Tilting mode stability of spheromak

Abstract

We analyze the fixed boundary tilting mode stability for a force-free spheromak in spherical and cylindrical geometries. For this purpose we follow Rosenbluth and Bussac who introduce two methods, one is by the $\delta_\mu$, the change of an energy eigenvalue, another is the $\delta$W method by computing $\delta W_{MHD}$ with the arbitary shaped cross sections. We fined that the first method by $\delta_{\mu}$ is inadequate for the determination of stability, while the 2nd method gives the correct results, namely that the oblate spheromak is essential for tilting mode stability. The inadequacy of the $\delta_{\mu}$ method is demonstrated by using various forms of boundaries. In cylindrical geometry we also find that a convex shaped cross section is better than a concave form against this tilting mode.