L. C. Woods
Physics of Plasmas
L. C. Woods
Physics of Plasmas
WILEY
VCH
WILEYVCH Verlag GmbH & Co. KGaA
Author
Prof Dr.Leslie C. Woods
University of Oxford and Balliol College
[email protected] with 69 figures
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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 Xray image of the sun Credit: ESA, NASA
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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 crossreference 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 phasespace 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 collisiondominated 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 magnetoionic 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 ionacoustic instability, which has important applications in solar physics. Chapter 5 , which is concerned with transport in magnetoplasmas, starts from the FokkerPlanck equation and gives an account of the theory of electronion 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 secondorder 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 nearcollisionless 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 phasespace 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 Forcefree 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 cutoff . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 . . . . . . . . . . . . . . . . . . . . . . . Magnetoionic theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6.1 Electrical conductivity . . . . . . . . . . . . . . . . . . . . . . . . . 3.6.2 The dielectric tensor . . . . . . . . . . . . . . . . . . . . . . . . . .
87 89 92 92 93
4
Magnetoplasma Stability 4.1 RayleighTaylor and KelvinHelmholtz 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 twostream 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 ionacoustic 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 crosssection . . . . . . . . . . . . . . . . 5.2 The FokkerPlanck equation . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.1 Friction and diffusion coefficients . . . . . . . . . . . . . . . . . . . 5.2.2 Scattering in velocity space . . . . . . . . . . . . . . . . . . . . . . 5.2.3 Superpotential 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 fullyionized plasma . . . . . . . . . . . . . . . . . 144 5.4 Friction and diffusion coefficients . . . . . . . . . . . . . . . . . . . . . . . 146 5.4.1 First superpotential . . . . . . . . . . . . . . . . . . . . . . . . . . 146 5.4.2 Second superpotential . . . . . . . . . . . . . . . . . . . . . . . . . 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 Secondorder transport . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1.1 Convection versus conduction . . . . . . . . . . . . . . . . . . . . . 6.1.2 The secondorder 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 Gyroaverages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 StefanBoltzmann 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 ml 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’ K4 8.3144 J K1 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 skindepth 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 WILEYVCH 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 microfields 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 microfields like collisional forces. The microfields 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 ‘meanfreepath’, 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 collisiondominated, 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 sphericallysymmetric force field. For neutral particles this field has a very short range, and the particles can be pictured as being almost rigid ‘billiardballs’, 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 billiardball 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 multiatomic particles that give rise to the largest discrepancies between the predictions of the simple billiardball 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 nearcontinuous 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 wellknown 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 meanfreepath.
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 7l = 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 24 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 smallangle 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, NorthHolland 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 nonzero 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 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 ith 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 ‘frameindifference’ 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 secondorder tensors X
In general a secondorder 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(AA),
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 righthand term in (1.13) can be expressed as 1 x 0. Let B denote another secondorder 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 righthand 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 righthand 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 lefthand 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'^,
= $mut2
m(uu’) = M(UU’), m ( u 2  u 12 ) =  M ( u ~  ~ ’
i.e.
~1.
+
+
From the second pair of equations it follows that u u’ = U U’, whence the relative velocity u  U is reversed by the collision, as required by perfect elasticity. This condition and the momentum equation gives
( m+ M)u’ = ( m M ) u
+ 2MU,
( m+ M)U’ = (m  M ) U + 2mu,
and hence the gain in the wall’s kinetic energy per collision is
iM(U‘2  U2) =
2mM { m u 2(m M)2
+
MU^ + ( M  r n ) u ~ } ,
Gas
Figure 1.2: Collision between gas and wall molecules.
As the wall is stationary, W Ioscillates about a mean position fixed in the wall, and since u and U are uncorrelated, over a large number of collisions along O X the product uU will have positive and negatives values with equal probability. Hence the average of UU is zero, so the net gain of wall energy is proportional to the average of (mu’  M U 2 ) ,or equivalently to the average of (mu2 muI2  M U 2  M U r 2 ) .We now extend the averaging to all directions of O X to find that the average gain of energy by the wall is proportional to Q, where
+
Q=
4mM ( m M ) 2 { ($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 1023JK1),
(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 offdiagonal 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 fullyionized 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 offdiagonal 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 lengthscale 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=np
(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 lengthscale 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 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, ++, 788 a z
(2.149)
and (2.150) For a scalar 4

F = v4 = P84 +e 84 dr rd#
+ i84
dz '
(2.151)
and (2.152)
Physics of Plasmas L. C. Woods Copyright 0 2004 WILEYVCH 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 lowfrequency 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’ magnetoacoustic 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 AppletonHartree magnetoionic 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 (nonzero) 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 nonlinear. 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 Fourieranalyzed into components proportional to exp[i(k * r  w t ) ] ,representing plane waves propagating in the kdirection. 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
weoSkka2~+~okcosBBBok~,, = 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 kdirection, (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 ioncyclotron 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 ith row and jth column of the matrix in this equation, then we identify the leading submatrices by the values of i and j at their upper lefthand comers and their lower righthand comers, e.g. the submatrix ( 2 , 4) extends from the element (i = 2 , j = 2) to the element (i = 4, j = 4). The waves are classified as follows: Submatrix Wavetype Eigenvectors Phase Velocity (1,1) entropy B 0 (21 3) acoustic itk . 8 &a (2,5) magnetoacoustic 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 magnetoacoustic 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 magnetoacoustic and Alfvenic branches are decoupled if X is negligible. From (3.15) this occurs when either 6' = 90" or & / A 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 lengthscale 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 OXaxis 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 shockwave theory for a gas such as air, thesejump conditions are called the RankineHugoniot relations; in MHD they are known as the HoffmannTeller relations. Unlike gasdynamic 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: vQV =
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
VB=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 [$I. Hence (3.55)
3
84
to (3.58) yield the jump conditions
n . [ p ] = 0, n . [_ow+ (p+B2)1 1 2Po
n . [B]= 0,
 BB] 1
= 0,
Po
I
Wavesin Magnetoplasmas
I
(3.61)
n x [El = 0,
+
where n is unit vector parallel to OX and h = u p / Q is the enthalpy. From (3.59), n x E = v,Bt  B,vt, where t denotes the component lying in the shock plane S. Suppose that vt is measured in a frame F , whose origin lies in S. Let F’ be a frame with axes parallel to F and whose origin 0’ moves in S at a steady velocity Ut relative to 0. Provided B, # 0, we can choose Ut such that in F’, the vector n x E’ is zero and hence v{/w,= B,/B,. By the last of (3.61) this relation will hold on both sides of the shock. It follows from wh/wI = B,/B, and (3.59) that Ek is also zero. Finally, by rotating F‘ about O’X’, a frame F“ can be found in which w: and B: are zero. Suppressing the dashes, we have now found a frame in which
E = 0,
W, = 0 ,
B, = 0 ,
w,/w, = B,/B,,
(3.62)
on both sides of the shock. Notice that the velocity and magnetic field vectors are now parallel. When B, is zero, the discontinuity is called aperpendicular shock. In this case the reduction just described is not applicable, so these shocks must be considered separately.
Figure 3.6: The shock front.
3.5 Shock waves
85
[v,]  m[v] = 0 m[v,I
1
+ [PI+  P y ) [ B y l P O
1 m[vy] B,[By] Po
~ [ v Z+v;] + [h] B+[UYl 
m[vI(By)  m ( 4 P y l
0
=
0
=
0
=
0
= P
mv+(p+B$)nB,B 1 1 2PO
=
PO = H
$?+h
' (3.63)
1
1
I
,
(3.64)
J
where B,, m, P, and H are constants and the lefthand sides are evaluated at either the upstream or downstream states. With the help of the state equations,
pv=RT,
h=
71
4
RT = RT,
(3.65)
we may express (3.63) as five homogeneous equations in the five jumps [v],[TI,[v,],ivy], and [B,]. If one of these jumps is specified, together with the upstream values of p, T , B,, and By,then the downstream values can be determined.
3.5.2 Thermodynamic constraint The entropy must increase across the shock wave (see 52.3.3) and to make use of this constraint we need the relationship between the thermodynamic state variables on each side of the shock. In gas dynamics this is known as the Hugoniot relation. On multiplying (3.63)s by (v,) and (3.63)4 by (v,), adding and using v, = m v ,we get
From vyBx = v,By and (3.60)s we find
t [%I [By]+ (%)(By) Using these formulae to eliminate [vZ + v$] 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 nonzero 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 lowfrequency 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)magnetoacoustic 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 nonnegative, 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 ‘superAlfvknic’ 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). Switchon and switchof 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
', \
;,/
switchoff 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 switchon 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 crossfield 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

switchon
v2
perpendicular shock shock
(v parallel to B)
(V perpendicularto
B)
(b)
(a)
Figure 3.9: Special types of shock:
(a) parallel and switchon 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 nondimensional 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
nondimensional 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 socalled 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 Magnetoionic theory 3.6.1 Electrical conductivity The propagation, reflection and scattering of radiowaves 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 magnetoionic 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 steadystate 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 Magnetoionic 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 + Eolff
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 bxlw2x2 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 Magnetoionic 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 ionneutral 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 1a
eomew2/e2. (Such plasmas are said to be ‘overdense’).
Physics of Plasmas L. C. Woods Copyright 0 2004 WILEYVCH 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 nonlinear 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 twostream 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 RayleighTaylor 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:

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 RayleighTaylor and KelvinHelmholtz 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:
vv=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
RayleighTaylor and KelvinHelmholtz 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 Dopplershifted 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
^
$, j,).
These equations reduce to
+ ebg) ijy  a i j ; + kJ3ijk + Gfj'
 2 (LzI2Qo  k%B
where k2 = k:
4.1.2
+ lcz and t3
= 0,
I
(4.5)
Bi/po.
Surface waves
To study surface waves we choose solutions of (4.5) that decay exponentially as y cy
+
fm: (4.6)
= fjy(0) exP(Fkyy) 1
where the minus sign holds for y > 0 and the plus sign for y < 0, and where k , is a positive number whose value will be determined shortly. Hence lim ijb = kyijg(0),
y++O
Denote by
lim ij; = kyijg(0).
y40
(4.7)
[ @]the jump, 'ZJZ @ I , in the function @ across the interface, then
[Gg] = 0,
[fig = 2k,ijg,(0),
(4.8)
100
Magnetoplasma Stability
4
Figure 4.1: Surface waves.
of which the first is an obvious boundary condition and the second follows from (4.7). We shall also assume that, like Ch, the tangential velocity perturbations Cx,ijz and the pressure perturbation 9 are odd functions of y. Let I$ = (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 nonzero 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 RayleighTaylor and KelvinHelmholtzinstabilities
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 nonzero 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 RayleighTaylor 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 KelvinHelmholtz 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 linearpinch 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 RayleighTaylor 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 selfmagnetic 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 RayleighTaylor instability. An example is shown in Fig. 4.2’. ‘Taken from F.L. Curson, et al., (1960). Proc. Roy. SOC.A. 257,388401.
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 crosssection 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 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 righthand 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 KruskalShafranov 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 (mnq); 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 nondissipative 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 > 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 fourthorder derivative being negligible because 6, 1); it also provides an exact algebraic theory for comparison purposes. For electronion 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 FokkerPlanck equation (5.42)
where V = d/dw and the subscript ‘ei’ indicates that only electronion 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 2l. 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 fullyionized 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 electronelectron 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 electronelectron collisions and to electronion collisions at small Z. For ionion 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 superpotential 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 sspecies 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 =(13. @(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)= rd’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 superpotential
The value of the second superpotential 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.
Slowingdown time This time is: (5.85)
Consider, for example, the slowingdown 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 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 muchquoted 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 =1a: 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 Rcurve 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 firstorder 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 lefthand 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 lefthand side of (5.105) is reduced to the vector form Klqm(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 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 secondorder 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 firstorder 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 Secondorder 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 timelapse 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 'frameindifference' 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 frameindifference the heat flux vector q must be determined relative to P.
6.1.2 The secondorder heat flux We shall start by illustrating the mechanism of secondorder 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 firstorder 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 righthand 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 meanfreepath methods, it was not immediately obvious how to remove the convective component from Q; see Kennard (1938, pp. 16272) and Woods (1993, pp. 5960).
6 Extensions of Theory
168
Hot
yLx
0
Sheared flow
cool
Figure 6.1: Fluid shear generating a secondorder 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 righthand 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 timescales 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 firstorder 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 Secondorder 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 crossfield 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 secondorder 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 meanfreepath 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 secondorder term about lo4 times larger than the firstorder 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 firstorder 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 secondorder 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 righthand 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 = 2p7e
(1  2 7 e ) .
Applying the solution in (5.124) we have

H =  2 p ~ W:e (1  27e)  T D(W : H )
.
(6.18)
The last term can be approximated by 7D(W :TI,) where
TI^ =  2 p ~ W e: ,
(6.19)
is the firstorder 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 secondorder 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 secondorder 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 selfcollision 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 firstorder 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 secondorder 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
*‘
$ ( 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 ionacoustic 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 6.26x10l6n,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 secondorder 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 secondorder 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 Wm2 required for the quiet corona and also the 104Wm2 required for the active corona. As our purpose here is merely to introduce an important application of secondorder 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