Having established some characteristics and constraints with respect to particle pressure and magnetic pressure in a magnetic confinement fusion device, we now further examine some properties of the magnetic field itself. In Ch. 5, we discussed the motion of individual charged particles due to a variety of constant and spatially dependent magnetic fields. Here we investigate the case of a simple axially symmetric В-field, the consequent particle motions then recalled with some results from Ch. 5.

Specifically, we consider particle motion in a cylindrically symmetric configuration, i. e. Э/Э9 = 0, with an axial magnetic field В = Bzk, as illustrated in Fig. 9.2. As suggested by electromagnetic theory, we take В to be represented by means of a vector potential A,

В = V x A. (9.12)

This vector potential can be shown to be determined from the electric currents which generate В and is, in the stationary case, given by

A(r) = ^f.^r * .d3r’ (9.13)

4kj r — rl

where r is to be taken in cylindrical coordinates (r,9,z). Since, in the geometry chosen here, the field generating currents point in the poloidal direction e0, it is obvious that A possesses only one non-zero component which is A0 and thus yields

В = V x Aete = + -|-K)k. (9.14)

oz r dr

In the absence of any other external field, the equation of particle motion is then, according to Eq.(5.1), described by

with Ед representing that electrical field which is consistent with temporal variations of В and hence satisfies Maxwell’s relation

ЭА_ .

Ел — ——— ‘Леев

Multiplication of the above equation by r leads to the angular momentum і due to the rotational motion about the z-axis, i. e.

т{г2в + 2ггв^+ r-^-(rAe} = ^-{тг2в + qrAg’j

<9Л9>

dt ‘

= 0.

wherein Ae needs to be determined from

Bz = (VxA)-k = -^(Me).

r or

That is, specifically we have

Ae = ~r’ Bz(r’)d’r’ ; r *

roughly assessing, Ae can be approximated to the lowest order by

Ae~LrBz(0) (9.22b)

and inserted into Eq.(9.20) which, recognizing that the rotation is such that sign(в) = sign(q) and further that r6 = vxand here mvjVlqlBz = rg, can then be

written as

These surfaces where rAe = constant may be labeled flux surfaces of the magnetic field and the particle’s guiding centres move on them in the absence of other forces; this follows as a consequence of angular momentum conservation. Even if field gradients exist, which cause particle drifts across В-lines, the particles would, in a cylindrically symmetric plasma, remain on their flux surface since the underlying symmetry here constrains all gradients into the radial direction and, correspondingly, all drifts appear in poloidal directions. In this case, only collisions or anomalous transport processes can drive particles across the flux surfaces, which then is described by the perpendicular diffusion coefficient, Eqs.(6.18b) and (6.19). We further note that, according to Eq.(9.5b), the surfaces of constant В must also be surfaces of constant pressure and consequently, in most cases also of constant density and temperature.