Mathematical and Computational Models for Transient Magnetohydrodynamic Flows of Magnetically Confined Plasmas

Abstract

This thesis is concerned with the development of a multiphysics and multimaterial computational model for the solution of the MHD equations in the presence of complex real material boundaries for highly transient case studies. The objective of building this model is to contribute in the design of an integrated infrastructure for whole-device modelling simulations of magnetically confined tokamak plasmas for fusion applications. Specifically, this work focuses on the solution of a nonlinear set of partial differential equations, using explicit time integration and a finite volume approach on a Cartesian grid taking advantage of hierarchical adaptive mesh refinement techniques both in space and time. A key element of this methodology is the discretisation of topologically complex rigid boundaries as well as material and state of matter interfaces, which represents a significant departure from current, segregated solutions on physics-driven mesh alignment. This is facilitated through the implementation of state-of-the-art multiphysics and multimaterial numerical techniques, which are new to magnetically confined fusion simulations. Overall, the numerical approach taken in this thesis is radically different to the current methodologies on the counts of time integration, equation discretisation, mesh generation and scalability prospects, in an attempt to address some of the current limitations and meet the modelling requirements for more detailed, accurate and efficient simulations of tokamak plasmas. In particular, this research explores highly unstable flows when a plasma is in direct contact with a vacuum or a region of partial vacuum, separating it from a confinement vessel. To this end, a modified flux approach is employed, which has been newly adapted to work within an MHD framework for the first time, resulting in a novel plasma-vacuum interface fitting model. As a result, this study evaluates the differences between true vacuum and partial vacuum modelling usually performed by adopting a low-density approximation. Additionally, a Riemann rigid body ghost fluid method (GFM) is employed to facilitate the interaction at material interfaces. The GFM is originally modified in this thesis such that the plasma behaviour is coupled to the externally applied magnetic fields. This provides a seamless solution across the various plasma regions and takes into effect the electromagnetic properties of the surrounding structures, which are either modelled as perfect conductors or bodies of finite electrical resistivity. Consequently, this study provides insights into the two-way communication between the various components of the system and the effect of varying electromagnetic properties on the flow of the plasma. Finally, the model is used to simulate vertical displacement instabilities within a straight tokamak configuration, which is a case study with a known analytic solution that has previously been performed in the literature. This is a type of instability which can lead to a major disruption known as a vertical displacement event and sufficient evidence is provided to demonstrate that this model is capable of simulating such phenomena, which is crucial for their prediction, prevention and mitigation.