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Kinetic investigation of magnetic islands in tokamaks [Elektronische Ressource] / Mattia Siccinio

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¨ ¨TECHNISCHE UNIVERSITAT MUNCHENMax-Planck-Institut f¨ur PlasmaphysikKinetic Investigation of MagneticIslands in TokamaksMattia SICCINIOVollst¨ andiger Abdruck der von der Fakult¨ at fur¨ Physik der Tech-nischen Universit¨ at Munc¨ hen zur Erlangung des akademischenGrades einesDoktors der Naturwissenschaftengenehmigten Dissertation.Vorsitzender: Univ.-Prof. Dr. R. Kruc¨ kenPrufer¨ der Dissertation:1. Hon.-Prof. Dr. S. Gun¨ ter2. Univ.-Prof. Dr. A. G. Peeters,Universit¨ at BayreuthDie Dissertation wurde am 11.01.2010 bei der Technischen Uni-versit¨ at Munc¨ hen eingereicht und durch die Fakult¨ at fur¨ Physikam 29.03.2010 angenommen.Contents1 Nuclear Fusion and Tokamaks 11.1 NuclearFusion .......................... 11.2 TheTokamak. 21.2.1 PropertiesofthePlasma...... 21.2.2 PrinciplesofaTokamak............. 41.3 TokamakGeometry............. 71.3.1 MagneticCoordinates ... 71.3.2 LargeAspect-RatioApproximation........... 91.3.3 Perturbed Equilibria ...............122 Particle Motion in an Inhomogeneous Magnetic Field 142.1 TheSmalLarmor-RadiusExpansion..............142.1.1 MotionwithConstantFields...........142.1.2 MotionwithSpace-andTime-VaryingFields .....172.2 MotionofParticlesinaTokamak.222.2.1 ParticleOrbits ......................232.2.2 ToroidalPrecesionofTrappedParticles....253TheTearingMode 293.1 Classical Stability of the Tearing Mode .............293.2 Modelling of a Magnetic Island . . ...........303.

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¨ ¨TECHNISCHE UNIVERSITAT MUNCHEN
Max-Planck-Institut f¨ur Plasmaphysik
Kinetic Investigation of Magnetic
Islands in Tokamaks
Mattia SICCINIO
Vollst¨ andiger Abdruck der von der Fakult¨ at fur¨ Physik der Tech-
nischen Universit¨ at Munc¨ hen zur Erlangung des akademischen
Grades eines
Doktors der Naturwissenschaften
genehmigten Dissertation.
Vorsitzender: Univ.-Prof. Dr. R. Kruc¨ ken
Prufer¨ der Dissertation:
1. Hon.-Prof. Dr. S. Gun¨ ter
2. Univ.-Prof. Dr. A. G. Peeters,
Universit¨ at Bayreuth
Die Dissertation wurde am 11.01.2010 bei der Technischen Uni-
versit¨ at Munc¨ hen eingereicht und durch die Fakult¨ at fur¨ Physik
am 29.03.2010 angenommen.Contents
1 Nuclear Fusion and Tokamaks 1
1.1 NuclearFusion ........... ........... .... 1
1.2 TheTokamak . .... 2
1.2.1 PropertiesofthePlasma . . .... 2
1.2.2 PrinciplesofaTokamak . . ........... .... 4
1.3 TokamakGeometry ......... .... 7
1.3.1 MagneticCoordinates ... .... 7
1.3.2 LargeAspect-RatioApproximation ....... .... 9
1.3.3 Perturbed Equilibria .... ........... .... 12
2 Particle Motion in an Inhomogeneous Magnetic Field 14
2.1 TheSmallLarmor-RadiusExpansion .......... .... 14
2.1.1 MotionwithConstantFields........... .... 14
2.1.2 MotionwithSpace-andTime-VaryingFields . .... 17
2.2 MotionofParticlesinaTokamak . .... 22
2.2.1 ParticleOrbits ....... ........... .... 23
2.2.2 ToroidalPrecessionofTrappedParticles .... .... 25
3TheTearingMode 29
3.1 Classical Stability of the Tearing Mode ......... .... 29
3.2 Modelling of a Magnetic Island . . ........... .... 30
3.3 ClassicalTearingModesandRutherfordEquation ... .... 35
3.4 TheNeoclassicalDrive ....... .... 37
i3.5 ThePolarizationCurrent...... ........... .... 41
4 The Drift-Kinetic Equation 45
4.1 DerivationoftheDrift-KineticEquation ........ .... 45
4.2 The Drift-Kinetic Equation for a Perturbed Tokamak Geometry 48
4.2.1 Analytical Solution: the Double Parameter Expansion . 55
4.3 NumericalSolution: theHAGISCode.......... .... 59
4.4 The\Standard"PolarizationCurrent .... 60
4.4.1 ThePerturbedDistribution ........... .... 60
4.4.2 ThePerturbedCurrent... .... 65
4.4.3 Summary .......... .... 71
5 Role of the Parallel Streaming of Passing Ions 74
5.1 SolutionoftheDrift-KineticEquation ......... .... 74
5.1.1 ThePerturbedDistribution ........... .... 74
5.1.2 ThePerturbedCurrent... .... 77
5.2 ConsiderationsontheResults ... .... 80
5.2.1 TheContributionofParallelStreaming ..... .... 80
5.2.2 TheRoleofResonatingParticles ........ .... 82
5.3 ThePerturbedDistributionintheTrappedSpace ... .... 83
6 Modi cation of the Polarization Current due to the Toroidal
Precession of Trapped Ions 86
6.1 SolutionoftheDrift-KineticEquation ......... .... 87
6.1.1 ScalingoftheIslandFrequency .... 87
6.1.2 ThePerturbedDistribution ........... .... 90
6.1.3 ThePerturbedPerpendicularCurrent...... .... 94
6.1.4 ThePerturbedParallelCurrent ......... .... 96
6.2 E ects on the Island Stabilization . ........... .... 98
6.2.1 ComparisonwiththePolarizationCurrent ... .... 99
6.3 DiscussionoftheResults ...... ....100
ii6.3.1 TheInterplayofToroidalPrecessions ...... ....100
6.3.2 The Role of Collisions . . . ........... ....105
6.3.3 TheDeviationfromParabolicDependence ... ....106
7 Gyrokinetic Investigation of Magnetic Islands 107
7.1 TheGyrokineticEquation ..... ........... ....107
7.2 GKWCoordinatesandLocalLimit ....113
7.3 ImplementationofaMagneticIslandinaFlux-Tube . ....116
7.4 FirstResults. . ........... ........... ....121
7.4.1 NonlinearFlatteningofthePressureProle .. ....121
7.4.2 TheElectrostaticPotential. ....124
7.5 Outlook .... ........... ........... ....126
8 Summary and Outlook 128
Bibliography 131
iii
Alle persone che mi sono state vicine negli ultimi 27 anni.
Grazie di cuore.
M.Acknowledgements
I am indebted to a great number of people for help and advice throughout
my time as a PhD. Of course, my rst and absolutely necessary thanks go to
my family, which always supported me and my choices all my life long. All
my gratefulness goes indeed to my wife Eva, who was so crazy not only to
marry me, but also to follow me in this long trip to Germany...
I gratefully acknowledge the friendly attitude of all my colleagues at the
IPP, referring especially to the large Italian community (Clemente Angioni,
Vittoria Belmondo, Nicola Bertelli (a.k.a. Bertelli the Netherlands), Alessan-
dro Biancalani, Roberto Bilato, Emiliano Fable, Tobias Hein (Italian citizen
honoris causa), Andrea Scarabosio, Giuliana Sias, Giovanni Tardini). In
particular, I am strongly indebted with Dr. Omar Maj (my favourite mathe-
matical handbook) and Dr. Alberto Bottino (my favourite computer problem
solver), which conjugate their immense knowledge to an uncommon kindness.
I had the opportunity during the last year to start a very interesting and
promising collaboration with the Plasma Physics Group to the University
of Warwick, Coventry, United Kingdom. For this reason, I would like to
thank Prof. Arthur G. Peeters and Dr. William A. Hornsby, not only for
their scienti c value but also for their hospitality during the weeks I spent in
Warwickshire.
Finally, last but indeed not least, my most grateful thanks go to my IPPsupervisor, Dr. Emanuele Poli. He is a person of undoubted competence,
able to transmit all his passion, and endowed at the same time with an
indescribable patience (I can be extreeeemely boring sometimes).
Thanks a lot to you all!!
Mattia
viEs durft¨ e uns gut tun, uns manchmal daran zu erinnern, dass wir zwar
in dem Wenigen, das wir wissen, sehr verschieden sein m¨ ogen, dass wir aber
in unserer grenzenlosen Unwissenheit alle gleich sind.
Karl PopperChapter 1
Nuclear Fusion and Tokamaks
1.1 Nuclear Fusion
Nuclear fusion represents an alternative energy source which o ers several
advantages compared to both fossil fuels and the conventional nuclear