Fundamentals of plasma physics

Fundamentals of Plasma Physics

Paul M. Bellan

to my parents

Contents

Preface xi

1 Basic concepts 1

1.1 History of the term “plasma” 1

1.2 Brief history of plasma physics 1

1.3 Plasma parameters 3

1.4 Examples of plasmas 3

1.5 Logical framework of plasma physics 4

1.6 Debye shielding 7

1.7 Quasi-neutrality 9

1.8 Small v. large angle collisions in plasmas 11

1.9 Electron and ion collision frequencies 14

1.10 Collisions with neutrals 16

1.11 Simple transport phenomena 17

1.12 A quantitative perspective 20

1.13 Assignments 22

2 Derivation of fluid equations: Vlasov, 2-fluid, MHD 30

2.1 Phase-space 30

2.2 Distribution function and Vlasov equation 31

2.3 Moments of the distribution function 33

2.4 Two-fluid equations 36

2.5 Magnetohydrodynamic equations 46

2.6 Summary of MHD equations 52

2.7 Sheath physics and Langmuir probe theory 53

2.8 Assignments 58

3 Motion of a single plasma particle 62

3.1 Motivation 62

3.2 Hamilton-Lagrange formalism v. Lorentz equation 62

3.3 Adiabatic invariant of a pendulum 66

3.4 Extension of WKB method to general adiabatic invariant 68

3.5 Drift equations 73

3.6 Relation of Drift Equations to the Double Adiabatic MHD Equations 91

3.7 Non-adiabatic motion in symmetric geometry 95

3.8 Motion in small-amplitude oscillatory fields 108

3.9 Wave-particle energy transfer 110

3.10 Assignments 119

viii

4 Elementary plasma waves 123

4.1 General method for analyzing small amplitude waves 123

4.2 Two-fluid theory of unmagnetized plasma waves 124

4.3 Low frequency magnetized plasma: Alfvén waves 131

4.4 Two-fluid model of Alfvén modes 138

4.5 Assignments 147

5 Streaming instabilities and the Landau problem 149

5.1 Streaming instabilities 149

5.2 The Landau problem 153

5.3 The Penrose criterion 172

5.4 Assignments 175

6 Cold plasma waves in a magnetized plasma 178

6.1 Redundancy of Poisson’s equation in electromagnetic mode analysis 178

6.2 Dielectric tensor 179

6.3 Dispersion relation expressed as a relation between n

2

x

and n

2

z 193

6.4 A journey through parameter space 195

6.5 High frequency waves: Altar-Appleton-Hartree dispersion relation 197

6.6 Group velocity 201

6.7 Quasi-electrostatic cold plasma waves 203

6.8 Resonance cones 204

6.9 Assignments 208

7 Waves in inhomogeneous plasmas and wave energy relations 210

7.1 Wave propagation in inhomogeneous plasmas 210

7.2 Geometric optics 213

7.3 Surface waves - the plasma-filled waveguide 214

7.4 Plasma wave-energy equation 219

7.5 Cold-plasma wave energy equation 221

7.6 Finite-temperature plasma wave energy equation 224

7.7 Negative energy waves 225

7.8 Assignments 228

8 Vlasov theory of warm electrostatic waves in a magnetized plasma 229

8.1 Uniform plasma 229

8.2 Analysis of the warm plasma electrostatic dispersion relation 234

8.3 Bernstein waves 236

8.4 Warm, magnetized, electrostatic dispersion with small, but finite k
239

8.5 Analysis of linear mode conversion 241

8.6 Drift waves 249

8.7 Assignments 263

9 MHD equilibria 264

9.1 Why use MHD? 264

9.2 Vacuum magnetic fields 265

ix

9.3 Force-free fields 268

9.4 Magnetic pressure and tension 268

9.5 Magnetic stress tensor 271

9.6 Flux preservation, energy minimization, and inductance 272

9.7 Static versus dynamic equilibria 274

9.8 Static equilibria 275

9.9 Dynamic equilibria: flows 286

9.10 Assignments 295

10 Stability of static MHD equilibria 298

10.1 The Rayleigh-Taylor instability of hydrodynamics 299

10.2 MHD Rayleigh-Taylor instability 302

10.3 The MHD energy principle 306

10.4 Discussion of the energy principle 319

10.5 Current-driven instabilities and helicity 319

10.6 Magnetic helicity 320

10.7 Qualitative description of free-boundary instabilities 323

10.8 Analysis of free-boundary instabilities 326

10.9 Assignments 334

11 Magnetic helicity interpreted and Woltjer-Taylor relaxation 336

11.1 Introduction 336

11.2 Topological interpretation of magnetic helicity 336

11.3 Woltjer-Taylor relaxation 341

11.4 Kinking and magnetic helicity 345

11.5 Assignments 357

12 Magnetic reconnection 360

12.1 Introduction 360

12.2 Water-beading: an analogy to magnetic tearing and reconnection 361

12.3 Qualitative description of sheet current instability 362

12.4 Semi-quantitative estimate of the tearing process 364

12.5 Generalization of tearing to sheared magnetic fields 371

12.6 Magnetic islands 376

12.7 Assignments 378

13 Fokker-Planck theory of collisions 382

13.1 Introduction 382

13.2 Statistical argument for the development of the Fokker-Planck equation 384

13.3 Electrical resistivity 393

13.4 Runaway electric field 395

13.5 Assignments 395

14 Wave-particle nonlinearities 398

14.1 Introduction 398

14.2 Vlasov non-linearity and quasi-linear velocity space diffusion 399

x

14.3 Echoes 412

14.4 Assignments 426

15 Wave-wave nonlinearities 428

15.1 Introduction 428

15.2 Manley-Rowe relations 430

15.3 Application to waves 435

15.4 Non-linear dispersion formulation and instability threshold 444

15.5 Digging a hole in the plasma via ponderomotive force 448

15.6 Ion acoustic wave soliton 454

15.7 Assignments 457

16 Non-neutral plasmas 460

16.1 Introduction 460

16.2 Brillouin flow 460

16.3 Isomorphism to incompressible 2D hydrodynamics 463

16.4 Near perfect confinement 464

16.5 Diocotron modes 465

16.6 Assignments 476

17 Dusty plasmas 483

17.1 Introduction 483

17.2 Electron and ion current flow to a dust grain 484

17.3 Dust charge 486

17.4 Dusty plasma parameter space 490

17.5 Large P limit: dust acoustic waves 491

17.6 Dust ion acoustic waves 494

17.7 The strongly coupled regime: crystallization of a dusty plasma 495

17.8 Assignments 504

Bibliography and suggested reading 507

References 509

Appendix A: Intuitive method for vector calculus identities 515

Appendix B: Vector calculus in orthogonal curvilinear coordinates 518

Appendix C: Frequently used physical constants and formulae 524

Index 528

Preface

This text is based on a course I have taught for many years to first year graduate and

senior-level undergraduate students at Caltech. One outcome of this teaching has been the

realization that although students typically decide to study plasma physics as a means to￾wards some larger goal, they often conclude that this study has an attraction and charm

of its own; in a sense the journey becomes as enjoyable as the destination. This conclu￾sion is shared by me and I feel that a delightful aspect of plasma physics is the frequent

transferability of ideas between extremely different applications so, for example, a concept

developed in the context of astrophysics might suddenly become relevant to fusion research

or vice versa.

Applications of plasma physics are many and varied. Examples include controlled fu￾sion research, ionospheric physics, magnetospheric physics, solar physics, astrophysics,

plasma propulsion, semiconductor processing, and metals processing. Because plasma

physics is rich in both concepts and regimes, it has also often served as an incubator for

new ideas in applied mathematics. In recent years there has been an increased dialog re￾garding plasma physics among the various disciplines listed above and it is my hope that

this text will help to promote this trend.

The prerequisites for this text are a reasonable familiarity with Maxwell’s equa￾tions, classical mechanics, vector algebra, vector calculus, differential equations, and com￾plex variables – i.e., the contents of a typical undergraduate physics or engineering cur￾riculum. Experience has shown that because of the many different applications for plasma

physics, students studying plasma physics have a diversity of preparation and not all are

proficient in all prerequisites. Brief derivations of many basic concepts are included to ac￾commodate this range of preparation; these derivations are intended to assist those students

who may have had little or no exposure to the concept in question and to refresh the mem￾ory of other students. For example, rather than just invoke Hamilton-Lagrange methods or

Laplace transforms, there is a quick derivation and then a considerable discussion showing

how these concepts relate to plasma physics issues. These additional explanations make

the book more self-contained and also provide a close contact with first principles.

The order of presentation and level of rigor have been chosen to establish a firm

foundation and yet avoid unnecessary mathematical formalism or abstraction. In particular,

the various fluid equations are derived from first principles rather than simply invoked and

the consequences of the Hamiltonian nature of particle motion are emphasized early on

and shown to lead to the powerful concepts of symmetry-induced constraint and adiabatic

invariance. Symmetry turns out to be an essential feature of magnetohydrodynamic plasma

confinement and adiabatic invariance turns out to be not only essential for understanding

many types of particle motion, but also vital to many aspects of wave behavior.

The mathematical derivations have been presented with intermediate steps shown

in as much detail as is reasonably possible. This occasionally leads to daunting-looking

expressions, but it is my belief that it is preferable to see all the details rather than have

them glossed over and then justified by an “it can be shown" statement.

xi

xii Preface

The book is organized as follows: Chapters 1-3 lay out the foundation of the subject.

Chapter 1 provides a brief introduction and overview of applications, discusses the logical

framework of plasma physics, and begins the presentation by discussing Debye shielding

and then showing that plasmas are quasi-neutral and nearly collisionless. Chapter 2 intro￾duces phase-space concepts and derives the Vlasov equation and then, by taking moments

of the Vlasov equation, derives the two-fluid and magnetohydrodynamic systems of equa￾tions. Chapter 2 also introduces the dichotomy between adiabatic and isothermal behavior

which is a fundamental and recurrent theme in plasma physics. Chapter 3 considers plas￾mas from the point of view of the behavior of a single particle and develops both exact

and approximate descriptions for particle motion. In particular, Chapter 3 includes a de￾tailed discussion of the concept of adiabatic invariance with the aim of demonstrating that

this important concept is a fundamental property of all nearly periodic Hamiltonian sys￾tems and so does not have to be explained anew each time it is encountered in a different

situation. Chapter 3 also includes a discussion of particle motion in fixed frequency oscil￾latory fields; this discussion provides a foundation for later analysis of cold plasma waves

and wave-particle energy transfer in warm plasma waves.

Chapters 4-8 discuss plasma waves; these are not only important in many practical sit￾uations, but also provide an excellent way for developing insight about plasma dynamics.

Chapter 4 shows how linear wave dispersion relations can be deduced from systems of par￾tial differential equations characterizing a physical system and then presents derivations for

the elementary plasma waves, namely Langmuir waves, electromagnetic plasma waves, ion

acoustic waves, and Alfvén waves. The beginning of Chapter 5 shows that when a plasma

contains groups of particles streaming at different velocities, free energy exists which can

drive an instability; the remainder of Chapter 5 then presents Landau damping and instabil￾ity theory which reveals that surprisingly strong interactions between waves and particles

can lead to either wave damping or wave instability depending on the shape of the velocity

distribution of the particles. Chapter 6 describes cold plasma waves in a background mag￾netic field and discusses the Clemmow-Mullaly-Allis diagram, an elegant categorization

scheme for the large number of qualitatively different types of cold plasma waves that exist

in a magnetized plasma. Chapter 7 discusses certain additional subtle and practical aspects

of wave propagation including propagation in an inhomogeneous plasma and how the en￾ergy content of a wave is related to its dispersion relation. Chapter 8 begins by showing

that the combination of warm plasma effects and a background magnetic field leads to the

existence of the Bernstein wave, an altogether different kind of wave which has an infinite

number of branches, and shows how a cold plasma wave can ‘mode convert’ into a Bern￾stein wave in an inhomogeneous plasma. Chapter 8 concludes with a discussion of drift

waves, ubiquitous low frequency waves which have important deleterious consequences

for magnetic confinement.

Chapters 9-12 provide a description of plasmas from the magnetohydrodynamic point

of view. Chapter 9 begins by presenting several basic magnetohydrodynamic concepts

(vacuum and force-free fields, magnetic pressure and tension, frozen-in flux, and energy

minimization) and then uses these concepts to develop an intuitive understanding for dy￾namic behavior. Chapter 9 then discusses magnetohydrodynamic equilibria and derives the

Grad-Shafranov equation, an equation which depends on the existence of symmetry and

which characterizes three-dimensional magnetohydrodynamic equilibria. Chapter 9 ends

Preface xiii

with a discussion on magnetohydrodynamic flows such as occur in arcs and jets. Chap￾ter 10 examines the stability of perfectly conducting (i.e., ideal) magnetohydrodynamic

equilibria, derives the ‘energy principle’ method for analyzing stability, discusses kink and

sausage instabilities, and introduces the concepts of magnetic helicity and force-free equi￾libria. Chapter 11 examines magnetic helicity from a topological point of view and shows

how helicity conservation and energy minimization leads to the Woltjer-Taylor model for

magnetohydrodynamic self-organization. Chapter 12 departs from the ideal models pre￾sented earlier and discusses magnetic reconnection, a non-ideal behavior which permits

the magnetohydrodynamic plasma to alter its topology and thereby relax to a minimum￾energy state.

Chapters 13-17 consist of various advanced topics. Chapter 13 considers collisions

from a Fokker-Planck point of view and is essentially a revisiting of the issues in Chapter

1 using a more sophisticated point of view; the Fokker-Planck model is used to derive a

more accurate model for plasma electrical resistivity and also to show the failure of Ohm’s

law when the electric field exceeds a critical value called the Dreicer limit. Chapter 14

considers two manifestations of wave-particle nonlinearity: (i) quasi-linear velocity space

diffusion due to weak turbulence and (ii) echoes, non-linear phenomena which validate the

concepts underlying Landau damping. Chapter 15 discusses how nonlinear interactions en￾able energy and momentum to be transferred between waves, categorizes the large number

of such wave-wave nonlinear interactions, and shows how these various interactions are all

based on a few fundamental concepts. Chapter 16 discusses one-component plasmas (pure

electron or pure ion plasmas) and shows how these plasmas have behaviors differing from

conventional two-component, electron-ion plasmas. Chapter 17 discusses dusty plasmas

which are three component plasmas (electrons, ions, and dust grains) and shows how the

addition of a third component also introduces new behaviors, including the possibility of

the dusty plasma condensing into a crystal. The analysis of condensation involves revisit￾ing the Debye shielding concept and so corresponds, in a sense to having the book end on

the same note it started on.

I would like to extend my grateful appreciation to Professor Michael Brown at

Swarthmore College for providing helpful feedback obtained from using a draft version in

a seminar course at Swarthmore and to Professor Roy Gould at Caltech for providing useful

suggestions. I would also like to thank graduate students Deepak Kumar and Gunsu Yun for

carefully scrutinizing the final drafts of the manuscript and pointing out both ambiguities

in presentation and typographical errors. I would also like to thank the many students who,

over the years, provided useful feedback on earlier drafts of this work when it was in the

form of lecture notes. Finally, I would like to acknowledge and thank my own mentors and

colleagues who have introduced me to the many fascinating ideas constituting the discipline

of plasma physics and also the many scientists whose hard work over many decades has

led to the development of this discipline.

Paul M. Bellan

Pasadena, California

September 30, 2004

1

Basic concepts

1.1 History of the term “plasma”

In the mid-19th century the Czech physiologist Jan Evangelista Purkinje introduced use

of the Greek word plasma (meaning “formed or molded”) to denote the clear fluid which

remains after removal of all the corpuscular material in blood. Half a century later, the

American scientist Irving Langmuir proposed in 1922 that the electrons, ions and neutrals

in an ionized gas could similarly be considered as corpuscular material entrained in some

kind of fluid medium and called this entraining medium plasma. However it turned out that

unlike blood where there really is a fluid medium carrying the corpuscular material, there

actually is no “fluid medium” entraining the electrons, ions, and neutrals in an ionized gas.

Ever since, plasma scientists have had to explain to friends and acquaintances that they

were not studying blood!

1.2 Brief history of plasma physics

In the 1920’s and 1930’s a few isolated researchers, each motivated by a specific practi￾cal problem, began the study of what is now called plasma physics. This work was mainly

directed towards understanding (i) the effect of ionospheric plasma on long distance short￾wave radio propagation and (ii) gaseous electron tubes used for rectification, switching

and voltage regulation in the pre-semiconductor era of electronics. In the 1940’s Hannes

Alfvén developed a theory of hydromagnetic waves (now called Alfvén waves) and pro￾posed that these waves would be important in astrophysical plasmas. In the early 1950’s

large-scale plasma physics based magnetic fusion energy research started simultaneously

in the USA, Britain and the then Soviet Union. Since this work was an offshoot of ther￾monuclear weapon research, it was initially classified but because of scant progress in each

country’s effort and the realization that controlled fusion research was unlikely to be of mil￾itary value, all three countries declassified their efforts in 1958 and have cooperated since.

Many other countries now participate in fusion research as well.

Fusion progress was slow through most of the 1960’s, but by the end of that decade the

1

2 Chapter 1. Basic concepts

empirically developed Russian tokamak configuration began producing plasmas with pa￾rameters far better than the lackluster results of the previous two decades. By the 1970’s

and 80’s many tokamaks with progressively improved performance were constructed and

at the end of the 20th century fusion break-even had nearly been achieved in tokamaks.

International agreement was reached in the early 21st century to build the International

Thermonuclear Experimental Reactor (ITER), a break-even tokamak designed to produce

500 megawatts of fusion output power. Non-tokamak approaches to fusion have also been

pursued with varying degrees of success; many involve magnetic confinement schemes

related to that used in tokamaks. In contrast to fusion schemes based on magnetic con￾finement, inertial confinement schemes were also developed in which high power lasers or

similarly intense power sources bombard millimeter diameter pellets of thermonuclear fuel

with ultra-short, extremely powerful pulses of strongly focused directed energy. The in￾tense incident power causes the pellet surface to ablate and in so doing, act like a rocket

exhaust pointing radially outwards from the pellet. The resulting radially inwards force

compresses the pellet adiabatically, making it both denser and hotter; with sufficient adia￾batic compression, fusion ignition conditions are predicted to be achieved.

Simultaneous with the fusion effort, there has been an equally important and extensive

study of space plasmas. Measurements of near-Earth space plasmas such as the aurora

and the ionosphere have been obtained by ground-based instruments since the late 19th

century. Space plasma research was greatly stimulated when it became possible to use

spacecraft to make routine in situ plasma measurements of the Earth’s magnetosphere, the

solar wind, and the magnetospheres of other planets. Additional interest has resulted from

ground-based and spacecraft measurements of topologically complex, dramatic structures

sometimes having explosive dynamics in the solar corona. Using radio telescopes, optical

telescopes, Very Long Baseline Interferometry and most recently the Hubble and Spitzer

spacecraft, large numbers of astrophysical jets shooting out from magnetized objects such

as stars, active galactic nuclei, and black holes have been observed. Space plasmas often

behave in a manner qualitatively similar to laboratory plasmas, but have a much grander

scale.

Since the 1960’s an important effort has been directed towards using plasmas for space

propulsion. Plasma thrusters have been developed ranging from small ion thrusters for

spacecraft attitude correction to powerful magnetoplasmadynamic thrusters that –given an

adequate power supply – could be used for interplanetary missions. Plasma thrusters are

now in use on some spacecraft and are under serious consideration for new and more am￾bitious spacecraft designs.

Starting in the late 1980’s a new application of plasma physics appeared – plasma

processing – a critical aspect of the fabrication of the tiny, complex integrated circuits

used in modern electronic devices. This application is now of great economic importance.

In the 1990’s studies began on dusty plasmas. Dust grains immersed in a plasma can

become electrically charged and then act as an additional charged particle species. Be￾cause dust grains are massive compared to electrons or ions and can be charged to varying

amounts, new physical behavior occurs that is sometimes an extension of what happens

in a regular plasma and sometimes altogether new. In the 1980’s and 90’s there has also

been investigation of non-neutral plasmas; these mimic the equations of incompressible

hydrodynamics and so provide a compelling analog computer for problems in incompress￾ible hydrodynamics. Both dusty plasmas and non-neutral plasmas can also form bizarre

strongly coupled collective states where the plasma resembles a solid (e.g., forms quasi￾crystalline structures). Another application of non-neutral plasmas is as a means to store

1.4 Examples of plasmas 3

large quantities of positrons.

In addition to the above activities there have been continuing investigations of indus￾trially relevant plasmas such as arcs, plasma torches, and laser plasmas. In particular,

approximately 40% of the steel manufactured in the United States is recycled in huge elec￾tric arc furnaces capable of melting over 100 tons of scrap steel in a few minutes. Plasma

displays are used for flat panel televisions and of course there are naturally-occurring ter￾restrial plasmas such as lightning.

1.3 Plasma parameters

Three fundamental parameters1

characterize a plasma:

1. the particle density n (measured in particles per cubic meter),

2. the temperature T of each species (usually measured in eV, where 1 eV=11,605 K),

3. the steady state magnetic field B (measured in Tesla).

A host of subsidiary parameters (e.g., Debye length, Larmor radius, plasma frequency,

cyclotron frequency, thermal velocity) can be derived from these three fundamental para￾meters. For partially-ionized plasmas, the fractional ionization and cross-sections of neu￾trals are also important.

1.4 Examples of plasmas

1.4.1 Non-fusion terrestrial plasmas

It takes considerable resources and skill to make a hot, fully ionized plasma and so, ex￾cept for the specialized fusion plasmas, most terrestrial plasmas (e.g., arcs, neon signs,

fluorescent lamps, processing plasmas, welding arcs, and lightning) have electron tem￾peratures of a few eV, and for reasons given later, have ion temperatures that are colder,

often at room temperature. These ‘everyday’ plasmas usually have no imposed steady state

magnetic field and do not produce significant self magnetic fields. Typically, these plas￾mas are weakly ionized and dominated by collisional and radiative processes. Densities in

these plasmas range from 1014 to 1022m−3

(for comparison, the density of air at STP is

2.7 × 1025m−3

).

1.4.2 Fusion-grade terrestrial plasmas

Using carefully designed, expensive, and often large plasma confinement systems together

with high heating power and obsessive attention to purity, fusion researchers have suc￾ceeded in creating fully ionized hydrogen or deuterium plasmas which attain temperatures

1

In older plasma literature, density and magnetic fields are often expressed in cgs units, i.e., densities are given

in particles per cubic centimeter, and magnetic fields are given in Gauss. Since the 1990’s there has been general

agreement to use SI units when possible. SI units have the distinct advantage that electrical units are in terms of

familiar quantities such as amps, volts, and ohms and so a model prediction in SI units can much more easily be

compared to the results of an experiment than a prediction given in cgs units.

4 Chapter 1. Basic concepts

in the range from 10’s of eV to tens of thousands of eV. In typical magnetic confinement

devices (e.g., tokamaks, stellarators, reversed field pinches, mirror devices) an externally

produced 1-10 Tesla magnetic field of carefully chosen geometry is imposed on the plasma.

Magnetic confinement devices generally have densities in the range 1019 −1021m−3

. Plas￾mas used in inertial fusion are much more dense; the goal is to attain for a brief instant

densities one or two orders of magnitude larger than solid density (∼ 1027m−3

).

1.4.3 Space plasmas

The parameters of these plasmas cover an enormous range. For example the density of

space plasmas vary from 106 m−3

in interstellar space, to 1020 m−3

in the solar atmosphere.

Most of the astrophysical plasmas that have been investigated have temperatures in the

range of 1-100 eV and these plasmas are usually fully ionized.

1.5 Logical framework of plasma physics

Plasmas are complex and exist in a wide variety of situations differing by many orders of

magnitude. An important situation where plasmas do not normally exist is ordinary human

experience. Consequently, people do not have the sort of intuition for plasma behavior that

they have for solids, liquids or gases. Although plasma behavior seems non- or counter￾intuitive at first, with suitable effort a good intuition for plasma behavior can be developed.

This intuition can be helpful for making initial predictions about plasma behavior in a

new situation, because plasmas have the remarkable property of being extremely scalable;

i.e., the same qualitative phenomena often occur in plasmas differing by many orders of

magnitude. Plasma physics is usually not a precise science. It is rather a web of overlapping

points of view, each modeling a limited range of behavior. Understanding of plasmas is

developed by studying these various points of view, all the while keeping in mind the

linkages between the points of view.

Lorentz equation

(gives xj, vj for each particle from knowledge of E
x,t,B
x,t)

Maxwell equations

(gives E
x,t, B
x,t from knowledge of xj

, vj for each particle)

Figure 1.1: Interrelation between Maxwell’s equations and the Lorentz equation

Plasma dynamics is determined by the self-consistent interaction between electromag￾netic fields and statistically large numbers of charged particles as shown schematically in