Periodic Orbits as a Basis for Fluid Turbulence

湍流仍然是经典物理学的突出问题之一。在我们的日常生活中无处不在，从天气模式到滔滔不绝的水龙头，理解和控制湍流仍然是科学界面临的一项重大国际挑战。尽管经过了几十年的研究，我们仍然没有一个被证明具有预测能力的基本理论。除了忠实地模拟昂贵的控制方程之外，标准的方法是参数化流场中的时间变化或小尺度，这两者都不符合表征湍流的时间和空间尺度的复杂相互作用。本提案旨在研究一种新的动力系统方法，允许湍流以嵌入湍流中的不稳定周期轨道（UPOs）的形式选择自己的“基函数”。每个UPO都代表了支撑湍流状态的某些动态过程的“完美”平衡。由于当今的计算机能力，围绕着湍流的UPO扩展的构建存在许多问题。本提案旨在回答这些问题，最终目的是评估UPO扩展对湍流的实用性和预测能力。

Turbulence remains one of the outstanding problems of classical physics. Ubiquitous in our every day lives, from weather patterns to agushing water tap, understanding and controlling turbulence continues to be a major international challenge for the scientific community. Despite decades of research we are still without a basic theory which has any proven predictive power. Beyond faithfully simulating thegoverning equations which is prohibitively expensive, the standard approach is to parametrize away temporal variations or small scalesin the flow field neither of which are consistent with the complicated interplay of temporal and spatial scales characterizing turbulence. This proposal seeks to investigate a new dynamical systems approach in which turbulent flows are allowed to select their own `basis functions' in the form of unstable periodic orbits (UPOs) embedded in the turbulence. Each UPO represents the `perfect' balance of somedynamical process which underpins the turbulent state. There are many issues surrounding the construction of a UPO expansion of turbulence which are now eminently addressable due to present-day computer power. This proposal is directed at answering these with the ultimate aim of assessing the practicality and predictive power of a UPO expansion for turbulence.