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LECTURE NOTES on IDEAL MAGNETOHYDRODYNAMICS By ASSOCIATIE EURATOM-FOM FOM-INSTITUUT VOOR PLASMAFYSICA RIJNHUIZEN - NIEUWEGEIN - NEDERLAND LECTURE NOTES ON IDEAL MAGNETOHYDRODYNAMICS by J.P. GoedbJoed Rijnhuizen Report 83-145 ASSCOCIATIE EURATOM-FOM Maart 1983 FOM-INSTITUUT VOOR PLASMAFYSICA RIJNHUIZEN - NIEUWEGEIN - NEDERLAND LECTURE NOTES ON IDEAL MAGNETOHYDRODYNAMICS by J.P. Goedbloed Rijnhuizen Report 83-145 Corrected version of the notes of March 1979, originally printed as internal report at Instituto de Ffsica, Universidade Estadual de Campinas, Campinas, Brazil This work was supported by the "Stichting voor Fundamentaal Onderzoek der Materie" (FOM), the "Nederlandse Organisatie voor Zuiw-WetenschappelijK Onderzoek" (ZWO), EURATOM, the "Fundacio de Amparo i Pesquiw do Ettado de Sao Paulo" (FAPÉSP), and the "Conaelho Nacione) de PesquisM" (CNPQ, Brazil). "Then I saw that all toil and skill in work come from a man's envy of his neighbour. This also is vanity and a striving after wind." Ecclessiastes 4:4 "Ever since the creation of the world his invisible nature, namely, his eternal power and deity, has been clearly perceived in the things that have been made." Romans I:20 "Remember then to sing the praises of his work, as men have always sung them." Job 36:24 PREFACE These notes were prepared for a course of lectures for staff and students of the Instituto de Flsica, Universidade Estadual de Campinas, Brazil. The course consisted of two-hour lectures twice a week during a period of 9 weeks in the months June-August 1978. It has been my intention to make the subject- matter as much as possible self-contained, so that all needed physical and mathematical techniques and derivations were pre­ sented in detail. The aim was to bring a physics graduate student with a little previous knowledge of plasma physics to the point where he could sense the possibility of contributing himself to modern developments in the field of n.agnetohydro- dynamics. It has been stated many times during the course that ideal MHD is still full of questions where answers remain to be given, whereas at the same time the framework of the theory is clear-cut enough to provide confidence that eventually a satisfactory picture will emerge. An open field like this should be a fruitful area for academic research. I wish to thank Prof. Paulo H. Sakanaka for the golden opportunity he offered me to visit UNICAMP and to teach this course. His personal help, the interest of Prof. Ricardo M.O. Galvao, and the effort of the students made the visit a very valuable and exciting experience for me. The diligence of Carmen typing the manuscript I have appreciated very much. I am indebted to the foundations CNPQ and FAPESP (Brazil) for the support of this work and to the foundation FOM (The Netherlands) for granting me a leave of absence. I welcome notification of errors, criticism, and suggest­ ions for improvement of these notes. Hans Goedbloed Association Euratom-FOM FOM-Instituut voor Plasmafysica Rijnhuizen, Nieuvegein The Netherlands CONTENTS p I. Introduction 1 II. Derivation of macroscopic equations 4 A. Boltzmann equation 4 B. Moments of the Boltzmann equation 6 C. Two-fluid equations T 12 D. One-fluid equations 13 III. The model of ideal MHD 23 A. Introduction 23 B. Differential equations 24 C. Boundary conditions 27 D. Equation of state 29 IV. Characteristics 33 A. Partial differential equations in two independent variables 33 B. Characteristics in ideal MHD 36 V. Conservation laws 49 A. Conservation form of the ideal MHD equations 49 B. Shocks 51 C. Global conservation laws 56 D. Energy conservation for models 2 and 3 59 VI. An example: Dynamics of the screw pinch 63 A. Pinch experiments 63 B. Mixed initial-value boundary-value problem — 66 C. Field-line topology 73 D. Reduction of the plasma equations 77 E. Circuit equations 79 F. Solution of the problem 82 G. Flux and energy conservation 86 VII. Lagrangian and Hamiltonian formulations of ideal MHD 92 A. Summary of some concepts of classical mechanics 92 B. Kinematic considerations ——-—-—-- 96 C. Lagrange and Hamilton equations of motion 101 VIII. Linearized ideal MHD 105 A. Introduction 105 B. Linearized equation of motion 108 C. Boundary conditions — 112 D. Self-adjointness of the force-operator 117 E. Mamilto.i's principle 123 IX. Spectral theory 128 A. Mathematical preliminaries 128 B. Rayleigh-Ritz variational principle 132 C. Initial value problem 135. D. Stability. The energy principle 138 E. o-Stability 145 X. Waves in plane slab geometry 149 A. Waves in infinite homogeneous plasmas 149 B. The continuous spectrum for inhomogeneous media 158 C. Damping of Alfvén waves 170 D. Stability of plane force-free fields. A trap 191 XI. The diffuse linear pinch 204 A. Equilibrium model 204 B. Derivation of the Hain-LÜst equation 208 C. Equivalent system of first order differential equations 216 D. Boundary condition at the plasma-vacuum interface 219 E. Oscillation theorem 222 F. Newcomb's marginal stability analysis. Suydam's criterion 2 33 G. Free-boundary modes 242 H. Fixed-boundary modes - 247 I. o-stable configurations 251 XII. Sharp-boundary high-beta tokamaks 254 A. Introduction B. Equilibrium 260 C. Vacuum field solution for the circle 267 D. Variational principle for stability 273 E. Numerical solution for circular cross-sections 233 .1. I. INTRODUCTION In these notes a cross-section through plasma theory is presented which is restricted to ideal magnetohydrodynamics (MHD). This cross-section will again be restricted to my lim­ ited personal point of view, which is that I wish to deal with a model which - respects the main physical conservation laws, - has a decent mathematical structure» - permits the analysis of plasma behavior in the complicated geometries considered for the confinement of plasmas for controlled thermonuclear reactions {CTR). Ideal MHD is the only model so far that satisfactorily combines these features. This theory treats the plasma as a perfectly conducting fluid interacting with a magnetic field. If we talk about the model of ideal MHD we mean: " the equations of ideal MHD, - boundary conditions on a prescribed boundary and initial data on and inside that boundary. .2. In order for the model to be complete both have to be consider­ ed simultaneously. Nevertheless, different persons put differ­ ent stress on these two points. The exposition tends to be more physical when the stress is on the first point, whereas consideration of the boundaries tends to lead to more involved mathematics. In the first part of these notes, where we consider simple geometries (homogeneous media, e.g. infinite space or homogeneous slab models), a relatively simple analysis will therefore lead to an abundance of physical phenomena (in par­ ticular the various kinds of MHD waves), whereas gradually more tedious analysis is needed to correctly treat these phenomena in more complex geometries (inhomogeneous media, e.g. diffuse linear and toroidal pinches). These complicated geom­ etries also provide interesting new physics, like equilibrium and stability properties, which cannot be analyzed in homo­ geneous media. Since MHD instabilities are a major threat to CTR confinement, it is essential to have a firm understanding of this subject if one wishes to contribute to this field. It is the aim of these notes to facilitate this understanding. There are two ways of introducing the equations of ideal MHD: - derive them by appropriate averaging of kinetic equations, - pose them as reasonable postulates for a hypothetical medium called "plasma". Since a satisfactory derivation of the ideal MHD equations does not exist, we basically choose for the second method (starting with chapter III). However, this approach will be supplemented .3. with a heuristic derivation (chapter II) in order to render some credibility to the equations and also to obtain some understanding of the domain of validity of the ideal MHD de­ scription. Strict minds may skip this chaotic exposition and start reading at chapter III. The MKSA system of units has been chosen for the next chapter, whereas starting with chapter III u will be put equal to 1 for convenience. The only operation needed to return to the conventional systems of units is then to divide B2 by v (MKSA system of units) or 4TT (Gaussian system of units). .4. II. DERIVATION OF MACROSCOPIC EQUATIONS A« BOLTZMANN EQUATION Consider a collection of charged particles in an electromagnetic field- Different species of particles, specifically ions and electrons, will be distinguished by a subscript a. We now define the time-dependent distribution function for particles of species a in six-dimensional phase space: f (r^y^t) . The probable number of particles in the six- dimensional volume element dJr d v centered at r,y will then be f (£,^,t) a3 r d3 v . The variation in time of the distribution function is found from the Boltzmann equation; 3f 3f q 3f 3f 3t £ 3r m v^ * *' 3v k3t 'coll v* l' 'v a *\» Here, E and £ are composed of the contributions of the external fields and the averaged internal fields originating from the long-range interparticle interactions. The PHS of Eq. (2-1) gives the rate of change of the distribution function due to short-range interparticle interactions, which are somewhat arbitrarily called collisions. Neglect of these collisions leads to the Vlasov equation; 3f 3f q 3f 2 2 jr + i-jf * IT<I***V'TT - °- < - > A closed system of equations is obtained by adding Maxwell's equations to determine E and B. In order to determine the charges and currents that occur in Maxwell's equations we take moments of the distri- .5. bution function. The zeroth moment gives the number of parti­ cles of species a per unit volume: v*-0 5 K(«-t)d3v' (2_3) whereas the first moment gives the average velocity: i r 3 ^ua (r.t^ ') * -vu = n— ;(£.t —r)- Ji \v,f a {r,y,t)'t'V ' d v.
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