Numerical Bifurcation Analysis of Heat Transfer by the Edge Plasma of a Tokamak

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Nuclear fusion provides one of the most encouraging possibilities for the generation of clean and plentiful energy. The effort to harness this immense power is an active and vital area of research worldwide, while tokamaks-toroidal magnetic confinement fusion reactors-pose one of the promising techniques for doing so.The design and construction of such reactors require a detailed and analytical understanding of how the plasma behaves inside the chamber. Requiring temperatures on the order of 100 million Kelvin to sustain the fusion reaction, this plasma is inconceivably hot, fast, and turbulent. Failure to understand and correctly manage its stability can lead to reduced performance, loss of control and damage to components. The problem of ensuring effective stability under the phenomenal temperature gradients therefore is not just complex, but crucial.The project will work heavily with deflation-a numerical technique for exploring multiple solutions of nonlinear differential equations-with particular application to models of fluid heat transfer. This technique will be extended to periodic orbits and applied to problems in ordinary and partial nonlinear differential equations.The aim is to produce techniques deployable in the Excalibur project NEPTUNE, exploring the different possible mechanisms by which heat might be transferred by the edge plasma-or plasma scrape-off layer-within a tokamak. This project will combine hydrodynamics, numerical analysis, and nonlinear dynamics in application to exascale-class computations.Bifurcation analysis within fluid dynamics-numerical or otherwise-is a tremendously well-developed and active topic, offering many potential avenues of research.By way of example, one paper that provides an excellent springboard for some of the ideas that could be used in tackling this project is Fryderyk Wilczynski's 2019 PhD thesis "Convective Motions in the Scrape-Off Layer of Magnetically Confined Plasmas". Many of the results and methodology in this work are notably relevant to our problem.Wilczynski's paper focuses on the analysis of turbulence in the edge plasma, characterised by intermittent fluctuations known as filaments or blobs. Applying certain technical assumptions, the Braginskii equations of plasma dynamics are used to develop a 2-dimensional model of the scrape-off layer, on which Wilczynski performs extensive stability and bifurcation analysis. The resulting model bears resemblance to Rayleigh-Bénard convection in classical fluid dynamics, one of the most comprehensively studied examples of a nonlinear system.