Toroidal Pinches

Figure 1.5: Schematic diagram of a toroidal pinch.

An obvious way of dealing with the problem of the end plates is to bend a linear pinch into a circle to produce a toroidal plasma. See Figure 1.5. This procedure allows us to carry over most of the results that we previously obtained for a linear pinch while obviating the need for the end plates. In a so-called toroidal pinch, the plasma is confined on a series of axisymmetric, nested, toroidal magnetic flux-surfaces [26]. As such, the plasma thermal energy is not able to reach the toroidal vacuum vessel surrounding the plasma by freely streaming along magnetic field-lines, which, as we have seen, is a very rapid process, but must instead diffuse across the magnetic flux-surfaces, which is a comparatively slow process. The magnetic field of a toroidal pinch consists of a poloidal component that circulates about the magnetic axis, and a toroidal component that runs parallel to the axis. See Figure 1.5. The poloidal magnetic field is generated by a toroidal current that is induced in the plasma via transformer action. In essence, the plasma forms the single-turn secondary winding of a transformer circuit. The toroidal magnetic field is generated by currents flowing in magnetic field-coils that surround the plasma [64]. The poloidal magnetic field is responsible for confining the plasma via the pinch effect (when it is crossed with the toroidal plasma current). The toroidal magnetic field is needed to stabilize the plasma against ideal kink modes, but does not contribute greatly to confinement (other than by reducing the gyro-radii of charged particles within the plasma).

Let $R_0$ and $a$ be the major and minor radii of the plasma torus, respectively. Consider an idealized magnetic flux-surface of circular poloidal cross-section. Let $r<a$ be the minor radius of this surface, and let $\theta$ and $\varphi$ be poloidal and toroidal angles, respectively. Furthermore, let $B_\theta(r)$ and $B_\varphi$ be the mean poloidal and toroidal magnetic field-strengths, respectively, on the surface. The safety-factor,

$$q(r) = \frac{r\,B_\varphi}{R_0\,B_\theta},$$ (1.76)

is the mean number of toroidal circuits of the plasma that a magnetic field-line within the flux-surface completes for every poloidal circuit [64]. A more exact definition of this important quantity is given in Equation (2.128). Incidentally, throughout this book, we shall make the conventional assumptions that the $r$, $\theta$, $\varphi$ coordinate system is right-handed, and that $B_\theta>0$ and $B_\varphi>0$. This implies that $I_p>0$ (in other words, the toroidal plasma current runs in the $+\varphi$ direction), and $q>0$.