of toroidal Alfvén eigenmodes in ﬁnite-β tokamaks. Physics of Fluids B: Plasma Physics, 5(10):3668–3690, 1993. 14, 55

46. A. Könies and R. Kleiber. A computational approach to continuum damping of Alfvén waves in two and three-dimensional geometry. Physics of Plasmas, 19(12):122111, 2012. 14, 101

47. D. Borba, A. Fasoli, Gorelenkov N. N., S. Günter, P. Lauber, N. Mellet, R. Nazikian, T. Panis, S. D. Pinches, D. Spong, D. Testa, and JET-EFDA contributors. The Inﬂuence of Plasma Shaping on the Damping of Toroidal Alfvén Eigenmodes. In 23rd IAEA Fusion Energy Conference, Daejeon, Republic of Korea, 11-16 October 2010, Daejeon, Republic of Korea, 2010. International Atomic En- ergy Agency. 15

48. M. N. Rosenbluth and P. H. Rutherford. Excitation of Alfvén waves by high- energy ions in a tokamak. Physical Review Letters, 34(23):1428, 1975. 15, 47, 49

49. R. R. Mett and S. M. Mahajan. Kinetic theory of toroidicity-induced Alfvén eigenmodes. Physics of Fluids B: Plasma Physics, 4(9):2885–2893, 1992. 15, 48, 50, 89, 94

50. D. Testa, D. Spong, T. Panis, P. Blanchard, A. Fasoli, and JET-EFDA contributors. Recent JET experiments on Alfvén eigenmodes with intermediate toroidal mode numbers: measurements and modelling of n = 3 toroidal Alfvén eigenmodes with the TAEFL code. Nuclear Fusion, 51(4):043009, 2011. 15 References 153

51. T. Panis, A. Fasoli, D. Testa, JET-EFDA Contributors, et al. Analysis of damping rate measurements of toroidal Alfvén eigenmodes on JET as a function of n: part I. Nuclear Fusion, 52(2):023013, 2012. 15, 16

52. H. L. Berk and B. N. Breizman. Saturation of a single mode driven by an energetic injected beam. III. Alfvén wave problem. Physics of Fluids B Plasma Physics, 2(9):2246, 1990. 16

53. L. Villard, K. Appert, R. Gruber, and J. Vaclavik. Global waves in cold plasmas. Computer Physics Reports, 4(3-4):95–135, 1986. 16

54. Ya. I. Kolesnichenko, V. V. Lutsenko, H. Wobig, and V. Yakovenko. Alfvén instabilities driven by circulating ions in optimized stellarators and their possible consequences in a helias reactor. Physics of Plasmas, 9(2):517–528, 2002. 16

55. B. D. Blackwell. Observations of Alfvénic MHD activity in the H-1 heliac. 2006. 16

56. B. D. Blackwell, D. G. Pretty, J. Howard, R. Nazikian, S. T. A. Kumar, D. Oliver, D. Byrne, J. H. Harris, C. A. Nuhrenberg, M. McGann, et al. Conﬁgurational Effects on Alfvénic modes and Conﬁnement in the H-1NF Heliac. arXiv preprint arXiv:0902.4728, 2009. 16

57. J. Bertram, M. J. Hole, D. G. Pretty, B. D. Blackwell, and Dewar R. L. A reduced global Alfvén eigenmodes model for Mirnov array data on the H1-NF heliac. Plasma Physics and Controlled Fusion, 53(8):085023, 2011. 16

58. J. P. Freidberg. Plasma Physics and Fusion Energy. 19

59. R. B. White. The Theory of Toroidally Conﬁned Plasmas. Imperial College Press, 2006. 20

60. R. L. Dewar and S. R. Hudson. Stellarator symmetry. Physica D: Nonlinear Phenomena, 112(1-2):275 – 280, 1998. Proceedings of the Workshop on Time- Reversal Symmetry in Dynamical Systems. 21, 141

61. J. P. Goedbloed, R. Keppens, and S. Poedts. Advanced Magnetohydrodynamics: With Applications to Laboratory and Astrophysical Plasmas. Cambridge University Press, 2010. 22, 49

62. A. H. Boozer. What is a stellarator? Physics of Plasmas, 5(5):1647–1655, 1998. 22

63. W. D. D’haeseleer, W. N. G. Hitchon, J. D. Callen, and J. L. Shohet. Flux Coordi- nates and Magnetic Field Structure: A Guide to a Fundamental Tool of Plasma Theory. Scientiﬁc Computation. Springer Berlin Heidelberg, 2012. 23

64. T. Fehér. Simulation of the interaction between Alfvén waves and fast particles. PhD thesis, University of Greifswald, November 2013. 23, 24, 33, 39, 47, 52, 54, 96 154 References

65. S. P. Hirshman and J. C. Whitson. Steepest-descent moment method for three- dimensional magnetohydrodynamic equilibria. Physics of Fluids, 26(12):3553– 3568, 1983. 23, 24

66. S. P. Hirshman, P. Merkel, et al. Three-dimensional free boundary calculations using a spectral green’s function method. Computer Physics Communications, 43(1):143–155, 1986. 23

67. S. P. Hirshman and H. K. Meier. Optimized fourier representations for three- dimensional magnetic surfaces. Physics of Fluids, 28(5):1387–1391, 1985. 24

68. O. P. Fesenyuk, Ya. I. Kolesnichenko, H. Wobig, and Yu V. Yakovenko. Ideal magnetohydrodynamic equations for low-frequency waves in toroidal plasmas. Physics of Plasmas, 9(5):1589–1595, 2002. 32, 33, 34

69. M. S. Chu, J. M. Greene, L. L. Lao, A. D. Turnbull, and M. S. Chance. A numerical study of the high-n shear Alfvén spectrum gap and the high-n gap mode. Physics of Fluids B, 4(11):3713–3721, 1992. 33, 41

70. V. D. Shafranov. Equilibrium of a toroidal pinch in a magnetic ﬁeld. Soviet Atomic Energy, 13(6):1149–1158, 1963. 37

71. A. Salat and J. A. Tataronis. Shear Alfvén mode resonances in nonaxisymmetric toroidal low-pressure plasmas. I. Mode equations in arbitrary geometry. Physics of Plasmas, 8(4):1200–1206, 2001. 40

72. Ya. I. Kolesnichenko, A. Könies, V. V. Lutsenko, and Yu. V. Yakovenko. Afﬁnity and difference between energetic-ion-driven instabilities in 2d and 3d toroidal systems. Plasma Physics and Controlled Fusion, 53(2):024007, 2011. 44, 50

73. M. S. Chu, J. M. Greene, W. Ling, A. D. Turnbull, and H. L. Berk. A numerical study of the Alfvén continuum damping of toroidal Alfvén eigenmodes. Physics of Plasmas, 1:1214–1225, May 1994. 44, 45, 69, 71, 75

74. E. Hameiri. Singular solutions and the continuous spectrum. Physics of Fluids, 24(3):562–563, 1981. 45

75. A. Salat and J. A. Tataronis. Shear Alfvén mode resonances in nonaxisymmetric toroidal low-pressure plasmas. II. Singular modes in the shear Alfvén continuum. Physics of Plasmas, 8(4):1207–1218, 2001. 45

76. G. Y. Fu, H. L. Berk, and A. Pletzer. Kinetic damping of toroidal Alfvén eigenmodes. Physics of Plasmas, 12(8):082505. 47, 48, 49, 96

77. N. N. Gorelenkov and S. E. Sharapov. On the collisional damping of TAE-modes on trapped electrons in tokamaks. Physica Scripta, 45(2):163, 1992. 49, 89 References 155

78. D. Borba, H. L. Berk, B. N. Breizman, A. Fasoli, F. Nabais, S. D. Pinches, S. E. Sharapov, D. Testa, and contributors to the EFDA-JET Work Programme. Mod- elling of Alfvén waves in JET plasmas with the CASTOR-K code. Nuclear fusion, 42(8):1029, 2002. 49

79. J. W. Connor, W. M. Tang, and J. B. Taylor. Inﬂuence of gyroradius and dissipation on the Alfvén wave continuum. Physics of Fluids, 26(1):158–163, 1983. 49

80. P. Rodrigues, A. Figueiredo, J. Ferreira, R. Coelho, F. Nabais, D. Borba, N. F. Loureiro, H. J. C. Oliver, and S. E. Sharapov. Systematic linear-stability assessment of Alfvén eigenmodes in the presence of fusion α-particles for ITER-like equilibria. Nuclear Fusion, 55(8):083003, 2015. 50, 93

81. G. Strang and G. Fix. An Analysis of the Finite Element Method. Prentice Hall, 1973. 51

82. C. Beattie. Galerkin eigenvector approximations. Mathematics of Computation of the American Mathematical Society, 69(232):1409–1434, 2000. 51

83. J. E., Roman, C. Campos, E. Romero, and A. Tomás. "SLEPc Users Manual". Technical report, Polytechnic University of Valencia, 2014. 52

84. A. Könies. Private Communication. 53

85. N. N. Gorelenkov. Double-Gap Alfvén Eigenmodes: Revisiting Eigenmode In- teraction with the Alfvén Continuum. Physical Review Letters, 95(26):265003, 2005. 55

86. P. A. M. Dirac. The quantum theory of the emission and absorption of radiation. Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, 114(767):243–265, 1927. 62

87. R. C. Grimm, R. L. Dewar, J. Manickam, S. C. Jardin, A. H. Glasser, and M. S. Chance. Resistive instabilities in tokamak geometry. In Plasma physics and controlled nuclear fusion research 1982. 1983. 77

88. J. Manickam, R. C. Grimm, and R. L. Dewar. The linear stability analysis of mhd models in axisymmetric toroidal geometry. Computer Physics Communications, 24(3):355 – 361, 1981. 77

89. A. D. Miller and R. L. Dewar. Galerkin Method for Differential Equations with Regular Singular Points. Journal of Computational Physics, 66(2):356–390, 1986. 77

90. A. Pletzer and R. L. Dewar. Finite element method for resistive plasma stability studies in cylindrical geometry. Technical report, Australian National Univ., 1989. 77 156 References

91. D. Borba and W. Kerner. CASTOR-K: stability analysis of Alfvén eigenmodes in the presence of energetic ions in tokamaks. Journal of Computational Physics, 153(1):101–138, 1999. 89

92. H. L. Berk, R. R. Mett, and D. M. Lindberg. Arbitrary mode number boundary- layer theory for nonideal toroidal Alfvén modes. Physics of Fluids B: Plasma Physics, 5(11):3969–3996, 1993. 89, 94

93. G. Y. Fu and C. Z. Cheng. Excitation of high-n toroidicity-induced shear Alfvén eigenmodes by energetic particles and fusion alpha particles in tokamaks. Physics of Fluids B: Plasma Physics, 4(11):3722–3734, 1992. 89

94. J. Candy and M. N. Rosenbluth. Nonideal theory of toroidal Alfvén eigenmodes. Physics of Plasmas, 1(2):356–372, 1994. 89, 94

95. F. L. Hinton. Collisional transport in plasma, volume 1. Amsterdam: North- Holland Publishing Company, 1983. 90

96. Könies, A. and Fehér, T. and Lauber, P. and Mishchenko, A. and Kleiber, R. and Borchardt, M. and Briguglio, S. and Vlad, G. and Gorelenkov, N. and Isaev, M. and others. Benchmark of gyrokinetic, kinetic MHD and gyroﬂuid codes for the linear calculation of fast particle driven TAE dynamics. Technical report, Max Planck Institute for Plasma Physics, 2012. 90

97. S. M. Hamberger, B. D. Blackwell, L. E. Sharp, and D. B. Shenton. H-1 design and construction. Fusion Science and Technology, 17(1):123–130, 1990. 97

98. S. R. Haskey, B. D. Blackwell, C. Nührenberg, A. Könies, J. Bertram, C. Michael, M. J. Hole, and J. Howard. Experiment-theory comparison for low frequency BAE modes in the strongly shaped H-1NF stellarator. Plasma Physics and Con- trolled Fusion, 57(9):095011, 2015. 97, 100

99. P. Xanthopoulos, W. A. Cooper, F. Jenko, Yu. Turkin, A. Runov, and J. Geiger. A geometry interface for gyrokinetic microturbulence investigations in toroidal conﬁgurations. Physics of Plasmas, 16(8):082303, 2009. 97

100. M. J. Hole, B. D. Blackwell, G. W. Bowden, M. Cole, A. Könies, C. Michael, F. Zhao and S. R. Haskey. Global Alfvén Eigenmodes in the H-1 heliac. 2016. 98, 134

101. S. Haskey. "Beta induced Alfvén eigenmode excitation in the strongly shaped H-1NF stellarator", 11 2015. DPP15 Meeting of the American Physical Society. 100

102. K. Toi, M. Takechi, M. Isobe, N. Nakajima, M. Osakabe, S. Takagi, T. Kondo, G. Matsunaga, K. Ohkuni, M. Sasao, S. Yamamoto, S. Ohdachi, S. Sakakibara, H. Yamada, K.Y. Watanabe, D.S. Darrow, A. Fujisawa, M. Goto, K. Ida, H. Idei, H. Iguchi, S. Lee, S. Kado, S. Kubo, O. Kaneko, K. Kawahata, K. Matsuoka, References 157

T. Minami, S. Morita, O. Motojima, K. Narihara, S. Nishimura, N. Ohyabu, Y. Oka, S. Okamura, T. Ozaki, K. Sato, M. Sato, A. Shimizu, T. Shimozuma, Y. Takeiri, K. Tanaka, T. Tokuzawa, K. Tsumori, I. Yamada, Y. Yoshimura, CHS, and LHD Experimental Groups. Energetic ion driven MHD instabilities observed in the heliotron/torsatron devices Compact Helical System and Large Helical Device. Nuclear Fusion, 40(7):1349, 2000. 116

103. M. Takechi, K. Toi, S. Takagi, G. Matsunaga, K. Ohkuni, S. Ohdachi, R. Akiyama, D. S. Darrow, A. Fujisawa, M. Gotoh, H. Idei, H. Iguchi, M. Isobe, T. Kondo, M. Kojima, S. Kubo, S. Lee, T. Minami, S. Morita, K. Matsuoka, S. Nishimura, S. Okamura, M. Osakabe, M. Sasao, M. Shimizu, C. Takahashi, K. Tanaka, and Y. Yoshimura. Energetic-Ion-Driven Toroidal Alfvén Eigenmodes Observed in a Heliotron/Torsatron Plasma. Phys. Rev. Lett., 83:312–315, Jul 1999. 116

104. A. Könies. Private Communication. 123

105. C. Beidler, G. Grieger, F. Herrnegger, E. Harmeyer, J. Kisslinger, W. Lotz, H. Maassberg, P. Merkel, J. Nührenberg, F. Rau, et al. Physics and engineering design for wendelstein vii-x. Fusion Science and Technology, 17(1):148–168, 1990. 123

106. G. J. Kramer, A. Bortolon, N. M. Ferraro, D. A. Spong, N. A. Crocker, D. S. Darrow, E. D. Fredrickson, S. Kubota, J. K. Park, M. Podestà, et al. Mitigation of Alfvénic activity by 3D magnetic perturbations on NSTX. Plasma Physics and Controlled Fusion, 58(8):085003, 2016. 134

107. C. Z. Cheng and M. S. Chance. NOVA: A nonvariational code for solving the MHD stability of axisymmetric toroidal plasmas. Journal of Computational Physics, 71(1):124–146, 1987. 135

108. C. Schwab. Ideal magnetohydrodynamics: Global mode analysis of three- dimensional plasma conﬁgurations. Physics of Plasmas, 5(9):3195–3206, 1993. 135

109. D. Yu. Eremin and A. Könies. Beta-induced Alfvén-acoustic eigenmodes in stellarator plasmas with low shear. Physics of Plasmas, 17(1):012108, 2010. 135

110. J. Bertram, B. D. Blackwell, and M. J. Hole. Ideal-magnetohydrodynamic theory of low-frequency Alfvén waves in the H-1 Heliac. Plasma Physics and Controlled Fusion, 54:1–11, 2012. 135

111. B. Van der Holst, A. J. C. Beliën, and J. P. Goedbloed. Low frequency Alfvén waves induced by toroidal ﬂows. Physics of Plasmas, 7(10):4208–4222, 2000. 135

112. Z. S. Qu, M. J. Hole, and M. Fitzgerald. Modeling the effect of anisotropic pressure on tokamak plasmas normal modes and continuum using ﬂuid approaches. Plasma Physics and Controlled Fusion, 57(9):095005, 2015. 135 158 References

113. A. Biancalani, L. Chen, F. Pegoraro, and F. Zonca. Continuous spectrum of shear Alfvén waves within magnetic islands. Physical Review Letters, 105(9):095002, 2010. 135

114. Sun, B. J. and Ochando, M. A. and López-Bruna, D. Alfvén eigenmodes including magnetic island effects in the TJ-II stellarator. Nuclear Fusion, 55(9):093023, 2015. 135

115. A. Salat. Collapse of Alfven modes into ballooning-type modes due to toroidal nonaxisymmetry. Plasma physics and controlled fusion, 34(7):1339, 1992. 135