Periodic Orbits as a Basis for Fluid Turbulence

湍流是经典物理学中的一个突出问题。湍流在我们的日常生活中无处不在，从天气模式到喷涌的水龙头，理解和控制湍流仍然是科学界面临的一个重大国际挑战。尽管经过几十年的研究，我们仍然没有一个基本的理论，具有任何证明的预测能力。除了忠实地模拟控制方程，这是昂贵的，标准的方法是参数化的时间变化或小规模的流场，这两者都不符合复杂的相互作用的时间和空间尺度的特点湍流。该建议旨在研究一种新的动力系统方法，其中湍流被允许以嵌入湍流中的不稳定周期轨道（UPO）的形式选择自己的“基函数”。每一个UPO都代表着某种物理过程的“完美”平衡，而这种过程又是动荡状态的基础。围绕着湍流的UPO扩展的构建有许多问题，由于当今的计算机能力，这些问题现在非常容易解决。本建议旨在回答这些问题，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.