Bending a solenoidal coil around until its ends meet deforms an initially straight axial magnetic field into an axisymmetric toroidal field, as shown in Fig. 10.1. The toroidal field В is produced by simply passing a current through the coil wound around the torus. The torus is topologically defined by its major radius R0 and the minor radius a. The central line along the toroidal ring is called the minor axis, while the major axis is the one pointing out from the centre of the plane display of Fig. 10.1, i. e. going perpendicularly through the torus ring, Fig. 10.2.

Recalling our discussion of drift motion in Ch. 5, an examination of the toroidal configuration in Fig. 10.1 suggests that some complications do arise. The coils around the torus are more closely spaced on the inboard side than on the outboard side, creating therefore a radial variation in the magnetic field of the form

(10.1)

R

as shown in Fig. 10.2, with R denoting the distance from the major axis. Recall that a gradient in the magnetic field creates a grad-B force which drives the positive ions in one transverse direction and the negative electrons in the other. Further, the curvature drift drives them apart in the same way. The net result is a local charge separation, thereby generating a vertical electric field causing simultaneous ion and electron drift in an outward direction perpendicular to the major torus axis, Fig. 10.2.

To demonstrate these drifts we consider a simple magnetic field generated only from the current in the toroidal coils; then the field possesses only a component in the toroidal direction, еф, that is, we have

(Ю.2)

The magnitude of the toroidal field can be calculated by employing Eq.(9.50) with I now representing the total current threading the hole of the torus. This then gives

(10.3)

and proves the proportionality introduced in Eq. (10.1). Considering a charged particle moving along B, with speed % we note that it will be subjected to the curvature drift and, if it possesses also a velocity component perpendicular to B, to the VB drift as well and hence, according to Eqs.(5.51) and (5.61), its guiding centre drift velocity (ІЗ) will be

(10.4)

Curn

Coil

Introducing Eqs.(10.2) and (10.3) into Eq.(10.4) we find the drift motion in the simple toroidal field here to be described by

This equation is called the toroidal drift, the result of which is a polarization of the plasma with an electric field vector pointing in the negative z-direction. The resulting E-field unfortunately causes both the ions and electrons to move radially outwards due to the ExB drift with a velocity of

ExB

В B*(Ro) Ro

as shown in the detail of Fig. 10.2. Thus, the plasma in a simple toroidal field drifts outward until it strikes the surrounding wall. In this case, there is no radial equilibrium and hence practically no plasma confinement.

A common approach for reducing this outward drift resulting from charge polarization is the following. In addition to the toroidal magnetic field Вф produced by the toroidal coil windings, a poloidal magnetic field B9 is introduced. Depending upon the relative magnitude of these two B-field components, B9 and Вф, a helical path of particle migration results, as suggested in Fig. 10.3. Then the particles spend equal time in the upper and lower halves of the toroid thus canceling out the undesired charge separation when the particles have moved along a twisted field line through a completed poloidal rotation, 0 = 2k.

The poloidal angle made by a В-field line after it has traversed through a 271- revolution in the toroidal direction (ф) is called the rotational transform, and represented by l (iota). If l is very large, the revolving field lines become too tight making the plasma unstable against kink-type perturbations, Sec. 9.6. Therefore it is useful to define a safety factor by

2k

q = — (10.7)

i

which measures the field line pitch and it is seen from stability considerations to

require q>l in the plasma and q>2.5 at the edge of the plasma in order to suppress kink-type instabilities (Kruskal-Shafranov criterion). Thus, the rotational transform is constrained by

к2я (10.8)

which means that, following a magnetic field line, its pitch has to be such that the number of revolutions around the major torus axis exceeds the number of revolutions around the minor axis.

While the rotational transform does provide for some spatial mixing of the charged particles and so greatly reduces the electric field and, consequently, the transverse outward drift, another interesting feature occurs. The effect of the twisted magnetic field lines-each of which completely traces out a magnetic flux surface by its revolutions around the toroidal and poloidal axes-is to create a system of nested toroidal flux surfaces which guide ion motion. Note, that in order to avoid the undesired charge separation, the rotational transform must not be an integer multiple of 2it. In that case, the field lines would recombine after some trips around the torus and hence not cover the entire magnetic flux surface on which they lie. Further, note that q may vary from one magnetic flux surface, as defined in Eq.(9.24), to the next, and this leads to a shear in the magnetic field since the В-lines point in a different direction as one proceeds radially onto different flux surfaces. We illustrate this in Fig. 10.4. In typical tokamak plasmas q ranges from 1 near the centre of the plasma to values of 3-4 at the plasma edge. It is this shear which is effective against the growth of kink and drift perturbations. A perturbation aligned with B(r) will, at a point with increased minor radial distance r + dr, encounter field lines at a different angle which again will vary as the perturbation grows to another distance r + dr’. Any helically resonant instabilities are thus radially localized.

The poloidal magnetic field can be established in two ways. One is by arranging outside the plasma a set of poloidal coils which carry a current in the toroidal direction. Devices based on this principle are called stellarators. Others, known as tokamaks, employ a toroidal current flowing in the plasma; the latter happen to be relatively less complex devices and are therefore of greater current interest.