Chaos and Structures in Nonlinear Plasmas

The following sections are included:

• Preface

• Contents

To appreciate the advances made in nonlinear plasma physics during the 1980s it is important to know the classical aspects of the theory of nonlinear oscillations. In addition the principal theoretical methods and the physical phenomena found in the study of the one-dimensional nonlinear oscillator are important aspects of many nonlinear problems in physics and engineering. A list of phenomena produced in nonlinear oscillations is given in Table 1.1. The work Nonlinear Oscillations by Minorsky (1962) presents a systematic development of the techniques developed for nonlinear ordinary differential equations by mathematicians, engineers, and physicists up to 1960. A more accessible introduction to the properties of nonlinear oscillators can be found in Chapter 3 of the textbook Newtonian Dynamics by Baierlein (1983). At a more advanced level the Landau and Liftschitz (1958) Classical Mechanics gives a concise treatment of nonlinear oscillators, while the standard reference for understanding the behavior of coupled nonlinear oscillators is the review article A Universal Instability of Many-Dimensional Oscillator Systems by Chirikov (1979)…

An important class of dynamical systems occurs when the flow of the system points conserves a measure such as phase space volume. Such measure preserving flows have distinctly different properties and theorems from the volume contracting flows that occur in dissipative systems. The constraint on the flow of conserving volume in the x_i-vector space is a strong restriction that allows the dynamics to be derived from the gradients of a scalar function called the Hamiltonian H(p, q, t)…

Predicting the trajectories of charged particles in specified magnetic fields is a central problem in both laboratory and space plasma physics. The basis for the fusion project, which has developed into a world-wide scientific-technological project over the past thirty years, is to understand and predict the particle orbits well enough to confine the ions and electrons with magnetic fields for periods of time sufficient to allow the slow nuclear reactions to take place. In space plasmas it is important to understand the trajectories of the charged particles in the magnetospheric fields. Near the earth or other planets the magnetic fields are dipole-like, but away from the planet they are weak and strongly varying. Near the magnetopause and in the magnetotail region on the night-side of the planets the magnetic field can form thin loops, reversed fields and cusps…

In Chapter 4 we develop in some detail the phase space structures of the mixed regions of chaotic orbits and integrable orbits that act to define the generic Hamiltonian system. While Chapter 3 deals with phase space flows arising from Newton's laws taken with the Lorentz force acting on the charged particles, here the emphasis is changed to describe the mappings created by the intersection of the flows with a reference plane in phase space. Obtaining the maps on the surface of section for the flows is generally a difficult physics problem as shown in Chapter 3. With a given area preserving map the dynamics can be investigated in much more mathematical detail as made clear in the work of Henon (1983, for example). In this Chapter we use maps to determine the structure of the dynamics, the ergodicity, the transport and the diffusion approximation with more clarity and accuracy than is possible for the flows of Chapter 3…

In this Chapter we present the first-principles derivations of several of the wellknown integrable nonlinear wave equations of plasma physics. The starting point is the collisionless kinetic equation for the one-particle phase space distribution F(x, v, t) determined in the self-consistent electric field E(x, t) = −∂_xø(x, t). This is the famous Vlasov-Poisson self-consistent field problem. In the first two sections the kinetic form of the Korteweg and deVries (KdV) equation and the cubic nonlinear Schrodinger (NLS) equation are derived from the Vlasov-Poisson field equations. Then the effects of the kinetic particle resonances are discussed, and the theory of the modulational instability (MI) is developed…

The theory developed in this Chapter is closely related to the dynamics of fluids on rotating planets and their simulations in the laboratory. Because of the economic importance of this area of geophysical and atmospheric research there is vast literature relevant to the subject of Chapter 6. The reasons why the low frequency transport processes in a magnetized plasma are closely analogous to the dynamics of rotating fluids are made clear in the first section, Sec. 6.1. The most useful source of up-to-date research and further references is to be found in the special issue of CHAOS 4(2) (1994) edited by M. Nezlin. In this issue one can find the appropriate references to both the plasma physics and geophysical literature for the subject of Chapter 6.

The subject of plasma turbulence is a large and complex field. Here we restrict the Chapter to that part of the subject that follows naturally from adding forcing from instabilities and dissipation from wave-particle interactions to the drift wave problem developed in Chapter 6. The introduction of growth and damping to the drift wave-Rossby wave spectrum is of key importance to both the problem of plasma confinement in tokamaks and to the description of Rossby wave turbulence. The techniques developed here are general ones, applying to many mode coupling problems. As an example, the problems of ionospheric turbulence (Sudan and Keskinen, 1977), as well as numerous other forms of plasma turbulence, are treated by the techniques presented in this Chapter…

The following sections are included:

• Bibliography

• Index