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。利用这种巨大能量的努力是全世界研究的一个活跃而重要的领域，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，这种反应堆的设计和建造需要对等离子体在反应室内的行为有详细的分析了解。维持聚变反应需要一亿开尔文的温度，这种等离子体是不可思议的热，快，和湍流。如果不了解和正确管理其稳定性，可能会导致性能下降、失控和组件损坏。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。该项目将大量使用放气-一种用于探索非线性微分方程多解的数值技术-特别适用于流体传热模型。这种技术将被扩展到周期轨道和应用于普通和偏非线性微分方程的问题，目的是产生可部署在神剑项目NEPTUNE的技术，探索不同的可能机制，通过这种机制，热量可能会被边缘等离子体或等离子体刮离层在托卡马克。本计画将结合联合收割机流体动力学、数值分析与非线性动力学应用于百万兆级的计算。流体动力学中的分叉分析--数值或其他--是一个非常成熟且活跃的课题，提供许多潜在的研究途径。举例来说，有一篇论文为解决这个项目的一些想法提供了一个很好的跳板，这篇论文是Fryderyk Wilczynski 2019年的博士论文。磁约束等离子体刮除层中的对流运动。在这项工作中的许多结果和方法是显着相关到我们的problem.Wilczynski的论文侧重于在边缘等离子体中的湍流的分析，其特点是间歇性波动被称为细丝或斑点。应用一定的技术假设，等离子体动力学的Braginskii方程被用来开发一个2维模型的刮削层，Wilczynski进行广泛的稳定性和分叉分析。由此产生的模型与经典流体动力学中的瑞利-贝纳德对流相似，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.