Toroidal Contributions to the Shear and thej∥-Kink Instability in the Tokamak and Screw Pinch

It is shown that, for a Tokamak plasma with fixed boundary, the toroidal contributions to the shear are sufficiently important to modify the $j_{\parallel }$-kink instability. Assuming that $\beta$ is small and using the energy principle, $\delta W$ is calculated to order ${\left(\frac{r}{R}\right)}^6$ and the toroidal corrections are shown to change the sign of the destabilizing term. One consequence is that the range of rotational transform angles for which the $m=1\mathrm{}$, $\nu =1\mathrm{}$ instability will occur, when the shear is weak, is below the Kruskal-Shafranov limit and not above it. Also, the new terms have an important stabilizing effect for the screw pinch.