Structure, Transport, and Chaos in Volume-Preserving Dynamics

用KAM理论解释了近可积保体积映射中余维一环上拟周期运动的持久性。然而，这些圆环的健壮性和破坏后残余物的存在只能在二维中被理解。PI建议研究三维映射的环面，推广Greene发现的剩余准则和Aubry发现的反可积极限。研究内容包括一般保持结构的映射和流的对称性约化、不变性和可积性损失。一个重要的应用是用于连续混合材料的明管流动中的混合优化。我们目前对混合过程的理解，在很大程度上局限于本质上是二维的流动，要么是封闭的，要么是循环的。三维系统中的输运可以通过破坏结构的通量来量化，并使用基于拉格朗日形式的广义作用量来计算，从而获得准确和计算高效的体积通量。PI和学生将使用瞬变动力学的概念来量化和优化开放流动中的运输。扩展到情节和更一般的时间依赖关系将澄清非周期动力学中拉格朗日相干结构的定义。通过在流体中混合被动标量获得的模式的复杂性，任何人在热咖啡中倒入奶油都可以观察到。这一过程没有被完全理解，或许就不那么明显了。如果气流足够湍急，那么混合就很快，而且均匀也不难实现。然而，如果流动很慢，规模很小，或者是粘性的，那么混合就困难得多。然而，这样的过程对许多应用都很重要，包括微米级生物反应器的开发以及聚合物和颗粒材料的有效混合。关于层流混合的预测理论也将有助于理解气候模拟和大气中的污染扩散以及海洋生物的营养扩散和产卵效率。层流中的混合是通过伸展和折叠进行的，这是由于混沌运动引起的，在扩散有效的地方形成了细小的结构。任何衡量混合的方法都需要对混沌及其伴随的传输进行量化。不可压缩流体中的混沌运动与保守动力学中的混沌运动有一些相似之处。后一种模型用于预测加速器中粒子的寿命，获得简单化学反应的速率，计算等离子体聚变设备中的限制时间，了解高激发原子系统的光谱，并设计高效的航天器轨迹。对于混沌动力学来说，很难预测具体的轨迹；然而，混沌可以被有益地利用，例如，通过明智地应用小航向校正来提高航天器轨迹的效率，或者提高限制装置中粒子的寿命和化学反应的速度。在这项研究中，将利用混沌来优化混合，目标是获得开放的、三维的混合设备的实用设计。

The persistence of quasiperiodic motion on codimension-one tori in nearly-integrable volume-preserving maps is explained by KAM theory. However, the robustness of these tori and the existence of remnants upon destruction are understood only in two-dimensions. The PI proposes to study tori of three-dimensional maps, and to generalize the residue criterion discovered by Greene and the anti-integrable limit discovered by Aubry. Studies will include symmetry reduction, invariance, and the loss of integrability for general, structure-preserving maps and flows. An important application is the optimization of mixing in open duct flows used in the continuous blending of materials. Our current understanding of the mixing process is, for the most part, limited to flows that are in essence two-dimensional and either closed or recycling. Transport in three-dimensional systems can be quantified by the flux through the destroyed structures, computed using a generalized action based on Lagrangian forms, thereby obtaining accurate and computationally efficient volume fluxes. The PI and students will use the concept of transitory dynamics to quantify and optimize transport in open flows. The extension to episodic and more general time-dependence will clarify the definition of Lagrangian coherent structures in aperiodic dynamics.The complexity of patterns obtained by mixing a passive scalar in a fluid can be observed by anyone pouring cream into hot coffee. That this process is not fully understood is perhaps less obvious. If the flow is sufficiently turbulent then mixing is rapid and uniformity is not hard to achieve. If, however, the flow is slow, on a small scale, or viscous, then mixing is much more difficult. Yet, such processes are important to many applications including the development of micrometer scale bioreactors and effective mixing of polymer and granular materials. A predictive theory for laminar mixing would also contribute to the understanding of climate modeling and pollution dispersal in the atmosphere as well as nutrient dispersal and spawning efficiencies for sea life. Mixing in laminar flows proceeds by stretching and folding due to chaotic motion that gives rise to fine-scale structure where diffusion is effective. Any measure of mixing requires quantification of chaos and its concomitant transport. Chaotic motion in incompressible fluids has some similarities to that in conservative dynamics. The later models are used to predict the lifetime of particles in accelerators, obtain rates for simple chemical reactions, calculate confinement times in plasma fusion devices, understand the spectra of highly excited atomic systems, and design efficient spacecraft trajectories. For chaotic dynamics, prediction of specific trajectories is difficult; nevertheless, chaos can be profitably utilized, for example, to improve efficiency of spacecraft trajectories, by judiciously applying small course corrections, or to enhance the lifetimes of particles in confinement devices and the rates of chemical reactions. In this study, chaos will be used to optimize mixing with the goal of obtaining practical designs for open, three-dimensional, mixing devices.