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Equilibrium and Stability of Tokamak Plasmas with Arbitrary Flow By Equilibrium and Stability of Tokamak Plasmas with Arbitrary Flow by Luca Guazzotto Submitted in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy Supervised by Professor Riccardo Betti Department of Mechanical Engineering The College Arts and Sciences University of Rochester Rochester, NY 2005 June 2005 Laboratory Report 340 Equilibrium and Stability of Tokamak Plasmas with Arbitrary Flow by Luca Guazzotto Submitted in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy Supervised by Professor Riccardo Betti Department of Mechanical Engineering The College Arts and Sciences University of Rochester Rochester, New York Shall we then live thus vile, the race of Heav'en Thus trampl'd, thus expell'd to sufler here Chains & these Torments? iii Curriculum Vitae The author was born in Turin (Italy) in 1975. He attended the "Politecnico di Torino" from 1994 to 2000. 'There he received a degree with honors in Nuclear Engineering. He came to Rochester in fall 2000. He worked under the direction of Profcssor Betti from 2000 to 2005. In that period, he received an LLE J. Horton fellowship. He earned a Master of Science in Mechanical Engineering in 2002. Publications and Conference Present at ions Numerical study of tokamak equilibria with arbitrary flow, L. Guaz- zotto, R. Betti, J. Manickam and S. Kaye, Phys. Plasmas 11, 604 (2004) Magnetohydrodynamics equilibria with toroidal and poloidal flow, L. Guazzotto and R. Betti, (invited) Phys. Plasmas 12, 056107 (2005) Stabilization of the Resistive Wall Mode by Diflerentially Rotating Walls in a High-Beta Tokamak, Sherwood Theory 2002 2C30 Axisymmetric MHD Equilibria with Arbitrary Flow and Applications to NSTX, APS-DPP 2002 FP1.079 Two-Dimensional MHD Simulations of Tokamak Plasmas with Poloi- dal Flow, APS-DPP 2002 FP1.087 MHD equilibria with arbitrary flow, Sherwood Theory 2003 1E04 High@ Tokamak Equilibria with Poloidal Flows Exceeding the Poloidal Alfve'n Velocity, APS-DPP 2003 P 1.129 Progress towards the development of a linear MHD stability code for ax-isymmetric plasmas with arbitrary equilibrium flow, Sherwood The- ory 2004 lE35 Quasi-omnigenous tokamak equilibria with fast poloidal flow, Sher- wood Theory 2004 1E48 Magnetohydrodynamic Equilibria with Poloidal and Toroidal Flow, APS-DPP 2004 E11.005 (invited) Progress in the development of a linear MHD stability code for axisym- metric plasmas with arbitrary equilibrium flow, Sherwood Theory 2005 P3-12 Acknowledgments Coming to the end of my experience in Rochester, I feel the need to thank several people, both here in the U.S. and in Italy. Going in temporal order, my thanks go to Prof. G. Coppa and Dr. G. Lapenta, friends and mentors, whose freely given support has been so impor- tant from the beginning of my experience in Rochester. I would then like to thank my thesis adviser, Prof. R. Betti, for showing me the way research is done, for sharing with me his enthusiasm and open- mindness about science. I am very grateful to Dr. J. Manickam and Dr. S. Kaye, and to the NSTX group in general, for their assistance and hospitality during my visits to Princeton, and for their help and encoura.gement in developing the code FLOW. I am also deeply thankful to Prof. J. P. Freidberg, for his input and par- ticipation in the development of the code FLOS, and for his kind assistance each and every time I had the opportunity to interact with him. The invaluable contribution of Prof. R. Perucchio to the finite element implementation of the code FLOS is also gratefully acknowledged. Finally, last but surely no least, I need to thank my family back in Italy, for their unfaltering support and encouragement for all these years, and for making it possible to me to come and study in this far away part of the globe. Abstract The purpose of the present work is the analysis of equilibrium and stability properties of tokamak plasmas in the presence of macroscopic flow. First, the formulation of the equilibrium problem is reviewed. Toroidal flow produces a modification in the equilibrium profile due to the centrifugal force produced by the rotation in curved geometry. The effects of poloidal rotation are more surprising, since poloidal flow can produce qualitative modifications in the equilibrium. If the flow is of the order of the poloidal sound speed, discontinuous equilibria develop due to the transition between the subsonic and the supersonic solution of the Bernoulli equation. In the presence of transonic (ranging from subsonic to supersonic values) flows, discontinuous equilibria are the only solution of the Grad-Shafranov-Bernoulli system of equations. If the flow is faster than the poloidal Alfv6n speed, the Shafranov shift can be inverted and if the plasma is kept in equilibrium by an external vertical field, the separatrix moves to the outboard side of the plasma. Since the poloidal rotation is damped by the neoclassical poloidal viscosity, a new quasi-omnigenous solution for equilibria with large rotation is also derived in order to minimize the flow damping. All equilibrium analytical results are confirmed by numerical solutions obtained with the code FLOW. The presence of flow also greatly alter the stability problem. A 6W formulation including the effect of arbitrary flow is introduced and implemented in the code FLOS. FLOS is used to study the effect of flow on internal kink modes. Finally, external kink modes are considered in the case of static plasma but flowing walls. The two-stream flow pattern is shown to stabilize the mode for wall velocities in a realistic range for future experiments. Contents viii CONTENTS 1. Introduction .............................. 1.1 Nuclear fusion ........................... 1.2 Magnetic confinement fusion and the tokamak concept .... 1.3 Particle drifts ........................... 1.4 Tokamak equilibrium ....................... 1.4.1 The basic problem of toroidal equilibrium ....... 1.4.2 The magnetohydrodynamics model ........... 1.4.3 The Grad-Shafranov equation .............. 1.5 Tokamak stability ......................... 1.5.1 Kink instability ...................... 1.5.2 The m = 1 internal kink mode .............. 1.5.3 The external kink mode ................. 1.5.4 Other plasma instabilities ................ 1.6 Thesis objectives ......................... 2 . Numerical study of tokamak equilibria with arbitrary flow 2.1 Introduction ............................ 2.2 General Equations ........................ 2.3 Numerical solution ........................ 2.4 Conclusions ............................ Contents ix 3 . Equilibria with pressure anisotropy and toroidal flow .... 40 3.1 Introduction ............................ 40 3.2 FLOW results for equilibria with pressure anisotropy and to- roidalflow ............................. 41 3.3 Conclusions ............................ 49 4 . Tokamak equilibria with poloidal flow .............. 51 4.1 Introduction ............................ 51 4.2 Numerical solution of t.he Bernoulli equation in the presence of poloidal flow .......................... 52 4.3 Equilibria with poloidal flow in the range oft.he poloidal sound speed ................................ 54 4.4 Equilibria with poloidal velocity exceeding the poloidal Alfv6n velocity .............................. 68 4.4.1 The analytic model .................... 71 4.4.2 Numerical results ..................... 77 4.4.3 Physical interpretation of the results .......... 85 4.5 Quasi-omnigenous and omnigenous equilibria with fast poloi- dalflow .............................. 89 4.5.1 Quasi-omnigenous equilibria with fast poloidal flow . 91 4.5.2 Fully-omnigenous equilibria with flow .......... 108 4.6 Conclusions ............................115 5 . Tokamak stability in the presence of macroscopic flow ...118 5.1 Introduction ............................ 118 5.2 Formulation ............................ 120 5.3 Numerical implementation of the eigenvalue problem .....126 5.4 Benchmark problems in slab geometry .............140 Contents x 5.5 Benchmark problems in cylindrical geometry ..........148 5.6 NSTX-like internal kink with purely toroidal flow .......155 5.7 Conclusions ............................ 168 6 . Resistive Wall Modes in the presence of rotating walls ...170 6.1 Introduction ............................ 170 6.2 Equilibrium ............................ 174 6.3 Stability .............................. 177 6.3.1 The plasma ........................ 178 6.3.2 The vacuum ........................180 6.3.3 The resistive rigid wall ..................180 6.3.4 The flowing wall .....................181 6.3.5 Matching conditions and dispersion relation ......183 6.4 Results and discussion ...................... 185 6.4.1 0 limits in the presence of a single static wall .....186 6.4.2 0 limits in the presence of a single, 1-stream rotating wall ............................ 188 6.4.3 0 limits in the presence of 2 1-stream rotating walls . 190 6.4.4 0 limits in the presence of 1 2-stream rotating wall . 194 6.4.5 0 limits in the presence of 1 2-stream rotating wall and a static wall ........................ 197 6.4.6 0 limits with a simplified approach ...........197 6.5 Conclusions ............................ 200 7 . Conclusions .............................. 201 Bibliography ............................... 204 List of Tables xi LIST OF TABLES 2.1 Physical meaning of the "intuitive" free functions ....... 36 2.2 Relation between the two sets of free functions . h$ is the position of the geometric axis of the plasma ............ 37 4.1 FLOW equilibrium
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