Bellan P.M. Fundamentals of plasma physics

CUP draft Sept. 2004.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 towards 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 conclusion 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 fusion 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 regarding 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 equations, classical mechanics, vector algebra, vector calculus, differential equations, and complex variables - i.e. , the contents of a typical undergraduate physics or engineering curriculum. 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 arc included to accommodate 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 memory 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 invariancc. 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.
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 introduces 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 equations. Chapter 2 also introduces the dichotomy between adiabatic and isothermal behavior which is a fundamental and recurrent theme in plasma physics. Chapter 3 considers plasmas 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 detailed 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 systems 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 oscillator)* fields: the 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 situations, 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 partial 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 Alfven 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 instability 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 magnetic 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 energy 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 Bernstein wave in an inhomogencous plasma. Chapter 8 concludes with a discussion of drift waves, ubiquitous low frequency waves which have important deleterious consequences tor 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 dynamic 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 with a discussion on magnetohydrodynamic flows such as occur in arcs and jets. Chapter 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 equilibria. 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 presented 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 enable 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 revisiting the Debye shielding concept and so corresponds, in a sense to having the book end on the same note it started on.