Three-dimensional peeling-ballooning theory in magnetic fusion devices

Resumen

Nuclear fusion is the fundamental process that generates heat and light in the stars, and is therefore a promising potential candidate for the generation of energy on earth. However, where in the stars the extreme pressure in their center is the process that drives light elements close enough together for them to fuse and release part of their combined energy as energy, on earth extreme temperatures are employed. Matter at these temperatures exists in the state of plasma, where the atoms are stripped clean of their electrons, which makes for a system that can react very violently. Therefore, the general question of confining it in a stable way through strong magnetic fields is of great importance and this work is about a particular class of particularly important instabilities, called high-n instabilities. high-n instabilities are instabilities that have strong localization around the magnetic field lines that confine the plasma, and they have previously been identified as possible culprits for important processes that occur in magnetic configurations, such as the periodic release of energy through Edge-Localized Modes (ELMs), or the disastrous complete loss of confinement during disruptions. However, the analysis of high-n instabilities in realistic 3-D geometries, including the important effects of the deformation of the plasma edge, has not been done yet in a systematic and dedicated manner. Therefore, in the first part of this work a suitable theoretic framework is developed that allows for this. Here, an important simplification can be made by considering the high-n nature of the modes, while at the same time not posing any limitations on the 3-D aspects of the instabilities. One of the results of the theoretical framework is a system of coupled ordinary differential equations that contain an eigenvalue, whose sign determines whether the mode formed by the corresponding eigenvector is unstable or not. The solution of these equations, however, is something that has to be done using numerical techniques, so to this end the numerical code PB3D is developed. This stands for Peeling-Ballooning in 3-D, which are two important modes that are described well through high-n theory. PB3D can treat the stability of various equilibrium codes such a VMEC and HELENA in a modular way, is parallelized making use of the message-passing interface (MPI) and is optimized for speed. The code is verified making use of other, established numerical codes, as well as physical criteria. The succesful introduction of PB3D paves the way for a multitude of potential applications concerning 3-D effects. It can be investigated, for example, how the many previous findings concerning peeling-ballooning modes in axisymmetric configurations change or not when 3-D effects are introduced, or whether there may be new regions of stability in the parameter space that opens up when considering these 3-D effects. These are all important considerations. To start with something, as a first concrete application, in this work the modification of the stability boundary by a toroidal field ripple is considered, due to the discreteness of the toroidal field coils. Good qualitative and even some quantitative agreement is found with experimental results. PB3D will be used in the future to provide answers to the other questions posed above.