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

核聚变为产生清洁和丰富的能源提供了最令人鼓舞的可能性之一。利用这种巨大能量的努力是世界范围内一个活跃而重要的研究领域，而托卡马克——环形磁约束聚变反应堆——是实现这一目标的有前途的技术之一。这种反应器的设计和建造需要对等离子体在腔室中的行为有详细和分析性的了解。需要1亿开尔文的温度来维持聚变反应，这种等离子体是难以想象的热、快和湍流。不了解和正确管理其稳定性可能导致性能下降、失控和部件损坏。因此，在惊人的温度梯度下确保有效稳定性的问题不仅复杂，而且至关重要。该项目将大量使用deflation-一种用于探索非线性微分方程的多个解的数值技术-特别应用于流体传热模型。该技术将扩展到周期轨道，并应用于常非线性和偏非线性微分方程的问题。其目的是开发可用于“王者之剑”海王星项目的技术，探索托卡马克内部边缘等离子体（或等离子刮擦层）传递热量的不同可能机制。该项目将结合流体力学、数值分析和非线性动力学应用于百亿亿级计算。流体动力学中的分岔分析-数值或其他-是一个非常发达和活跃的主题，提供了许多潜在的研究途径。例如，Fryderyk Wilczynski 2019年的博士论文《磁约束等离子体刮擦层中的对流运动》为解决这个项目提供了一些很好的出发点。这项工作中的许多结果和方法都与我们的问题密切相关。Wilczynski的论文侧重于分析边缘等离子体中的湍流，其特征是被称为细丝或斑点的间歇性波动。采用一定的技术假设，利用等离子体动力学的Braginskii方程建立了刮擦层的二维模型，Wilczynski对该模型进行了广泛的稳定性和分岔分析。所得到的模型与经典流体动力学中的瑞利-巴姆纳德对流相似，后者是非线性系统中研究最全面的例子之一。

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.