Bifurcation approach for ballooning and tearing instabilities in a toroidal plasma

An exact representation of MHD in terms of stream functions and potentials is constructed in such a way that, instead of the velocity V, the product V x Bo is expressed by stream functions, where Bo is a reference magnetic field. From this, we derive a set of reduced equations which keep the distinction between paralled and toroidal motion. This representation is suited for the bifurcation theory of MHD instabilities. To illustrate this, we construct a class of analytical equilibria and we study its linear MHD stability in order to formulate the problem of the bifurcation of time-asymptotic solutions of the non linear MHD equations for a tearing and ballooning unstable toroidal plasma