Three-dimensional peeling-ballooning theory in magnetic fusion devices

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Nuclear fusion is the fundamental process that generates heat and light in the stars but it is also a promising potential candidate for the generation of energy by man. However, where in the center of stars the combination of extreme temperatures with extreme pressure is what drives light elements close enough together for them to fuse and release part of their combined mass as energy, on earth only extreme temperatures can be employed. Matter at these temperatures exists in the state of plasma, where the atoms are stripped clean of their electrons. In the resulting physical system the presence of long term electromechanical forces between the charged particles can lead to violent collective behavior. Therefore, the general question of confining hot plasma in a stable way is crucial in order to achieve fusion. One strategy of doing this is by employing powerful magnetic fields to guide the charged particles around a toroidal configuration. This work is about a class of instabilities that these configurations are susceptible to, 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 some significant processes that occur in magnetic configurations, such as the periodic release of energy through Edge-Localized Modes (ELMs), or even the complete loss of confinement during disruptions, during which a large amount of energy is released to the reactor walls, damaging them. However, whereas much work has been performed in this field, the analysis of high-n instabilities in realistic 3-D geometries, including the effects of the deformation of the plasma edge, has not yet been done yet in a systematic and dedicated manner. Therefore, in the first part of this work a suitable theoretical framework is developed. Here, the simplification can be made that only modes pertaining to the same field line couple, through their high-n nature. This reduces the dimensionality of the problem by one, but at the same time does not pose any limitations on the 3-D aspects of the instabilities. One of the results of the theory is a system of coupled ordinary differential equations that can be solved for an eigenvalue, the sign of which 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, two 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 physical criteria and by comparisons with two other, well-established numerical codes that have ranges of applicability bordering on that of PB3D. The first one, MISHKA, is a general-n stability code for axisymmetric equilibria, whereas the second one, COBRA, can treat general 3-D cases, but only in the n→ ∞ limit, with a static edge. The successful introduction of PB3D paves the way for a multitude of potential applications concerning 3-D edge effects. It can be investigated, for example, how many previous findings concerning peeling-ballooning modes in axisymmetric configurations change or not when 3-D effects are introduced. The theory of high-n stability of axisymmetric equilibria, for example, in the past has shed light on the dynamics of ELMs, and how this changes by including 3-D effects is a topic of interest. This is true even more so as recently the relevance of ELM control has risen due to the potentially dangerous behavior of ELMs in the next generation nuclear fusion reactors. A strategy for controlling them also intrinsically relies on applying 3-D resonant magnetic perturbations. The study of these effects with PB3D is planned in the near future in the ITER Organization. Before that, in this work, as a first concrete application, the modification of the stability of the pedestal of a High-confinement plasma equilibrium configuration by a toroidal field ripple is considered. These so-called H-mode configurations are characterized by a steep pressure gradient near the plasma edge, called the pedestal, which increases the temperature and pressure attainable in the core. Therefore, they are often seen as vital in order to achieve fusion. In practice, however, a degradation of the pedestal size is often observed, due to 3-D modifications of the equilibrium, such as the periodic ripple in the toroidal magnetic field due to the discreteness of the toroidal field coils. It was observed here that the application of a toroidal ripple in the shape of the poloidal cross section in the order of a percent, lead to a substantial decrease in the highest possible pedestal pressure, in the order of 30-40%. This substantiates good qualitative agreement with experimental results, where degradations of similar magnitude were observed.
Física del plama, Fusión nuclear, Electromagnetismo, Plasma physics, Electromagnetism, Simulation
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