Hitching the subcritical branch of convection

流体中的温差引起的热对流驱动了从行星和恒星内部到金属铸造和热提取的多种过程。在最简单的形式中，当浮力与粘性力的比率瑞利数Ra超过临界值Rac时，对流就开始了。在临界值Rac处，流动动力学放大了对非对流平衡的微小扰动。在更复杂的例子中，可能存在亚临界对流，即RAc以下的Ra。这不仅发生在热交换器的管道中，而且最近在合金的连铸过程中也被发现，它可能导致凝固合金中不希望看到的偏析缺陷。最近，亚临界对流也出现在行星内部的数值模拟中，在行星内部的数值模拟中，热的固体核被较冷的液态金属包围，通过浮力、行星自转诱导的科里奥利力和由于磁场而产生的洛伦兹力之间的相互作用，通过复杂的对流过程从固体核吸收热量。在这三种情况下，亚临界对流的性质都是至关重要的：在RAC以下点燃对流显著增强传热：这是冷却应用的圣杯，也是核聚变反应堆设计中的技术僵局之一。挑战是将非对流流动扰动到亚临界对流状态。相反，在连铸中必须避免亚临界对流引起的缺陷。对流也是行星跳动的心脏，在其他过程中，驱动着维持其磁场的发电机行动。偏离一个潜在的亚临界对流状态可能会关闭行星核心中的对流，这是行星可能“死亡”的方式之一。在这些问题中，核心问题是“对流能在多大程度上低于临界？”以及“什么扰动会点燃或熄灭亚临界对流？”此外，在行星磁场存在的情况下，亚临界对流是否存在甚至是未知的。控制方程的直接模拟不能回答这些问题，因为它们不能可靠地判断对流是否稳定。延拓方法可以捕获对流状态而不考虑其稳定性，但不能直接应用，因为到达或离开对流状态需要不连续的“跳跃”，正如这里所寻求的。本项目将利用稳定性理论的最新发展，在所有三个例子中回答这些数学问题。对于第一个问题，通过采用目前用于研究剪切流中向湍流转变的钩子步长和时间延迟控制方法，可以从模拟或远距离状态获取不连通分支上的精确解。然后，可以使用延拓方法将这些状态追溯到次临界分支的起源。对于第二个问题，我们将使用具有最佳瞬时能量增长的微扰来破坏非对流平衡进入亚临界分支(或相反的分支)，并找到通向对流熄灭或引燃的途径。虽然亚临界对流在地球物理和铸造问题中的重要性直到最近才被曝光，但阐明其真正作用的技术也是如此。随着冶金学家越来越多地转向严格的数学来控制他们的过程，在工业上开发它们的机会也是如此。与冶金专家的持续合作以及这项工作与核聚变反应堆的相关性为这些新方法提供了直接的机会，开始在这些行业和其他行业用量身定制的优化方法取代目前的反复试验的设计实践。为此，我们将把这些方法应用到一个开源的数值程序包中，该程序包能够在尽可能广泛的问题范围内发现或点燃全范围的次临界对流。

Thermal convection due to temperature differences in fluids drives multiple processes from planetary and stellar interiors, to the casting of metals and heat extraction. In its simplest form, convection sets in when the ratio of buoyancy to viscous forces, the Rayleigh number Ra, exceeds the critical value Rac at which an infinitesimal perturbation to the non-convective equilibrium becomes amplified by the flow dynamics. In more complex examples, subcritical convection may exist, i.e. for Ra below Rac. This not only happens in the ducts of heat exchangers, but also was recently discovered during the continuous casting of alloys, where it could lead to unwanted segregation defects in solidified alloys. Recently, subcritical convection also appeared in numerical simulations of planetary interiors, where a hot solid core is surrounded by a colder liquid metal extracting heat from it through a complex convective process controlled by the interplay between buoyancy, the Coriolis force induced by the planet's rotation and the Lorentz force due to its magnetic field.In all three cases, the subcritical nature of convection is crucial: igniting convection below Rac significantly enhances heat transfer: this is the holy grail of cooling applications and one of the technological deadlocks in the design of nuclear fusion reactors. The challenge is to perturb a non-convective flow into a subcritical convective state. Conversely, defects incurred by subcritical convection must be avoided in continuous casting. Convection is also the beating heart of planets, driving amongst other processes, the dynamo action that sustains their magnetic field. An excursion away from a potentially subcritical convective state could shut down convection in planetary cores, one of the ways planets may "die".In these problems the central questions are "how far below criticality can convection exist ?" and "what perturbation either ignites or extinguishes subcritical convection ?". Furthermore, whether subcritical convection even subsists in the presence of planetary magnetic fields is not even known. Straight simulations of the governing equations cannot answer these questions because they cannot reliably tell if convection is stable. Continuation methods can capture convective states regardless of their stability, but do not directly apply as reaching or leaving the convective state requires a discontinuous 'jump', as sought here.This project will answer these mathematical questions in all three examples, by taking advantage of recent developments in stability theory. For the first question, exact solutions on disconnected branches will be captured from either simulations or distant states by adapting the hook step and Time Delay Control methods currently used to study the transition to turbulence in shear flows. These states can then be traced back to the origin of the subcritical branch using continuation methods. For the second, we will use perturbations with optimal transient energy growth to destabilise the non-convective equilibrium into the subcritical branch (or the reverse) and find paths to the extinction or the ignition of convection.While the importance of subcritical convection in geophysical and casting problems only came to light very recently, so did the techniques to elucidate its true role. So too did the opportunity to exploit them in industry, as metallurgists increasingly turn to rigorous mathematics to control their processes. Ongoing collaboration with metallurgists and this work's relevance to nuclear fusion reactors offer direct opportunities for these new methods to start replacing current trial-and-error practice in design by tailored optimisation methods in these industries and potentially others. To this end, we will implement these methods into an open-source numerical package capable of finding or igniting the full range of subcritical convective flows in the widest possible range of problems.