Metastability of Geophysical Turbulent Flows

地球大气和海洋等地球物理湍流的动态在决定我们这个星球的长期气候方面发挥着关键作用。因此，对它们的长期平均动态的理解和建模对于研究气候变化的影响变得越来越重要。然而，这些系统的运动方程非常复杂，需要经过长时间计算的高分辨率数值模拟才能获得长期的流动统计数据。我提出了一个博士项目，它建立在我的硕士导师Jason Laurie博士开发的关于预测这些地球物理流动中的平均流动轮廓的理论[1，2]的基础上。该项目的计划将是开发一种动力学理论方法[3]，以确定平均流动动力学的随机雷诺方程，该方程取决于基本湍流涨落的雷诺应力。我们将超越典型的准线性近似，包括与底层湍流相关的随机反馈。这将产生一组高精度、计算效率高的方程，适用于使用大偏差和瞬子方法研究亚稳性[4]。对于气候科学家来说，这种方法的好处可能是巨大的。事实上，建模这些现象的问题通常需要使用并行计算，因此我们计划使用Jason Laurie博士的计算工具开发新的支持CUDA的高性能并行代码。该项目将进一步使我能够在我在阿斯顿大学攻读硕士期间与劳里博士合作的点涡理论的基础上继续发展。我们的点涡计算可以用来研究二维经典和量子湍流的理想化设置的动力学，并可以用来进一步深入了解关于涡旋散射和声发射的数学。在阿斯顿大学攻读硕士学位期间，我曾与劳里博士一起研究点涡旋动力学。这个项目让我对流体力学和理论物理作了广泛的介绍；包括哈密顿力学、相平面分析、椭圆积分的解、湍流、涡旋动力学等主题。在这个项目中，我在数值方法和C++编程方面积累了丰富的经验，这些经验在这个博士职位上将被证明是无价的。博士项目将使我对流体力学和地球物理湍流的研究兴趣自然发展，并进一步发展我的科学计算技能。在阿斯顿大学取得数学学士学位后，我在阿斯顿大学攻读了研究型硕士学位，并以优异的成绩获得了一流的高分成绩。正是在这段时间里，我对高等数学的各个不同方面有了更深的理解，例如，混沌和动力系统、概率建模、数值方法等等。

The dynamics of geophysical turbulent flows, such as the Earth's atmosphere and oceans play a critical role in determining the long-term climate of our planet. Therefore, the understanding and modelling of their long-term mean dynamics becomes increasingly important for studying the effect of climate change. However, the equations of motions of these systems are incredibly complex and require high resolution numerical simulations computed over long computation times in order to acquire long-term flow statistics. I propose a PhD project that builds upon theory developed by my MSc supervisor Dr Jason Laurie concerning the prediction of mean flow profiles in these geophysical flows [1,2]. The plan of the project will be to develop a kinetic theory approach [3] to determine a stochastic Reynolds equation for the mean flow dynamics that depends on the Reynolds stresses of the underlying turbulent fluctuations. We will go beyond the typical quasi-linear approximation to include stochastic feedback correlated to the underlying turbulence. This will yield a set of equations that will be highly accurate, and computationally efficient and applicable to studying metastability using large deviation and instanton methods [4]. The benefits of this approach could be substantial for climate scientists. Indeed, the problem of modelling these phenomena typically necessitates the use of parallel computing, and as such we plan to develop new CUDA-enabled high-performance parallel codes using the computational facilities of Dr Jason Laurie. The project will further allow for me to build upon the point-vortex theory I have been working on with Dr Laurie during my MSc by Research at Aston University. Our point-vortex calculations can be used to study the dynamics of idealised setups of 2D classical and quantum turbulence and can be used to gain further insight into the mathematics regarding vortex scattering and sound emission. I have worked with Dr Laurie during my MSc by Research at Aston University on point-vortex dynamics. This project has given me an extensive introduction to fluid dynamics and theoretical physics as a whole; including topics such as Hamiltonian mechanics, phase-plane analysis, solutions of elliptic integrals, turbulence, vortex dynamics, etc. For this project, I have developed substantial experience in numerical methods and programming in C++ that will prove invaluable in this PhD position. The PhD project will provide a natural progression of my research interests in fluid mechanics and geophysical turbulence and further develop my scientific computing skills. I undertook the MSc by Research at Aston University following my undergraduate BSc in Mathematics, also at Aston University, in which I achieved the result of a high first class with honours. It was during this time that I developed my understanding of various different aspects of advanced mathematics, for example; chaos and dynamical systems, probabilistic modelling, numerical methods, and more.