Nonlinear instabilities in magnetized plasmas: a geometrical treatment

The objectives of this paper are four-fold. The first, and main concern, is the development of an alternative approach to the description of plasma physics using methods of differential geometry. These methods have long been used in many other areas of physics, such as general relativity, or quantum field theory, but do not seem to have seen extensive application in plasma physics, and in particular in magnetohydrodynamics (MHD). The second objective is to employ this formalism for perturbation calculations, particularly to nonlinear processes in MHD. The use of differential geometry for variational calculations in ideal MHD allows a self-consistent, and compact calculation of the Lagrangian, and yields results valid for arbitrary topologies of the magnetic field. The third objective is to outline the use of this formalism in analyzing several plasma processes that occur in systems with complex magnetic-field topologies. We specifically focus on the nonlinear stability of plasmas in the magnetotail-like configuration of the magnetic field, such as found in the Earth's magnetosphere. Finally, we utilize previous results to present a self-consistent method for the investigation of the nonlinear stability of magnetized plasmas and for the investigation of the transition between linear and nonlinearbehavior for systems close to equilibrium. This method is based on the analysis of potential energy density, using results for plasma displacement from a linear model to calculate the second- andthird-order energies. We demonstrate this method on an example of a force-free field with magnetic-field lines stretched from dipolar configuration. In this example, we can clearly identify the transition between linear and nonlinear instability. PACS Nos.: 52.30.–g, 52.35.–g