On the theory of symmetric MHD equilibria with anisotropic pressure

Abstract

In this thesis we discuss the theory of symmetric MHD equilibria with anisotropic pressure. More specifically, we focus on gyrotropic pressures, where the pressure tensor can be split into components along and across the magnetic field. We first explore 2D solutions, which can be found using total field type formalisms. These formalisms rely on treating quantities as functions of both the magnetic flux function and the magnetic field strength, and reduce the equilibrium equations to a single Grad-Shafranov equation that can be solved to find the magnetic flux function. However, these formalisms are not appropriate when one includes a shear field component of magnetic flux, since they lead to a set of equations which are implicitly coupled. Therefore, in order to solve the equilibrium problem with a magnetic shear field component, we introduce the poloidal formalism. This new formalism considers quantities as functions of the poloidal magnetic field strength (instead of the total magnetic field strength), and yields a set of two equations which are not coupled, and can be solved to find the magnetic flux function and the shear field. There are some situations where the poloidal formalism is difficult to use, however, such as in rotationally symmetric systems. Thus we require a further formalism, which we call the combined approach, which allows a more general use of the poloidal formalism. One finds that the combined formalism leads to multi-valued functions, which must be dealt with appropriately. Finally, we present some numerical examples of MHD equilibria, which have been found using each of the three formalisms mentioned above.