Computational Methods in Plasma Physics

364 Pages·2010·2.617 MB·English

by  Stephen Jardin

Computational
Methods in
Plasma Physics
Chapman & Hall/CRC
Computational Science Series
SERIES EDITOR
Horst Simon
Associate Laboratory Director, Computing Sciences
Lawrence Berkeley National Laboratory
Berkeley, California, U.S.A.
AIMS AND SCOPE
This series aims to capture new developments and applications in the field of computational
science through the publication of a broad range of textbooks, reference works, and handbooks.
Books in this series will provide introductory as well as advanced material on mathematical, sta-
tistical, and computational methods and techniques, and will present researchers with the latest
theories and experimentation. The scope of the series includes, but is not limited to, titles in the
areas of scientific computing, parallel and distributed computing, high performance computing,
grid computing, cluster computing, heterogeneous computing, quantum computing, and their
applications in scientific disciplines such as astrophysics, aeronautics, biology, chemistry, climate
modeling, combustion, cosmology, earthquake prediction, imaging, materials, neuroscience, oil
exploration, and weather forecasting.
PUBLISHED TITLES
PETASCALE COMPUTING: Algorithms and Applications
Edited by David A. Bader
PROCESS ALGEBRA FOR PARALLEL AND DISTRIBUTED PROCESSING
Edited by Michael Alexander and William Gardner
GRID COMPUTING: TECHNIQUES AND APPLICATIONS
Barry Wilkinson
INTRODUCTION TO CONCURRENCY IN PROGRAMMING LANGUAGES
Matthew J. Sottile, Timothy G. Mattson, and Craig E Rasmussen
INTRODUCTION TO SCHEDULING
Yves Robert and Frédéric Vivien
SCIENTIFIC DATA MANAGEMENT: CHALLENGES, TECHNOLOGY, AND DEPLOYMENT
Edited by Arie Shoshani and Doron Rotem
COMPUTATIONAL METHODS IN PLASMA PHYSICS
Stephen Jardin
Chapman & Hall/CRC
Computational Science Series
Computational
Methods in
Plasma Physics
Stephen Jardin
Boca Raton London New York
CRC Press is an imprint of the
Taylor & Francis Group, an informa business
A CHAPMAN & HALL BOOK
On the cover is a computed iso-contour surface for the toroidal current density in the nonlinear phase of an
internal instability of a tokamak plasma. Contours on one toroidal plane and one mid-section plane are also
shown. (Courtesy of Dr. J. Breslau and the M3D team.)
CRC Press
Taylor & Francis Group
6000 Broken Sound Parkway NW, Suite 300
Boca Raton, FL 33487-2742
© 2010 by Taylor and Francis Group, LLC
CRC Press is an imprint of Taylor & Francis Group, an Informa business
No claim to original U.S. Government works
Printed in the United States of America on acid-free paper
10 9 8 7 6 5 4 3 2 1
International Standard Book Number-13: 978-1-4398-1095-8 (Ebook-PDF)
This book contains information obtained from authentic and highly regarded sources. Reasonable efforts
have been made to publish reliable data and information, but the author and publisher cannot assume
responsibility for the validity of all materials or the consequences of their use. The authors and publishers
have attempted to trace the copyright holders of all material reproduced in this publication and apologize to
copyright holders if permission to publish in this form has not been obtained. If any copyright material has
not been acknowledged please write and let us know so we may rectify in any future reprint.
Except as permitted under U.S. Copyright Law, no part of this book may be reprinted, reproduced, transmit-
ted, or utilized in any form by any electronic, mechanical, or other means, now known or hereafter invented,
including photocopying, microfilming, and recording, or in any information storage or retrieval system,
without written permission from the publishers.
For permission to photocopy or use material electronically from this work, please access www.copyright.
com (http://www.copyright.com/) or contact the Copyright Clearance Center, Inc. (CCC), 222 Rosewood
Drive, Danvers, MA 01923, 978-750-8400. CCC is a not-for-profit organization that provides licenses and
registration for a variety of users. For organizations that have been granted a photocopy license by the CCC,
a separate system of payment has been arranged.
Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used
only for identification and explanation without intent to infringe.
Visit the Taylor & Francis Web site at
http://www.taylorandfrancis.com
and the CRC Press Web site at
http://www.crcpress.com

to Marilyn
Contents
List of Figures xiii
List of Tables xvii
Preface xix
List of Symbols xxi
1 Introduction to Magnetohydrodynamic Equations 1
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Magnetohydrodynamic (MHD) Equations . . . . . . . . . . . 4
1.2.1 Two-Fluid MHD . . . . . . . . . . . . . . . . . . . . . 6
1.2.2 Resistive MHD . . . . . . . . . . . . . . . . . . . . . . 8
1.2.3 Ideal MHD . . . . . . . . . . . . . . . . . . . . . . . . 9
1.2.4 Other Equation Sets for MHD . . . . . . . . . . . . . 10
1.2.5 Conservation Form . . . . . . . . . . . . . . . . . . . . 10
1.2.6 Boundary Conditions . . . . . . . . . . . . . . . . . . 12
1.3 Characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . 14
1.3.1 Characteristics in Ideal MHD . . . . . . . . . . . . . . 16
1.3.2 Wave Dispersion Relation in Two-Fluid MHD . . . . . 23
1.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2 Introduction to Finite Difference Equations 27
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.2 Implicit and Explicit Methods . . . . . . . . . . . . . . . . . 29
2.3 Errors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
2.4 Consistency, Convergence, and Stability . . . . . . . . . . . 31
2.5 Von Neumann Stability Analysis . . . . . . . . . . . . . . . 32
2.5.1 Relation to Truncation Error . . . . . . . . . . . . . . 36
2.5.2 Higher-Order Equations . . . . . . . . . . . . . . . . . 37
2.5.3 Multiple Space Dimensions . . . . . . . . . . . . . . . 39
2.6 Accuracy and Conservative Differencing . . . . . . . . . . . 39
2.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
vii
viii Table of Contents
3 Finite Difference Methods for Elliptic Equations 45
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
3.2 One-Dimensional Poisson’s Equation . . . . . . . . . . . . . 46
3.2.1 Boundary Value Problems in One Dimension . . . . . 46
3.2.2 Tridiagonal Algorithm . . . . . . . . . . . . . . . . . 47
3.3 Two-Dimensional Poisson’s Equation . . . . . . . . . . . . . 48
3.3.1 Neumann Boundary Conditions . . . . . . . . . . . . 50
3.3.2 Gauss Elimination . . . . . . . . . . . . . . . . . . . . 53
3.3.3 Block-Tridiagonal Method . . . . . . . . . . . . . . . 56
3.3.4 General Direct Solvers for Sparse Matrices. . . . . . . 57
3.4 Matrix Iterative Approach . . . . . . . . . . . . . . . . . . . 57
3.4.1 Convergence . . . . . . . . . . . . . . . . . . . . . . . 59
3.4.2 Jacobi’s Method . . . . . . . . . . . . . . . . . . . . . 60
3.4.3 Gauss–Seidel Method . . . . . . . . . . . . . . . . . . 60
3.4.4 Successive Over-Relaxation Method (SOR) . . . . . . 61
3.4.5 Convergence Rate of Jacobi’s Method . . . . . . . . . 61
3.5 Physical Approach to Deriving Iterative Methods . . . . . . 62
3.5.1 First-Order Methods . . . . . . . . . . . . . . . . . . 63
3.5.2 Accelerated Approach: Dynamic Relaxation . . . . . 65
3.6 Multigrid Methods . . . . . . . . . . . . . . . . . . . . . . . 66
3.7 Krylov Space Methods . . . . . . . . . . . . . . . . . . . . . 70
3.7.1 Steepest Descent and Conjugate Gradient . . . . . . 72
3.7.2 Generalized Minimum Residual (GMRES) . . . . . . 76
3.7.3 Preconditioning . . . . . . . . . . . . . . . . . . . . . 80
3.8 Finite Fourier Transform . . . . . . . . . . . . . . . . . . . . 82
3.8.1 Fast Fourier Transform . . . . . . . . . . . . . . . . . 83
3.8.2 Application to 2D Elliptic Equations . . . . . . . . . 86
3.9 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
4 Plasma Equilibrium 93
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
4.2 Derivation of the Grad–Shafranov Equation . . . . . . . . . 93
4.2.1 Equilibrium with Toroidal Flow . . . . . . . . . . . . . 95
4.2.2 Tensor Pressure Equilibrium . . . . . . . . . . . . . . 97
4.3 The Meaning of Ψ . . . . . . . . . . . . . . . . . . . . . . . . 99
4.4 Exact Solutions . . . . . . . . . . . . . . . . . . . . . . . . . 102
4.4.1 Vacuum Solution . . . . . . . . . . . . . . . . . . . . 102
4.4.2 Shafranov–Solov´ev Solution . . . . . . . . . . . . . . . 104
4.5 Variational Forms of the Equilibrium Equation . . . . . . . 105
4.6 Free Boundary Grad–Shafranov Equation . . . . . . . . . . 106
4.6.1 Inverting the Elliptic Operator . . . . . . . . . . . . . 107
4.6.2 Iterating on J (R,Ψ) . . . . . . . . . . . . . . . . . . 107
φ
4.6.3 Determining Ψ on the Boundary . . . . . . . . . . . . 109
4.6.4 Von Hagenow’s Method . . . . . . . . . . . . . . . . . 111
4.6.5 Calculation of the Critical Points . . . . . . . . . . . 113
Table of Contents ix
4.6.6 Magnetic Feedback Systems . . . . . . . . . . . . . . 114
4.6.7 Summary of Numerical Solution . . . . . . . . . . . . 116
4.7 Experimental Equilibrium Reconstruction . . . . . . . . . . . 116
4.8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
5 Magnetic Flux Coordinates in a Torus 121
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
5.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
5.2.1 Jacobian . . . . . . . . . . . . . . . . . . . . . . . . . 122
5.2.2 Basis Vectors . . . . . . . . . . . . . . . . . . . . . . . 124
5.2.3 Grad, Div, Curl . . . . . . . . . . . . . . . . . . . . . 125
5.2.4 Metric Tensor . . . . . . . . . . . . . . . . . . . . . . . 127
5.2.5 Metric Elements . . . . . . . . . . . . . . . . . . . . . 127
5.3 Magnetic Field, Current, and Surface Functions . . . . . . . 129
5.4 Constructing Flux Coordinates from Ψ(R,Z) . . . . . . . . . 131
5.4.1 Axisymmetric Straight Field Line Coordinates . . . . 133
5.4.2 Generalized Straight Field Line Coordinates . . . . . 135
5.5 Inverse Equilibrium Equation . . . . . . . . . . . . . . . . . 136
5.5.1 q-Solver . . . . . . . . . . . . . . . . . . . . . . . . . . 137
5.5.2 J-Solver . . . . . . . . . . . . . . . . . . . . . . . . . . 138
5.5.3 Expansion Solution . . . . . . . . . . . . . . . . . . . . 139
5.5.4 Grad–Hirshman Variational Equilibrium . . . . . . . . 140
5.5.5 Steepest Descent Method . . . . . . . . . . . . . . . . 144
5.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
6 Diffusion and Transport in Axisymmetric Geometry 149
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
6.2 Basic Equations and Orderings . . . . . . . . . . . . . . . . 149
6.2.1 Time-Dependent Coordinate Transformation . . . . . 151
6.2.2 Evolution Equations in a Moving Frame . . . . . . . 153
6.2.3 Evolution in Toroidal Flux Coordinates . . . . . . . . 155
6.2.4 Specifying a Transport Model . . . . . . . . . . . . . 158
6.3 Equilibrium Constraint . . . . . . . . . . . . . . . . . . . . . 162
6.3.1 Circuit Equations. . . . . . . . . . . . . . . . . . . . . 163
6.3.2 Grad–Hogan Method . . . . . . . . . . . . . . . . . . . 163
6.3.3 Taylor Method (Accelerated) . . . . . . . . . . . . . . 164
6.4 Time Scales . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
6.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168
7 Numerical Methods for Parabolic Equations 171
7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 171
7.2 One-Dimensional Diffusion Equations . . . . . . . . . . . . . 171
7.2.1 Scalar Methods . . . . . . . . . . . . . . . . . . . . . . 172
7.2.2 Non-Linear Implicit Methods . . . . . . . . . . . . . . 175
7.2.3 Boundary Conditions in One Dimension . . . . . . . . 179