Physics of Plasmas

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 [email protected]

with 69 figures

This book was carefully produced. Nevertheless, author and publisher do 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

0 2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

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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 1 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 2. 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 3 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 5 , 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

1 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 . 5 Boltzmann's distribution law . . . . . . . . . . . . . . . . . . . . . .

2 3 4 4 4 5 9 11 12 13 14 15 17 18 18 19 21 22 23 23 24 26 28 29

2 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 1 1

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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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3 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

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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

4

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

5

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

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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

6 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

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6.6.2 Solar physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.6.3 Engineering applications . . . . . . . . . . . . . . . . . . . . . . . . 6.6.4 Dusty plasmas . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

194 196 197

Bibliography

207

Index

209

Lists of physical constants, plasma parameters and frequently used symbols In SI units, the constants required in plasma theory are:

I Physical Quantity

Value

Electron mass Proton mass Electron charge Boltzmann constant Permittivity (Free Space) Permeability (Free Space) Speed of light (Vacuum) Protordelectron mass ratio Temperature at 1 eV Planck constant Stefan-Boltzmann constant Gas constant

units

9.1095~10-~~ kg 1.6726 kg 1.6022x 10- l9 C 1 . 3 8 0 7 ~ 1 0 - ~J~K - l 8.8542~10-” F m-l 4n x1OP7 Hm-’ 2.9979~10’ m s-’ 1.8362~10~ 1.1605~10~ K 6 . 6 2 6 2 ~ 1 0 - ~ J~s 5.6703~10-’ W rn-’ K-4 8.3144 J K-1 mol-’

The important plasma parameters are: Parameter

Symbol ~

Formula

see pages

~~~~

Resistivity Cyclotron frequency (electrons) Thermal speed Larmor radius Coulomb logarithm Collision intervals Thermal conductivity (B = 0) Magnetic diffisivity Magnetic Reynolds number Plasma frequency Collisionless skin-depth Debye length

43,152

77

39

Wce

C

26 40 145 156 153 52 53 41 62 42

TL

In A

r e

, ‘Tt

I€

E

R,

Wpe

6e AD

Commonly used symbols are defined on the pages indicated below:

h P

84 12 V

fl 77 Ve

Q

74 43 43 55

y

771

E

2 !

22 153 52 6

he

77

TT

wc

62 52 14 40

6,

72 46 5 upe 41 K Q

6 43 AD

u

42 34

€0

A

34 144 43

7%

6 34 156

Physics of Plasmas L. C. Woods Copyright 0 2004 WILEY-VCH Verlag GmbH & Co KGaA

1 The Equations of Gas Dynamics

1.1

Molecular models and fluids

A plasma is a mixture of positive ions, electrons and neutral particles, electrically neutral over macroscopic volumes, and usually permeated by macroscopic electrical and magnetic fields. In addition to these ‘smoothed’ or averaged electromagnetic fields, which with laboratory plasmas are often imposed from outside the plasma volume, there are the localized micro-fields due to the individual particles. The trajectories of the charged particles are thus continuously modified by a range of electromagnetic forces, the average fields acting like body forces and the micro-fields like collisional forces. The micro-fields are responsible for the transmission of pressure and viscous forces, for the conduction of particle energy, and for the friction forces between diffusing components of the plasma. Some care is needed in dividing the continuum of electromagnetic forces into their macroscopic and microscopic components, but with this achieved, there is little formal distinction between the theory of the macroscopic behaviour of neutral gases and that of magnetoplasmas. The aim of this book is to describe the various physical processes that underpin plasma theory and the equations representing these processes. The distinction between what we shall term the ‘mechanisms’ and the equations based on them - symbolisms- is particularly important in a complex subject like plasma physics. Definitions of physical properties can be taken either from the mechanisms or the symbolisms, but one must take care not to mix the two, e.g. to adopt a purely mathematical definition of a property and then to assume that this automatically entails the usual physical attributes of that property. Our method is to commence with the macroscopic description of the individual components of the plasma, that is we shall treat the collections of electrons, ions and neutrals as comprising separate fluids, and their fluid properties developed. In the next chapter they will be combined to make a plasma, an approach with the merit of making a clear distinction between the fluid and electrical properties of a plasma. Readers already familiar with fluid mechanics might skip to Chapter 2, although in § 1.S there is an introduction to kinetic theory that will be required in later chapters.

1.1.1

Introduction

Except for the basic concept of a ‘mean-free-path’, the trajectories of individual particles will be described in a later chapter. In this chapter we shall introduce the standard macroscopic variables of gas dynamics, such as pressure, temperature, fluid velocity and entropy, and derive the equations relating them. Excepting entropy, these physical properties are best defined

2

I

The Equations of Gas Dynamics

in terms of mechanisms although sometimes synthetic definitions have a place. Consider temperature for example; either it is defined physically via thermometers and the mechanism of thermal equilibrium, which requires close physical contact through molecular particle collisions, or it may be defined symbolically as a kinetic temperature, which is a property of the distribution of molecular velocities and collisions are not explicitly involved. The danger of employing the second definition is that it is too easy to adopt properties of temperature that really depend on the first definition. For example the conduction of heat, which depends on the temperature gradient, is a collisional process in which the gradient of the kinetic temperature would be misplaced without the additional constraint that the medium is collision-dominated, the precise meaning of which will be discussed later in 1.3.4.

1.1.2 Microscopic particles A substance in the gaseous state consists of an assembly of a vast number of microscopic particles that, excepting when they collide with each other, move freely and independently through the region of physical space available to them. The nature of the particles depends largely on the temperature of the assembly. At low temperatures, but above the critical value at which liquefaction can occur, they are molecules. At higher temperatures the molecules dissociate into atoms, and at still higher temperatures the atoms become ions by shedding some of their electrons. The resulting assembly is termed a ‘plasma’. Partially ionized plasmas consist of a mixture of neutral atoms, electrons, and ions, requiring at least three distinct species of microscopic particles to be included in a complete mathematical representation of their collective behaviour. The simplest model of a microscopic particle is a small featureless sphere, possessing a spherically-symmetric force field. For neutral particles this field has a very short range, and the particles can be pictured as being almost rigid ‘billiard-balls’, with an effective diameter equal to the range of the force field. As they have no structure, these particles have only energy of translation. The gas is usually assumed to be sufficiently tenuous for collisions involving more than two particles at a time to be ignored, i.e. only binary collisions are considered. The model is appropriate for monatomic uncharged molecules. Diatomic and more complex molecuIes do not have symmetric force fields, but for many purposes they are also well represented by the billiard-ball model. Their relative orientations at collisions may be assumed to be randomly distributed, so that averages taken over a large number of encounters will have values independent of orientation, just as with symmetric force fields. It is the internal vibratory energies possessed by multi-atomic particles that give rise to the largest discrepancies between the predictions of the simple billiard-ball model and observation, a phenomenon that is easily included in kinetic theory by adding an average internal energy to the translatory energy of each molecule. Kinetic theory is concerned mainly with the connection between the motions and interactions of microscopic particles comprising a gas and the transport of macroscopic properties like fluid momentum and energy through that gas. The oldest example relating macroscopic properties to microscopic behaviour is provided by the pressure force acting on the walls of a gas container. That it is due to the near-continuous bombardment of the walls by the vast number of neighbouring molecules, is a concept dating back to Boyle and Newton. The more subtle relationship between heat and the energy of molecular agitation required more than an-

1.I

3

Molecular models andjuids

other century before it was revealed with increasing detail in the works of Waterston, Clausius, and Maxwell’. Clausius’ main contribution to kinetic theory was the concept of the mean free path, which is the average distance travelled by a molecule between successive collisions, and which led to Maxwell’s introduction of the velocity distribution function, to be discussed in $1.5. The intermolecular force law plays a central role in kinetic theory and classical kinetic theory proceeds on the assumption that this law has been separately established, either empirically or from quantum theory, except with charged particles, when the well-known Coulomb force law applies. We shall return to this topic in Chapter 5; for the present it is sufficient to understand the concept of the mean-free-path.

1.1.3 The mean free path Two microscopic parameters play a leading role in our account of the collective behaviour of an assembly of particles. These are the mean free path A, which is the average distance moved by a particle between successive encounters with other particles, and the collision interval T , which is the average time taken by a particle to move this distance. The reciprocal of r is known as the ‘collision’ frequency, v = T - ~ .The terminology is particularly fitting for ‘hard’ molecules, i.e. those with force fields abruptly falling to zero outside a molecular diameter CT, say. An approximate formula for X can be found as follows. Suppose there are n molecules per unit volume, and we assume that all are stationary, save one that has a velocity v, relative to the others. In a tenuous or dilute gas, X >> CT and hence mr2vris a good approximation to the volume swept in one second by the sphere of influence of the moving particle. Those molecules with centres lying within this volume will experience a collision and therefore the collision frequency per molecule is 7-l = m 2 n v r . Replacing v, by the average molecular speed E relative to the centre of mass of all the similar molecules within a macroscopic volume element, and writing X = re, we arrive at the estimate X x 1/(7r&)

(7-

=X/E).

(1.1)

The accurate formula for X is 2-4 times this value. ‘Soft’ molecules have extended force fields that make only slight changes in the momentum and energy of most passing molecules, so many such ‘grazing’ collisions are required to accumulate significant changes in these properties for a given test particle. However, by modifying to denote an ‘effective’ diameter, we can extend (1.1) to the case of soft molecules. Then X becomes the average distance that a sequence of small-angle collisions takes to stop a test particle moving in a given direction, i.e. to give a 90” deflection, and T is the time it takes for this change to happen. Even with hard molecules, a small sequence of collisions is required to ‘stop’ a particle. Another consequence of this cascade process is that momentum and energy require related but slightly different times to be transported in a specified direction. The Coulomb force fields of electrons and ions have ranges extensive enough to influence great numbers of nearby particles, so that purely binary collisions are very rare. The billiardball model and the associated concept of a mean free path are not strictly relevant, although ‘See The kind of motion we call heat by Stephan Brush, North-Holland Publishing Company, 1976.

4

I The Equations of Gas Dynamics

it is usual to describe the distance required for a 90” deflection of a test particle as being a ‘mean free path’. More precisely, X is defined to be the distance over which this particle loses its momentum along its initial direction of motion.

1.1.4 Fluid particles A fluid is sometimes described as being a ‘continuum’, that is a substance that has a continuous rather than a discrete structure, but since nature is particulate, obviously this is an approximate model, valid only on a length scale so large that the mean free path appears negligible2. Familiar properties of a fluid are density, pressure, temperature and velocity, which we shall discuss in detail shortly. However, there are other important properties of fluids which depend on X having a non-zero magnitude and that would vanish in a genuine continuum, e.g. fluid viscosity, electrical conductivity and thermal conductivity, all of which are proportional to relevant mean free paths. The mechanism that qualifies an assembly of particles to be described as being afluid is the frequency of collisions between the particles, which in turn depends on the size C of the assembly. Evidently if C is smaller than the mean free path A, there will be few collisions and the assembly will lack the continuum properties required of a fluid. If L << A, the assembly is certainly not a fluid and is described as being a ballistic system; in plasmas the word ‘collisionless’ is adopted with the same meaning. The smallest physical element that can be called a fluid is termed a ‘fluid particle’ and such elements have the usual properties of thermodynamic systems. Neighbouring fluid particles interact with each other via collisions, e.g. by exerting a force on each other and by exchanging particle energy, whereas adjacent ballistic elements can not do this. It follows that a key dimensionless parameter in fluid mechanics is the ratio of a microscopic length (or time) to a macroscopic scale length (or time), a ratio known as the Knudsen number, k,. We shall return to this basic concept in 31.3.4.

1.2 Macroscopic variables 1.2.1 Number density The first task in the application of thermodynamics to continua is to specify precisely what is meant by the ‘thermodynamic system’ at apoint (r,t ) in the medium. A ‘point’ in fluid mechanics is not a mathematical point, i.e. an object in space that has position but no magnitude; by the point P(r,t ) is meant an infinitesimal volume element dr (= dx dy dz), centred at the mathematical point M(r, t ) , and with sufficient extension so that dr contains a vast number of microscopic particles of the species under consideration. This is termed a macroscopic point. Let the number of particles in dr be n dr, then the fluctuations in n due to particles entering or leaving dr merely due to the discreteness of matter, will be negligible. We may therefore introduce the number density n(r,t ) as a continuum variable provided the size dC = (dr)1/3 of the fluid particle is very much larger than the inter-particle distance, n--1/3. On the other hand, we must not take d! to be too large, otherwise significant variations *A ‘fluid’ is to be distinguished from a ‘liquid’, which is a special case of a fluid.

1.2 Macroscopic variables

5

in n(r, t ) may be smothered. Thus de should satisfy

np1l3 << de << C,

(C, = (dlnn/dx)-'),

(1.2)

where L, is a typical distance over which n changes by a significant fraction of its local value3. TheJfuid density is the macroscopic variable

where m is the particle mass, and for the present we are assuming that only one molecular species is present. Let w be the velocity of a typical particle at P , measured relative to a frame L (the 'laboratory' frame), and use ( . . ) to denote average values taken over the particles in dr. The fluid velocity at P is the macroscopic variable v(rlt ) = (w). It is the velocity of the centre of mass of the particles at P. For a more precise account, we need a statistical treatment. The basic variable for statistical mechanics was introduced by Maxwell at the end of the eighteen fifties; this is the number density f ( r ,w, t ) in six-dimensional phase space, usually termed the 'velocity distribution function'. Thus f ( r , w , t) dr dw is the number of molecules that at time t have positions lying in a volume element dr (= dz dy dz) about the position r, and velocities (of translation) lying within the velocity-space element dw (= dv, dv, dv,) about w. The physical number density is +

where the integral is over the whole of velocity space. It follows that f dw/n is the probability that a given molecule at the (macroscopic) point (r, t ) has a velocity in the element dw at w. We shall sometimes adopt the notation

where the average on the left-hand side is the macroscopic variable corresponding to the phase-space function 4.

1.2.2 Fluid velocity The fluid velocity v is defined by the average

nv(r, t) =

s

w f ( r , w , t) dw ,

(1.6)

3Let 6n be the root mean square fluctuation in n, then it is a standard result in fluctuation theory that the N - ' l 2 , where N is the total number of particles in the system. For a system of volume condition that 6n/n << 1 requires that 71(6.!?)~ >> 1, as in (1.2). 6n/n

N

1 The Equations of Gas Dynamics

6

and similarly the acceleration of the fluid at P(r, t) is

1

fi = w f(r,w,t)/ndw, dt

(1.7)

where w is the acceleration of the typical particle relative to L. When there are several distinct components present, the i-th component of which is a fluid with a velocity vi and density pi, the velocity of the fluid as a whole is

and we can therefore interpret v as being the velocity of mass flux. An obvious property of v is its dependence on the choice of the laboratory frame L, and as the equations describing the behaviour of fluids must be independent of this choice, v can not appear alone in these equations. A ‘frame-indifference’ molecular velocity will be introduced shortly. There is another fluid motion of considerable importance in the theory. This is the ‘spin’ n of the fluid at P ( r , t), the meaning of which is that the fluid circulates around the mathematical point M(r, t) with an angular velocity a.Below we shall show that it is related to the fluid vorticity 6 = V x v by4

a = ;v

x v.

(1.9)

A macroscopic point that coincides momentarily with P(r,t), that has the same velocity, acceleration, rate of change of acceleration and so on, as the fluid, is called a convectedpoint, and its locus is termed a path line. If axes are fixed relative to the fluid at this point and allowed to rotate with the local fluid spin 52, then the point thus augmented, say P,, is a convectedframe. Viewed from this frame, the fluid near P, will appear to be almost stationary, and without spin. This is a frame in which the ambient fluid is stationary and therefore it is the appropriate frame in which to specify the local thermodynamic system. By referring to a fluid property ‘p ‘at Pc’, we shall mean that value of ‘p as observed in a convected, spinning macroscopic point at r , t. The spin and acceleration of P, are important when time derivatives of vectors and tensors at P, are required in the theory. Spatial changes in the fluid velocity v(r, t ) influence the transport of properties between adjacent fluid particles. To calculate the effects on transport we need the following analysis. Suppose that a convected point Pc(r,t) moves with a velocity v(r, t), then a neighbouring convected point Qc(r R, t) has the fluid velocity

+

v’(r

+ R, t) = v(r, t) + R.Vv(r, t) + O ( R 2 ) ,

~

~

41n Cartesian coordinates, with unit vectors i , j, k,

V

X

v

=

a

a

(iz+ j ay

a + k-)8% x (v,i

+v,j

+ vkk)

(1.10)

7

1.2 Macroscopic variables

Figure 1.1: Strain of a fluid element.

where R is an infinitesimal displacement vector (see Fig. 1.1). The combination Vv is a second order tensor known as the velocity gradient tens03 , which can be analyzed into three distinct components, known as pure rate of strain, dilatation and spin (or vorticity). For this purpose we require the following mathematical analysis. ~~

~

Mathematical note 1. The decomposition of second-order tensors X

In general a second-order tensor A has a symmetric part A", an antisymmetric part A", a trace A, a 0

vector A", and a deviator A defined by

-

As = i ( A + A ) , A"

-

= a(A-A),

x

A

= 1:A,

A" = flx l : A , A 0

.

= A " - g l i, (1.11)

where 1 is the unit tensor, i.e. 1 A = A and A 1 = A, and the tilde denotes the transposed ten-

i

has zero trace. With sor. Since 1 : 1 = 3 and 1 : A = 1:A, it follows that the deviator of double products like ab :A we shall adopt the convention that ab :A = b A . a = A :ab, e.g. if A = Kij, ab :A = K ( a .j ) ( b . i). Hence

-

-

-

1 :ab = b 1 - a= b a,

1 x 1 :ab = b 1 x 1 . a = b x a,

-

r 1 x 1 x 1:ab = -r x (a x b) = r (ab - ab) = 2 r . (ab)",

or

(ab)" = -1 x (ab).

I The Equations of Gas Dynamics

8

Since a tensor A can always be expressed as the sum of three dyads, e.g. A = ab + cd + ef, it follows that X

A = a . b + c . d + e . f , 2A" = a x b + c x d + e x f , A" = -1 x A" = -A" x 1 = -1 x 1 .A",

and

0

A

= A -A" x 1

In particular

+ 4 2 1.

(1.12)

X

vv=v.v, (Vv)"=~Vxv=n,

and

0

v v = v v - s1 x 1

+91v.v.

(1.13)

It is easily verified that for any vector a, a x 1 = 1 x a, hence the second right-hand term in (1.13) can be expressed as -1 x 0. Let B denote another second-order tensor, then as A :B = A :8, it follows that A " : B" = A" : (-B") = 0.

Also

A": 6" =A" x 1 : 1 x B" = -2A" *B",

and therefore expanding each tensor, we obtain x x

A :B = A : B - 2A" B" + 9 A B . 0

0

(1.14)

Also note that 0

0

0

0

A:B=A:BS=A:B.

(1.15)

Using R . (52 x 1 ) = R x 52.1 = R x fl = -52 x R, a n d R . 1 = R, we find from (1.10) and (1.13) that

d

=

+ 52 x R + SRV

.v + R . vV+ o(P). 0

(1.16)

A rigid body motion about an axis 1, rotating through a small angle 0, changes a position vector R fixed in the body to R 01 x R. The velocity of the point is therefore fl x R, where fl is the angular velocity el. Hence the second right-hand term of (1.16) represents a rigid body motion of the fluid element with an angular velocity 51. Such motion does not strain (i.e. deform) the element, and it will not induce a stress, except in materials of unusual microstructure. The term 52 x R can be removed from (1.16) by transforming to the convected frame P,. Let R be the unit vector along R, then by (1.16) the 'outward' speed of Qc relative to P, is

+

IRI times gV - v + RR :G v . If R is distributed isotropically, the average of RR taken over

1.2 Macroscopic variables

9

i

all directions radiating from P, is 1 (cf. (1 30)) and as 1 : V v = 0,the average fluid speed outwards from P, on the sphere IRI = a is i a V v. Thus the third right-hand term in (1.16) is due to the changing volume of the fluid element; this type of strain is called dilatation. The remaining term in (1.16), representing pure straining motion without volume change, is called the deviatoric rate of strain and it plays a central role in transport theory. The symmetric part of the velocity gradient tensor, viz.

e = ;(vv

+ V v ) = V v +$I V -v 0

-

0

(1.17)

is called the rate ofstrain tensor. If the short vector R is a material line, i.e. is convected with the fluid, its rate of change in the laboratory frame is dR/dt = v’ - v = R V v , and therefore

dR _ s1 x R = R . e . dt

(1.18)

The left-hand side of this equation is the rate of change of R in a frame that is both convected and spinning with the fluid element. We call the infinitesimal vector R an embedded vector, since no fluid crosses it. The velocity w of a typical particle p can be divided into two components, (i) the fluid velocity v = (w) and (ii) the peculiar velocity c , peculiar that is to the particular particle under consideration. Thus w = v c, and by definition ( c ) = 0. The distinction between v and c is fundamental in kinetic theory. In particular notice that c is independent of the choice of reference frame, which of course is not true of w. The fluid particle P, is the basic thermodynamic system in fluid mechanics and the kinetic theory approach to the thermodynamic variables defines them as averages over functions of c .

+

1.2.3 Temperature The basic thermodynamic variable is temperature; it is usually introduced with the aid of the concept of thermal equilibrium between two contiguous macroscopic systems, say G and W . Such systems are said to be in thermal equilibrium if no net energy transfer occurs between them when they are in physical contact. At the microscopic level ‘physical contact’ means that the molecules of G and W are colliding with each other. One of the basic laws of thermodynamics - known as the ‘zeroth’ law - states that

Two systems in thermal equilibrium with a third are in thermal equilibrium with each other. The third system can be regarded as being a ‘thermometer’, and the three systems are said to be at the same temperature. Now choose G to be a gas and W to be the rigid boundary wall confining it. Take the line of impact at the collision to be the axis O X , not necessarily perpendicular to the wall (see Fig. 1.2). Let the velocity components of a gas particle GI and a wall molecule W , be u,v,w, and U , V,W just before the collision and u’, d,w’and U’, V’,W’just after it, then from momentum and energy conservation, the velocity components parallel to OY and 02

1 The Equations of Gas Dynamics

10

will be unchanged, whereas in the O X direction,

mu

imu2 +

+ MU

MU^

= mu‘

+ MU’,

+ MU'^,

= $mut2

m(u-u’) = -M(U-U’), m ( u 2 - u 12 ) = - M ( u ~ - ~ ’

i.e.

~1.

+

+

From the second pair of equations it follows that u u’ = U U’, whence the relative velocity u - U is reversed by the collision, as required by perfect elasticity. This condition and the momentum equation gives

( m+ M)u’ = ( m- M ) u

+ 2MU,

( m+ M)U’ = -(m - M ) U + 2mu,

and hence the gain in the wall’s kinetic energy per collision is

iM(U‘2 - U2) =

2mM { m u 2(m M)2

+

MU^ + ( M - r n ) u ~ } ,

Gas

Figure 1.2: Collision between gas and wall molecules.

As the wall is stationary, W Ioscillates about a mean position fixed in the wall, and since u and U are uncorrelated, over a large number of collisions along O X the product uU will have positive and negatives values with equal probability. Hence the average of UU is zero, so the net gain of wall energy is proportional to the average of (mu’ - M U 2 ) ,or equivalently to the average of (mu2 muI2 - M U 2 - M U r 2 ) .We now extend the averaging to all directions of O X to find that the average gain of energy by the wall is proportional to Q, where

+

Q=

4mM ( m M ) 2 { ($mc2)G- ( $ M c ~ ;) ~ }

+

I1

1.2 Macroscopic variables

here c is the peculiar speed (the fluid velocities are zero) and the subscripts denote gas and wall molecules. Thermal equilibrium therefore requires that

(+mc2)&? = (aMC2)w, in which case the gas and wall are at the same temperature. If a second gas is present, also in thermal equilibrium with the wall, then

the subscripts denoting the first and second gases. Hence

and the gases are in thermal equilibrium with each other; this is the zeroth law described earlier. We have now established the result: When two gases at the same temperature are mixed, the average kinetic energy of their molecules is the same.

The above suggests that we could define the absolute temperature T ( r ,t) at a point Pc(r,t ) as being proportional to the average energy of translation of the particles in P,. Hence we take

%kBT= rn(;c2)

(IC, = 1.3807 x 10-23JK-1),

(1.19)

where the constant of proportionality IC, is termed Boltzmann’s constant. However, the definition is based on the fact that energy can be transported between adjacent systems, so it is important not to forget the role of collisions; (1.19) does not apply in a collisionless gas.

1.2.4 Equations of state

+

At P, a particle has the energy rn(;cz E ) , where me is the energy due to its internal motions and its intermolecular potential. The average particle energy per unit mass will be denoted by u;thus

u=

(1.20)

($C2+€)

is a macroscopic variable u(r,t ) , termed the speciJic energy (i.e. energy per unit mass of the medium). From (1.19) and (1.20),

u = iRT + ( E )

( R= k B / m ) ,

where R is termed the gas constant. As ( E ) is found to depend on formof(1.21)is

u = .U(e,T)7 a relation known as the caloric equation of state.

(1.21)

e and T ,a more general (1.22)

12

1

The Equations of Gas Dynamics

If the particles have no internal structure, i.e. possess energy of translation only, (1.2 1) gives

u = c,T

(cv =

$1 ,

(1.23)

where c, is the spec$c heat at constant volume. In 51.3.3 we shall show that with such particles the pressure, i.e. the force per unit area, acting normal to a convected surface, has an average value of p = ie(c2) = ieu = RQT = nkBT.

(1.24)

This basic relation depends on the definition of temperature in (1.19) and is therefore not valid in a collisionless gas. A gas to which (1.23) and (1.24) apply is said to be aperfect gas. More generally the pressure is related to e and T by a relation P = P(Q1T),

(1.25)

known as the thermal equation of state. One mole is a mass of gas in grams numerically equal to its molecular weight M ; e.g. one mole of 0 2 is 32 grams of oxygen. From (1.24) applied to a volume V of gas containing N molecules,

pV = NkBT = ( N m / M )M ( k B / mT ), where ( N m / M )is the number of kilomoles in the volume. When this number is lo3, the equation is written in the form p v =~ NAkBT = RT

(R kBNA; N A lo3 M / m ) ,

(1.26)

where R is termed the universal gas constant, VA is the molar volume and N A is called Avogadro 's number. In a perfect gas R and N Aare constants with the values

R = 8.3144 JK-'mol-';

N A = 6.0220x1023mol-'

At 'standard' conditions, 0, = 1 . 0 1 3 3 ~ 1 Pa, 0 ~ T = 273.16K = OOC), 2.2415 x ~ O -m3 ~ mole-'.

vA= R T / p =

1.3 Pressure Of all the macroscopic variables in plasma theory, pressure is the least understood despite its 'everyday' familiarity. In a macroscopic description pressure is the force per unit area acting on a surface, which may be either the surface of a fluid element or the walls of a confining vessel. The microscopic picture defines pressure as being due to the change of momentum of the particles due to collisions either with other particles or with the wall molecules. To quote Maxwell (1 860),

1.3 Pressure

13

“Daniel Bernouilli, Herapath. Joule, Kronig, Clausius, etc. have shown that the relations between pressure, temperature and density in a perfect gas can be explained by supposing the particles to move with uniform velocity in straight lines, striking against the sides of the containing vessel and thus producing pressure. It is not necessary to suppose each particle to travel any great distance in the same straight line: for the effect in pmducingpressure will be the same if the particles strike against each other; so that the straight line described may be very short.”

1.3.1 Macroscopic definition of pressure We start with the macroscopic interpretation of pressure. To include the action of viscosity, we need to introduce the concept of a pressure tensor.

Figure 1.3: Pressure components.

Consider the force acting on one face of a small parallelepiped as illustrated in Fig. 1.3. For the face lying orthogonal to the OX axis the force is (px2i p,,j p,,k) dy dz, where the first subscript indicates the orientation of the face and the second the direction of the force and i, j , k are unit orthogonal vectors in a Cartesian coordinate system. Similarly the force acting on the rectangle orthogonal to the OY axis is (pyxi+pyyj+py,k) dz dx, and so on for the remaining rectangle. Thus to define the force acting on the surface of the volume element we need to define the nine components p,, ,p,, , ..., or equivalently the second order tensor

+

+

(1.27)

-

Notice that the pressure acting on the surface normal to i is now i p and more generally, if n is a unit vector normal to a fluid surface, the pressure force acting on this surface is n p. For the fluid particle to experience a net force, there needs to be a change in the pressure across the width of the parallelepiped. Considering the rectangles normal to OX, the force is i p dy dz on one surface and -i ‘ ( p dy dz ( d p / d x )dx dy dz) on the opposite surface.

-

+

-

1 The Equations of Gas Dynamics

14

-

Therefore the net force in the OX direction is -i (&)/ax) d r , where d r = d x d y d z is the volume element, and similarly for the two other directions. Hence the total force is

a

a + k-)a dz

- (i- + jdx d y

- p d r = -V

pdr

.

(1.28)

At the microscopic level the pressure force is due to the uneven molecular bombardment of a fluid element. The off-diagonal components like pzy, p,,, pyz,etc. are due to the tangential forces applied to the surfaces by the colliding particles and therefore the net force they apply to the element is of a shearing nature, which at a macroscopic level is termed a ‘viscous’ force. It is readily shown that for the simple microscopic particles occurring in a fully-ionized plasma the pressure tensor is symmetric, i.e. p,, = py,, etc. In general the normal components, p,,, pYy,p , , are equal in magnitude, otherwise it would mean that the normal pressure on a surface would depend on the orientation of that surface. They are also much larger than the off-diagonal components and it is therefore convenient to separate them by writing

p=p(ii+jj+kk)+n = p l +TI,

(1.29)

where p is the usual thermodynamic pressure, the tensor 1 defined by the equation is the unit tensor introduced in Mathematical note 1 (page 7) and 1T is known as the viscosity tensor. A fluid is said to be ‘ideal’ if TT is zero.

1.3.2 Kinetic definition of pressure In a frame P,, consider the flux of momentum across a small (convected) surface n d C where n is the unit normal (see Fig. 1.4).The mass flux per particle is mc and its momentum flux is mcc, where c is now a generic velocity, that is in each appearance it ranges over all the possible values accessible to the particles in an infinitesimal volume containing dC. Averaging over all these values, we find that the total momentum flux parallel to unit normal is F = n ~ ( c c )where , ( . . ) denotes the average taken over the full range of values of c with allowance for the frequency of occurrence of particular values.

-

Figure 1.4: Momentum flux and pressure.

1.3 Pressure

15

Divide the momentum flux parallel to the unit normal n into

then F+ is the momentum flux crossing unit area in the positive sense, from fluid @ to @) as shown in Fig. 1.4. This momentum is absorbed in fluid @) by collisions, and by allowing for all the angles of the particles incident on dC, it can be shown that the transfer is completed on average at a perpendicular distance g X from the interface, where A is the mean free path. Hence on a length-scale for which X is small, we can say that fluid @experiences a ‘surface’ force F+ per unit area due to the particles arriving from 0 Also, to set up the return flux -F-, @will suffer a further surface force, F-. It follows that N

F++F-=n-p

(P

= e(cc)),

is the force per unit area exerted by fluid @ on fluid @ across their interface. From ( lS), e = rnn and w = v c, it follows that

+

p(r, t ) = m

s

cc f ( r ,c , t ) dc .

(1.30)

When X is comparable with the length-scale of interest, C say, the net normal momentum flux across n dC, namely n p dC, remains the pressure force transmitted across n dC, although, of course, the particle collisions through which the force is manifest in the fluid, no longer all lie in the close neighbourhood of n dC. A moderate change of scale does not alter our interpretation of n p as being a surface collisional force. However when the mean free path greatly exceeds C, i.e. the system is collisionless, it is no longer a fluid. In this case the pressure vanishes and the formula m = e(cc) merely defines the (unchanging) momentum flux, which is not a force (cf. Newton’s first law). It follows from the above that the length scale for the pressure gradient must be much larger than A, i.e.

.

-

XIVlnpl << 1.

1.3.3 Vanishing pressure gradient

-

Without collisions in the infinitesimal volume element at P,, the force V p is zero. This is obvious from the physical description given above, and it can also be verified from the kinetic definition of V p, namely

.

where 6V is a small volume enclosing the macroscopic point P, at which the divergence is required, and dV is its surface. Hence from (1.30)

1 The Equations of Gas Dynamics

16

Figure 1.5: Particle flux across a fluid element P,.

Consider a collinear stream of particles crossing 6V, entering at the surface element nl dCl and departing across n2dC2, as shown in Fig. 1.5. Because this stream is defined in a convected frame, the particles will be relieved of any body force, like gravity etc., that acts on the fluid in the laboratory frame. Hence we may assume that the entering and leaving particles have the same velocity c. It follows that (cc n dC)1 = -(cc n dC)z. Hence the contribution to the surface integral in V p from this stream is (f2 - f1)cc n dC, where the factor is required to avoid double-counting, An increment to V * p is therefore possible only if f 2 # f l , i.e. the particles enter and depart in different numbers, which requires that some particles either join or leave the stream en route across the element. The only physical mechanism for this process is particle collisions within the fluid element. Hence a necessary condition for V p to be non-zero is the occurrence of collisions; it is not sufficient since gains and losses may balance. Pressure is transmitted through a neutral gas by the impulsive forces occurring at particle collisions. Between collisions neutral particles are free of the influence of pressure gradients. The impulsive collisional forces have two distinct components, namely a purely random component that scatters the particles isotropically, and a non-random component proportional to -V * p. Without the latter, particles would not be driven (on average) down the pressure gradient, i.e. their collective behaviour would differ from that of the fluid they comprise - an evident contradiction. The same considerations apply to charged particles, with the important modification that since abrupt collisions are rare (see §5.1.1), the non-random force is not applied impulsively, but is present as a continuous influence on particle motions. The normal pressure between the fluids is n p n. Let i, j,k denote the unit Cartesian vectors, and take n to be parallel to each in turn. The average value of the normal pressures at some point in the fluid is

.

-

3

-

-

. -

which equals the thermodynamic pressure p . Hence p = 13.Q (c2) = f l : p = $trace p .

(1.31)

1.3 Pressure

17

1.3.4 Local thermodynamic equilibrium For a given continuum variable cp, the macroscopic scales are defined by (1.32) the minimizing being over all relevant values of (r, t ) and all orientations at a point in the gas. The Knudsen numbers kN,

f X/C,

km

T/T,

ICN

E maX{kNL,

km},

(1.33)

are a measure of how nearly the medium may be regarded as being a continuum. In a ‘true’ continuum k, is zero, but in this limiting case, diffusion is completely suppressed by collisions, making it impossible to transmit fluid momentum and energy through the gas except by the collective process of convection. A thermodynamic system P, must have the possibility of reaching uniform values for its macroscopic variables in a relaxation time Tth much smaller than the macroscopic time-scale 7 for, as will be explained shortly, only then can precise values be assigned to Pc’s pressure and temperature. In a gas the mechanism that tends to produce equilibrium is the interaction of particles via collisions. Since 7 t h M 7, for these thermodynamic variables we must have k , << 1. And as the molecular speed, c = X/r, and the speed C / 7 at which macroscopic perturbations propagate are usually comparable, this constraint entails k,, << 1. In continuum mechanics pressure is force per unit area and in a gas the existence of such a force requires the particles to collide either with each other or with confining walls. Away from boundaries, the ambient particles around a point P, play the role of the confining ‘walls’. This implies that our local thermodynamic system, P,, must have a typical dimension that is at least a mean free path in length. In this case adjacent systems, say PCland Pc2, can interact, with each one exerting a pressure on the other. If PClhas a slightly higher pressure than Pc2, a net force will result, with P c 2 experiencing more numerous or more energetic collisions with than PCldoes with particles from Pc2. On a continuum description, particles coming from PC1, the force is said to be due to the component of the pressure gradient directed from P c 2 to P,1, but the underlying mechanism is an imbalance in the particle collisions. In a ‘collisionless’ gas, however great the difference between the values of Q ( c 2 ) at two neighbouring points, there would be no pressure force in the gas. We therefore define the pressure as being the force transmitted across a unit surface. The concept of ‘equilibrium’ is not required in this definition but collisions are essential. Gas temperature appears to be a variable that does not depend on the presence of collisions. Its definition in terms of the average kinetic energy of particles (see 51.2.3) is one of the most famous results of the early kinetic theory. But in classical thermodynamics the notion of ‘thermal equilibrium’ plays a central role in the definition of empirical temperature, a necessary preliminary to the introduction of the absolute temperature. Thermal equilibrium between P,1 and P c 2 requires a collisional interchange just as already described for the pressure. Now suppose that PClis hotter than P,2. The transfer of energy between these systems involves two stages. First the more energetic particles from PClmove through a free path and then they deposit their excess energy in P c 2 by collisions. It is important to distinguish between mere energyjux, which like radiation, need not be deposited locally and heat flux, which does require collisions.

1

18

The Equations of Gas Dynamics

The precise definition of pressure and temperature is therefore determined by the force and energy transmitted between neighbouring fluid elements. When the laws of conservation of momentum and energy are obtained using these variables, their physical properties are invoked in the formulation. Alternatively, these and related variables like the viscous stress tensor TI and the heat flux vector q are defined implicitly by their roles in the conservation laws. Macroscopic equilibrium is an unnecessary restriction in the definition, but we must ensure that the gas (or fluid) has an internal structure such that the symbols p , TI , T, q, etc. really do have the properties implied by in their appearance in the conservation laws. The local transport of momentum and energy sets a lower bound on the size of the thermodynamic system P,. If d l is a typical dimension of P,, then it follows from the above discussion that we need X 2 de. Similarly, the macroscopic time element dt cannot be much less than T , for this would imply that the pressure and temperature could respond to changes that occur much faster than the collisional mechanism effecting these changes. We can now define our thermodynamic system at P, to be an element in which, for each of the thermodynamic variables of interest,

k,

<< 1,

k , << 1, X

2 de << C,

T

5 dt << 7.

(1.34)

It is not possible to be more precise than this.

1.4 Macroscopic conservation laws 1.4.1 Convection and diffusion Convection is the transport of a macroscopic property, such as density, momentum, energy, the concentration level of a contaminant, and so on, by the fluid motion. Let @(r,t ) be such an attribute, specified as an amount per unit mass of fluid - known as a ‘specific’ property then a volume element of mass e d r will possess an amount e@dr of it. As the volume of fluid crossing a stationary surface n dC in one second is v n dC, it follows that the transport of @ due to convection occurs at the rate e@v n dC across this surface. This description can be generalized by introducing a specific property $(r,w, t ) ,whose value may depend on the velocity w = v c of the particles involved in its transport. Suppose that the average value of 4 taken over particles at a macroscopic point is @, i.e.

-

.

+

@ ( r , t )= ( 4 ( r , w , t ) ) ,

.

(1.35)

then the local transport of across n d S is the average of e4w n dS. This gives a total flux of e(q5w) * n = e(q5(v c ) ) n:

+ .

Total flux of @ = Q@V

- n + J+ - n,

(1.36) (1.37)

We term J+ the diffusion vector for 4. Diffusion is thus the transport of a property by the purely random component of molecular motion. A central task in kinetic theory is that of obtaining the diffusive flux J+ n for various properties 4.

.

1.4 Macroscopic conservation laws

19

In some circumstances an expression for J+ of the form (1.38) can be found; K+ is termed the coeficient of dzffusion for a. It follows from (1.38) that has the dimensions: (length)2/time, and since it is due to particle transport, we may write K+

= ff

x2/7,

K+

(1.39)

where Q is a constant of order unity. Sometimes there exist several ‘routes’ for the diffusion of a, i.e. a number of distinct processes each contribute to J+. If these are independent, the total flux is obtained by summation, and K+ becomes (1.40)

the subscript j denoting a particular process. The convection term in (1.36), viz. v n, is dependent on the choice of reference frame in which the velocity is measured, whereas the diffusion term is not. In fact we shall adopt ‘frame-indifference’ as being the essential property that distinguishes diffusion from convection. In some circumstances it is not evident from the physics where to draw the line between these two transport processes, and a mathematical definition is useful. The physical assumption involved in (1.37) is that 4 is a property transported by individual particles, whose random motions are independent of the amount of 4 that they carry. In a simplified description, if we take eq5 to be the density of molecules labelled in a particular way, the intensity 4 becomes the fraction of labelled molecules at a given point. If 4 is initially uniform, variations in the density Q cannot affect 4, since labelled and unlabelled molecules will have the same propensity to migrate at each point in the fluid. Thus J+ cannot depend on Ve,at least in a linear model of the phenomenon. The random molecular motions will disperse any non-uniformity in @, and the first spatial derivative of is the dominant term driving the system towards equilibrium. For example when 4 is the particle energy ( f e z E ) (see (1.20)), by (1.37) its diffusion vector is

-

+

q

= Q(C($C2 + E ) ) ,

(1.41)

which is known as the heatflux if collisions are dominant and the energyflw otherwise. The transport of energy between adjacent fluid elements requires a non-zero, finite value for the mean free path A.

1.4.2

A general balance equation in physical space.

Consider a volume V of the medium, with a surface dV across which 4 is being convected and diffused. Let S+(r, t ) be the rate per unit volume at which is being introduced at points within V due to physical or chemical processes, the details of which will be specified later.

I

20

The Equations of Gas Dynamics

per unit volume. Its increase is due to its flux across The amount of 4 within V is (&) = dV and its creation within V .Hence by (1-36)we obtain the balance equation

By Gauss’ divergence theorem, this can be expressed as

1{ G(e@)+ d

V

-

V (e@v+ J,)

- S,} dr = 0.

Granted that this equation holds for all subdivisions of V - or equivalently that V may be chosen arbitrarily -we deduce that

d

+ V (e@v+ J,) -(@@I at

= S,

,

(1.42)

which is the required balance equation for @. There may be several gas components present, in which case it is necessary to distinguish the macroscopic variables 0, v, T , p, q, etc. by a different subscript for each species. Thus for the ‘ith-fluid’, Pi: (1.43) The source term is partly due to mass transfer between the fluid components resulting from chemical or physical interactions between them. Let hij be the rate at which Pi gains mass from Pj, then h i j d j is the rate at which the property 4 is received from Pj by Pi.Conversely, Pi loses 4 at the rate hji4i to Pj . The net gain of 4 by Pi due to mass transfers is therefore

There may also be transfers of 4 without mass exchange. Let &jej be the rate per unit volume of such a transfer from Pj to Pi,then by conservation of 4 during this transfer,

The total source due to internal and external transfers is therefore (1.44) where S g is the rate at which Pi receives 4 per unit volume from sources outside the system.

1.4 Macroscopic conservation laws

21

1.4.3 Conservation laws for a simple fluid To write down the conservation laws of mass momentum and energy, we use equation (1.42) with (1.20), (1.30), (1.37), (1.41) and (1.44): ma^^ :

4 = 1, 0 = 1, J4

ae

-

at

= 0,

S$ = 0. Hence

+ V . (p)= 0 .

(1.45)

mmentum : 4 = w, @ = v, J , = e(wc) = ~ ( c c=) p, S$ = QF,where F is the body force per unit mass acting on the fluid. Hence

a

- (QV) at

+V

*

(Qvv

+ p) = QF. = ( + ( v+ c ) +~ E ) = 12 v 2 + u,

4 = (zw 1 2 +E), J+ = e(($2 + € ) C ) = @( +w2 energy : Sd =

c + v * cc + ($2

+

€)C)

pF v. Hence

(1.46)

. + q,

=p v

(1.47) There is another expression of the conservation equations known as the Lagrangian form, which can be obtained by transforming to a convected frame. From (I .42) and (1.45),

where

d D- -+v*V,

(1.49)

at

is the rate of change in a frame moving with velocity v relative to the laboratory frame. In this convected frame ( I .45), (1.46) and (1.47) become mass:

DQ + Q V * V = 0,

(1.50)

momentum:

QDv+V-~=QF,

(1.51)

QDu

energy:

+ p :VV = -V

q.

(1.52)

In the reduction of the energy equation, we have used the symmetry of p (see 51.3.1) and the expansion

v .(p .v)= ( V

*

By (1.29) p :v v = p l : v v

. +v v :p =v

p) v

*

(V.p) + p :v v .

+ TT : v v = p v . v + Tr :v v ,

I The Equations of Gas Dynamics

22

whence from (1.50) (1 .53) Now (1.52) can be written (1.54) which is the first law of thermodynamics for a fluid element.

1.4.4 Specific entropy Another form of the energy equation (1.54) is (1.55) where

T ds

= du + p d ( l / Q ) ,

(1.56)

defines the specific entropy s. Hence from (1.23) and (1.24),

s = c, 1nT - R l n e

+ const.

(1.57)

which formula is often written in the form (1.58) For a monatomic gas (1.23) gives y = 5/3. If the conditions at the convected point are adiabatic, i.e. no heat is removed or supplied to the volume element dr at P,, then q is zero. Furthermore, if the fluid is also ideal (see 5 1.3. l), lT is zero, and (1.55) reduces to

Ds = 0

(isentropic flow).

(1.59)

Thus s is constant at P, and if this holds throughout the fluid, the motion is said to be isentropic. For such flows (1.58) reduces to P = Key 1

(1.60)

where K is another constant. Equation (1.52) can be rearranged as (1.61)

(1.62)

1.5 Introduction to kinetic theory

23

=

is the entropy production rate per unit volume. With Q, = s in (1.48), J, q/T in (1.61) is identified as being the difision vector for entropy. Here q is interpreted as being the energy flux into the fluid element (see last paragraph of 5 1.4.l), whereas in O, q is the heat flux, which by Fourier's law (see $5.5.3) is q = -tcVT, where tc is the thermal conductivity. I f the motions of the particles are reversed, it follows from (1.41) that energy flux changes sign and is therefore a reversible phenomenon, whereas heat flux does not. Hence (1.54) can be expressed in the form (1)

(2)

pDu - { -p@Dp-' - TV (q/T)} = T O ,

in which the terms labelled ( I ) and (2) represent the power supplied reversibly to the macroscopic point P, by compression and heating. The stability of the process requires the internal energy of P, to increase at a rate not less than that provided by the reversible power supply. Hence to secure stability it is necessary to have a non-negative entropy production rate, ff

L 0,

(1.63)

at all points in the gas. With Q, = s , J4 = q/T, S4 = O, (1.42) yields d

-(es)

at

+V.(esv+q/T)

=ff.

(1.64)

Hence the total entropy within a volume V ,

S

=Iv

esdr,

(1.65)

satisfies (1.66)

1.5 Introduction to kinetic theory 1.5.1 Kinetic entropy By (1.24) and (1.57) the entropy density of a monatomic gas is given by ps= -kBnlnn+$kBnlnT+~so,

where SO is an arbitrary constant. We shall generalize this expression to obtain a relation between the entropy density ps and the phase space density function, f (r, c , t ) defined in $1.2.1. (In the convected frame P,, w is replaced by c.) From the additivity of entropy it follows that a mixture with all components having the same mass, has an entropy density defined by (1.67)

I The Equations of Gas Dynamics

24

Now suppose that the i-th component is specified to be those particles of a monatomic gas that lie in the volume element dci of velocity space, at a point where the phase space density is f i = f (r, ci, t ) . Then the number density of this component is ni = fi dci. The spread in velocities of the i-th component is proportional to ldci 1 l I 3 , and hence by (1.19) the associated temperature Ti is proportional to l d ~ i 1 ~Thus / ~ . (1.67) gives es =

-kBcIn i

fi

fi

dci

+ kBC ni( $ InTi - In Idci()+ @so. i

The second right-hand term reduces to a constant times Q, and hence with a particular choice for SO (which has no significant consequences), and allowing the components to form a continuum so that the sums can be replaced by integrals, we arrive at the formula QS=

- k €3

/

f(lnf -

dc*

(1.68)

(We could have introduced this equation as being the dejnition of entropy in phase-space, but it is instructive to see its connection with fluid entropy.) Now we change the interpretation and take (1.68) to define the entropy density of a single-component perfect gas. As s is prescribed only to within an additive constant, the term -f in the integrand is sometimes omitted. The entropy within a volume V of the fluid is

S=

s,

Q S dr

= -kB

f (In f

- 1) dc dr

,

(1.69)

which changes at the rate given by (1.66). In an isolated system the surface term is zero, in which case S steadily increases with time until when equilibrium is attained, it will be at its maximum value. In this state small variations in S due to changes in f will be zero. We shall use this principle to determine the equilibrium value o f f .

1.5.2 Equilibrium distribution function Let fo denote the equilibrium value ofthe distribution function, then at f = fo, the entropy S of the system is a maximum. From (I .62) and (1.63) this requires the gradients to vanish and therefore maximizing S is equivalent to maximizing Q S subject to any constraints applying to the distribution function. We shall define fo to be that distribution function which applies to the same values of the number density n and energy density QU as in the non-equilibrium case. Hence, as the gradients relax towards zero and f -+ fa, the functions n=

1

f dc,

QU

=

J $rnc2fdc,

(I .70)

are held constant, which restrictions are accommodated by introducing Lagrangian multipliers a and p. Thus 6(es) = -kB

J

6{ f (In f - 1)

+ a f + pc2f}dc= -kB

J

(In f

+ a 4-pc2)6fdc.

1.5 Introduction to kinetic theory

25

Since the variation in f is now arbitrary, d(es) is zero only if the integrand vanishes, i.e. i f f has the value fo given by

fo = exp(-a - p c 2 ) .

(1.71)

The constants a and /3 are determined by using the constraints in (1.70). The first constraint requires that

1

C t 3 0 0 0 3

n = exp(-a)

J’ J’ -02

-02

-ca

exp { -P(c:

+ ci + c:)} dc, dc, dc,,

and since (1.72) the expression for n yields (1.73) Also by (1.24) the second constraint in (1.70) becomes

-ca--M-m

The average value of c: is

whence by (1.72), 3 = 1/(2P). Similarly k B T = m/(2P),so that P = m / ( 2 k B T ) . The formula in (1.7 1) now reads

3=c2 =

1/(2/3). Thus from (1.73),

(1.74)

It remains to show that the stationary point is actually a maximum; this is left as an exercise.

1

26

The Equations of Gas Dynamics

Figure 1.6: Spherical coordinates for the velocity vector.

1.5.3 Averages over velocity space First suppose that the particles are in an equilibrium velocity distribution, then the probability that a particle chosen at random has a velocity in the range c , c f dc is

(fo/n) dc = n-1 C3exp(-v2) dc , where (1.75) and i: denotes unit vector along c. Transforming from the Cartesian coordinates ( c ~ cy, , c,) to the spherical coordinates (c, 8, $), we have dc = c2dc sin 8 d8d4 = c2 dc4r d n ,

(1.76)

where 4n dS2 is the element of solid angle subtended at the origin. Thus the probability that the relative speed v falls in v, v dv and the unit vector C lies in dCl is

+

fo 4 - dc = -v2 exp(-v2) dv dR n J;r

(O
(1.77)

The average value of a function d(v) is therefore

(i; = 4

/ Srn

f i n o

$ ( v ) v 2exp(-v2) d v d R .

(1.78)

Shortly we shall need the values of the integrals J1

=

S,EdR,

J2 E

LEedS2.

From Fig. 1.6, C = i sin 8 sin $

+ j sin 8 cos $ + k cos 8, and therefore

(1.79)

1.5 Introduction to kinetic theory

i%

=

27

ii sin2 8 sin2 4 j j sin2 8 cos2 4 kk cos2 0 +(ij ji) sin2 8 sin 4 cos 4 +(jk kj) sin 8 cos 8 cos q5 (ki ik) sin 8 cos 8 sin 4.

+

+ +

+

+

+

From these expansions we find that

and

2n

i.e.

n

i%sinOdOd4=5(ii+jj+kk),

J 2 = & l

edQ=O,

eEdR=

il,

(1.80)

where 1 is the unit tensor (see page 7). We shall also need the standard integrals: 1

-$

l m u v exp(-v2)

dv =

( r = 0)

-& ( i ( r - 1))

( r = 1,3,5, ...)

1 . 3. . . .

( r = 2 , 4 , 6,...)

2

2

(1.81)

Two important applications of (1.78) are

-c = - C 4 l and

03

(1.82)

J;;

4 cc=-c2

JR JO

J;;

M

u4 exp( - v 2 ) du i% dR =

;C2 1,

(1.83)

the trace of which is

-

c2 = ;C2 = 3kBT/m= 3RT.

(1.84)

It follows from (1.30), viz. p = e E , that in equilibrium the pressure tensor is

Also

c2.

p = l .2 @

(1.85)

_ j0 -- F ( u ) dudS2,

(1.86)

p=pl,

n

where

4 F ( v ) = -u2 e-vz

J;;

(u

z c/C,

c =(21c~~/m)l/~)

(1.87)

is the Maxwellian speed distribution, plotted in Fig. 1.7. Its maximum occurs at v = 1, identifying C as the most probable molecular speed.

28

1

"0

The Equations of Gas Dynamics

0.4 0.8 1.2 1.6 2.0 2.4 2.8

v

= c/c

Figure 1.7: The Maxwellian speed distribution.

1.5.4 Evolution of the phase-space density Consider a volume V of phase space. Let dS = ( d S , , dS,) be a surface element of V , with dSr referring to physical space, and d S , to velocity space. Suppose that a bunch of particles, moving with velocity w and acceleration w pass through dS, then the number flux across the surface element is f w d S , f w dS,. If d S is directed outwards from V, this flux will be a loss from V .Let (aflat), d v , be the net rate at which collisions cause particles to appear (or disappear, if this number is negative) in the 6-D volume element d v = d w dr, at some point within V .Then particle numbers are conserved if

. +

By Gauss' theorem the surface integrals can be written as phase-space volume integrals,

and as the volume V is arbitrary in extent, it follows that

in which the precise nature of the partial derivatives has been made explicit. It follows that, provided

- a. w = o ,

(1.88)

aw

the equation for f can be written in the 'gradient' form

Df=C,

cs(%), C

(1.89)

1.5 Introduction to kinetic theory

where ]D-

a -

d d -+w.-+W,

at

29

(1.90)

aw

dr

is the rate of change following the cloud of particles through phase space. A useful approximation to the collision term is 1

c = -(f o-f), 7-

(1.91)

where r is the time for equilibrium to be achieved in the absence of driving forces and is called the relaxation time. Equation (1.89) with C given by (1.91) is known as the BGK kinetic equation (Bhatnager, Gross and Krook, 1954).

1.5.5 Boltzmann’s distribution law Under steady state conditions Maxwell established that the distribution function is given by (1.74): fo =n

(:Yiz

2nkBT

exp( -m(w - v)’/2kBT)

(1.92)

(The complete proof is due to Boltzmann.) Its derivation does not restrict the variables n, v, or T to being uniform in space. Since lDf0 = 0, it follows from (1.90) that o = w . -afo SF.-afo

(F = W ) ,

bW

dr

where F is the body force per unit mass. With v = 0 in (1.92), its logarithm reads

2

Info = Inn - In T so that

-

(m/2kBT) w2 + const.,

This requires that V T = 0, i.e. that the temperature is uniform throughout the gas and that 771

V l n n = -F. kBT Let the force have a potential $, i.e. F = -V$, then $ = - ( kBT/m) In n + const.,

whence n = no exp(-rn4/kBT),

(1.93)

I

30

The Equations of Gas Dynamics

where no corresponds to zero potential. This is known as Boltzmann’s distribution law. Equation (1.92) now reads (1.94) Since w and r are independent variables, we may augment (1.88) to

-d. w + dr

-.w dW

-

=0,

(1.95)

which implies that the six-dimensional phase-space ‘fluid’, with density f and ‘velocity’ ( w ,w) is incompressible, i.e. a given volume element du of this fluid remains constant in magnitude as it is convected through phase-space by a fixed number of particles. Thus ID(du) = 0 , which is Liouville’s theorem for six-dimensional space. While collisions transfer momentum and energy between particles, they cannot alter the totality of these properties at any point Pc(r, t ) .And, of course, particle mass is also conserved at collisions. These constraints require the collisional term C to satisfy

I

Ij)lr@dw =0,

(Ij)

= (1, w , .I”>>

(1.96)

7

where the integral is taken over the whole of velocity space and the constant particle mass m has been deleted from Ij). Given the solution f = f (r, w , t ) of the kinetic equation (1.89), the fluid variables defined earlier with the notation ( . . . ) are obtained from the following velocity-space integrals (see (1.4), (1.6), (1.24), (1.30) and (1.41)):

e

= m

s

fdw,

ev = m

1

wfdw ,

p = nkBT=$pu=$m/c2fdw, ( c = w - v ) , p = mJ c c f d w ,

1 (1.97)

q = ?2 / c 2 c f d w . Exercises 1.

1. I A fluid motion along OY has a velocity uy dependent only on the distance x measured along OX. The random molecular motions transfer momentum between the planes 2 = a, 2 = a + A, where X is the mean-free-path. Show that this transfer occurs at the rate p dv,/dx, ( p N $@A) per unit area. Interpret this as a fluid force.

1.2 Show that the fluid stress acting on a small element P of volume d-r,with centre G(r), supplies P with a moment -(p * V) x r d-r about G. Prove that this moment is zero only if p is a symmetric tensor. [Use (1.27) and express V and r in terms of the unit vectors i, j, k].

1.3 Starting with the equation of continuity, (1.43, show that the steady flow of a fluid, in which the 02-component is everywhere zero, can be described by a stream-function @(x, y) related to v by QV = -k x V@,where k is unit vector along 02.Prove that the rate of change of @ along a streamline is zero.

1.5 Introduction to kinetic theory

31

1.4 Prove that the fluid acceleration a = Dv is related to the vorticity c‘ by

av + V ~ V ’ + 6 x v ,

a=-

at

and that in an ideal, isotropic fluid with a conservative body force (F = -V+),

1.5 The ‘circulation’ around a closed contour C is defined by

where d s is an element along C. Show that, subject to conditions that should be stated, if C is convected with the fluid in a system rotating with an angular velocity S2, the circulation changes at the rate dr -=-LV(f) dt

s

~ V p - n d C - 2 C2xv.d~.

Under what conditions does Kelvin’s theorem, dI’/dt = 0, apply? 1.6 Deduce from (1.62) and (1.63) that for small gradients the viscosity tensor will have the form lT = -2pVv, where p is a positive parameter (known as the viscosity). Show that in incompressible flow the equation of motion is dv

e-d t

= -Vp+ pV2v

+ eF,

where F is the body force per unit mass, and expressed it in the form aV

- - v x c‘ at

=

-v+-v(’e + i . 2 )

+VV’V,

=

p / e . Show that in an inviscid fluid the parameter where F is of the form -V$ and u H iv’ $ p/e is constant. (This is known as Bernoulli’s equation.)

=

+ +

. + q as momentum and energy diffusion vectors [cf. ( I .46),

1.7 Interpret J, = p and JE = p v (1.47), and (1.42)].

1.8 The velocity relative to a point P,, has the same direction everywhere, and varies linearly with distance in one direction j i , perpendicular to 9. Sketch the decomposition of this shearing motion into a rotation and a straining motion, and show that the latter has its principal axes at 45” to 2 and

9,

1.9 Use the divergence theorem in velocity space and the independence of r, w, t to establish that

provided that as IwI --t 00, Iq5Ffl obtain (1 SO), (1.5 l), and (1 S2).

-+

0 sufficiently rapidly. By choosing

4 = m, mw, fmw’,

1.10 Would you expect a gas consisting of neutrinos to have a measurable pressure?

Physics of Plasmas L. C. Woods Copyright 0 2004 WILEY-VCH Verlag GmbH & Co KGaA

2 Magnetoplasma Dynamics

A magnetoplasma is a gaseous mixture comprised of electrons, ions, and neutrals, permeated by a magnetic field. The basic dependent variables are those of ordinary fluid dynamics, i.e. the density (e),temperature (T), and velocity (v) for each component, plus the magnetic induction (B),the electric field (E) and the electric current density (j). In some circumstances the macroscopic variables change on length- and time-scales long enough for an adequate description of the mixture to be given by the one-fluid equations introduced in 5 1.4.3. In this case the system of equations relating these variables defines what is termed ‘magnetohydrodynamics’ (MHD) despite compressibility often being an important feature. With a fully ionized plasma a two-fluid description is often required, and is essential if the temperatures of the ions and electron can differ. In this case we distinguish between the electrons and ion fluids, using subscripts “e” and “i” for this purpose. Magnetic and electric fields are divided into large-scale or ‘macroscopic’ values, denoted by B and E, which are slowly varying functions of r and t and rapidly fluctuating, local or ‘microscopic’ values. The former are responsible for body forces like j x B, while the latter are involved in the interaction between charged particles and are therefore responsible for the transmission of pressure and energy through the plasma. The distinction between the fluid velocities, v,, vi, and the frame-indifferent peculiar velocities, c,, ci is also that between macroscopic and microscopic variables. As the MHD equations give the simplest model for a magnetoplasma, it is tempting to use them without verifying their validity for a given phenomenon. In fact, it is sometimes the case that only an a posteriori check is possible-one uses MHD to find a solution and checks afterwards that it is consistent with the MHD constraints. Therefore our first concern is to give a full list of MHD equations, with the criteria of their applicability. This is folIowed by some important applications and the chapter closes with an account of boundary conditions in plasma fluids.

2.1 Electromagnetic fields 2.1.1

Maxwell’s equations

The four basic electromagnetic fields are the magnetic induction B, the magnetic field H, the electric field E and the electric displacement D. To these are added the macroscopic charge density Q and the current density j defined by the rate n .j at which particle charge is convected across unit surface with n as its normal. These are functions of position (r) and time ( t ) and satisfy the following experimental laws, collectively known as Maxwell’s

2 Magnetoplasma Dynamics

34

electromagnetic equations:

(Faraday 's law)

fH . d s = l ( j + s ) . n d C , C

(AmpBres law) (Charge conservation).

(2.3)

In (2.1) and (2.2) C is a fixed contour, C is a surface with C as edge and n is unit normal to C, and in (2.3) dV is the surface of a volume V . Replacing the contour integrals by surface integrals and the surface integral in (2.3) by a volume integral, and using the arbitrariness of C, C, and V, we obtain Maxwell's equations in their differential forms: aB V X E = --,

at dD

VxH = jiand

at

'

dQ V . j =---.

at

With the assumption that at some past time the vector fields were zero, the divergences of (2.4) and (2.5) give

V-B=O,

V-D=Q.

(2.7)

To complete the set of equations constitutive relations for j, B, and D are required; these depend on the thermodynamic state of the medium and on its relative motion. We shall start with the simplest case, namely a stationary, isotropic medium, M . In this case the constitutive equations are j=u-E,

B=pH,

D=EE,

(2.8)

where u is the conductivity tensor, p is the permeability and E is the dielectric constant or permittivity. In general, the scalars p and E are functions of the thermodynamic state of M , which includes the strength \HI of the magnetic field. The conductivity tensor not only depends on the thermodynamic state, but it is also influenced by the direction of the magnetic field vector H relative to j, as shall be explained in 52.4.3. In free space the values of p and E are

p o = 47r x 10-7Hm-',

E,,

= 8.8542 x 10-12Fm-1.

(2.9)

These constants are related to c, the speed of light, by copo = c - ~ ,which can be deduced from the above equations by showing that E satisfies the wave equation when Q and j are zero.

35

2.1 ElectromagneticJields

2.1.2

Galilean transformations

Equations (2.1) to (2.3) are invariant under the Lorentz transformation of special relativity, but as we shall be concerned only with velocities very much less than the speed of light, it is sufficient to consider Galilean transformations. A convected frame P,(r, t ) moves with a non-relativistic velocity v relative to the laboratory frame L in which the medium is stationary and in which the electromagnetic vectors are E, B, and j. Let E*, B*, j* denote their values as observed in P,, then the relations between the vectors E and E*,B , and B * , and j and j* are the required Galilean transformations. To establish the electric field transformations we start with Faraday's law (2. l), written for P, in the differential form

1

( D = da t + v . V ,

n V x E* dC = -D(B* n dC)

(2.10)

where n d C is a convected surface and D is the time derivative in P,. Our method is to transform this equation into the L-frame, where it should read

n . V x E dC

=

dB

-at

.n d C ,

(2.1 1)

since, provided Iv I << c, Maxwell's equations are frame-indifferent under Galilean transformations. To make the transformation we need the following theory. Mathematical note 2. Convection of a j u x element Let X(r, t) denote a well-behaved (continuous and sufficiently differentiable) vector field, then its flux X n dC through a convected surface element n dC changes at the rate

-

.

D(X.ndC) = { g + v V . X - V x ( v x X ) } . n d C ,

(2.12)

where D is the convective derivative D = d / d t + v V. This can be proved as follows. Let d r denote a volume element, then the volume V of fluid within a surface dV changes at the rate 9

=

$1, 1" dr =

n.vdC =

s,

V .vd7

Interpreting the change as being due to changes in the volume elements, we have

/,

-

(D(d7) - V v d7) = 0,

or, as V is arbitrary, D(d7) = V * v d r

(2.13)

Next consider a material line element dx, namely one that is convected with the fluid, then the rate of change of dx is just the difference between the two velocities at each end of the element. Hence (see Fig. 2.1) D(dx) = dx * V V .

(2.14)

2 Magnetoplasma Dynamics

36

dx

Figure 2.1: Convected elements. Let m be a vector associated with a given fluid element, then D ( d x - m) = d x - (Dm + Vv m) .

(2.15)

-

Take m to be the surface element n dC, then d x m becomes a convected volume element dr, whence by (2.13) and (2.13, D(dx. n d C ) = V . v d x . n d C = d x . {D(ndC) dx.{D(ndC) + V v . n d C - n V . v d C }

Or

+ Vv .ndC},

=O.

As dx is arbitrary, it follows that D(ndC) = ( n V . v - V v . n ) d C .

(2.16)

Hence D(X-ndC) = DX.ndC+(XV.v-X.Vv).ndC =

( $ + V . V X + X V . ~ - X . V ~

1

. n d ~ ,

which with the help of the vector identity

v x (v x x) = vv . x - v . vx + x.vv xv . v , -

yields (2.12).

Applying (2.12) with X = B*,we obtain

-

n V

x E* = -{ -+ v V .B* - V x dB* dt

(v x B*)} - n d C .

By the frame-indifference of the basic equations, V * B' = 0, and hence

n.V

x (E*- v x B*)dC = - = - n d C . at

As this must be identical to (2.1 l), we conclude that

B*=B,

E*=E+vxB.

(2.17)

2.1 Electromagneticfields

37

A similar transformation of Ampire’s law yields the rules D* = D and H* = H -v x D, and the transformation for the current density is evidently j* = j + QV. Thus the first equation in (2.8) becomes j

+ Qv = 6 .( E + v x B) ,

(2.18)

where in magnetohydrodynamics, as we shall shortly explain, Q is zero; the appropriate form for the conductivity tensor will be derived in 52.4.3, 55.3.4 and s5.5.2.

2.1.3 Polarization On a microscopic length scale the motions of the charged particles in a plasma generate locally fluctuating electromagnetic fields, which satisfy Maxwell’s equations with p and E equal to their free-space values, po and c0. To suppress the unwanted details of these microjields it is necessary average the fields over the region occupied by a macroscopic point (see 51.3.1). The resulting averages are the macroscopic electric and magnetic fields, E(r,t ) and B(r,t ) ,and the price paid for the reduction is that po and c0 are replaced by p and E , which are functions of the thermodynamic state of the medium. On a macroscopic scale the average field can be said to induce responses in the medium known as the electric and magnetic polarizations. These are:

P

= D - foE,

M = ~ ~ - ~ B - H .

(2.19)

In an isotropic medium it follows from (2.8) that

P = xee0E,

(x. = -E - 1);

M = xmpO-lB,

€0

(xm

-)P

1 - PO ,

(2.20)

where xe and xm are called the electrical and magnetic susceptibilities. Introducing (2.19) into (2.5) and (2.7)2 we obtain

V

X

B = po(j

+ dPt + V x M) + p o c odEz ,

(2.21) (2.22)

The term V X M is called the ‘magnetization current’. If the magnetic field is weakened by the presence of the medium, the material is said to be ‘diamagnetic’ and if strengthened, the material is ‘paramagnetic’. In a plasma the contributions to P and M from bound charges are negligible. For example, in non-polar and atomic gases, xe (e.g. see Bleany and Bleany, 1985) so even if the plasma consists largely of neutral particles, it is a good approximation to ignore this polarization. Similar remarks apply to xm.Neither do the free charges contribute significantly to x e , so we shall adopt

-

E=Eo

(P=O)

as being an accurate approximation for a plasma.

(2.23)

2 Magnetoplasma Dynamics

38

Finally there is the magnetic susceptibility due to free charges. Despite the frequent and misleading use of the term ‘diamagnetic’ in texts on plasma theory, xm is extremely small. That charged particles are classically non-magnetic was established by Miss van Leeuwen in a theorem that was partially anticipated by Bohr. The Bohr-van Leeuwen theorem states that when (classical) Boltzmann statistics are applied completely to any dynamical system, it is found to have zero magnetic susceptibility. This is a surprising result in view of the gyratory motion of the charged particles about the magnetic field lines. In 92.2.2 it will be shown that a particle with mass m,and charge Q cycles around a magnetic field of strength B at a frequency w, = QB/m. Consider the electron gas. As each electron describes a circle about the field, it has an angular momentum about the centre of its orbit and behaves like a magnetic dipole whose magnetic moment is opposite to the field, which should yield diamagnetism. However, while each electron behaves like a dipole weakening the macroscopic field, it is not true that their collective effect is obtained by merely summing over the separate dipoles. Langevin’s theory of magnetism for solids and neutral gases is certainly based on such a summation, but only because the molecular dipoles involved are so far apart that their mutual interactions can be largely neglected. In a magnetoplasma each gyrating electron generates a toroidal micromagnetic field opposing the macro-field B within its orbit, but augmenting B outside its orbit. The mutual interference of millions of overlapping orbits, cancels the micro-fields, resulting in zero magnetization, a description that also holds when the orbits are fragmented by collisions, As van Vleck (1932) observes: However, the magnetic moment involved in the susceptibility is not the magnetic moment of each electron with respect to the center of its particular orbit, but instead the combined magnetic moment of all the electrons with reference to some one common point chosen as the origin for measuring angular momentum.

Quantum effects make a plasma either very slightly paramagnetic or very slightly diamagnetic depending on whether or not the spin of the electron is taken into account (see e.g. Ginsburg 1970, p. 27). We shall therefore accept that P=Po

W=0)

(2.24)

is a good approximation for a plasma. Equations (2.23) and (2.24) allow us to eliminate H and D and write (2.5) and (2.7)2 as

V and

x B = poj

+ P ~ EdE ~---, at

(2.25) (2.26)

Magnetohydrodynamics (MHD) is a special case of a plasma in which spatial and temporal rates of change are small enough to allow us to set IE,VIand E,b/dt in the above equations equal to zero. In a non-dimensional form, the constraints are (see 52.2.3 and 52.2.4)

(2.27) where the length A, and time w;: are each proportional to c0 f . Sometimes, in applications to conducting liquids, the additional constraint of incompressibility is appropriate.

2.2 Basic plasma parameters

39

2.2 Basic plasma parameters Plasma neutrality Let a = 1 , 2 , . .. denote the several species of fluid present in a plasma, then for a mixture

2.2.1

of perfect gases the density, momentum density, energy density, and pressure are defined by the sums over a: (2.28) The charge and current densities are given by (2.29)

where denotes the charge on the particles of the a-species. For example, in a plasma consisting of electrons (Q, = -e) and one species of ion with a charge number 2 ( Q = ~ Ze), Q = e(2ni - n,),

j = e(Zniv, - neve).

(2.30)

Most plasmas of interest are electrically neutral over sufficiently long distance and timescales. In this case, adopting the subscript zero to indicate averaged or ‘background’ values, (2.30) are reduced to the frame-indifferent forms (see 1.4.1) neo

2.2.2

j = eneo(vi - v,).

= znio,

(2.31)

The cyclotron frequency

Consider the motion of a particle P of mass m, charge Q, and velocity w moving in (macroscopic) fields E and B. By (2.17) the electromagnetic force acting on P is the Lorentz force, Q(E w x B), and if in addition there is a body force f per unit mass but no collisions, the acceleration of P is

+

W =

Q

-(E m

+ w x B) + f .

(2.32)

The fluid acceleration Dv is the average of the particle accelerations, i.e. Dv = G, where w = v c ; also W = Dv C. The peculiar velocity c is frame-indifferent, being measured in the convected frame P,. The average of (2.32) is

+

+

Q

Dv = - (E m

+ v X B) + f ,

(2.33)

-

where, since we are ignoring collisions, the pressure force, V p/e, is absent (see 51.3.3). Subtracting (2.33) from (2.32) we get C = W,C x b,

(2.34)

where wc=

QB/m.

(2.35)

2 Magnetoplasma Dynamics

40

0 Figure 2.2: Gyration of a charged particle.

It follows from (2.34) that the particle P moves with a velocity c about a point G of position vector X (see Fig. 2.2). The point G(X,t ) is termed the guiding centre of P, and the angular velocity wc is the cyclotronfrequency. The radial distance between G and P, a = la/,where a = r - X, is called the Larmor radius of P and it is related to P’s perpendicular speed by a = IcI/Iwcl. The integral of (2.34) is c = d = wca x b

+ cII

(a=r-X,

cI,= b b - c ) ,

(2.36)

where cIIis the integration constant. These properties are illustrated in Fig. 2.2. When a particle average is required for the Larmor radius, we replace c by the thermal speed C = ( 2 l ~ ~ T / mand ) l /denote ~ the radius by r,, 1.e.

rr, = C / l W C l .

(2.37)

2.2.3 Plasma frequency For electrons (2.32) reads mew = -e(E

+ w x B) + m,f,.

(2.38)

The particle average of this equation gives dve me- = -e(E dt

+ ve x B ) + mefe,

(2.39)

where ve is the electron fluid velocity. Our present interest is in the case that B and f, are zero. Thus from (2.39) and (1 S O ) written for the electrons,

me-dve = -eE, dt

dne = -neV dt

- v,.

41

2.2 Basic plasma parameters

+

Let ne = neO fie, V, = G e , where ne, is the uniform background density and the circumflex indicates a small perturbation. Then to first order in these perturbations,

ace = -eE

me-

at

a

-A, - - -n,,V

at

- 9,.

(2.40)

By (2.26), (2.30)1, and (2.31)1,

-

c0V E

=

0 = eZi$

- efi,,

Znio= neO.

(2.41)

The ion mass greatly exceeds the electron mass (mi 2 1836 m e ) ,so with sufficiently high frequency waves it may be assumed that the ions remain stationary, in which case

4 = 0,

j = -en,+.,.

(2.42)

By (2.40) to (2.42),

a me d2fi, e2 m e - V . 3, = --__ = -eV E = -fie, at ne0 at2 €0 (2.43) where

(2.44) is called the plasma frequency. This is evidently the natural frequency at which the highly mobile electrons (relative to the heavy ions) move to maintain charge neutrality.

2.2.4

The Debye length

In steady-state conditions (2.4) becomes V x E = 0, permitting the introduction of a scalar potential 4 via E = -V$. Consider the small perturbation due to a point charge at the origin. On assuming the ion density to be unchanged by the charge, we obtain from (2.41)

4

tffv24 = efie,

(2.45)

The force per unit mass is -eE/me = eV(q5/me), and therefore to apply Boltzmann’s distribution law to the present case, we replace $ in (1.93) by -eq5/me. Also, since it is the distribution function for the electrons that is relevant here, we set T = T,, which gives ne = neo

exp(e4/lc,Te).

Hence for a small perturbation, Ce = n,e$/lc,T,.

(2.46)

2 Magnetoplasma Dynamics

42

(2.47) where

(2.48) With spherical symmetry about the origin, the solution of (2.47) is

(2.49) where Q~ is the charge at the origin. This result shows that the movement of the electrons has masked the Coulomb potential Q ~ / Tdue , to the fixed charge at the origin, almost completely in a distance A., This shielding distance is known as the Debye length-it is the scale length below which average charge neutrality cannot be guaranteed.

2.3 Magnetohydrodyamic equations The MHD equations are based on the assumption of charge neutrality, which we showed in the previous section permits us to adopt the simplifications of (2.27), i.e.

? D / C << 1;

<<

w;:/7-

(2.50)

1 1

where C and 7- are the macroscopic length and time scales. Maxwell’s equations now reduce to V x B = poj, 0 - j= 0 , (2.51) and

2.3.1

dB VxE=--, at

Ohm’s law

V*B=O

(2.52)

+

With the Lorentz force, Q(E v x B), and the force g due to gravitation in place of eF in (1.5 l), and adopting charge neutrality from (2.3 l), we obtain the equations of fluid motion for the electron and ion fluids,

+V

dve

e e d t

pi-

dvi dt

+V

*

*

pe = -ene(E pi =

+ v e X B) + Re + eeg +

ene(E vi x B) - Re

+ Qig

i.

(2.53)

where Re is the force density acting on the electron fluid due to collisions with the ions. By Newton’s third law, -Re is the corresponding force acting on the ion fluid. The time derivatives on the left-hand side of (2.53) are the appropriate convective derivatives, i.e. dldt = d / d t v, V, where Q = e, i for the electron and ion fluids.

+

-

2.3 Magnetohydrodyamic equations

43

The time it takes for an electron to be scattered by a sequence of collisions with ions until its velocity relative to the average velocity of the ions is reduced to zero is called the ‘collision interval’, r,i, and its reciprocal v,i = reT1is the ‘collision frequency’. There is also a small contribution due to electron-electron collisions and the resultant collision interval and collision frequency are denoted by re and v,. In Chapter 5 we shall derive expressions for re and other collision intervals that arise in transport theory. By Newton’s second law the average ‘slowing-down’ force on an electron is m,v,(vi - v,), and therefore the total impeding force per unit volume is

Re = n,m,v,(v,

- vi) = eneqj

,

where

(2.54) is termed the ‘resistivity’. A gradient in T, generates an additional force on the electrons, which is known as the thermoelectric force. When the temperatures of the ions and electrons are comparable, it follows from (1.19), viz. m,(icz) = $lC,T,, that the electrons will be moving at speeds about (mi/m,)1/2 times faster than the ions. Hence the ions are relatively stationary and if there is a temperature gradient in the electron gas with the temperature increasing from left to right as illustrated in Fig. 2.3, then the electrons arriving at a given point P from.the right side will be more energetic than those arriving from the left. We shall anticipate the relation nere 0: T,”” between the average electron collision time 7, and the electron temperature T, (see (5.64)). Then the right-side electrons have longer collision times and mean free paths than the left-side electrons, so that on average the former experience a smaller drag force than the latter with the result that the electron gas experiences a net force directed to the left. This thermal force drives the electrons down the temperature gradient, producing an electric current in the opposite direction, proportional to VT,. The thermoelectric force per unit volume is -en,& VT,, where b is a constant. There is a corresponding thermoelectric force on the ions, but this is too small to be of interest. Another effect of the thermoelectric phenomenon is that additional kinetic energy is transported by the electrons as they flow down the temperature gradient. This augments the heat flux vector q by an amount equal to -T,b j. A further point is that in the general case of a magnetoplasma, which is an anisotropic medium, the scalars q and b are replaced by q and 6 * , where 71 and 6 are second-order tensors whose structure will be described in 52.4.3. Hence the total collisional force on the electrons is

-

-

Re = en,(q j - 6 VT,).

(2.55)

The plasma density e and velocity v are the baricentric (centre of mass) properties,

e = @e f Q i ,

ev

=

eivi

+ @eve I

(2.56)

whence from j = ene(vi - v,), and ee << e i , it is sufficiently accurate to write

vi

E

v,

v,

M

v - j/en,.

(2.57)

2 Magnetoplasma Dynamics

44

.-. e

a thermal force on electrons

0-

,-

.,.+a.

0 -0

o-,

i

0-

cg

cg

o1

de

e

heat flux due to electric current

temperature gradient

Figure 2.3: Thermoelectric effect.

The first of equations (2.53) can now be rearranged into the form of a generalized Ohm’s law as follows, r].j=E+vxB--

j x B + %+ 6 . V T , + 2 m dv 2. en, en, e dt

In most practical applications it is usual to omit the very small electron inertial and viscosity terms and (too often) the thermoelectric term. In this text we shall take (2.58) and for the reason to follow, omit the last three terms only if Iwce~,l << 1. Comparing qlj] with lj x Bllen,, we find from (2.35) and (2.54) that unless j and B happen to be parallel, the second of these terms (known as the ‘Hall’ term) is more important than the resistive term if I w ~ ~ >T>~ 1.~ With solar and laboratory plasmas this is usually the case. Also, it follows fromequation (2.62) below that IVp,I may be comparable with Ij x BI, so the term containing this gradient must also be retained if Iwce-re1>> 1. Finally, comparing the last two terms in (2.58) (replacing 6 by 61 for the purpose) we find that b has the form akB/e,where a is a constant. It follows from kinetic theory that a 1 and hence the last two terms may be comparable in magnitude and should be retained together. In a conducting liquid Ohm’s law takes the simplest form, N

q.j =E + v

x B,

which as remarked above, applies in a plasma only if

2.3.2

(2.59) IWceTel

<< 1.

Conservation laws in MHD

When complete details of the individual behaviour of the fluid species comprising a plasma are not required, we may sum the conservation laws to obtain ‘one-fluid’ equations, a reduction that is useful only if the component temperatures T,, LY = e, i have the same value T , and if the quadratic terms in the diffusion velocities (va - v) can be ignored. The usual equations of magnetohydrodynamics (MHD) are reduced equations of this type.

45

2.3 Magnetohydrodyamic equations

Equation (1.45) is valid for both species, i.e. -d + e a+ - ( ( e , v , ) = ~

(a=i,e).

at

Summing over a, by (2.56) we obtain the plasma continuity equation

Be

-++-(ev)

at

=O

(2.60)

Similarly the left-hand side of (1.46) yields

+ V - (evv + P) , provided p is defined by (2.61) where the approximation holds if quadratic terms in the difision velocity are small enough to be omitted. The right-hand side of (1.46) for the plasma case can be obtained by adding the right-hand sides of the equations in (2.53). This yields the equation of motion for a plasma, (2.62)

where the second form follows from (2.60). When the energy equation (1.47) is applied to a single species, it is necessary to add terms to allow for the interaction between the fluid components. One such term that we shall ignore here is the transfer of energy between the species due to the temperature difference (T,- Ti), since there is no point in adopting one-fluid equations if this difference is significant. Another is due to the collisional drag force R, and there are hrther terms if there are mass transfers between the species due, for example, to ionization. We shall assume here that the plasma is hlly ionized. Applying (1.47) to the species P, of a plasma we have

(2.63) B)+ R, + p a p } * V, + 3 , p , where we have added the collisional drag force (-Ra= Re = -Ri)to the Lorentz force =

{ Q,Y.,(E+ V,

X

and 3 & p is the energy dissipated in Pa by the friction between the species Pa and Pp. As Pp moves with a velocity (vp - v,) relative to P, and exerts a force R, on it, the friction force does work at the rate R, (vp - v,), which is dissipated in both P, and Pp. If we assume that the collisions are isotropic and elastic, the dissipated energy will be distributed inversely as the particle masses, i.e.

.

(2.64)

2 Magnetoplasma Dynamics

46

To sum the left-hand side of (2.63) into a one-fluid form we adopt the definitions

eu =

c a

@a%

q = C (9, a

1

eu’ =

c;ea(v, a

(2.65)

-VY,

+ ~ a ( v a- .)[ha + ;(va - v ) ~+]

*

(va - v)} ,

(2.66)

+

where we have introduced the specific enthalpy, h, z u, p a / @ , and expanded p, to p = p a l ll, (see (1.29)). Adopting the diffusion approximation, ignoring the viscosity tensor and using vi x v and ecru, = $pa,we can reduce the heat flux vector to

+

q zz qe

+ qi + eehe(ve -

V)

= qe

+ qi + gpe(ve - V)

or by (2.3 I), (2.67) The term containing j represents the transport of electron energy by diffusion and is sometimes important. (Although we are assuming that the ion and electron temperatures are equal, we have written Te rather than T so that the physical significance of the temperature in a particular context is made clear.) We need constitutive relations for qe and qi. These will be derived in Chapter 5, but for the present we shall take: qe = -ICe

*

VTe

- Te6 j , *

qi = - I C ~ * VTi ,

(2.68)

where the additional term in qe is due to the thermoelectric effect discussed in 52.3.1; the appearance of the same tensor 6 in both (2.55) and (2.68) is an example of the reciprocal relations of irreversible thermodynamics (e.g. see Woods, 1996). The sum over a: of the right-hand terms of (2.63) yields E . j pv * g, hence the energy equation for a plasma reads

+

d dt

-{p(u

+ fw’)} + V - { Q V ( U + fv’) + pav + q} = E . j + e v - g+ C,

(2.69)

where the additional term, C,represents possible contributions to the plasma energy due to radiation.

2.3.3 Lagrangian form and entropy production The Lagrangian form for the energy equation is often more usefkl. To derive this we take the scalar product of (2.62) with v and subtract the result from (2.69). The result is (cf. (1S2)) pDu+ p : Vv = -V

*

- + v x B) + C ,

q + j (E

(2.70)

where D is the convective derivative defined in (1.49). Following the route used to obtain

2.3 Magnetohydrodyamic equations

47

(1.54) and using (2.58) and (2.67) we find

1 -j.Vp,. ene

5 kB +-j.VT,2e

(2.71)

From (1.23), (1.24), and (1.57) applied to the electron fluid, we get meVs, = Ic,(:VlnT, - Vlnp,)

.

where s, is the specific entropy of the electron fluid. Therefore, since V j = 0,

Shortly we will require the constitutive relation for the viscous stress tensor, TI. Here we shall just quote the form of this equation, leaving the details to 55.6. Let u denote the kinematic viscosity (the shear viscosity divided by the density), then TI is given by

n

= -2QUW:Vv,

(2.72)

where W is a fourth-order tensor whose detailed structure (see (5.125)) depends on the parameter w,r. Let es denote the plasma entropy density, then its differential form is given by the lefthand side of (2.7 I ) (see (1 .%)) and with the aid of (2.68), we can arrange (2.7 1) as an equation for the evolution of entropy:

(2.73) where

Tu = 2 ~ u V vW: : Vv

+ V T - (K,+ &)

-

.VT/T + j 7 - j .

(2.74)

Since -m,s, j / e M Q ~ s , v ,this , term in (2.73) is the reversible transport of entropy by the electron fluid. By the Second Law of thermodynamics, in normal circumstances, the rate of entropy production is a non-negative quantity. The condition u 2 0 imposes constraints on the signs of the phenomenological scalars appearing in (2.74), e.g. the viscous dissipation represented by the first right-hand term is positive only if the kinematic viscosity, u, is positive. However, there are circumstances in plasma physics in which the entropy production rate depends on second-order terms (in Knudsen number) in transport and can be negative (see Woods 1996, p. 159). To complete the magnetohydrodynamic equations we need detailed expressions of the constitutive relations for the fluxes j, q, , qi and TI introduced above; these will be derived in Chapter 5.

48

2 Magnetoplasma Dynamics

2.4 Electromagnetic forces Stress tensor and Poynting vector

2.4.1

To preserve generality in this section, we shall initially retain the charge density Q, and then show that the electromagnetic force density acting on a plasma can be written in the form

f

= j x B f QE = V - T - topo-as at

T

1 = -BB + E,EE- (- 1

where

2PO

PO

B2 +

€

(2.75)

E2)1

(2.76)

is the electromagnetic stress tensor and 1

(2.77)

SE-EXB PO

is called the Poynting vector. We require the following vector relations which are obtained by expanding the vector triple products:

-

(V x E) x E = E VE - VE * E = V * (EE) - EV * E - V ( i E 2 ) , and

(V x B) x B = B . VB - VB . B = V . (BB) - v ( p 2 ) .

Then, using the electromagnetic field equations (2.4), (2.7), (2.25), and (2.26), we get

aB - E o a- ( ~ - + -1 (V x B) x B + EOEX at at

f = toEV E

x B)

P O

1 + -(V

= eoEV-E

PO

x B) x B

+ t o ( V X E) x E - t o p oadst .

Equation (2.75) follows since for any scalar 'p we have V (1'p) = V ~ J . The electromagnetic work rate per unit volume can be treated similarly: 1 aE j.E = --E.VXB-E~E.PO

1

= - -V PO =

at

. (E x B) + -B . v x E - to-ata ( $ E ~ ) 1

P O

i a at

-V.S---(iB2) Po

8

-E0,,(fE2),

49

2.4 Electromagneticforces

(2.78) where p = - ~12 + - ~ 2t o1

2PO

(2.79)

2

p being called the electromagnetic pressure. In MHD (2.50) is equivalent to co N 0 and in this case p is termed the magnetic pressure.

2.4.2 Magnetic forces in MHD In MHD(2.75),(2.76),(2.78), and(2.79)become (2.80)

(2.81) One particular advantage of the expression for f is that it enables us to express the momentum equation in divergence form. When the gravitational term can be ignored, (2.62) and (2.80) yield

-8( ~ v ) + V . { ~ v v +

(2.82)

dt

P O

\ lateral pressure

Figure 2.4: Forces acting on a flux tube.

5? --+

50

2 Magnetoplasma Dynamics

Another usefkl form for the force density f can be obtained by separating the magnetic field B = Bb into its scalar and vectorial components. Thus

f = j x B = -BxVxB/p, = -Bb

x {V x (Bb)}/po

= -B2b =

x V X b/po - Bb x V B x b/po

B2&/p0- !jb X VB2 X b/po

where (2.83) is the field line curvature; here s is the distance measured along a field line, n is the unit normal and R is the radius of curvature. Also

b

x VB2 X b = VB2 - b b * V B 2= (1 - bb) * V B 2= V,B2.

j

x B = - 0, (- 1

Hence

2PO

B2)

1 +B2&.

(2.84)

P O

The force (B2/p0)& can be interpreted as being due to a tension B2/poacting along the lines of force, while the perpendicular gradient is due to a lateral magnetic pressure (see Fig. 2.4).

2.4.3 The induction equation To obtain the expression for the rate of change of the magnetic induction, which is known as the ‘induction equation’, we start from Ohm’s law (see 52.3.1) in the form q . j = E + v e x B + - V 1p + d - V T + - - , 2ene

me dve e dt

(2.85)

where we have replaced p , by half the total plasma pressure, p . Now dve dt

-=

where

dv, + v, ‘ v v , at

dve

=-

at

+ !jvv,2 - v, x c, ,

c, = V x v, is the electron fluid vorticity. Hence (2.86)

Therefore by (2.52)1, the curl of (2.85) yields

a m V x ( q - j ) = - - ( B - r C ee ) + V x at

{

Vex

(

B--c,

me e

>>

-

mi -VeXVp, 2ee2

(2.87)

51

2.4 Electromagnetic forces

-

where we have omitted the very small term V x (6 VT). The thermoelectric tensor is dominated by its parallel component in a strong magnetic field and in the weak field limit, 6 is a constant. Because the heat flows more readily along the field lines than across them, in the strong field case b VT is small, which allows us to omit the term from (2.87). An obvious source for the generation of electron vorticity is in the current sheets occupying discontinuities of the magnetic field. At these discontinuities the spin f (see (1.9)) due to the electron fluid motion coincides with the spin -wceb = eB/m, due to the electron gyrations, i.e. 4, = 2eB/m,. In 54.6.5 we shall show that this results in a filamented magnetic field. An alternative expression for V x (77 j) follows from (2.53)2, the curl of which is

-

c,

d (77-j)= - -(B+?<~)+v at

vx

x {v x (B+:c,

)>+zee.Ve x Vp, (2.88) mi

where by (2.57) we have replaced v iby v. At this stage we need expressions for the tensors 77 and 6 , to obtain which the following mathematical note is directed.

Mathematical note 3: Laterally isotropic tensors We shall start by describing transverse isotropy in terms of the unit Cartesian vectors i, j, k and various polyadic combinations ofthem like ii, ikj, jiji, , . .,with k chosen to be the preferred direction. Tensors of second order may be represented as sums over 9 dyads like ab, where a = i, j, or k and similarly for b; likewise tensors of higher order are sums over polyads like abc (third order, 27 terms), abcd (fourth order, 81 terms), etc., where a, b, c, d, . . . = i, j, k. First consider the transformation of a polar vector a into a‘ by a rotation about a unit vector k through a small angle 8. The rotation will displace the point represented by a in a direction at right angles to both a and k. Therefore a’ = a Bk X a. The dyad ab transforms according to

+ (ab)’ = a’b’ = (a + Bk x a)(b+ Bk x b) x ab + B(k x ab + ak x b),

and in general, for small 0,

( a b c . ..)’ = a b c . . .+B (k x a b c . . .+ak x b c . . .+abk x c . . .+abck x

. . .).

If the polyad is invariant under the transformation, then (abc . . .)’ = abc . . ., so that

k x a b c . . . +ak x bc . . . + a b k x c . . . + a b c k x ... = 0 .

(2.89)

Admittedly this condition for the invariance of abc . . . is based on small angles of rotation, but as a large rotation can be divided into a large number of small rotations in each of which (2.89) holds, we must expect this equation to hold generally. Suppose that we reverse the magneto-fluid system P by a 180” rotation about any axis normal to the preferred direction k. Transverse isotropy ensures that changes in the orientation of P in the plane transverse to k cannot affect the constitutive relations, so apart from a change in the sign of B, phenomenological tensors will be invariant under the proposed transformation. Thus choosing a rotation about j, we have

i

-+

-i‘,

j

-+ j’,

k

-+

-k’; C 4 C’ :

C(B)= C’(-B),

(2.90)

52

2 Magnetoplasma Dynamics

where L: denotes a phenomenological tensor of any order. Scalars are unaffected by coordinate transformations, so are isotropic. Among vectors, k itself is the only one satisfying (2.89). The set of dyads possessing lateral isotropy about k satisfy k x ab - ab X k = 0, and it is readily verified that only the dyads kk,

ji-ij=kxl,

(2.91)

ii+jj=l-kk,

meet this constraint. It follows that the most general second-order tensor possessing k-symmetry is the linear combination (2.92)

where the dependence of the coefficients all, a I , and a,, on the sign of B follows from (2.90). If, instead of k, an arbitrary unit vector b is the preferred direction, then (2.92) is replaced by

It follows from the above that the resistivity tensor has the form

and similarly for the other second-order phenomenological tensors, K , , ~i and 6. The anisotropy is due to the effect of the magnetic field on the particle trajectories and its strength is measured by the ratio of the cyclotron frequency w, to the collision frequency, u = 1/r. Thus for the electrons, if (w,,~,) >> 1, the magnetic field is said to be ‘strong’ and the crossfield mean free path (viewed as a displacement rather than a distance travelled) is of the order of the Larmor radius (see Fig. 2.2). The perpendicular and transverse resistivities, v1 and q,,, are functions of w,,~,, whereas the parallel resistivity, v,,is unaffected by the presence of the field with the value given in (2.54). If Iw, T,~ << 1, the approximation 7) M 771 is adopted; >> 1, kinetic theory shows that qL M 2vI1= 277 and q,, E 0 (see in the other limit, Iw, T,~ 55.5.2). For present purposes, to avoid an algebraic complexity that adds little to the physics of the problem, we shall adopt 7) = 71, in which case the left-hand term of (2.88) becomes

v[ x (Vx B) +[v x v x B = VE x (Vx B) - < v ~ B , where E = v/po)is termed the magnetic diffusivity.

While the ion-fluid vorticity may be important in strongly sheared flows, its coefficient

(mile)in (2.88) is very small, allowing us to omit Ci from the equation, giving - = V x (v x B) + @‘B dB dt

-0 6x

(Vx B) + 2 V e x Vp 2e~2

(2.95)

The last term, being independent of B, is called the ‘battery term’; it is quite small and in most circumstances can be omitted. When E can be assumed to be constant and the battery term is

53

2.4 Etectromugneiicforces

omitted, (2.95) takes the form (2.96) the left-hand side of which (equal to D B in the notation of (1.49)) is the rate of change of B in a reference frame following the fluid motion. From (2.60), V v = e d e - l / d t , and therefore an alterative form for (2.96) is

-

(2.97)

2.4.4 Diffusion of magnetic fields The ratio of the first right-hand term of (2.96) to the last is

R.,

E

we/(,

(2.98)

where U denotes a characteristic velocity and C is the macroscopic length scale, which is assumed to be the same for B and v. By analogy with the ordinary Reynolds number UC/u, where v is the kinematic viscosity, R, is called the magnetic Reynolds number. When R, << 1, the term containing J dominates the right-hand side of (2.96), reducing the equation to

dB _ at - [V2B.

(2.99)

In one spatial dimension the equation becomes

one solution of which is

where A is a constant; more general solutions can be based on this formula. Consider a particular point a distance C from the origin. After a time t = C2/4J the magnetic field will have decayed to l/e times its initial value. It is usual to omit the factor 1/4 and take the time-scale for the decay of the magnetic field to be T,

= C2/t = CR,/U.

(2.100)

This is also the time it takes for a magnetic field to diffuse into a plasma that initially has no field, and written as C = (T,J)~/~,gives the distance C that a field will have penetrated a plasma after a time T,, a distance that is termed the “skin” depth. If 5 is very small, i.e. the resistivity is negligible, the magnetic field is unable to penetrate adjacent plasma.

54

2 Magnetoplasma Dynamics

2.4.5 Conservation of magnetic flux

-

With X = B in (2.12) we find that the magnetic flux B n dC through a convected surface element n dC changes at the rate

1

-(B.ndC) d = {O ZB+ V . V B + B V . V - B . V V - n d C =
(2.101)

fL B .ndC = 0 ,

(2.102)

B ndC = const.

(2.103)

L-

and therefore

Hence the flux through a convected surface in an infinitely electrically conducting plasma remains constant. Unfortunately this has become known as the frozen jield condition with the implication that the field lines are somehow “tied” to the fluid and are swept along with it. However, there is no way of labelling field lines, which possess no properties beyond that of direction. All that has been established is that the magneticflux is unchanged through a convected surface; identifiable field lines do not exist and hence they cannot be tied to the fluid. There is some analogy between (2.103) and Kelvin’s theorem, which states that in an inviscid fluid in which the pressure depends only on the density and the body force is conservative (F = V@),the circulation defined by

r=

-

v ds =

-

C n dC

(C = V x v) ,

is constant (see exercise 1.5). However, unlike field lines, vortex lines are identifiable material lines swept along with the fluid, so the analogy can be misleading.

2.5 Magnetostatics 2.5.1

Steady state equations

To obtain the steady state equations we set v and a / d t equal to zero in the MHD equations of 552.3 and 2.4. We shall assume that the body force is negligible and ignore viscosity, in which case from (2.62) and (2.84), Vp = j x B = - V,

1 1 (-2PO B 2 ) + --B’K, Pa

(2.104)

2.J

Magnetostatics

55

plasma

Figure 2.5: Nested magnetic surfaces confining a plasma.

which by V = V,

+ V,,may be written

V ( p + 2 -1B 2 ) = V , , ( r1B 2 ) + - B12 n = - B -1V B P O

P O

P O

(2.105)

P O

We also have the equations

p0j=VxB, V.B=0, V.j=O, and since V

(2.106)

x E = 0, we can introduce an electric potential function Q, so that

qj = E = -V4.

(2.107)

.

-

It follows from (2.104) that B Vp = 0, and j Vp = 0 and therefore B and j lie on constant pressure surfaces, which if closed, appear as continuous windings of intersecting magnetic field and current lines. These are termed magnetic surfaces and p is said to be a surface quantity. Figure 2.5 shows a set of nested surfaces with a limit line at their centre known as the magnetic axis. If p increases towards the axis, its negative gradient is balanced by a j x B force directed inwards and the plasma is therefore confined by the magnetic force. Such confinement is the basis of various devices which may ultimately liberate hsion energy in a controlled process. Three important geometrical configurations will now be described.

2.5.2 The theta pinch Straight and parallel magnetic field lines are generated by the sudden discharge of a current about a cylindrical conductor in the azimuthal or @-direction. Opposing induced currents appear in the plasma and because the resulting axial magnetic field, B,, cannot penetrate the plasma (assumed to have perfect conductivity), it lies in the outer region between the conductor and the plasma (Fig. 2.6). The magnetic force compresses or 'pinches' the plasma until its pressure is large enough to balance this force. As the field lines are straight, IC. = 0 and (2.105) gives p

+ -B:1

2P0

= const..

(2.108)

2 Magnetoplasma Dynamics

56

The field B, is zero within the plasma and p is zero in the magnetic field region, so a balance of forces across the interface requires that (2.109) where now p and B , apply to opposite sides of the pinch surface. metal conductor

induced current

discharge current

\

\

\

/

/

\

Figure 2.6: Theta pinch.

2.5.3 The linear pinch When the directions of the current anL nagnetic field of :.e theta pinc.. are interchanged, i.e. j = j,2 and B = Bee, the resulting configuration is called a linear or z-pinch. With F = Bee in (2.150) of Appendix 1 (p. 67), we get

. 1 a(rB0) Po32 = -___ r

ar

(2.1 10)

’

so that the current flowing in a cylinder of radius

T

is

j z r dr = 27rrBe/po. The field line curvature is

--PIT,

(2.111)

so (2.105) gives

(2.112)

+

If we let T = a denote the edge of the plasma (the radius at which p = Ic,(ni n,)T is zero), then integrating the left-hand side of (2.1 12) by parts we find

2.5 Magnetostatics

57

B.

Figure 2.7: The linear pinch.

where I , is the total current. Hence we arrive at Bennett’s pinch condition,

poi: = 8.rrNkBT,

(2.1 13)

where N is the total number of particles per unit length of the plasma column. Unfortunately, as shall be shown later in 54.3, the z-pinch is unstable and has no merit as a fusion device except possibly at very high densities. On the other hand, unless it carries an axial current, the theta pinch is stable, but plasma losses from the ends are considerable unless eliminated by bending a theta pinch into a torus.

2.5.4 Axisymmetric toroidal equilibrium To derive the equilibrium condition for a toroidal magnetoplasma it is convenient to employ the two coordinate systems shown in Fig. 2.8, namely the cylindrical coordinates ( R ,Z,‘p) and the ‘local’ coordinates (11,, (, 9). The corresponding triads of orthogonal unit vectors are (R, Z, @) and i,@). As we shall only be concerned with axisymmetric systems, alacp = 0 and cp is an ignorable coordinate. It follows from an application of (2.150) to the vector potentiaI A and to the magnetic induction B (set T = R, B = -cp, z = Z ) that

(4,

(2.1 14) (2.115) We define functions $ ( R , Z ) , and F ( R , Z)by

11, = RA,,

F

= RB, ,

(2.1 16)

then from (2.1 14) and (2.1 15) obtain (2.1 17)

2 Magnetoplasma Dynamics

58 Z

t

’R Figure 2.8: Cylindrical and local coordinates.

and

-

dF - d F Rpoj =R- - Z+Rpoj&= V F x @+Rpoj&. dZ dR

(2.1 18)

.

As 8 . V2c, = 0, it follows from (2.1 17) that B 01) = 0, therefore (see Fig. 2.5) q!I = constant is a magnetic surface. Also, as there is no pressure gradient in the toroidal direction, 8 * V p = 0, and since from (2.104) and (2.1 17), 0 = R B V p = V p x V $ * J8, it follows that V p and Vzl, are parallel. Therefore the magnetic surfaces are also isobars, which makes p = p ( $ ) a surface quantity. Turning to the local coordinates, we choose i to be orthogonal to 8 and to lie in the magnetic surface, so that

.

B = B~~+B,G,,

(2.1 19)

where B, and Be are called the toroidul andpoloidul components of B. It follows from (2.1 17) that

RBc = -5.8 x Vzl, = -$*02c,,

(2.120)

4

where is the unit vector i x 8. The function @ ( R 2 , ) is called thepo1oidulJu.q or sometimes by analogy with classical fluid dynamics, the stream function. Similarly

Rpojc =

-4VF.

(2.121)

The equilibrium condition V p = j

=1

x B can now be written as

(jp4.vq!I- -Bp4 1

R P O or since the gradients are parallel to

4, (2.122)

59

2.5 Magnetostatics

By(2.114) to (2.117)

so that 11 satisfies what is known as the Grad-Shafianov equation: (2.123) This is a non-linear elliptic equation, to solve which it is necessary to prescribe functions p ( $ ) and F ( Q ) and suitable boundary conditions for $, usually in the form of a given boundary curve & ( R ,Z)= constant. The periodicity of the toroidal system imposes two integral constraints on p(+) and F ( $ ) . Many solutions have been computed for toroidal fusion machines.

2.5.5 Force-free magnetic fields If the pressure is negligible, then in steady conditions, and without gravity, the equation of plasma motion, (2.62), reduces to j x B = 0, which defines a "force-free'' magnetic field. Hence the vectors j and B are parallel and we can write V x B = aB,where in general a is a function of position. With axial symmetry and no changes along the axis, it follows from (2.150) that 1d v x B = --dB, 8 + --(rBo) dr r dr 1

i = aBee + C Y B , ~ , (2.124)

and therefore (2.125)

+

The solution of (2.125) is of the form Be" B: = f ( r ) , where f ( r ) is a generating function. Given f ( r ) ,the magnetic field components follow from

BZ = - i r f ' ( r ) ,

B: = f ( r )

+ arf'(r),

(2.126)

+

provided f ( r ) satisfies the constraints f'(r) < 0 , f ( r ) i r f ' ( r ) L. 0 . For example the function f ( r ) = B i / ( 1 ( r / r o ) 2 )gives the magnetic field,

+

(2.127) which has the form of a set of nested helical flux surfaces, with the pitch of the helix, Be/B,, increasing linearly with radius r.

2 Magnetoplasma Dynamics

60

Figure 2.9: Force-free magnetic fields with a current sheet.

When the force-free parameter a is constant, it follows from (2.124) that Bessel's differential equation, 1d

dB,

;% ( r F ) + Q ~ B=, 0 ,

B, satisfies (2.128)

ofwhich the solution is B, = Jo(ar); then (2.124)l gives BB= J I ( a r ) . Electric currents tend to flow in streamers since in some circumstances the ohmic heating of the electrons occurs at a rate greater than that at which the electrons can divest their energy. For example, if we assume that temperatures are initially uniform and that radiation losses are negligible, the only mechanism for energy loss is by direct transfer to ions. In this case a thermal instability occurs that heats the electrons up to about T, 15Ti (see exercise 4.14). And as the resistivity is proportional to T;1.5(95.3.4), small temperature perturbations will cause the current to concentrate into a collection of relatively thin streamers. To simplify the case, we shall suppose that these streamers are close enough to form a current sheet, which would correspond to a discontinuity-or at least to a rapid change-in the magnetic field. A suitable function to described rapid changes at x E r/ro = 1 is N

j(x) = B ; { I - (1 - z - 2 ) ~ ( z- I)},

(u(t)=

A + + -arctan(At)) lr

,

where U(t) becomes the unit function in the limit A + 0. Its derivative is the approximate delta function, 6 ( t ) = A/(A2t2 + 1), and the width of the rapidly changing region is proportional to A. From (2.126) we obtain

B,/Bo = (1 - U ( z - 1) - i(z - x-')6(x - 1))' and

,

Be/& = { z - 2 u ( x- 1)+ +(x - z-')d(z - l ) } * . Figure 2.9 shows the B,/Bo and Bo/Bo curves for X = 10 and 100.

2.6

Transition equations across surface layers

61

2.6 Transition equations across surface layers 2.6.1 Surface intensities We will now describe a method of deriving transition equations across a narrow surface layer C by integrating the differential equations over the volume of a small ‘pill box’ lying on it. Boundary conditions for C then follow when its physical properties are assigned, properties that depend on its thickness cr and the time-scale on which it can respond to disturbances reaching it. One can think of C as being a relative discontinuity, by which we mean that its thickness Q is much smaller than any of the intrinsic length-scales retained in the field equations. A surface layer C of thickness a separates regions ( I J and @ of a magnetoplasma and moves with a velocity V, not necessarily the same as the fluid velocity v. Let E ( 6 > a ) be the thickness of a cylinder of cross-section 6C and volume 6V = E SC placed at some point (r, t ) on C (Fig. 2.10), then the sutjiuce intensity +* of a variable +(r, t ) is defined by the limit (2.129)

+

where dr is a volume element and may be a scalar, vector or tensor function. Let V O denote any possible product between and the gradient operator V and write V = V, V,,,where V,,is a gradient lying in the surface C and V, is normal to it. Then by Gauss’ theorem

+

+

+

where n denotes unit normal, the subscripts 1 and 2 refer to the domains on either side of C and it has been assumed that bC is small enough for variations in this element of surface to be

0’

Figure 2.10: Transition layer.

62

2 Magnetoplasma Dynamics

neglected. On choosing the unit normal n to be parallel to n2, i.e. n2 = n and nl = -n, we amve at the transition equation

(v@ $)*

=n@

[+I + v,, @ $*

([+I = $2

(2.130)

- $1) .

To evaluate the limiting integral of d$/dt we first transform to a time derivative 6/6t following the surface, which is assumed to be moving with a velocity V relative to the laboratory frame (cf. (1.49)),

then, since 6/6t is a rate of change in the frame of the surface C, it can be interchanged with the integral over the cylinder, so that by the definition in (2.129), (&+/st)* = 6$*/6t. Applying (2.130) we get

(v.v$)*=v.(v+)*=v-~[+]+v.v,,+*

and therefore

(2.131)

With each variable $ there may be associated length (A+) and time (T+) scales such that $ can only change by a significant amount in a distance C 2 A+, or in a time 7 2 T+. Thus if C << A*, 'T << T+, both $* and [$I must be zero, if C A+, 'T T+, @* will be zero, but [$]may be non-zero, and finally if C >> A+, 7 >> T+, both $* and [$] may be non-zero. N

N

2.6.2 Boundary conditions Boundary conditions are obtained by applying the transitions rules of (2.130) and (2.13 1) to the field equations and then applying physical reasons for setting the surface intensities equal to zero. For example integrating the field equation V x E = - d B / d t , across a surface layer we get SB* (2.132) n x [El V,, x E* = -- + V . n[B] V - V , , B * , 6t which is simply a mathematical transformation of the field equation. To turn it into a boundary condition we need to be able to set at least one of the intensities equal to zero and to do this we need to know the shortest distances over which E and B can change significantly. For E this distance is the Debye length A, (see §2.2.4),whereas for B there are two possibilities, namely the collisionless screening length 6, and the resistive thickness A, defined

+

+

(2.133)

where u,, is the fluid velocity of the electrons perpendicular to the layer. These lengths can be inferred from the equation of motion for the electron fluid, (2.53)1, (2.134)

2.6 Transition equations across surface layers

63

We shall compare the inertial term Qeve * Vv,, the Lorentz term -eneve x B and the resistive term Re - en:q(ve - vi) = n,q j,using the approximation where C is the macroscopic length-scale (V Inertial

Lorentz

N

C-').

Resistive

-

we) and LXfieresistive term is important when Hence the inertial term is important if L A,. The largest of 6, and A, will determine the spatial rate of change of B, therefore

C

N

A, = A,,

A, = max(A,,S,).

(2.135)

It remains to assign a value for the thickness of the transition layer. It is usually the case that A, << A, which leaves the five possibilities for a set out in Table 2.1. Table 2.1: Types of discontinuity

Sizeofa

B*

IB) E* IE1

( @+ not necessarily zero) The choice of a is a matter of being consistent with the field equations. In MHD A, is negligible, but A, is usually not small, so row 4 of the Table would be appropriate, which reduces (2.132) to

n x [El

+ V,, x E* = V . n [B].

(2.136)

The physical origin of E* is an electric dipole of strength T * , which gives E* = n T * , and its contribution to (2.136) can be removed only if V,, = 0, i.e. if conditions are uniform over the surface. In a perfectly conducting fluid, A, = 6,, and as electron inertia is negligible in MHD, row 5 of the Table is the proper choice. This leaves (2.132) unchanged, so no boundary condition can be deduced. Of course the surface intensities could vanish for physical reasons not covered by the length-scale arguments, but unless these can be introduced at the outset, they should not be used to obtain boundary conditions.

64

2 Magnetoplasma Dynamics

2.6.3 Current sheets and surface charge In what follows it will be assumed that (Y is at least small enough to suppress B* (row 4 of Table 2.1). In this case the equation V B = 0 yields

-

n . [B]= O .

(2.137)

From V x B = p oj we get (2.138)

n x [B]= poj*

which makes the condition for zero current sheet the same as that for continuity in B. By Table 2.1 the current sheet j* is zero if A, is large enough to admit the choice cy << A, for the surface layer thickness. If the conductivity in MHD (8, = 0) is large enough, cy A, and j* need not vanish, in which case (2.138) only serves to define its strength. Equation (2.6), viz. N

V . j + -BQ = 0 , at

yields

n.

60' 11+ V,,* j* + - V . ~ [ Q-]V . V , , Q = * 0. dt

(2.139)

While it is true that charge separation is negligible in MHD, if one side of C, say C z , is a perfect insulator capable of accumulating and maintaining a surface charge Q*, we must retain 6 ~ * / 6 in t (2.139). On the other hand if CZ is a perfect conductor, we retain V,, j* unless conditions on the conductor are uniform, in which case (2.139) reduces to

.

(2.140)

n.j=O.

From eoV* E = Q, we obtain eon [El = Q*

,

.

-

(2.141)

since the possible term V,,E* = n V,,T*is zero, n and V,, being orthogonal. If C is a conductor, the surface charge will vanish, leaving

n . [El = 0 .

(2.142)

2.6.4 Fluid equations Turning now to the fluid equations, from (2.60) and the reasonable assumption that the intensities (QV)' and e* are suppressed by viscosity and gas pressure, we get

n

- [QV]= V

*

n [Q].

(2.143)

Similarly (2.82) yields the jump condition (2.144)

2.6

Transition equations across surface layers

65

With a stationary interface through which no fluid passes and on which viscosity is not important (2.143) and (2.144) reduce to

n . V = 0,

n

-n.

-

1

[-BB] PO

(2.145)

=0.

On a magnetic surface n B = 0 and (2.145) becomes

4’1

[P. 1

=0.

(2.146)

2PLo

Finally, there is the equation C = V n x [v] = C*

x v defining the vorticity. The transition equation is

,

in which C’ is known as a vortex sheet. Vorticity is diffused by viscosity and so if C has a thickness less that the viscous boundary layer on a solid surface, C’ = 0, and we obtain the no-slip condition (continuous tangential velocity) (2.147)

nxv=O,

for the surface. Exercises 2. 2.1 Denote by Pc a frame being convected by the fluid so that v is its velocity in the laboratory frame and let p be a charged particle a distance x from P, moving with the velocity k = c . Show that x satisfies 2 = w,k x b,where wc is the cyclotron frequency and b = B/IBI. Show that if B is uniform, c = w c ( x - X) x b cII, (cII = bb . c ) , is a solution. Choose XII so that a = x - X is always perpendicular to b, and show that X = Xll= c11b,ell = 0 and that a = -wCa. Give a physical interpretation of the vector a.

+

2.2 If v is the velocity of a fluid comprised of particles of mass m and charge Q, show that its value perpendicular to the magnetic field is

e

xb

r

E x b - -Vp, 77 B2

where the second form follows from (2.104) and the simplest form of Ohm’s law and D denotes the material derivative. What is the physical meaning of the vector R?

2.3 Show that the induction equation for a plasma can be expressed in the form dB =

dt

B.v~-BV.~+V~(EV~B),

where the magnetic difisivity, (, is variable. A cylindrical plasma has an axial magnetic field B = B, ( T , t ) i ,a radial velocity v = vr(r,t)P and a radial gradient, V = i a/&-. Use the formula for V x F given in Appendix 1 to show that

1

B I , r where the dashes denote radial derivatives. Under what conditions might the magnetic field increase? dB

-= dt

-

(T[B’)’

-

-(TVT)

66

2 Magnetoplasma Dynamics

2.4 Let the heat flux vector be given by Fourier’s Law, q = - K V T . Express the energy equation in theformec,DT+epDQ-’ = --TI :V v + V . ( K V T ) + v j 2 , WhereDisthematerialderivative; show that the lefl-hand side can be written as QTDs, where s = cv In T - R In e const. is the specific entropy. Also show that in isentropic flow

+

( K = const., y = 1

p = K e yexp(s/cv)

+ R/cv = cp/cv .)

In what circumstances does the energy equation reduce to the diffusion equation

2.5 Explain what is meant by the magnetic Reynolds number R,. Show that in liquid conductor a magnetic field with a characteristic length scale L will decay in a time of order p o a L 2 ,where a is the electrical conductivity. Estimate this time for the Earth’s magnetic field (a x lo4 mhom-l) and draw conclusions from your answer. 2.6 An axisymmetric magnetoplasma has a magnetic field (0, Be, B z ) .Show that

and given that the plasma is stationary with negligible pressure, that Be is zero on the axis, and that B, + 0 as r + ro, sketch possible radial distributions of the magnetic fields. 2.7 The vector A represents a physical property that is “frozen” in the fluid. Prove that DA = A . Vv = A . e

+ fl x A ,

where e is the symmetrical part of the tensor Vv. Hence show that the rate of change of A viewed in a convected frame that is spinning with the fluid is e A. 2.8 An axi-symmetric force-free field in cylindrical geometry is defined by p o j z = a B z , poje = aBe, where cy is a function of r. Show that these equations are consistent with poj = V x B if

.

{

(k)’}‘+

rj, = 0,

where the dash denotes the radial derivative, and B, = 0. 2.9 Distinguish between transition conditions and boundary conditions and illustrate your answer by reference to the field equation V x E = - dB/dt. 2.10 The vector potential A is defined by B = V x A. Prove that the electric field vector is given by E = -Vq5 - dA/& , where q5 is the electric potential and prove that on an infinitely conducting, stationary boundary, A satisfies the boundary condition n x A = 0, where n is the unit normal. 2.1 1 Let K = A B; use the result in the previous question to prove that +

_ dK - -2E. B + V . (A x E + Kv - q5B). -

dt The magnetic helicity in a volume V is defined by K = K d r . Prove that in ideal MHD (zero viscosity, thermal conductivity and electrical resistivity) the helicity in a closed flux tube is constant. 2.12 An ideal magnetoplasma is bounded by infinitely conducting walls. Show that subject to the constraint of constant helicity, its magnetic energy density, B 2 / 2 p o ,is a minimum if V x B = aB, where a is a constant (Woltjer’s theorem).

s,

2.6 Transition equations across surface layers

67

Appendix 1: Divergence and curl in cylindrical coordinates

Figure 2.11: Cylindrical coordinates, Let (i,8 , i )be unit vectors in the radial, azimuthal and axial directions, then for a vector F, As i,8,P are independent of r and z, and

di

-

-

a8 ae

-di= o

80-@, -=-F

>a0

'

we find that

(2.148)

+ 22 "a';]

dFr + ii az

-

In order to calculate V F and V pairs appearing in (2.148) and find 18 V F = -(rF,) a

r dr

x F we introduce the operators ' ' and ' x ' between the vector

dFe dF, ++, 7-88 a z

(2.149)

and (2.150) For a scalar 4

-

F = v4 = P-84 +e- 84 dr rd#

+ i-84

dz '

(2.151)

and (2.152)

Physics of Plasmas L. C. Woods Copyright 0 2004 WILEY-VCH Verlag GmbH & Co KGaA

3 Waves in Magnetoplasmas

The propagation of small amplitude waves through a magnetoplasma is a topic of central importance for several practical reasons: first, it offers a relatively simple means of comparing theory with experiment; secondly - granted that the theory is confirmed - there is the possibility of using waves for diagnostic purposes; thirdly, the low-frequency waves are closely related to MHD shock waves of interest in their own right; fourthly, in certain cases waves can be used to heat and accelerate plasmas; fifthly, the growth of unstable waves generates the plasma turbulence believed to have an important role in transport phenomena in plasmas, and finally the waves do occur naturally in the solar system magnetoplasmas. Here our principal concern is with the three basic MHD modes, known as the ‘fast’ and ‘slow’ magneto-acoustic waves and the Alfven wave, and also with the large amplitude shocks corresponding to these waves. Dissipation is usually treated as being a small effect, superimposed on the basic modes, but in this introductory account we shall deal with ideal plasmas in which the viscosity, resistivity and thermal conductivities are negligible. Because of laboratory studies of MHD waves, propagation in bounded plasmas is given a brief introduction. Group velocity is an important concept since the wave energy is transmitted at this speed, so an account of this topic is included. Shock waves are frequently generated in plasmas, especially in solar and planetary plasmas, so we shall describe their main characteristics viewed as discontinuities; for the structure and stability of shocks the reader is referred to larger books. Our account of plasma waves is but a brief introduction to an extensive topic, having a history that started over sixty years ago with the Appleton-Hartree magneto-ionic theory with which the chapter concludes.

3.1

MHD waves in an unbounded plasma

3.1.1

Introduction

At low frequencies MHD waves can be viewed as being an extension of sound waves with the magnetic field adding two more modes of propagation. Wave damping occurs when the plasma has finite (non-zero) values for the viscosity and the electrical and thermal conductivities. We shall avoid this complexity and deal only with ideal plasmas, i.e. plasmas for which the transport coefficients 7 , K , and v (see (2.54), (2.68), and (2.72)) equal to zero. Thus from (2.58) 1 E + v x B - -jxB=O, ene

70

3

Waves in Magnetoplasmas

where we have omitted the gradient terms from Ohm’s law since they would contribute nothing to the linear theory to follow. By (2.5 l), (2.52), and (3.1) the electromagnetic equations are aB _ -VX

at

and

(

p0j=VxB,

(3.3)

V.B=O.

The ideal MHD equations follow from (2.60), (2.62), (2.73), and (1 -60):

pDv

and DS

+ Vp = j x B ,

=0 ,

p = KQ*

(I( = const.).

3.1.2 Linearization Most of the above equations are non-linear. To find the modes of propagation of smallamplitude waves through a uniform background plasma we linearize them as follows. A typical dependent variable, X(r, t) say, is separated into a uniform part XOand a perturbation X1 (r, t), which is Fourier-analyzed into components proportional to exp[i(k * r - w t ) ] ,representing plane waves propagating in the k-direction. The propagation vector k and frequency w are found to satisfy a constraint F ( w , k ) = 0, called a dispersion relation, the determination of which is a principal objective of the analysis. For a given mode

+ 2 exp[i(k.r - w t ) ] ,

X(r, t) = XO

(3.7)

where the wave amplitude 2 depends on the initial conditions; complete solutions are found by summing over all possible modes. Linearization is the omission of all terms that are not linear in the perturbations, for example (3.6)~gives

hence Pl = a 2 &

(

a2

- 5Po).

3e0

(3.8)

’

the last term of which can be omitted. We shall choose a steady state in which vo and jo are zero, so that j x B x j1 x BOand Dv z &,/at = -iwvl. Since V r = 1, for gradients we have VX = XlV [i(k. r - w t ) ] = Xlik Vr = ikXl,

-

3.1 MHD waves in an unbounded plasma

71

hence we can adopt the replacements V + ik and D + -iw . Figure 3.1 illustrates the geometry with which we are concerned. Linearizing (3.2) to (3.6) and using (3.8) (altering the equation order) we have w ; = 0,

w p - e0k.C = 0 ,

1

1

P O

PO

weoS-kka2~+-~okcosBB--Bok~,, = 0 ,

w B + k B o c o s B C - B o k . C - - c okB0 sBj en0

=

0,

(3.9) (3.10) (3.1 1) (3.12)

where B is the angle between Bo and k, the subscript denotes a component parallel to Bo, no denotes nee, and (3.12)1 has been used in the derivation of (3.10). Since (3.1 1) has no component in the k-direction, (3.9) to (3.12) give seven scalar equations for seven independent variables and because they are homogeneous, they will yield nonzero values for these variables only if the determinant comprised of the coefficients appearing in the equations is zero; this constraint is the required dispersion equation.

3.1.3

The dispersion equation

Let z’ = Bo/Bo denote the unit vector parallel to the steady state magnetic field and adopt the variables:

As will be shown below, the scalar b = Ibl, known as the Alfven speed, is the speed at which magnetic field disturbances propagate along the field lines, rather like transverse waves along stretched wires. Two scalar equations involving 8,, and j,,can be derived from (3.1 1).

Figure 3.1: Propagation vector.

3

72

To describe the fluid motion, the fluid vorticity function and we therefore introduce the variables c=ikxC,

tII=Z.t,

C

j,=Z.ikxc,

= V

Waves in Magnetoplasmas

x v proves to be a convenient

3=-k.+sinO.

4

Two more independent scalars are required; for (3.9) we shall use 2 and = fikasinO/eo, the entropy wave, g = 0, being included for completeness only. The seven equations are now arranged in a matrix form involving the seven independent variables defined above, together with the speeds: (3.13) and the parameters (3.14) where u is the phase velocity, a is the adiabatic sound speed and S, is termed the ion inertial length. In terms of the wave length, X = 27r/k and the ion-cyclotron frequency, w,i = e B / m i ,

X = 27rbcosO(Si/X) = (b2/u)cosO(W/u,i),

(3.15)

where we have used (3.13), (3.14), and PO = mino. With the variables defined above, the equations can be written in the symmetric matrix form:

u o o

0

0

0

0

O u a

0

0

0

0

O a u

0

b,O

u b,O O O b , b , u X 0 0 0 0 X u 0 0 0 0 0 b , 0

0

0

0

0

(3.16)

O

b, u

Let (i,j ) denote the i-th row and j-th column of the matrix in this equation, then we identify the leading sub-matrices by the values of i and j at their upper left-hand comers and their lower right-hand comers, e.g. the sub-matrix ( 2 , 4) extends from the element (i = 2 , j = 2) to the element (i = 4, j = 4). The waves are classified as follows: Sub-matrix Wave-type Eigenvectors Phase Velocity (1,1) entropy B 0 (21 3) acoustic itk . 8 &a (2,5) magneto-acoustic 6, k +, Gll, ill CSl Cf (6, 7 ) Alfvknic el, fbx (2,7) coupled plasma waves as above Us, VI, V f

-

4

1

3.1 MHD waves in an unbounded plasma

13

where c, and c j are known as the slow and fast magneto-acoustic speeds and are defined in (3.19) below and w,, wI,w j are the roots of (3.17). It follows from the form of the matrix that unless b, = 0, i.e. 6' = 0, pure acoustic waves cannot exist in the presence of a magnetic field and that the magneto-acoustic and Alfvenic branches are decoupled if X is negligible. From (3.15) this occurs when either 6' = 90" or & / A << 1 ; w/wci << 1. The inequality & / A << 1, or equivalently w/w,i << 1, is usually taken to be the constraint that defines the MHD wave regime.

3.1.4 MHD waves Returning to (3.16), the condition that the variables have non-zero values is given by the vanishing of the determinant of the matrix, which yields

+

(a2 - b;) [u4 - (a2 b2)u2+ a2b:] =

2

(2) b;b2(u2 - a z ). Wci

(3.17)

There is also the entropy wave, u = 0, which is no more than a local temperature perturbation that is convected with the fluid. In the MHD regime the right-hand side of (3.17) is zero, in which case the phase velocities are given by u=o,

u=&c,,

(3.18)

u=+cCf,

?A=*!),,

where c2f = 12 ( u2

4

+ b2)

-

+

${a4

+ b4 + 2a2b2(1- 2 c o s 2 0 ) } f .

(3.19)

Notice from the list of eigenvectors that the fast and slow waves produce perturbations in B and v lying in the plane containing Bo and k, whereas for the Alfvkn waves the perturbations are confined to the normal to this plane. From (3.17) in the MHD regime

u4 - (a2

+ b 2 ) U 2 + a%:

=

(2- c ; ) ( u ~- c;)

=0

)

whence (u2 - a2)(u2- b:) = u2b2sin2 6' 2 0 ,

and it follows that the magneto-acoustic velocities satisfy c,

I min(a, b,) 5 max(a, b,) 5 c f .

(3.20)

The equality signs hold only for waves propagating along the field lines, i.e. for 0 = 0, in which case

c, = min(a, b) ,

c f = max(a, b ) .

(3.21)

3

74

Waves in Magnetoplasmas

3.2 Coupled plasma waves 3.2.1 High frequency waves then the In order of increasing magnitude, let the solutions of (3.17) be fus, fu I , fuf, equation can be expressed

(u -c,)(u

w2 b4b2sin28u2 w2 b2(u2 - a 2 ) 2 -2 0, (a2 - b 2 ) 2 wEi u2sin2o

-Cf)=2 WCi

(3.22)

which shows that us and ufsatisfy us

I cs I C f I U f ,

(3.23)

i.e. the term on the right-hand side of (3.17), known as the Hall term, reduces the speed of the slow wave and increases the speed of the fast wave. We may therefore adopt the terminology slow,intermediate and fast to identify the three wave modes regardless of frequency. For certain values of the parameters the phase velocity may tend to zero or to infinity; the first case is called a resonance and the second a cut-of. Setting u = 0 in (3.17), we find that the slow wave has a resonance if w2

= w;i

C O S ~8

.

(3.24)

At w/wci M 0, us x cs, and as w/w,i increases to fcos 8, us decreases to zero. In the neighbourhood of this resonance the slow wave is termed the second ion cyclotron wave; at frequencies exceeding w,i cos 8 this wave does not exist. Convenient non-dimensional variables for the dispersion curve coordinates are w/w,( and k&. As b = wC&, when non-dimensionalized, (3.17) depends on two parameters, the angle 8 and the ratio of the plasma pressure to the magnetic pressure, which is known as the plasma (3.25) Typical dispersion curves for a low P hydrogen plasma are drawn in Fig. 3.2. The usual MHD range of frequencies has w 5 w,i and does not extend beyond the resonance at w = wci cos 8. However, for completeness the curves have been extended beyond the range of validity of (3.17) and show a high frequency resonance at w = w,, cos 8. At low frequencies the intermediate wave has the phase velocity b, and is called the Alfvtn wave. The zero in the denominator of the second expression in (3.22) is matched by w/wci x 0 in the numerator, holding the left-hand side of the equation at a finite value. As w/w,i increases, the intermediate wave speed, vl,increases if a > b, and decreases if a < b,, since it is evident from the third expression in (3.22) that at high frequencies (w/wci >> 1) vI must be asymptotic to the sound speed a in order to hold the left-hand side of the equation at a finite value. Since the equation represents a cubic curve (in u2),the transition between the tangent u = b, at low frequencies to the asymptote u = a at w/w,i >> 1 is monotonic. In the special case, often found in laboratory plasmas, where a << b, (3.17) reduces to (3.26)

3.2 Coupledplasma waves

75

lo4 lo3

lo2

w -

mi 10 1

0.1 0.01

0.1

1

10

lo2

lo3

lo4

k 6i

Figure 3.2: Dispersion curves. fast

0 1 fast MHD 1I1 whistler If 111 electron cycl.

intermediate

0 1 Alfven I I1 1st ion cycl. I1 111 acoustic

slow

0 1 slowMHD

I I I 2nd ion cycl

Near w = wCi the roots are

(3.27) The fast wave is hardly affected by the resonance experienced by the intermediate wave at w = wci, but this is not a true resonance since V, >> a. Near w = wci the intermediate mode is called thefirst ion cyclotron wave.

3.2.2 Whistlers At high frequencies (3.26) has one positive root for u2,namely bZbw/wci,or Vf =

b2 -kcosB, Wci

(3.28)

and one negative root corresponding to the non-propagating slow wave. Waves of the latter type, which have large imaginary values for either w or k , are said to be evanescent. If generated at a source, they would be completely damped in one or two wavelengths by kinematic rather than dynamic processes. As remarked earlier, at these frequencies the intermediate wave is an acoustic wave. Fast waves with the speed given in (3.28) are known as whistlers or helicons, the first name deriving from the characteristic whistle heard at receivers detecting such waves in the

76

3

Waves in Magnetoplasmas

geomagnetic field, while the second comes from the fact that the end of the B vector traces a helix. As whistlers are the fastest plasma waves, they are important in shock wave theory. Fast shock waves (see 53.5.3) have whistler waves standing upstream as part of their structure; observations show that the Earth's bow shock has a thickness of the order of the whistler wavelength (see Woods 1987).

3.2.3 Propagation of wave fronts Because a magnetoplasma is anisotropic, a disturbance propagating through it from a point source may have a non-spherical wave front W ,in which case the tangent plane to W will not be normal to the phase velocity vector ku,where 1;is unit vector parallel to k. In the following we shall be concerned only with the low frequency (MHD) waves. For present purposes a polar diagram, known as a hodograph, in which the radius T is proportional to the speed and the angle 0 specifies the wave direction, provides the best representation of the dispersion relation. Using the speed (a2 b 2 )4 to non-dimensionalize the hodograph, we find from the magneto-acoustic dispersion relation,

+

+

+

u4 - (a2 b2)u2 a2b: = 0

that

T4

-

T

3

+ +r2cos2 6 = o ,

where U

(a2

+b2)f

(3.29)

r=-a22ab + b2

'

The hodographs for the magneto-acoustic waves are therefore the curves = i[+{ 1f (1 - r2cos20) f

13 3 .

(3.30)

The Alfven wave hodographs are the circles (actually spheres) defined by r=*

b

(a2

+ b 2 ) f cos 0 .

(3.31)

Figure 3.3 illustrates these curves for the choice r = 0.94 with the ratios, (i) a/b = 1.45 ( p = 2.54) and (ii) a / b = 0.69 ( p = 0.57). The vector 1; drawn from the origin in a given direction intersects the curves at three radii, proportional to the values of c, , b,, and cf for waves propagating in the direction k. Notice that only the fast wave is able to propagate across the magnetic field (0 = 7r/2). The plane C shown dotted at P(r, 8 ) on the fast wave hodograph is orthogonal to k and in the immediate neighbourhood of P represents a plane wave propagating outwards from a point source at the origin. Let ( R , 4)be the polar coordinates of a point on C, then its equation is

Rcos(4 - 0) = r ( 0 )

(3.32)

where 6 is now in a parametric role. As 0 varies, (3.32) defines a family ofplanes, the envelope to which is obtained by finding the locus of the points of intersection of consecutive planes in

3.3 MHD waves in cylindrical plasmas

77

Figure 3.3: Hodographs or Friedrichs diagrams for MHD waves.

for (i) a / b = 1.45, (ii) a / b = 0.69.

+

the limit as the difference in their parameters, say 8 and 8 68, tends to zero. This envelope of plane waves is called the wavefront for obvious physical reasons and to find it we eliminate 8 between (3.32) and its derivative with respect to 8, namely Rsin(q5 - 8) = d r / d 8 . We find that r2sin 28 (3.33) Rsin(4 - 8 ) = 8r(2r2 - 1) ' Solving (3.32) and (3.33) for R and q5 and making use of (3.29) to simplify the results, we arrive at equations defining the wave front due to a point source at the origin; in parametric form these are:

R2 = r2 +

r4sin2 28

6dr2 (1 - r2 cos2

e) '

T4

t a n 4 = T4 - lr2 tan 8 ,

(3.34)

4

where r ( 8 ) is given by (3.29).

3.3 MHD waves in cylindrical plasmas 3.3.1

The dispersion relation

Experiments involving MHD waves are most conveniently undertaken in cylindrical magnetoplasmas, with the steady magnetic field Bo parallel to the axis of the cylinder OZ. The waves may be generated in a plane orthogonal to Bo by an oscillating discharge between a ring electrode and a ball electrode on the cylinder axis, which produces an azimutha1 force on the plasma. The torsional waves generated in this way propagate along the cylinder, which acts as a wave guide. The speed and damping of the waves can then be measured from the responses from magnetic probes spaced along the cylinder. The theory is algebraically complicated, requiring a knowledge of Bessel functions, so here we shall just give an introductory account. Full details can be found in Woods (1987).

I8

3

Waves in Magnetoplasmas

In cylindrical coordinates ( T , 8, z ) a typical variable X ( T ,8, z , t ) is divided into a steady uniform component X Oand a wave-like perturbation thus: ~ ( rB,, 2, t ) = XO 2 ( r )exp i ( m ~k z - w t ) , (3.35)

+

+

where for continuity of 2 the azimuthal wave number m must be an integer. Because of the radial dependence of 2 ( r )we must retain the operator V in the perturbation equations and not make the replacement V --+ ik as in $3.1.1. In the following we shall adopt the independent variables,

l j = $/&lo, h = B/Bo,

=j/p&

=

v x h, t = v x 9 ,

and the parameters kb = w / b , R = w/w,i, P’ = a2/b2,where a is the sound speed and b is the Alfvkn speed. Restoring V in place of ik in (3.10) we get “9 + i a 2 V $ - i b 2 ( Z *V h - Vh,) = 0 ,

(3.36)

where Z is unit vector parallel to Bo and h, = I * h. From (3.9)2,$ = -iV 9 / w , and since by (3.35),Z*V -+ ik, (3.36), its curl and its divergence are

+ iwP’V$ + i w V h , + w k h

ki+

= 0,

(3.37)

k z t + w k j = 0,

(3.38)

(kt P‘V2) 4 + V2L, = 0 .

(3.39)

+

and

We shall also require the curl of (3.38), which by

vx is

j =v x

kzV x

v x fi = vv.h-0%= -026,

t =wkV2h.

(3.40)

A similar treatment of (3.1 1) gives

k 9 - wZ4 and

kt

+wh +Re = 0

(3.41)

+ w z x v7j + w j + wk -RV2h kb2

=0,

(3.42)

where (3.38) has been used in (3.41) and (3.40) has been used in its curl. In the present case there is one less equation since the divergence of (3.41) is zero. Finally we take the Z components of (3.37), (3.38), (3.41), and (3.42), then eliminate tZ and 5, to arrive at the three scalar equations contained in

I

(k? +P’V2)

- (k,2 - p’k2) 0

v2

0

k; kRV2 kz - k2

(3.43)

3.3 MHD waves in cylindrical plasmas

79

Now we seek solutions in which each dependent variable satisfies the reduced wave equation

v2x= -ax,

(3.44)

where the eigenvalue LY is determined by the condition that the homogeneous equations possess non-zero solutions. This condition is obtained by the replacement V2 + -a in (3.43), and then equating the determinant of the resulting matrix equal to zero, we obtain the quadratic (3.45)

+

It proves convenient to write a = k2 k z , where k , is a constant to be chosen so that appropriate boundary conditions on the surface of the magnetoplasma are satisfied. From (3.35), (3.44) and the Laplacian operator given in (2.152), we find that 2 satisfies the m-th order Bessel equation, 1d d k m2 - ( 7 - k:)k = o (3.46) r d r dr As r = 0 lies within the plasma domain, we need a solution that is finite at the axis; this is

- _r-

k(r)= d J , ( k , r ) ,

(3.47)

where d is a constant and J m ( k c r )is the m-th order Bessel hnction of the first kind. Since (3.45) is a quadratic with roots aq,az, the general solution will be the sum of two terms like that in (3.47), with constants dl and d2 and corresponding radial wave numbers k,l and k,2. The boundary conditions from which the values of the radial wave numbers are obtained can be quite complicated, so this aspect of the problem will be omitted; for the present we shall assume that k , is known.

3.3.2 Fast wave cut-off In a weak magnetic field kb and R are relatively large, in which case the O ( k t ) terms in (3.45) are dominant and give k2 = ( w / u )-~ k:, so the phase velocity is i.e. a sound wave cut-off at frequencies below k,u. In the strong magnetic field limit the dominant terms occur in the coefficient of a and yield another sound wave w p = f a . There is no cut-off because the strong field confines the waves to be entirely longitudinal, which removes the influence of the boundaries. In the limit p’ --+ 0, (3.45) reduces to a quadratic in k2 with the solutions k 2 =-- k? 1-R2

-kz 1 f 2 2(1 - R2)

(3.48)

where the positive sign applies to the intermediate wave and the negative sign to the fast wave (cf. (3.26) with b, = b, which corresponds to the case k , = 0 of (3.48)). The intermediate wave has a resonance ( k = 00) at R = w/w,i = 1, while the fast wave is cut-off ( k = 0) at kb = k,, i.e. at w = kcb, which properties are illustrated in Fig. 3.4.

80

3

Waves in Magnetoplasmas

Figure 3.4: Fast wave cut-off.

3.4 Group velocity 3.4.1 Transmission of energy The group velocity vg is the velocity of propagation of a group or 'packet' of waves within a narrow frequency range. The concept has particular importance because the energy in the waves is propagated at the group velocity. In this section we shall give a general account of the topic and later apply it to MHD waves. We start by showing that progressive plane waves transmit energy in a direction along unit vector 6 at the speed 4 * vg,where vg

=

dw

=

dF/dk

dF/dw

(3.49)

7

in which F ( w , k) = 0 is the dispersion equation. Let C be the plane passing through the origin with unit normal fi, then on C, fi vg = 0. The velocity of energy transport across C is Q / E , where E is the local energy density and Q is the energy flux across C. The perturbation in a typical variable X is

.

XI= k e x p i { k . r - w ( k ) t } .

(3.50)

Assume that as we proceed from the origin along the normal 4, a non-conservative force slowly attenuates 2 by a factor exp(-&q r), where q = 1771 5 Ikl. The energy density ~ ( 0 ) at the origin is proportional to the square of the wave amplitude,

-

E(0) = a x 2

( u = const.) ;

-

hence at a point P(r) in the region 4 r 2 0

c(r) = a$'

exp(-q. r) .

The total energy in this region is therefore (3.51)

81

3.4 Group velocity

-

i j r being the distance measured normal to the plane. The attenuation is equivalent to changing k in (3.50) to k i i v and so w must also be altered, the relation w(k) being unchanged with weak attenuation. Thus w is replaced by w &iv * (dw/dk), which in turn modifies the inertia force acting on each particle of the system from -mi: = mw’r to

+

+

This change amounts to an additional force

acting on the particles. Since the waves in i j r > 0 are attenuated and yet have a fixed total energy d ’ / q per unit area, the rate of working of these additional forces must be balanced by the energy Q crossing C, i.e.

where

T = iCmi-’

-

is the total kinetic energy per unit area of the region i j r > 0. In progressive waves the average kinetic and potential energies are equal, i.e. 2T = E , whence by (3.51)

-

The speed at which energy is transported across C is Q / E = i j vg,from which it follows that the wave energy is propagated with the. group velocity.

3.4.2 Wave packets Now consider a wave packet. For a spread of wave numbers k, (3.50) is replaced by the Fourier integral x1=

-

l,,k

(3.52)

A(k) exp{i(k r - wt)} d k ,

+

where the amplitude function A(k) is negligible outside a small range k’ - 6k < k’ < k’ 6k say. The expansion

k - r - w t = k’. r - w‘t

+ ( k - k’) - {r -

(2,”)+

... ,

where w’ = w(k’), enables us to write (3.52) in the approximate form XI

= Xexp{i(k’

- r -w’t)} ,

82

3

Waves in Magnetoplasmas

Figure 3.5: A wave packet.

where

1

k’+bk

X=

k’-bk

A(k) exp{i(k - k’) (r - vgt)} d k .

The wave amplitude is constant on the surfaces r - vgt = const., i.e. the wave packet is propagated with the velocity vg (see Fig. 3.5). Since dkldk is the unit vector i; and w = kv,, (3.49) can be written

~ , = k v , + k -dv, . dk

(3.53)

In an isotropic medium v p is independent of the direction of propagation, so

dVP - Edv, ,l;d”. dk dk dk dk and vg

=k(VP

+ k-)dv,

dk

’

= ‘((v,

-

(3.54)

where X (= 2.rrlk) is the wave length. If dv,/dX > 0, a group of waves progressively spreads out or ‘disperses’, with those of the longest wave length moving ahead. In this case vg < v p and we have normal dispersion. In the opposite case, ug > zip, and the dispersion is said to be anomalous.

3.5 Shock waves 3.5.1 Jump conditions across an MHD shock On a sufficiently large length-scale a shock appears to be a thin transition region across which the macroscopic variables leap from their ‘upstream’ values to their ‘downstream’ values, ‘upstream’ being the direction from which the plasma is flowing. We shall take the

3.5 Shock waves

83

shock front to be planar and orthogonal to the OX-axis and choose the flow direction so that z = -co is upstream and z = co is downstream. Strictly the shock extends from one limit to the other, but on a macroscopic scale almost all the change occurs in a thin region of thickness A, which is typically only a few mean free paths. In the following we shall take the shock thickness to be negligible and deal only with the relations between the variables upstream of the shock and those downstream of it. In the classical shock-wave theory for a gas such as air, thesejump conditions are called the Rankine-Hugoniot relations; in MHD they are known as the Hoffmann-Teller relations. Unlike gas-dynamic shocks, in MHD several types of shock waves are possible, although some of them are found to be unstable to small disturbances. In a reference frame fixed in the shock front, the plasma flow will appear stationary and allow us to omit the time derivatives from the basic MHD equations. We shall also omit acceleration due to gravity and plasma heating due to radiation. Thus in steady conditions (2.60), (2.82), (2.69), and (2.52) are: v-QV =

0,

v . { ~ ~ + ( ~ + -1 B ~ ) I + I T 1- - B B=} 0, 2P0

PO

V.{ev(u+iv2)+p.v+q+-E 1 xB} PO

= 0,

(3.55) (3.56) (3.57)

and

V-B=O,

VxE=O,

(3.58)

where (2.78) is used in the derivation of (3.57). The constitutive equations appropriate to the uniform states upstream and downstream of the shock wave are TT = 0, q = 0, and

E+vxB=O.

(3.59)

The shock front S lies in the OYZ plane as illustrated in Fig, 3.6; the subscripts 1 and 2 will be used to denote values at z = -co (upstream) and at z = co (downstream) respectively. We shall also adopt the notation

[XI =

X2

-X1,

(X)= i(X2 + X I )

so

[XYI = (X) [YI+ [XI( Y 7)

and

(XU) = a[X][Y]+(X)(Y).

(3.60)

It will be assumed that the shock is uniform, so that the origin 0 can be chosen to be at any point on it. In the notation of $2.6.1,as V,,is zero, (V 0$J)* = n 0 [$I. Hence (3.55)

3

84

to (3.58) yield the jump conditions

n . [ p ] = 0, n . [_ow+ (p+-B2)1 1 2Po

n . [B]= 0,

- -BB] 1

= 0,

Po

I

Wavesin Magnetoplasmas

I

(3.61)

n x [El = 0,

+

where n is unit vector parallel to OX and h = u p / Q is the enthalpy. From (3.59), n x E = v,Bt - B,vt, where t denotes the component lying in the shock plane S. Suppose that vt is measured in a frame F , whose origin lies in S. Let F’ be a frame with axes parallel to F and whose origin 0’ moves in S at a steady velocity Ut relative to 0. Provided B, # 0, we can choose Ut such that in F’, the vector n x E’ is zero and hence v{/w,= B,/B,. By the last of (3.61) this relation will hold on both sides of the shock. It follows from wh/wI = B,/B, and (3.59) that Ek is also zero. Finally, by rotating F‘ about O’X’, a frame F“ can be found in which w: and B: are zero. Suppressing the dashes, we have now found a frame in which

E = 0,

W, = 0 ,

B, = 0 ,

w,/w, = B,/B,,

(3.62)

on both sides of the shock. Notice that the velocity and magnetic field vectors are now parallel. When B, is zero, the discontinuity is called aperpendicular shock. In this case the reduction just described is not applicable, so these shocks must be considered separately.

Figure 3.6: The shock front.

3.5 Shock waves

85

[v,] - m[v] = 0 m[v,I

1

+ [PI+ - P y ) [ B y l P O

1 m[vy]- -B,[By] Po

~ [ v Z+v;] + [h] B+[UYl -

m[vI(By) - m ( 4 P y l

0

=

0

=

0

=

0

= P

mv+(p+-B$)n--B,B 1 1 2PO

=

PO = H

$?+h

' (3.63)

1

1

I

,

(3.64)

J

where B,, m, P, and H are constants and the left-hand sides are evaluated at either the upstream or downstream states. With the help of the state equations,

pv=RT,

h=-

7-1

4

RT = RT,

(3.65)

we may express (3.63) as five homogeneous equations in the five jumps [v],[TI,[v,],ivy], and [B,]. If one of these jumps is specified, together with the upstream values of p, T , B,, and By,then the downstream values can be determined.

3.5.2 Thermodynamic constraint The entropy must increase across the shock wave (see 52.3.3) and to make use of this constraint we need the relationship between the thermodynamic state variables on each side of the shock. In gas dynamics this is known as the Hugoniot relation. On multiplying (3.63)s by (v,) and (3.63)4 by (v,), adding and using v, = m v ,we get

From vyBx = v,By and (3.60)s we find

t [%I [By]+ (%)(By) Using these formulae to eliminate [vZ + v$] from (3.63)4, we obtain the required thermody(VyBz)

= (Vy)Bz =

(VzBy)

=

'

namic relation,

[h - ( 4 P ]

1 +[v][By] = 0. 4P0

(3.66)

3

86

Waves in Magnetoplasmas

By (3.60) and (3.65)

therefore (3.66) can be written (3.67) By u = l/e and (3.60) the relation can be written in an explicit form as

(3.68)

el (3.69) From (1 S8)

+

s = c, l n ( p ~ - ~ / ~const., )

hence for the entropy to increase across the shock wave we need

:2(fy.

(3.70)

From (3.69) this requires that

for r in the range r 2 0. Since dF/dr 2 0 for r L 0, F ( 0 ) = -00, F ( l ) = 0 , and F ( m ) = 00, it follows that F ( r ) 2 0 if and only if T 2 1. Hence the thermodynamics constraint is satisfied ifp2 2 pl, i.e. the shock waves is compressive. Including the constraint on the density ratio implied by the denominator of (3.68), we have Pl 5 P2 7

el I ez I 4e1.

(3.71)

There is no essential loss in generality by assuming that u', and B, are positive and then (3.63) and (3.71) yield the inequalities:

[PI and

either [vY]

2 0,

2 0,

[el 2 0,

[By] 0,

[v] 5 0, or

[vy] 5 0,

[%I

5 0, [By]5 0.

(3.72)

3.5 Shock waves

87

Classification of MHD shocks

3.5.3

Omitting the second and fifth equations from (3.63), we are left with

Hence for non-zero jumps the mass flux must satisfy (3.73) Let m; and m:, where m; L m:, denote the roots of (3.73), then since by (3.72), [p]/ [v] _< 0, it follows from (3.73) that (3.74) The velocity u,1 of the plasma into the stationary shock front has the same magnitude as the velocity -us at which the shock would propagate into stationary plasma, i.e. 21x1 = -us,

(3.75)

and therefore when the shock is very weak, v, will equal the speed of one of the low-frequency MHD waves. This can be verified as follows. For weak shocks we can write [v] M bv, [u,] M 6u,,. . . , where 6v, 6vz,.. . , are small compared with (v) w v, (u,)M v,, . . .. Thus

where a is the sound speed. Also by (3.60),

hence (3.73) becomes U: -

(a2

+ b2)u: + a2bz = 0 ,

(3.76)

the roots of which give the slow (c,) and fast (cf)magneto-acoustic speeds. Similarly (3.74) reduces to (3.20), viz. c,

5 min(a, b,) 5 max(a, b,) 5 cf .

(3.77)

3

88

Waves in Magnetopiasmas

0 BY2

B,

shock front

Figure 3.7: Fast shock wave vectors.

Hence in the weak shock limit, the roots m,, m f correspond to the speeds c,, cf,making it natural to adopt the terminology slow (m,)andfast ( m f )shock waves. From (3.63)s and (3.63)s we obtain (3.78)

(3.79) In the rest of this section we shall adopt the convention that Byl is non-negative, allowing shocks to be classified according to the value of By2. Equation (3.79) shows that By2 is positive provided that vxl,vx2 are either both ‘super-Alfvknic’ vast shock) or both ‘subAlfvknic’ (slow shock). Those shocks for which By2 < 0, v,1 > b x l , and vX:! < bx2 are termed intermediate shocks, but it is unlikely that such shocks occur in Nature (Woods 1987, p. 368). Switch-on and switch-of shocks can occur in certain conditions; these appear as the limiting cases Byl= 0 of the fast shock and By2 = 0 of the slow shock. The limiting intermediate shock occurs where Byz = -Byl, or (By) = 0, for which case (3.73) gives the root m: = b$/po(v).The velocity and magnetic field vectors are parallel on each side of the shock and as B, is constant, [vx] = 0. Hence by (3.63)1, [ Y ] is zero, i.e. there is no density change across these shocks, and their speed is the same as the Alfvkn speed, (3.80) These discontinuities are called transverse or sometimes Alfvdn shocks. Shocks with BY2< -Byl are not thermodynamically possible since they have [vx] > 0 (see (3.72)).

3.5 Shock waves

89

shocks

,'/'

,' slow /

,,'

shocks

', \

;,/

switch-off shock

Figure 3.8: Classification of shocks.

In Fig. 3.7 is shown velocity and magnetic field vectors for the fast shock. A similar figure for the slow shock would show these vectors rotated towards the normal. Figure 3.8 indicates possible shock types for a given upstream B1, with B,1 # 0. If Byl= 0, we can either have aparallel shock, with no disturbance of the magnetic field, or a switch-on shock, as shown in Fig. 3.9. For the parallel shock (3.73) gives (3.81)

As in classical gas dynamics, these shocks can occur only if vxl is supersonic and subsonic.

vx2

is

The general theory of shock wave calculations is algebraically complicated so we shall leave the subject at this stage, referring the reader to the account in Woods (1987) where further references can be found. There also remains the important question of the stability of MHD shocks, which can be tested by considering their reactions to an impinging disturbance. For a general account of this subject the reader is referred to the text by Jeffrey and Tanuiti (1964).

3.5.4 Perpendicular shock waves When the magnetic field has no 3, component, we have the perpendicular shock illustrated in Fig. 3.9. These relatively simple shocks are important because they have been investigated in laboratory plasmas much more than any other type and their thickness provides an important check on cross-field transport of momentum and energy. For these shocks we may choose an origin such that vy and v, are zero, and then rotate the plane about OX until B, is zero, then (3.62) are replaced by E = -v,Byz', v = v X 2 ,B =

3

90

I

Waves in Magnetoplasmas

Bz

-

switch-on

v2

-perpendicular shock shock

(v parallel to B)

(V perpendicularto

B)

(b)

(a)

Figure 3.9: Special types of shock:

(a) parallel and switch-on shocks, (b) perpendicular shock. by?, where usBy is constant. Applying (3.61) to this case, we arrive at (cf. (3.64)) pv, = m ,

uzBy = -Ez,

I (3.82)

and

$ u z + h + - B; = H, PO e

where m, E,, P,, and H are constants. As will become clear below, in actual magnetoplasma shocks it is important to distinguish between the electron and ion temperatures, so at this point we shall depart from MHD theory by writing

(3.83) The following non-dimensional variables will be adopted (omitting the subscripts z, y on

u, and By):

(3.84) (3.85) The parameter b varies from unity upstream to b2 = B2/& downstream of the shock; the shock strength is measured by E = bz - 1. On eliminating the constants from (3.82) in favour of upstream values, we arrive at the set of

3.5 Shock waves

91

non-dimensional equations:

bw = 0, M2iW2

-W(Mi1

+ 9 1 + 81 + d - $b2) + 'p + 0

= 0,

(3.86)

and It follows that (3.87) (3.88) The ratio MA^ is known as the Alfikn Mach number.

,

Ti . /

0

distance

o x

Figure 3.10: Typical temperature profiles in a perpendicular shock wave.

In perpendicular shock wave calculations it is convenient to take b2 = ,92/,91 = B 2 / B 1 as the parameter defining the shock strength. Notice the limited range of b 2 , 1 5 b2 < 4. Upstream of the shock it is often the case that the electrons and ions are in thermal equilibrium, i.e. T,I = Til, ( 8 1 = P I ) . The downstream relation between T, and Ti depends on how the equilibrium state is defined. In many of the experiments on perpendicular shocks, temperature profiles are obtained that comprise (i) a narrow region 0-, @ in which the electrons are heated to quite high temperatures, mainly by ohmic dissipation, and in which the ions are heated to modest temperatures by compression and viscous dissipation, and (ii) a region Q+ 0, orders of magnitude wider than (i) in which the electrons and ions slowly reach mutual thermal equilibrium. Figure 3.10 shows typical profiles for a laboratory plasma. A similar divergence of temperature occurs in the shock wave due to the impact of the solar wind on the Earth's magnetic field (see 56.6.2) the so-called bow shock - but with the ions being heated at Qand not the electrons. A likely explanation is that some ions are initially reflected back upstream from the potential jump in the shock front and while gyrating about the magnetic field, they gain sufficient energy from the transverse electric field to overcome the potential barrier on the second approach. Downstream these fast ions become thermalized, heating the whole ion fluid.

3

92

Waves in Magnetoplasmas

3.6 Magneto-ionic theory 3.6.1 Electrical conductivity The propagation, reflection and scattering of radio-waves in the ionosphere provides an important application of plasma theory that has attracted attention since the early days of wireless. The standard mathematical model, called magneto-ionic theory, is based on Maxwell's equations for the electromagnetic field, plus the fluid equations for the electrons, ions and neutrals present in the ionosphere. The main difficulty is the great variability of the electron number density, which is a complicated function of height, time of day and season. The phenomenon involves wave frequencies rather higher than those usual in MHD theory, so that both charge separation and electron inertia play important roles. The following introductory theory is an extension of the account in 52.2.3 of the plasma frequency, wpe. Our treatment is based on (2.39) in the form

(3.89) in which the term, -mew,, represents the drag force on the electron fluid due to electron collisions with the background ions and neutrals, which are assumed to be relatively stationary. Thus u is a composite collision frequency for momentum transfer, a phenomenological parameter to be assigned from experiment, or in a deeper theory to be calculated from the mechanics of particle collisions. Pressure gradients are assumed to be negligible, but could be readily included in a more general theory. The steady-state conditions are assumed to be uniform, and the small amplitude electromagnetic waves taken to be proportional to exp{i(wt - k r)}. Hence dv,/dt = ZWV,, and (3.89) becomes

.

(U

+ i w ) ~ ,+ w,,b

x V, = -(e/m,)E

(uce

= -eB/m,).

(3.90)

To solve this for v,, we shall apply the general result given in the following mathematical note. Mathematical note 4. Solution of a vector equation

Let vectors A and B satisfy [allbb + 0 , b x l

+ a i ( 1 - bb)]

*

A = B,

(3.91)

where b is unit vector and all, a,,, al are scalar constants, then the solution of this equation is A = [Pllbb+PAbxl+Pi(l - b b ) ] * B ,

where This is readily verified by direct substitution. Proof. The scalar constants must satisfy

[allbb+aAbxl+aL(l -bb)]*[Pllbb+P,-,bxl + p , ( l - b b ) ] = 1 ,

(3.92)

3.6 Magneto-ionic theory

93

~~

On comparing (3.90), viz. [(v

+ i w ) b b + wceb X 1 + (v + i w ) ( l - bb)] -v,= -(e/rn,)E,

(3.93)

with (3.91) and (3.92), we arrive at the solution j = -enve = u . E ,

(3.94)

where u is the conductivity tensor, ff=

e2ne

me(v

+

bbi ~ )

+ iw) b x l + + (v + iw)2

wce(v WZe

where lI = 1 - bb.

3.6.2

The dielectric tensor

Equations (2.4), (2.25), and (2.26) can be reduced to a single equation for E as follows. In vucuo it follows that

and therefore the plasma has an effective displacement vector given by

aD

.

-=J+Ec,-=

at

aE

at

so that

.

Q ~ E vv E =

l a --{ c2 at

Suppose that waves afthe form

-

a

(1- + Eo-lff

E = Eoexp{ i(wt - k r)}

at

(3.95)

94

3

Waves in Magnetoplasmas

are propagating through the plasma, then (3.95) gives (3.96) where IC. is called the dielectric tensor, since the effective displacement vector is D = K E . Transverse waves propagate in a direction orthogonal to the field, i.e. k E = 0. For these waves (3.96) gives

.

{k21 - ( t ) 2 ~ } - E = o , If E is not zero, this requires that w and k satisfy the dispersion relation (3.97)

-

Longitudinal waves have the electric field parallel to k, in which case (3.96) reduces to

k K = 0, and the dispersion equation is ( K (= 0 .

(3.98)

In the general case it follows from (3.96) and (3.92) that the dielectric tensor is bb -

where

iw,,wx w2x2 bxl-w2x2 wZe - w 2 x 2

WZ,

(3.99)

x = 1- i v / w .

The dispersion relations obtained by substituting (3.99) into (3.97) and (3.98) are algebraically complicated; hrther details can be found in Budden (1966), Clemmov and Dougherty (1969) and Ginzburg (1970). Additional physics, easily included, add ion motions and gas pressures, but the dispersion relations become very complicated. Exercises 3.

3.1 Use (2.58) to show that the effect of retaining electron inertia in the wave theory is to add -(rn,/e)ac,/at to (3.2), where the electron fluid vorticity C, has the perturbation amplitude C

e

=

C

+

G

kx(kxB).

3.2 Verify that the inclusion of electron inertia replaces (3.17) by

(ru4 - (Pa2 + b2)u2+ a2b2)(ru2- b:) =

+

(-)

w

Wct

2

b2b2(u2 - a 2 ) ,

where I? = 1 k26:, and 6, = [m,/(e2pono)]f . Show that the intermediate wave now has a resonance at w = wc, cos0 (see Fig. 3.2).

95

3.6 Magneto-ionic theory

3.3 Find expressions for the phase velocity of the fast, intermediate and slow waves in the neighbourhood of w = wCland w = wCicos8. 3.4 What happens to the sound speed in incompressible flow? Show that in such a flow the phase velocity of an MHD wave is given by

3.5 To allow for the presence of a neutral gas of negligible pressure it is necessary to add to the equations of $3. l an equation of motion for the neutral gas, namely

where vln is the ion-neutral collision frequency, en is the neutral gas density and vn is its fluid velocity. Now there is an opposite drag force on the equation of motion for the plasma,

e (:t- + v . V ) v + Vp = j x B - evzn(v - vn) . Show that the effect of the neutral gas is to replace the first term in (3.10) by we& e = i + - -vin ,

E

w 1-a<

where

E = weno - - _ -W vine0

vni

3.6 Show that a neutral gas modifies the MHD dispersion equation to

(e2ii4- (2+ b2)eii2 + A:) ( e d - b:) = 0 ,

+

where ii = w / k . Write k = k,(l ia), ii = u(1 - ia), u = w / k , where k, is real, u is the phase velocity and the attenuation a is assumed to be small compared with unity. Let C = el i&, where C1 and C2 are real and show that for the Alfven wave, u2 = b;/C1, a = C2/el . 3.7 Retain thermal conductivity and resistivity in the wave theory of $3.1.2. Show that with 77 = vllbb vl (1 - bb) and similarly for 6, the entropy and pressure perturbations are given by wj. = - i w ~ c K ~ / T ofl, =a2(1 -icn)c,

+

+

where

and quadratic terms in the components of Q and 6 are ignored. Also show that the second term in (3.10) is replaced by -k2a2$( 1-icn) and that the first term in (3.1 1) acquires a factor (1fic,,) andtheequationhasanadditionalterm ( ~ / k ~ ) ( c , , - c , ~ ~ )Xk bpo711,wherec,, = k 2 q s / ( p ~ w ) and cqI,= k2v~l/(p0w). 3.8 Write the wave number as k,(l + ia),where k, and a are real and a << 1. Use the theory of the previous exercise to show that for the Alfven wave, the attenuation a is k2qil/(2wpo). 3.9 Show that with ion cyclotron waves, the electric field rotates in the same direction as the ions, accelerating them. 3.10 Find the group velocity of the slow and intermediate MHD waves.

3

96

Wavesin Magnetoplasmas

3.11 In the notation of (3.64) show that in ( T ,v, By)-space the Hoffmann-Teller relations for MHD shock waves lie on the surfaces 2 2

RT = (Px- i B z / p 0 ) v- m v , and

(V - v * ) B y / p o= P , ,

ZRT 2 = H - $m2v2- f ( B ; / p o )( v 2 / v * ,)

where v* = B:/pom2and P, = P,B - ”/porn2. Show that the first and last of these relations have a curve of intersection whose projection on the (By, v)-plane is

where v1, v2 are the roots of the numerator. 3.12 Referring to the previous exercise, show that the roots vl, v2 are the values of v upstream (vl > v2) and downstream of a shock with B, = 0, that is of an ordinary gas dynamic shock. Also show that the specific volume v* corresponds to the A l f h speed mu* = B,”/(poevz) = b;/w, . 3.13 Deduce the MHD generalization of the Rayleigh line of gas-dynamic theory,

and also show that 2 2

Po2

p;

h+$mv + y =H 2v ( v - v * ) 2

3.14 Use the results of the previous exercise to show that on the MHD Rayleigh line R, dh- v d p = dH and therefore that on R, T ds = d H . Also show that there is only one point on each branch of R at which the entropy reaches a maximum. 3.15 Prove that the entropy jump across a weak MHD shock is third order in any measure of the shock strength.

3.16 A 1 eV hydrogen plasma of electron density 7 ~ 1 0 mP3 ~ ’ is in a 1.2 KG magnetic field. Show that a perpendicular shock of density ratio 2.47 will move across the field at a velocity of 24 cm ps-l and will heat the plasma to 48 eV. Is the plasma velocity behind the shock supersonic or subsonic?

3.17 Show that if the collision frequency v in (3.89) is zero, the dispersion equation reduces to w = wpe for longitudinal waves and to k2c2 = w2 - wEe for transverse waves. Show that if w < wpe,transverse waves are suppressed in a distance 6, = c/wpe (known as the ‘collisionless screening depth’) and that waves of frequency w cannot propagate through a plasma of density neo> eomew2/e2. (Such plasmas are said to be ‘overdense’).

Physics of Plasmas L. C. Woods Copyright 0 2004 WILEY-VCH Verlag GmbH & Co KGaA

4 Magnetoplasma Stability

Magnetoplasmas are notoriously prone to instabilities of one kind or another, the main division being between macroscopic and microscopic processes. The first type can be investigated from the MHD equations without reference to the transport of charge or heat, whereas the second type usually requires a kinetic theory approach and is often related to the transport of charge or heat. We shall deal first with MHD stability, by which we mean that when a given equilibrium configuration is subject to all possible ‘small’ amplitude disturbances that can be specified through the MHD equations, the amplitudes decay with time. If they do not, the configuration is unstable. Care is need with ‘small’, for sometimes larger or non-linear disturbances are able to destroy a configuration of plasma and field that is stable on a linear perturbation analysis. Instabilities also occur on much shorter length and time scales than those relevant for MHD; for example when electric current densities exceed certain limits, the streaming of the electrons past the ions is able to excite an electrostatic response known as a two-stream instability. There are many such ‘microinstabilities’. They generate turbulence, which in turn may modify the transport coefficients, such as the resistivity and the thermal conductivity, K . Their possible influence on the macroscopic variables is therefore indirect. There are two distinct approaches to MHD stability theory, namely normal mode analysis and potential energy analysis. The first method resembles the wave theory of 53.1.1, since we test the stability of an assumed equilibrium state X Oby finding the dispersion relation, i.e. the relation between the wave frequency w and the wave number k . Then if either for real k , the frequency w in X = X O+ 2 exp [i(k r - w t ) ] has a positive imaginary part, or if for real w , the wave number k has a negative imaginary part, the state X Ois unstable. The first type of instability - termed absolute - remains local and grows with time; the second is a convective instability, in which the wave amplitude increases with displacement from a source. Normal mode analysis yields both the conditions that X Omust satisfy for stability and the rate of growth of a disturbance if these conditions are not satisfied. A classical example of the method - the Rayleigh-Taylor instability of an interface - is presented in the following section. The potential energy method tests the permanence of an assumed ‘equilibrium’ state by examining the change in this energy for all allowable small displacements away from equilibrium. If U is the potential energy and a vector entailing all physically possible displacements, then the conditions are:

-

<

dU

-= 0

at

(equilibrium);

d2U

- > 0 (stability).

at2

4

98

Magnetoplasma Stability

The classical relation between the entropy S, the internal energy U and the volume of a system P within an impermeable surface C is

T d S = dU + p d V ,

(4.1)

where p is the pressure and T is the temperature of P. Let p , and T, be values of the pressure and temperature just outside P , then a positive value of the energy difference 6U = (T T,) 6s - ( p - p , ) 6V means that the interface is stable. For example if P changes by an expansion, so that 6V > 0, there will be a negative change to ( p - p , ) and the energy U will be increased. Similarly if P is heated, both (T - T,) and the entropy will be increased and again the energy will be increased. We conclude that stability of P requires that

+

(T - ‘ I 65’ , ) ( p , - p ) 6V > 0 .

(4.2)

This is one of the many thermodynamic instabilities of continuum physics, for an account of which the reader is referred to Woods (1996).

4.1 Rayleigh-Taylor and Kelvin-Helmholtz instabilities A classical stability problem in fluid mechanics is posed by the incompressible, isothermal, inviscid flow of two horizontal infinite streams, one above the other, in a gravitational or acceleration field. In the basic or unperturbed state the interface, say y = 0, is horizontal and the streams on either side of it flow in parallel directions with different velocities and densities. Thus (see Fig. 4.1) =

{

vV 2l 2

’ e={

i:

,

p={

PO - S ~ Z Y PO - gel9

Y > 0, y < 0.

In addition it will be assumed that the fluids are perfect conductors and that in the equilibrium state there is a uniform horizontal magnetic field, say Bo = BOB.

4.1.1 Linearized equations The ideal MHD equations are:

v-v=o,

d e = 0, -

dt

dB V x ( v x B )= -, at

poj = V x B,

1

which follow from the equations given in $3.1.1 with the addition of the gravitational force. Notice that it appears that we have made two equations out of the single equation for the conservation of mass, which of course would not be valid. In fact the relation d e l d t = 0 is a degenerate form of the energy equation (2.70) since in isothermal conditions (q = 0), the internal energy u is constant.

4. I

99

Rayleigh-Taylor and Kelvin-Helmholtz instabilities

Our interest lies in the behaviour of surface waves propagating along the interface, so we shall assume perturbations of the form

x1= 2 ( y ) exp i(k,z + k,z d

whence

-4

at

-

-+v V at d

and

-

-iw, V +

-wt),

(ikx,

-iw

rk,) ,

+ ikzvo = -aw, .-

I

If

1

where vo is u2 if y > 0 and u1 if y < 0, and 3 is the Doppler-shifted frequency, (w - k,vo). The subscript '0' will be used with e with a similar meaning. In the following we shall assume that the interface is not an abrupt discontinuity, but has a small, finite width across which the steady state density varies continuously from el to e2. Let derivatives with respect to y be denoted by dashes, then the linearized forms of (4.3) are as follows: ij;

- iGeo(cx, 3y, cZ)

+ ikxijx + ik,ij, = 0, - iW6 + @hijy = 0,

+ (ikxjj, jj',

ik,jj) = ( . ? y ~ o G ,

X ~-o89, o),

Bo(k,ijx, kzijg,iijh - kxijx) = -LzI(Bxl By, B z ) ,

(BL- i k Z ~ yik,Bx r - Zkxh,,ikxBy - A:)

.

=

.

I

^

$, j,).

These equations reduce to

+ ebg) ijy - a i j ; + kJ3ijk + Gfj'

- 2 (LzI2Qo - k%B

where k2 = k:

4.1.2

+ lcz and t3

= 0,

I

(4.5)

Bi/po.

Surface waves

To study surface waves we choose solutions of (4.5) that decay exponentially as y cy

-+

fm: (4.6)

= fjy(0) exP(Fkyy) 1

where the minus sign holds for y > 0 and the plus sign for y < 0, and where k , is a positive number whose value will be determined shortly. Hence lim ijb = -kyijg(0),

y++O

Denote by

lim ij; = kyijg(0).

y4-0

(4.7)

[ @]the jump, 'ZJZ- @ I , in the function @ across the interface, then

[Gg] = 0,

[fig = -2k,ijg,(0),

(4.8)

100

Magnetoplasma Stability

4

Figure 4.1: Surface waves.

of which the first is an obvious boundary condition and the second follows from (4.7). We shall also assume that, like Ch, the tangential velocity perturbations Cx,ijz and the pressure perturbation 9 are odd functions of y. Let I$ = (41 42), then for the product &J! we have

+

[4@]= $[@]+ 144.

Hence

[G2poGx] = ijZQo [cx],

(4.9) [G2poGJ = w2Qo [G;] = - 2k, Z5Gii, ( 0) ,

since G’fi = -kzuhfi is an odd function. Using these results, it follows by integrating (4.5)l and taking the differences of (4.5)~and (4.5)s across the interface, that

ifiy(0) -

[fi4

=

[o]

*

(4.10)

[Gfi]

4.1.3 The dispersion equation A non-zero solution for the set of homogeneous equations in (4.10) requires that the determinant of the square matrix vanishes, i.e.

-

( k i B - Gz@o) ( 2 k y G- 2k,k:B

+ (kf + k:) [ ~ ] g =) 0 ,

(4.1 1)

4.1 Rayleigh-Taylor and Kelvin-Helmholtzinstabilities

101

where to make the configuration clear we have replaced k,, k , by k,, k,, (see Fig. 4.1). To find k, we seek a non-zero solution of (4.5) at a sufficiently large distance from the interface for the density gradient to be negligible. On assuming that f i x and j decay at approximately at the same rate as fiv, we obtain from (4.5) and (4.6) the condition

kY

which reduces to

( k i - k 2 )( k t B g / p o- G2po)2 = 0 .

(4.12)

As k,, and w may be chosen independently, it follows that k , = k (> 0). From (4.1 1) the surface waves therefore satisfy the dispersion relation

e i ( w - kllvi)2 + Q ~ ( W- k I l ~ 2 = ) ~g k ( Q i- ez)

4.1.4

+ 2kiBg/pO.

(4.13)

Special cases

When there is no velocity discontinuity, (4.13) reduces to 2

w =gk-

el - e 2 el + e2

+

2 k fBi Po(&

+ e2)

(4.14)

If either Bo = 0, or k,, = 0, and ~2 > el, i.e. the heavier fluid is on top, w 2 < 0, hence w is imaginary and we have the classical Rayleigh-Taylor instability. Taylor’s contribution was in recognizing the importance of accelerations other than gravity. Notice that a magnetic field will stabilize a given mode if (4.15) that is short wave lengths are stabilized, whereas sufficiently long waves are not. Also, very long waves representing a disturbance lying parallel to the magnetic field, i.e. with k,, M 0, are always unstable when the acceleration is directed from the heavier to the lighter fluid. Let (4.16) then wo is the growth rate of the instability in the absence of the magnetic field, W, is the speed of propagation of an A l f v h wave along the interface, and rAis the time it takes for this wave to move a distance AI,/27r.From (4.14) the condition that w2 be positive - the stability condition - can be expressed as w0rA 5 1, which means that the growth rate must be slow

102

4 Magnetoplasma Stability

enough for MHD waves propagating along the interface to connect regions where the fluid is falling to regions where it is rising. Returning to (4.13), we find from (4.16) that its roots are (4.17) In the absence of acceleration (wg = 0), the flow is unstable if (4.18) When Bo is zero, the velocity discontinuity always destabilizes the flow; this is the classical Kelvin-Helmholtz instability. According to (4.16)1 the growth rate is proportional to k f , so the shorter waves should grow more rapidly. However viscosity and surface tension impede the growth of the very short waves and with an initial perturbation having a full spectrum of harmonic components, one particular wave number will dominate, and after a little while will appear to exclude all others. It is observed that the lighter fluid develops a set of ‘fingers’ that progressively penetrate the heavy fluid. Examples of the instability of interest for plasma physicists occur on the surface of the linear-pinch discharge (see 52.5.3) during its radial oscillations, and on the accelerated surface of the plasma ablated from a target by a laser beam and in sunspot magnetoplasmas.

. . . . . . . . . . . . . . . .... . . . . . . . .. . . . . .-:.; . .. .. .. . . . . . . . . . .. . . .. .. ... ..plasma . . . . . . . . .. .. .. .

* .

* .

* . . * .

e l

*.

*

*

.

*

.

Figure 4.2: Shape of the Rayleigh-Taylor instabilities

The steady linear pinch has a volume current, but in the early stages of the discharge, the current is confined to a cylindrical surface shell, which collapses towards the axis under the pressure of its self-magnetic field. The neutral gas is swept inwards by the shell until the axis is reached when the temperature is large enough to form a thin plasma column. This column then expands against the magnetic pressure, which, exerting an inward force, brings it to rest and returns it to the axis. During this phase the plasma inertia is represented by an outwards force, i.e. one directed from the plasma to a region of zero density, and we therefore have the conditions for a Rayleigh-Taylor instability. An example is shown in Fig. 4.2’. ‘Taken from F.L. Curson, et al., (1960). Proc. Roy. SOC.A. 257,388-401.

103

4.2 Interchange instabilities

4.2 Interchange instabilities 4.2.1 Flute instability

t

+

+

Figure 4.3: Thermodynamic instabilities;

(a) flute instability, k,, = 0, (b) interchange instability. In disturbances with crests parallel to the magnetic field, that is with k,, = 0, the plasma is displaced without twisting or bending the magnetic field lines. And therefore they offer no resistance to the process (cf. (4.14) with k,, = 0). In Fig. 4.3(a) the dashed curve CD represents an initial fluid surface. If after a short time it is convected into the fluted surface indicated by the solid line and the corrugations continue to develop, we have what is termed aJlute instability. In Fig. 4.3(b) the fluting is distorted into a convective overturning of the initial state (i); in the final state (iii), the flux tubes 1 and 2 have been interchanged and the instability is named accordingly. If there is no dissipation, we may a'ssume that there is no flow across the field lines and, provided the magnetic energy is unaltered by the field exchange, the question of stability can be decided by thermodynamic considerations alone. Let C be a magnetic surface enclosing a magnetoplasma of cross-section A ( l ) ,where 1 is the distance measured along a field line L on C (see Fig. 4.4). The volume within C is J

since the flux $0 = A B is constant (cf. $2.5.4). The volume change due to a fluting of the surface near L is

6V = $0 68,

B

(4.19)

4

104

Magnetoplasma Stabiliv

t ?I/ increasing

Figure 4.4: A perturbed flux surface.

If this is an isentropic perturbation, 65'= 0, and it follows from (4.2) that the configuration is stable if and only if

(P,- P) 60 > 0 .

(4.20)

Let $ denote the unperturbed flux variable then (see Fig. 4.5)

from which it follows that 68 and a0/a$ have the same sign. And since p , is at a larger value of $ than p , we can write (4.20) as (4.21)

Pressures usually fall away with increasing distance from the axis of a magnetically confined plasma; in this case the stability condition becomes (4.22)

/

, * ' /

\

.J/

=

const.

Figure 4.5: Evaluation of 68.

\

\

'\

z

4.2 Interchange instabilities

105

Figure 4.6: Effect of curvature on stability.

When this is satisfied, the field strength increases outwards from the plasma and we have what is termed a ‘minimum B configuration’. From (2.104)

where n is unit vector orthogonal to B and directed towards the centre of curvature, and R is the radius of curvature of the field. Hence, when the field is convex towards the plasma, B increases outwards and the contribution to 0 is negative, which is stabilizing; fields that are concave towards the plasma increase 0 and are therefore destabilizing. In Fig. 4.6 we have illustrated this feature of field line curvature for mirror geometry.2 Because the two stabilizing contributions to 0 occur at large B , they are unable to offset the destabilizing effect of the central region at low 8. Thus, as a whoIe, the mirror geometry is unstable unless other stabilizing effects can be included to modify the situation.

4.2.2 Thermal stability Turning now to the question of thermal stability, we first assume that it can be separately considered. Then by (4.2) and the perfect gas relations in 5 1.2.4, - M c V 6 T ( 6 l n T - i61nQ) > 0 , the mass IM of the plasma being unchanged by the perturbation. Hence thermal stability requires that O<-

dlnT 2
or O < -

dlnT 2 < -. dlnp 5

Another way of expressing this result is to assume a polytropic law of the form p which case 6 In T I 6 In e = n - 1,and stability requires that O
(7 =

$1.

(4.23) 0;

en, in (4.24)

*A magnetic mirror is a region of increasing magnetic field strength and two such regions between which the field is weaker is called a magnetic bottle. Charged particles can be trapped between the mirrors, although leakage through the mirrors makes such configurations poor for plasma confinement (see 56.4.2).

106

4 Magnetoplasma Stability

Finally, considering both the mechanical and thermal terms of (4.2) together, we obtain the stability condition

go6 p 68 + Mc, 6T( In Q - 6 In T ) > 0 , so that

WOP

(4.25)

6 In Q

for Sp, 6T negative. Granted that (4.23) holds, the right-hand side of (4.25) is positive, making the interchange less likely to occur. Certainly mirror geometries do not appear to be as unstable as predicted by (4.22).

4.3 Instabilities of a cylindrical plasma 4.3.1 The sausage instability In cylindrical geometry the perturbation in a typical dependent variable is expressed as

X ( r , 8, z , t ) = X O

+ 2 ( r )exp(i(m8 + k z - w t ) } , ( m= 0, *I, ~ 2 .,.).,

where m is the azithmuthal wave number. Two easily understood instabilities of the linear pinch are illustrated in Fig. 4.7. If the plasma column narrows or necks-in symmetrically, we have an m = 0 perturbation, which increases the local current density and hence increases the external Be field (see (2.110) and Fig, 2.6). The local magnetic pressure acting inwards on the column is increased, which is further constricted and eventually ‘strangled’, disrupting the discharge. To stabilize the pinch against this sausage instability it is necessary to maintain an axial component in the magnetic field and provided that B, is large enough, the constricting force will be opposed by an internal magnetic pressure that increases sufficiently to suppress the perturbation. Consider the case when the plasma pressure is very much smaller than the magnetic pressure. The stability condition is obtained by considering the balance of magnetic pressures at the surface of the column. In equilibrium, B:0/2p0 = BZ0/2po,where Boo is the azimuthal field immediately outside the surface of the column (Fig. 4.7) and BzOis the axial field within the column. Let the radius r change from ro to ro 6r, then by conservation of magnetic flux (see $2.4.5),

+

7rrgBz0 = 7r(ro

+ Sr) B, = ~ ( ~ 0+22r06r)B, . 2

As no current flows outside the plasma column, it follows from (2.1 10) that d(rBg)/dr = 0. From these equations we find that (4.26) Stable equilibrium requires that d

d

-dr (B10) > - dr (B&)

7

(4.27)

4.3 Instabilities of a cylindrical plasma

107

i.e. that with an inwards radial displacement, the internal axial field pressure increases more rapidly than the external azimuthal field pressure. Combining (4.26) and (4.27) we arrive at the stability condition for the sausage perturbation,

B, > B e / J Z .

(4.28)

4.3.2 The kink instability The kink instability occurs because the external magnetic field increases on the inside of the bend and decreases on the outside, The net magnetic pressure is therefore in the direction of increasing the m = 1 perturbation. To stabilize the pinch against this, it is necessary to surround it by a highly-conducting solid cylinder, which compresses the Bo field of the approaching column and prevents the kink reaching the solid wall (see Fig. 4.7). Tayler (1957) applied normal mode analysis to investigate the stability of a plasma cylinder of radius TO within an infinitely conducting cylinder at r = R. It is algebraically complicated by the need for Bessel function theory, so here we shall just give an outline of the theory; for more details see the original paper or Woods (1 987). In the steady state it is assumed that there is a uniform axial field, Bo = B&, inside the plasma, a current sheet on its surface C and a combined axial and azimuthal field in the vacuum region (Fig. 4.8). The wave theory outlined in 53.3.1 for waves in bounded plasmas is relevant for this case. Modified Bessel fhctions of the first ( I m ( x ) )and second ( K m ( s ) ) kinds arise in the combination, (4.29) where

x = kro,

m=O

A = R/ro

I1

conducting wall

,

m=1

Figure 4.7: Sausage ( m= 0) and kink ( m = 1) instabilities.

4 Magnetoplasma Stability

108

Figure 4.8: Stabilized pinch.

The stability condition involves parameters defined in s3.3.1 and also (4.30) where the subscript 0 denotes steady state conditions, BOis the axial field within the plasma and B&, B& are the axial and azimuthal fields in the vacuum at the edge of the plasma.

4.3.3 Stability condition With the restrictions m L 0, k 1 0, which do not affect the final result and s1 = w/wci = 0, we find that the marginal stability surface is defined by

x

+ (rn +

teT)2ym

i e;xF,(gx),

( ~ , ( z ) = z ~ , ( . z ) / ~ ; ( z ) ),

(4.31)

where F,(z) is an even function that increases monotonically ylith z, and g is related to the speeds defined in (3.13) byg = (1 - a 2 / b 2 ) / ( 1 a2/b2). To (4.3 1) we must add two physical constraints, namely that pressure is always positive and that R > T O . From (2.108) and (4.30) these conditions can be written

+

p = - -1 +e;

eq

1 2 0,

(p 5 2p0p/B,2).

A21

(4.32)

When p’ s a2/b2(= gp) is either zero or infinity, g reaches its maximum value of unity, and (4.3 1) becomes

x + ( m+ f e z ) Y, 5 e ; x F m ( z )

.

(4.33)

Since (4.31) implies (4.33) but not vice versa, both compressibility (viz. changing p’ from infinity to a finite value) and finite beta (increasing p’ from zero) are destabilizing effects. Larger negative values of the hnction Y, are found with reducing values of A, and it follows from (4.3 1) that this increases the stability of the plasma column. Similarly, increasing

4.3 Instabilities of a cylindrical plasma

109

Figure 4.9: The stability diagram for a cylindrical plasma. (The stable regions lie above the labelled curves and below the upper curve)

z = lcro increases stability, in other words it is the long waves that are likely to become unstable. In Fig. 4.9 we have reproduced Tayler's stability diagram based on (4.32) and (4.33). Complete stability occurs when a point lies in the stability region of the diagram for both m = 0 and m = 1, as it is found that the higher modes are not critical. If A > 5, complete stability is not possible and further, there is no stability when the axial field is uniform across the whole of the tube (a, = &). It has been assumed in the theory that the axial and azimuthal fields do not penetrate each other, which requires the conductivity to be infinite. By (2.100) the diffusion time is 7, = pOrg/vand when this diffusion is complete, the pinch is certainly unstable. In fact very little diffusion is required to destabilize the pinch. Sharp boundaries are unattainable, even in the early stages of a discharge and diffuse boundaries must be taken into account. In order to stabilize such boundaries, it proves essential to have a reversed axial field external to the plasma. At m = 0, (4.33) becomes 1 l2z2Y05 lTFo. The smallest value of FO is 2 and as YO is negative, it follows that 1 5 2C,: which is equivalent to Bo 2 Be"/& in agreement with (4.28).

+

4.3.4

Stability of an unbounded flux tube

In the limit as the walls recede and disappear, i.e. as A

m 5 (m Let

+ &x)2 + e y .

6, Z denote unit

+ co,we

find that (4.33) becomes (4.34)

vectors in the azimuthal and axial directions, then on the plasma

4

110

Magnetoplasma Stabiliw

surface at T = rg the propagation vector k and field B are

mk = -0 + kz',

(C,

B = Bet?+ BzZl

TO

B,/Be) ,

so that

m Bk,,= k . B = --Be To

Be + kB, = -(m + l,x) . TO

(4.35)

It follows that when k,, = 0, (4.34) reduces to m 5 C:x2 and if this is not satisfied, we have an instability with the perturbation and magnetic field helices coinciding, which is an example of the interchange instability of $4.2.1. The right-hand side of (4.34) is a minimum at z = -me,/(Cz

+!a>

(c, < 0) ,

(4.36)

+

and with this value of x, the stability condition is satisfied provided that m 2 1 l;/lf. Suppose that l, and li are large, i.e. from (4.30) that the azimuthal field at the boundary is much smaller than the axial field, then (4.32) gives x 1 p, and the condition for stability becomes m 2 2 p. Then with ,f3 < 1, only the modes rn = 1 and m = 2 will be unstable. Even the m = 1 mode is stabilized if the axial dimension of the plasma column, L say, is small enough; the wave length 2 r / k cannot exceed L, that is k 2 2r/L. With k,,= 0, and m = 1, (4.34) reduces to x = lcrg 2 l i M IBe/Bz or

+

+

1,

(4.37) which is known as the Kruskal-Shafranov stability condition. In toroidal systems, L = 2rR, where R is the major radius and the stability condition can be expressed: q > l

(4

3

Ir&IRBol);

(4.38)

q is appropriately known as the safety factor. If n waves are excited around the torus, k = n / R

and (4.35) gives

k - B = -Be (m-nq); r

(4.39)

hence interchange instabilities (k,,= 0) are likely when q is a rational number.

4.4 The energy principle The normal mode analysis is not convenient to study the stability of diffuse initial states, nor for the discovery of general stability criteria. In these matters the energy principle has the advantage, although it can be applied only to non-dissipative systems.

4.4

The energy principle

111

4.4.1 Potential energy We start from (2.70) for an ideal plasma and without radiation,

eDu

+ pV - v = 0 ,

(4.40)

and add to it the identity

to obtain

.

= B Vv, and therefore From (2.97), QD(B/Q)

(

B2

QD u + - 2,,,)

-k Q ( p + g ) D

1 (f) = -BB PO

: Vv

Let edr be the mass of a convected volume element, then conservation of mass can be expressed as D(ed7) = 0, so that

B2 G )dr] + ( ( p +

B2 -)V.vdr

D[(Q~+

2 P O

1 = -BB:Vvdr.

(4.41)

P O

Let P denote the total pressure tensor, (4.42) (see 52.4.2) then (4.41) can be written (4.43) Let 6X denote the change X(r0 + El t) - X(r0, to) obtained by following an element of plasma over a small displacement <,where X(r0, t) is a macroscopic variable. Then

i=v, and

t=r-ro= roJ

6x=/

ro,to

ax

-dt+/ at

r,to

v-VXdt.

r o $0

= X(r0,t) - X(ro,to), then to first order in I<[, 6X = ax + < * v x .

Let ax

(4.44)

4

112

Magnetoplasma Stability

At equilibrium, vo = 0, so 6v = dv = v, which means that 6 or d operating on a product containing v makes no change. From the above definitions it follows that

l:DXdt =

l:(”

at + v . V X

Therefore, integrating (4.43) over (t - t o ) , we obtain (4.45) The total potential energy of the system is given by

U(t) =

/

V

(eu

+ -B2) 1 2Po

dr

(4.46)

where V is its volume. Hence integrating (4.45) over this volume, we obtain the potential increment due to the displacement <,

aUv=-kP:V
(4.47)

4.4.2 Surface term Suppose that the volume V is separated into volumes V’ and V” by a magnetic surface S (see Fig. 2.5), across which there is a discontinuity P” - P’ = [ P I in the total pressure. Let n dC be the surface element, with n directed from V’ to V”,then the force -n [ P ] dC

-

-

on this element does work on the system at the rate -n [ P I i dC. Hence there is a surface term to be added to (4.47). The total first-order perturbation in the potential energy is therefore

dU = -

l

P : Vedr -

s,

n . [ PI . [ d C .

(4.48)

.

In equilibrium this is zero; in particular as n B = 0 on the magnetic surface,

n.[P]=[npt]=O

(4.49)

At the beginning of this chapter we described the energy principle, which states that for all physically possible perturbations of the system, the equilibrium and stability conditions are Equilibrium : dU = 0;

Stability : d2U > 0 ,

(4.50)

To find a2U we operate on (4.48) with a, i.e. under the integral signs we hold the position vector r fixed and allow the displacement vector to vary. At an intermediate displacement <’,a2U is quadratic in <‘,and as the effect on a2U at is obtained by integration from <‘= 0 at t = to to <’at the final value oft, a factor !j appears. Our expression for d2U is therefore

<

a2u= - L2

a P :V
s,

<

a[ n - P I - < dC >

to which has been added the stability condition of (4.50).

0

(4.51)

4.4

113

The energy principle

4.4.3 General stability condition Stability requires (4.5 1) to be satisfied for all physically possible displacement fields <, so we need to know the dependence of dP and a[ n P I on E . To find these the first step is to integrate the basic equations for ap/at and dB/dt,

.

(4.52) to obtain expressions for dp and dB. Finally, to reduce the effort of testing d2U over all 6, we seek displacement fields that minimize a 2 U , and decide the question of stability from the sign of (d2U),,. From (2.60) and ( 3 . Q

-i - vp - y p v . i , whence a p = -< vp - y p v c .

we get bp/at =

*

(4.53)

*

Similarly

dB _ - -V x E = V x at

(ix B)

yields

dB

B1= V

X

(< X B) = B . V < - E - V B - B V - 5 .

(4.54)

The equation of motion, (3.5), has the perturbation form

e l = -Vdp + dj x B + j x dB , where dj = V x dB/po. Hence

e l = F(5)

(4.55)

1

where

F(t) = V(<.Vp+ypV.<)

1 + -(V PO

X B I )x B + j x BI

(4.56)

From (2.80) and (4.42) the electromagnetic stress tensor is 1 1 T=-BB--B21 P O

2PO

=pl-P.

Hence ~0dT = BIB + B B 1 - B1 * B 1 ,

(4.57)

114

4 Magnetoplasma Stability

1

so

dT :V< = -(B1-V< B PO

+B

V< B1- B1- BV <) ,

*

-

which with the help of poj= V x B, V B1 = 0, and (4.54), can be written 1

dT : V t = -B; PO

1

- j x B1 .( + -v PO

. ( B ~ Bs). .

It now follows from (4.53) and (4.57) that in the volume V’, 1

-8P:VE = -Bf-j P O

1

xB1-<+-V.(BIB.<)+V.t(<.Vp+ypV.<). (4.58) PO

If it is assumed that outside of C the volume V” is a vacuum field, both j and p are zero and (4.58) reduces to

v” :

1 1 -aP : VE= -(q2 + -v P O

P O

. (B;B“.t ),

(4.59)

the superscript ‘e‘ denoting ‘external’. Following integration over V’ and V”,the divergences in (4.58) and (4.59) give rise to a surface integral on C with the integrand 1

C = --n.[B1B-<]. PO

Turning to the surface term in (4.5 I), by (4.42), (4.44), and n * B = 0, we have

a[P.n]*< = a[np,].<= 6[npt I * E -

1

-a[n*BB].< PO

e - [ Vpt 1 n - E + C.

(4.60) As the boundary condition (4.49) holds on the surface C at all times during its displacement, the term 6[ npt ] 6 is zero. Also, integrating V x Vpt = 0 across C (cf. (2.125)) we get n x [ Vp,] = 0. Thus with = i$ nn (n x <) x n in (4.60), we obtain

<

-

+

where the term -C cancels the surface integrals obtained from (4.58) and (4.59). Finally, substituting the above expressions into (4.5 l), we arrive at the form of the energy principle obtained by Bernstein, Frieman, Kruskal and Kulsrud (1 958):

a2u= a2up+ a2u, + a2uv> 0 ,

with

a2U, =

$/ V‘

1

{ - B f - j xB1.E+V.E(E.Vp+^IPV.E)}d7, Po

(4.61) (4.62) (4.63) (4.64)

4.5 Resistive instabilities

11.5

4.4.4 Cylindrical plasma with a volume current Many applications of the energy principle can be found in the literature; the algebra is usually extensive, so here we shall leave the subject, quoting one important result due to Suydam (1958). With a cylindrical magnetoplasma an interchange instability is possible if the displacements have crests running parallel to the steady state magnetic field, B = Bod 33. We shall assume a standing wave

+

< = < ( r ) e x p { i ( m d +k z ) } so that the wave-number vector is k = (rn/r)e + k4. Then the condition for an exchange instability is f = 0, where (4.65) The ratio v-l is the pitch of the spirals in the magnetic field and with volume currents it is a continuous function of the radius r. At a given radius, r = ro, k and m may be chosen so as to satisfy f ( r o ) = 0, making an instability possible. But provided the pitch changes sufficiently rapidly with r , in distorting the lines of force an interchange would require an investment of magnetic energy sufficient to offset the loss in potential energy due to the change in the magnetic pressure. Thus a configuration in which the magnetic field lines are sheared is more stable than one in which they are not. Suydam (1958) showed that a necessary condition for the stability of a sheared magnetoplasma is

(4.66) The first term in this criterion is likely to be negative, so there is a minimum value for the radial variation in the pitch below which the plasma is unstable. The application of linear perturbation theory is not straightforward because it reveals that there are a sequence of singular points, say r = ri, i = 0, 1, 2 , . . . ,n at which f ( r ) is zero i, needs separate consideration. Newcombe (1960) proved a and each sub-interval, ( ~ri+l), theorem relevant to this situation that is necessary to complete the theory (see Woods (1987). Suydam’s criterion has been generalized to a toroidal system by Lava1 et al. (1971) in which geometry it is replaced by (4.67) where q is the safety factor defined in (4.38).

4.5 Resistive instabilities Finite electrical conductivity not only modifies the ideal plasma instabilities considered above, but it also generates additional growing perturbations known as resistive instabilities. In mathematical terms new modes become possible because the inclusion of dissipation raises

4 Magnetoplasma Stability

116

the order of the basic differential equation. There are several types of resistive instability, those known as the gravitational, rippling and tearing modes being considered the most important. Only the tearing mode will be described here.

4.5.1

The tearing mode

The tearing mode instability occurs in the current layer separating opposing magnetic fields. Let a steady-state magnetic field, B (y) = B, ( y ) k B, (y)%, be sheared in the ?direction, and let a small amplitude, standing wave of wave-number vector k = k,% k , i be parallel to B on the surface y = 0. Then the perturbations in a dependent variable X(xl y, z , t ) are of the form

+

X1(xl y,

21

t ) = @Y) exp(i(k,a:

+ b)+ rt}

I

+

(4.68)

where y (> 0) is the growth rate of the waves. On linearization we can adopt the replacements (4.69) As the steady-state velocity is zero and we shall simplify the account by assuming that the perturbations are incompressible (V v1 = 0), the linearized form of the induction equation, (2.96), is

dB1 =B at

*

V V -~~

1 VB -

+ EV2B1

(B = B,; 5 = q / p o ) .

(4.70)

By (4.69) the ?-component of (4.70) is Thy= i f G y

.

( k 2 = k2 + k,2)

+ [(f$!- PBY)

(4.71)

where f G k B and the prime denotes dldy. A second relation between Gy and fly is obtained from the curl of the linearized, ideal form of the equation of motion, (3.9, viz. avl

1

QvX = V x (h x B) = -v at PO

.

x (B VB1 + B~. VB) .

(4.72)

By (4.69) the Q-component of the curl of (4.72) is

-

--poe7V2Gy= ? V{ V (ifB + ByB’)}- V2(ifBy)

since for a vector X,V x obtain

v X X = VV . X - V2X. With V .B = 0and k. B’ = f’, we

ipop~(fi;-k%y)= - f B ; + f ” h y + k 2 f B y .

(4.73)

The fourth-order differential equation satisfied by Byis obtained by using (4.71) and its second derivative to eliminate ijy from (4.73).

4.5 Resistive instabilities

117

In this calculation two important lengths arise. The first is the skin depth 6 (see $2.4.4) defined by

6

(
(Td

e2/<) ,

(4.74)

where C is the shear length, IB/B'I and T d is the difision time for this distance. The second length is

6*

(d&TA/k)3

(TA

= C/vA,

vA

= Bo/(j%@)i)

,

(4.75)

where the constant Bo is a typical value of B and rAis the time for an Alfvkn wave to travel the shear length; usually T~ is much smaller than T d . We shall adopt as a suitable function for f ( y ) = k B,

-

f ( y )= kBoF(y)

( F = tanh(y/C), F'

4.5.2 Differential equation for

=

( 1 - F 2 ) , F"

=

-2F(1-F2))

.

(4.76)

B?/

In the tearing mode the skin depth is much smaller than either the wave length or the shear length, i.e. h2k2 << 1 and b2/C2 << 1. In this case the fourth-order equation for B, = X say, takes the form

where

+ (2h2C2/6:)F2,

(4.78)

B = ( k 2 + F " / F ) (2b212/6:)F2 + k2 + F " / F - 2 ( F ' / F ) 2 .

(4.79)

A = 1 - 2h2(F'/F)' and

Boundary conditions for (4.77) may be assumed to be given values of X and X' on planes y = fa,where a > C, or using the symmetry about y = 0, it could be solved in 0 5 y 5 a, with X',X"' zero on y = 0 and X , X ' specified on y = a. For given values of k l and 6/C it is found that solutions satisfying these boundary conditions exist only when 6, is constrained to have an eigenvalue, say

6, = eqlce, d i e ) ,

(4.80)

and when the function @ is known, the grow rate y of the instability can be deduced. Accurate results will require a numerical integration of (4.77), but for present purposes it will be sufficient to estimate from the form of the coefficients A and B. A physical meaning can be attached to the distance 6, as follows. The curvature of the By(y) profile changes sign where the coefficient of X " vanishes, i.e. at A = 0. At small values of y , F x y/C, so that A = 1 - 2a2/y2+ and anticipating that 6/6, >> 1, we find that A is zero at y = 6,. At large values of y , F M 1 and A = 1 + 262C2/6,4 x 26212/6,4as 6/6, >> 1. Also B = k2 (2d2t2/6:),which reduces (4.77) to

X" = k 2 X ,

(4.81)

118

4 Magnetoplasma Stability

the fourth-order derivative being negligible because 6, << l. At small values of y, F M y/C, ) in which case (4.77) from which it follows that A and D are dominated by their O ( Y - ~terms, becomes

x”= 62x.

(4.82)

The ‘inner’ solution satisfying X’ = 0 at y = 0 is X = Xo cosh(y/6), and while this is not accurate up to y = 6,, for purposes of estimation we shall assume that with a small error X ’ / X = (1/6) sinh(y/6) M y/h2. Therefore, given a value for the gradient of the magnetic field at y = 6,, we can apply this formula to find (4.83) The gradient Bb is positive and therefore the profile is concave upwards near y = 0, as shown in Fig. 4.1 1. The growth rate is found from the above equations to be (4.84) A fuller theory gives A 0: l/k12, with kt restricted to the range fastest growth rate occurs when kl is at its lowest limit; this is

( ~ ~ / ~ d< )kl i

< 1. The (4.85)

Intuitively one might have anticipated y theory yields a much large rate of growth.

N

7;’

for a resistive instability, but as T~

<< T d , the

4.5.3 Physics of the tearing mode The dotted lines shown in Fig. 4.10(a) show the steady state field configuration; in (b) this has developed to the stage when the field lines have ‘reconnected’, creating stationary loops that slowly fade away. The instability appears when the magnetic field lines come close together at points like R. This produces a large positive B{ in the neighbourhood of R, which in turn enhances the speed of convection towards R, hrther increasing B:, and so on, until reconnection occurs, forming what are termed ‘magnetic islands’. The diffusion process tends to reduce the curvature of the B, profile, so it is essential that there be an inwards flow, convecting the magnetic field, as shown in Fig. 4.10(b). The reconnection of sheared field lines in the manner described above, i.e. with the aid of the tearing-mode instability, requires a plasma flow strongly converging towards the neutral points in the magnetic field, a process known as ‘flux pile-up’. Magnetic energy is transformed into thermal energy via this forced ohmic dissipation at rates much higher than with ‘normal’ ohmic dissipation, and has been considered as a possible mechanism for the creation of the very large and sudden release of magnetic energy in solar flares. The difficulty with this

4.5 Resistive instabilities

I19

@?

B=O

wp@-bx ....................................

.......... .....B .........+9y+..’

0 ............................... v,

-1 -..

Y.

...............................

.............................

.....................................

..

(b) Figure 4.10: The tearing mode; (a) initial perturbation, (b) a developed perturbation, showing

‘magnetic islands’ model is that for the rate of heating to reach the values implied by observation, the flow must convergence with velocities of at least 1% of the AlfvCn speed, which in plasma loops with a field of about 100 G is 3 x103 km s-l. Litvinenko (1999) observes that the actual rate of pile-up, which is limited by either the plasma or magnetic pressure in the pile-up region, is some three orders of magnitude too small to explain the most powerfit1 flares. In Fig. 4.1 1 we have sketched a typical unstable &-profile, showing the points B and C where changes sign. N

B:

I

6.

Y

a

Figure 4.11: Unstable profile for the tearing mode

120

4 Magnetoplasma Stability

4.6 The two-stream instability The instabilities considered so far in this chapter are macroscopic phenomena, that is they can deduced from the MHD equations, the independent variables for which are (r, t). Next we shall give an account of an important microinstability, that is an instability which requires phase space (r, w, t ) (see 51.5.4) for its calculation. The phenomenon is generated by the relative motion of the electron fluid through the ion fluid and is switched on when the velocity difference (v,-vi) exceeds a certain value proportional to the speed at which sound waves can be transmitted through the plasma. It is therefore excited when strong electric currents flow though the plasma and since the turbulence that it creates increases the effective resistivity, it has the effect of setting an upper limit to the current density.

4.6.1 Vlasov theory of plasma waves We follow the method employed with the macroscopic instabilities, that is we adopt the normal mode analysis to obtain a dispersion equation, the roots of which will enable us to determine whether or not the assumed steady-state configuration is stable or not. The starting point is the kinetic equation given in (1.89), (4.86) where f (r, w, t ) is the number density of particles in 6-D phase space, usually called the distribution function and F is the acceleration force acting on the particles. The collisionless form of this equation in a magnetoplasma, which by (2.32) and (4.86) is (4.87) is called the Vlusov equation. It is adopted in plasma physics in situations when the mean free paths greatly exceed the macroscopic length-scale, or equivalently when the macroscopic time-scale is much smaller than a typical particle collision time. When there are several gases present, which we shall distinguished by subscripts a = 1, 2, . . . , n, (4.87) is replaced by the set

a +w a +Qa a (at (E + w x B) -)fa dr m, dW *

= 0 ( a = 1 , 2 , . . . , n) ,

(4.88)

since in the absence of collisions, the evolution of f a is independent of the evolution of the other gases. The number density in configuration space is (4.89)

-

the integral being taken over all physically realistic regions of velocity space. By (2.26) and (2.30), viz. coV E = eZni - en, = ~ ~ n~ , in , we get

+

(4.90) L

Y

J

121

4.6 The two-stream instabiliry

which provides the link between the species. Unlike the collisional form of the kinetic equation, which has Maxwell’s distribution (1.92) in the steady state, the steady state form of Vlasov’s equation has many solutions. For example Maxwell’s distribution is one solution and if E = 0, then f a o = fa0(w)is another. We shall not be specific at this stage, but simply write fa0 to indicate such a solution. Now suppose that fal(r, w , t ) , E l ( r , t ) and B l ( r , t ) are small deviations from the steady state fao(r, w), Eo = 0,Bo = 0, then neglecting quadratic terms, we find from (4.88) and (4.90) that (4.9 1) (4.92) where from (2.4) (4.93) We shall assume that the perturbations are entirely electrostatic, by which we mean that the movement of the charged particles generates a perturbed electric field but no corresponding magnetic field. It follows from (4.93) that V x El = 0 and therefore that there exists a potential function 41 such that El = -V41 . Equations (4.91) and (4.92) become (4.94) (4.95) With the wave forms defined by fal(r, w, t ) = fa(w)exp[i(k.r 41(r, w , t ) = $exp[i(k. r - w t ) ] , (4.94) yields

-

w t ) ] and

-

Then, since V241 = -k2$exp[i(k r - w t ) ] ,when this expression is substituted into (4.95), the result is

where (see (2.44)) (4.96)

122

4

Magnetoplasma Stability

The dispersion equation, D ( k , w ) = 0, is the condition that I$ has a non-zero value; hence

D(k, w ) f 1+

afao k. -

& / w - k . w dw d w = 0 ,

a

(4.97)

and to determine the stability or otherwise of the plasma we need to find the roots of this equation.

4.6.2 Solution of the dispersion equation We shall simplify D ( k , w ) to a one-dimensional form by taking k to be parallel to the particle velocity and writing k w = ku. Also the frequency is divided into real (w,) and imaginary (wi) parts, w = w, iwi, and it will be assumed that wi/w, << 1, otherwise the waves would be damped in too few oscillation periods to be of interest. With these changes, (4.97) becomes

-

+

(4.98) where Fa0 (u)is a one-dimensional equilibrium distribution. To evaluate the integral in (4.98) we adopt the Taylor expansion

D ( k , w ) M D ( k , w,)

( k , wr) + i W i a D awr ’

(4.99)

By the Plemelj formulae (see (4.126) in the Appendix to this chapter) this complex fimction can be written D ( k , w,) = D,(k, w,) i D i ( k , w,) where

+

(4.100) and

&(%)

D i ( k , w,) = 7 r c a

.

(4.10 1)

u=w,/k

Substituting these expressions into (4.99) and equating the result to zero, we arrive at

w. -

- D i ( k , wr) a D r ( k , wv)/awr

’

(4.102)

where W r satisfies

D,(k, w,) = 0 .

(4.103)

4.6

The two-stream instability

123

The form of the solution to (4.103) varies with the choice ofthe phase velocity, up = w,./k, and in particular with its relation to the thermal speeds

ci = (kBTi/mZ)'

7

cs = ( k B ~ / m 3i ) 7

c e

= (kBTe/me)3

7

(4.104)

where C, - which mixes electron temperature with ion mass - is called the ion-sound speed. We shall first assume that Fe0 and Fi0 are overlapping Maxwellian distributions, i.e. that (4.105)

4.6.3 Landau damping Although our concern is presently with instabilities, as an introduction to two-stream instabilities, we shall now explain an interesting and unexpected phenomenon known as "Landau damping", unexpected because as collisions are not included in the Vlasov theory, there is no evident means of damping waves. The root-mean-square value of u taken over the electron Maxwellian distribution equals the thermal speed Ce,so if, for example, our interest is in the behaviour of waves with a phase velocity w r / k >> C,, in evaluating the electron integral in (4.100) it is a good approximation to assume that w r / k >> u,even though there will be some values of u for which this is not true, but only relatively few electrons will have these higher speeds. We can therefore adopt the expansion

In the velocity range of interest here we can treat the ions as being stationary and can therefore ignore the ion integral in (4.100). Hence (4.106) from which it follows that to a first approximation, w,. x wpe. In terms of the Debye length, A D = C,/wpe (see (2.48)), the expression for w,. can be written W,

=~

p e1 {

+ 2k2AX2,}

(kXD

<< 1 ) .

(4.107)

The imaginary part of w now follows from (4.101), (4.102), and (4.106), which give (4.108) The electrostatic potential is damped according to

4

124

Magnetoplasma Stability

a result first obtained by Landau, and named after him. The explanation usually offered is that in a distribution F ( u ) that decreases with increasing Iu(, there are more particles moving slightly slower than the wave than there are particles moving slightly faster than it, and as the excess of slower particles have to be accelerated by the wave, the energy in the wave is reduced and thus the wave is damped. Returning to (4.102), we note that the sign of wi depends on the sign of the gradient dF,o/du, which with a Maxwellian distribution is always negative and as a result the electrostatic waves are damped. However, if the equilibrium distribution has two or more peaks, there is the possibility of an instability resulting, but we shall not pursue this furthe3.

4.6.4 The ion-acoustic instability Suppose that the Maxwellian distributions of (4.105) are replaced by (4.109) that is the electrons are drifting relative to the ions with a (fluid) velocity w. Our concern is with waves having phase velocities w T / k lying between the thermal speeds for the ions and the electrons: Ci<< w T / k << C, . In the electron term of (4.100) we may assume that u >> w,/k, whence aU2

For the ion term, since u (4.106) and obtain

<< w , / k ,

--

c:'

we may adopt an expansion like that used in deriving

k2 and hence from (4.100) and (4.103), (4.1 10) so that

(4.1 11) It follows from (4.101), (4.102), and (4.110) that wi = -w;Di/2wii, where 'A full account of the subject is to be found in b a l l and Trivelpiece (1973)

(4.112)

4.6 The two-stream instabiliy

125

(4.1 13) Since C, >> ( w r / k - v ) , the first exponential in (4.1 13) can be replaced by unity with little error. Then from (4. I 11) and (4.112) we obtain the formula

where

cy

= kX,.

Let (4.115)

where R

= Te/Ti,then (4.1 14) yields (4.1 16)

(wi

It follows that for ion waves (i.e. waves with a phase velocity .-+ C,) to become unstable > 0), we need both TJ

> Cs/(l + a2)f, and

Te >> T i .

(4.117)

If plasma pressure is relatively small so that j x B x 0, the electric current flows nearly parallel to the magnetic field, and the effect of magnetic fields on this theory can be ignored. The velocity TJ of the electron fluid relative to the ions is proportional to the electric current density, i.e. TJ = TJ, - T J ~= Ij/ene I = IB'/poene1 , where B' is the gradient of the magnetic field. Suppose that T,/Ti >> 1, then in circumstances where the gradient tends to steepen, it will do so until 2) satisfies (4.1 17) and the ion-acoustic instability is switched on. The waves will rapidly grow in amplitude and since they are orthogonal to the current, they will add to the resistivity and tend to reduce the current below the critical value. The instability will then be switched off and provided that the reason for the initial steepening of the magnetic field gradient remains, the current will increase again and the process will repeat. This switching on and off of the instability will hold the current close to the critical value. It follows from (4.1 16) that the maximum growth rate occurs where a / ( l a 2 ) 3 /is 2a maximum, which is at a = 1/aTherefore . if we assume that an average value for TJ is at this maximum, we get

+

(4.118)

4 Magnetoplasma Stability

126

which is the maximum current possible in the circumstances described. When T, = Ti the critical value for the relative velocity is much larger. At the maximum growth rate, (4.116) gives a value of u = FC,,where 7 M 0.604, above which relative speed the plasma is unstable. When this instability is switched on by a very steep gradient in the magnetic field, provided that wire 5 1, the electrons will be rapidly heated, the resistivity greatly increased and the current reduced. But as the time-scale for the transfer of energy from the electrons to the ions is very much longer than that for the growth of the instability, the electrons will remain hot long enough for the ion-acoustic instability to take over and reduce the current even further.

4.7 Fibrillation of magnetic fields

Figure 4.12: Unstable magnetic field due to edge vorticity

In $2.4.3 we mentioned the generation of electron vorticity at discontinuities in the magnetic field. In Fig. 4.12 this is illustrated for a cylindrical plasma with a discontinuity in B at r = rg. The magnetic field is upwards (from the page) and as the gyratory velocity of the electrons at the edge is not distinguishable from the fluid velocity u,, there is a vorticity vector C, parallel to B with a magnitude given by [, = 2eB/m, (see 52.4.3). Inwards from the discontinuity, where B is more or less uniform, the velocities involved in the overlapping gyrations of the electrons cancel out on summation, giving v, M 0. It follows from (2.87), the approximation 7 * j M vj and neglect of the battery term that

and hence in the edge region B satisfies the backwards heat equation (cf. 2.99),

dB _ -V

at

x (v, x B) - cV2B

(r M

To)

1

(4.1 19)

4.7 Fibrillation of magneticjelds

127

which is unstable unless there exists mechanism to switch off the instability (e.g. see Ockendon et al., 1999). In the edge region -JV2B > 0, and the magnetic field grows as indicated in Fig. 4.12b. The peaking of the field at r x ro produces a positive value for EV2B in 0 < r < 7-0,where normal diffusion occurs, so again we have an increasing value of B. Collisional difision of the vorticity inwards from the edge will increase the instability, so that instead of the expected trend to a uniform magnetic field, the field develops an evolving fibrous structure, a good example of which is to be found in the fields within solar plasma loops (Bray et al., 1991). Exercises 4. 4.1 If the light fluid lies above the heavy fluid in Fig. 4.1, find the condition for unstable flow. 4.2 A cylindrical plasma has a steady-state magnetic field specified by B = B&J B,i. Given a perturbation of the form k = ( m / r ) 8 ki, where m is the azimuthal wave number, show that an interchange instability is possible if k B = 0. The pitch of the spirals in the magnetic field is g = rB,/Be. Explain why a large value of a@/& would tend to stabilize the plasma. 4.3 What is the connection between thermal stability and interchange stability? In tokamaks the temperature and density profiles are sometimes modelled by T = To(1 - y)", p = po(1 - y)O, where y = (T/u)', Experimental values for a and fi average about 2.5 and 1.25. Why would it be wrong to conclude from (4.23) that these profiles are not stable? 4.4 Show that a magnetoplasma with the field lines curved towards the plasma is unstable even if Ss dl/B = 0. Prove that the linear pinch (Fig. 2.7) is unstable.

+

+

4.5 An ideal magnetoplasma is bounded by infinitely conducting walls. Show that subject to the constraint of constant helicity (see exercise 2.1 I , p. 66), its magnetic energy is a minimum if V x B = aB,where a is a constant. [Hint: Use the relation a[(V x A)' - aA V x A] = 2aA- (V x V x A - a V x A) - V . (2V x A x dA - (YAx aA)].

.

4.6 Express the equation of fluid motion in the form p$ = F(i$),and show that an alternative form for the energy principle of 54.4 is

a'U = - i L i $ - F ( e ) d r . Also derive this formula from the equation of motion and the fact that the system is conservative. [Hint: Use integration by parts and vanishing surface integrals.] 4.7 Show that in a toroidal plasma the safety factor on the minor axis (T = 0) is given by 2B,/(pojvpoR),where j,o is the toroidal current density on the minor axis. A tokamak has a major radius of 3 m, a minor radius of 1.26 m and is operating with a toroidal magnetic field of B, = 3.4 T. Assuming a parabolic form for the current density, show that the maximum allowable current is I w 4.5MA. 4.8 A stationary plasma fills a conducting cylinder; there is an azimuthal magnetic field Be(?-)and no axial magnetic field. Show that the equilibrium condition is d p / d r = - ( B e / p o ) d ( r B e ) / r d r . Test its stability by assuming a perturbation ofthe form ( ( T , 0, z ) = t 0 ( r ,z ) exp(im0). Consider the case of an azimuthal, symmetric perturbation ( m= 0) and show that

a'U = 3 where a = y p

L

[a (V * t)'

+ b&V .i$ + cE']

dr ,

+ B i / p o , b = -2Bi/(pO7-),c = d p / d r + Bi/(p0r).

4 Magnetoplasma Stability

128

4.9 The integrand of a2U in the previous question is a positive quadratic form provided a b2 < 4ac. Deduce the stability condition

> 0, and

4.10 A collisionless plasma has distributions Fi0 = nib(u - V ) , F,o = n,b(u), where b ( z )is the delta function and V is the ion drift speed relative to the electrons. (These distributions describe a beam of cold ions drifting with velocity V through a cold electron background.) Show that the dispersion relation is

Let Fm(wm)be the minimum value of F ( w ) ,when W , lies in the range 0 < w < k V . Show that if F, > 1 the flow is unstable and that this “two-stream instability” condition can be written

kV

< W p e { 1 + (wpi/wpe)2/3 } 3 / 2 .

[Hint: To evaluate the improper integrals involving 6’Flb.u first integrate by parts then use the property of the delta function.]

4.1 1 Generalize the theory of exercise 4.10 to the case FZO= n i 6 ( u - V),Fe0 = n e 6 ( u ) ,and show that the maximum growth rate of an electron-ion streaming instability occurs when k * V M wpe. Solve the dispersion relation for the maximum growth rate to obtain

4.12 Show that with stationary ions the dispersion relation in a collisionless plasma can be expressed

where 2~0= w r / k . Deduce that for an instability to exist we need F ( u ) to be a minimum at 2~ = uo and

which conditions are known as the Penrose criteria. 4.13 A magnetic field B has a jump in magnitude over a distance bB, which steepens until bB reaches its maximum value. Show that if the electrons are much hotter than the ions, a useful estimate for

where ( B )is the average value of B across the jump. 4.14 Show from (5.97) and (4.116) that if losses of electron energy occur only through electron-ion collisions, ohmic heating will cause a thema1 instability if

F ( a , R )> { mz q - R ) - l } h .

4.7 Fibrillation of magneticjelds

129

Appendix 2: The Plemelj Formulae. We start with Cauchy’s integral formula, giving the value of complex function F ( z )at a point z = z +iy either within or outside a domain D enclosed by a contour C on which F ( z ) takes the value F ( t ) : (4.120) where D’ is the complementary domain outside C. A more general form of the Cauchy integral appearing in (4.120) is (4.121) where in general f ( t ) is a complex function of position defined on the contour C and the complex number z represents any point in the Argand plane not on C. The symbol t in f ( t ) represents the point t ( z ,y) rather than the corresponding complex number t = z iy, for f ( t ) is defined only on C and cannot be regarded as being an analytic function of a complex variable. We shall adopt the notation

+

lim

z-+t;

z€D

~ ( z=)F+(to),

lirn

z+t;

ZED’

F(r)= F-(to),

(4.122)

as the two limit values of F ( z ) as z tends to an ordinary point P ( t 0 ) on C (see Fig. 4.13). The value of the integral at z = to shall be denoted by 3 ( t o ) ,i.e. (4.123) A meaning can be given to this improper integral as follows. Let to be a point lying on a smooth arc t l t z of C, where t l , t2 are the points of intersection of C and the small circle Iz - to1 = 6 shown in Fig. 4.13. Denote this arc by 7,then ifthe limit

exists, it is called the principal value of the improper integral F(to).An alternative definition is readily shown to be (4.124) where h is a small complex number such that its vector in the Argand plane is not parallel to the tangent to C at to. In the following it will be assumed that if the integral does not exist in the ordinary sense, its principal value is to be taken. The first of Plemelj’s formulae can be written

F+(to)

+ F-(to) = 2F(to),

(4.125)

one proof of which is as follows. Let z and 2’ be the points to - h, t o + h shown in Fig. 4.13, then by (4.121), 1

}dt

130

4 Magnetoplasma Stability

't Figure 4.13: Defining principal values Assuming for the moment that F + , F - do exist, we now obtain (4.124) by taking the limit h using the definition in (4.123). The second Plemelj formulae is

F+(to) - F - ( t o ) = f ( t 0 ) I

--$

0 and

(4.126)

which can be proved as follows. In the notation just introduced,

}dt

1

+

The restriction on arg h means that one of the points t = to h, z = to - h must lie in D and other must lie in D'. Therefore by an application of (4.120) the last term of this equation reduces to f ( t 0 ) The penultimate term is uniformly convergent with respect to h provided that f ( t ) is continuous at t o and it vanishes in the limit lhl + 0, which establishes (4.124). Finally the Plemelj formulae can be written in the form (4.127)

Physics of Plasmas L. C. Woods Copyright 0 2004 WILEY-VCH Verlag GmbH & Co KGaA

5 Transport in Magnetoplasmas

To complete the system of macroscopic equations in plasma physics we need to determine values for the transport tensors, Re, ~ i 6,, 77,and euW appearing in the following equations:

-

-

g , = -Re VT, - T,6 j ,

and

-

gi = --& VTi ,

H = - 2 ~ u W :VV,

which are equations (2.68), (2.59), and (2.72). The tensorial character of the transport coefficients is due to the anisotropy caused by the presence of the magnetic field, the measure of which is the magnitude of the dimensionless number a = w,r, where w, is the cyclotron frequency and r is the appropriate collision interval. In the simplest treatment of transport it is possible to separate the effect of anisotropy from that of isotropic transport ( a = 0 ) . Hence we shall begin by determining the scalar coefficients K,, ~ i 6, , 17, and eu, for particles following linear paths, the first step being to calculate the value of T for the transport in question. Then the scalar coefficient, which is a function of r , is obtained and finally it is modified to accommodate the influence of the magnetic field on the particle trajectories.

5.1 Coulomb collisions 5.1.1

Particle diffusion in electric microfields

As remarked in $2.1.3, the electric field E(r,t) appearing in Maxwell's equations is a macroscopic variable, defined by averaging over a macroscopic point, by which we mean an infinitesimal volume containing a vast number of particles. We shall now reduce the timeand length- scales of the description, and consider the effect of random microfields on the motion of a typical particle, moving in a plasma without a (macroscopic) magnetic field. With neutral particles, the forces at collisions are due to highly localized electric microfields, but in a plasma the long range of the Coulomb force between charged particle extends the range of the microfields to a Debye length AD, with an associated time-scale equal to the reciprocal of the plasma frequency, w;:. These parameters, namely (see $52.2.3 and 2.2.4)

play central roles in microfield processes.

5

132

Transport in Magnetoplasmas

Figure 5.1: Binary encounter, repulsive force

A basic parameter in collision theory is the so-called 90” deflection impact parameter bo. By an ‘impact parameter’ is meant the perpendicular distance of a pre-scattered particle trajectory from the stationary reference particle in a binary encounter. In Fig. 5.1 ‘1’ is the reference particle, and ‘2’, the scattered particle, has an initial velocity g relative to ‘1’ and, when scattered through an angle x,a final velocity g’. It is easily verified that Igl = lg‘l = g, say. The impact parameter b is defined in the figure; the azimuthal angle E specifies the plane in which ‘2’ moves. Clearly x depends on both g and b. When x = 90” for an average value of g, b is equal to 80. To find 80 one equates the electrostatic potential energy between colliding particles at their closest, to their average kinetic energy. If the particles carry charges Z e and Z‘e, the potential energy is Z’Ze2/(4nf,&) (see 55.1.2). By (1.19) the average kinetic energy for the two particles is 3kBT.Hence -

The number of particles in a ‘Debye sphere’ is

n,= +rX3,n,

(5.3)

and so an alternative expression for 50 is

(5.4) For particles carrying the same charge, say Z = 2’ = 1, we can interpret 80 as being the diameter of an equivalent solid, neutral particle, since colliding particles can penetrate no

5.1 Coulomb collisions

133

closer. By (1.1) a particle P will therefore move a distance

between 90" collisions. The average inter-particle distance d is

The plasmas of interest to us have temperatures high enough and densities low enough for the following inequalities to be well satisfied:

i.e.

-bo << d << AD << A900 .

(5.6)

It follows that a particle must travel a relatively long way to experience a 90" deflection, and since it is influenced via its electric field by all the particles within a Debye sphere, the vast majority of its interactions with other particles will involve quite small deflections. In fact we shall find that the average electron collision frequency for momentum transfer is typically more than a hundred times the 90" collision frequency, (21cBTe/m,)f &&,which shows how much more important are the accumulated effects of grazing collisions than is the occasional abrupt collision. The Debye cylinder swept by a particle P in the distance 10-2Xg~~ required for this momentum transfer, contains rA~10-2A900n nz particles ( 2 1013 in a typical laboratory plasma). Thus P interacts with a vast number of particles per momentum transfer time, and the binary event called a 'collision' in a neutral gas is replaced by an essentially continuous process in a plasma. N

N

5.1.2 Particle orbits We start by considering the relative motion of two particles pl, p z of masses ml, m2 moving in each other's field of force. Let the particles be at position vectors rl , r 2 and exert forces F, -F on each other. Then mli'1 = -F, m2F2 = F, so that mlmz(P2 - F1) = (m1f mz)F, or MI: = F (r = 1 2 - r1, A4 = mlrnz/(rnl + m2)) . Thus the motion of p z relative to pl is the same as the motion of a particle of mass M (the 'reduced' mass) about a fixed centre of force F. Specifying r by poIar coordinates r, 6 in the plane of the orbit, we can write the conservation laws for momentum and energy as

~ ' 8= const. = gb,

$ M ( F+ ~ T'P)+ v

= const. =

:Mg2,

(5.7)

where b is the impact parameter defined in Fig. 5.1, g is the constant relative velocity and V is the potential energy (zero at T = 00) of the force F.

134

5

Transport in Magnetoplasmas

Figure 5.2: Hyperbolic orbit of an electron (2’ = - 1) in the Coulomb field of an ion.

Let the electric charges on p l , p z be Ze, Z’e, then since (2.26), viz. eoV* E = Q, has the solution

for the electric field at a vector distance r from a charge q, the force on pz due to p l is

F = Z’eE = Z Z ’ e 2 r / ( 4 m 0 r 3 .) Hence

V = ZZ’e2/(47rc0r). The impact parameter bo is the positive number

bo

(5.9)

E IZZ‘Ie2/(4.ircoMg2).

To be definite in the following, we shall assume that Z and 2’ have opposite signs; for the other case it is necessary only to change the sign of bo in the final expressions. Eliminating 4 from (5.7) we get

the positive (negative) sign applying to the outgoing (incoming) trajectory, and using (5.7)1 to remove the time dependence, we find that dB

-=

dr

f b dr

r2[1- b2/r2

+ 2bo/r]f

*

5.1

Coulomb collisions

135

Choosing OX to be an axis of symmetry for the orbit of p , we obtain the integral b2 _ - 1+ E C o d TbO

(E

= (1 + ( b / b o ) 2 ) a )

(5.10)

Equation (5.10) describes a conic with eccentricity E and focus at the origin. Since E > 1, it is a hyperbola, as illustrated for the case of an electron being scattered by an ion in Fig. 5.2. Let 00 be the angle between OX and the asymptotes of (5. lo), then for the upper branch of the conic, B 4 T - Bo as T -+ 00, whence cos Bo = c1 or tanBo = b/bo.

(5.1 1)

The scattering angle is given by (see Fig. 5.2) =

(5.12)

- 2e0,

so that if b < bo, x > ~ / and 2 we have ‘close’ collisions, while if b > bo, then x < x / 2 and we have grazing or ‘distant’ collisions. The centre of mass C of the particles is at a distance T , = mlr/(ml+ m 2 )from p z , so if T in (5.10) is replaced by (rnl m2)Tc/ml,it becomes the polar equation of the orbit of p 2 in the centre-of-mass frame. The limit B 4 x - 80 as rc -+ 00 is unchanged, so x is also the scattering angle in the CM frame.

+

5.1.3 The Rutherford scattering cross-section Now suppose that instead of precise knowledge of the impact parameter b of p2, we know only that it is incident on an element of area b db dc as shown in Fig. 5.3. The probability 0 dCl that it is deflected into the solid angle d R = sin x d x dc is termed the diferential cross-section for the scattering collision. Of N incident particles per unit area per second, lbdb &IN will be scattered into d R . By definition this number also equals u d R N . Hence (T d R = -b db dc, the negative sign being necessary since d b / d x is negative. From (5.1 1) and (5.12) (5.13) therefore

(5.14)

This is the Rutherford scattering cross-section; it is evident that small-angle scattering is far more probable than large deflections.

5

136

Transport in Magnetopiasmas

dQ = sin X dX de

de

Figure 5.3: Coulomb scattering

5.2 The Fokker-Planck equation 5.2.1 Friction and diffusion coefficients Consider the particles comprising a fully-ionized plasma. Choose one of them to be a ‘test’ particle and suppose that the species to which it belongs has a velocity distribution f ( r , w, t). The long-range Coulomb forces between the test particle p and the ‘field’ particles within a Debye distance will cause p to experience a multiplicity of ‘distant’ collisions, which as noted above, will be far more numerous than close collisions. Consequently, almost all the changes in direction and speed experienced by p will be small. Let p have an initial velocity w, then after a small time interval At, distant collisions will generate a random walk motion in p, producing a cumulative change Aw satisfying [Awl << IwI. We are ignoring here the non-random collisional contributions to Aw, allowing them to enter in the usual way via the convective terms in the kinetic equation (see remarks in 3 1.3.3). Let P(w1Aw) denote the transition probability density that p experiences the change Aw in At, then f ( r , w - Awl t - At) x P ( w - AwlAw) d(Aw)

is the number of particles like p that are deflected from the range (w - Aw, w ) into the element (w, w+dw) owing to interactions occurring in (t -At, t ) .These particles contribute to the number f(r, w, t) dw, and assuming that the process is Markovian, i.e. that no earlier time-intervals contribute to this number, we obtain the Chapman-Kolmogorov equation, f ( r l w, t) =

s

f ( r , w - Aw, t - At) P ( w - AwlAw) ~ ( A w,)

(5.15)

where the integration is over all possible changes in the velocity vector. The time At must

137

5.2 The Fokker-Planck equation

be small enough for Aw to remain quite small compared with w, but long enough compared with the transit time of p over the correlation length for microfield fluctuations. For electrons this lower limit is about w ; : (see 52.2.3 and 54.6.3). Expanding the integrand in (5.15) in a Taylor series to first order in At and to second order in Aw, we obtain the approximate form f ( r , w , t) =

/{(f

-

a aw

At lDf)P(wlAw) - AW* -[f~ ( W I A W ) ]

where lDf is the rate of change of f following the bunch of particles through phase space (see (1.90)) and the terms in the integrand are evaluated at (r, w, t). As the probability of a transition of any kind occurring is unity,

/

P ( ~ ~ d(aw) A ~ )= 1 .

Hence the leading term in the expression for f cancels with the lefl-side and the remaining terms can be arranged as a kinetic equation for f :

lDf

=C F

a --.aw

(Af)

+ --*1

a2

2 awaw

'

(Bf)

7

(5.16)

in which

A w P ( w ~ A w d(Aw), )

at and

B = (AwAw) E

At

AWAWP(w1Aw) d(Aw).

(5.17)

(5.18)

The expression in (5.16) for the collision term C is known as the Fokker-Planck equation. Notice that it may be expressed as the divergence in velocity space of a 'flow' vector J,

@ = - - -d. J ,

aw

J = A f - - - I- (dB f ) 2 aw

(5.19)

This vector describes the continuous flow of phase plane points due to the accumulation of many small-angle collisions. The averages (Aw) and (AwAw) are termed thefriction and dzflusion coefficients for reasons that will become clear below. To apply (5.16) to a plasma we must calculate these averages for the case of Coulomb collisions. In general there will be several types of field particles or scatterers influencing the test particle p, including the species to which p itself belongs. We shall start by considering just one type of scatterer, denoting its distribution function by fs ( r , w,, t ) . The probability that a single scatterer deflects p into the solid angle dS2 = sin x d x dc (see Fig. 5.3) is a ( g , x) dR, where g = Iw - w,I is the relative speed between the interacting particles and a ( g , x ) is the Rutherford scattering cross-section. The corresponding scattering rate is gcr(g, x)dR, i.e. of a group of N incident particles, Nga(g, x) dR will appear in dR. The assumption of small

5

138

Transport in Magnetoplasmas

scattering angles allows us to superimpose linearly the contributions of all the scatterers lying in the appropriate element dw of velocity space. Thus the total probability that p is scattered into dS1 per second per unit volume is fs dw, ga(g, x ) do. Hence the averages in (5.17) and (5.18) are equivalent to ( ~ w= )

(AWAW) =

JJ

JJ AwAw gu(g, x) dS1

,=

[AwAw]

Aw gn(g, x) d o fs dw, fs

=

dw, =

J [awl

J

1’1

AwAw ga(g, x) sin x dx dt .

fs

[AwAw]

dw,

, (5.20)

fs

dw,

I

,

(5.21)

When there are several types of scatterer, say s = 1, 2, . . ., the integrals in (5.20) are required for each value of s and must be summed over s to give the averages. In the next section we shall express these averages as velocity space gradients of two ‘super-potential’ functions.

5.2.2 Scattering in velocity space Let the test particle p have mass m and velocities w, w’ before and after an elastic collision with a scatterer having mass m, and velocities w,, w: before and after the collision. Then with g = w - w,, g‘ = w‘ - w:, denoting the relative velocities and

G = (mw

+ m,w,)/(m + m,),

G’ = (mw’

+ m,wL)/(m + mS),

denoting the centre of mass velocities, it is readily shown from the conservation of momentum and energy that

G = G ’ , g=g‘,

M w=G+-g m

(ME

m+m,

(5.22)

The angle between g and g’ is the scattering angle x and since g is unchanged in rnagnitude by the collision, l a g ( = Ig’ -gl = 29 sin Resolving the vector Ag into a component Ag, parallel to g and components Ag2 and Ag3 perpendicular to g as shown in Fig. 5.4, we have

ix.

A g = 2 g s i n ~ x ( - s i n ~ x 1 c o s ~ x c ocsot s, i x s i n ~ ) . BY(5.221, w‘ - w = (M/m)(g’ - g ) , i.e. Aw = (M/m)Ag.

(5.23)

5.2 The Fokker-Planck equation

139

Figure 5.4: Scattering in velocity space

The first average in (5.21) is now calculated using (5.14) and (5.23). The integrals containing cos E and sin E vanish, leaving only a component parallel to the unit vector g / g . Thus with x lying in xmin5 x 5 xmax,we get (5.24)

As x is assumed to be small, k7r >> xmax>> xmin.The largest impact parameter b for which the Coulomb force is effective is the Debye length A, ($2.2.4), so a reasonable assumption is that xmincorresponds to b = A., Then by (5.13)1, cot ;xmin M 2/xmin = X,/bo. Therefore

ins in$^]^^" M (1-cr)lnA

(a = -ln$xmax/lnA)

(5.25)

Xmin

(5.26) and we have removed the weak dependence of the logarithm on particle velocity by replacing bo by an average value $0.

5.2.3 Super-potential functions Provided that and (5.26) that

4xmaxis large enough, a << 1, when it follows from (5.9), (5.24), (5.25), (5.27)

5

140

Transport in Magnetoplasmas

Finally, by integrating over the field particles we obtain (5.28)

(5.29) As

ag = g- ag = g , g.-

aw

aw

we can write (5.30) and therefore (5.3 1) where (5.32) This scalar function 3-1 is the first of the two super-potentials, introduced by Rosenbluth, MacDonald and Judd (19 57). A similar method is applied to the average ( A w A w ) . First we calculate [ A ~ i A w j ]i,~ j, = 1, 2, 3. The integration over E eliminates the terms in which i and j are not equal, and it is readily found that

[AW~AW = ~[Aw3Aw3], ]~ = r / g = [AwlAwl]nlnA. As l n h is typically between 10 and 20 in the plasmas of interest to us, we can neglect [ A ~ ~ A compared W ~ ] ~with the other components, and as this quantity is the coefficient of the tensor component gg/g2, by removing it we get

[AwAw]

I’ = - (1 - g g / g 2 )

9

.

(5.33)

(5.34) (5.33) can be expressed

(5.35)

5.3 Lorentzian plasma

141

Now we introduce the second super-potential,

65

J

fs(ws) Iw - WsI dws ,

(5.36)

then the average of (5.35) over the field particles gives (AWAW)

=

a26 r-aw aw .

(5.37)

By (5.17), (5.18), (5.31), and (5.37) the Fokker-Planck equation may be expressed

(5.38) In general, when there are several scatterers, we write 7f = C ,?is and 6 = C , Gs, and a separate kinetic equation, Dfs = Cs,is required for each species. As 7-1 and 6 are integrals, we now have a set of rather complicated, coupled, integro-differential equations, which can be solved accurately only by numerical computation. Finally we note two useful relations involving 7-1 and 6. The first is

mv2 6 = 27-1 M W

d (v, = -)dW

7

(5.39)

which follows on integrating V k g = 2/g over the field particles. The second is the Poisson equation

m Vk7-1 = -47r--f,(ws), M

(5.40)

which is implied by (5.28) and (5.3 1).

5.3 Lorentzian plasma 5.3.1

Collisional loss rate

A hypothetical, fully-ionized gas with ions of infinite mass and no electron-electron interactions, is known as a ‘Lorentzian’ plasma. This ideal medium approximates the situation when the positive ions have a very high nuclear charge ( 2 >> 1); it also provides an exact algebraic theory for comparison purposes. For electron-ion collisions, M = m,mi/(m, + mi) --t m, as mi --t 00. The ions have negligible random velocities, otherwise their temperature would tend to infinity with mi. We shall take the laboratory frame to be fixed in the ion fluid so that fi(wi) = nid(O), where ni is the ion number density and S(wi) is the delta function. In this case with m = m, and fs = fi in (5.32) and (5.36), we get ‘H = ni/w and 6 = niw, where w is the electron speed. Therefore (5.31), (5.33), and (5.37) yield (5.41)

5 Transport in Magnetoplasmas

142

*

where is unit vector parallel to w and 1,= 1 (5.38) becomes

-+*. By (5.41) the Fokker-Planck equation (5.42)

where V = d/dw and the subscript ‘ei’ indicates that only electron-ion collisions are involved. As 1 1,= 0, v w = v* = -1 A, v = -2

*.

and

*,

+

W

W’

.w - )w- - + . v + = - - + 2

V.l.=-(V

W

we obtain

i.e.

1 2 v . (-l1 f ) = - -G W W2

1

f + -1, W

- Of;

(5.43)

Expansion of the distribution function

5.3.2

To make further progress we need to expand the distribution function f in the form

f

,

=fo+G*ffi+++:f,+*..

(5.45)

where the tensors fo, fi , f 2 , . . . have the order indicated by their subscripts and depend only on r, t, and w = IwI. Also fo is the local Maxwellian distribution given in (1.77), and therefore by the theory in 8 1S . 3 (with w in place of c ) we have (5.46) The electron fluid velocity is the average of the particle velocities,

ww2fdwdf2.

v , = - /1w f d w = - / / 471

ne

ne

R

o

(5.47)

Hence by (5.44) and (5.46) 471

I

v, = 3ne

O0

w3fl dw .

(5.48)

5.3 Lorentzian plasma

143

Denoting afo/aw, dfl/dut,. . . by

aw afo = Wf' Vf0 = -__ 0, aw

aw

fh, f i t . .., we have v(w.fl) = -W1l l .

fl

+ ww

*

fl' ,

whence 1 = wf;+wG.f;+-l,.fl,

-df =Vf dW

W

l,.Vf and

i.e.

1 v . (--I1. W

=

(5.49)

1

-l..f1, W

Vf) = -v 1

W2

1 . (llfl).= -(-3c. fl + 1, :af:) w2 w

,

1 2 v . (-L .Vf)= - - - p f l . W

(5.50)

5.3.3 Electrical conductivity In the absence of a magnetic field, the kinetic equation (1 39) for the electron gas is (5.51) where from (5.44) and (5.50) Cei = -ueiw * f l ,

(5.52)

vei being the collision frequency,

(5.53)

obtained by setting 2' = -1 in (5.29). In a Lorentzian gas Ceeis zero. It follows from (5.49), (5.51), and (5.52) that in steady, uniform conditions

-

+

eE (wf; I%++

- f, +

W

(5.54)

Multiplying this equation by G, integrating over f and using (5.46) we get

eEfA = v,ifl.

(5.55)

The current density is proportional to the electric field, i.e. j = -eneve = rLE,

(5.56)

5

144

Transport in Magnetoplasmas

where 0; is the Lorentzian conductivity. Hence by (5.48), (5.52), and (5.53,

Jdm

- fAdw = 4Te2 3

fo-

a

w3 (-) aw v,i

dw,

on integrating by parts. Thus w 5f o dw With the approximation w - v,

M

w in the Maxwellian distribution (1.92),

we arrive at

(5.57) In a neutral plasma n, = Zni, so (ne/Z2ni) in (5.57) equals 2-l. For a reason that will become clear in $5.3.4, a convenient average value for the collision frequency is defined by (5.58) then in a neutral plasma 0,

=

32 e2n,r, --

3~ me

(5.59)

The factor In A is known as the Coulomb logarithm. Adopting the average 3kBT,for m,g2 (see (5.2)), we find from (5.9) and (5.26) that for scattered electrons, (5.60)

where nDis the number of electrons in a Debye sphere.

5.3.4 Conductivity in a fully-ionized plasma The above theory is readily modified to accommodate several types of ion, specified by the subscript s,in the plasma. In place of the frequency v,i in (5.53) we take the sum over the scattering ions,

(5.61)

5.3 Lorentzian plasma

145

where (5.62) S

S

and the weak dependence of In A on 2 has been ignored. Therefore niZ2 = n,Z in (5.53) is replaced by neZ*,where the number Z* is known as ' 2 effective'. The collision interval defined in (5.58) becomes 7,

3 / 2 E: ma/2(kBTe)3/2 = 3 ( 2 ~ ) -e4 1nA Z*n, '

(5.63)

and from the table of values given after the list of contents we find that 7,

=

2 . 7 5 ~ 1 0T:/2 ~ 1nA 2*n, '

(5.64)

When the electron-electron collisions are taken into account, the electron distribution function is distorted by the electric field, which has the effect of increasing the collision frequency between the ions and electrons and the conductivity is reduced from its Lorentzian value, with only the functional dependence on 2' changed. The complete theory is complicated and beyond the scope of this text (see Spitzer, 1962). Spitzer's conductivity, us, is related to the Lorentzian value by a, = ?DO,,where -yE is a slowly varying function of Z* closely approximated by "iE

=

3T

32a,'

Q,

= 0.295

+ 2" 0.39 + 0.85

'

(5.65)

From (5.59) and (5.65) we get

a,=-,

e2n,r,

(5.66)

%me

where T, is given by (5.63); at 2' = 1, 0 ,

e2nere

= 1.98 -.

(5.67)

me

A further modification is required for the Coulomb logarithm at high temperatures, since above T, = 4.2 x105 K quantum mechanical effects increase the effective impact parameter and hence reduce A. According to Spitzer this can be allowed for by replacing A by (4.2x105/T,) 'A whenever T, > 4.2 x105 K. We shall therefore redefine In A to include this modification, then from (5.60) we find that lnh =

{

16.34+ 1.51nTe -0.51nne 22.81 In T, - 0.5 Inn,

+

(T, < 1.16x105K) (T,> 1.16x105K) '

(5.68)

which apply to both electron-electron collisions and to electron-ion collisions at small Z. For ion-ion collisions the first of these equations with Ti and niZ3 in place of T, and n, is sufficiently accurate.

5 Transport in Magnetoplasmas

146

5.4 Friction and diffusion coefficients 5.4.1 First super-potential Returning to the general theory of $5.2.3, we shall now assume that the field particles have Maxwellian distributions. Upon integrating (1.77) over the solid angle R, we find that the probability that a particle of the s-species has a velocity w, in (0,m) is given by

4 n, f, dw, = -exp(-w:/Cz)w:

J;;c:

dw,

(C, = ( 2 k B T 8 / m , ) f ) .

(5.69)

The function 'H defined in (5.32) is analogous to the gravitational potential due to a symmetrically distributed mass of density mf , / M centred on w, = 0. Using a theorem dating back to Newton, we shall first calculate the increment d?i due to the 'mass' in a spherical shell C of radius w, and thickness dw,. Within C the increment d'H is zero, while outside C it is the same as that due to a concentrated mass at the origin. Hence

and d'H = 0 if w 5 w,. Therefore by (5.69) and making the substitutions y = w,/C,,

lx(: i)

x = W/C,,

%(x) = -mns MC.3 J;;

-

exp(-y2) y2 dy

+ const.

(5.70)

An integration by parts now yields (5.71) where (5.72)

is the error function. To evaluate the constant of integration we note that at w = 0 (a: = 0), (5.32) and (5.70) yield 'H(o0)= 0, and mns X(m) - const. = - -l m e x p ( - y 2 ) ydy = - -mn, 2 MC,fi' MCS J;;

and hence (5.71) becomes %(a:) =

mns @ ' ( X I -. MC, x

(5.73)

For small values of x,expanding the integrand of the error function leads to -=-(1-3. @(a:) 1

x

f

i

1

2

+...).

(5.74)

5.4 Friction and dzjiision coejicients

147

By (5.3 1) the friction coefficient is

I? dwd‘FI mrn d Q(x) (aw)= r-d’FI = _ _ - = -2%-

C,dw dx

dw

dx( x

M C,2

> ’

Therefore (5.75) where by (5.29) the ‘difision constant’ A , is defined by

A,

f 2rn,

= (

Z,Z’)*e4nSIn A 2m:m2

(5.76)

1

and G ( x )is the function

@(x) G ( x ) = - -1 - d 2dx( x

@(x) -x@’(x)

1-

5.4.2

-

(5.77)

2x2

Second super-potential

The value of the second super-potential can be deduced from (5.39) and (5.71). As the functions are spherically symmetric we find that

whence

the constant of integration being zero in order to keep dG/dx finite at x = 0. Integration by parts yields

&lX

z’@(x’)dx’ = =

also Q’ - G’ = @’

2

- (XZ 2l2 {

@(x) 1

).a()

1 +z exp(-x2)}

fi

- G ( x );

+ -G + -Q” x 22

=

2

-G(x) X

Hence

86 a29 G(x) = n,C, { Q ( x )- G ( x ) }, - = 2n,Cs-. dX 6x2 X

(5.78)

148

5

Transport in Magnetoplasmas

Table 5.1: Values of G(z) and p(z) = @(x) - G ( x ) X

G(x) p(x)

0 0 0

0.4 0.6 0.8 1.2 1.4 1.6 1.0 0.2 0.137 0.183 0.208 0.214 0.205 0.186 0.163 0.073 0.149 0.292 0.421 0.534 0.629 0.708 0.766 0.813

5.0 6.0 8.0 2.5 3.0 3.5 4.0 1.8 2.0 0.041 0.031 0.020 0.014 0.008 0.119 0.080 0.056 G ( x ) 0.140 p ( x ) 0.849 0.876 0.920 0.944 0.959 0.969 0.980 0.986 0.992 X

Since

then (cf. (5.34))

The diffusion coefficient now follows from (5.33, (5.76), and (5.78):

{

( A w A w ) = -AD CP 2w

AD

(5.79)

Its parallel and perpendicular components are

we:(AwAw) = ((Aw,,)') = -WG ( C )w,

(5.80)

A D

3

and

A

( ~ - ~ ~ ) : ( A W A W ) ~ ( ( A W ,w ) ~ ) = ~ {- G@( (" c )~s }) c3

5.4.3

(5.81)

Limiting cases

From (5.74) and (5.77) we find the limits

(5.82)

It follows from (5.75), and (5.80) to (5.82) that

x =0 :

(Awl,)= 0,

2 AD

( ( A W , ~= ) ~~) ( ( A W ~ = )-~) .' 3&C3

(5.83)

5.4 Friction and dzjiision coejicients

149

(5.84)

For intermediate values of x Table 5.1 is adequate. Equations (5.82) confirm what is physically evident, namely that stationary test particles experience no friction and that their diffusion is isotropic. On the other hand (5.84) show that quite fast test particles mainly diffuse transversely to their original direction. Also note that heavy test particles ( m >> m,) tend to be dominated by friction. The theory for ( ( A W , , ) ~ ) given above is not valid if x2 > In A (Spitzer 1962).

5.4.4

Relaxation times

A ‘relaxation’ time is the time it takes for collisions to effect a substantial change in a given initial velocity or energy distribution. Several relaxation times are important in plasma physics, some of are listed below.

Slowing-down time This time is: (5.85)

Consider, for example, the slowing-down time of cold ions drifting through a background of thermal electrons. In this case the electrons are the scatterers and the ions are the test particles, so that m, = me,m = mi and C, = C, = (2kBTe/me) ‘j2. In a frame fixed in the electron fluid the ions have a drift velocity w = wi. Provided that wi << C,,(5.63), (5.79, (5.76), (5.82)1, (5.85), and me << mi yield T,

3/2 e;m;/’ (kBTe)”’ =3(2~) e41nA Zne

mi mi = 7,Zm, Zm,

(5.86)

which can be used to obtain an estimate for the electrical conductivity. Deflection time This time is:

(5.87) since this is approximately the time it takes grazing collisions to deflect a typical test particle through 90”. For example the deflection time for electrons moving at the r.m.s. speed

150

5

Transport in Magnetoplasmas

= (3kBTe/m,)1/2 and then being scattered by ions at a similar temperature (z = (3mi/2m,)1/2 x m) is

w

(5.88)

Particle energy exchange time This time is:

rE

w3 E2 ( ( A E ) 2 ) - 4ADG(w/C,) '

(5.89)

where the second expression follows from (5.80) and the approximation 2

( A E ) = [+m{(w

+ a w , , l 2+ (awl)2)

-

kmw2j2= ( m w ~ w , , ). 2

At large velocities, rE= W2X2/2A, = (w2/2c,")T~>> rD,showing that deflection is the dominant process. From Table 5.1, G((1.5)f) = 0.203, so for the conditions in which (5.88) applies, we find that rE x O.85re2*.

Self-collision time When the test particles and scatterers are identical, a 'self-collision' time rc can be defined by rc = rDat w = (3kBT/rn)'/2. At z = (1.5)'12 Table 5.1 gives @ - G = 0.714, G = 0.203. Hence from (5.87), rc=--

2 ~ 3 cg~ (mPA)ll2 1 ~ (kBT)3/2 Z4n ' 0.714 e4 1nA

(5.90)

where A is the particle mass number (A, = 1/1836) and mp is the mass of the proton. For this case (5.89) gives rc = 1.14~,, so rc is a measure of the relaxation times, both for the velocity distribution to become isotropic and for the energy distribution to thermalize. Notice that the electrons have a self-collision time M 1/43 of that for protons.

Equipartition time Suppose that each of two groups of particles are in internal thermal equilibrium, i.e. have Maxwellian velocity distributions, but have different temperatures, say T and T,. Then the equipartition time, reg is implicitly defined by -dT = - T, - T dt Tell

An expression for reg can be found as follows.

(5.91)

5.4 Friction and dijiision coefficients

151

A particle of the species at temperature T changes its energy at the rate

+

+

( A E )= fm((w Aw11)2 (Awl)’ - w2)

+

+

= ;m{2w(Aw,,) ((AW,1l2) ((AWL)”,).

From (5.79, (5.80),and (5.81) this is (5.92) Hence by (5.69) this species increases its internal energy pu per unit volume at the rate

n

exp(-w2/C:)w2

dT dw = $kBndt ’

(5.93)

By (5.92) the left-hand side of this equation is

where a!

= C/Cs=

(Tms/Tsm)1/2. Using (5.77) and integration by parts we find that

and 4 L m d ( a x ) e - z z dx =

a! (1 + a2)1/2’

We can now evaluate the integral in (5.93) and rearrange that equation in the form of (5.91) to deduce that (5.94)

When the species involved are electrons and ions at comparable temperatures, the last factor in (5.94) reduces to (kBTe/me)1/2. Comparing this result with (5.63), we obtain the equipartition time res = r:i say, where (5.95) Let T~,,, rii denote the self-collision times for electrons and ions, then from (5.90) and (5.95), Tee

: ~ i i: T : ~ = 1 :

(mi/me)+: mi/2m,.

(5.96)

5

152

Transport in Magnetoplasmas

For example, in a hydrogen plasma the electrons reach thermal equilibrium 43 times more rapidly than the ions, whereas thermal equilibrium between the ions and electrons is 9 18 times more slowly attained then equilibrium in the electrons alone. Let (p, (pi denote the rates at which energy is transferred into the electron fluid from the ion fluid and visa versa, per unit volume, then (p, = $n,kB dT,/dt = -(pi. Hence from (5.91) and (5.99, (5.97) There is also electron heating due to ohmic dissipation.

5.5 Transport of charge and energy 5.5.1 Ohm’s law In the introductory paragraph to this chapter we cited the constitutive relation,

~ - j = E + v x B -j -x B en,

Vpe $6. VT,. +en,

(5.98)

and in 55.3.4 showed that in the absence of a magnetic field, when Q = q l ,for singly-ionized particles, the resistivity, q = l/a is given by ?1 = 0.51-

me

7,

e2n,re

=

2 . 7 5 ~ 1 0T,”/2 ~ 1nA ne ’

(5.99)

where the Coulomb logarithm, In A, is 1nA =

i

16.34+ 1.51nTe - 0.51nne 22.81 lnT, - 0.51nne

+

(T, < 1.16x105K) (T, > 1.16x105K)

(5.100) ‘

It remains to find the scalar components vl and q,, in (see (2.94))

5.5.2 Resistivity in a magnetoplasma The theory of resistivity in a magnetoplasma has been the subject of many research papers and several books. Shkarofshy, Johnson and Bachynsky (1966) give a full account and there is also a much-quoted article by Braginskii (1969, whose results are listed in the Appendix to this chapter. As a function of a, = w,,r, the ratio R = ql/ql, at Z* = 1 has been calculated by Robinson and Bernstein (1962), and is shown in Fig. 5.5(a), labelled ‘R & B’. Also shown is Braginskii’s value, labelled ‘Br’, for this ratio, which differs somewhat from the R & B value. The curves agree only at the limits, a, = 0 where R = 1, and a, = co where R = 1.98.

5.5 Transport of charge and energy

153

We have also drawn the (dashed) curve, R = Po/a,, where a, is defined in (5.65) and Po is defined in (5.102)~below, an empirical equation that is a very good fit to Braginskii’s formula for the perpendicular resistivity and with a much simpler functional dependence on a, (cf. (5.132)). From (5.66), 7711 =

771 =

%me i

a0

P O me

, Po e2n,re

= 0.295

+ z*0.39o.85, +

5.88 =1-a: 12

+

(all

I z*). J >

(5.102)

+

Chapman and Cowling’s (1970) approximate theory yields the factor (a: 1) in place of (a: 12) (cf. (5.109)), and gives a R-curve steeper than those shown in Fig. 5.5(a) and closest to the ‘Br’ curve, which implies that Braginskii’s theory is the more likely to be correct. As a check on this, in Fig. 5 3 b ) we compare X = ui/uII derived from Braginskii’s theory (see exercise 5.14) with that obtained by Ferziger and Kaper (1972). There is fair agreement, but some caution is required with published values of the resistivity and other transport coefficients in strong magnetic fields. The component 77, is quite small, with a maximum value of about -0.19r],, at we M 2 and it vanishes at the limits we = 0 and 00, so we shall ignore it. The thermoelectric tensor 6 has the parallel value 6,, = 0.71kB/e; for the w edependence of 6, and 6, see the Appendix.

+

5.5.3

Fourier’s Law

Fourier’s law for the heat flux vector is q(r, t ) = - K

. VT(r, t ) ,

(5.103)

and with different values for the thermal conLxtivity tensor K , applies to both the ion and electron fluids. In the absence of a magnetic field, K = ~ 1which , is the case we shall first consider. In a theory correct to first-order in Knudsen number (see 5 1.3.4) K is proportional to the mean free path X = TC,say where C is an average particle speed. It is usual to deduce the formula for K from kinetic theory, but a physical approach is more instructive. (a)

(b)

11 ’0

25

50

75 I00 125 I50 175 200 me

Figure 5.5: (a) R= qL/qll; (b) X=

al/all,

both at 2’ = 1

5

154

Transport in Magnetoplasmas

The vector q is the diffusive transport of the internal energy u and therefore can be expressed in the form of (1.38) and (1.39): u = c,T,

q = -pxVu,

x = aX2/r= a C 2 7 ,

where y, is the thermal difisivity and a is a constant of order unity. It follows that K = a C , p c 2 T = 2cYC,pT = 3 a ( k B / r n ) p 7 ,

(5.104)

where we have used (1.23) and (1.85). In the classical theory of heat transport through a neutral gas, if T is the collision interval for the transport of momentum (see Fj5.6.1), then a = 5/4, giving the classical formula K = 2 . 5 ~ ~However, 7. in plasmas kinetic theory shows that larger values of a are required ) (5.134) and (5.135)). for K,, (T = 7,) and for ~ i (T, = ~ i (see

5.5.4

Thermal conductivity in a magnetoplasma

To deduce the effect of the magnetic field, we write (5.103) as K-'

- q(r, t)

= -VT(r,

t) ,

(5.105)

and evaluate the left-hand side by using the fact that it takes a collision time T for q to respond to the thermodynamic force -VT(r, t ) . It follows from (2.36), i.e. i- = (-wcb) x (r - X), that the charged particles spin about the field lines with an angular velocity -web. Hence in a frame F spinning at this rate, the effect of the magnetic field on the particles vanishes, so that after a time T -the instant at which collisions transfer the energy - the left-hand side of (5.105) is reduced to the vector form K-lqm(r T V , t T),where q, is the value of the heat flux vector as observed in the frame F at time (t T ) . Hence (5.105) is equivalent to

+

+ TV, t +

qm(r

T)

= -KVT(r,

T)

+

+

,

(5.106)

where a = W,T. By an application of (1.18) the rate of change of a vector a in F is Da Da is the usual convective time derivative. Therefore

+ wcb x a where

+ T V ,t + 7) = q(r, t ) + a b x q(r, t) + TDq + ;.r2D2q+ . . .

qm(r

We now impose the constraint lTnDnql<< Iql, i.e. ( 7 / 7 ) " << 1,where 7 is the macroscopic time scale, which by (1.33) requires that the Knudsen number k , be much less than unity. The vector q is O(k,) and therefore TDq is O(?C,)~,hence correct to O ( k N )(5.106) , reduces to q + w b x q = -KVT (W = W,T) . (5.107) The mathematical note in 53.6.1 applied to (5.107) with the values a,, = 1, a, = a, aL = 1, yields (5.103) in which K has the following components: K,, = K

7 0 kEi

= -pr,

m

K,

K

-WK

= - K L = - (70 = 3 4 .

1

+ W2'

1+

a 2

(5.108)

5.6 Transport of momentum

155

\\ \

Figure 5.6: The ratio X = Kl/KII for the electron fluid

Br = Braginskii (1965); C & C = Chapman & Cowling(1970); F & K = Ferziger & Kaper (1972) Alternatively (5.103) may be expressed as’

q,-nk.VT

k-bb--

W

1+w2

bx 1

1 + -(1 1+w2

- bb)) .

(5.109)

In Fig. 5.6 the ratio X = I E ~ / K for~transport ~ in the electron gas is plotted for three different theories; the curve labelled ‘F & K’ is from the theory given by Ferziger and Kaper (1972, p. 454), the curve ‘C & C’ is from Chapman and Cowling’s kinetic theory (1970), which gives the same results as (5.108), while the ‘Br’ curve is from Braginskii (1965). In addition there is a dashed curve, which shows the conductivity ratio for the plasma (electrons plus ions) obtained from Braginskii’s theory. Notice the dominance of the ion thermal conductivity perpendicular to the field. These results show that there is some uncertainty in magnetoplasma transport theory. The simplest theory, i.e. the one leading to (5.108), is physically lucid, gives values lying between those from the other two theories, and we shall adopt it.

5.6 Transport of momentum 5.6.1

Classical formula for the viscous stress tensor

The transport of momenturn through a plasma is determined by the pressure tensor p, which was defined in 51.3.1. It is comprised of two parts, p1 due to the thermodynamic pressure p and TT due to viscosity: (5.1 10)

p=pl+TT.

Like the flow of energy, the transport of momentum takes a collision interval T to complete and therefore the momentum transport actually obtained at time t is due to a cause m say, at _______~~ ~~~

‘For an alternative derivation ofthe tensors

k of (5.109) and w of (5.129) see Woods (1993, p. 232).

156

5

Transport in Magnetoplasmas

(r - TV, t - r ) . Thus, to a first approximation, p(r, t ) M m(r - rv, t - r ) = m(r, t ) - TD*m(r, t ) ,

where the time derivative D* is calculated in convected frame spinning with the angular velocity 0 = V x v. In 1.2.2 we showed that for a vector a embedded in a fluid element,

i

D*a = Da - f2 x a = a - e ,

(5.111)

where D is the usual material derivation and e is the rate of strain tensor. Like p, the tensor m may be expressed as a sum of dyads, C(ab),where the vectors a and b are chosen from the unit vectors i, j,and k. Now

+

D*(ab) = (D*a)b aD*(b) = a . eb

+ ab .e = (ab + ba) - e ,

where in the last expression we have used the symmetry of e. The tensor p and its cause m are symmetric (see (1.30)) and therefore D*m = 2 e . m = 2 m - e .

(5.1 12)

p(r, t ) = m(r, t ) - 2rm. e(r, t ) ,

(5.1 13)

Hence

and it remains to identify m. The dominant component of p is p l and in the limit as the delay time tends to zero, the viscosity vanishes and (5.1 13) reduces to p l = m. It follows that correct to O(T),(5.1 13) becomes p = p l - 2ppe

(5.114)

( P P = PT) 7

and the viscous stress tensor is therefore given by TT = -2ppe.

From (1.17)

e = i(vv

(5.1 5)

-

+ v v ) = g + + I V.v

(5.1

where g is the deviator of Vv. In fact we could replace e in (5.1 15) by g, since dilatation is important only when the medium can exhibit bulk viscosity, which plasmas do not. The above theory holds for both the ion gas and the electron gas. According to calculations made by Braginskii (see Appendix), for the ion gas, ppi = 0.96piri and for the electron gas, ppe = 0.73pere, where the collision interval T, is defined in (5.64) and ~i = ( 2 m i / r n e ) ~ Tife T, = Ti (cf. (5.90)). Thus

re =

2 . 7 5 ~ 1 0T ~j s, l n h Z*n,

Ti

=

1 . 6 7 ~ 1 0AfT) ~ s, 1nA Z3n,

where A = mi/mpis the particle mass number.

(5.117)

5.6 Transport of momentum

157

5.6.2 The viscous stress tensor in a magnetic field The general principle that we adopted in 55.5.4 for the heat flux vector, q, can also be applied to (5.1 15), that is the value of TT actually obtained applies at a time T later than the strain tensor e to which it is the response. However, the strain tensor itself is the result of earlier collisions. Hence, applying these two time-delays, we replace (5.1 15) by (cf. (5.106)) H,(r

+ rv, t + r ) = -2ppe(r - rv, t -

(5.118)

7).

The time delay applied to e leads to transport theory accurate to second-order in the Knudsen number, which theory we shall develop in 56.1. For present purposes we need to solve

n,(r

+ T V ]t +

7 ) = H(r,

t)

+ TD*TT(~,t) = -2ppe(r1 t ) ,

(5.1 19)

+

where for a vector a, D*a = Da w,b x a is a time derivative in a frame F in which all ordered motions have been eliminated (see 55.5.4). Applying (1.18) to the dyad ac we have

+

+

D*(ac)= (D*a)c a(D*c)= (Da)c w,b x (ac)

+ a(Dc) + w,ab x c

=D(ac)+w,bxac-w,acx b , and more generally for a tensor A, D * A = DA

+web x A -wcA x b .

(5.120)

Hence (5.119) yields2 7DTI+n+abxTT--aaTIxb=-2pTe.

(5.121)

In a first-order treatment the time derivative can be omitted, leaving TT$abx~1-aTTxb=-2p~e,

(a=w,r).

(5.122)

To solve this equation for TT we need the following analysis. Mathematical note 5. Solution of a tensor equation. Let tensors A and D satisfy the equation

A

+ wb x A - wA x b = D ,

(5.123)

then its solution is

A=W:D, ~

~

(5.124) ~~

*Equation (122) of Maxwell's (1867) seminal paper The Dynamical Theory of Gases was probably the first constitutive equation to include time derivatives. His equation was a particular component of ( 5 121), without the rril terms.

5

158

Transport in Magnetoplasmas

where w is a fourth-order tensor with five distinct components:

W=W1+-

1 1+Q2

W2+-

+

1 4w2

w3+-

2Q

W

+

1 4.r~~

(5.125)

w5.

The tensors w i , i = 1,2, . .. ,5 , can be represented as open products of the following second-order tensors,

111

G

bb,

1 A

11

bxl,

EE

1 - bb,

(5.126)

with the convention that a diamond implies that inner products are required, i.e. that for any second-order tensor A, 1 io 1j :A = 1 i A 1j. The basic fourth-order tensors are:

W1

=

w2

=

w4

=

+ i(1P 1 1 a(l[p 111 + IA), 111 111

1A0

lli),

W1 = 111 111 - $ 1 1 111,

lAo

i(1.L'

1 A

-

lAo

II),

lip 1 1 + 1 P l,l,

1

(5.127)

w3

=

w5

= l~plA-lA~llll

where the second form for WI holds only for its contraction with a deviator, it being readily deduced that

~.lA=1111:~=-111~~:~.

11';'1J.-1A*

The solution in (5.124) can be derived directly from (5.123); alternatively we can confirm it by substitution. From

1A*lII=O,

l A * l L = l h ,

1A'1A=-lL)

which are readily verified, and (5.127) we can show that

Hence by (5.125) and (5.127),

W + w b x W - WWxb

=

W1+ W2 + W3

= 11p 11,

+

lLO

1 1 + 11p 1 L

+ 110 111 = 101,

which is the unit fourth-order tensor, and it follows that when (5.124) is substituted in (5.123), we get

W : D + a b x W : D - n W x b : D =1'1

: D = D.

and the solution in (5.124) is verified.

It follows from (5.123) and (5.124) that the solution of (5.122) is H = -2ppW: e

(PP = PT).

(5.128)

I59

5.6 Transport of momentum

Accepting that bulk viscosity in a plasma is zero, we can replace e by its deviator and write the pressure tensor as 0

(5.129)

p=pl-2prW:Vv.

The formulae for p is more intelligible in component form. Defining five coefficients of viscosity, p i , i = 1 , 2 , . . .5, by

(5.130)

taking b along OZ and introducing the tensor S

pxx = P - {2P2SXX PYY = P - (2p2Sxx P z z = p - 2pSzz,

= V v ,we find that 0

+ ( p - p q ) ( S x x+ SYY) +p4Sxy}, + ( p - p 2 ) ( S x x+ SYY)- p4Szy},

= P Y X = -2p2sxy - p4(Syy - S X X ) , PYZ - P z y = - 2 M s y z 2p5szx,

PXY

Pzx

(5.131)

+

=

Pxz

= -2p3szx - 2p5Syz.

Notice that as B + 0, a --+ 0, and p 2 ) p3 -+ p, p4, p5 -+ 0, which reduces (5.131) to the isotropic pressure tensor. While the above theory is a satisfactory first approximation to the transport of momentum, it is not exact since it is based on the assumption that the collision interval r is independent of the speed of the particles. The basic fourth-order tensors Wi are correct, but in a more accurate theory it is necessary to determine the values of the five coefficients of viscosity from kinetic theory (e.g. see (5.136) and (5.137)). Exercises 5. 5.1 A stationary dust particle is approximated by a sphere of radius a and has accumulated a

charge - 2 o e . It lies in a uniform stream of ions moving towards it with a speed v. Explain why ZOis positive and show that those ions with impact parameters less than b,, where bz = u2 2 ~ 2 0 e ~ / ( 4 m ~ ,mwill ~ vcollide ~ ) with the dust particle.

+

5.2 In the solar wind at a distance of 0.3 AU from the Sun, T, M 2x105 K and n, = 3 ~ 1 0 ' m - ~ . Determine the values of AD, n D , the 90" impact parameter and the average distance that an electron moves to experience a 90" deflection. Explain why, despite the fact that A900 exceeds the diameter of the solar system, the electrons can be regarded as being a fluid at 0.3 AU.

5.3 Prove that for heavy Maxwellian scatterers,

where A and B are defined in (5.17) and (5.18).

S

160

Transport in Magnetoplasmas

5.4 Suppose that there are several scatterers and that the distribution function for the test particle depends only on t. By multiplying (5.38)by w and then integrating, show that for this case

s

where U = w(f/n) dw. 5.5 In the previous question let the test particles have a mass mT and a charge qT and assume that they are in a beam, i.e. they have a delta function distribution f(w, t = 0) = n6(w - wg).Show that with stationary Maxwellian ions and electrons the initial velocity U decays according to

where C, = (2kBTe/me)lI2, C, = (2kBTi/m;)'l2, Z is the ionization number of the ions (use

n, = Zn,) and @(y) is defined in (5.72). 5.6 Defining the slowing-down time byr, = -U(aU/at)-l, show that unlessTi/T, > (rni/me)'I3,

the slowing down of an electron beam is mainly due to scattering off the ions. 5.7 A beam of fast ions of mass mb and speed 2)b is being slowed down by collisions with thermal ions ofmass mi. Show that this takes a time

5.8 In 55.4.4.1we calculated the 90" deflection time rD.Show that for a deflection through a smaller angle a the time required is re = rD4a2/r2 . [Hint: Use the constancy of the difisivity - (di~placernent)~/ftirne)]. 5.9 The Boltzmann collision term (see (1.89))for the rate of change of the distribution f l (r, w1, t) due to collisions between particle of type 1 and type 2 is given by

where the dashes denote properties after the collision, k ,k' are unit vectors defined by w2 -w1 = gk, wa - w: = gk' and 0 is the differential cross section defined in $5.1.3. By interchanging the roles of direct collisions, k -+ k', and indirect collisions, k' -+ k, show that for an arbitrary function Q(wI),

5.10 In the previous question let Aw = w' - w be a small change in the velocity due to collisions. Then employing the expansion

aQ 8W

Q' = Q ( w ) + - A w

+ -AwAw+..., 2 awaw

and integrating by parts, obtain an expansion for

Use the fact that Q(w) is an arbitrary function to deduce the Fokker-Planck collision term (see

(5.16)).

5.6 Transport oSrnornenturn

5.11 The velocity w,(z)ji tensor is

TI =

161

+ w,(z)y lies orthogonal to the magnetic field. Show that the viscous stress

- p T ~ ~ { ~ ( g ? + y y - 2 ~ ~ ) + F i ( Q ) ( ? ~- F - yz (ym) ) ( ? ? + y ? ) } -PT{

where FI ( P )= 1/(1

Fl(a)(?y f 9%)+ Fz(Q)(g? - SP)} ,

+ 4w2) and F2(w) = 2 w / ( 1 + 4 ~ ' ) .

5.I2 Show that plasma flowing parallel to a magnetic field BE and sheared in the OX-direction is subject to a retarding force of magnitude - 2 p P 4 ' / ( l vanish in the limit Q + oo?

+ 4 a 2 ) . For what physical reason does this force

5.13 By applying the method of 55.5.4,show that the thermoelectric heat flux (see (2.68)) in a strong magnetic field is approximately (cf. (5.133)) qth

= Ted11 ( j I I

+b X

j/Qe

+j,/&)

.

+

5.14 In the definition of Ohm's law, Braginskii uses the electric field (E Ve x B), whereas some authors (e.g. Ferziger and Kaper), use (E v x B)(cf. (5.98)). Neglecting the thermoelectric

+

force, show that

(me = -eB/me

x B = (TVx B - web x j

(TV~

Apply the method of 55.5.4to show that

+

jm(r

t+

T ~ v ~ , 7,)M j + w e b

and hence that

[In an accurate treatment uII= 1.9601.]

5.15 Show that if Ohm's law is written in the form

- +v x B +Vp,/en,)

j = B (E

--toaswe +m.

+

x j = a(E v x B

(T

= nee

2

T ~ / v L ~ ) .

+ Vpe/ene),

162

5

Transport in Magnetoplasmas

Appendix 3: Braginskii’s transport coefficients. Braginskii’s (1965) frequently used formulae are based on approximate solutions of the twocomponent Fokker-Planck equations. Braginskii states that the largest errors in his coefficients are 1, but larger errors have been between 10 and 20 per cent and occur in the intermediate range Q found in this range (Epperlein 1984). In very strong magnetic fields the errors are small, but if electric currents are passing through the magnetoplasma, transport is dominated by the second-order effects, described in the next chapter. The collision times adopted by Braginskii are given in (5.1 17). N

Thermoelectricity (5.133)

Electron thermal conductivity

(5.134)

=

where Q, Iwcerel, A, 2 are given in Table 5.2.

=

Q:

+ 6 1 ~ ;+ 60, and the coefficients a, P, y, 6 for various choices of

Ion thermal conductivity

K:

where Q,

i

= KO

+

~ i ( 2 . 5 ~ :4.65) , Ai

’

K:

,2Q:

=Kg

+ 2.65

Ai

(5.135)

’

= wciri and A, E a: + 2 . 7 0 ~ :+ 0.677.

Electron viscosity ( 2 = 1) p? = 0.73p,re,

where A = w:

+ 1 3 . 8 ~ :+ 11.6.

1

(5.136)

5.6 Transport of momentum

163

(5.137)

For a physical explanation of the doubling of the argument in vz, p i , p i , and &, see the derivation in 55.6.2. Table 5.2: Dependence of transport coefficients on Z

z cxo

Po 70 60

61 a:

ah

a: a:

1 0.51 0.71 3.16 3.77 14.8 6.42 1.84 1.70 0.78

2 3 4 0.44 0.40 0.91 1.02 4.89 6.06 1.05 0.58 10.8 9.62 5.52 5.23 0.60 0.35 1.70 1.70 0.34 0.24

0 0.38 1.09 6.92 0.41 9.06 5.08 0.26 1.70 0.20

0 0.29 1.52 12.5 0.10 7.48 4.63 0.07 1.70 0.09

1

2

3

4

0

1.44 3.72 3.53 2.5 13.5

1.29 3.60 2.84 2.5 12.7

0

pi 5.10 4.45 4.23 4.12 3.80 pl, 2.68 0.95 0.59 0.45 0.15 Py 1.5 1.5 1.5 1.5 1.5 /?:

3.05 1.78 3.96 5.12 2.5 15.4

7: 4.66 7; 11.9 7:’ 2.5 7; 21.7

0.88 3.25 1.20 2.5 10.2

Strong magnetic fields It follows from the above expressions that when w eand a, are much larger than unity, good approximations for the case Z = 1 are:

Kfl

= 3.94,

K’,

54pi = -

2eB

’

.;

(5.138)

= 2-

Kb

(Wci‘7-i)’.

For the validity of the above formulae we require the time-based Knudsen number, k N T ,to be much less than unity (see (1.33) and $5.5.4).

Physics of Plasmas L. C. Woods Copyright 0 2004 WILEY-VCH Verlag GmbH & Co KGaA

6 Extensions of Theory

In this final chapter we shall explain some important aspects of plasma theory that did not fit naturally into the preceding chapters. These are second-order transport theory, thermal instabilities, particle orbit theory, magnetic mirrors, partially ionized gases and a brief introduction to the many applications of plasma physics. By second-order transport we mean, for example, that in addition to the familiar heat flux --K VT (the first-order term), there is a component proportional to the product of two gradients, VT and Vv. Normally this second-order effect is small and can be neglected, but when there is a strong magnetic field present, the second-order term can exceed IK VT/ and even drive heat up the temperature gradient, thus creating a thermal instability. Our account of particle orbit theory extends the introduction of $2.2.2,to the case when there are gradients in the magnetic field which modify the motion of the guiding centres, causing them to drift across the field at right angles to the gradient, rather like the motion of a top spinning on an inclined plane. Magnetic mirrors occur where the magnetic field lines converge to form a strong local field and they reflect some of the charged particles moving towards them; of practical interest is the trapping of particles between two such mirrors (see Fig. 4.6). An introduction to Saha’s equation, giving the dependence of the ionization level of a partially ionized plasma on temperature and pressure, is important not least because we need to know the conditions under which the plasma equations are valid. Finally we outline some of the more important applications of plasma theory, each of which would require a book to itself. The applications include fusion research using magnetic confinement of Deuterium plasma, solar physics with its plasma loops reaching high into the corona, its remarkable prominences, sunspots and solar wind, engineering applications to metallurgy, MHD direct generation of electricity, and railguns, and final we touch on the relatively new and difficult subject of dusty plasmas, Saturn’s rings providing a familiar example of dusty plasmas.

6.1

Second-order transport

In the previous chapter our concern was mainly with the microscopic transport of momentum, energy and charge through a plasma, the process beginning and ending with successive particle collisions. This is coupled to the macroscopic description through an intermediate equation of which Fourier’s law in $5.5.3 and $5.5.4provides the simplest example:

6 Extensions of Theory

166

where the equations are arranged in the order ‘macroscopic’, ‘mesoscopic’ and ‘microscopic’. The mesoscopic equation in (6.1) links the macroscopic description (&) to the microscopic description ( T ~on ) the assumption that only one collision interval is required to give an accurate accounting of the transport of energy. It is this assumption that we shall reconsider in this section, but first we need a non-dimensional parameter to characterize the situation. By a macroscopic time-scale 7 is meant the time required for an ‘appreciable’ change to occur in a macroscopic variable; for example it could be the time it takes a fluid particle to move through a shock wave, or the time J(Vv)-lI required for the fluid velocity to change in magnitude, or for heat to be convected through a macroscopic length L: = IT(VT)-ll,and so on. The non-dimensional number that we are seeking is the Knudsen number defined in 51.3.4, which is the ratio of a microscopic time-scale, say T , to the macroscopic time-scale for a given process, e.g. k , = TIVV~. An equivalent definition is that k , is the ratio of a mean free path, X = ET, where E is the average particle speed, to a macroscopic length scale. In (6.1) k, = XIV 1nTI and since the transport coefficient is linear in X (or T ) , the equation is the first-order term in the Knudsen number expansion for the total heat flux, namely q = q,

+ q, + . . . ,

(qn = O(k:))

For q, to be sufficiently accurate, i.e. for q M ql, we need k , << 1,which, for example, will not be the case in a very strong shock wave, where the shock thickness is often less than about four mean free paths. Also, as we shall show, for the transport of heat across strong magnetic fields in the presence of fluid shear, we have the surprising result that lqzl >> lqll,so that q x q, and if q, has a component lying in the same direction as VT,a thermal instability will result.

6.1.1 Convection versus conduction In order to calculate the heat flux vector correct to second-order in k , it is essential to distinguish carefully between convection and conduction. One of the very few errors in the work of the great 19th Century scientist, James Clerk Maxwell, was an initial failure to make this distinction in first-order transport. When M. Clausius drew Maxwell’s attention to the mistake in his pioneering (1860) paper on the transport of heat, Maxwell immediately appreciated the point and wrote:’

“In applying these results to the case of the conduction of heat through a stratus of air from a hot sudace to a cold one we must introduce the conditions that the transfer shall be of heat only, and not of mattel; and that every intermediate slice of air shall be in equilibrium. In my former paper Ipaid little attention to this subject as I had no experimental data to compare with theory, but the errors of principle into which I fell are worth correcting in order to compare the results of my method of calculation with those of M. Clausius.“ Maxwell’s error in his first treatment of heat flux amounted to not distinguishing between the flux of enthalpy, h = u p / Q , and the flux of internal energy, u. Therefore instead of

+

‘See Garber, Brush & Everitt (1986, p. 346).

6.1 Second-order transport

167

+

obtaining the heat flux vector q, he found the total energy flux Q = q pbv, where bv is the increment in the fluid velocity v accumulated over the time-lapse r between successive collisions. That the conduction of heat must always be reckoned in a frame convected with the fluid is clearly understood, but what is not often appreciated is the true meaning of the word convected. It is that in a truly convected frame P all ordered motions of the molecules (or other particles) are absent and remain absent with increasing time. This means that the molecular motions viewed from this frame are completely random. It also means that since forces that accelerate the fluid cannot be felt in P, neither the pressure gradient Vp nor the density gradient Ve can appear in the formula for the heat flux vecto? q, unless of course there is also a temperature gradient present to exactly cancel the pressure gradient. Another way of expressing the point is that diffusion can be distinguished from convection by noting that unlike convection, diffusion is independent of the choice of reference frame in which the fluid velocity is measured. Thus 'frame-indifference' is an essential property of diffusion and not only must P be moving with the fluid velocity v to eliminate all ordered particle motions from P, it must also have the acceleration and spin (see (1.9)) of the fluid element in question. And to maintain frame-indifference the heat flux vector q must be determined relative to P.

6.1.2 The second-order heat flux We shall start by illustrating the mechanism of second-order heat flux with the simplest possible example. Let ji. and 9 denote unit vectors in the OX and OY directions and assume that the temperature T increases in the direction at the rate d T / d y = T' as indicated in Fig. 6.1. Then the first-order heat flux is q, = -~cT'g,where K is proportional to the mean free path, A. If in addition the plasma is flowing with a sheared velocity v = w'yji., where v' = d v / d y , there will be a secondary heat flux along k proportional to the product T'v', which is due to the difference between the heat carried to the right in the top half of the figure and that carried to the left in the bottom half; its magnitude is usually quite small compared with the primary heat flux gl. It is readily deduced from the figure that q, = - 1.721'q,k= ~ T V ' T ' K ~ ,

(6.3)

where the dimensionless number TW' is the Knudsen number k N ,and since KT' cx AIL is also proportional to k N ,q, = O(k:). To deal with the general case, we start from Fourier's law in a magnetic field, which from (5.106) is .rDq(r, t )

+ q(r, t ) t wb x q(r, t ) = -KVT(r, t ) ,

(6.4)

where a = W ~ Tand we have included the O ( k N ) 2term TDq (see derivation of (5.107)). The right-hand side of (6.4) is O ( k N ) so , the next step is to increase its accuracy to the next order in k,. *In the 19th Century when kinetic theory was dominated by mean-free-path methods, it was not immediately obvious how to remove the convective component from Q; see Kennard (1938, pp. 162-72) and Woods (1993, pp. 59-60).

6 Extensions of Theory

168

Hot

yLx

0

Sheared flow

cool

Figure 6.1: Fluid shear generating a second-order heat flux, qz

The heat flux q(r, t ) is a consequence of there being a temperature gradient and because the collisional transport of energy takes a collision interval T to complete, the temperature gradient to which it is the response is the gradient at time (t - T ) , i.e. VT(r - T V , t - T ) . Hence to obtain a more accurate form of (6.4) we replace the right-hand side by

VT(r - T V ,t - T ) M VT(r, t ) - TDVT(r, t ) ,

(6.5)

where

DVT

DVT - !2 x V T ,

in which !2 is the fluid spin. We have used the derivative D because on collisional time-scales temperature gradients are embedded in the fluid, that is they are convected with the fluid and spin with any vorticity present. The spinning of the charged particles about the magnetic field lines is on much too small a scale to influence the spatial temperature distribution during a collision interval. Therefore from (1,18),

VT(r - T V , t - T ) M VT(r, t ) - 7 8 VT(r, t ) , a

(6.7)

where e is the rate of strain tensor. Hence (6.4) becomes TDq+ q + a b x q = -KVT

+T K e . VT.

(6.8)

In $5.5.4 we obtained the solution of

which by (5.109) is the first-order heat flux

q, = - K ~ . V T , where k=bb--

W

1+w2

bXl+-

(6.10) 1 (1 - b b ) . 1+w2

(6.11)

6.1 Second-order transport

I69

Similarly it follows that the solution of (6.8) can be written as

q = -K

k VT + r~ Ik .em VT - r k Dq.

(6.12)

Our interest is in cross-field gradients so we shall ignore parallel gradients, which in any case are relatively small as q,,is unimpeded by the magnetic field. Also with strong magnetic fields we may omit terms O(w'), in which case (6.1 1) becomes 1

Ik = - - b x W

1,

(6.13)

and the solution in (6.12) reduces to K

TK

W

W

q = -b x VT - -b

x e . VT.

(6.14)

the coefficient yo in (6.1) is exactly 2.5 (see $5.5.3 and (5.138)), so in (6.14), 5 k B p / ( 2 ~ Bwhere ), Q is the particle electric charge. Also e can be replaced by its deviator because the term omitted is parallel to the leading term in (6.14) and an order O ( k N ) smaller. Hence the second-order heat flux takes the form For

K,,

K/W =

q2=-51c,?7rbx v"v - V T . 2QB

(6.15)

It applies to both the ion and electron gases, and was first derived by a simple mean-free-path argument (Woods 1983). In strong magnetic fields from (6.1 1) and (6.19, (6.16) where k , = r1]eIIis the Knudsen number. Hence

-

which is an important result. For example the electron gas in a typical tokamak (see $6.6.1) has w e lo6 and k , lo-', making the second-order term about lo4 times larger than the first-order term. What is more interesting is that q,, may either be down the temperature gradient (as to be expected) or, depending on the structure of the fluid flow determining e, the heat can even flow up the temperature gradient, apparently defying Clausius' form of the Second law of thermodynamics. The w-' cx B-' dependence of the first-order heat flux q,, encouraged the belief that plasma energy could be confined by strong toroidal magnetic fields long enough for fusion reactions to liberate useful energy. This was the basis of the tokamak machine (e.g. see Wesson 1997), but after many years of gradual improvement in the energy confinement of these machines, success remains elusive, very likely because of the dominance of second-order transport and the instabilities for which it is responsible (see Woods 1987). N

6 Extensions of Theory

170

6.1.3 The viscous stress tensor The same approach can be applied to the viscous stress tensor, starting from (5.1 18), the strain tensor on the right-hand side of which reads (cf. (6.7))

e ( r - TV, t - 7) = e ( r , t ) - TDe(r,t ) = e ( r , t ) - 27e. e ,

by an application of (5.1 12). Hence in place of (5.121) we have TDTT

+ TI + a b x TI - a m x b = -2p7-e

(1 - 2 7 e ) .

Applying the solution in (5.124) we have

-

H = - 2 p ~ W:e (1 - 27-e) - T D(W : H )

.

(6.18)

The last term can be approximated by -7-D(W :TI,) where

TI^ = - 2 p ~ W e: ,

(6.19)

is the first-order solution. If we ignore parallel gradients and ignore O ( a ) 2terms in the time derivative, it can be omitted from (6.18), in which case the second-order solution becomes TI,

(6.20)

= 4 p ~ ~ ~ : e - e .

A relatively easy way of dealing with the complicated algebra is to adopt (5.13 l), with S replaced by

s = s * ( l- 2 T s ) = s - 2 7 - s . s .

(6.21)

6.2 Thermal instability 6.2.1

Heat flux in a cylindrical magnetoplasma

Take the case of a cylindrical magnetoplasma, with a strong, helical magnetic field, 6 , 2) is the triad of unit vectors. It will be assumed that properties are independent of the axial and azimuthal variables, so that VT = i TI,where the dash denotes the radial derivative. The radial component of the second-order heat flux follows from (6.15):

B = B , ( T )+~ Be(r)6,where (P,

(6.22) where

H

= bxP.

-F.

(6.23)

For the unit vector parallel to B we have b = bz2

+ be8

(b,

= B , / B , be

&/B),

(6.24)

171

6.2 Thermal instabilify

where B is the field strength. The radial velocity wT of either the ion or electron fluid is suppressed by the strong field to values much smaller than either the azimuthal component v g , or the axial component v,. With axial symmetry and uniform conditions along the axis, we find that vg vv= @e- - -ep +v;p~. r

(6.25)

Therefore = a(i4

+ &) + C ( E + E),

(6.26)

I t c=p ,.

(6.27)

where

a

= i(uh - v g / r ) ,

Hence

H = ab, - cbg.

(6.28)

From (6.22) applied to the electron gas (Q = -e), the radial heat flux is (6.29) and a similar equation applies to the ion gas. On comparing the equations of motion for the ion and electron fluids in (2.53), we see that the terms on each side of the equations have similar magnitudes and therefore meve m,ui. Also from (5.90) the self-collision time for a given species is proportional to the square root of its particle mass, i.e. re/ri (rne/mi)’/’.From (6.27) and (6.28), H J H i we/vi milme and therefore (HeTe/(Hiri) (v~i/rn~)~/’, from which it follows that the secondorder radial heat flux in the electron gas is much larger than that in the ion gas. On the other hand from (5.109), ( q e / 4 i ) A cx ( K , e / a z ) / ( K i / t i $ ! ) = ( m e T i ) / ( m i T e ) cx (rne/mi)’/’,SO that in a strong magnetic field, the first-order radial heat flux in the ion gas is much larger. Observations of tokamak energy losses show that this is mainly via the electron component of the plasma, which supports the dominance of second-order heat transport. The assumption that the electron and ion fluids have roughly the same momentum allows us to simplify the formula for the current density, viz. j = en,(Vi - ve),to j = - e n e v e , and write N

N

N

-

N

We shall assume that, at least in the present context, the viscosity is negligible and that the plasma is in equilibrium. Let p , = pi +pe denote the total plasma pressure, then (2.104) gives V p t = j x B = (jgB, - j,Be)f,

so that (6.30)

6 Extensions of Theory

172

Eliminating j e from He, we get

He = -;TbE{-u’+ j, ene

P:

(6.31)

(-)I},

T e n , B,

where (6.32) The equation for a field line is r dfJ/Be= dz/B,. Integrating this over 0 obtain the pitch 63 = 2 ~ r B , / B eof the helical magnetic field. Hence u = 27r/*,

VI

27r

= -*2

*‘-

<05

2 7 ~we ,

(6.33)

Thus if the pitch decreases with increasing radius, u’ is positive. And if in addition, the pressure is negligible, He will be negative, in which case it follows from (6.29) that unless the first-order heat flux is dominant, heat will flow up the temperature gradient and the magnetoplasma will be unstable. We shall pursue this possibility further in $6.2.3.

6.2.2 Unstable current profiles In axisymmetric cylindrical geometry (2.5 1)1 yields (6.34) We shall choose the 02-direction to make j , positive. The axial current is 27ra

I = 27r r a j z r d r = -Bea, Jo PO

(6.35)

where a is the boundary radius of the cylinder. From (6.32) (6.36) with

J ( r )=

-$lTjz(r’)r’dr’

2B6,T

=TPO

(6.37)

in which Be,Tis the value of Be at the radius r . With the help of (6.30) and (6.34) we find the derivative

and hence (6.3 1) becomes

-{

He = Po32 J-j,-2eneB

(6.38)

6.2 Thermal instability

173

To obtain the total cross-field heat flux in a strong magnetic field, it follows from the account in the previous section that we add the first-order ion heat flux to the second-order electron heat flux: (6.39) where for we have adopted the value in (5.135). Assuming that T, = T, = T and pi = p,, we find that (6.40) The parameter He has the dimensions of a frequency and we can identify T , He as a Knudsen number. For a thermal instability we require that

He < 0 ,

T~\H,I> $ ( w . i q ) - ' .

(6.41)

To simplify the account, we shall assume that the pressure gradient is negligible, in which case (6.38) reduces to

He = 2eneB The equilibrium condition is j,Bo = jeBz,SO that j2= j," (1

+ B i / B : ) , whence (6.42)

It follows that He will be negative if j2 > j , J , which is a necessary condition for an instability. As j > j,, then He will certainly be negative if j , > J . By (6.37) this condition can be

Figure 6.2: Stable and unstable current profiles

6 Extensions of Theory

174

written

which requires the local value of j z ( r )to be greater than the average value of the current up to that radius. Assuming w C i ~toi be large enough to satisfy (6.41), then the curves in Fig. 6.2 show stable and unstable current profiles with the arrows indicating the direction of the heat flux. The convex section of the profile evidently satisfies J > j,, so if Bo,,/B, is small, it is stable. The concavelconvex profile has J < j , up to some point 5’ on the profile. The heat flows up the temperature gradient in this region, so the temperature profile becomes more peaked. The reduced resistivity at the temperature peak allows the current density to increase, making the distribution even more unstable. And so on, until the steepening process is checked either by the ion-acoustic instability or by radiation losses, or by an MHD instability that destroys the temperature profile.

6.2.3 Planar geometry From (6.37) J ( r ) -+ 0 as T

4

m, reducing (6.42) to

-PLlj2 (6.43) He = 2en, B ’ which is exact in planar geometry. This is a good approximation for flux tubes that carry electric currents in narrow annular regions on their surfaces. The reduction in resistivity due to ohmic heating tends to channel electric currents, so this is often an accurate model. From (6.41) and (6.43) it follows that heat will flow inwards, i.e. up the temperature gradient if 5 ~ ~ ~ i p o j ’ / ( 8 n , m>i )1, i.e. if

(z)

4 . 6 4 ~ 1 0 T,Ti ~~ S In A j>l.

At Ti = Tethis constraint can be expressed j

> jCT

j,,

= 2.15x10-’71nh(n,/T,)S.

(6.44)

The streaming instability (see 84.6.4) limits the current density to a maximum value of j , = en,(kBT,/m,)f z 6.26x10-l6n,Te 4

,

(6.45)

and it follows that heat will flow up the temperature gradient if j,,, > j C r . For example at l n h = 20 (a value typical of the solar atmosphere), the condition for a thermal instability becomes

T, > 0.83~1, a.

(6.46)

Thus provided T, >> 0.83na, from (6.39)and (6.43) the heat flux is (6.47)

6.2 Thermal instability

175

6.2.4 Heating the solar corona A typical temperature for the quiet solar corona is

-

1.5%lo6K, whereas the photosphere

- the likely source of the thermal energy - has a temperature less than 6x103 K. Although many theories have been advanced to explain why the corona is so much hotter than the photosphere, this old problem remains unsolved. However, there is a mechanism based on second-order transport that may provide the answer, or at least part of the answer. From observations we find that the condition in (6.46) is satisfied in the solar atmosphere extending upwards from the upper chromosphere. Plasma loops in the solar atmosphere (see Fig. 6.14) exist with a wide range of temperatures and in particular there are loops that are hotter than ambient in the chromosphereltransition region (lo4K < T < 5 x105 K) and cooler than ambient in the corona (106K < T < 2x106K)(see Bray et al., 1991). There is relatively little variation in the temperature around a given loop. The theory of the previous section suggests the mechanism illustrated in Fig. 6.3; heat is transported upwards from the upper chromosphere and transition regions, using appropriate flux tubes as conduits. In the region below the transition region, the loop temperature is greater than that in the ambient plasma and by second-order transport heat flows up the temperature gradient, across the magnetic field into the loop. This causes a local increase in temperature, and a relatively small temperature gradient is established around the loop that conducts the thermal energy up to that part of the loop high in the corona. This stage is ‘normal’ heat flux, i.e. along the magnetic fields lines and down the temperature gradient. High in the loop the situation is the reverse of that in the chromospheric region, that is the temperature in the loop is less than that in the ambient corona and heat therefore flows out of the loop into the hotter corona. This mechanism can easily supply the power of about 3 x102 Wm-2 required for the quiet corona and also the 104Wm-2 required for the active corona. As our purpose here is merely to introduce an important application of second-order transport theory, we shall omit the details. Piasma loops are sometimes found at temperatures greatly in excess of the ambient coronal temperature, a phenomenon that could be explained by an accumulation of energy N

corona

chromosphere

Ta <

Tp <

in

T,

Figure 6.3: Transporting heat to the corona via plasma loops

6 Extensions of Theory

176

via the second-order thermal instability. This energy cannot be transported into the cooler corona plasma, so it is trapped until either removed by conduction to the photosphere or perhaps released in a flare. Thermal flares occur in plasmas with temperatures in the range 30 - 50 x lo6 K (Tandberg-Hanssen and Emslie, 1988), and it is a reasonable assumption that at least some flares are a consequence of the thermal instability described in this section.

6.3 Particle orbit theory 6.3.1 Rate of change of the peculiar velocity The peculiar velocity, c = w - v, was introduced in § 1.2.2 and we now introduce the ‘agitati~n’~ acceleration, F,defined as being the difference between the particle acceleration w and its value averaged over all the particles of the same species: (6.48) Long-range forces like those due to gravity, macroscopic electric potentials and so on, affect w and % alike and cannot contribute to 3.This acceleration is due to the short-range, impulsive forces effective during molecular collisions. The fluid acceleration Dv is equal to the average value of the particle accelerations, so that (6.48) can be written

-

F=W-Dv,

W=DV

d (DF-+v*V). at

(6.49)

A particle p leaves the convected frame P, with an initial peculiar velocity c relative to it. Let dc denote the change in P’s velocity after a time dt, as seen in P,, a frame that rotates with angular velocity 52 and moves with a velocity v relative to the laboratory frame L. Thus, measured in L, the total change in c is dv dc 52 x (c dt), which equals dw, i.e.

+ +

d c + f l x cdt = dw-dv = wdt -

dV (at +w.Vv)

dt

= wdt-(Dv+c*Vv)dt,

or

dc

=

(&-DV)dt-c*(VV+C! x 1)dt.

By (6.49), (1.13), and (1.17) this can be written

dc = -c * e dt

+ F dt

( e = $ (Vv + %)) ,

(6.50)

whence &= -c-e+F,

.

(6.51)

gives the rate of change of c in the frame P,. Notice that the term -c e arises from the dejnition of c; as P moves through the sheared fluid, there are continuous changes in the velocity of the ambient fluid, so that the origin from which c is measured is likewise changing. 4This was the term used by Maxwell to describe molecular motions relative to the mass motion.

177

6.3 Particle orbit theory

With charged particles, in the presence of a macroscopic magnetic field, the Lorentz force adds a term to the right-hand side of (6.51), since by (2.32) w includes the Lorentz term .

Q

W, = - (E

m

Thus

Q + (V + C) x B) = WL+ -C m

x B.

Q C = -c x B - c . e + 3 .

(6.52)

m

As 3is proportional to the collision frequency, 7-l and (6.53) and r, where 7 is the macrothe time-scales for the right-hand terms in (6.52) are w;' 7, scopic time-scale for changes in the fluid velocity. The Knudsen number is k, = r / T = T I lei 1, hence the orders of the three terms in TC are W,T, k,, and 1, from which it follows that provided w,r >> ( k , l),we may adopt the approximation )

)

(6.54) By (2.37) the constraint w,r >> ( k N )1) is equivalent to T, << ( 7 C ) r C ) , where T, is the Larmor radius and C is the thermal speed. Sound waves move at about this speed, so 7 C is a macroscopic length scale and the constraint requires the Larmor radius to be much smaller than this length.

6.3.2 Guiding centre drifts Let x denote the position vector of the particle p relative to P,, then f = c and (6.54) can be expressed

2 = w,?

x b

+ xll .

(6.55)

Now assume that B is steady and uniform, in which case the solution of (6.55) is

c = jC = W,(X - X)x b + cII

(cII

where XI, is arbitrary. Choose X,, so that

bb * c),

a=x-X

(6.56)

(6.57)

is always perpendicular to b, then A,,= 0, i.e. XI,= c,,b, and the derivative of (6.56), i.e. X

= wc(k

-

X) x b

+ C,,

)

is the same as (6.55) if X = XI,and C,, = c l , b = 0. Hence X, = 0 and X = 0. It now follows from (6.55) to (6.57) that

a = -wca,

a = w,lb

x c,

a = cl/w,

,

(6.58)

6 Extensions of Theory

178

which describe the motion of a vector rotating about an end point G at X (relative to P,), with angular velocity w,. The point G is the guiding centre, introduced in $2.2.2. The situation is depicted in Fig. 6.4. The point G follows p along the magnetic field lines, but relative to the fluid particle centred on P,, has no motion perpendicular to the field. Because r, << d!, the particle motions are effectively collisionless and hence pressure gradients have no place in particle orbit theory (see $ 1.3.4). Instead of a ‘fluid’ velocity, we now have a ‘particle average’ velocity in its place. The motion of P, perpendicular to the field follows from the collisionless form of (2.53), namely eDv = ~ n , ( E + vX B)

+ Qg,

for each of the components. Hence the velocity perpendicular to the field is

m vL = E x b f -(g-Dv) B QB

xb

(6.59)

Let (6.60)

u=X+P=X+v

denote the guiding centre velocity in the laboratory frame, then, since X, = 0, we have established that in a steady, uniform magnetic field,

The guiding centre motion perpendicular to the magnetic field is independent of the particle velocity and is equal to the fluid velocity, v,.. The physical reason for this is that each guiding centre is never more than a Larmor radius, r,, from its particle and because r, is relatively very small, the speed at which particle mass is transported, i.e. the ffuid velocity, must be the same as the transport of guiding centres. Therefore, provided the constraint on r, spin =: - w b

c_3‘

0

Figure 6.4: Guiding centre motion in a uniform field

179

6.3 Particle orbit theoly

applies, the condition ul = vl is generally true whatever forces are responsible for the fluid and guiding centre motions. Perpendicular motions of guiding centres are called ‘drifts’, and these are named according to the forces that cause them. Thus the first two drifts in (6.59) are ‘electric’ and ‘gravitational’. In general a force F‘ causes a drift u:, and conversely a drift u: requires the presence of a force F‘, where

FI = -Q~Bx u ’

m

: U

m

= QBF’ x b.

(6.62)

6.3.3 Drifts due to variations in the magnetic field There are also drifts caused by magnetic field inhomogeneities that depend on the particle’s peculiar velocity. The physical reason why a gradient in the magnetic field strength B can cause a charged particle p to drift becomes apparent on tracing the orbit of p in a plane orthogonal to the field. Assuming that in a frame Po the electric field is zero and p moves with constant speed, we see that the reduction of the Larmor radius in the stronger field and its subsequence increase in the weaker field, periodically repeated, drives the guiding centres in a direction orthogonal to both b and VB (see Fig. 6.5). From this description it seems that the ions and electrons drift in opposite directions and could therefore generate an electric current in addition that required by Amptre’s experimental law. However j must equal ene(vi - ve) by definition and equal V x H by the law, i.e. for the geometry of Fig. 6.5,

j =ene(vi - v e ) = V x H = - b x V H ,

(6.63)

and the ion and electron drifts cannot alter this basic relation. In a frame P,i convected with the ion fluid, vi = 0 and from (6.63)

(6.64) Comparing this equation with (6.62), we conclude that in P,i there is an electromagnetic force F:, acting on the electrons and no comparable force acting on the ions. Hence in determining B (weaker)

0

0

0

0

0

0

electrons

0

0

0 e

00000000 000000000 B (stronger)

Figure 6.5: Grad B drifts in PO

6 Extensions of Theory

180

Figure 6.6: Relative ion drifts in the convected frame

the additional influence of magnetic field inhomogeneities on particle drifts, it is essential first to remove the electromagnetic force by choosing an origin convected with P,. Let P, be a frame convected across the magnetic field at the fluid velocity and suppose that ZL denotes the average orbital speed for a particle that has no drift relative to P,, then particles with cL > ZI will have larger Lannor radii than average and hence be more affected by the gradient in the magnetic field than the particles for which cI < ZI.Relative to P, the two groups will therefore drift in opposite directions, as depicted in Fig. 6.6 for ions.

6.3.4 Gyro-averages Referring to Fig. 6.7, we have for the value of B at the particle p,

B = B,

+ a -VB + O(Icz),

where B, is its value at the guiding centre and Ic, is the small number defined by

IcL

= / --1nB w, dt

I

= Ja.VlnBI << 1

(6.65)

Thus (6.54) gives .

Q

Q

cI = -c x B, - -a.VB x c + O(Ic2). m m

(6.66)

To solve (6.66) we adopt the method of variation of parameters, that is we start with the solution for the case B = const., which by (6.56) may be written

C = -Q ( X - X ) X B , + C ~ ~ + ~ , ,

(6.67)

m

where X and d, are constants for B uniform, but will depend on (xlt ) otherwise. In a frame following the particle, B = DB c VB = -V x E' c VB M c VB if IE'1 << cjB1, where E' is the electric field at P,. This approximation, known as the

+

-

+ -

6.3 Particle orbit theory

181

Figure 6.7: Motion of a particle about its guiding centre ‘weak electric field’ constraint, requires that in a gyro-time wCi,the change in the particle velocity due to the electric field E’ is small compared with the peculiar speed c, i.e. ( ~ E ’ / m ) w ; l= E’/B << c. Differentiating (6.67) and adopting this constraint, we obtain Q Q Q i: = -c x B - -X x B + -a x (c * VB) + C,, + d, , m m m

(6.68)

where the location subscript G is not longer essential. Subtracting (6.66) from (6.68) we find

Q * Q F = -B x X = - [ c m m

VB x a - a . VB x c] -el,

-

d,,

where by (6.62)1 F is the force on the particle due to the inhomogeneity. By (6.62)2 the corresponding drift is

m X, = $ { c - V B x a - a - VB x c} x b + -b QB

X

(ell +d,),

(6.69)

of which the average over one complete orbit, i.e. over all values of 8, will give the average drift velocity due to the magnetic field gradient. The Larmor vector a = x - X is given by (6.67),

m m a = -b x ( c - d,) x -b x c , QB QB

(6.70)

where the approximation holds if Id, I << IcI I, which is equivalent to the constraint in (6.65). Let i be unit vector parallel to a and hence orthogonal to 8 and b (see Fig. 6.7), then for present purposes it is sufficiently accurate to assume that i and 8 are distributed isotropically, i.e. that their values averaged over one complete gyration are zero. We shall denote gyroaverages by (. . . ) and averages over all speeds by a bar. From the average of 1 A = (Pi 88 bb) A , where A is any second-order tensor, it follows that (cf. the similar averages in 51S.3.)

.

(ti)- A = (88) A = +(A a

+ +

- bb * A) =

.

$(l - bb) . A

=

+A,.

(6.71)

6 Extensions of 7'heoly

182

- Vb = 0, we have

.

By (6.70), (6.71), a = (cl/wc)P, and P Vb = 0,

2

= 2 ( ( ? f ) -VB wc

- (Pi):VBb)

= -{V,B-V,.Bb}, c', 2WC

and since V,

*

-

B b = -Vll B b = -VllB, we get

c: (a.VB x c) = -VB 2%

.

(6.72)

-

A similar calculation yields (c VB x a) = 0. We also require the gyro-average of ell x b. Adopting the weak electric field constraint, we have

x b = ~llbx b = cII (Db x b + c * Vb X b) x Hence, as ( c , ) = 0, we find

(~llbx b) = (cII (c,

+ CIIb)) - Vb x b =

C:

CIIC

*

Vb x b .

b *V x b .

(6.73)

(6.74)

It follows from (6.72) and (6.74) that the gyro-average of (6.69) is 2

(a,)= 2%

C2 1 b x VlnB- LbmVb x b + - b WC

WC

xd,

(6.75)

If this equation is further averaged over all speeds, according to the discussion in $6.3.2 and 36.3.3, the result must be zero in order to satisfy vI = 0 and (6.61)1, which constraint enables us to evaluate the constant d, . For this purpose we introduce the kinetic definitions,

-

gI = L2ec,, 2

-

6311= eci ,

(6.76)

where we have used g to imply a resemblance with the perpendicular and parallel pressures, p , , pll,which are defined similarly for a collisional plasma (see (1.30)). Then (6.75) becomes 1 QnB

6 u = -{ ( ~ e c ~ - g , ) b x V l n B (- e c ~ - p l I ) b . V b x b } ,

(6.77)

which applies to both the ion and electron components.

6.3.5 The grad B and field curvature drifts The first right-hand term in (6.77) is (6.78)

6.3 Particle orbit theoty

183

which is called the ‘grad B’ drift. Notice that its direction relative to the fluid particle P, depends on the sign of (iec? - p,), and that if c, = ZL, where ZL = (2pL/e)h there is no drift at all. As 6u, is zero, this drift has no effect on electric currents or fluid velocities. In the usual account of this drift p1 is neglected, allowing the misnomers ‘diamagnetic’ drift or ‘diamagnetic’ currents to be introduced (see exercise 6.8). The second term on the right-hand side of (6.77) is (6.79) By (2.83) the field line curvature is

where n is unit normal to the field line and R is its radius of curvature. For this reason (6.79) is termed the ‘field curvature’ drift. A more convenient form for 6u, follows from b - Vb = -b x V x b = b x V h B x b - B-’b x V x B

= V, 1nB

+ B p 2 p 0j x B ,

so that

.

-b Vb x b = b x V In B

+pojL/B.

Adding (6.78) and (6.79), we obtain the total relative drift,

6u =

1 -{QnB e(3c; + cp) - (pL+ Qn)}bx V In B + ”{ eci - pS}j, . (6.81) QnB2

This drift is in addition to that calculated in 56.3.2, so in place of (6.32) we now have u=v

+ 6u + cllb

(6.82)

for the relation between the guiding centre velocity u and the fluid velocity v. The mean guiding centre velocity, ii, is obtained by averaging u over all particle speeds. It follows from 6u = 0 and C,l = 0, that in a theory correct to first order in k, (see (6.65)), ii = v,

(6.83)

confirming what is otherwise physically obvious, namely that particles and their guiding centres remain close together when k, is small.

6 Extensions of Theory

184

6.4 Magnetic mirrors 6.4.1 Constants of motion of gyrating particles Because of their gyratory motion, each charged particle behaves like a dipole, with a magnetic moment' M given by

M = - mc2, 2B '

M=-Mb,

M=.&

nB'

(6.84)

where we have included an expression for the average value of the moment for a given species of particle. Provided the magnetic field changes slowly enough, the particles tend to move so as to enclose a constant magnetic flux within their Larmor orbits, i.e. T
cL

Q Q - el = -cI - (c, + c S )x B - -a - V B x (c, + c S )- c, m m

+

Q -a. V B x cI - c I I . m m Therefore by (6.72) the gyro-average is

- -- cII - c L x B

.ell) = -" cll.VB.

(6.85)

d l 2 d -(~mc,)=~mc~cll~VInB=~mc~-(lnB), dt dt

(6.86)

(cL .C,)

Q

= - c I I . ( a -V B

m

x c,

m 2w,

Therefore

whence

dM - -0, dt

M--

mc2, 2B

The force acting on a particle in the convected fluid frame is always at right-angles to its motion, from which it follows that its energy is constant. Thus there are two constants of the particle motion, namely

- const., M = - mc2, 2B

E = imc2= const..

(6.87)

In general neither of these quantities are exact constants. They require variations of B across the Larmor radius a to be small, i.e. k, << 1 (see (6.65)), and of course collisions must be relatively rare, which in this context means that k, << W,T (see (6.54)). $The parameter M is the magnitude of the magnetic moment vector M = -Mb, so named for the following reason. By definition a loop carrying a current I around an area A has a magnetic moment M = IA, where A has the sense of a right-handed screw following the current. From Fig. 6.4 we see that A and b are antiparallel, i.e. A = - r a 2 b for a current circulating at the Larmor radius a. As Q/%ra is the corresponding charge per unit length distributed around the current loop, I = c l Q / 2 r a , and the value given for M follows.

6.4 Magnetic mirrors

185

It follows from (6.86) and (6.87) that

and therefore

6,, = -frmc:V,, In B = - M V , , B . Hence we have the so-called mirror force, F,, = - M V , , B ,

(6.88)

which repels particles from regions of increasing magnetic field strength. It is important to notice that as F,, is calculated in the convected fluid frame, this force has no effect on the fluid velocity. It follows that neither the electron nor the ionfluid velocities can be affected by the mirror force and as the electric current density depends only these velocities (see (2.30)), magnetic mirrors cannot influence j.

6.4.2 Magnetically trapped particles An immediate consequence of (6.87) is that as a particle p approaches a region of magnetic field of increasing strength, the increase in c, required to balance the increase in B can be found only at the expense of cI,. Thus p’s guiding centre G has a reducing value of cI,and eventually it may stop, reverse its motion and move away from the strong magnetic field region. For this reason the term ‘mirror’ is an apt description of the region where B has its maximum value. Two mirrors, as depicted in Fig. 6.8, make a ‘magnetic bottle’ and some of the particles within the bottle will be trapped, with their guiding centres oscillating between the mirrors.

I,

BO

Figure 6.8: Magnetic bottle

mirror

6 Extensions of Theory

186

By (6.87) c,, and B are related by E = imcp

+ M B = E(sin26 + M B / E ) ,

(6.89)

where cos26 = c",g

+ cp).

Here 19 is the pitch angle of p's trajectory, measured from a plane perpendicular to the magnetic field. Thus as p approaches the mirror field, B may increase sufficiently to reduce its value of cII to zero, at which point p is reflected from the mirror. Let B, be the maximum field strength of a mirror, then particles with E < MB, will be reflected, and hence in a symmetrical field as shown in Fig. 6.8 they will be trapped. It follows that those particles in any region where the field strength is B , and having pitch angles satisfying

will escape through the magnetic 'throat' at the mirror. The boundary between captured (0 < 19 < a ) and passing ( a < 6 < x / 2 ) particles is determined by the pitch angle a.

6.4.3 Fraction of trapped particles Consider the magnetic bottle of Fig. 6.8. Let a, be the critical pitch angle at the point of minimum magnetic field Bo,then by (6.90) the bottle holds all particles for which

0 < 6 < a,,

C O S ~C Y ~= R, 1

(R,

E

B,/Bo),

(6.91)

where R, is termed the 'mirror ratio'. Collisions will steadily scatter these particles into one of the two loss cones, a, < 6 < n/2, and -n/2 < 6 < -ac, allowing the bottle to leak slowly on the collision time-scale. A simple expression for the fraction of trapped particles can be deduced by assuming that they are part of an equilibrium distribution. With a velocity-space, spherical coordinate system (c, 8, p), oriented so that cII lies along the axis from which B is measured, the element of solid angle is sin 8 d8 dp, and the trapped particles lie in n/2 - a, < 8 < n/2 a,. Then by (1.77) the fraction of trapped particles is

+

sin 8 dB

1"4

fi

v2evy2dv, (6.92)

That particles can be trapped by magnetic fields has important consequences for energy confinement in toroidal magnetic fields.

6.5 Partially ionizedplasmas

187

6.5 Partially ionized plasmas 6.5.1

Degree of ionization

To this stage we have not dealt with the problem of determining whether or not the gases being studied actually constitute a plasma. Of course the gases have to be hot, but how hot? To answer we need to know about the physics of partial ionized plasmas. Unlike ions, neutral particles are not directly confined by magnetic fields, but provided there is a sufficient proportion of ions in the plasma, the neutrals may be adequately confined by collisions with these ions. A preliminary answer can be obtained from results given in exercises 3.5 and 3.6, on page 95, where the propagation of waves through a partially ionized gas was described. For waves of frequency w the parameter linking the ion and neutral gases (of the same species) is found to be (6.93) where ui, and v,i are the ion-neutral and neutral-ion collision frequencies, e is the plasma density, en is the neutral gas density and i = If 5 << 1, k‘ M 1 ,on/@,which means that at least as far as waves are involved, the medium behaves as if the mass of the neutrals were added to that of the ions. In this case, v,i >> w and collisions lock the neutrals into the plasma, which is thus confined along with the plasma by the magnetic field. In the other limit, 5 >> 1, in which case C x 1 ivi,/w and the ions and neutrals are nearly independent, To obtain this limit we need v,i << w,but heat could be lost across a magnetic field by being passed from the ions to the neutrals and thence out beyond the flux tube. To simplify the account of ionization we shall assume that the plasma is comprised of electrons, hydrogen ions and hydrogen atoms. The relatively small energy required to dissociate the molecules will be ignored. Let n,, ni, and n, be the number densities of the electron, ion and neutral gases; by charge neutrality, n, = ni. The degree of ionization, x , and total gas pressure are given by

m.

+

+

x = ni/(n, + ni),

p = nt$T,

(nt

n,

+ ni + n,) .

(6.94)

We shall write the ionization energy per particle as kBT*.For hydrogen this energy is approximately 13.60 electron volts ( l e v x 11605K), so that T* = 1 . 5 7 8 ~ 1 K 0 ~. Applying the law of mass action between hydrogen molecules, hydrogen ions and electrons, we can find the conditions under which recombination and ionization occur at equal rates. The resulting equilibrium equation is due to Saha (1921) and named after him. To a very good approximation the steady-state Saha equation can be written

(~7’~

ni = 1 2xm, nn

Pe

(lc,T)5’2exp(-9?),

(6.95)

where p , = n,kBT is the electron pressure, me is the electron mass, h is Planck’s constant and % 5 T * / T .From (6.94) (6.96)

188

6 Extensions ofTheory

Figure 6.9: Ionization degree in Hydrogen for several pressures

and hence (6.95) can be expressed

Let

M

G

kBT

-r2

(

h2 2rm, Ic,

(6.97)

T-3/2exp(T*/T)

x 30~T-~/'exp(T*/T), then

X = (l+M)-'.

(6.98)

For example, in the solar photosphere, p x 104Pa, T x 6,00OK, therefore = 1.87Pa, and ni = 1 . 8 7 ~ 1 0 - ~ n , .In Fig. 6.9 the function x ( T )is plotted for a range of pressures, the values of which (in Pascals) are shown in the curves. The remarkable feature is how short is the temperature range between zero and complete ionization.

M = 2.95x107, x = 1 . 8 7 ~ 1 0 - ~pe,

6.5.2 Ratio of the specific heats The internal energy per unit volume of the mixture of gases is

189

6.5 Partially ionizedplasmas

where @ = mini + m,n, Hence by (6.94) =RT{i

and

+ meneM

+ x($ + 3))

p = ReT(1-t

The enthalpy is h = u

m(ni

+ n,), m being the mass of a hydrogen atom.

( R = Ic,/m)

(6.99)

]

x) .

(6.100)

+ p / e , thus

h=RT{$+x($+%)}. It follows from (6.96) that

TXi- = i X ( 1 - X 2 ) ( i +%)]

(6.101)

exe = - $ x ( I- x')

(6.102)

From (6.99) to (6.101) du

4( 1 + x)+ (4 f % ) T X T ) dT } -t RT(

= R{

f

%)Xe d@

+ (2 + % ) T X T ) }dT + +RT($+ % ) X e d @ ] d p = RQ(1+ X + T X T )dT + R T ( l + X + QXe)d e . The entropy density s is defined by T d s = du + p de-' = dh - e-l d p dh = R{2(1+ x )

and

and the ratio of

specific heats follows from

From the above equations we find that

from which it follows that

5 y(X1'i) = 3

+ ~ ( +52%) - 2 X R + Z ( 3 4- 2%) - 2 x 8

+

(3

X ( 1 - X 2 ) ( 1 + Z ) ( 5 + 2%) + Z ( 3 f 2%) - 2X8}{2 + X ( 1 - t X z ) } ' (6.103)

where z

= x + iX(1-

+ %).

In Fig. 6.10 we have plotted y as a function of temperature for several values of the pressure; as to be expected, its value at x = 0 and x = 1 is 513 and in between these limits it falls close to its minimum possible value of unity.

6 Extensions of Theory

190

Figure 6.10: Ratio of specific heats for several pressures

6.5.3

Resistivity

When n,/n, is so small that collisions between charged particles can be ignored, the electrical resistivity is given by (cf. (2.54))

q = - meVen e2n, ’

(6.104)

where uen is the collision fi-equencybetween electrons and neutrals. This frequency is related to the collision cross-section, Qen by6 Ven =

37r

8

(-)8kBT 7rme

’I2

nnQ,,

M

annnT’12,

(6.105)

where a, x 7 . 4 1 6 ~ 1 Q0 e~n and a typical value for Q,, is 2.5 ~ 1 O - rn2. l ~ The total collision frequency for the electrons is ( V e n V e i V e e ) . In a fully ionized hydrogen plasma, vei+vee = where Ve = 1/Te (see 55.3.4); therefore the total collision frequency is approximately (Ven $ y e ) , where from (5.64),

+ +

+

3ve 1 = a,n,TL3’2

(a,

= 1.818~10-~lnA).

(6.106)

The total resistivity is

From (6.98) it follows that the diffusivity, viz. = q/po,is given by (6.107) %ee pp. 258,360 of Chapman & Cowling, (1970).

6.6 Applications of plasma physics

191

Figure 6.11: Magnetic diffisivity for a range of pressures

and from (6.102) its derivative is

m2s-1, ~ For example in the photosphere, where T M 6, OOOK, (6.107) gives [ M 5 . 8 ~ 1 0 while from (6.108) we find that E 0: T-13.4,e.g. a ten percent reduction in temperature increases the difisivity by a factor of 4.1. This extreme sensitivity of difisivity on the temperature is of great importance in the photosphere and the cool chromosphere above it (see exercise 6.10).

6.6 Applications of plasma physics In a small book there is space only to outline some of the important areas to which plasma theory has been applied. Most of the Universe is comprised of plasma, so there is no question as to its great importance in astrophysics, especially so with our nearest star, the Sun with its remarkable plasma loops and sunspots. Laboratory plasmas have been intensely studied in the last 50 years, with the aim of holding very hot deuterium and tritium ions within a toroidal magnetic field long enough for fusion energy to be released. While liquid metals are not plasmas, the equations determining their behaviour in magnetic fields are closely related to those of plasma theory, so we may add metallurgy as being an important field of application. In this category there is also the Earth’s magnetic field and its evolution due to fluid motion in the Earth’s core. Plasmas in which a significant number of dust particles are present is a relatively new field of growing importance; astrophysical examples of what are called ‘dusty plasmas’ include Saturn’s rings and the tails of comets.

6 Extensions of Theory

192

Figure 6.12: Tokamak currents and fields (see text).

6.6.1 Tokamak research A tokamak7 is essentially a toroidal chamber through which passes a strong toroidal magnetic field B, for the purpose of confining a high temperature plasma. This field is produced by external electric currents flowing in coils wound around the torus, as shown in Fig. 6.12(a). Superimposed on the toroidal field is a much weaker poloidal field Be, generated by an electric current I p flowing in the plasma around the torus. The plasma forms the secondary circuit of a transformer, so I p is induced by changing the magnetic flux BT through the torus, usually carried by an iron core, as indicated in Fig. 6.12(b). In a plasma consisting of deuterium, or deuterium mixed with tritium, various fusion reactions that transform binding energy into particle kinetic energy will occur - e.g. the D-T reaction is D2 T3 --+He4 n1 17.6 MeV; it will occur frequently if the ion temperature and density are high enough. For afusion reactor high values of Ti and ni must be maintained long enough for the energy liberated by fusion to more than balance the energy losses due to radiation, conduction, convection and neutron flux. Let T~ be the time it takes these processes to remove all the energy from the system, then for a given value of ni7, there is a minimum temperature at which the plasma is said to ignite, i.e. at which the liberated fusion energy is just sufficient to balance all losses. With n i ~ > , 1.5 x1OZo m-3 s, the ignition temperature is Ti 3.5 x108 K 30 keV. A slightly lower bound (niTE > 6 m-3 s) suffices if power is supplied from outside the system to compensate the losses, which gives the triple product for marginal success as niTi7-E > 2 7 ~ 1 keV 0 ~m-3 ~ s. As D-D plasmas require temperatures about ten times higher to achieve ignition, almost all reactor proposals have concentrated on D-T fusion. Figure 6.12 shows the largest tokamak in the world, the JET machine (Joint European Torus), which was built near Oxford almost 20 years ago. A first approximation to the equilibrium fields and currents for a given geometry follows

+

-

+ +

N

'This word is derived from the Russian for 'toroidal chamber-magnetic': Toroidal Kamera Magnitnaya. The machine was invented in the Soviet Union in the late 1950s.

6.6 Applications ofplasma physics

193

Figure 6.13: The Joint European Toms

from the Grad-Shafranov equation, (2.123), with choices made so as to avoid the various MHD instabilities described in Chapter 4. The more demanding theory is that of calculating the transport properties of the toroidal magnetic field, whose variation in strength creates magnetic mirrors (56.4.2) that trap a large fraction of the particles, collisions being relatively rare. The trapped particles execute banana shaped orbits that allow them to transport energy across the width of these orbits rather than across the much smaller Larmor radii of classical theory (see (6.16)l). This non-local process, which greatly increases the loss of energy, is termed ‘neo-classical’ transport. Unfortunately tokamaks (and similar machines) lose their energy hundreds of times more rapidly than predicted by neoclassical theory and the best triple product achieved in JET is more than a factor of ten smaller than the break-even value quoted above. Besides this rapid energy loss, they are also subject to instabilities termed ‘disruptions’ which impose a serious operational limit on the number density. These phenomena are believed to result from turbulence and many theoretical studies have been concerned with finding the relationship between the instabilities thought to be responsible for the turbulence and the transport of energy and mass across the magnetic fields. An alternative approach, based on the second-order transport described in $6.1.2,was developed by Woods (1987). It explains both the high values of the thermal conductivity and the disruptive instability as well as several other tokamak phenomena. An extensive account of tokamaks and their perplexing behaviour is to be found in Wesson‘s (1996) volume.

6 Extensions of Theory

194

6.6.2 Solar physics The Sun consists mainly of ionized hydrogen and helium with small quantities of heavier elements and except for a shell a thousand kilometers or so thick, concentric with the photosphere, it is fully ionized. The detection of magnetic fields in the Sun dates back to 1908, when Hale published a paper with the cautious title ‘On the probable existence of a magnetic field in sunspots’. His discovery that sunspots contained magnetic fields of the order of several kilogauss, made him the founder of the science of solar magnetism. The magnetic field averaged over the solar surface is of the order of one gauss, but actual fields deviate greatly from the average. Against one’s expectation that magnetic fields should diffuse into low energy, uniform distributions, the magnetic flux intersecting the photosphere, appears in discrete bundles or flux tubes as shown in Fig. 6.13. These tubes form an hierarchy of magnetic elements, which in increasing size and darkness, are known as flux fibres, faculae, magnetic knots, pores and sunspots. Despite the large electrical conductivity of the partially ionized plasma atmosphere in and just above the photosphere, these elements can appear and disappear in hours, rather than in the months that classical magnetic difisivity would predict (see (2.100)).

Figure 6.14: Plasma loops in the solar corona

The Sun’s magnetic activity, notably sunspot numbers and their average latitude, change appreciably in an 1 1-year period, due it is believed to a self-exciting dynamo acting in an alternating current mode. Understanding how the fluid motions deep in the solar interior can create magnetic fields and then interact with them to drive such a dynamo is a major topic of research. The Earth’s periodic magnetic field is also due to a dynamo, although the nature of the fluid involved is quite different. Perhaps the most remarkable discovery in solar physics is the curious temperature profile over the few thousand kilometers above the photosphere (the visible surface). The temperature

6.6 Applications ofplasma physics

195

Sun

Figure 6.15: The Earth’s magnetosphere and bow shock

of the photosphere is a little less than 6 000 IS,and with increasing height it falls to a little over 4 000 K in the lower chromosphere, it then slowly increases with height until at about 2 500 km above the photosphere it Ieaps in what is termed the ‘transition layer’ to about 1500 000 K in the corona. One of the long standing problems of solar physics is what process heats the corona considering that the energy must come from the relatively cool photosphere? Magnetic fields are assumed to be the source of this energy, perhaps via small flares below the resolution scale or maybe by AlfvCn waves propagating up along the flux tubes. A related problem is the nature of the transition layer; what prevents the coronal energy flowing down the very steep temperature profile? There is no shortage of theories, one of which depends on this author’s second-order transport theory, which shows that in some circumstances heat can flow up temperature gradients (see $6.2.4). Much of the research in solar physics is concerned with the structure and dynamics of vast plasma loops that rise up into the corona to heights of 50 000 km or more. Concerning the mechanism of their support against gravity, the consensus is that magnetic fields concentrated into threads play the key role, although the precise configuration of these fields remains a matter for debate. Collections of relatively cool loops sometimes form enormous arches known as quiescent prominences, which can remain almost stationary for many weeks. But if they creep above about 50 000 km they are liable to become unstable and start to rise rapidly, flinging parts of their substance out into the interplanetary space. These missile-like blobs of plasma are called ‘coronal mass ejections’ and on interacting with the Earth’s magnetic field, are the cause of magnetic storms. The solar wind and its interaction with the Earth’s magnetosphere, illustrated in Fig. 6.15,

196

6 Extensions of Theory

is another area of research that requires an understanding of plasma physics. Many books have been written on solar physics, but there are few that emphasize the relevant plasma physics. Priest’s (1982) text is one that does, although the observational side of solar physics has greatly advanced since the advent of scientific satellites such as SOH0 and Ulysses. Bray et al. (1991) is a comprehensive work that deals with plasma loops and also provides an excellent introduction to solar plasma physics.

6.6.3 Engineering applications Although not strictly plasma physics, MHD theory applied to metallurgical problems should be included in our list of important applications. Electrical currents are used in industry to heat metals to the liquid state when these metals can be stirred, levitated and pumped with magnetic fields; electric-arc welding with magnetic fields imposed to control the shape of the liquid pool at the weld is another example; metallurgical MHD is an old and established branch of engineering. The magnetic Reynolds number (see 52.4.4) is quite small, so unlike the case in solar physics, the diffusion of currents and magnetic fields is dominant feature of the subject. Flow in ducts, known as Hartmann flow, have velocity profiles ranging from the classical Poiseuille flow of a viscous fluid to rather abrupt boundary profiles known as Hartmann layers. For an introduction to this subject the recent text by Davidson, (2001) is recommended. Faraday performed the first experiment in MHD when he attempted to measure the voltage induced by the Thames flowing through the Earth’s magnetic field. A related subject of some interest a few decades ago was the ‘direct generation of electricity’ from hot gas, made electrically conducting by seeding it with Caesium (see Rosa (1987)). Instead of passing through a gas turbine in the usual way of power stations, the gas is heated well above the usual gas turbine temperatures and driven by a pressure gradient along a duct across which a magnetic field is applied. This produces a v x B electric field, which generates current directly before the gas passes on through a gas turbine beyond the duct. The extended temperature range yields a higher Carnot efficiency, but the much-too-rapid erosion of the electrodes in the hot gas has proved to be a serious practical problem. A railgun is just about the simplest form of electric motor imaginable. It is a device for the acceleration of an electrically conducting member that is free to slide along the rods or “rails” supplying the current to it. The bare essentials are illustrated in Fig. 6.16, which shows two parallel, conducting rods, electrically connected to a free-sliding shunt, which is described as being the “armature”. An electrical current flows along one rod, across the armature and back along the second rod. The current results in a Lorentz force on the armature that drives it away from its starting position near the breech, towards the muzzle at the opposite end of the rails at which point the armature and any load it may be carrying leave the electrical circuit, or “gun” and move into free flight. The armature may be either a metal block shaped for maximum efficiency or simply a plasma conducting the current between the rails behind the load. Metal armatures and the adjacent rails are melted by the current and in the very small gap a plasma completes the circuit, so here we have a case of MHD and plasma processes occurring in the same engineering device.

6.6 Applications ofplasma physics

197

Figure 6.16: The Railgun Principle

6.6.4 Dusty plasmas If, besides its ions, electrons and neutral molecules, a plasma also contains appreciable quantities of charged dust grains, it is termed a ‘dusty plasma’. Typical grains range in size from nanometers to millimeters, a dimension conveniently characterized by an effective radius T d , and they may be positively or negatively charged, with a charge qd = *&e, where the charge number, z d , can be as large as lo5 or even greater for large grains. There are several possible grain charging mechanisms, including impact ionization, photoemission, thermionic emission and radioactivity, and a central part of the subject is that of determining qd and the charge rate d q d / d t under various ambient conditions. Given the value of qd, the dust number density n d and the average velocity Vd of the grains, the Lorentz force acting on the grain component can be obtained and an equation of motion written for the grain ‘fluid’ similar to that given in (2.53) for the ions. A dusty plasma consists of at least four fluid components, each of which requires conservation and constitutive equations, with the further complication of an equation for the grain charge rate and Saha’s equation for the ionization level of the plasma component. Analytical progress requires some simplifying approximations. Small amplitude waves, shock waves and various instabilities can be studied, much as for the simple case of a fully ionized plasma. Dust is ubiquitous and becomes particularly important when the number density ?zd is large enough to reduce the intergrain distance a to values less than the plasma Debye length A, ($2.2.4). There are several astrophysical examples of dusty plasmas, the zodiacal light due to interplanetary dust being a classical example. Interstellar clouds are also dusty plasmas, where typical parameters are ne zz lo3 m-3, n, M lo1’ m-31 n d M 10-1 mW3, T, w 10 K, Td E 0.2 pm and a/A, I0.3. The rings around Saturn and the other outer planets and the tails of comets are further examples of dusty plasmas. As comets approach the Sun, the collisional pressure of the solar wind generates tails extending away from the comet in a direction opposite to the Sun. The interaction depends on the mass of the particles being collided, so that ion tails are almost straight along the Sun-comet radius, while the dust tails lie at a small angle to this and are slightly curved (see Fig. 6.17). Laboratory plasmas usually contain substantial quantities of dust. For example spluttering,

6 Extensions of Theory

198

Figure 6.17: Comet Kohoutek, 1974

arcing and evaporation at the walls of tokamaks can create substantial quantities of impurities with high values of 2, which greatly increase the radiation losses. But there are also engineering applications,described as 'plasma processing', in which dusty plasmas are useful employed, a notable example being their use in the manufacture of semiconductor devices. Chemical vapour deposition can be enhanced by using low temperature plasmas, ash can charged and thence removed from chimneys and so on - the number and variety of applications are steadily increasing. For a recent text describing the subject see Shukla and Mamun, (2002). Exercises 6. 6.1 A planar flow is sheared in a direction parallel to the temperature gradient, which is orthogonal to the magnetic field B. The fluid velocity is v = u(s)G, where G = b x 1,the inflexions marking unit vectors. In the notation of (6.23) show that H = iv' where the dash denotes alas. Also

show that the second-order heat flux is

and find the condition that this exceeds the first-order heat flux. 6.2 The magnetic field in a cylindrical plasma has the helical structure b M 2

Show that the angular velocity defined by H

EE

.

b x v"v P equals

+ s8, where s << 1.

$ { r(ue/r)' - su:}, where

Prove that the second-order radial heat flux in the electron gas is almost

199

6.6 Applications ofplasma physics

6.3 A cylindrical plasma has a uniform magnetic field B, over 0 < 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 2 = 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.] 4

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)).

6 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=

1

Bo

+ Ecose

(E

= r/Ro),

where 0 is the angle defined in the figure. Show that the mirror ratio defined in (6.90) is ( 1 + ~ ) / (1-6) and deduce that the fraction of particles trapped in -B0 < 0 c: 00 at minor radius T is

6.10 The equation dB = [B” dt

- a[T’B’/T - Bv:, ( a = 3 / 2 ) ,

given in exercise 2.3, p. 65, applies to a fully ionized plasma. Show that for a plasma at T < 6 000 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 1 0

~ ~ - 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= 0 ,

and deduce that f o ( 1

+ 9)is a possible form for the distribution function f.

A3 Show that the energy flux can be expressed as q =pc-

4

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 =

0

/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), 1

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).

6 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 ( - - tanh M Bo M

tanh 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 0 and 2 = T sin 8. 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 1 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 0 = 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 7 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

+

1 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. 1

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 6 divergent waves (upstream and downstream).

6 Extensions of Theory

204

A17 Equation (3.18) lists the 7 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 << IB, I, show that cII M RoqdO/dt,where q = rB,/RoBe. Starting from the relation c i = c2{1 - (cL/c)’}, prove that

where M is the magnetic moment (see $6.4.1). Show that if E << 1, where f00 are the reflection points.

A20 Let n be the time for a particle to travel between the reflection points described in the previous exercise (the “bounce” time); show that

where K is the complete elliptical integral of the first kind.

6.6 Applications ofplasma physics

205

A2 1 Use the results in exercises A 17 and A 1 8 to show that the displacement A r of the particle p satisfies

c l , n r ( e )= -2q {c

2

WC

- -(PA 2

e

+ PII)}{sin2

$00

- sin2

;el,

and show that the average over all the particles of the given species is = O ( 2 )M 0 .

Figure 6.21 shows adjacent banana orbits. At the point Q the different number densities in the counter-streaming particles generate an electric current: bn Arb?The average value of this current over all the particles of the given species is known as the ‘bootstrap’ current, j b t . Show that j b t = 0. Qcll Ar (n1 - R Z )

M

--&cll

Figure 6.22: ‘Bootstrap’ current A22 Two cold electron beams are drifting through a stationary, cold ion background with a relative speed V, i.e. (see exercise 4.10),

Felo = ne16(u-

iv),F,Zo

+ iv),Fio = n i 6 ( u ) .

= nez6(u

Deduce the dispersion relation 1=

/2

Wp2e

z +

(w- lcv/2)

WEe / 2

(w+ k v / 2 ) 2

w& +--. W2

Show that the maximum growth rate occurs when the imaginary part of w equals wpe/J8 and the corresponding wave number is kV = 3w,,/& A23 Show that in planar geometry with a sufficiently strong magnetic field, the heat flux across the

plane OYZ is

Show that the heat will flow up the temperature gradient if

Physics of Plasmas L. C. Woods Copyright 0 2004 WILEY-VCH Verlag GmbH & Co KGaA

Bibliography

[The pages on which the references are cited are given in square brackets. General texts that amplify parts of the subject matter of this book are marked by asterisks and two asterisks mark those that are particularly relevant.]

Bernstein, I.B., Frieman, E.A., Kruskal, M.D., and Kulsrud, R.M. (1958). Proc. Roy. SOC.A 244, 1 7 4 0 . [ 1 141 Bhatnager, P.L., Gross, E.P., and Krook, M. (1954). Phys. Rev. 94,5 11-25. [29] Bleany, B.I., and Bleany, B., (1985). Electricity and Magnetism, (3rd Ed.) Oxford University Press. Braginskii, S.I. (1965). Reviews of plasma physics, Vol. 1 (ed. M.A. Leontovich) pp. 205-3 11. Consultants Bureau, New York. [ 1521, [ 1551, [ 1621 *Bray, R.J., Cram, L.E., Durrant, C.J., and Loughhead, R.E. ( 1 99 1). Plasma loops in the solar corona. Cambridge University Press. [ 1951 Brush, S.G. (1976). The kind of motion we call heat North-Holland Pub. Co. [3] Budden, K.G. (1966). Radio waves in the ionosphere. Cambridge University Press. [94] *Chapman, S. and Cowling, T.G. (1970). The mathematical theory of non-uniform gases. Cambridge University Press. [ 1531, [ 1551, [ 1891 *Clemmov, P.C. and Dougherty, J.P. (1990). Electrodynamics of particles and plasmas. Addison-Wesley, New York. [94] *Cercignani, C. (1988). The Boltzmann equation and its applications. Springer-Verlag, New York. *Chen F.F. (1 984). Introduction toplasma physics and controlledfusion. 2nd edn. Plenum Press, New York. Curson, F.L., Folkierski, A., Latham, R., and Nation, J.A. (1960). Proc. Roy. SOC.A. 257, 388-401. [lo21 **Davidson, P.A. (200 1). An introduction to magnetohydrodynamics. Cambridge University Press. [ 1951 *Dendy R.O., (Ed.) (1995). Plasma Physics: An Introductory Course, Cambridge University Press. Epperlein, E.M. (1984). J. Phys. D:Appl. Phys. 17, 1823-7. [ 1621 *Ferziger, J.H. and Kaper, H.G. (1972). Mathematical theory of transport in gases. North Holland, Amsterdam. [ 153, 1551 Gaber, E., Brush, S.G., and Everitt, C.W.F. (1986). Maxwell on Molecules and Gases. The MIT Press, Cambridge, Mass. [ 1661 *Ginsburg, V.L. (1970). The propagation of electromagnetic waves in plasmas. 2nd edn. Pergamon Press, Oxford. [38], [94] *Goldston R.J. and Rutherford P.H. (1995). Introduction to Plasma Physics. IOP Publishing Ltd., Bristol. Jeffrey, A. and Tanuiti, T. (1984). Non-linear wave propagation. Academic Press, New York. [89]

208

Bibliography

*Kennard, E.H. (1938). Kinetic rheoiy of gases. McGraw-Hill Book Co. Inc., New York and London. [ 1671 **Krall, N.A. and Trivelpiece, A.W. (1973). Principles of plasma physics. McGraw-Hill Book Co. Inc., New York and London. [ 1241 Laval, G., Luc, H., Maschke, E.K., Mercier, C., and Pellat, R. (1971). Plasma physics and controlled nuclear fusion research, Vol 11, p. 507, IAEA Vienna. [ 1151 Litvinenko, Y.E. (1999). Solar Physics. 186,291. [ 1191 Maxwell, J.C. (1860). Phil. Mag. 19, 19-32; 20,21-37. [12], [166] Maxwell, J.C. (1867). Phil. Trans. Roy. SOC.157, 49-88; (See also ScientiJicpapers of James Clerk Maxwell, W.D. Niven, ed. New York: Dover, (1965), vol. 2,43.) [ 1571 Newcombe, W.A. (1960). Ann. Phys. 10,232. [115] Ockendon, J., Howison, S., Lacey, A. and Mochivan, A. (1999). Applied partial dzfferential equations. Oxford University Press. [ 1271 *Present, R.D. (1958). Kinetic theory of gases. McGraw-Hill Book Co., New York. **Priest, E.R., (1982). Solar Magnetohydrodynamics. D. Reidel Pub. Co. [ 1951 *Raizer, Y.P. (199 1). Gas discharge physics. Springer-Verlag, New York. Robinson, B.B. and Bemstein, I.B. (1962). Annals of Phys. 18, 110. [ 1521 *Rosa, R.J. (1987). Magnetohydrodynamic energy conversion. Hemisphere Pub. Corp., New York. [ 1951 Rosenbluth, M.N., MacDonald, W.M., and Judd, D.L. (1957). Phys. Rev. 107, 1. [140] Saha,M.N. (1921). Proc. Roy. SOC.A99, 135. [186] Schrijver, C.J. and Zwann, C. (2000). Solar and stellar magnetic activity. Cambridge University Press. *Shkarofsky, I.P., Johnston, T.W., and Bachynsky, M.P. (1966). The particle kinetics of plasmas. Addison-Wesley, Reading, Mass. [ 1521 Shukla, P.K. and Mamun, A.A. (2002). Introduction to dusty plasma physics. Institute o f Physics Publishing Ltd. [ 1971 **Spitzer, L. (1962). Physics of fully-ionized gases., 2nd. edn. Interscience, New York. [1451, [I491 Suydam, B.R. (1958). Proc. 2nd U N Conf on the Peaceful Uses ofAtomicEnera, Vol. 3 1, p. 157. United Nations, Geneva. [ 1151 Tayler, R.J. (1957). Proc. Phys. SOC.B 70, 1049-63. [lo71 *Tandberg-Hanssen, E. and Emslie, A.G. (1988). The physics of solarflares. Cambridge University Press. [ 1761 *Wesson, J. (1997). Tokamaks,2nd edn. Clarendon Press, Oxford. [ 1691, [ 1921 Woods, L.C. (1983). J. Plasma Phys. 29(1), 143-154. [169] *Woods, L.C. (1986). The thermodynamics offluid systems. Oxford University Press. **Woods, L.C. (1987). Principles of magnetoplasma dynamics. Oxford University Press. [761, [771, [891, [1071, [1111, [llgl, [1451, [1691, [1731, [I971 *Woods, L.C. (1993) An introduction to the kinetic theory of gases and magnetoplasmas. Oxford University Press, Oxford. [ 1551, [ 1671 *Woods, L.C. (1996). Thermodynamic inequalities in gases and magnetoplasmas. John Wiley & Sons, Chichester. [46], [98] Van Vleck, J.W. (1932). The theoiy of electric and magnetic susceptibilities. Oxford University Press. [38]

Physics of Plasmas L. C. Woods Copyright 0 2004 WILEY-VCH Verlag GmbH & Co KGaA

Index

adiabatic conditions 22 Alfvh waves 71,73,74, 101 hodographs for 76 AmpBre’s law 34, 37 anomalous dispersion 82 Avogadro’s number 12 Bennett’s relation 57 Bohr-van Leeuwen theorem 38 Boltzmann’s collision term 160 constant 11 distribution law 29 boundary conditions 62 for fluid equations 64 no-slip 65 Braginskii’s transport formulae 152, 162-163 bulk viscosity 156, 159 Chapman-Kolmogorov equation 136 charge separation 64 collision frequency 144 electron-ion 133, 143, 145 collisionless plasma 120 collisionless screening length 62 collisions binary 132 electron-electron 145 electron-ion 43, 145 grazing 3, 133, 135 comets, dusty plasma in 197 conduction of heat 167 conservation laws 19-22, 34 convection 18, 167 convective derivative 35 corona, temperature of 195 Coulomb collisions 132-135, 137 Coulomb logarithm 139, 144, 145

current sheets 64 cylindrical coordinates 67 Debye length 41, 62, 131, 139 deflection time 149 diamagnetic drift, absence of 183 dielectric tensor 93 diffusion coefficient 19, 137, 146, 148 v. convection 167 diffusivity, magnetic 52 discontinuities, types of 63 dispersion curves 74 dispersion equation 70,7 I , 78,80, 97 for MHD waves 76 for surface waves 100 in colIisionless plasma 122 electrical conductivity 143, 144 electromagnetic forces 1,48-50 pressure 49 stress tensor 48 electron inertia 63 orbits 135 vorticity 126 electrostatic perturbations 121 energy conservation of 21 equation, Lagrangian form of 46 exchange time 150 principle 1 10- 1 14 transmitted in waves 8 1 transport of 165 enthalpy, flux of 166 entropy and stability 23 density of 189

210 kinetic 24 production rate of 23 specific 22 equipartition time 150 evanescent waves 75 Faraday’s law 34 fast MHD waves 74 fibrillation instability 126 flares 176 fluid definition of 4 particle 4 spin 7 velocity 5 , 6 vorticity 6, 72 fluid element convection of 35 dilatation of 9 rate of strain of 9 flute instability 103 Fokker-Planck equation 136-138, 141, 142, 160 force-free magnetic field 59,60 Fourier’s law 153-155,165,167 frame-indifference 19, 35, 167 friction coefficient 137, 146, 147 frozen field condition 54 fusion reactor 192 Galilean transformation 35 grad B drift 183 Grad-Shafranov equation 59, 193 group velocity 80-8 1 guiding centre 40 drifts of 177-1 83 and Ampbre’s law 179 Hall term 44, 74 Hartmann flow 196 heat flux second order 167 up temperature gradients 172, 174, 195 v. energy flux 17 vector 19,46 heat, conduction of 166 helicity 66, 127 helicon waves 75

Index

Hoffmann-Teller jump conditions 83 impact parameter 132, 133 induction equation 50, 53 and backwards heat 126 instability absolute 97 convective 97 effect of magnetic fields on 101 entropy, role of 98 fibrillation of magnetic fields 126 flute 103 in cylindrical plasmas 108 interchange 103, 110, 115 ion-acoustic 124, 125 Kelvin-Helmholtz 102 kink 107 of current profiles 173 of cylindrical plasma 106-109 pinch 109 potential energy 97 Rayleigh-Taylor 97, 98 resistive 1 I5 sausage 106 streaming 97 tearing mode 1 17 two-stream 120 internal energy, flux of 166 ion cyclotron wave 74,75 ion inertial length 72 ion-acoustic instability 124-125, 174 growth rate of 126 ion-cyclotron frequency 72 ion-sound speed 123 ionization 145, 187 isentropic flow 22 Kelvin’s theorem 54 Kelvin-Helmholtz instability 102 kinetic entropy 24 kinetic equation BGK 29 collisionless form of 120 Fokker-Planck 136 kinetic theory 23-30 kink instability 107 Knudsen number 4, 17, 166, 169, 173, 177 Kruskal-Shafranov stability 110

Index Landau damping 123, 124 Larmor radius 40,52, 177, 179 linearpinch 56, 102 Liouville’s theorem 30 Local Thermo. Equil. 17-1 8 Lorentz force 39, 177 Lorentzian conductivity 144 plasma 141-143 magnetic axis 55 bottle 185 diffusivity 52, 191 flux, conservation of 54 helicity 65 mirrors 105, 184-186 particles trapped in 185 stability of 105 moment 184 Reynolds number 53 storms 195 surfaces 55,58 tension 50 magnetic fields diffusion of 53 force-free 59 islands 118 toroidal 186 magnetization current 37 magneto-acoustic waves 73, 76 magneto-ionic theory 92-94 magnetoplasma, definition of 33 magnetostatics 54-59 mass, conservation of 2 1 Maxwell and convection 166 and viscosity 157 EM equations of 33 statistical mechanics 5 Maxwellian distribution 27,29 mean free path 3 MHD conservation laws in 44-46 definition of 38,63, 73 dissipation 45 equations 33,42 ideal 70 in metallurgy 196

21 I instabilities 97-1 15 MHD shock waves 76 classification of 87, 88 fast 88 parallel shocks 89 perpendicular 89-91 slow 88 stability of 89 switch-off 88 switch-on 88 weak 87 MHD waves 73-82 cut-off 74 experiments involving 77 Friedrichs diagrams for 76 fronts of 77 group velocity of 80 hodograph 76 in cylindrical plasmas 77-79 intermediate 74, 75, 79 resonance in 74 torsional 77 types of 69,72 whistlers 75 microfields, electric 131 minimum B configuration 105 mirror force 185 molecules, soft and hard 3 momentum conservation of 2 1 transport of 155, 165 neo-classical transport 193 number density 5 Ohm’s law 4 3 4 4 , 1 5 2 particle collisions 133 orbit theory 133, 176-183 peculiar velocity 9 rate of change of 176, 177 perfect gas I2 permeability 34 permittivity 34 perpendicular shock waves 89,90 phase space, defined 5 pinch instability 109 plasma beta 74

212 plasma collisionless 4 dusty 197-198 frequency 40, 131,137 loops 127 neutrality of 39 not diamagnetic 38 partially ionized 2, 187 physics applications of 191-198 processing 198 waves Vlasov theory of 120-123 waves, coupled 74 Plemelj formulae 122, 129, 130 polarization 37-38 Poynting vector 48 pressure impulsive forces in 16 kinetic definition of 14-15 magnetic 49 role of collisions 2, 13 tensor 13, 155 in magnetic field 159 railgun plasma 196 Rankine-Hugoniot jump conditions 83 rate of strain tensor 156 Rayleigh line 96 Rayleigh-Taylor instability 98-1 02 reconnection of magnetic fields 118 relaxation times 149-152 resistive instability 115-1 18 thickness 62 resistivity 43, 52, 152, 190 Rutherford cross-section 135, 137 Saha equation 187 sausage instability 106-107 scattering cross-section 135 Second Law of thermo. 169 second-order heat flux 167 dominance of 169,17 1 equation for 169 second-order transport I95 self-collision time 150, 151 shock waves 82-91 entropy increase across 86

Hugoniot relations 85 jump conditions in 83-84 perpendicular 84 thermo. constraint 85-86 slowing-down time 149 solar corona, heating of 175 flares I 18, 176 magnetic fields 127, 194 plasma loops 195 prominences 195 wind 91, 159, 195 sound speed, adiabatic 72 specific enthalpy 46 specific heat 12 specific heats, ratio of 188-190 Spitzer conductivity 145 stability effect of curvature on 105 of an unbounded flux tube 109 of cylindrical plasma 1 15 safety factor for 1 10 state equations 12 stream function 58 streaming instability 97, 126 super-potential functions 139-141 first 140, 146 second 141,147 surface charge 64 intensities 6 1, 62 waves 99-101 switch-off shocks 88 switch-on shocks 88 tearing mode instability 1 16 temperature kinetic 11 role of collisions 1 1 tensors analysis of 8 in transport 43 laterally isotropic 51-52 thermal conductivity 172 equilibrium 18, 152 instability 170-172, 174-176

Index

in planar geometry 174 stability 105-1 06 thermodynamic instabilities 98, 103 thermodynamic system 4, 6, 18 thermoelectric force 43 heat 43 tensor 51 theta pinch 55 tokamak 169 energy, loss of 193 research in 192 toroidal equilibrium 57 transition layers 6 1, 62 transport coefficients, tensorial nature of 13 1 ofenergy 165 of momentum I65 trapped particles 186 two-stream instability 120

213 velocity distribution equilibrium 24-25 function 5 velocity space averages over 26-27 scattering in 138 velocity, peculiar 9 viscosity tensor 14, 155-1 57, 170 second-order 170 Vlasov equation 120 solutions of 121 vortex sheet 65 vorticity 50, 65 wave dispersion 82 fronts 76, 77 packets 81 theory, electron inertia in 94 waves in infinite plasmas 70-73 whistlers 75