Structure, Transport, and Chaos in Volume-Preserving Dynamics

用KAM理论解释了准周期运动在近可积保体映射的余维1环面上的持续性。然而，这些环面的鲁棒性和破坏后残余物的存在仅在二维中被理解。该方法提出研究三维映射的环面，推广了Greene发现的残差判据和Aubry发现的反积极限。研究将包括对称减少，不变性，和可积性的损失一般，结构保持映射和流。一个重要的应用是用于物料连续混合的明渠流的混合优化。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通过破坏结构的通量来量化，使用基于拉格朗日形式的广义作用来计算，从而获得准确且计算效率高的体积通量。PI和学生将使用过渡动力学的概念来量化和优化开放流中的运输。扩展到偶然性和更一般的时间依赖性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而获得的图案的复杂性可以通过向热咖啡中倒入奶油来观察。这一过程尚未被完全理解，这一点或许不那么明显。如果流动足够湍流，那么混合速度很快，均匀性不难实现。然而，如果流动缓慢、规模小或粘稠，那么混合就困难得多。然而，这些过程对许多应用都很重要，包括微米级生物反应器的开发和聚合物和颗粒材料的有效混合。层流混合的预测理论也将有助于了解气候模型和大气中的污染扩散，以及海洋生物的营养物质扩散和产卵效率。层流中的混合是通过混沌运动的拉伸和折叠进行的，混沌运动产生了精细尺度的结构，扩散是有效的。任何混合测量都需要对混沌及其伴随的输运进行量化。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。后来的模型被用来预测加速器中粒子的寿命，获得简单化学反应的速率，计算等离子体聚变装置的约束时间，理解高激发原子系统的光谱，以及设计有效的航天器轨迹。对于混沌动力学，具体轨迹的预测是困难的；然而，混沌可以被有益地利用，例如，通过明智地应用小的航向修正来提高航天器轨迹的效率，或提高约束装置中粒子的寿命和化学反应的速率。在本研究中，混沌将用于优化混合，目标是获得实用的开放式、三维混合装置设计。

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.