A complete mathematical model of a plasma requires three basic elements: first, the motion of all particles must be determined for some assumed electric and magnetic field configuration; second, the current and charge densities must be computed from the particle trajectories; and third, the electric and magnetic fields must be self-consistently determined from the currents and charges, taking into account both internal and external sources. To be self-consistent, the electric and magnetic fields obtained from the last step must correspond to the fields used in the first step. It is this self-consistency requirement that makes the analysis of a plasma difficult.

To develop an understanding of the processes occurring in a plasma, a useful first step is to forget about the self-consistency requirement and concentrate on the motion of a single particle in a specified field configuration. This approach can be useful in a variety of situations. If the external fields are very strong and the plasma is sufficiently tenuous, the internally generated fields are sometimes small and can be safely ignored. This situation arises, for example, in radiation belts at high energies and in various electronic devices such as vacuum tubes and traveling wave amplifiers. In other situations, the self-consistent electric and magnetic fields may be known from direct measurement. In this case, it is often useful to follow the motion of individual tracer particles in the known electric and magnetic fields in order to gain insight into the physical processes involved, such as particle transport and energization. Finally, in some cases it is possible to use the general solution for the particle motion in an assumed field geometry to determine a fully self-consistent solution in which the currents and charges produce the assumed fields.

Motion in a Static Uniform Magnetic Field

The simplest field configuration of importance in plasma physics is a static uniform magnetic field. The equation of motion for a particle moving at non-relativistic velocities in a static uniform magnetic field is given by

where q(v×B) is the Lorentz force.

This textbook is intended for a full year introductory course in plasma physics at the senior undergraduate or first-year graduate level. It is based on lecture notes from courses taught by the authors for more than three decades at the University of Iowa, Columbia University, University of New Hampshire, and Princeton University. During these years, plasma physics has grown increasingly interdisciplinary, and there is a growing realization that diverse applications in laboratory, space, and astrophysical plasmas can be viewed from a common perspective. Since the students who take a course in plasma physics often have a wide range of interests, typically involving some combination of laboratory, space, and astrophysical plasmas, a special effort has been made to discuss applications from these areas of research. The emphasis of the book is on physical principles, less so on mathematical sophistication. An effort has been made to show all relevant steps in the derivations, and to match the level of presentation to the knowledge of students at the advanced undergraduate and early graduate level. The main requirements for students taking this course are that they have taken an advanced undergraduate course in electricity and magnetism and that they are knowledgeable about using the basic principles of vector calculus, i.e., gradient, divergence, and curl, and the various identities involving these vector operators. Although extensive use is made of complex variables, no special background is required in this subject beyond what is covered in an advanced calculus course. Relatively advanced mathematical concepts that are not typically covered in an undergraduate sequence, such as Fourier transforms, Laplace transforms, the Cauchy integral theorem, and the residue theorem, are discussed in sufficient detail that no additional preparation is required. Although this approach has undoubtedly added to the length of the book, we believe that the material covered provides an effective and self-contained textbook for teaching plasma physics. MKS units are used throughout.

This chapter is devoted to the analysis of MHD equilibria and stability. By equilibria, we mean a plasma state that is time-independent. Such states may or may not have equilibrium flows. When the states do not have equilibrium flows, that is, U = 0 in some appropriate frame of reference, the equilibria are called magnetostatic equlibria. When the states have flows that cannot be simply eliminated by a Galilean transformation, the equilbria are called magnetohydrodynamic equilibria. When we introduce small perturbations in a particular equilibrium which is itself time-independent, the time dependence of the perturbations determines the stability of the system. If an equilibrium is unstable, the instability typically grows exponentially in time. The mathematical problem for the stability of magnetostatic equilibria is made tractable due to the formulation of the so-called energy principle. It turns out that when MHD equilibria contain flows that are spatially dependent, the power of the energy principle is weakened significantly, and there has been a general tendency to rely on the normal mode method, for which we provide simple examples.

In nature and in the laboratory, plasmas can be stable according to the equations of ideal MHD. However, even ideally stable plasmas can become unstable in the presence of small departures from idealness, such as a small amount of resistivity. This may appear counter-intuitive upon first glance unless one takes into account the fact that in the presence of even small dissipation the frozen field theorem discussed in Chapter 6 is violated, which enables the plasma to access states of lower potential energy through motions that would be forbidden for ideal plasmas, i.e., by allowing magnetic field lines to slip with respect to the plasma fluid. Such instabilities are called resistive instabilities. These instabilities are part of a general class of phenomena called magnetic reconnection, which is a subject of great interest for space, laboratory, and astrophysical plasmas.