Physics of Plasmas_L.C.Woods

L C Woods Physics of Plasmas L C Woods Physics of Plasmas WILEY- VCH WILEY-VCH Verlag GmbH & Co KGaA Author Prof Dr.Leslie C Woods University of Oxford and Balliol College leslie.woods@balliol.oxford.ac.uk with 69 figures This book was carefully produced Nevertheless, author and publisher not warrant the information contained therein to be free of errors Readers are advised to keep in mind that statements, data, illustrations, procedural details or other items may inadvertently be inaccurate Library of Congress Card No.: applied for British Library Cataloging-in-PublicationData: A catalogue record for this book is available from the British Library Bibliographic information published by Die Deutsche Bibliothek Die Deutsche Bibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data is available in the Internet at Cover Picture Left: "Lightbulb" CME A coronal mass ejection Credit: NASA Upper right: Heating coronal loops Credit: M Aschwanden et al (LMSAL).TRACE NASA Lower right: A soft X-ray image of the sun Credit: ESA, NASA 2004 WILEY-VCH Verlag GmbH & Co KGaA, Weinheim All rights reserved (including those of translation into other languages) No part of this book may be reproduced in any form - nor transmitted or translated into machine language without written permission from the publishers Registered names, trademarks, etc used in this book, even when not specifically marked as such, are not to be considered unprotected by law Printed in the Federal Republic of Germany Printed on acid-free paper Printing Strauss Offsetdruck GmbH Morlenbach Bookbinding GroObuchbinderei J Schaffer GmbH & Co KG Grunstadt ISBN 3-527-40461-9 Preface This text gives an account of the principal properties of a tenuous gas, hot enough for some of the molecules to shed electrons and become ionized In general a macroscopic volume of such a gas consists of a mixture of free electrons and the ions and neutrals of several molecular species and is called aplasma If the temperature is high enough, e.g 10 000 K at a pressure of Pascal, a hydrogen plasma will be fully ionized, which is the case of most interest in this book If there is also a magnetic field present, the ions and electrons will gyrate about the field lines, producing an anisotropic medium with some very interesting properties Because of the orbiting motions, it is more difficult for the plasma to flow across the magnetic field lines than along them and with very strong fields both the plasma and its energy are said to be ‘confined’ by the field, although some leakage across the field lines does occur Examples of naturally occurring magnetoplasmas are found in the Sun’s corona, the solar wind and comet’s tails; laboratory examples include the plasma created in the fusion research machines known as tokamaks and in the application of what is termed ‘plasma processing’ to the manufacture of semiconductor devices Although molten metal is not a plasma, it is a conductor of electricity and therefore subject to magnetic forces; its behaviour is described by the equations of magnetohydrodynamics (MHD), which are a limiting case of the magnetoplasma equations Electric currents are used in industry to heat metals to the liquid state, when these metals can be stirred, levitated and pumped with magnetic fields New applications of plasma physics arise from time to time; however, in a short book such as this there is space for little more than the basic principles of the subject One of the attractions of plasma physics is the range of subjects required for its understanding; these include fluid mechanics, electricity and magnetism, kinetic theory and thermodynamics, although for this text relatively little experience in these topics is assumed There are many equations, so some effort has been made to cross-reference them at each stage of developing the theory To help the reader with mathematical points, I have included ‘mathematical notes’ at appropriate stages in the chapters, and I have also have added some appendices covering standard analyses With a subject like plasma theory, subscripts are essential to distinguish between the properties of the several fluid components, so to avoid doubling up on subscripts, I have followed the common practice of employing the dyadic notation for tensors and where the vector and tensor analysis is complicated, I have filled in the steps involved Many texts on plasma theory begin with a description of the collisionless motion of individual charged particles known as particle orbit theory Particles at a given time t and at a point r in physical space are then grouped according to their velocity w and a ‘kinetic’ equation describing the evolution of the number density of particles at a point P = P(r, w, t) in phase-space is found It is at this stage that particle collisions enter the model via a collision operator @, which removes particles from P or introduces particles into P by collisional scattering Finally, integrals of the kinetic equation over velocity space yield the fluid or MHD equations However, these moments representing the conservation of mass, momentum and N VI Preface energy, are independent of @, the term containing which vanishes in each integration Hence C could in fact be zero The standard account thus precedes from a microscopic description to what purports to be a collisional macroscopic model, without collisions playing any role at all Terms corresponding to pressure and temperature appear in the moment equations and yet these properties are essentially continuum concepts that require the existence of local thermodynamic equilibrium, a state for which particle collisions are essential To avoid the confusion and occasional errors that the standard approach has introduced into plasma theory, in this text the subject is developed in the reverse order from that described above, that is we start with collision-dominated classical fluid mechanics in Chapter 1, adding the effects of electromagnetic fields in Chapter At this stage we only need sufficient knowledge of particle orbit theory to determine the length and time scales below which a fluid or continuum description is not valid Chapter presents the theory of small amplitude plasma waves and shock waves, and finishes with a brief introduction to magneto-ionic theory, required in studying the reflection and scattering of radio waves in the ionosphere Stability of plasmas is treated in Chapter 4, covering the usual macroscopic instabilities of ideal plasmas, and also an important instability that depends on the electrical resistivity Finally we remove collisions entirely from the model and introduce the Vlasov theory of plasma waves, applying it to Landau damping and the ion-acoustic instability, which has important applications in solar physics Chapter , which is concerned with transport in magnetoplasmas, starts from the Fokker-Planck equation and gives an account of the theory of electron-ion collision intervals and several other relaxation times of important in the transport of particle energy and momentum The final chapter collects a miscellany of important topics, including second-order transport theory, thermal instabilities, particle orbit theory, magnetic mirrors, partially ionized plasmas and a brief introduction to some important applications of plasma physics By secondorder transport is meant, for example, the transport of heat in the presence of strong fluid shear, when the heat flux vector depends not only on the temperature gradient as in Fourier’s law, but also on the rate of strain of the fluid This proves to be very important in the presence of magnetic fields and leads to the thermal instabilities next described in the chapter Particle orbits in the presence of magnetic field gradients is a particularly important phenomenon in near-collisionless plasmas, with applications to transport in tokamaks Partially ionized plasmas add the complexity of a third fluid comprised of the neutral particles, to the model, so a brief introduction to Saha’s equation for the dependence of the degree of ionization on the temperature and pressure is included The final section briefly describes a few important applications of the theory - fusion research, solar physics, metallurgy, MHD direct generation of electricity and dusty plasmas The treatment ispitched at a level suitable for graduate students in mathematics, engineering and physics who need an introductory account of plasma physics It is recommend that the reader should aim to get a clear physical picture of the mechanisms at each stage before checking through the analysis Most of the exercises are straightforward extensions of the theory and therefore worthy of attention L C Woods Oxford, 1st August, 2003 Contents The Equations of Gas Dynamics 1.1 Molecular models and fluids 1.1.1 Introduction 1.1.2 Microscopic particles 1.1.3 The mean free path 1.1.4 Fluid particles 1.2 Macroscopic variables 1.2.1 Number density 1.2.2 Fluid velocity 1.2.3 Temperature 1.2.4 Equations of state 1.3 Pressure 1.3.1 Macroscopic definition of pressure 1.3.2 Kinetic definition of pressure 1.3.3 Vanishing pressure gradient 1.3.4 Local thermodynamic equilibrium 1.4 Macroscopic conservation laws 1.4.1 Convection and diffusion 1.4.2 A general balance equation in physical space 1.4.3 Conservation laws for a simple fluid 1.4.4 Specific entropy 1.5 Introduction to kinetic theory 1.5.1 Kinetic entropy 1S.2 Equilibrium distribution function 1S.3 Averages over velocity space 1S.4 Evolution of the phase-space density 1S Boltzmann's distribution law 4 11 12 13 14 15 17 18 18 19 21 22 23 23 24 26 28 29 Magnetoplasma Dynamics 2.1 Electromagnetic fields 2.1.1 Maxwell's equations 2.1.2 Galilean transformations 2.1.3 Polarization 2.2 Basic plasma parameters 33 33 33 35 37 39 1 VIII Contents 2.2.1 Plasma neutrality 2.2.2 The cyclotron frequency 2.2.3 Plasma frequency 2.2.4 The Debye length 2.3 Magnetohydrodyamic equations 2.3.1 Ohm'slaw 2.3.2 Conservation laws in MHD 2.3.3 Lagrangian form and entropy production 2.4 Electromagnetic farces 2.4.1 Stress tensor and Poynting vector 2.4.2 Magnetic forces in MHD 2.4.3 The induction equation 2.4.4 Difision of magnetic fields 2.4.5 Conservation of magnetic flux 2.5 Magnetostatics 2.5.1 Steady state equations 2.5.2 The theta pinch 2.5.3 The linear pinch 2.5.4 Axisymmetric toroidal equilibrium 2.5.5 Force-free magnetic fields 2.6 Transition equations across surface layers 2.6.1 Surface intensities 2.6.2 Boundary conditions 2.6.3 Current sheets and surface charge 2.6.4 Fluid equations Waves in Magnetoplasmas 3.1 MHD waves in an unbounded plasma 3.1.1 Introduction 3.1.2 Linearization 3.1.3 The dispersion equation 3.1.4 MHDwaves 3.2 Coupled plasma waves 3.2.1 High frequency waves 3.2.2 Whistlers 3.2.3 Propagation of wave fronts 3.3 MHD waves in cylindrical plasmas 3.3.1 The dispersion relation 3.3.2 Fast wave cut-off 3.4 Group velocity 3.4.1 Transmission of energy 3.4.2 Wave packets 3.5 Shock waves 3.5.1 Jump conditions across an MHD shock 3.5.2 Thermodynamic constraint 39 39 40 41 42 42 44 46 48 48 49 50 53 54 54 54 55 56 57 59 61 61 62 64 64 69 69 69 70 71 73 74 74 75 76 77 77 79 80 80 81 82 82 85 IX Contents 3.6 3.5.3 Classification of MHD shocks 3.5.4 Perpendicular shock waves Magneto-ionic theory 3.6.1 Electrical conductivity 3.6.2 The dielectric tensor 87 89 92 92 93 Magnetoplasma Stability 4.1 Rayleigh-Taylor and Kelvin-Helmholtz instabilities 4.1.1 Linearized equations 4.1.2 Surface waves 4.1.3 The dispersion equation 4.1.4 Special cases 4.2 Interchange instabilities 4.2.1 Flute instability 4.2.2 Thermal stability 4.3 Instabilities of a cylindrical plasma 4.3.1 The sausage instability 4.3.2 The kink instability 4.3.3 Stability condition 4.3.4 Stability of an unbounded flux tube 4.4 The energy principle 4.4.1 Potential energy 4.4.2 Surface term 4.4.3 General stability condition 4.4.4 Cylindrical plasma with a volume current 4.5 Resistive instabilities 4.5.1 The tearing mode 4.5.2 Differential equation for By 4.5.3 Physics of the tearing mode 4.6 The two-stream instability 4.6.1 Vlasov theory of plasma waves 4.6.2 Solution of the dispersion equation 4.6.3 Landau damping 4.6.4 The ion-acoustic instability 4.7 Fibrillation of magnetic fields 97 98 98 99 100 101 103 103 105 106 106 107 108 109 110 111 112 113 115 115 116 117 118 120 120 122 123 124 126 Transport in Magnetoplasmas 5.1 Coulomb collisions 5.1.1 Particle diffusion in electric microfields 5.1.2 Particle orbits 5.1.3 The Rutherford scattering cross-section 5.2 The Fokker-Planck equation 5.2.1 Friction and diffusion coefficients 5.2.2 Scattering in velocity space 5.2.3 Super-potential fknctions 131 131 131 133 135 136 136 138 139 X Contents 5.3 Lorentzian plasma 141 5.3.1 Collisional loss rate 141 5.3.2 Expansion of the distribution function 142 5.3.3 Electrical conductivity 143 5.3.4 Conductivity in a fully-ionized plasma 144 5.4 Friction and diffusion coefficients 146 5.4.1 First super-potential 146 5.4.2 Second super-potential 147 5.4.3 Limiting cases 148 149 5.4.4 Relaxation times 5.5 Transport of charge and energy 152 5.5.1 Ohm’slaw 152 5.5.2 Resistivity in a magnetoplasma 152 5.5.3 Fourier’s Law 153 5.5.4 Thermal conductivity in a magnetoplasma 154 5.6 Transport of momentum 155 5.6.1 Classical formula for the viscous stress tensor 155 5.6.2 The viscous stress tensor in a magnetic field 157 Extensions of Theory 6.1 Second-order transport 6.1.1 Convection versus conduction 6.1.2 The second-order heat flux 6.1.3 The viscous stress tensor 6.2 Thermal instability 6.2.1 Heat flux in a cylindrical magnetoplasma 6.2.2 Unstable current profiles 6.2.3 Planar geometry 6.2.4 Heating the solar corona 6.3 Particle orbit theory 6.3.1 Rate of change of the peculiar velocity 6.3.2 Guiding centre drifts 6.3.3 Drifts due to variations in the magnetic field 6.3.4 Gyro-averages 6.3.5 The grad B and field curvature drifts 6.4 Magnetic mirrors 6.4.1 Constants of motion of gyrating particles 6.4.2 Magnetically trapped particles 6.4.3 Fraction of trapped particles 6.5 Partially ionized plasmas 6.5.1 Degree of ionization 6.5.2 Ratio of the specific heats 6.5.3 Resistivity 6.6 Applications of plasma physics 6.6.1 Tokamak research 165 165 166 167 170 170 170 172 174 175 176 176 177 179 180 182 184 184 185 186 187 187 188 190 191 192 199 6.6 Applications ofplasma physics 6.3 A cylindrical plasma has a uniform magnetic field B, over < T < a, which is kept in balance by a comparable azimuthal field in a 68 < r The plasma pressure is negligible (see exercise 2.6) Show that heat will flow across the transition region towards the higher temperature provided the width 6, of the transition region satisfies + Would you expect the width to increase or decrease with time? 6.4 Show that in a frame moving with the mass motion of either the electrons or the ions, the acceleration of a particle is given by 2: = wcc x b - c e , where e is the rate of strain tensor Explain the origin of the last term and the circumstances in which it can be omitted Show that the solution of the equation can be expressed = wca x b - (a 151,) e , where a = x = X and X = ell and interpret the vector a + 6.5 A plasma has crossed electric and magnetic fields Describe the motion of an individual particle, explaining why it drifts parallel to E x B A suitable Galiiean transformation reduces this drift to zero Explain both the resulting drift and how it can be removed by another Galilean transformation, even though the gravitational force remains 6.6 Show that the perpendicular force acting on the guiding centres in a magnetic field with spatial variations is FI = $ ( ~ - c 2L ) V ~ l n B$+(c i - c l2l ) n / R , where n/R is the field line curvature vector 6.7 Are the ion and electronfluids repelled by magnetic mirrors If your answer is ‘no’, explain how the individual particles of these fluids can systematically behave differently from their collective motion [Distinguish between convected mirrors and mirrors localized in space.] t Vn Figure 6.18: (a) ‘Diamagnetic drift’ (b) Toroidal coordinates 6.8 Suppose there is a density gradient along the OX-axis, with uniform B along OZ In a frame with stationary guiding centres, at any elemental volume P(0,yo), the electrons passing through P in the OX-direction have come from the denser gas in y > yo, whereas those passing in the opposite direction have come from the less dense gas in y < yo The two groups of electrons have the same orbital speed Comment on the conclusion that because of the imbalance in particle numbers, there is a fluid drift (known as ‘diamagnetic drift’) along OX relative to the guiding centres (see Fig 6.18(a)) Extensions of Theory 200 6.9 The magnetic field strength in a tokamak varies inversely with distance R measured from the major axis (see Fig 6.18(b)): B= Bo + Ecose (E = r/Ro), where is the angle defined in the figure Show that the mirror ratio defined in (6.90) is ( + ~ ) / (1-6) and deduce that the fraction of particles trapped in -B0 < c: 00 at minor radius T is 6.10 The equation dB = [B” dt - a[T’B’/T - Bv:, ( a = / ) , given in exercise 2.3, p 65, applies to a fully ionized plasma Show that for a plasma at T < 000 K,(Y > 13.4 A flux tube is cooler than the ambient plasma and its magnetic field distribution has a point of inflexion at T = r S Describe the evolution of B in the neighbourhood ofrs Additional Exercises Al Consider the efflux of molecules through an element of area dS in the OXY-plane, as shown in Fig 6.19 Show that the mass flux of a collisionless gas per unit area in the negative 02-direction is given by When the upper plane consists of air under pressure, it can be shown that the constant 0.40 is replaced by 0.68 How would you explain this? “t Figure 6.19: Effluxthrough a small hole Now suppose that instead of a neutral gas, the upper half plane is initially occupied by a plasma and the lower half is a vacuum Assuming that the electrons and ions are at the same temperature, what would you expect to happen upon the opening of the hole? 6.6 Applications ofplasma physics A2 Let 20 ~ ~ - T z ( v ~ - ~ / ~ ) c *: A B,- T ~ v v where A is a constant vector and B is a constant tensor Prove that J fop{l, c , c2)dc= , and deduce that f o ( + 9)is a possible form for the distribution function f A3 Show that the energy flux can be expressed as q =pc- JJ J;; n o m 9v5exp(-v2)duedi2, (f - f o ) / f o and e is unit vector parallel to c Use the result in A2 and Fourier’s law, q = -nVT,to show that A = V 1nT and n = 5kpr2/2m Also show that B = Vv and where cp = /I = pr,,where B is the tensor introduced in A2 A4 Show that in a collisionless plasma and in conditions to be specified, where vA is the Alfvkn speed and c is the speed of light A5 Prove that V x ( E t v x B) = -eD(B/e)+ B * V V , and hence eD(u+ -)B2 2Po e +@(P+ +?-’ B2 PO PO + v x B) x B) A6 A viscous, conducting liquid flows along the OX-axis between parallel plates Show that in steady, incompressible flow the equations for B and v are: [V x B = E + v x B and (E = q/po), Vp - evV.(Vv+%) = -(V x B) x B PO A7 In A6 assume that v = (u,(z), 0,0) and B = (B,(t), 0, Bo),where BOis constant (see Fig 6.20) Show that u, satisfies the equation where u = 1/77 and the pressure gradient, -PO, is assumed to be constant Adopt a no-slip boundary condition to obtain the solution where A4 = Bon(a/ev)1’2 (known as the Hartmann number) Extensions of Theory 202 z=a z = -a Figure 6.20: Flow of a viscous, conducting liquid between parallel plates A8 Show that in A7 the total current is I = - 2Poa ( - - M Bo M M How would you distinguish between (a) insulting walls and (b) conducting walls? A9 The coordinates defined in Fig 6.18(b) are related by R = & that the Grad-Shrafranov equation transforms into + + T cos and = T sin Show = -po ( R ~T cos e)’p’($) - F ( $ ) F / ( + ) = Let $o(r) denote the value that $ would take in the limit as the aspect ratio E r / & tends to zero and write $ = $o(r) $ I ( T , e), where $1 is a small ‘correction’ added to $0 to allow for toroidal effects, then show that + and write down the equation satisfied by $1 Assume that the magnetic surfaces are circular and that the centre of the circle specified by $ = const is displaced outward from the major axis to R = & A(T).Show that in this case + A10 Let cp be a scalar, vector or tensor function and let O denote any possible product between cp and V Show that the surface intensities of V O p and acp/at can be calculated from (VOcp)* = nO[cp] +VllOp* and where V is the velocity of the discontinuity Apply these equations to V x E = -aB/at; under what circumstances is n x [El = V n[B]a boundary condition? 6.6 Applications ofplasma physics 203 A1 Let n be the unit normal to a p1asma:vacuum interface, C, which has a velocity un at a point P Show that the unit normal at P changes at the rate dn -=nn.Vu-Vu, dt and that with the perturbations specified in (3.39, in (i,8, 2) notation, the perturbation in the unit normal is given by where = ijr on T = r g A12 Verify that when electron pressure is important, the right-hand side of (3.90) acquires a term ia:kk.v,/w, where a, = d p e / e e )is the sound speed in the electron gas Deduce the dispersion relation w' = w;e k2u2 A13 Starting from the equation of motion for the electron gas, describe the conditions under which it may be reduced to Ohm's law in its simplest form, viz qj = E v x B Solve + 77 j = E v X B - -j X B ene forj, write your solution in the formj = u A14 Adopting the notation of exercises 3.7 and acoustic waves is - + + +Vpe en, - (E + ) and find the relation between 77 and u 3.8, show that the dispersion relation for magneto- + b2)u2+ a2b2 + ia2(u2- b2)cn + iu2(u2- a ' ) ~ , , =~ 0, u4 - (a' where u = w / k and show that the attenuation is where ur is the real part of u A15 Show that in a neutral gas the pressure ratio across a shock wave is given by where y is the ratio of the specific heats Compare this with the corresponding expression in a magnetoplasma A16 A discontinuity S is moving with the velocity V relative to the laboratory frame; show that the jump condition for the momentum equation (2.82) across S is: n -[ p v + p - TI = V n [p] Suppose that S is initially in a steady state, and is disturbed by a train of small amplitude waves etc., where the impinging on it Write the variables as e = eS 6, v = v, C, V = subscript s denotes a steady state (upstream or downstream) and show that the linearized version of the continuity equation across the discontinuity is + + 8vsI + ii- IeSvsI = V [ e s ] n, * [esC + Show that eight boundary conditions of this type can be obtained for S and that there are just two degrees of freedom in the response of S to the impinging waves [ V and one component of A.] Conclude that for S to be stable it must be able to emit divergent waves (upstream and downstream) Extensions of Theory 204 A17 Equation (3.18) lists the possible waves that could be emitted from the discontinuity S described in the previous question Let N1 denote the number of waves propagating in the upstream direction relative to S, then there will be Nl waves leaving S and moving towards x = -m (see Fig 3.6) and (7 - Nl)waves moving towards x = 00 Show that for stability there has to be one more wave overtaking the shock than moving ahead of it and deduce that there are three possible shock structures A18 Consider the motion of a particle p trapped in the tokamak field described in exercise 6.9 Show that, as illustrated in Fig 6.21, it will drift away from the magnetic surface r = const at the velocity i Su,where Su is defined in (6.81) and i is unit vector in the radial direction Show that i b x V In B M - sin B/Ro,and hence - - Show that the average radial drift over all particles is zero Figure 6.21: Banana orbit + where cp is the A19 The element of distance along a banana orbit is d s = {(Rdcp)’ (rdB)2}1/2, angular displacement about the major axis (see Fig 6.18(b)) Given that 1Be I