Nonlinear growth of Neoclassical Tearing modes in a TokamakPlasma

Joao Vasco Ribeiro Ferreira Gama

Dissertacao para obtencao do Grau de Mestre em

Engenharia Fısica Tecnologica

Juri

Presidente: Prof. Joao Seixas

Orientador: Prof. Bernardo Brotas

Co-orientador: Doutor Rui Coelho

Vogal: Prof. Fernando Serra

Novembro de 2009

Resumo

O trabalho desenvolvido na presente tese e baseado num codigo numerico criado para resolverequacoes diferenciais que descrevem genericamente o comportamento dos modos tearing neoclassicos,cuja derivacao e apresentada.

Em primeiro lugar o programa e simplificado para estudar o comportamento linear do modo tearing.Os resultados sao comparados com a solucao da equacao tearing, e com as relacoes de dispersao daliteratura. E proposta uma dependencia diferente da taxa de crescimento com a viscosidade.

Posteriormente, e estudada a evolucao quasi-linar dos modos tearing, tendo como base varios di-agramas de fase obtidos pelo programa. E analisada a influencia do campo externo ressonante, e ofenomeno de mode flipping, e seguidamente a rotacao dos modos.

Sao entao apresentados os resultados relativos ao estudo dos NTMs, que confirmam a teoria ex-posta. Sao obtidos resultados inovadores atraves da activacao da corrente externa.

Palavras chave: Tokamak, Modelo MHD, Neoclassico, NTM, modo Tearing, Bootstrap

i

Abstract

The work developed in this thesis is based on a numerical code created to solve differential equationswhich generically describe the behavior of neoclassical tearing modes. Derivation of these equations ispresented.

Firstly, the computer program is simplified in order to study the linear behavior of the mode. Resultsare compared with the tearing equation solution, as well as with theoretical predictions for dispersionrelations. A new scaling for the growth rate, as a function of viscosity, is proposed.

The quasi-linear evolution of tearing modes is subsequently analyzed, based on several phase por-traits obtained by the program. The influence of the resonant external field is examined, along with themode flipping and mode rotation.

Results concerning the study of NTM’s are finally presented, as confirming the previously expoundedtheory. Innovative results are obtained through external current activation.

Keywords: Tokamak, MHD model, Neoclassical, NTM, Tearing mode, Bootstrap

ii

Acknowledgements

Este trabalho resultou de um desafio do Doutor Rui Coelho, que desde entao teve a porta sempreaberta, perdendo varias tardes para me esclarecer qualquer duvida que eu tivesse, e - sempre comentusiasmo - falar sobre varios outros aspectos associados a este interessante campo das intabilidadesmagnetohidrodinamicas. Obrigado pela assistencia, disponibilidade e simpatia.

Desejo tambem agradecer ao Professor Bernardo Brotas, pela disponbilidade em supervisionar otrabalho efectuado.

Os meus agradecimentos para o Instituto de Plasmas e Fusao Nuclear, nao so pelo financiamento,atraves de bolsa, para a realizacao desta tese, como pelo ambiente de trabalho estimulante, e pelaoportunidade de investigar num tema que me e caro.

Nao posso deixar de agradecer a toda a minha famılia, em particular a minha mae, pela forca quetodos me deram. Gostaria de agradecer aos meus amigos mais proximos, por todo o apoio; e ao AntonioAfonso, Andre Mourao, Duarte Alvim e Fernando Patrıcio em particular, pois a sua ajuda reflecte-se naspaginas que se seguem.

Por fim, por varias razoes diferentes, um agradecimento muito especial para a Ana.

iii

Contents

List of Tables vi

List of Figures x

1 Introduction 11.1 Thermonuclear fusion in the face of global energetic needs . . . . . . . . . . . . . . . . . 11.2 The Deuterium-Tritium reaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.3 Tokamak . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.4 MHD stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.5 The Tearing mode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131.6 Outline of the thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2 Tearing Mode Numerical Model 182.1 Equations used . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182.2 Magnetic flux evolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192.3 Vorticity and velocity evolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222.4 Temperature evolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242.5 Bootstrap current . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

3 Linear dynamics of the tearing mode 303.1 The Tearing Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303.2 Dispersion relation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323.3 MHD Linear model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363.4 Effects of Nonvanishing Viscosity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 403.5 MHD Linear model with viscosity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

4 Quasi-linear evolution of the tearing mode 484.1 Effects of third order non-linear forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 484.2 MHD quasi-linear model without rotation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 494.3 Interaction with resonant external magnetic fields . . . . . . . . . . . . . . . . . . . . . . . 574.4 MHD model for toroidally rotating modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

5 Quasi-linear evolution of neoclassical tearing modes 645.1 Effects of the bootstrap current on mode growth and saturation . . . . . . . . . . . . . . . 645.2 Neoclassical MHD model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 675.3 Interaction with resonant external magnetic fields . . . . . . . . . . . . . . . . . . . . . . . 705.4 Mode stabilization through ECCD and ECRH . . . . . . . . . . . . . . . . . . . . . . . . . 75

6 Conclusions 78

iv

A Normalizations 81A.1 Auxiliary definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81A.2 Normalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

B Values for Contour Plots 82B.1 Growth rate in linear regime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82B.2 Island width in the end of the linear regime . . . . . . . . . . . . . . . . . . . . . . . . . . 83B.3 Tearing Layer size . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

C Numerical details 85C.1 Finite diferences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85C.2 Explicit step . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88C.3 Implicit step . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89C.4 Implementation of Runge-Kutta method . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90C.5 Spatial grid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90C.6 Adaptive Stepsize . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91C.7 Computational details . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

D Boundary conditions 94D.1 Boundary conditions in time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

D.1.1 Equilibrium Magnetic flux as a function of Safety Factor . . . . . . . . . . . . . . . 94D.1.2 Equilibrium Electric Current and Electrical resistivity . . . . . . . . . . . . . . . . . 94D.1.3 Perturbed Magnetic flux and stream function . . . . . . . . . . . . . . . . . . . . . 95D.1.4 Equilibrium velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95D.1.5 Pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

D.2 Boundary conditions in space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96D.2.1 axis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96D.2.2 edge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

v

List of Tables

1 Estimated world energy resources in 2005 [5] [6] . . . . . . . . . . . . . . . . . . . . . . . 32 Growth rates γ × 106 as a function of magnetic Prandtl number Γ and Reynold’s number S. 823 Island width in the end of the linear regime w′ × 103 as a function of magnetic Prandtl

number Γ and Reynold’s number S. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 834 Tearing layer L × 103 calculated using expression (9) and values of section B.1, as a

function of magnetic Prandtl number Γ and Reynold’s number S. . . . . . . . . . . . . . . 84

vi

List of Figures

1 Increase in world energy consumption [1]. . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 GDP and electricity consumption [2]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 Energy consumption per type of fuel in 2006 according to [3] and [4]. . . . . . . . . . . . . 24 Potential energy as a function of nuclear separation [5]. . . . . . . . . . . . . . . . . . . . 45 Reactivity as function of the temperature [8]. . . . . . . . . . . . . . . . . . . . . . . . . . 56 progress in the achieved value for npτETp [9]. . . . . . . . . . . . . . . . . . . . . . . . . . 67 Structure and operation of the tokamak[10]. . . . . . . . . . . . . . . . . . . . . . . . . . . 78 Tokamak geometry [5]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 Evolution over time of electron density, magnetic signal (

∼Bθ, m=2), electron temperature

and total plasma current during a disruption [5]. . . . . . . . . . . . . . . . . . . . . . . . . 910 Flux surfaces [5]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1011 Field line in a q = 2 flux surface [5]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1012 Stabilizing effects for MHD modes: magnetic field line bending (left); magnetic field com-

pression (middle); negative pressure gradient in the direction of the curvature (right)[5]. . 1113 Stability diagram for a safety factor profile between qo ≈ 1 and qa ≈ 3 [5]. . . . . . . . . . 1214 Poloidal section showing magnetic islands of modes (2, 1) (left) and (3, 2) (right) [15]. . . . 1315 Magnetic island geometry [5]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1616 Perturbation with (2, 1) mode projected over the flux surface. . . . . . . . . . . . . . . . . 1917 Helical ribbon of a given flux surface [21]. . . . . . . . . . . . . . . . . . . . . . . . . . . . 2118 Diagram of plasma confinement method through employment of magnetic mirrors[26]. . . 2619 Diagram depicting banana orbits of two ions with opposite poloidal velocities at the starting

point[25]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2720 Numerical solution of the tearing equation for a (2, 1) mode with q(x) = qo + (qa − qo)x

2,qo = 1.3, qa = 3.5 and a perfect conducting wall at the edge. . . . . . . . . . . . . . . . . . 30

21 Numerical solutions of the tearing equation for a (2, 1) mode with q(x) = qo + (qa − qo)x2,

qo = 1.3, qa = 3.5 and positive external current (ψI+ ); vacuum (ψv); and negative externalcurrent (ψI− ) at the edge. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

22 Radial derivative of the solution found for the tearing equation. . . . . . . . . . . . . . . . 3123 Calculation of ∆′: difference between the derivative of the magnetic flux in the two points

closest to the resonant surface in function of the minimal spatial step on the grid. ∆′

estimate will be the value of b on the linear fit (y = ax+ b). . . . . . . . . . . . . . . . . . . 3224 Calculation of A: applying the definition to the two points closest to the resonant surface

and plotting it as a function of the minimal spatial step on the grid used. A estimate willbe the value of b on the linear fit (y = ax+ b) using the tow smaller grid distances. . . . . 35

25 Growth rate γ as a function of Reynold’s number S, both in logarithmic scale. Points aresolutions of equation 69, and the line represents the standard dispersion relation (61). . . 35

26 k, ψo, J and η as function of x. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

vii

27 Sum of the absolute values of the differences between consecutive points (in time) dividedby the respective time interval, and time. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

28 Normalized profiles of∼ψmn for different time values. . . . . . . . . . . . . . . . . . . . . . 38

29 Normalized profiles of∼ψmn for the tearing equation solution (points), the initial profile in

the MHD simulation (blue line), and the profile in the MHD simulation at t = 575 τa (red line). 3930 Magnetic flux on the resonant surface as a function of time, with different boundary con-

ditions: external current I+

Ip= 7.43 × 10−5 (purple); vacuum (black); perfectly conducting

wall (blue); and external current I−

Ip= −7.43× 10−5 (green). . . . . . . . . . . . . . . . . . 40

31 Dispersion relations obtained using expression (69) for different values of ∆′ andA (lines);and growth rates computed by the numerical code for different values of Reynold’s number(red points). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

32 Numerical solution of equation (76) (full line); and analytical approximation given by equa-tion (77) (dot line) [31]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

33 Growth rate γ as a function of the logarithm of the magnetic Prandtl number Γ. Each pointis a solution of equation (79). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

34 Vorticity profiles at t = 25τa (red line); t = 45τa (purple line); t = 117τa (blue line) andt = 575τa (black line in the larger graph). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

35 Growth rates γ as a function of the logarithm of the magnetic Prandtl number Γ, withS = 105. The dot line is the asymptotical limit, obtained by using Γ = 0. . . . . . . . . . . 45

36 Growth rates γ as a function of the logarithm of the magnetic Prandtl number Γ, withS = 108. The black line is the solution of equation (79), the blue line is the solution ofequation (82), and red points are results obtained by the simulation program. The dot lineis the asymptotical limit, obtained by using Γ = 0, which is the same in all cases. . . . . . 46

37 Growth rates γ × 106 as a function of the logarithm of magnetic Prandtl number Γ and thelogarithm of magnetic Reynold’s number S. Values are on table B.1. . . . . . . . . . . . . 47

38 Tearing mode structure in the resistive layer; plasma flow pattern and non-linear thirdorder forces [18]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

39 Phase portrait of island width. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4940

∼ψs ×107 as a function of time, with S = 105, Γ = 0.01 and: a) αo = 10−7, in the initialprofile, qo = 1.3 and qa = 3.5 in the parabolic safety factor profile; b) αo = 10−7, qo = 1.0

and qa = 3.5; c) αo = 1.7× 10−4, qo = 1.3 and qa = 3.5; d) αo = 10−4, qo = 1.0 and qa = 3.5. 5041 Radial profile of the equilibrium current in the absence of perturbations (blue), and when

the island is near its saturated size (red). Apart from the region near the resonant surface,the profiles are mostly identical. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

42 Phase portrait, with S = 105, Γ = 0.01 and: a) αo = 10−7, in the initial profile, qo = 1.3

and qa = 3.5 in the parabolic safety factor profile; b) αo = 10−7, qo = 1.0 and qa = 3.5; c)αo = 1.7× 10−4, qo = 1.3 and qa = 3.5; d) αo = 10−4, qo = 1.0 and qa = 3.5. . . . . . . . . 52

43 Phase portrait, with S = 105, Γ = 0.01, αo = 10−7, in the initial profile, qo = 1.3 andqa = 3.5 in the parabolic safety factor profile. Four regimes of growth may be observed. . 53

viii

44 Partial phase portrait, with S = 107, Γ = 0.01, αo = 10−7, in the initial profile, qo = 1.3 andqa = 3.5 in the parabolic safety factor profile. . . . . . . . . . . . . . . . . . . . . . . . . . . 54

45 Island width when linear growth regime ceases, as a function of the logarithm of magneticPrandtl number Γ and the logarithm of magnetic Reynold’s number S. Values are on tableB.2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

46 Tearing Layer calculated using expression (9), as a function of the logarithm of magneticPrandtl number Γ and the logarithm of magnetic Reynold’s number S. Values are on tableB.3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

47 Profile of∼ψmn, subject to a resonant helical field generated by a current sheet at x = 1.3.

The green line is the rational surface, the blue line is the plasma edge, and the red line isthe current sheet radial location. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

48 Phase portrait of the island width w, showing different external currents at x = 1.3. Thered dash corresponds to the saturated island width in the absence of external current.The colored dashes are values of stable equilibrium island size for different values of theexternal current bigger than Itr, with the exception of the purple line which correspondsto an unstable equilibrium with IE = Itr. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

49 Perturbed magnetic flux in the resonant surface as a function of time. The branchesrelated to time dependence of the external current are signaled in different colors, respec-tively gray, red and blue. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

50 Perturbed magnetic flux in the resonant surface as a function of time showing mode flip-ping. The branches related to time dependence of the external current are signaled indifferent colors, respectively gray, red and blue. . . . . . . . . . . . . . . . . . . . . . . . . 60

51 Mode frequency as a function of time, with q(x) = qo + (qa − qo)x2, qo = 1.3, qa = 3.5,

vzi = vzo + (vza − vzo)x3, vzo = 5 kHz, vza = 0.2 kHz, S = 105, Γ = 0.01, ϵ = 0.3,

ρ/mp = 1019 m−3, and perfectly-conducting wall boundary conditions. . . . . . . . . . . . 6152 Perturbed magnetic flux in the resonant surface as a function of time. The branches

related to time dependence of the external current are signaled in different colours, re-spectively gray, red and blue. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

53 Phase space diagram of the neoclassical MHD tearing mode [39]. . . . . . . . . . . . . . 6754

∼ψs x 107 as a function of time, for αo = 128.0 x 10−7 and αo = 88.0 x 10−7 . . . . . . . . . 69

55∼ψs x 107 as a function of time, for αo = 128.0 x 10−7 and αo = 88.0 x 10−7, when changesin the bootstrap current are turned of. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

56 Phase portrait for a simulation with αo = 110.0 x 10−7 . . . . . . . . . . . . . . . . . . . . 7057 Phase diagram associated with equations(110) and (119). The red dot marks the unstable

equilibrium corresponding to the threshold for the NTM’s growth. Its value diminishes oncethe effect of a destabilizing resonant external magnetic field is taken into account . . . . . 71

58∼ψs x 107 as a function of time, for αo = 98.0 x 10−7 (above) and αo = 1.0 x 10−7 (below),with an external current in the form IE = Ci t turned off when t = 4000 τa. The timelinewith non-null external field is signaled in red. . . . . . . . . . . . . . . . . . . . . . . . . . 72

ix

59∼ψs x 107 as a function of time, for αo = 1.0 x 10−7 with an external current in the formIE = Ci t and turned off at t = 10000 τa. The timeline with non-null external field issignaled in red. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

60∼ψs x 107 as a function of time, for αo = 10.0 x 10−7 with an external current which startsas a ramp and stabilizes at t = 80000τa with a ratio Ie

Ip= 7.43× 10−5. . . . . . . . . . . . . 74

61∼ψs x 107 as a function of time, with an external current which starts as a ramp and thenstabilizes[40] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

62 Gyrotron, high powered source of electromagnetic waves with frequencies reaching anorder of magnitude of hundreds of GHz [41] . . . . . . . . . . . . . . . . . . . . . . . . . . 75

63 Effect of preventive ECCD inhibiting the occurrence of (3, 2) modes. Green dots depictpotentially unstable situations which resulted stable with ECCD, and red crosses portraysituations which revealed unstable. The red line marks the prevision of meta-stability withpreventive ECCD [44] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

64 Scheme of the asymmetrical grid considered . . . . . . . . . . . . . . . . . . . . . . . . . 85

x

1 Introduction

1.1 Thermonuclear fusion in the face of global energetic needs

Several studies and projections show that global energy needs will increase drastically within the next

decades (Figure 1). There are both demographic and economic reasons behind this escalation (Figure

2).

Figure 1: Increase in world energy consumption [1].

Three kinds of sources may be used in order to provide for human energetic needs:

fossil energy - oil, natural gas, and coal;

renewable energy - hydric, eolic, solar, biomass, geothermal, waves, tides, among others;

nuclear energy - nuclear fission and fusion;

The distribution profile of energy use per source in 2006 is shown in Figure 3 [3][4].

Fossil fuels account for more than 80% of global energy consumption. However, their availability is

limited. Table 1.1 shows the estimated amount of energy still available for remaining world resources

(figures are only indicative).

On the other hand, according to Intergovernmental Panel on Climate Change (IPCC) reports[7] there

1

Figure 2: GDP and electricity consumption [2].

Figure 3: Energy consumption per type of fuel in 2006 according to [3] and [4].

is a strong possibility that the excessive exploitation of coal, oil and natural gas will threat world climate

by increasing the greenhouse effect.

Renewable energies may contribute to the mitigation of this energetic problem, and technological in-

vestigation and improvement in that field is an ongoing effort. However, some structural problems dictate

a high elasticity of production costs: their cost may be competitive only insofar as they fulfill a small frac-

2

Divided by presentResources Energy world energy consumption

(109 joules) per year - 4.8 ×1020 joules(years)

Coal 1.0 ×1014 208Oil 1.2 ×1013 25

Natural Gas 1.4 ×1013 29Uranium 235 1 ×1013 21

Uranium 238 and Thorium 232 1.0 ×1016 20833

Table 1: Estimated world energy resources in 2005 [5] [6]

tion of global energy needs. Currently, the only viable alternative with no greenhouse gas emissions and

high power availability is nuclear fission. However, generated radioactive residues with millenary decay

times represent a serious hazard for the environment. Additionally, the underlying physical concept in a

nuclear fission reactor core is that of an unstable system where adequate feedback control is needed.

On the other hand, investing in nuclear fusion as an alternative energy source could, in principle,

overcome the obstacles and harmful effects conventionally associated with nuclear energy. In the first

place, fusion energy production is inherently safe1 and there are no grounds to fear accidents as those

which have occurred in fission power plants in the past[5][11]. Moreover, the fuel necessary for nuclear

fusion is virtually unlimited in nature. Deuterium is easily extractable from sea water, while Tritium may

be generated at low cost from Lithium, also plentiful2. It is equally worth noting that nuclear fusion,

not unlike fission, also generates radioactive waste material. However, fusion residues are scant when

compared to those of fission - thus avoiding nuclear waste transportation, as well as its associated

hazards - and have decay times around a few decades (in contrast to thousands of years required for

fission waste material)[5][11].

The use of this energy source entails additional global advantages, such as: the large production

capacity of each single fusion power plant, particularly suitable for urban energetic needs3); the dis-

semination of its fuel reserves throughout the globe, which could allow for national energy autonomy

worldwide, with subsequent decline in international conflict and increased security; its dependence on1Deuterium and Tritium gas are totally ionized (in a plasma form), and as it will be shown in section 1.3, this plasma is not

perfectly confined. Therefore, eventual perturbations, internal or external, lead to a deterioration in confinement, and work as anegative feedback which may result in the plasma extinction. In addition, it is possible to cease gas injection at any time, thusinhibiting additional reactions.

2Energy obtainable from present Lithium land reserves divided by world energy consumption indicates a 20833 year time limit,while sea reserves would result in a 21×106 year time line limit.

3This factor becomes increasingly relevant as city population tends to grow worldwide.

3

technological progress and innovation, source of highly qualified employment; among other examples.

1.2 The Deuterium-Tritium reaction

Nuclear fusion is the process by which multiple atomic nuclei with positive charge combine to form a

heavier nucleus. If occurring between nuclei with lower mass than iron, the process will release energy4;

and it will absorb it otherwise.

Figure 4: Potential energy as a function of nuclear separation [5].

In order for nuclear fusion to take place, an energetic barrier caused by electrostatic repulsion be-

tween the positively charged nuclei must be overcome (Figure 4). A nuclear reaction will occur if the

nuclei are able to come within enough proximity from each other to allow nuclear force - stronger at

shorter distances - to surpass repulsive forces.

Reactivity measures the product between the probability5 of a reaction taking place and the energy

it delivers. Excluding impractically high temperatures, fusion reactions will reach maximum reactivity

between two specific isotopes of hydrogen: Deuterium (D) and Tritium (T) (Figure 5):

D +D → T + p+ 4.03MeV

4The difference between the nuclear binding energy of the reactants and products, if positive, will increase kinetic energy of theparticles, or else it will decrease it.

5This probability is proportional to the cross section (σ) of the collision at that temperature, calculated using the Rutherfordmodel for collisions.2

4

Figure 5: Reactivity as function of the temperature [8].

D + T → α+ n+ 17.6MeV (1)

where p, n and α stand for proton, neutron and helium nucleus respectively, and 1 eV ≃ 1.60

×10−19 J ∼ 11000oK.

Figure 5 shows that, in order to attain an optimum regime, temperatures of the order of 10 keV

should be reached. At those high temperatures Deuterium and Tritium are completely ionized: they are

in a plasma state. From the fusion power produced, approximately one fifth goes to α-particles, which

will heat the plasma through collisions. The remaining power produced goes to neutrons. In order to

maintain the appropriate temperature, difference between power losses and α-particle heating power

must be supplied from the outside. The ratio between the amount of energy generated by the reactions

and the amount of supplied heat is commonly referred to as Q. Ignition is considered to happen when

Q = ∞. The requirement for ignition to take place may be expressed approximately by6[5]:6This inequality establishes the condition that fusion heating by α-particles exceeds power losses.

5

npτETp = 5× 1021 m−3s keV (2)

where np and Tp stand, respectively, for density and ionic peak temperature, and τE represents

energy confinement time7.

Since the early years of fusion research, there has been a significant increase in the value reached

for this parameter, although a prominent decline in the increase rate has been noticeable over the past

decade (Figure 6).

Figure 6: progress in the achieved value for npτETp [9].

1.3 Tokamak

Two main approaches to fusion research strive to fulfill the ignition requirement: inertial confinement and

magnetic confinement. The former consists of a rapid pulse of energy irradiating (directly or indirectly8)7Energy confinement time measures the rate at which a system loses energy to its environment. It is given by the ratio between

energy content and power loss.8In the case of indirect drive, the fuel pellet is placed inside a container composed of a high atomic number material, which

will convert the driver beams into x-rays. Although the container takes up a considerable amount of energy to heat, absorption of

6

a fuel pellet, causing it to simultaneously implode and heat up to very high pressure and temperature

levels. The latter consists of using magnetic fields to confine the fusion fuel - already in a plasma form.

This is a more advanced line of research. Several confinement schemes have been studied (among

others, mirror machines, stellarators, tokamaks, spheromaks, reversed field pinches). Tokamak is widely

considered the most promising model, at least in the short/medium term[5].

The word tokamak transliterates the Russian acronym for ”toroidal chamber with magnetic coils”. The

term stands for a toroidal-shaped machine with external coils producing a magnetic field for confining

plasma, which also works as the secondary winding of a transformer9. The current inside the chamber

will generate the poloidal component of the magnetic field which, together with the toroidal field created

by the external coils, will result in the helicoidal magnetic field necessary for the equilibrium of the

configuration (Figure 7).

Figure 7: Structure and operation of the tokamak[10].

In spite of the high temperature and pressure levels inside the chamber’s core, this configuration

enables equilibrium as the plasma pressure may be balanced by the magnetic pressure10.

Conventional tokamak geometry is shown in the scheme of Figure 8. The ϕ and θ directions are

named, respectively, toroidal and poloidal directions. Ro and a are termed, sequentially, major and

minor radius. The Ro and a ratio is referred to as aspect ratio.

A reactor capable of electricity production in an industrial scale is yet to be built. DEMO, intended

to become the first prototype capable of industrial-scale electrical production in a few decades-time,

is only still being devised. Its conception and construction will benefit from the decisive contribution

thermal x-rays by the target is more efficient than direct absorption of laser light.9In a transformer, a varying current passing through the primary winding creates a varying magnetic field whose flux will pass

through the secondary winding, therefore inducting an electrical current there.10The cross product between the current density and the magnetic field is at the origin of this magnetic pressure.

7

Figure 8: Tokamak geometry [5].

of research allowed by ITER: International Thermonuclear Experimental Reactor. ITER (also the Latin

term for ”path”), in Cadarache, France[11], is the largest nuclear fusion reactor in construction today,

aimed at attaining 500 MW of generated thermal power. The program associated with its construction

will have an estimated cost of 9.3 billion dollars, and include participation by China, India, South Korea,

Japan, Russia, USA and the European Union.

Experimental research in tokamak systems started in 1956, and several technological improvements

have been achieved since then: introduction of a limiter, and later of a divertor; plasma heating and

current drive by Neutral Beam Injection and Radio Frequency Waves; and plasma shaping (elongation

and triangularity).

These breakthroughs enabled plasma operation in H-mode: the development of a region where the

transport coefficients are reduced by up to one order of magnitude, resulting in a pedestal in the plasma

density and an improvement in global confinement[12].

The two major obstacles to higher confinement are turbulence/transport and plasma instabilities. The

latter, besides hindering confinement, may also lead to disruptions in the plasma, i.e. sudden losses in

confinement, which may generate significant mechanical strain on the tokamak chamber. Figure 9

shows a sudden drop in the plasma current and an abrupt increase in particle loss as a consequence of

8

a disruption.

Figure 9: Evolution over time of electron density, magnetic signal (∼Bθ, m=2), electron temperature and

total plasma current during a disruption [5].

1.4 MHD stability

In a tokamak device, plasma equilibrium relies on the existence of a set of nested toroids where the

magnetic field lines lie, opportunely referred to as magnetic surfaces (Figure 10).

Along different flux surfaces, the ratio between the poloidal and toroidal magnetic fields changes.

This relation is paramount in studying the stability of the magnetic configuration. Each field line follows

a helical path as it goes around the torus, each turn differing by an angle of ∆θ when it returns to the

same poloidal plane. Function q11, termed safety factor due to its role in determining stability, is defined

by [5]:11function ι, termed rotational transform, is the inverse of q, defined as ∆θ/2π, giving a measure of the pitch of the magnetic field

lines as they twist around the torus. It is commonly used in stellarators and other toroidal plasma confinement systems besidesthe tokamak.

9

Figure 10: Flux surfaces [5].

q =2π

∆θ(3)

Using ψ to label a magnetic surface, where q(ψ) = mn , with m and n integers, a magnetic field line

joins up on itself after m toroidal and n poloidal turns (an example is depicted in Figure 11).

Figure 11: Field line in a q = 2 flux surface [5].

A measure of the efficiency in plasma pressure confinement by the magnetic field is given by the

following ratio[5]:

β =2 µo p

B2(4)

10

where p is plasma pressure, B is the norm of the magnetic field and µo the magnetic constant.

Plasma instabilities may be divided into two general categories: magnetohydrodynamic (MHD) and

kinetic instabilities. The former may pose the most stringent limitations to plasma performance and

detrimental effects on plasma confinement[5], and are described by a magnetohydrodynamic model

of the plasma. Instability drive may arise predominantly from plasma current or pressure gradients,

combined with adverse magnetic field curvature.

Perturbations may be decomposed into different modes [13]. If all of them are stable in the plasma,

they will attenuate, and the perturbation will vanish. On the contrary, if some modes are unstable, they

will grow in time.

Figure 12: Stabilizing effects for MHD modes: magnetic field line bending (left); magnetic field compres-sion (middle); negative pressure gradient in the direction of the curvature (right)[5].

Stability for each mode in a given flux surface will depend on stabilizing effects generated by the

compression or curvature of the field on that surface and those surrounding it (Figure 12). These effects

will be stronger for higher m modes, but will be minimized when the m and n of the mode are matched

by the safety factor q of the surface (q = mn )[14]. This flux surface is called resonant surface for this

mode. Therefore, for lower values of m, a mode can develop inside the plasma chamber in case there

is a resonant surface inside the plasma or in the vacuum region12.

Tailoring of the q profile (limiting its range, for instance) can secure some stability in tokamaks.

However, it is useful to accept dealing with some unstable modes in order to achieve otherwise better

conditions. In the diagram of Figure 13 the typical boundaries for q are presented, as well as the unstable

modes for normal tokamak operation.

Plasma instabilities that can occur even when resistivity (η) is null are called ideal instabilities. Pre-12in which case it is called an external mode

11

Figure 13: Stability diagram for a safety factor profile between qo ≈ 1 and qa ≈ 3 [5].

diction of the instability of a mode by ideal MHD theory means the mode is expected to be unstable

and evolve in a very fast Alfven time scale (τa =√µoρ a/B, where ρ is the plasma density and µo the

magnetic constant). This does not imply, however, that resistivity will cease to influence the mode’s

development.

On the other hand, even when ideal MHD theory predicts a plasma to be stable, it may be unstable

nevertheless. When resistivity plays a role in determining the stability of a mode, it is denominated

resistive mode.

The tearing mode is a resistive mode characterized by topological change (the tear ) of the flux

surfaces, and formation of magnetic islands (Figure 14). Even though it follows a hybrid time scale,

faster than the resistive diffusion time scale (τr = µoa2/η) but slower than the Alfen time scale[17],

it is considered responsible for major plasma disruptions13 [15]. There may exist two scenarios that

cause the stochastization of the whole field line topology and subsequent confinement degradation:

the nonlinear coupling between two or more chains of islands; and the mode locking, where an island

rotation is slowed by the resistive wall, thus breaking the plasma rotation.13A major plasma disruption is said to occur when the loss in confinement is severe to the point of interrupting the fusion reaction.

12

Figure 14: Poloidal section showing magnetic islands of modes (2, 1) (left) and (3, 2) (right) [15].

Because of the numerous destabilization mechanisms which may trigger the tearing mode’s growth,

finding a suitable region free of instabilities seems unattainable. Effort is rather focused on controlling

the growth, magnitude and consequences of the mode[17].

1.5 The Tearing mode

Taking into account the Ohm law for a plasma (−→E +−→v ×

−→B = η

−→J ), the evolution of the magnetic field in

a resistive-MHD plasma is governed by the following equation[16]:

∂−→B

∂t= ∇× (−→v ×

−→B ) +

η

µo∇2−→B (5)

where −→v is the plasma velocity,−→B the magnetic field and η the resistivity.

The first term describes the convection of the magnetic field with the plasma flow. Taking the rate of

change of the magnetic flux ψ passing trough any moving surface S, given by dψdt = −

∫c(−→E+−→v ×

−→B ).d

−→l ,

and assuming resistivity to be null, will yeld dψdt = 0. Its immediate implication is that the magnetic field

lines are frozen in the fluid.

The second term describes the resistive diffusion of the field through the plasma. It derives from

the fact that a spatial change in the magnetic field is related to a current (Ampere’s law) which, in case

resistivity is not null and the plasma is static, will implicate the existence of an electric field, that will

promote, in its turn, a temporal change in the magnetic field (Faraday’s law of induction).

13

In this manner, if the first term dominates the frozen flux constraint will prevail and the topology of the

magnetic field will not be able to change. On the other hand, if the second term dominates there will be

little coupling between the field and the plasma flow and the topology of the magnetic field will be free to

change.

The relative magnitude of these two terms is conventionally expressed in terms of the magnetic

Reynolds number:

S =τrτa

=B a

õo

η√ρo

(6)

where S is the magnetic Reynolds number, τr and τa are the resistive and Alfven times, respectively,

B is the norm of the magnetic field, a is the minor radius of the tokamak, and ρo is the plasma density.

Because the magnetic Reynolds number is very high for normal tokamak operation, one could say

that the motion conforms to ideal MHD. However, if there is a tearing instability in the thin layer around

the resonant surface associated with a mode (designated as tearing layer ), the parallel to the equilibrium

field −→v ×−→B contribution to Ohm’s law goes to zero, and the η

−→J term becomes important in balancing

the induced parallel electric field[5].

As the perturbation on−→B diffuses around the resonant surface, a similar diffusion occurs for the

perturbation on−→J . The cross product of the former with the equilibrium current density, and of the latter

with the equilibrium poloidal field will both promote the occurrence of a perturbation in the fluid velocity

field.

Because the equilibrium magnetic field is not constant across the radius, the convection caused by

the perturbation of velocity will cause an increase (or decrease) in the magnetic field and its perturbation

in a given point in space. In its turn, the increase in the magnetic perturbation will generate an increase

in the velocity perturbation. This process will account for an exponential growth in the early development

of the perturbation.

When the growth of the mode occurs in timescale longer than the diffusion, it is appropriate to con-

sider the perturbed field inside the tearing layer to be a constant. However, matching to the ideal plasma

region obliges the existence of a discontinuity in the spatial derivative of the function (a footprint of the

current sheet). This jump in the logarithmic derivative of the perturbation flux is commonly designated by

stability parameter, defined by ∆′ ≡ (∼ψ′

+ −∼ψ′

−) / (a∼ψ (rs)) and will depend on the boundary conditions

14

[17]. Its value is related to the growth rate of the mode by[5]:

γ = 0.55 τ− 3

5r τ

− 25

a

(n a2 q′R q

) 25 (a∆′) 4

5 (7)

where γ is the growth rate of the mode (assuming∼ψ=

∼ψr e

(γ+iω)t), q′ is the radial derivative of q, and

n refers to the mode number (m,n). The expression above shows that the development of the instability

takes place in a hybrid time scale, faster than the resistive diffusion time scale but slower than the Alfven

time scale[17].

The reconnection taking place inside the tearing layer will generate a set of magnetic islands whose

dimensions will grow with the mode. The geometry of the magnetic islands is exposed in the scheme of

Figure 15. The width of the islands in the poloidal plane is given by:

w =

√rsLs

∼Brs

mBz(8)

where rs is the radius of the ressonant surface,∼Brs is the perturbed radial magnetic field, and

Ls =q2Ro

q′rs, both evaluated at the rational surface of the mode[5].

As for the tearing layer width, it can be obtained through the condition that inertia balances the torque

driven by the eddy currents triggered by the magnetic perturbation, which in the case of a tokamak

yields[16]:

L ≈ (γρoηo)14

√rsLs

mBz(9)

where ρo is the density in the axis, and ηo the resistivity in the axis.

As the perturbation grows, however, its influence on equilibrium values is no longer neglectable, and

the linear approximation loses its applicability. Second-order eddy currents become significant, and the

third-order forces they produce will slow the growth of the mode [18]. At this time, the growth of the

island becomes related to ∆′ by[5]:

d w

dt= 1.66

η

µo(∆′ − αw) (10)

where w is the island width, and α a term related to the curvature of the current density profile.

15

Figure 15: Magnetic island geometry [5].

A more comprehensive understanding of the tearing mode must also consider the influence of tem-

perature. In fact, perturbations of velocity will convect the plasma and cause perturbations on the tem-

perature as well. Temperature will not diffuse isotropically, because the magnetic field lines will conduct

heat more readily. Its perturbations will generate the perturbation of the so-called bootsprap current.

1.6 Outline of the thesis

The work developed in the present thesis will be based on a numerical code created to solve differential

equations which generically describe the behavior of neoclassical tearing modes according to approxi-

mations considered and discussed throughout the text. Results obtained by this code will be compared

to literary findings regarding linear approximation (first neglecting viscosity; and thereafter assuming a

very high S value); and subsequently compared with results concerning non-linear evolution of tearing

modes (reduced MHD), and neoclassical tearing modes (NTMs).

The previously described model is expected to be useful for studying the development of tearing

modes in the context of assumed simplifications, particularly in examining dispersion relations, stability,

saturation, and response to external fields. Such an examination is of paramount importance for the

development of feedback systems as well as other methods devised for controlling or stabilizing these

modes.

Previous sections in this chapter have attempted to argue the relevance of research conducted in the

16

last decades in the field of thermonuclear fusion, particularly with tokamaks. MHD instabilities, one of the

main obstacles to a higher and more reliable tokamak performance, have been mentioned. Specifically,

aspects associated with the physical mechanism and growth of tearing instabilities, which will be the

focus of the present thesis, have been reviewed.

In chapter 2 the reduced MHD model widely used in subsequent chapters will be derived.

Chapter 3 will review relevant literature on the subject of tearing mode dispersion relations, as far

as growth rates in linear approximation are concerned, regarding both Reynolds and Prandtl numbers.

Results of simulations following the MHD derived model and considering the linear approximation will

be presented and discussed attending to selected literature on the subject. Dispersion relations will also

be produced through performed simulations, as well as compared to those identified in the reviewed

literature.

Chapter 4 will approach the quasi-linear evolution of tearing modes. First, the mechanism preclud-

ing linear description will be outlined, followed by a presentation and discussion of simulation results

ignoring plasma rotation, particularly phase portraits where growth regimes cited in the literature may

be identified. Results concerning interaction with an external resonant field will be subsequently ex-

pounded, as well as its influence on saturated magnetic island width and on mode flipping. Rotation will

thereafter be considered, and simulation results will be discussed along with some model limitations.

Chapter 5 will begin by reviewing some theoretical issues regarding neoclassical modes. A presen-

tation of the equations implemented in the code devised in this work to simulate NTMs will follow, as

well as simulation results and its discussion. On activating the external current, original results may be

obtained, with implications regarding a possible mechanism for the creation of seed islands. A summary

approach to ECCD and ECRH methods will then be made.

The last chapter will be devoted to final remarks and conclusions, including a summary of results

presented throughout the text, as well as a discussion of some of its implications and limits.

Appendixes will include a brief description of relevant details concerning the numerical code devel-

oped for this thesis. Some calculations regarding assumed normalizations and boundary conditions will

also be mentioned in appendix, as well as numerical and computational details regarding the simulation

program.

17

2 Tearing Mode Numerical Model

2.1 Equations used

A fluid model of the plasma may help to understand phenomena related to instabilities, disruptions and

magnetic reconnection. The Resistive Magnetohydrodynamics (MHD) model presented in this work

includes Maxwell’s equations for electric and magnetic fields, a plasma momentum balance equation,

and the temperature evolution equation:

∇×−→E = − ∂

−→B

∂t(11)

∇.−→B = 0 (12)

∇×−→B = µo

−→J (13)

−→E +−→v ×

−→B = η

−→J (14)

ρd−→vdt

=−→J ×−→

B −∇.↔P +ν∇2−→v (15)

3

2

dT

dt= ∇.(χb∇bT + χ⊥∇⊥T ) +Q (16)

where−→E ,

−→B ,

−→j , −→v , ∇.

↔P , T, Q are, respectively, the electric and magnetic fields, the plasma

current density, the moving plasma velocity field, the stress tensor term, the plasma temperature and

the heating power density. The scalars ρ, η, ν, χb, and χ⊥ are, sequentially, mass density, electrical

resistivity, isotropic viscosity, heat conductivity along the magnetic field lines, and perpendicularly to

those lines. The total derivative ddt is defined as d

dt = ∂∂t +

−→v .∇; the operator ∇b is defined as ∇ba =

((∇a).−→b )

−→b , where

−→b ≡

−→B

||−→B ||

is the magnetic field normalized vector; and the operator ∇⊥ is defined

as ∇⊥a = ∇a−∇ba.

18

This model also includes an equation for the bootstrap current (section 2.5), which will relate evolution

of the temperature profile to evolution of the magnetic field.

This work will focus on perturbations perpendicular to the equilibrium field lines, assuming that the

helicity of this mode (m,n) matches the helicity of one flux surface in the chamber - the resonant surface.

Figure 16 shows a perturbation with a single mode (2, 1) projected over a flux surface.

Figure 16: Perturbation with (2, 1) mode projected over the flux surface.

In a large aspect ratio tokamak with circular cross section, it is a valid approximation to write the Re-

sistive MHD equations (11)-(16) in cylindrical geometry. In this case, the ’toroidal z-direction’ is periodic

(the approximate cylindrical tokamak length is given by 2πRo), thus ϕ = zRo

.

2.2 Magnetic flux evolution

It is possible to merge equations (11) and (14) in order to obtain:

19

∂−→A

∂t= −→v ×

−→B − η

−→J −∇ϕ (17)

where−→A is the potencial vector (

−→B = ∇×−→

A ) and ϕ is the electrostatic gauge potential.

Considering the type of perturbations under analysis, the perturbed component of the velocity field

will always be contained in the plane perpendicular to the magnetic field lines of the resonant surface

(with versor−→b mn =

−→B rs

||−→B z||). As such, it is possible[19], for any given plane, to establish a stream function

u which satisfies the following condition:

∼−→v⊥≡ g(r)∇ ∼u ×

−→b mn (18)

where the function g(r) corresponds to ||−→b ||−2. Its value equals approximately 1, and it will be thus

considered in the following calculations.

As for the equilibrium velocity, its poloidal component is known to be strongly damped in a tokamak

[20], allowing to write the total velocity field as:

−→v ≃ −→vzo +m∑ n∑

∇ ∼u ×

−→b mn (19)

where −→vzo is the equilibrium toroidal velocity (assumed to vary over the minor radius) and−→b mn

corresponds to−→b in the resonant surface.

Regarding the magnetic field, similarly to the velocity field description, one adopts a description of a

field potential, namely the magnetic flux:

∼ψr≡

∫ ∫s

∼−→B .−→n dS (20)

ψo ≡∫ ∫

s

−→Bo.

−→n dS (21)

where S corresponds to the ribbon of the flux surface (depicted in figure 17).

In this case, the following relation applies:

∼−→Bmn= ∇

∼ψrmn ×

−→b mn (22)

20

Figure 17: Helical ribbon of a given flux surface [21].

Therefore:

−→B = Bθ

−→eθ +Bz−→ez +

m∑ n∑∇

∼ψrmn ×

−→b mn (23)

The subscript ’r’ used in∼ψrmn derives from the fact that the value of the perturbed magnetic flux is

necessarily a real number. In fact it is half the sum of a complex conjugate pair of Fourier harmonics for

that mode. The same applies to the stream function:

∼ψrmn=

∼ψmn ei(mθ−

nR z)+

∼ψ∗

mn e−i(mθ−nR z)

2(24)

∼urmn=

∼umn ei(mθ−

nR z)+

∼u∗mn e−i(mθ−

nR z)

2(25)

where∼ψmn,

∼ψ∗

mn,∼umn and

∼u∗mn are all functions of r.

Applying this decomposition into Fourier harmonics to equation (17), and considering a single mode

(m,n), one obtains:

∂−→Ao∂t

= −→vo ×−→Bo +

∼−→v mn ×∼−→B

∗

mn

4+

∼−→v∗

mn ×∼−→Bmn

4− η

−→Jo −∇ϕo (26)

21

∂

∼−→Amn

∂t= −→vo×

∼−→Bmn +

∼−→v mn ×−→Bo − η

∼−→J mn (27)

According to equation (13), it is possible to establish the following relations:

µo−→Jo =

(−∇2ψo −

2n

mRBo) −→ez (28)

µo

∼−→J mn= −∇2

∼ψmn

−→b mn (29)

Taking that into account, and since Bzo is constant in time, it is possible to determine the dot product

with −→ez on both sides of equation (26), and the dot product with−→b on both sides of equation (27), yielding

the following result:

∂ψo∂t

=1

2

m

r

∂

∂rIm(

∼u∗mn

∼ψmn) +

η

µo

(∇2ψo +

2n

mRBo)+ Eo (30)

∂∼ψmn∂t

=i

Rk(r)

∼umn Bo +

in

Rvz

∼ψmn +

η

µo∇2

∼ψmn (31)

where ∇2o ≡ 1

r∂∂r (r

∂∂r ) and ∇2 ≡ 1

r∂∂r (r

∂∂r ) +

1r2

∂2

∂θ2 . Eo is the equilibrium toroidal electrical field

which guarantees ∂ψo

∂t = 0 in the absence of perturbations. Function k(r) is defined as k(r) ≡ m−nqq .

2.3 Vorticity and velocity evolution

It is possible to apply the continuity equation to each fluid velocity dimension. If the plasma is considered

incompressible, adding the resulting equations will yield the following vectorial form:

ρd−→vdt

=−→f (32)

where the total derivative ddt is defined as d

dt =∂∂t +

−→v .

Replacing−→f by the Lorentz component (

−→J ×

−→B ), the stress tensor term, and a friction term resulting

from the viscosity of the plasma, it is possible to obtain equation (15).

In this work, ρ and ν will be considered constant along the minor radius, which is a reasonable

22

approximation when operating in H-mode [12]. The equilibrium plasma density and viscosity will be

named ρo and νo respectively.

To deal with the stress tensor term, it is useful to introduce−→U , the symmetrical of vorticity of plasma

velocity, defined as−→U ≡ −(∇×−→v ). Thus, if the curl operator is used on both sides of equation (15), the

result will be:

ρo

(∂−→U

∂t+ (−→v .∇)

−→U − (

−→U .∇)−→v

)= (

−→J .∇)

−→B − (

−→B.∇)

−→J + νo∇2−→U (33)

where (−→a .∇) is an operator over the following vector, and not a dot product between (−→a and ∇).

Considering a single mode (m,n), decomposition into Fourier harmonics is again possible. Expand-

ing−→U , −→v ,

−→J and

−→B will give rise to the set of equations:

∂−→Uo∂t

= (−→Uo.∇)−→vo +

1

4(

∼−→Umn .∇)

∼−→v∗

mn +1

4(

∼−→U

∗

mn .∇)∼−→v mn

− (−→vo .∇)−→Uo −

1

4(∼−→v mn .∇)

∼−→U

∗

mn − 1

4(∼−→v

∗

mn .∇)

∼−→Umn

+1

ρo((−→Jo.∇)

−→Bo +

1

4ρo(

∼−→J mn .∇)

∼−→B

∗

mn +1

4ρo(

∼−→J

∗

mn .∇)

∼−→Bmn

− 1

ρo(−→Bo.∇)

−→Jo −

1

4ρo(

∼−→Bmn .∇)

∼−→J

∗

mn − 1

4ρo(

∼−→B

∗

mn .∇)

∼−→J mn +

νoρo

∇2−→Uo (34)

∂

∼−→Umn

∂t= (

−→Uo.∇)

∼−→v mn +(

∼−→Umn .∇)−→vo − (−→vo .∇)

∼−→U mn −(

∼−→v mn .∇)−→Uo

+1

ρo(−→Bo.∇)

∼−→J mn +

1

ρo(

∼−→Bmn .∇)

−→Jo −

1

ρo(−→Jo.∇)

∼−→Bmn − 1

ρo(

∼−→J mn .∇)

−→Bo

+νoρo

∇2∼−→Umn (35)

It is useful to notice that:

∼−→Umn= ∇2 ∼

umn−→b mn (36)

It is then possible to replace the velocity perturbations by the corresponding term of the stream

23

function, and also to replace the magnetic perturbations as well as the current by the corresponding

terms of the magnetic flux. If one makes the dot product with −→eθ on both sides of equation (34), and the

dot product with−→b mn on both sides of equation (35), the result will be:

∂Uo∂t

= − 1

2µoρo

n

R

d

dr

(Im(∇2

∼ψmn .

∼ψ∗

mn))+

1

2

n

R

d

dr

(Im(

∼Umn .

∼u∗mn)

)+

νoρo

∇2Uo (37)

∂∼Umn∂t

= in

Rvz

∼Umn −i n

R∇2ovz

∼umn +i

Boµoρo

k(r)∇2∼ψmn −im

r

Boµoρo

∼ψmn

∂jz∂r

+νoρo

∇2∼Umn (38)

Because−→U ≡ −(∇×−→v ), it is possible to rewrite equation (37) as:

∂vzo∂t

=1

2µoρo

n

RIm(∇2

∼ψmn .

∼ψ∗

mn)−1

2

n

RIm(

∼Umn .

∼u∗mn) +

νoρo

∇2(vzo − vzi) (39)

2.4 Temperature evolution

If one applies the perfect gas law to the plasma, the relation between plasma pressure and temperature

will be:

p

T= n (40)

where n is the particle density of the plasma. Because of the constant plasma density approximation,

this means that p = αT , and thus that equation (16) is valid for both temperature and pressure.

Replacing the operators, and considering χb and χ⊥ constant[23], it is possible to rewrite equation

(16) as:

∂T

∂t=

2

3χ⊥∇2T +

2

3χ⊥

( χbχ⊥

− 1)(∇.((∇T.

−→b )

−→b )) +

2

3Q−−→v .∇T (41)

24

As the divergence on the magnetic field is null, ∇.((∇T.−→b )

−→b ) will equal

−→b .(∇(∇T.

−→b )), which is

given by:

∇(∇T.−→b ) = ∇T × (∇×

−→b ) + (

−→b .∇)∇T + (∇T.∇)

−→b (42)

The following approximation is convenient:−→b ≃ 1

||−→Bo||

−→B . Considering a single mode (m,n), T as

T = To+∼Trmn, and

−→b as

−→b =

−→Bo+∇

∼ψrmn×

−→b mn

||−→Bo||

, this result will follow:

B2o ∇2

bT =1

r2∂

∼ψrmn∂θ

∂∼Trmn∂θ

(Bθo∂r

− Bθor

− ∂2∼ψrmn∂r2

+1

r

∂∼ψrmn∂r

)+ (Bθo −

nr

mRBzo −

∂∼ψrmn∂r

)

( 1

r2∂2

∼ψrmn∂θ2

∂T

∂r+

2

r2∂

∼ψrmn∂θ

∂2∼Trmn

∂θ ∂r

+Bθo − nr

mRBzo −∂

∼ψrmn

∂r

r2∂2

∼Trmn∂θ2

− 1

r2∂2

∼ψrmn

∂θ ∂r

∂∼Trmn∂θ

)+

1

r2∂

∼ψrmn∂θ

∂2∼ψrmn

∂θ ∂r

∂T

∂r

+1

r2∂

∼ψrmn∂θ

∂∼ψrmn∂θ

(∂2T

∂r2− 1

r

∂T

∂r) (43)

Once again, decomposition into Fourier harmonics is possible, according to:

∼Trmn=

∼Tmn ei(mθ−

nR z)+

∼T

∗mn e−i(mθ−

nR z)

2(44)

Considering a single mode (m,n), and expanding T , −→v and ψ, will give rise to the set of equations:

∂To∂t

=2

3χ⊥∇2(To − Toi)−

1

2

m

r

∂Im(∼u∗mn

∼T

∗mn)

∂r

+2

3

(χb − χ⊥

)( 1

2 Rk(r)

((mr

+n

R

))Re( ∼Tmn

∂∼ψ∗

mn

∂r

))(45)

25

∂∼Tmn∂t

=2

3χ⊥∇2

∼Tmn − iϵnvzo

∼Tmn + i

m

r

∼umn

∂To∂r

+2

3

(χb − χ⊥

)(( m

2 r+

n

2 R

)2∂

∼ψmn∂r

∂∼ψ∗

mn

∂r− 1

2ϵ2 k(r)2

)∼Tmn (46)

2.5 Bootstrap current

One of the earliest methods to confine plasma has been the employment of so-called magnetic mirrors.

As shown in figure 18, the magnetic field gradient prevents certain particles - those for which the ratio

between their velocity on directions parallel and perpendicular to the field lines is below a certain thresh-

old (dependent on maximum and minimum field intensity) - from crossing the area where field lines have

the highest intensity.

Figure 18: Diagram of plasma confinement method through employment of magnetic mirrors[26].

In an ideal tokamak with infinite aspect ratio, this phenomenon would lose its relevance. However,

in real operating tokamaks the gradient of the magnetic field generates a set of particles trapped in

so-called banana orbits. The number of trapped particles is proportional to the product between particle

concentration and the square root of the inverse of aspect ratio. The fact that the magnetic field is not

homogeneous also prevents these particles from following the field lines of the flux surface where they

are, occurring a drift instead.

26

Figure 19 takes the example of two ions within the same flux surface. Starting at the same point,

their toroidal velocities have opposite directions. Poloidal projections of their banana orbit trajectories

are signaled respectively in red and blue, as well as the direction the particle follows in its trajectory.

Figure 19: Diagram depicting banana orbits of two ions with opposite poloidal velocities at the startingpoint[25].

The mean radius of each particle does not match the radius of the flux surface where it has set from.

This implies to a certain extent that this surface is being populated with ions of either lower or higher

mean radius depending on the direction they follow. If there isn’t any radial gradient of either density or

temperature, both effects are annulled, and this phenomenon becomes irrelevant.

However in case such a gradient exists, the linear momentum added due to particles from higher-

radius surfaces is not symmetrical to the one created by particles of lower-radius surfaces. This addi-

tional linear momentum generates an electric current.

It should be clarified that not only trapped particles are at the origin of this electric current. Due to

the particle’s multiple collisions the added linear momentum is also distributed among traveling particles

from the flux surface in question. The resulting electric current is called bootstrap current.

In order to derive this current’s value the aforementioned banana orbits - resulting from the trapping

of particles through the effect of magnetic mirrors which cause spatial variations in the magnetic field

inside the tokamak - need to be taken into account. Insofar as the drift connected to these orbits is at

the root of the bootstrap current, it becomes important to know the length of their poloidal projections,

27

which is given by[25]:

w ≈m c v∥

eBθ(47)

Where w stands for the length of the orbit’s poloidal projection, m is the particle’s mass, v∥ represents

the particle’s velocity component parallel to field lines, e stands for the elementary charge, and Bθ is the

poloidal magnetic field.

On the other hand, the concentration of trapped particles within a given flux surface is given by[25]:

nt ≈√r

Rn =

√ϵx n (48)

Where nt stands for the concentration of trapped particles, r represents the distance from the particle

to the minor axis, R is the major axis, x represents the ratio between r and the chamber’s minor radius,

ϵ stands for the inverse of the aspect ratio and n corresponds to particle concentration.

Taking the formerly described drift into consideration and assuming a small w, the linear momentum

added through that effect due to temperature and density gradients may be approximated by:

δpt ≈ wdptdr

= w(mv∥

∂nt∂r

+∂v∥

∂rmnt

)(49)

In order to assess the current density generated by that effect, the following expression should be

reminded:

J = e n v = ep

m(50)

Therefore, by introducing a frictional factor[27] and assuming a null density radial gradient, the boot-

strap current for electrons will be given by:

Jb = −e Fδp

m≈ −e w n F

∂v∥

∂r(51)

Where F is the frictional factor due to collisions between passing electrons and ions or trapped

electrons. This factor is given by F = veeϵ vei

and may be approximated by F ≈ ϵ−1.

Thus, by replacing w:

28

Jb ≈ −1

ϵ

m c v∥

Bθ

√ϵx n

∂v∥

∂r(52)

For trapped particles, thermal velocity is slower than parallel velocity (v∥ =√ϵ vth) approximately

by the root of ϵ. This thermal velocity is a function of temperature expressed as vth =√

2Tm . Thus the

relation between parallel velocity and temperature may be expressed as:

v∥ =

√2 ϵ T

m(53)

By replacing in (52) the following will be obtained:

Jb ≈ −1

ϵ

m c√

2ϵTm

Bθ

√ϵx n

∂√

2ϵTm

∂r(54)

Thus:

Jb ≈ − c

Bθ

√ϵx n

∂T

∂r(55)

Therefore, the bootstrap current due to the equilibrium temperature gradient is given by:

Jbo = − c

Bθ

√ϵx n

∂To∂r

(56)

On the other hand, the bootstrap current due to (m,n) mode temperature perturbations, will be given

by:

∼Jbmn= − c

Bθ

√ϵx n

∂∼Tmn∂r

(57)

Equations (30) and (31) should therefore be corrected taking this bootstrap current into account,

which results in:

∂ψo∂t

=m

r

∂

∂rIm(

∼u∗mn

∼ψmn) +

η

µ

(∇2ψo +

2n

mRBo)+ Eo + ηJbo (58)

∂∼ψmn∂t

=i

Rk(r)

∼umn Bo +

in

Rvz

∼ψmn +

η

µ∇2

∼ψmn + η

∼Jbmn (59)

29

3 Linear dynamics of the tearing mode

3.1 The Tearing Equation

In the early phase of instability development, the effect of the perturbation on equilibrium values can

be neglected. To attain a simplified approach to this problem, one might also neglect the influence of

temperature evolution, plasma rotation, viscosity and inertia. If so, in the ideal region, the perturbed flux

may be obtained through the so-called tearing equation[5]:

ϵk(r)∇2∼ψmn −m

x

∂jz∂r

∼ψmn= 0 (60)

With fixed∼ψs= 1 and the appropriate boundary conditions, equation (60) may be solved numerically,

and a code has been developed to this purpose. Figure 20 shows the perturbed flux profile when the

edge is a perfectly conducting wall, whereas Figure 21 shows the tearing equation solution for vacuum

boundary conditions as well as for different external currents.

Figure 20: Numerical solution of the tearing equation for a (2, 1) mode with q(x) = qo + (qa − qo)x2,

qo = 1.3, qa = 3.5 and a perfect conducting wall at the edge.

Section 1.5 introduced ∆′, stressing its dependence on boundary conditions. To calculate this value

one may use different spatial grids in the calculation of the tearing equation, and its radial derivative

(see figure 22). For each minimal spatial step on the grid, one may find the difference between the

30

Figure 21: Numerical solutions of the tearing equation for a (2, 1) mode with q(x) = qo + (qa − qo)x2,

qo = 1.3, qa = 3.5 and positive external current (ψI+); vacuum (ψv); and negative external current (ψI− )at the edge.

Figure 22: Radial derivative of the solution found for the tearing equation.

derivative of the magnetic flux in the two points closest to the resonant surface. The limit as this spatial

step goes to zero is a good approximation of ∆′. In this case, the limit is estimated by a linear fit using

the aforementioned points (see figure 23)

31

Figure 23: Calculation of ∆′: difference between the derivative of the magnetic flux in the two pointsclosest to the resonant surface in function of the minimal spatial step on the grid. ∆′ estimate will be thevalue of b on the linear fit (y = ax+ b).

3.2 Dispersion relation

The resistive tearing-mode instability was first investigated by Furth, Killeen, and Rosenbluth [28]. A

plane slab magnetic field model was used, and an eigenvalue problem for the flux perturbation resulted

from the assumption that there was a perturbed component on the magnetic field growing exponentially

in time. In order to solve the problem analytically, the plasma was divided into two regions: a narrow

inner region for which the resistivity could not be neglected and an outer region, where the infinite-

conductivity assumption holds (see section 1.5). Two characteristic times arose: the resistive diffusion

time and the Alfven time. By matching the solutions, it is possible to obtain the standard approximation

for the dispersion relation:

α1

√q(xs)

n ϵ q′(xs)γ

54 S

34 = ∆′ (61)

where γ is the growth rate of the instability, and α1 ≈ 2.1.

However, strong local asymmetries on the equilibrium current density introduce corrections that be-

come important for moderate values of S (104 − 105), such as those used in some of the simulations in

this work.

32

Introducing the coordinate variable xo = x− xs, where xs is the normalized radius of the resonance

surface, and expanding the tearing equation around xo = 0 up to terms O(x):

∂2∼ψmn∂x2

+1

xo + xs

∂∼ψmn∂x

−(m2

x2s+ b+

a

xo

) ∼ψmn= 0 (62)

where:

a =mxs

djodx

nR

q′

q

(63)

b =a(q′′n2qR

)− d

dx

(mxdjodx

)nR

q′

q

(64)

with a and b calculated at the resonant surface.

It is possible to expand the solution in the vicinity of the rational surface[29]:

∼ψmn

±=

∼ψmn

o(1 +

A±∆′

2xo +

A±∆′

2

a

2

(1− 1

axs

)x2o +

(m2

x2s− a2

2(3− 1

axs) + b

)x2o2

+ ...+ a(xo +

a

2(1 +

1

axs)x2o + ...

)Ln(xo)

)(65)

∼ψmn

o

is the fixed value for∼ψs,

∼ψmn

+

and∼ψmn

−correspond to the positive and negative values

for xo, respectively. Expansion is continuous at xo = 0. A is defined as A ≡ limσ→0(∼ψ′

+ +∼ψ′

−) /

(a∼ψ (rs))−K(1 + Log(σ)), where [30]:

K = 2∂j∂x

jz

(1− 2

s

)(66)

and

s =dLog(q(x))

dLog(x)(67)

Solving the inner layer equations in order to match the solution for the outer layer above one gets[29]:

33

∼ψmn=

∼ψmn

o(1 +

A

2xo +

(Aa

4(1− 1

axs) +

m2

x2s− a2

2(3− 1

axs) + b

)x2o

+a

2xoLn(x

2o + δ2) +

a2

4(1− 1

axs)x2Ln(x2o + δ2)

)+

∼ψmn

1

(x

δ) (68)

where δ = 4

√γS

(Rnqsq′s

)<< 1 is assumed, as well as

∼ψmn

1

∼ψmn

o = O(δ∆′), being∼ψmn

1

a small correction.

Finally, the dispersion relation may be obtained using this solution[29]:

α1 γ S δ = ∆′ + α1δ(aA

2+ a2Ln(δ)− a2

(α2 + π)

α1+ b)

(69)

where α2 ≈ −0.378.

A code has been developed to solve this equation by Newton’s method, in order to obtain the relation

between γ and S. The safety factor’s profile was considered to be q(x) = qo + (qa − qo)x2 with qo = 1.3

and qa = 3.5, and at the edge the wall was assumed to be perfectly conducting.

To obtain ∆′ and A the results of the tearing equation were used. The method to obtain A is anal-

ogous to the method used for the stability parameter: one applies the definition for a spatial grid with

a given smaller step σ, and does so for several grids with smaller steps; until attaining the limits for

the finite differences method. The limit is extrapolated by a linear fit using the points obtained. To A

calculation, however, only the last two points were used (see figure 24).

∆′ and A were calculated as 7.24 and −25.6 respectively. Using these parameters, the dispersion

relation could be calculated. Results are shown in Figure 25.

34

Figure 24: Calculation of A: applying the definition to the two points closest to the resonant surface andplotting it as a function of the minimal spatial step on the grid used. A estimate will be the value of b onthe linear fit (y = ax+ b) using the tow smaller grid distances.

Figure 25: Growth rate γ as a function of Reynold’s number S, both in logarithmic scale. Points aresolutions of equation 69, and the line represents the standard dispersion relation (61).

35

3.3 MHD Linear model

The model described in the previous section was simplified according to the linear approximation put

forth. A simplified version of equations (31) and (38) normalized according to appendix A is presented:

∂∼ψmn∂t

= ϵk(x)∼umn +

η(x)

S∇2

∼ψmn (70)

∂∼Umn∂t

= iϵk(x)∇2∼ψmn −im

x

∂jz∂r

∼ψmn (71)

A numerical code that solves equations (70) and (71) using fourth order Runge-Kutta method (ap-

propriate for second-order differential equations) has been developed, enabling the study of the time

evolution of the flux perturbation profile, as well as the stream function profile. The same safety factor

function has been used, as well as identical parameters. k(x), ψo(x), J(x) and η(x) were calculated

accordingly and are shown in Figure 26.

Figure 26: k, ψo, J and η as function of x.

36

The initial profile used in subsequent simulations, emanating from vacuum assumption (∼jmn= 0), is:

∼ψmn=

αo(

xxs)d , x < xs

αo , x = xs

αo(1−x1−xs

)d , x > xs

(72)

where αo = 10−7 and d = m.

Whatever the initial perturbation profile, one expects the normalized profile to conform to the tearing

equation solution when the mode is destabilized. Therefore, as time passes, the difference between

consecutive steps divided by the time interval between those steps should tend to zero.

The calculation displayed in Figure 27 consists of normalizing each∼ψmn profile to its value in the

rational surface before adding the absolute values of all the differences between each point and the

previous, and then dividing this sum by the time difference between both profiles. This calculation

confirms the prediction, as this value decreases more than one order of magnitude in just the first

few hundred Alfven times. Figure 28 depicts some normalized profiles, as they change from the initial

function to the point where changes become neglectable.

Figure 27: Sum of the absolute values of the differences between consecutive points (in time) dividedby the respective time interval, and time.

Once this happens one would expect no difference between this function and the tearing equation

37

Figure 28: Normalized profiles of∼ψmn for different time values.

solution. As shown in Figure 29, both profiles are remarkably similar. The small discrepancy may be

attributed either to the numerical errors associated with Runge-Kutta method or to the fact that neither

profile was calculated using an infinite number of points.

Once the normalized profile conforms to the solution of the tearing equation, the flux in any of its

points should be given by∼ψmn (t) =

∼ψo e

γt. If this is the case, the second time derivative of the function

logarithm should be null. In four simulations made with S = 105 and different boundary conditions

(conductive wall; vacuum; positive current I+ at x = 1.3; and negative current I− at the same position)

the second derivative of the logarithm of the flux in the rational surface was more than nine orders of

magnitude bellow the logarithm of the perturbation in that point. Figure 30 shows, for each of these

simulations, the value of the magnetic flux on the resonant surface for θ = 0 and z = 0 as a function of

time.

In varying S one can compare the growth rates obtained by the numerical code with the dispersion

relation (69). If both ∆′ and A are obtained through the use of the numerical solution of the tearing

equation, the resulting dispersion relation will significantly differ from the values obtained in the numerical

38

Figure 29: Normalized profiles of∼ψmn for the tearing equation solution (points), the initial profile in the

MHD simulation (blue line), and the profile in the MHD simulation at t = 575 τa (red line).

simulation.

It is possible, however, to use the values obtained through the numerical simulation and those com-

puted by the program designed to solve equation (69) as a function of the calculated ∆′ and A, in order

to calculate A by the least-squares estimator (A = −36.6 with this method). In this manner, the discrep-

ancy between the values will be much smaller. As an alternative, it is also possible to calculate ∆′ by

assuming relation (61) and using the simulation result for γ at S = 108 (∆′ = 6.68 with this method).

These three dispersion relations are drawn in Figure 31.

These results suggest that the latter method described for calculating ∆′ is more reliable than the

former. The fact that the first method entails some errors should be expected, since the logarithm of the

magnetic flux derivative closest to either side of the resonant surface does not display a constant slope.

Therefore, the estimated value for ∆′ will inevitably depend on the minimal set distance, which should

be kept above a lower bound in order to avoid numerical errors.

39

Figure 30: Magnetic flux on the resonant surface as a function of time, with different boundary conditions:external current I

+

Ip= 7.43×10−5 (purple); vacuum (black); perfectly conducting wall (blue); and external

current I−

Ip= −7.43× 10−5 (green).

3.4 Effects of Nonvanishing Viscosity

Nonvanishing viscosity plays a significant role in the stability of perturbations, sometimes stabilizing

unstable ones, sometimes, when in conjunction with plasma rotation, destabilizing stable ones[31].

By linearizing equations (31) and (38) around the resonant surface, ignoring the kinetic effects related

to the difference between the drift velocities of electrons and ions, assuming the thin layer approximation

( |r−rs|a << 1) and using the coordinate variable xo = x− xs, it is possible to obtain[31]:

γ∼ψmn + i

n q′(rs) Boθ(rs)

rsxo

∼umn=

η

µo

∼ψ′′

mn (73)

ν∼u′′′′mn −γ ∼

u′′mn= i

(n q′(rs) Boθ(rs)

rs ρo µoxo

∼ψ′′

mn +mJ ′

rρo

∼ψmn

)(74)

From these equations one can conclude that the vanishing viscosity limit is singular. Under conditions

of today’s tokamaks, viscous effects can be comparable to resistive ones, and the correction caused by

viscosity in the dispersion relation is not negligible. Physically, viscosity will diffuse the vorticity caused

40

Figure 31: Dispersion relations obtained using expression (69) for different values of ∆′ and A (lines);and growth rates computed by the numerical code for different values of Reynold’s number (red points).

by the tearing instability and consequently also the current density.

Equations (73) and (74) may be merged into a single equation for the stream function[31]:

iγrs

n q′(rs) Boθ(rs)xo

∼ψs= x2o

∼umn +i

1

γ

( rs τan q′(rs) a

)2(η

µo)2

∼u′′mn +

( rs τan q′(rs) a

)2( ηµo

)2(νγ

)2 ∼u′′′′mn (75)

In dimensionless form, this equation is:

υY ′′′′ − Y ′′ + z2Y = −z (76)

where z is obtained through the coordinate change z = xo

δ , Y is given by Y = δ γ rsn q′(rs) Boθ(rs)

∼umn,

and υ is defined as υ = ν√ei

3π2

n q′(rs) Boθ(rs)rs

√ηρo

.

An analytic expression which approximates (Figure 32) the solution of this equation is [31]:

41

y(z) =1

2z

∫ π2

0

e−z2

2 cos(θ)√sin(θ) dθ +

√υ

1√π√2

(√z K 1

4

(z2

2

)−

√2

2Γ(

1

4) e

− z√υ

)(77)

where K 14

is a modified Bessel function.

Figure 32: Numerical solution of equation (76) (full line); and analytical approximation given by equation(77) (dot line) [31].

In order to calculate the dispersion relation one must match the inner ∆′ with a given exterior ∆′,

considering only the odd parity solutions of (75):

∆′ = limϵ→0

∼ψ−1

s

∫ +ϵ

−ϵ

∼ψ′′(xo) dx = −i δ µoγ

η

∫C

(1− z Y (z)

)dz (78)

where C is a suitable contour in the complex plane.

Retrieving equation (61), valid in the limit of high S, it is possible to obtain the dispersion relation for

nonvanishing viscosity[31]:

γ5

√√√√1 +√ν

4√2 ei

3π8 Γ(14 )√π

√n q′(rs) Boθ(rs)

rs√ηρo

(iγ)−34 = α

− 45

1

(√ q(xs)

n ϵ q′(xs)

)− 45

S− 35∆′ 45 (79)

A code has been developed to solve this equation by Newton’s method, in order to attain the relation

between γ and Γ, the magnetic Prandtl number. The safety factor’s profile was considered to be q(x) =

qo+(qa− qo)x2 with qo = 1.3 and qa = 3.5, Reynold’s number was S = 108, and at the edge the wall was

considered to be perfectly conducting. Results are shown in Figure 33.

42

Figure 33: Growth rate γ as a function of the logarithm of the magnetic Prandtl number Γ. Each point isa solution of equation (79).

3.5 MHD Linear model with viscosity

To account for the effect of viscosity, equations (31) and (38) were normalized according to appendix A

without setting ν to zero:

∂∼ψmn∂t

= ϵk(x)∼umn +

η(x)

S∇2

∼ψmn (80)

∂∼Umn∂t

= iϵk(x)∇2∼ψmn −im

x

∂jz∂r

∼ψmn +

Γ

S∇2

∼Umn (81)

A numerical code that solves equations (80) and (81) using fourth order Runge-Kutta method has

been developed, enabling the study of the time evolution of the flux perturbation profile, as well as the

stream function profile. The same safety factor function has been used, as well as the same parameters,

except for the magnetic Prandtl number, set as Γ = 0.01.

Figure 34 displays four snapshots of the perturbed vorticity profile. The smaller ones show the

transition between the initial null profile and a later stage when the perturbed magnetic flux has already

conformed to the tearing equation. This scenario still preserves a steady exponential growth, and its

43

vorticity profile is depicted in the same figure.

Figure 34: Vorticity profiles at t = 25τa (red line); t = 45τa (purple line); t = 117τa (blue line) andt = 575τa (black line in the larger graph).

By varying the magnetic Prandtl number in the simulations, it is possible to graph the growth rates

obtained by the numerical code as a function of that parameter. Figure 35 shows these values in a

logarithmic scale for Γ. The value for γ when Γ = 0 is shown as an asymptotical limit (dot line).

Figure 36 shows the same relation, but with S = 108, where equation (79) applies. Solutions of that

equation are drawn in black, and the values obtained by solving equations (80) and (81) are in red. The

asymptotical limit is the same in both cases.

There is some discrepancy between the values obtained by the simulation program and those ob-

tained using equation (79). The scaling of this equation with ν seems inaccurate. A different scaling was

tried:

γ5

√√√√1 + 6√ν

4√2 ei

3π8 Γ( 14 )√π

√n q′(rs) Boθ(rs)

rs√ηρo

(iγ)−34 = α

− 45

1

(√ q(xs)

n ϵ q′(xs)

)− 45

S− 35∆′ 45 (82)

The new set of values obtained using this modified equation was compared to the simulation results,

as shown in figure 36. Indeed the discrepancies are small, and these values suggest this is a more

accurate scaling. This subject should be object of further investigation.

44

Figure 35: Growth rates γ as a function of the logarithm of the magnetic Prandtl number Γ, with S = 105.The dot line is the asymptotical limit, obtained by using Γ = 0.

In order to relate both S and Γ with the growth of the mode, one hundred simulations were made

using the same q(x) profile, and identical boundary conditions. Results are shown in Figure 37 and

table B.1. Boundary results are coherent with the aforementioned analytical dispersion relations, and

the overall results seem to show a continuous function γ(S,Γ) with those boundaries.

The red line in the figure shows the boundary for stability. Because its slope is lower than the unit, it

is possible to conclude that the viscosity necessary for a mode to be stable increases as the resistivity

decreases. As a result, it is unlikely that viscosity turns to be a key factor on stability, since experimentally

Γ < 20.

45

Figure 36: Growth rates γ as a function of the logarithm of the magnetic Prandtl number Γ, with S = 108.The black line is the solution of equation (79), the blue line is the solution of equation (82), and red pointsare results obtained by the simulation program. The dot line is the asymptotical limit, obtained by usingΓ = 0, which is the same in all cases.

46

Figure 37: Growth rates γ × 106 as a function of the logarithm of magnetic Prandtl number Γ and thelogarithm of magnetic Reynold’s number S. Values are on table B.1.

47

4 Quasi-linear evolution of the tearing mode

4.1 Effects of third order non-linear forces

As noted in section 1.5, the linear approximation loses applicability once the perturbation grows to the

point where its influence on equilibrium values becomes apparent. The cross product between the

second order perturbed current density driven by the plasma flow and the perturbed magnetic field will

give rise to third order non-linear forces (Figure 38).

Figure 38: Tearing mode structure in the resistive layer; plasma flow pattern and non-linear third orderforces [18].

These forces will oppose the plasma flow, and increase rapidly with the amplitude of the perturbation.

When both island width and tearing layer width are within the same order of magnitude, the island enters

the nonlinear regime[16]. Following Rutherford’s demonstration in slab geometry, once this happens the

island will grow linearly with time[18]:

d w

dt≈ 1.16

η

µo∆′ (83)

In cylindrical geometry this expression needs to be adjusted[16]:

48

d w

dt≈ 1.16

Zef η

µoRe(∆′) (84)

where Zef is a correction to the resistivity.

Afterwards, second-order perturbed current density will alter the equilibrium current density in such

a way as to cause a saturation of the mode. As mentioned in section 1.5, the growth of the island will be

given by:

d w

dt≈ 1.66

η

µo(∆′ − αw) (85)

Figure 39 shows the phase portrait of the island width where the regime described by equation (85)

applies. The island will grow by itself in case its width is under ws (given by ws = ∆′

α ), and decrease

otherwise. Thus, the saturated size of the island does not depend on S, in spite of its importance in

affecting the time it takes the island to achieve such size.

Figure 39: Phase portrait of island width.

4.2 MHD quasi-linear model without rotation

The model described in section 2 has been simplified neglecting the influence of temperature evolu-

tion and plasma rotation. A reduced version of equations (31), (38) and (30) normalized according to

49

appendix A is presented as follows:

∂∼ψmn∂t

= ϵk(r)∼umn +

η(x)

S∇2

∼ψmn (86)

∂∼Umn∂t

= iϵk(r)∇2∼ψmn −im

x

∂jz∂r

∼ψmn +

Γ

S∇2

∼Umn (87)

∂ψδo∂t

=1

2

m

x

∂

∂xIm(

∼u∗mn

∼ψmn) +

η(x)

S∇2ψδo (88)

where ψo ≡ ψoi + ψoδ and ψoi ≡ ψo(t = 0).

A numerical code to solve these equations using fourth order Runge-Kutta method has been devel-

oped, enabling the study of time evolution of the flux perturbation profile, the stream function profile and

the equilibrium magnetic flux profile.

Figure 40 shows the value of the magnetic flux in the resonant surface for θ = 0 and z = 0 as a

function of time for two q profiles and different initial perturbations (αo = ψs(t = 0)).

Figure 40:∼ψs ×107 as a function of time, with S = 105, Γ = 0.01 and: a) αo = 10−7, in the initial profile,

qo = 1.3 and qa = 3.5 in the parabolic safety factor profile; b) αo = 10−7, qo = 1.0 and qa = 3.5; c)αo = 1.7× 10−4, qo = 1.3 and qa = 3.5; d) αo = 10−4, qo = 1.0 and qa = 3.5.

50

Since the gradient of the equilibrium current in the rational surface is precisely what destabilizes the

mode, one should expect that, viscous effects apart, it would take a decrease in the radial derivative of

this current for the mode to saturate.

This was tested in the simulations mentioned. When the island is near its saturated size, the change

in the equilibrium current profile is noticeable near the resonant surface, since its derivative changes to

a value near zero. These results are shown in Figure 41.

Figure 41: Radial profile of the equilibrium current in the absence of perturbations (blue), and when theisland is near its saturated size (red). Apart from the region near the resonant surface, the profiles aremostly identical.

Numerical problems related with the Runge-Kutta method cause some numerical noise associated

with the time derivative of the island size. To deal with these problems, the program was changed to

associate each point for w with the average of the previous 100 points, and the same was done to time

steps in order to calculate the time derivative. This way it became possible to trace the phase portrait of

the four performed simulations, which are shown in Figure 42.

Phase portraits of simulations a) and b) show the four regimes mentioned so far. This is apparent in

Figure 43, where the diagram of simulation a) is depicted again, and four regions have been highlighted.

In region I, the normalized profile is changing so that the unstable eigenmode emerges from the initial

perturbation profile, as already explained in section 3.3. In region II, where the perturbation eigenfunc-

tions are clearly set, the growth of∼ψ is exponential in time, and thus the slope in the graph should be

51

Figure 42: Phase portrait, with S = 105, Γ = 0.01 and: a) αo = 10−7, in the initial profile, qo = 1.3 andqa = 3.5 in the parabolic safety factor profile; b) αo = 10−7, qo = 1.0 and qa = 3.5; c) αo = 1.7 × 10−4,qo = 1.3 and qa = 3.5; d) αo = 10−4, qo = 1.0 and qa = 3.5.

given by m = γ2 . In fact its value is m = 5.2×10−4, which is quite similar to m = 6.6525×10−4, calculated

using the value for γ obtained in a simulation with identical parameters in the previous chapter.

In region III, the slope becomes null. This means a constant dwdt , or a linear growth in time. One can

notice, however, that this regime ceases soon after it started. The slope is approximately null only for

a narrow range in w. That should be expected: in cylindrical geometry the Rutherford growth regime is

more circumscribed than in slab geometry.

Equation (84) could be normalized according to appendix A:

d w

dt≈ 1.22

∆′

S(89)

This equation enables the calculation of ∆′ through the use of the phase portrait. In the present case,

∆′ = 3.361. This value is of the same order of magnitude than the other values calculated for ∆′ in a

52

Figure 43: Phase portrait, with S = 105, Γ = 0.01, αo = 10−7, in the initial profile, qo = 1.3 and qa = 3.5in the parabolic safety factor profile. Four regimes of growth may be observed.

simulation with the same parameters and boundary conditions (∆′ ≈ 7.24 with the tearing equation, and

∆′ = 6.68 using the standard dispersion relation) but it still remains quite different. Such difference is

explained by the high resistivity used in this simulation. In fact, a simulation made with S = 107 resulted

in a value for the stability parameter much closer (see results in figure 44)

In region IV, there is once again a constant slope. This region is perfectly described by equation 85.

This same behavior may be found in the phase portrait of simulation b). As for simulations c) and d),

they are both in the regime of region IV.

In order to relate both S and Γ with tearing-layer width, 59 simulations have been conducted using

the same q(x) profile and identical boundary conditions. As soon as the second time derivative of the

logarithm of the flux in the rational surface reaches a value nine orders of magnitude smaller than the

flux logarithm, the island is deemed to be in the linear growth regime, and this ratio will be expected to

decrease further before increasing again. Island development ceases to be in the linear growth regime

once this second time derivative increases one order of magnitude - as explained above, its width should

be within the same order as the tearing layer. Results for island size when this happens (as a function

53

Figure 44: Partial phase portrait, with S = 107, Γ = 0.01, αo = 10−7, in the initial profile, qo = 1.3 andqa = 3.5 in the parabolic safety factor profile.

of Reynolds and Prandlt numbers) are shown in Figure 45 and table B.2.

As equation (9) relates tearing-layer width with the growth rate of the mode, results of section 3.5

may be retrieved in order to obtain an estimate of the tearing layer for different values of S and Γ. This

is shown in Figure 46 and table B.3, confirming that, for small viscosity, when the island ceases to be

in the linear regime, its width is in fact within the same order of magnitude as tearing-layer width, both

decreasing as S increases.

With an increase in the Prandtl number, however, these calculations show a different evolution. This

is due to the fact that the calculation of the tearing layer using expression (9) does not account for the

effects of viscosity. In fact, one may consider the existence of a viscous layer increasing with Γ[31],

therefore changing the width in which linear growth regime ceases.

54

Figure 45: Island width when linear growth regime ceases, as a function of the logarithm of magneticPrandtl number Γ and the logarithm of magnetic Reynold’s number S. Values are on table B.2.

55

Figure 46: Tearing Layer calculated using expression (9), as a function of the logarithm of magneticPrandtl number Γ and the logarithm of magnetic Reynold’s number S. Values are on table B.3.

56

4.3 Interaction with resonant external magnetic fields

Toroidal field coils may be used to create a current sheet that interacts resonantly with the tearing mode.

In fact, inevitable misalignments of the field coils used for normal tokamak operation produce a magnetic

error-field which may have a (m,n) component that will interact with the mode.

Figure 47: Profile of∼ψmn, subject to a resonant helical field generated by a current sheet at x = 1.3. The

green line is the rational surface, the blue line is the plasma edge, and the red line is the current sheetradial location.

As a consequence of this current sheet, the (m,n) perturbation of the magnetic flux will have a

discontinuity at the location of the coils, proportional to the intensity of the current (Figure 47). The

growth of the island is affected by this external field as[17]:

d w

dt= 1.66

η

µo

(∆′vac +

µors

( brs

)−m IEk w2

ei(ϕE−ϕo) − αw

)(90)

where ∆′vac is the stability parameter in the absence of any external field, r = b is the location of

the current sheet, ϕo is the phase of the mode (zero, since rotation is not yet being considered) and

k = xsϵ16

q′(xs)q2(xs)

Bo.

If the phase difference is between 90o and 270o, dwdt may decrease, enabling a stabilization scheme

57

based on the use of the external field. However, there is a value of island width below which it cannot be

reduced by this method, and any attempt to do so by using an external current IE with higher absolute

value will cause the mode to flip14, thus bringing the field to further destabilize the mode.

Figure 48: Phase portrait of the island width w, showing different external currents at x = 1.3. The reddash corresponds to the saturated island width in the absence of external current. The colored dashesare values of stable equilibrium island size for different values of the external current bigger than Itr,with the exception of the purple line which corresponds to an unstable equilibrium with IE = Itr.

Equation dwdt = 0 can have one, two, or no real roots, depending on the value of IE , as shown in

Figure 48. The value of the stable saturated island width will correspond to the highest root in case

there is more than one, or to the only root if that is the case, with the exception of IE = Itresh, for which

the equilibrium will be unstable. There will be no roots for IE < Itresh, which means the phase portrait

seems failing to show mode flipping. In fact, when the flipping occurs, there is a phase shift in the mode,

so that the new current interacting with the mode acts like its value is symmetrical. Consequently, island

size will not decrease indefinitely as the graph would indicate, but instead grow to a larger saturated size

than it would otherwise achieve15, as if destabilized by a positive IE with the same absolute value.

14When a mode flips, there is a shift of π in the position of the ”O”, effectively causing the value of the perturbed flux∼ψs at the

resonant surface to change sign[32]15This will no longer hold when plasma rotation is considered: when a steady phase difference is maintained, an IE slightly

58

Figure 49: Perturbed magnetic flux in the resonant surface as a function of time. The branches relatedto time dependence of the external current are signaled in different colors, respectively gray, red andblue.

Figure 49 shows the results of two simulations performed with the following parameters: ϵ = 0.3,

S = 105; Γ = 0.01; αo = 10−7, in the initial profile; qo = 1.3 and qa = 3.5 in the parabolic safety factor

profile. The perturbed flux on the resonant surface is shown as a function of time, and the dependence

of the saturated value on the external field is highlighted.

The external current varies in time as:

I(t) =

0 , t < 9000 τa

t−90002000 I±E , 9000 τa > t > 11000 τa

I±E , t > 11000 τa

(91)

One can clearly observe that the external current (with module |IE | < |Itresh|) changes the saturated

equilibrium width, decreasing it in case there is phase opposition, or increasing it otherwise.

If |IE | > |Itresh|, one would expect to find a mode flipping. Figure 50 depicts this situation: after the

saturation of the mode, the external field leads the perturbed field in the singular surface to saturate at a

negative value. A negative value for∼ψs corresponds to an imaginary value for w. The physical meaning

of this imaginary width consists precisely of the shift in the phase that characterizes the flipping of the

higher than Itresh may in fact be stabilizing on a small scale[33]

59

mode.

Figure 50: Perturbed magnetic flux in the resonant surface as a function of time showing mode flip-ping. The branches related to time dependence of the external current are signaled in different colors,respectively gray, red and blue.

4.4 MHD model for toroidally rotating modes

Equilibrium bulk plasma rotation and drift velocities of the particles may contribute to a propagating

tearing mode. Since vθ is, here, considered null, the rotation of the mode follows the toroidal direction.

Considering any given profile section and a uniform toroidal rotation, the set of magnetic islands in

the plasma will appear to be rotating, in which case the mode frequency will be ωo = nRovz.

To account for these effects, the model described in section 2 has been simplified neglecting the

influence of temperature evolution. A reduced version of equations (31), (38), (30) and (39), normalized

according to appendix A is presented as follows:

∂∼ψmn∂t

= ϵk(r)∼umn + i nϵ vzo

∼ψmn +

η(x)

S∇2

∼ψmn (92)

∂∼Umn∂t

= iϵk(r)∇2∼ψmn −im

x

∂jz∂r

∼ψmn + i n ϵ

(vzo

∼Umn −∇2vzo

∼umn

)+

Γ

S∇2

∼Umn (93)

60

∂ψδo∂t

=1

2

m

x

∂

∂xIm(

∼u∗mn

∼ψmn) +

η(x)

S∇2ψδo (94)

∂vδz∂t

=1

2nϵIm(

∼ψ∗

mn .∇2∼ψmn − ∼

u∗mn .∇2 ∼

umn) +Γ

S∇2vδz (95)

where vz ≡ vzi + vδz and vzi ≡ vz(t = 0).

A numerical code to solve these equations using fourth order Runge-Kutta method has been devel-

oped, enabling the study of time evolution of the flux perturbation profile, the stream function profile, as

well as the equilibrium magnetic flux and equilibrium velocity profiles.

In the numerical simulations carried out, the parabolic q-profile q(x) = qo+ (qa− qo)x2 is considered,

with qo = 1.3 and qa = 3.5, as well as ϵ = 0.3, ρ/mp = 1019 m−3 (mp is the proton mass), S = 105

and Γ = 0.01. The initial equilibrium velocity profile is vzi = vzo + (vza − vzo)x3, with vzo = 5 kHz and

vza = 0.2 kHz.

Figure 51: Mode frequency as a function of time, with q(x) = qo + (qa − qo)x2, qo = 1.3, qa = 3.5,

vzi = vzo + (vza − vzo)x3, vzo = 5 kHz, vza = 0.2 kHz, S = 105, Γ = 0.01, ϵ = 0.3, ρ/mp = 1019 m−3,

and perfectly-conducting wall boundary conditions.

Figure 51 shows simulation results for∼ψs as a function of time, with perfectly-conducting wall bound-

ary conditions. The change in the mode frequency should be expected - a toroidally rotating island is

subject to two kinds of torques: electromagnetic and viscous[17]:

61

∂

∂t(Iϕω) = n.(TEM + TV c) (96)

where Iϕ = Cϕρ(rs)w is the island toroidal momentum of inertia, Cϕ is a geometrical constant and

ρ(rs) is the local plasma density.

The electromagnetic torque is given by[17]:

TEM =2 π2

µor2s n Im(∆′)

Bz w2 q′(r)

16 q(r)2(97)

The role of viscosity is that of a restitution force promoting natural mode frequency16 [17]:

TV c ∝ νo(ωo − ω) (98)

Consequently, after saturation of the absolute value of the magnetic flux occurs, the mode frequency

may differ from the natural mode frequency:

ω =1

1 + 1Kpνo

dIϕdt

(ωo +

1

Kpνo

2 π2

µor2s n Im(∆′)

Bz w2 q′(r)

16 q(r)2

)(99)

where Kp is the proportionality constant between TV c and νo(ωo − ω).

Recalling equation (90), it is clear that the external field affects mode development, stabilizing and

destabilizing its growth alternately. The frequency of this oscillation equals the difference between both

their frequencies. The pendular behavior of phase difference expresses its dominant attractor for positive

IE . The term corresponding to plasma inertia (dIϕdt ∝ dWdt ) adds a weaker attractor to the system at a

90o phase[17].

If the mode locks in phase with the external field, it will be destabilized. This hinders implementing

a stabilization scheme based on a resonant external field, since it becomes apparent, once plasma

rotation is considered, that phase opposition between the field and the mode is not easy to maintain.

Figure 52 shows simulation results for∼ψs as a function of time, with an external current |IE | < |Itresh|.

The current is static, since only differential rotation matters.

16frequency in the absence of any torques

62

Figure 52: Perturbed magnetic flux in the resonant surface as a function of time. The branches relatedto time dependence of the external current are signaled in different colours, respectively gray, red andblue.

63

5 Quasi-linear evolution of neoclassical tearing modes

5.1 Effects of the bootstrap current on mode growth and saturation

In a high pressure tokamak, modes that would otherwise be stable regarding the density gradient of

the equilibrium plasma current may nonetheless be unstable[34]. The drive for these instabilities is

the bootstrap current, a toroidal neoclassical effect described in section 2.5, which results from plasma

pressure perturbations arising from the perturbed convection of the equilibrium temperature.

Due to the heterogeneity in temperature diffusion, which is faster along the magnetic lines, when the

magnetic island is large enough, plasma pressure becomes flattened inside the island. The resulting

local bootstrap current loss is equivalent to driving a helical current in the counter direction - which

is destabilizing. In this case, non-linear growth and saturation of these so-called neoclassical tearing

modes (NTM) is usually described by the generalized Rutherford equation[35]:

A1dw

dt=ηoµo

(∆′ −A2 cb

√ϵp′0s Lq0sB2θs w

)(100)

where Bθs stands for the equilibrium poloidal magnetic field on the rational surface and p0s for plasma

pressure on the same surface in the absence of perturbations. cb is a constant of the order of the unity,

A1 = 0.823 and A2 = 6.34. Lq0s is given by:

Lq0s =q0q′0

(101)

where q0 is the safety factor in the absence of perturbations, evaluated on the rational surface, and

the same applies to its derivative q′0. Similarly, Lp0s is calculated on the rational surface in the absence

of perturbations:

Lp0s =p0p′0

(102)

Taking into account the effect of change in the local magnetic shear due to the loss of the bootstrap

current inside the island leads to the following extension of equation (100)[36]:

64

A1dw

dt=ηoµo

(∆′ −A2

jbs

Bθs w ( 1Lq0s

+ jbsBθs

)

)(103)

where jbs is the bootstrap current on the rational surface in the absence of perturbations.

The method for calculating ∆′ for classical tearing modes has been presented in section 3.1. For

NTMs∼ψmn (rx) - where rx is the minor radius of the island’s x-point - may be employed in the definition

of ∆′ instead of∼ψmn (rs)[37].

However, when island width is not too large, the difference between∼ψmn (rx) and

∼ψmn (rs) may be

neglected. In NTMs, ∆′ is often considered to be the same as that of the classical tearing mode, since

it is determined by the region outside the island, and the bootstrap current is only taken into account

within the island region[35]. As such, it is possible to calculate ∆′ as described in section 3.1 in order to

use its value in equations (100) or (103).

As the island grows, equations (100) and (103) lose their validity since there are several nonlinear

effects on island growth which are not taken into account, such as bootstrap current density fraction,

transport coefficients, and viscosity[35]. Nonetheless, a small change in these equations allows for a

good prediction of the island’s saturated width[35] when the mode is unstable:

A1dw

dt=ηoµo

(∆′ − αw −A2 cb

√ϵp′0s Lq0sB2θs w

)(104)

and

A1dw

dt=ηoµo

(∆′ − αw −A2

jbs

Bθs w ( 1Lq0s

+ jbsBθs

)

)(105)

where α is the term mentioned in sections 1.5 and 4.1.

The above analysis assumes the island to be above the critical size necessary to make diffusion

along the magnetic field lines significantly more important than the diffusion perpendicular to those lines.

This does not hold when the island is very small: since the perimeter of the circuit is reduced, perpen-

dicular transport across the island dominates over parallel transport, and the temperature perturbation

vanishes.

By equating characteristic times for perpendicular transport (τ⊥ ≈(w2

)2 1χ⊥

) and parallel transport

(τ∥ ≈(2Lsmr w

)2 1χ∥

) one obtains a value for wd, a characteristic transport effect island width[27]:

65

wd =

√2Lsr

m

(χ⊥

χ∥

) 14

(106)

where Ls = qLq0s

ϵ .

This value is related to the critical width above which the mode is unstable by[27]:

wc =wsat2

(1−

√1−

(2wdwsat

)2) (107)

where wsat is the saturated island size, obtained from equating (100) to zero, given by [27]:

wsat =√ϵLq0sLp0s

βs(−∆′)

(108)

where β is always referring to βθ.

Richard Fitzpatrick studied the small island limit, where perpendicular transport is significant when

compared to parallel transport and temperature is not a function of island flux surfaces. In the small

island limit, island growth is shown to obey[38]:

dw

dt= C1

ηoµo

(∆′ − C2 ϵ

32β′s

Ls

w

w2d

)(109)

where C1 = 1.2 and C2 = 2.88, and β′s is the value of the derivative of β calculated on the rational

surface.

A more general expression, valid in both small and large island limits is also presented[38]:

dw

dt= C1

ηoµo

(∆′ − C3 ϵ

32β′s

Ls

w

w2d + w2

)(110)

where C3 = 9.26.

The dynamics of island evolution according to this model is summarized in the phase space diagram

shown in figure 53, where ∆′ < 0. Wc is the first fixed point shown on the diagram, and Wsat, the

saturation width of a destabilized island, is the second one. Both may be obtained by setting dwdt = 0 on

equation (110).

66

Figure 53: Phase space diagram of the neoclassical MHD tearing mode [39].

5.2 Neoclassical MHD model

The model described in section 2 is presented. Equations (59), (38), (58), (39), (46), (45), (56) and (57),

normalized according to appendix A, are presented as follows:

∂∼ψmn∂t

= ϵk(r)∼umn + i nϵ vzo

∼ψmn +

η(x)

S∇2

∼ψmn +

η(x)

Sjbmn (111)

∂∼Umn∂t

= iϵk(r)∇2∼ψmn −im

x

∂jz∂r

∼ψmn + i n ϵ

(vzo

∼Umn −∇2vzo

∼umn

)+

Γ

S∇2

∼Umn (112)

∂ψδo∂t

=1

2

m

x

∂

∂xIm(

∼u∗mn

∼ψmn) +

η(x)

S∇2ψδo +

η(x)

Sjbo (113)

∂vδz∂t

=1

2nϵIm(

∼ψ∗

mn .∇2∼ψmn − ∼

u∗mn .∇2 ∼

umn) +Γ

S∇2vδz (114)

∂po∂t

=2

3

X⊥

S∇2(po − poi)−

1

2

m

x

∂Im(∼u∗mn

∼p∗mn)

∂x

+2

3

X⊥

S

(f − 1

)(1

2ϵ k(x)

(cbβ

(mx

+ nϵ))

Re( ∼pmn

∂∼ψ∗

mn

∂x

))(115)

67

∂∼pmn∂t

=2

3

X⊥

S∇2 ∼

pmn − iϵnvzo∼pmn + i

m

x

∼umn

∂po∂x

+2

3

X⊥

S

(f − 1

)((cbβ

)2(m

2 x+n ϵ

2

)2∂

∼ψmn∂x

∂∼ψ∗

mn

∂x− 1

2ϵ2 k(x)2

)∼pmn (116)

∼Jbo= − β

2

q(x)√ϵ x

∂∼po∂r

(117)

∼Jbmn= − β

2

q(x)√ϵ x

∂∼pmn∂r

(118)

A numerical code that solves the above system of equations using fourth order Runge-Kutta method

has been developed, enabling the study of the time evolution of the magnetic flux, temperature and

fluid velocity. In the numerical simulations carried out, the parabolic q-profile q(x) = qo + (qa − qo)x2 is

considered, with qo = 0.9 and qa = 4.5, as well as ϵ = 0.3, ρ/mp = 1019 m−3 (mp is the proton mass),

S = 105, Γ = 0.01, β = 0.03, X⊥ = 50, f ≡ χb

χ⊥= 104, and there’s no fluid velocity. The initial profile

of the perturbations on the magnetic flux, temperature and stream function are obtained by multplying a

constant αo by the normalized values obtained on the linear regime on a previous simulation where the

magnetic flux as already conformed to the tearing equation.

Figure 54 shows the evolution of∼ψs in time with perfectly-conducting wall boundary conditions for

values of αo of 128.0 x 10−7 and 88.0 x 10−7. Because with this boundary conditions and safety factor

profile ∆′ is negative, without bootstrap current both perturbations would vanish, regardless of the initial

island size (this may be seen in fig 55). However, once the changes in the bootstrap current are taken

into account, the data show there is a threshold for the initial perturbation beyond which the mode is

destabilized and saturates at a higher island size. Bellow this threshold, the perturbation still vanishes.

As explained in section 4.2, in order to obtain the phase portrait the program was changed to asso-

ciate each point for w with the average of the previous 100 points, and the same was done to time steps

in order to calculate the time derivative. Figure 56 shows the results for αo = 2.78 x 10−5, just above the

threshold for instability.

68

Figure 54:∼ψs x 107 as a function of time, for αo = 128.0 x 10−7 and αo = 88.0 x 10−7

Figure 55:∼ψs x 107 as a function of time, for αo = 128.0 x 10−7 and αo = 88.0 x 10−7, when changes in

the bootstrap current are turned of.

This simulation, in spite of the absence of higher harmonics, describes the main features expected

to be seen by the theory presented in section 5.1.

The main result which should be emphasized is the confirmation of the widely accepted notion that

69

Figure 56: Phase portrait for a simulation with αo = 110.0 x 10−7

neoclassical tearing modes need a seed island wseed > wc to be driven by the bootstrap current in order

to grow towards a large saturated state.

5.3 Interaction with resonant external magnetic fields

As stated in section 4.3, toroidal field coils may be used to create a current sheet that interacts resonantly

with the tearing mode. It is important to address the relation between this external field and the NTM

growth, among other reasons because this interaction may be behind the triggering of some NTMs[40].

Once the equation (110) is changed to include the external current effect (in the form ciIEw2 e

i(ϕE−ϕo),

according to section 4.3), it will yield:

dw

dt= C1

ηoµo

(∆′ − C3 ϵ

32β′s

Ls

w

w2d + w2

+ C4IEw2

ei(ϕE−ϕo))

(119)

with C4 = µo

rs1k

(brs

)−m.

Considering a null phase difference between the mode and the external current, it is possible to find

three positive roots in the phase diagram of equation (119). A diagram which compares equation (110)

and (119) is depicted in figure 57, showing a decrease in the value of wc, the threshold beyond which the

island will grow towards the large saturated state. wsat, the large saturated state width, also increases.

Two simulations were made, both with an external current as a ramp (IE = Ci t), which was turned

off when a maximum value was attained at t = 4000 τa. The initial values were αo equal to 98.0 x 10−7

70

Figure 57: Phase diagram associated with equations(110) and (119). The red dot marks the unstableequilibrium corresponding to the threshold for the NTM’s growth. Its value diminishes once the effect ofa destabilizing resonant external magnetic field is taken into account

and 1.0 x 10−7 respectively. Results are shown in figure 58.

When initial island width is above the new threshold but below the one corresponding to the absence

of external current, it will grow towards the large saturated state (which should be slightly higher than

the one without the resonant current, had the external current remained on). Even when the external

field disappears, because the island is now above the neoclassical critical width, the growth will continue

until saturation occurs.

When starting island width is much smaller, on the other hand, it will grow to a small saturated size,

as shown in figure 57. Once the external current is turned off, because the island has not attained the

critical size, the perturbation will stabilize.

According to equation (119) it is possible to set such a high value for IE that the threshold value

for instability becomes null. This was tested on a simulation with no rotation and αo equal to 1.0 x

10−7. External current was once again a ramp, but this time turned off at t = 4000 τa. Results are

shown in figure 59. They confirm the hypothesis and suggest a possible explanation for seed islands:

misalignments of the field coils used for normal tokamak operation produce a magnetic error-field which

71

Figure 58:∼ψs x 107 as a function of time, for αo = 98.0 x 10−7 (above) and αo = 1.0 x 10−7 (below), with

an external current in the form IE = Ci t turned off when t = 4000 τa. The timeline with non-null externalfield is signaled in red.

may have an (m,n) component that drives the mode above the threshold value for neoclassical growth.

Figure 60 shows results for a simulation where differential rotation was considered, by setting the

initial equilibrium velocity profile as vzi = vzo, with vzo = 20Hz and the external current starts as a ramp,

but stabilizes at t = 80000τa with a ratio IeIp

= 7.43×10−5. Similarities may be found with figure 60, taken

from ref. [40] where a reduced MHD model is used and the bootstrap current is simulated through a

72

Figure 59:∼ψs x 107 as a function of time, for αo = 1.0 x 10−7 with an external current in the form IE = Ci t

and turned off at t = 10000 τa. The timeline with non-null external field is signaled in red.

dependence on the size of the island. In both cases the bifurcation can be seen clearly.

73

Figure 60:∼ψs x 107 as a function of time, for αo = 10.0 x 10−7 with an external current which starts as a

ramp and stabilizes at t = 80000τa with a ratio IeIp

= 7.43× 10−5.

Figure 61:∼ψs x 107 as a function of time, with an external current which starts as a ramp and then

stabilizes[40]

74

5.4 Mode stabilization through ECCD and ECRH

A method widely known to influence a mode’s amplitude operates through electron cyclotron resonant

heating (ECRH), as well as through electron cyclotron current drive (ECCD). Electromagnetic waves

with a frequency often reaching an order of magnitude of hundreds of GHz traveling in an approximately

toroidal direction both induce a current and heat up the surrounding plasma.

Heating not only changes the bootstrap current due to its dependence on the temperature profile, but

also changes the ohmic current because of the change in the resistivity caused by temperature variation.

The total current’s variation resulting from these factors operates a stabilizing effect.

Figure 62: Gyrotron, high powered source of electromagnetic waves with frequencies reaching an orderof magnitude of hundreds of GHz [41]

Regarding the effects of current drive, its effect diminishes with the size of the island. One could say,

therefore, that electron cyclotron current drive (ECCD) should not, in itself, lead to mode suppression

[17]. However, attending to the tearing mode dynamics, if the current is capable of driving the island

bellow the tearing mode threshold, it should stabilize.

It is possible also to combine both effects in order to more effectively obtain complete mode suppres-

sion. This has been experimentally verified, first in ASDEX Upgrade[42] for a (3,2) mode, and then in

several tokamaks and stelerators for that as well as other modes[43].

Frequencies employed to generate ECCD and ECRH are within an order of magnitude of hundreds

of GHz and the high powered sources of those frequencies, used for this as well as other ends, are

called gyrotrons. These devices are characterized by their power, pulse frequency and duration, as well

75

as deposition width, which ranges necessarily between that of electron cyclotron resonance and that of

the beam’s wave length[43].

This parameter assumes vital importance as the efficacy of both methods described depends on

deposition width and localization in a decisive way. Its importance is easily explained in theory by

the bootstrap current ’s dependence on the temperature derivative, which causes the effect of a more

localized heating to overcome the wider availability of energy to the region.

The location of the deposition region may be controlled by toroidal field adjustment; the gyrotron’s

movement; or variation of emitted frequency, without disregarding that the plasma density profile may

entail cut-off frequencies.

Propagation and absorption of electro-cyclotron waves is reasonably predicted by ray tracing codes,

and experimental results for efficiency are consistent with models based on Fokker-Planck equation[43].

Operation of ITER will depend on the efficacy and accuracy of these techniques. ITER’s project

has been changed to alter its implementation, allowing for the presence of a moving mirror of electro-

cyclotron waves near the plasma, which would predictably obtain higher efficacy in the suppression of

tearing modes[44] - even though a misalignment of 2 cm would be enough to make efficacy drop to zero.

There are basically three alignment techniques[44]: toroidal field variation between shots, employ-

ing Mirnov coils to identify the location of signal decay; step-by-step toroidal field variation (through a

feedback control system) aiming to identify optimal location at each single shot; and use of real-time

computing in order to infer the security factor’s radial profile from plasma diagnoses, and thus accurately

locate the resonant surface just before the occurrence of a mode (active tracking). The latter method,

striving to obtain real-time results, also employs a code to attend to electro-cyclotron wave refraction,

which compares fluctuations with a pre-calculated reference scenario for a specific source location.

One of the advantages of the active tracking method is its increased stabilization efficacy when island

size is very small, as observable in the following expression relating island growth with deposited heating

power[45]:

dw

dt=ηoµo

(∆′(w)− Ch

Prfχ⊥ Te

δ2∥

w

)(120)

Where Prf is the deposition power, and δ∥ is given by:

76

δ∥ =(χ⊥

χ∥

) √Rq2

mq′(121)

This expression applies when w < δ∥ as long as both values remain below deposition width. It is

apparent that the stabilizing effect will tend to infinity when island size tends to zero, predicting high

efficacy for the active tracking method in case alignment is accurate enough. Experimental procedures

have demonstrated the possibility of preventing (3, 2) modes from ever occurring through the use of this

method (see figure 63).

Figure 63: Effect of preventive ECCD inhibiting the occurrence of (3, 2) modes. Green dots depictpotentially unstable situations which resulted stable with ECCD, and red crosses portray situations whichrevealed unstable. The red line marks the prevision of meta-stability with preventive ECCD [44]

In this manner, ECCD and ECRH have proven to be adequate methods to deal with the effects of

MHD instabilities. Its importance makes, however, all the more necessary to understand the role of

temperature evolution in the behavior of these instabilities.

77

6 Conclusions

The present dissertation has striven to explain in chapter 1 the physical mechanism underlying the

development of tearing instabilities, underscoring the importance of understanding and controlling these

instabilities in nuclear fusion research. Chapter 2 has presented a detailed derivation of the MHD model

guiding the creation of the computer program used for performed simulations. The construction of this

computer program from scratch was an important part of this thesis.

Chapter 3 has examined the tearing mode within the scope of linear approximation. The tearing

equation has been reproduced alongside the results of a numerical code created to solve it. These

results were subsequently compared to those obtained by the code simulating the development of the

mode according to MHD equations using regarded simplifications (linear approximation and null viscos-

ity). Despite the difference in method, it has been verified that obtained results were similar in both

cases.

The standard dispersion relation has also been presented in this chapter, as well as the work devel-

oped by Militello et al. [29], whose analysis considers the corrections required by asymmetries in the

current density of the tearing layer which are significant for S values within the same order of magni-

tude as those used in numerical simulations. The results of a numerical code developed to solve that

dispersion equation using Newton’s method have been reported, and subsequently compared to those

obtained by MHD simulations. This comparison evinces the fallibility of calculating A through the solu-

tion of the tearing equation - which are due to the limitations of the finite differences method once the

minimal spatial step goes to zero - and suggests another approach to A calculation, by using the values

of the MHD simulation and using a least-squares estimator.

Considering thereafter a non-null viscosity, the results by Lazzaro et al. [31], valid for high S values,

have been also been reported in this chapter, describing the train of thought leading to the equation

relating the mode’s growth rate to Γ. The results of a program conceived to solve this relation by Newton’s

method have also been presented, and compared to those obtained by MHD simulations. It has been

concluded that a different scaling on ν (given by equation(82)) would lead to smaller discrepancies

between the two sets of values.

The simulation program was used in order to run 100 MHD simulations with different values of S and

Γ. Values for γ were coherent with a continuous function whose boundaries coincide with those studied

78

previously (no viscosity and different values of S, and high S and different values of viscosity). It has

been shown that the value of viscosity necessary for the mode to become stable increases as resistivity

decreases, and that for the values of both parameters used in real tokamaks viscosity is not a key factor

on stability.

Chapter 4 begun by explaining the physical mechanism behind the inapplicability of linear description

once the magnetic island reaches a certain width. Results of MHD simulations were presented subse-

quently, especially the phase portraits which display the aforementioned growth regimes. These phase

portraits have enabled the calculation of some parameters associated to the mode’s development, which

have thus been compared to the values produced in the former chapter. It has been verified that cal-

culating the growth rate within the linear regime through the phase portrait enables an estimate which

is quite close to the value computed straightforwardly. On the other hand, it has been shown that com-

puting ∆′ for Rutherford’s regime generates a result rather different from the previously calculated one,

although remaining within the same order of magnitude, if resistivity is high (S < 105). Such differences

have been explained both by the low resistivity assumption present on the model used for ∆′ calculation.

Sixty simulations for various values of S and Γ have subsequently been used in order to assess the

island’s width once the growth of the perturbed flux ceases to be exponential. Those values have been

compared with the dimensions of a purely resistive tearing layer. It has been concluded, as expected,

that for identical parameters at low Γ both values belong to the same order of magnitude. For higher Γ,

the results are compatible with a viscous resistive layer, instead of a purely resistive one.

Some aspects concerning the relation between saturated island width and external resonant fields

were reviewed, mentioning the limitations associated with a stabilizing method based on the existence

of an external current, which are due to mode flipping when this current overcomes a certain threshold.

Values obtained by MHD simulations illustrate both stabilizing and destabilizing possibilities of external

currents, as well as mode flipping. Results for a set of MHD simulations with plasma rotation have been

subsequently produced. The evolution of mode frequency has been displayed, including a discussion

of its underlying physical mechanism and a brief mentioning of external current influence on rotation,

apparent in results of an MHD simulation.

Chapter 5 begun by reviewing some theoretical issues regarding neoclassical modes. It was followed

by a presentation of the equations implemented in the code devised in this work to simulate NTMs, as

well as by results obtained by some of the effected simulations, which confirm the most relevant theoret-

79

ical aspects introduced. Original results are then produced by activating the external current. It can be

concluded that, following theoretical predictions, the external field may reduce the threshold necessary

for the mode to saturate at a high value, and that for sufficiently high external current values the pertur-

bation may be driven above that threshold. This mechanism is a possible source of seed islands. One

should note that for high β plasmas this seeding resonant magnetic field may be considerably low since

the threshold of instability is also lower. As it was shown, even in the presence of differential rotation

between the driving field and the natural mode frequency, excitation can always occur.

Lastly, a summary approach is made to ECCD and ECRH methods, which operate by either mod-

ifying temperature or driving helical currents, and present an additional argument for the importance

of understanding the role of temperature in MHD instability dynamics. It is also mentioned that the

implementation of these methods has led to the complete suppression of different tearing modes.

80

A Normalizations

A.1 Auxiliary definitions

τa ≡a√µoρo

Boτv ≡

a2ρoνo

τr ≡a2µoηo

τ⊥ ≡ a2

χ⊥(122)

S ≡ τrτa

Γ ≡ τrτv

X⊥ ≡ τrτ⊥

(123)

ϵ ≡ a

Rk(r) ≡ m− nq(r)

q(r)f ≡ χb

χ⊥(124)

A.2 Normalizations

t −→ t τa (125)

r −→ x a (126)

∼ψmn−→

∼ψmn Bz a (127)

∼umn−→

∼umn

a2

τa(128)

η(x) −→ ηo η(x) (129)

P (x) −→ Po P (x) (130)

81

B Values for Contour Plots

B.1 Growth rate in linear regime

Γγ(Γ, S)× 106

0.01 0.0359381 0.129155 0.464159 1.6681 5.99484 21.5443 77.4264 278.256 1000

108 43.735 40.390 31.354 29.130 18.586 16.378 11.312 7.593 6.624 4.918

46415900 61.988 55.964 52.641 38.557 29.216 24.525 14.089 13.434 9.879 7.071

21544300 97.975 90.474 76.186 60.350 46.942 39.677 26.696 19.985 14.370 9.896

107 151.752 138.369 119.137 94.833 74.567 55.303 40.376 28.861 19.789 12.687

S 4641590 230.332 210.782 183.356 138.010 110.585 81.333 57.829 39.254 24.753 13.569

2154430 336.125 316.005 273.932 220.374 162.553 115.401 77.584 47.967 25.151 7.871

106 469.803 453.746 405.789 317.758 227.911 152.410 92.296 45.842 10.734 -15.148

464159 747.288 687.990 574.262 435.900 295.188 175.678 81.687 10.511 -41.745 -78.697

215443 1020.939 935.901 771.719 549.712 327.158 140.701 -2.862 -108.199 -182.076 -230.708

105 1304.805 1178.006 925.528 576.862 227.822 -56.646 -269.406 -416.882 -512.581 -571.388

Table 2: Growth rates γ × 106 as a function of magnetic Prandtl number Γ and Reynold’s number S.

82

B.2 Island width in the end of the linear regime

Γw′(Γ, S)× 103

0.01 0.0359381 0.129155 0.464159 1.6681 5.99484

108 3.8360 3.8318 3.8389 3.8393 3.8381 3.8370

46415900 3.8478 3.8484 3.8485 3.8467 3.8453 3.8431

21544300 3.8626 3.8628 3.8600 3.8655 3.9592 3.8747

107 3.8795 3.8877 3.8804 3.8782 3.9228 3.8937

S 4641590 3.9314 3.9184 3.9221 3.9091 3.9421 3.8933

2154430 3.9763 3.9888 3.9465 3.9458 120.8329 56.1027

106 5.4379 5.2297 4.2261 37.8054 52.2845 59.8497

464159 11.8922 14.5289 28.4550 46.2915 58.0472 97.8231

215443 14.8511 31.8985 32.3248 52.6727 52.4482 110.2865

105 33.0723 38.6315 70.4288 52.7269 119.4123 -

Table 3: Island width in the end of the linear regime w′ × 103 as a function of magnetic Prandtl numberΓ and Reynold’s number S.

83

B.3 Tearing Layer size

ΓL(Γ, S)× 103

0.01 0.0359381 0.129155 0.464159 1.6681 5.99484

108 5.29971 4.28824 3.32240 2.69234 1.98613 1.58833

46415900 5.78257 4.65253 3.78188 2.88781 2.22390 1.75703

21544300 6.48371 5.24617 4.14807 3.23009 2.50381 1.98158

107 7.23315 5.83405 4.63862 2.99878 2.81091 2.15311

S 4641590 8.02847 6.4814 5.16656 3.97212 3.10195 2.37107

2154430 8.82408 7.17190 5.71200 4.46513 3.41554 2.58779

106 9.59452 7.85080 6.30162 4.89292 3.71665 2.77416

464159 10.77500 8.71177 6.87314 5.2953 3.96493 2.87446

215443 11.6491 9.40845 7.40018 5.61149 4.06818 2.71927

105 12.38600 9.96547 7.74417 5.67953 3.71629 -

Table 4: Tearing layer L× 103 calculated using expression (9) and values of section B.1, as a function ofmagnetic Prandtl number Γ and Reynold’s number S.

84

C Numerical details

C.1 Finite diferences

To solve systems of differential equations, one may use the finite differences method to replace a spatial

derivative in a point Aj by a sum∑j+pi=j−p ciF (Ai)[46], function of the values of some of the grid points.

In this case, tree points will be considered to approach the first and second derivative, in the form[∂F∂x

]Aj

≈ α′F (Aj−1) + β′F (Aj) + γ′F (Aj+1) and[∂2F∂x2

]Aj

≈ α′′F (Aj−1) + β′′F (Aj) + γ′′F (Aj+1).

The general case of an asymmetrical grid will be considered, as shown in the scheme presented in

figure 64:

Figure 64: Scheme of the asymmetrical grid considered

To obtain values α′, β′ and γ′ with O(h3) precision (the best obtained with tree points [48]), one

considers the following condition:

α′F (Aj−1) + β′F (Aj) + γ′F (Aj+1)−[∂F∂x

]Aj

= O(h3) (131)

Or:

α′F (Aj − h1) + β′F (Aj) + γ′F (Aj + h2)−[∂F∂x

]Aj

= O(h3) (132)

It is possible to make a Taylor expansion around Aj , and obtain:

α′(F (Aj)−

[∂F∂x

]Aj

h1 +1

2

[∂2F∂x2

]Aj

h21 +Oa(h3)

)+ β′F (Aj)

+ γ′(F (Aj) +

[∂F∂x

]Aj

h2 +1

2

[∂2F∂x2

]Aj

h22 +Ob(h3)

)−[∂F∂x

]Aj

= Oc(h3) (133)

85

Or:

F (Aj)

(α′+β′+γ′

)+[∂F∂x

]Aj

(−h1α′+h2γ

′−1

)+[∂2F∂x2

]Aj

(h21α

′

2+h22γ

′

2

)+Oa(h

3)+Ob(h3) = Oc(h

3)

(134)

and thus, it is possible to get the following system of equations:

α′ + β′ + γ′ = 0

h2γ′ − h1α

′ − 1 = 0

h21α′

2+h22γ

′

2= 0 (135)

Solving the system, one obtains the values of α′, β′ and γ′ as functions of h1 and h2:

α′ =−h2

h1(h1 + h2)(136)

β′ =h2 − h1h2h1

(137)

γ′ =h1

h2(h1 + h2)(138)

An analogous reasoning may be used to obtain values α′′, β′′ and γ′′ with O(h3) precision:

α′′F (Aj−1) + β′′F (Aj) + γ′′F (Aj+1)−[∂2F∂x2

]Aj

= O(h3) (139)

Or:

α′′F (Aj − h1) + β′′F (Aj) + γ′′F (Aj + h2)−[∂2F∂x2

]Aj

= O(h3) (140)

And once again, making a Taylor expansion around Aj :

86

α′′(F (Aj)−

[∂F∂x

]Aj

h1 +1

2

[∂2F∂x2

]Aj

h21 +Oa(h3)

)+ β′′F (Aj)

+ γ′′(F (Aj) +

[∂F∂x

]Aj

h2 +1

2

[∂2F∂x2

]Aj

h22 +Ob(h3)

)−[∂2F∂x2

]Aj

= Oc(h3) (141)

Or:

F (Aj)

(α′′+β′′+γ′′

)+[∂F∂x

]Aj

(−h1α′+h2γ

′)+[∂2F∂x2

]Aj

(h21α

′

2+h22γ

′

2−1

)+Oa(h

3)+Ob(h3) = Oc(h

3)

(142)

and thus, obtain a new system of equations:

α′′ + β′′ + γ′′ = 0

h2γ′ − h1α

′ = 0

h21α′

2+h22γ

′

2− 1 = 0 (143)

whose solution are:

α′′ =2

h1(h1 + h2)(144)

β′′ =−2

h2h1(145)

γ′′ =2

h2(h1 + h2)(146)

These coefficients allow the use the finite differences method on the asymmetrical grid considered.

87

C.2 Explicit step

If a differential equation in the form:

∂F (x, t)

∂t= D(x)

∂2F (x, t)

∂x2+ C(x)

∂F (x, t)

∂x+ E(x)F (x, t) +Q(x, t) (147)

is considered, depending on D(x), C(x), E(x) and Q(x) an analytical solution may be impossible.

That is the case with the differential equations presented in the model considered in this work.

The finite differences method allows the possibility of calculation to an approximation of function

F (x, t). Two kinds of steps may be used in conjunction to other methods for solving differential equations:

explicit and implicit.

For the explicit step, one uses the coefficients calculated in the section above, to turn analytical

equation (147) into a discrete system of equations, in the following way:

Fnj − Fn−1j

dT= Dj

(α′′Fn−1

j−1 +β′′Fn−1j + γ′′Fn−1

j+1

)+Cj

(α′Fn−1

j−1 +β′Fn−1j + γ′Fn−1

j+1

)+EjF

n−1j +Qn−1

j

(148)

or:

(149)

Therefore, if one knows the boundary conditions in time (F (x, 0)) and space (F (0, t), F (1, t)) and

computes the sets of values F 01 , F

02 , F

03 , . . . , F

0N−2, F

0N−1, F 0

0 , F10 , F

20 , . . . , F

n0 , F

n+10 and F 0

N , F1N , F

2N ,

88

. . . , FnN , Fn+1N , the calculation of the remaining points is straightforward.

This explicit method is known to be numerically stable and convergent whenever the following relation

holds:

dT

δ2≤ 1

2(150)

where δ is the minimum distance in the space grid. The errors are linear over the time step and

quadratic over δ.

C.3 Implicit step

If a differential equation in the form (147) is considered, depending on D(x), C(x), E(x) and Q(x) an

analytical solution may be impossible. The finite differences method allows the possibility of calculation

to an approximation of function F (x, t). For the implicit step, one uses the coefficients calculated in

section C.1, to turn analytical equation (147) into a discrete system of equations, the following way:

Fnj − Fn−1j

dT= Dj

(α′′Fnj−1 + β′′Fnj + γ′′Fnj+1

)+ Cj

(α′Fnj−1 + β′Fnj + γ′Fnj+1

)+EjF

nj +Qn−1

j (151)

or:

(152)

89

Thus, if one knows the boundary conditions in time (F (x, 0)) and space (F (0, t), F (1, t)) and com-

putes the sets of values F 01 , F

02 , F

03 , . . . , F

0N−2, F

0N−1, F 0

0 , F10 , F

20 , . . . , F

n0 , F

n+10 and F 0

N , F1N , F

2N ,

. . . , FnN , Fn+1N , in order to calculate the remaining values the presented system of equations must be

solved.

The implicit step is always numerically stable and convergent. However, since it requires solving the

system equations, it is more numerically intensive than the explicit method. The errors are linear over

the time step and quadratic over δ.

C.4 Implementation of Runge-Kutta method

It is best to solve nonstiff problems using explicit steps in order to achieve acceptable accuracy with

minimal costs. However, as problems become increasingly stiff, stability rather than accuracy becomes

the dominant consideration, and implicit steps become the more appropriate choice[48].

In this work a compromise between both approaches was tried, using a semi-implicit fourth order

Runge-Kutta method (appropriate for the second-order partial differential equations of this work) with

the following Butcher tableau:

12

12 0 0 0

12

12 0 0 0

12 0 1

2 0 0

1 0 0 1 0

16

13

13

16

(153)

The implementation is quite similar to the traditional explicit fourth-order Runge-Kutta method, but

the first step is implicit. This is enough to guarantee stability even for small δ values, and still achieve a

good balance between accuracy and computational effort.

C.5 Spatial grid

The equations developed in this work are expected to be quite sensitive around the resonant surface

q = mn of the mode, in the so-called tearing layer. There are, consequently, numerical advantages in the

use of non uniform spatial grid, concentrated around the rational surface.

90

To implement the grid with N points, one considers N = N1 +N2 +1 where N1 is the number of grid

points from x = 0 to x = xs. N1 is calculated as simple proportion N1

N = xs

1 , and rounded off.

Starting from both sides of the N1 grid point, the remaining points are calculated according to:

xN1−i+1 − xN1−i =δ

(α1)i(154)

xN1+i+1 − xN1+i =δ

(α2)i(155)

with x1 = 0, and xN = 1. For calculations where a point after the grid is needed (xN+1 = Ω) it was

calculated assuming Ω− 1 = 1− xN−1.

δ is the difference between the resonant surface and the near points, and its value regulates the

concentration of the grid points around this surface. α1 and α2 are calculated, by Newton’s method, from

the non linear equations that result from relations:

∑i

xN1−i+1 − xN1−i = xs (156)

∑i

xN1+i+1 − xN1+i = 1− xs (157)

Once determined all the grid points, α′, β′, γ′, α′′, β′′, and γ′′ are calculated accordingly.

C.6 Adaptive Stepsize

The program used to solve the equations of the MHD model exerts adaptive control over its own

progress, making changes in its stepsize. With fourth-order Runge-Kutta, the most straightforward tech-

nique is step doubling[47] - each step is taken twice, once as a full step, then, independently, as two half

steps. The difference between the two numerical estimates is a convenient indicator of truncation error:

∆ −→ ||y2 − y1y1

|| (158)

This notation hides the fact that y2 and y1 are actually vectors of solutions. In fact, the calculation of

the truncation error was not so straightforward:

91

∆ =1

N

N−1∑i=1

||y2i − y1iy1i

|| (159)

Since the truncation error of a fourth-order Runge-Kutta scales as dT 4[47], it is possible to estimate

the time step that corresponds to a desired accuracy by:

dT0 = dT1

(∆0

∆1

)0.25(160)

where ∆0 is the desired accuracy, dT1 is the present time step and ∆1 the corresponding truncation

error. This equation would be used in two ways: if ∆1 is larger than ∆0 in magnitude, the stepsize should

decrease to dT0 when the present (failed) step is tried again. If ∆1 is smaller than ∆0, the stepsize should

increase to dT0 for the next step.

However, since all the matrices in the program must be recalculated once dT changes, it is not very

efficient to change it at every time step. Equation (160) may be used instead as a reference to establish

boundaries on ∆ that once surpassed should increase or decrease the time step.

As such, a ratio R = ∆2

dT 20

is calculated. If the ratio is higher than one, the step is recalculated with

a new dT1 1.5 times smaller. If the ratio is lower than 0.8, dT1 used in the next step will be 1.5 times

higher.

C.7 Computational details

The implementation of the mentioned methods is computationally quite demanding: a program that

solves differential equations for a reasonable time window is considerably long. For this reason, the

execution speed of the algorithm was considered as an important priority in choosing the language and

the libraries used.

To write the code, C language has been chosen. To make simple operations between matrices,

BLAS library has been selected. To solve systems of equations, the choice fell on the libraries LAPACK

and ATLAS, both an expansion of the previous library.

The library BLAS (Basic Linear Algebra Subprograms [49]) was originally programmed in FORTRAN

and later ported to C. To ensure greater efficiency of this library, it was compiled in the computers used

to run the simulations.

92

Afterwards, LAPACK library [50] (dependent of BLAS) was compiled, once again in the computers

used to run the simulations, in order to ensure greater efficiency and computational optimization.

The library ATLAS (Automatically Tuned Linear Algebra Software [51]) is implemented in FORTRAN

and C and has all the functions of BLAS and some of LAPACK. The purpose of this library is to further

optimize the functions of the two earlier libraries.

Some of the simulations were conducted on Windows XP SP2 on a Pentium II 450 MHz with 384MB

of RAM and others on Linux Ubuntu 8.10 on a Pentium IV 2.4 GHz with 512MB of RAM. In Windows,

CYGWIN [52] was used to simulate a Linux interface. To compile and link in Windows, the libraries used

were BLAS and LAPACK, whereas in Linux BLAS and ATLAS were selected.

Among these libraries, functions suitable for operations with complex band matrices were used.

One difficulty in dealing with these libraries was linked to the lack of documentation, especially for

the version in C, concerning the compilation process and implementation of the code. This shortage

is partly due to the fact that these libraries are very old, and originally done in FORTRAN, with little

documentation ported to C.

The code required to solve the differential equations was built from scratch, and its design is an

important part of this work. Because the length of the code is over 5000 lines, its inclusion in this thesis

was considered unsuitable.

A full simulation in the Pentium II computer, as shown in figure 60, where S = 105 and 100 points are

used in the spatial grid, takes about 12min for each 10000τa.

93

D Boundary conditions

D.1 Boundary conditions in time

D.1.1 Equilibrium Magnetic flux as a function of Safety Factor

From the profiles of the safety factor and the toroidal magnetic field, it is possible to calculate the poloidal

magnetic field:

Bθ(r) = Bϕ(r)r

R q(r)(161)

From these values, one may determine the equilibrium magnetic flux for t = 0:

ψo(r) =

∫ 2πR

o

∫ r

o

Bϕ(τ)τ

R q(τ)dτ dl (162)

Normalizing, and attending that Bϕ is constant, it is possible to obtain:

ψo(x, t = 0) =

∫ x

o

ϵ τ

q(τ)dτ (163)

D.1.2 Equilibrium Electric Current and Electrical resistivity

The relationship between the equilibrium current and magnetic flux, as described in section 2.2, is given

by:

Jo(r) = − 1

µo∇2ψo(r) (164)

Normalizing:

Jo(x, t = 0) = −∇2ψo(x) (165)

Thus, assuming a constant electric field along the radius, and the verification of Ohm’s law, it is

possible to obtain the normalized electrical resistivity from the initial equilibrium current:

η(x) =Jo(0)

Jo(x)(166)

94

D.1.3 Perturbed Magnetic flux and stream function

In most cases, the initial perturbation of the magnetic flux is given by the vacuum condition (∇2∼ψmn≡ 0):

∼ψmn (x, t = 0) =

αo(

xxs)d , x < xs

αo , x = xs

αo(1−x1−xs

)d , x > xs

(167)

where αo = 10−7 and d = m.

The initial perturbed stream function is zero. Therefore the perturbation on the vorticity is also zero.

D.1.4 Equilibrium velocity

The equilibrium velocity is zero in most of the simulations.

In simulations where there was rotation, the equilibrium velocity corresponded to the following ex-

pression:

vzi = vzo + (vza − vzo)x3 (168)

Where vzo = vz(x = 0) e vza = vz(x = 1).

D.1.5 Pressure

Considering the equation of pressure balance:

−→J ×

−→B = ∇p (169)

which implies, in equilibrium, when considering both the toroidal ohmic current and the bootstrap

current, the following equality:

∂p

∂r= −

(jo(r)Bθ(r) + jbo(r)Bθ(r)

)(170)

Or:

95

∂po∂r

= −(jo(r)Bθ(r)− c

√r

R

∂po∂r

)(171)

After normalizing, it is possible to obtain:

∂po∂x

= −CBp(jo(x)

x ϵ

q(x)− β

2

√ϵx

∂po∂r

)(172)

Where CBp ≡ Bo

p(0) , computed before normalizing. Hence:

∂po∂x

= −CBp jo(x)x ϵ

q(x)(1 + β2

√ϵx)

(173)

Thus, by knowing jo(x), it is possible to perform the numerical integration that results in po(x, t = 0).

The function obtained by this numerical integration is very similar to the following function:

po(x, t = 0) =1

(1 + x2)3(174)

It is therefore possible to use the latter for simplicity.

The initial pressure perturbation is null:

∼pmn (x, t = 0) = 0 (175)

D.2 Boundary conditions in space

D.2.1 axis

Considering the physics of the problem and the cylindrical geometry in which it is described, it becomes

clear that all the perturbed values (magnetic flux, velocity and pressure) shall be zero:

∼ψmn (x = 0, t) = 0 (176)

∼umn (x = 0, t) = 0 (177)

96

∼pmn (x = 0, t) = 0 (178)

Concerning the equilibrium values, it is the radial derivative which should be null:

∂ψo(x = 0, t)

∂x= 0 (179)

∂vz(x = 0, t)

∂x= 0 (180)

∂po(x = 0, t)

∂x= 0 (181)

D.2.2 edge

The perturbed magnetic flux in the plasma edge depends on the problem being studied. For most of the

simulations considered, with perfectly conducting wall, its value is considered null:

∼ψmn (x = 1, t) = 0 (182)

If there is vacuum after the edge, one considers a perfectly conducting wall in the infinity (∇2∼ψmn≡

0), and a x−m scaling for the solution in vacuum. To implement this result, a point ψN+1 after the edge

is considered, which is calculated in each step through the following condition:

∼ψN+1=

∼ψN

(xN+1

xN

)−m(183)

Where xN+1 is calculated by xN+1 = 2− xN−1.

When there is a current sheet in vacuum xb ≡ ba , the solution will have a discontinuity in the derivative

at that point which is given by:

[∂

∼ψmn (xb, t)

∂x

]xb+

xb−

= −IE(t)xb

(184)

which results in the following condition:

97

∼ψN+1=

∼ψN

(xN+1

xN

)−m+µoIE(t)

2mx−mb

(xmN+1 − x−mN+1

)(185)

With regard to the perturbed stream function, as a consequence of the fact that the particle density

outside of the edge is residual, its value is considered null:

∼umn (x = 1, t) = 0 (186)

Considering the perturbed pressure, the residual particle density inhibits the heat diffusion outside

the edge. As such:

∂∼pmn (x = 1, t)

∂x= 0 (187)

Regarding the equilibrium magnetic flux and the toroidal velocity in the edge, the equations consid-

ered in the MHD model adopted lead to the following conditions:

ψo(x = 1, t) = ψo(x = 1, t = 0) (188)

vz(x = 1, t) = vz(x = 1, t = 0) (189)

Concerning the equilibrium pressure, the inexistence of heat diffusion in the edge allows once again

to find the necessary condition:

∂po(x = 1, t)

∂x= 0 (190)

98

References

[1] Energy Information Administration, International Energy outlook, US Department of Energy, Wash-

ington DC, 2001

[2] Energy Information Administration, International Energy Annual, US Department of Energy, Wash-

ington DC, 2004

[3] BP, BP Statistical Review of World Energy 2008, BP, London, 2008

[4] Eric Martinot, Renewables Global Status Report, Renewable Energy Policy Network for the 21st

Century, 2006

[5] J. Wesson, Tokamaks, Clarendon Press, Oxford, 1997

[6] Claude Mandil, World Energy Outlook 2005, International Energy Agency, 2005

[7] K. Aki, P. Richards, Climate Change 2007: The Physical Science Basis - Summary for Policymak-

ers, Intergovernmental Panel on Climate Change, New York, 2007

[8] S. Bennett, M. Sene, O. C. Jones, G. F. Phillips, Kaye and Laby, National Physics Laboratory, 2008

[9] http://fusionforenergy.europa.eu/ , 29-9-2008

[10] http://upload.wikimedia.org/wikipedia/commons/4/4b/Tokamak fields lg.png acedido em 7-6-2008

[11] http://www.iter.org/pics/ITER% col.jpg , 5-6-2008

[12] J. Hugill, Brazilian Journal of Physics 27, 407 (1997)

[13] R. Dilao, Tecnicas Matematicas da Fısica, IST, Lisboa, 1995

[14] H. P. Furth, Phys. Fluids 28, 1595 (1985)

[15] B. V. Waddell, B. Carreras, H. R. Hicks, J. A. Holmes, Phys. Fluids 22, 896 (1979)

[16] R. Coelho, Non-linear mode coupling between tearing modes in a Tokamak plasma, Instituto Su-

perior Tecnico, Lisboa, 2001

99

[17] R. Coelho, Modelling of the tearing mode response to electrodynamic plasma flow and mode cou-

pling perturbations in Tokamaks, Instituto Superior Tecnico, Lisboa, 2001

[18] P. H. Rutherford, Phys. Fluids 16, 1903 (1973)

[19] B. S. Massey, J. Ward-Smith, Mechanics of Fluids, Nelson Thornes, 1998

[20] S. Hirshman, D. J. Sigmar, Nucl. Fusion 21, 1079 (1981)

[21] R. B. White, Reviews of Modern Physics 58, 183 (1998)

[22] http://eqseis.geosc.psu.edu/∼cammon/HTML/RftnDocs/rftn01.html , 29-9-2008

[23] T. A. Gianakon, C. C. Hegna, J. D. Callen, Phys. Plasmas 3, 4637 (1996)

[24] R. J. Goldston, P. H. Rutherford, Introduction to Plasma Physics, IOP, 1995

[25] A G Peeters, Plasma Phys. Control. Fusion 42, B231 (2000)

[26] http://farside.ph.utexas.edu/teaching/plasma/lectures1/node22.html , 20-8-2009

[27] R. J. La Hayeb, Phys. Plasmas 13, 055501 (2006)

[28] H. P. Furth, J. Killen, M. N. Rosenbluth, Phys. Fluids 6, 459 (1963)

[29] F. Militello, G. Huysmans, M. Ottaviani, F. Porcelli, Phys. Plasmas 11, 125 (2004)

[30] P. Maget et al, Nuclear Fusion, 125 (2009)

[31] E. Lazzaro, G. Lampis, A. Bernardinello, Journal of the Physical Society of Japan 63, 4373 (1994)

[32] D. A. Monticello, R. B. White, M. Rosenbluth, Plasma Physics and Controlled Nuclear Fusion Re-

search , 1978 (Proc. 7th Int. Conf Tokyo, 1978) IAEA-CN-37/K-3 Vol.1 IAE, Viena, 1978

[33] R. Coelho, E. Lazzaro, Nuclear Fusion 41, 813 (2001)

[34] Q. Yu, S. Gunther, K. Lackner, Nuclear Fusion 40, 2031 (2000)

[35] Q. Yu, S. Gunther, K. Lackner, Phys. Plasmas 11, 140 (2003)

[36] Q. Yu, S. Gunther, Plasma Phys. Controled Fusion 40, 1989 (1998)

100

[37] R. B. White, D. A. Monticello, M. N. Rosenbluth, B. V. Waddell, Phys. Fluids 20, 800 (1977)

[38] R. Fitzpatrick, Phys. Plasmas 2, 825 (1995)

[39] T. A. Gianakon, C. C. Hegna, J. D. Callen, Phys. Plasmas 3, 4637 (1996)

[40] R. Coelho, E. Lazzaro, Phys. Plasmas 11, 1440 (2004)

[41] http://crppwww.epfl.ch/tcv/ecrhb.html , 1-9-2007

[42] G. Gantenbein et al, Phys. Rev. Lett 85, 1242(2000)

[43] Manfred Thumm, Plasma Phys. Control. Fusion 45, A143 (2003)

[44] R.J. La Haye et al, Nuclear Fusion 48, 054004 (2007)

[45] G. Kurita, T. Tuda, M. Azumi, T, Takizuka, T. Takeda, Nuclear Fusion 34, 1497 (1994)

[46] R. J. LeVeque, Finite Difference Methods for Ordinary and Partial Differential Equations, University

of Washington, Washington D.C., 1998

[47] W. H. Press, B. P. Flannery, S. A. Teukolsky , W. T. Vetterling, Numerical Recipes in C: The Art of

scientific computing, Cambridge University Press., Cambridge, 1988

[48] H. Pina, Metodos Numericos, McGraw Hill, Lisboa, 1995

[49] http://www.netlib.org/blas/ , 29-8-2009

[50] http://www.netlib.org/lapack/ , 29-8-2009

[51] http://math-atlas.sourceforge.net/ , 29-8-2009

[52] http://www.cygwin.com/ , 29-8-2009

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