| Explosive instabilities in plasmas are one of the most frequent, yet least understood phenomena in physics. In this thesis we present a method of investigating non-linear (explosive) plasma instabilities based on the combination of a variational approach with differential geometry. This method makes it possible to resolve questions about non-linear stability without solving the full non-linear dynamical equations. We demonstrate this method for the case of ideal magnetohydrodynamics (MHD) which is an excellent approximation of plasma behavior for many physical systems. At the end we apply this method to two examples of plasma instability investigation, in rectilinear, and in a curved magnetic field topology.; In the first example, plasma in a box, we know apriori that the system must be stable in the absence of gravity, since there is no source of free energy that could drive the instability. This simple example illustrates the simplicity of the method we developed.; The second example deals with plasma in a curvilinear, stretched magnetic field topology. Since the stretching of the field lines provides a source of free energy, it is possible for an instability to develop. In the case we present we demonstrate that initial linear instability evolves into an explosive behavior on a very short time scale. This result is consistent with the results obtained from global MHD modeling.; Possible applications for this method are space weather prediction, and the improvement in our understanding of explosive processes in magnetically confined plasmas. |
