Introduction

Chapter 12 is concerned with the MHD stability of toroidal systems, specifically tokamaks and stellarators. As might be expected the complexities associated with multidimensional systems make analysis quite difficult. Consequently, in practice stability results are often obtained numerically. There are, however, a variety of simple models which do provide physical insight into qualitative geometric features that lead to favorable and unfavorable stability behavior. This chapter describes several of these simple models plus summarizes several of the major numerical studies.

The discussion begins with a general analysis of ballooning modes. These are radially localized modes with short perpendicular wavelengths and long parallel wavelengths. They arise because most toroidal configurations have magnetic geometries with alternating regions of favorable and unfavorable curvature. The most unstable perturbations are those whose amplitude “balloons” out in the region of unfavorable curvature. The modes are important because they set one limit on the maximum stable β in toroidal configurations.

Introduction

As already mentioned in earlier chapters, the eikonal approximation can become invalid in local regions of the plasma. The most common problems are caustics (see Chapter 5), tunneling, and mode conversion. Both tunneling and mode conversion are processes where one incoming ray splits into two outgoing rays, a transmitted ray and a converted ray. The matched asymptotic methods are therefore more complicated than for caustics. Tunneling concerns only one eigenvalue of the N × N dispersion matrix, while mode conversion entails two. It follows that tunneling involves only one polarization, while mode conversion is associated with a pair. Therefore, tunneling can be reduced by Galerkin projection locally to a scalar formulation, while mode conversion is inherently a vector problem. An important point we should emphasize is the following: For caustics, it is always possible to find a local representation where the eikonal approximation is valid. In contrast, in tunneling and mode conversion regions, there is no representation in which the eikonal approximation is valid. It is only when we consider points far from the conversion region that we recover eikonal behavior. This leads to two important questions:

1. If the eikonal approximation is not valid within the conversion region, why persist in using ray tracing there?

2. Although the eikonal approximation is valid for the incoming wave field (by assumption), what justifies the assumption that the transmitted and converted wave fields become eikonal once more?

One of the major goals of this book is to develop the reader's geometrical intuition as it applies to the study of waves in plasmas, while at the same time developing useful methods for quantitative analysis. The great advantage of a geometrical approach is that it brings our visual intuition into play. We can sometimes develop a deeper physical understanding by drawing simple pictures. That deeper intuition can then guide us to new analytical approaches, or help us see the way through a complicated calculation. In the end, of course, we must be able to calculate solutions of wave equations, so in this chapter we provide a brief overview of methods for visualizing solutions and we provide some examples of field construction.

The outline of the chapter is as follows: We begin by summarizing why it is so challenging to visualize eikonal solutions in phase spaces of dimension higher than two. We first introduce the Poincaré surface of section, which is a commonly used tool for the study of higher-dimensional dynamical systems. While straightforward to understand and easy to use, the surface of section does not provide a global view of the Lagrange surface of rays, so it still leaves something to be desired. We then present some novel ideas for global visualization in two spatial dimensions (four-dimensional ray phase space) and consider two examples. The first example concerns electromagnetic waves in two spatial dimensions propagating from a vacuum into a dense plasma with a cutoff.

This is the central chapter of the book. We emphasize that in this chapter we introduce the eikonal approximation without discussing the accuracy of the approximation, or how to deal with situations where it breaks down (for example, at caustics or in mode conversion regions). Those are matters for later chapters. The great advantage of eikonal methods is that they reduce the solution of systems of PDEs, or systems of integrodifferential equations, to the solution of a family of ODEs. This often results in a substantial increase in computational speed in applications. In addition, the ray trajectories themselves often provide useful physical insight. The outline of topics follows.

Eikonal theory for multicomponent wave equations is first developed in x-space where we derive the eikonal equation for the phase and the action conservation law (in the form of a nonlinear PDE). It should be noted that the dispersion function that arises from the variational principle is one of the eigenvalues of the dispersion matrix, restricted to its zero locus in phase space. The polarization is its associated eigenvector. We discuss the fact that the interpretation of these results and the method of solution of the eikonal and action transport equations are most natural when viewed in ray phase space.

We then discuss how to relate the x-space and phase space viewpoints, the key ideas being lifts and projections. The notion of a Lagrange manifold arises naturally in this context as a lifting of a local region of x-space into phase space. Singularities that appear under projection are related to caustics, which are dealt with in Chapter 5.

Waves exist in a great variety of media (in all phases of matter) as well as a vacuum (in the case of electromagnetic waves). All simple waves, on the one hand, share some basic properties such as amplitude, frequency and period, wavelength, and wave velocity (both phase and group velocity). Waves in a turbulent medium, or waves generated by random sources, on the other hand, are more appropriately described in terms of probability distributions of amplitude, and spectral densities in frequency and wavelength. In this book, we focus primarily upon coherent waves that are locally plane wave in character. That is, they have a well-defined amplitude, phase, and polarization at most (but not all) points. The regions where this local plane-wave approximation breaks down are important, and the development of appropriate local methods to deal with them is an important topic of the book. We include a very short discussion of phase space approaches for incoherent waves, for completeness.

This is the first book to present modern ray-tracing theory and its application in plasma physics. The emphasis is on methods and concepts that are generally applicable, including methods for visualizing ray families in higher dimensions. A self-contained exposition is given of the mathematical foundations of ray-tracing theory for vector wave equations, based upon the Weyl theory of operator symbols. Variational principles are used throughout. These provide a means to derive a Lorentz-covariant ray theory, along with related conservation laws for energy, momentum, and wave action.