Renormalization and the breakups of magnetic surfaces

Containment of field lines is a fundamental problem for magnetically controlled fusion. Transport along field lines is much faster than transport across field lines, so that containment of field lines is the first step toward containment of plasma in toroidal devices. Thus this problem has been studied for many years. A critical point in its solution was the recognition by Feigenbaum that renormalization provided a fruitful way of studying period doubling in one-dimensional maps. This was very far from the problem of interest to fusion, but some years later Kadanoff5 and co-workers showed that these techniques could be applied to problems of more-direct relevance, namely, the existence of containing magnetic surfaces. MacKay then carried Kadanoff's program to conclusion. The purpose of this chapter is to gather these results together and to set them into a fusion context. Sections 2 and 3 discuss the magnetic-containment problem. First, magnetic field lines in axisymmetric systems are treated, and then the effect of nonaxisymmetric perturbations is introduced. Period doubling in a magnetic-field-line context is discussed in Section 4, primarily as an introduction to renormalization in such systems. The breakup of magnetic surfaces is treated in Section 5 and Section 6 contains conclusions