In an axisymmetric case, the flux surfaces (Fig. 10.4) embracing the magnetic field lines are, to a good approximation, annular toroidal surfaces with r = constant. The charged particles follow the helical field lines resulting from the combination of B6 + B0 and hence move on the flux surfaces, except for excursions of the order of the gyroradius.

While most of the particles are free to spiral around the helical field lines as they encircle both the major and minor axis of the torus, there is a class of particles which appear to be trapped in a magnetic well formed by the field variation between the inboard and outboard side of the torus. Both the toroidal and poloidal fields are stronger on the inside than on the outside of the torus, which results in an overall field variation as illustrated in Fig. 10.6 and at length amounts to a sequence of magnetic mirrors. Some of the plasma particles exhibiting a lower vjj in comparison with the particles moving completely about the torus (known as passing particles), are trapped by these mirrors according to the effects discussed in Ch. 9. If the particle trajectory from a number of toroidal journeys is projected onto a transverse plane of the torus (ф = const), a kind of banana-shaped orbit results for trapped particles, i. e. for particles encountering the mirror reflection. This is displayed in Fig. 10.7, where the trajectory of an untrapped particle is also illustrated. Its guiding centre motion is seen on a curve not quite coinciding with the corresponding magnetic flux surface, which is due to first order drift motion across B.

Recalling from Chapter 9 that the fraction of particles trapped in a mirror field is

	(10.18) (10.19)
and assuming Be « Вф such that B-B. °e
1 Ra + rcosd

we find here the ratio

В min Routboard ^ B( Ґ Cl)

В max R/nboard В(г — ~Cl)

1

R„+ а _ 1-е 1 l + e

R„ — a

where Є = a/R0 is the inverse aspect ratio of the tokamak. Thus, the fraction of trapped particles in a tokamak is given by

For a tokamak reactor with a/R0 * 1/3, approximately 70% of the plasma particles appear to be trapped, and thus their enhanced radial diffusion across the confining magnetic field is viewed to be a significant process in tokamak particle leakage.

The trapped particles bounce back and forth between poloidal angles of ±9b, thus avoiding the inner torus region. In combination with the toroidal drift this bounce motion results in the poloidal cross sectional orbit shown in Fig. 10.7, from where the name ‘banana orbit’ becomes self evident.

Apparently, the poloidal rotation of the magnetic field lines tends to suppress the vertical drifts due to field curvature and VB for trapped particles as well; they would spend equal times in the upper and lower halves of the torus if there were no toroidal electric field which, however, is necessary for driving the plasma current. As a consequence of this E-field, the banana orbits are no longer symmetric. It is seen that an ion spends a longer time period in the lower half of

the torus drifting radially inward than in the upper half where it drifts radially outward. In total, the trapped particles should be subjected to a net inward drift towards smaller minor radii, which thus pinches the plasma column. This phenomenon was predicted by A. Ware and is since called the Ware-pinch effect.

Another effect associated with banana trapping is the enhanced particle leakage. As previously discussed, collisions can cause particles to jump across the confining magnetic flux surfaces and thereby determine the diffusion losses. While the maximum distance which an untrapped particle can be displaced as a result of a single collision is of the order of the gyration radius rg, a trapped particle can be subjected to a maximum excursion as great as the banana width Ar, rap indicated in Fig. 10.7. Since typically Arttap> rg and the co-efficient for diffusion perpendicular to the magnetic field, D, increases with r2g = 2 / CO2,

(Eq.(6.16b)) and Ar2ttap, respectively, the trapped particles may escape more rapidly from the tokamak plasma than the untrapped ones. Evidently, if the fraction of trapped particles is large, this leakage enhancement constitutes a substantial problem in tokamak confinement.