Introduction to MHD Instabilities

In this paper we will discuss the theoretical framework of ideal MHD stability. We first show that the Alfvén wave is of central importance for the understanding of macroscopic plasma dynamics and, hence, of MHD stability. In section 2 we introduce the elements of spectral theory needed for our study, and in section 3 we apply these methods to inhomogeneous plasmas in cylinder geometry. Conclusions are drawn in section 4.