P/2170 USA The Stellarator Concept By L. Spitzer Jr. The confinement of a very hot ionized gas under, scopically, this same result is obtained from the fluid laboratory conditions would appear to require the use equations discussed below. of a strong magnetic field. While an enormous variety Basically, the confinement scheme in the stellarator of magnetic configurations is possible, two relatively consists of modifying the magnetic field so that a single simple geometries may be set apart at the outset, line of force, followed indefinitely, generates not a both involving infinite cylinders with axial symmetry. single circle but rather an entire toroidal surface, In the first of these, the magnetic field is produced by called a "magnetic surface". The tube enclosing the an axial current flowing through the gas. This con- gas is, ideally, one of these surfaces, enclosing an figuration is the so-called "pinch effect", which has entire family of such magnetic surfaces. It is shown been extensively discussed in the recent literature. In below that such surfaces can be produced, to a high the second simple geometry, the magnetic field is approximation. A later section discusses the confine- parallel to the axis, and is produced by external ment of a plasma in a system characterized by currents, flowing in solenoidal windings encircling the magnetic surfaces. Heating of the gas is treated in plasma. Such a straight cylinder, with an externally another section; the techniques considered involve produced field, forms the basis of the "pyrotron" heating to intermediate temperatures (about 106 °K) proposed by R. Post.1 by currents flowing parallel to the confining magnetic The stellarator, like the pyrotron, utilizes an ex- field, and subsequent heating to very high tempera- ternal magnetic field, produced by coils encircling a tures by means of a pulsating magnetic field, with a tube containing the heated gas. However, instead of a wide variety of possible pulsation frequencies. The finite cylindrical tube, the stellarator employs a tube present paper constitutes an introduction to the bent into a configuration topologically like a torus, series of papers from Project Matterhorn, printed in without ends. Such a tube will be referred to as the Proceedings of the 1958 Geneva Conference. "toroidal". Relative to the pyrotron, this configura- tion has the advantage, in principle, of permitting ROTATIONAL TRANSFORM AND MAGNETIC more complete confinement, since end losses are SURFACES eliminated. Relative to the pinch discharge, the stellarator offers the advantage, again in principle, of In this section we consider certain properties of a permitting equilibrium in a steady state. Both these stellarator magnetic field which do not depend advantages might be of importance in a controlled directly on the presence of a plasma in the system. In thermonuclear reactor. The present paper outlines particular, we are interested in how far a line of force the basic concepts involved in the confinement and in a stellarator tube may be followed before it inter- heating of a gas in a stellarator. sects the tube wall. If we could show in all rigour that only one magnetic surface passed through each One basic complexity in the stellarator results from point within the tube, then no line of force would the fact that the simple torus, in which the magnetic ever intersect the tube wall. While this result is not lines of force are circles centered at the axis of sym- exactly true, it is apparently true to a high order of metry, does not permit equilibrium confinement of a approximation. The demonstration of this result de- plasma in a straightforward manner. Microscopically pends on a certain abstract property of the stellarator this result follows at once from the particle drifts magnetic field, called a "rotational transform". associated with the inhomogeneity of the magnetic We first discuss the properties of such transforms, and field. These drifts, first pointed out by Gunn,2 and 3 postpone until later a discussion of how they are analyzed in detail by Alfvén, produce motions per- produced. pendicular both to В and to VB; these motions are in opposite directions for electrons and positive ions. The resultant separation of charges produces electric fields Properties of Rotational Transforms which sweep the ionized gas towards the wall. Macro- Let us pass a plane through the stellarator tube at some point. For convenience in representation, we shall take the plane to be perpendicular to the mag- * Project Matterhorn, Princeton University, Princeton, netic field, B, in the central region of the tube, New Jersey. although this restriction is not required. We assume 181 182 SESSION A-9 P/2170 L. SPITZER Jr. that the magnetic field is non-zero at all points in the addition to a general rotation. In such- systems there cross-sectional plane. Then through any point, such will be only one point that transforms into itself, and as Pi, for example, a line of force will pass. This line only one line of force that is closed after a single of force may now be followed, in the direction of the circuit around the stellarator. This line is called the magnetic field, along the stellarator tube, until it has "magnetic axis". completed one circuit of the toroidal tube and inter- A second basic result on measure-preserving rota- sects the cross-sectional plane at a point P2, as shown tional transforms, also established by Kruskal,6 is that in Fig. 1. In the ideal torus, P2 coincides with Pi and any point, other than one of the fixed points, when every Une of force is closed after one circuit. If the followed through successive transformations, will not degeneracy of the torus is removed, P2 no longer move far from a single closed curve. This result is coincides with Pi. illustrated in Fig. 1, where the points P2, P3, P4 One might raise some question as to whether any generated by successive magnetic transforms of the of the points Pa lie inside the tube wall. It is not point Pi, all lie close to a single closed curve. Thus a difficult to construct a toroidal tube with a solenoidal single line of force, after many circuits around the winding such that most of the lines of force stay tube, generates a magnetic surface. within the tube for at least one circuit. The interesting More precisely, let us introduce coordinates г, в in question is rather which lines intersect the tube wall the cross-sectional plane depicted in Fig. 1; v may be after many circuits, and we may safely assume that measured from the magnetic axis, denoted by the for most points Pi, P2 win exist and will lie inside the point O. The value of Д0 between Pi and P2 is denoted tube wall. If we exclude, for the moment, those areas by i, and is called the "rotational transform angle". for which P2 does not lie within the tube wall, then Let в equal 0 at Pi, and let us assume that в equals for every point Pi, we have a point P2; the trans- 2ir for the point Рй. The distance Ar from Pi formation of the set of points Pi into the set of points to Pn is called the "deviation from closure" of the P2 is called a "magnetic transform" of the cross- point n. Evidently Ar measures how far the line of sectional plane. force has strayed from a closed curve. Kruskal has A magnetic transform of this type is characterized shown that Ar decreases more rapidly than any power by an important property. Let us take the density of of \\n. Hence one may surmise that Ar varies about as points Pi to be proportional to the component of В exp(-Kn), where К is some dimensionless constant. normal to the cross-sectional plane. Since the plane is The physical reason for this result can best be under- transformed into itself by the magnetic transform, the stood in the special case when the normal component of density of these points must be the same function of the magnetic field is constant over the cross-section position before the transform as after. A transform plane. The analysis of more general systems may be with this property has been called "measure preserv- reduced mathematically to a consideration of this ing" by Kruskal. special situation. In this case, the density of points in Let us now assume that the magnetic transform is the plane must remain constant in successive trans- also primarily rotational. This condition states that at formations. Let us now draw, in the cross-section least, the outer portions of the plane all rotate in the plane, a closed curve connecting point 1 and its same direction in a single transformation. As we shall successive transformed points as smoothly as possible. see subsequently, there are several ways of achieving Since the H-transform now preserves areas, the total this result in a toroidal system. According to Kruskal,4 area enclosed within this curve must remain constant it follows from this assumption and from the Brouwer in successive transformations. Hence all points on the fixed-point theorem that there must be at least one curve cannot move inwards with successive trans- point in the plane which is transformed into itself. In formations. If some move in, others must move out. some types of stellarator, the magnetic transform In the special case in which the в coordinate of every involves only small deformations of the plane, in point returns to its original value after n transforms, it TUBE WALL is possible for some points on the curve, together with all their transformed points, to move steadily in, while the points between move steadily out.