Spectra and growth rates of a generalized screw pinch

The theory of Weitzner for the bumpy $\theta$ pinch is extended to the case of a generalized screw pinch in which the equilibrium magnetic field is ${\mathrm{B\hspace{0.17em}=\hspace{0.17em}(\epsilon B}}_r{\mathrm{,\hspace{0.17em}\epsilon B}}_{\theta }{\mathrm{,\hspace{0.17em}B}}_z)$ . $\epsilon$ represents the slow periodic $z$ variation of the axisymmetric equilibrium and the stability relative to their $r$ variation. Perturbation expansions in $\epsilon$ and in the bumpiness of the flux surfaces are performed on the linearized equations of ideal magnetohydrodynamics. From the theory of Grad there should be four stable continua, two of which are found for modes with ${\omega }^2{\mathrm{\hspace{0.17em}=\hspace{0.17em}O\hspace{0.17em}(\epsilon }}^2)$ , while for the transverse modes a generalized Suydam criterion for local instability is derived. The (global) growth rates of the most unstable transverse modes are computed numerically for various $B_{\theta }$ and the results are compared to those of the bumpy $\theta$ pinch (where $B_{\theta }\text{\hspace{0.17em}≡\hspace{0.17em}0)}$ ) as well as to the ordinary screw pinch (where $B_r\text{\hspace{0.17em}≡\hspace{0.17em}0}$ ).