Transport of Particles in Heterogeneous Media

本研究的目的是从数学的角度研究非均质介质的物理性质。异质媒体无处不在。一些例子包括海洋流动、大气湍流、含油砂和生物组织。对这种介质性质的了解几乎涉及到科学和工程的每一个分支(例如材料科学、化学工程、地球物理、医学成像、流体动力学)。该项目的一个组成部分涉及自适应光学。自适应光学是一种提高天文望远镜、激光通信系统和其他光学设备精度的方法。这些设备中失真的主要来源是大气湍流。该项目这一部分的目的是了解这种扭曲的机制。这项研究的另一个组成部分涉及海洋表面盐度或氯氟烃的变化。海洋漩涡可能会极大地改变各种化合物的混合速度。这个项目的目的是在更简单的数学模型中估计这些速率，以阐明整个问题中存在的机制。波在非均匀介质中传播的数值模拟仍然超出了现代计算机的能力范围。在实践中，它导致了各种简化近似的使用。这些近似值的推导和证明是微妙的。在非均匀介质中粒子传播的更简单的设置中，已经发现了这种推导的许多困难。这项研究的目标之一是开发工具，更好地理解粒子在非均匀介质中传播的近似模型。这个项目主要研究粒子在慢速去相关介质中的传输，因为最近观察到在慢速去相关介质中的传播具有在快速去相关介质中所没有的类似的性质。该项目的成果之一将是对随机旅行时间近似的严格证明--这是自适应光学中使用的一种方法。第二个项目集中于示踪剂粒子在强大的相反旋涡阵列中的异常扩散。在20世纪80年代，海洋学家观测到了存在强涡时颗粒的异常快速输运。这个项目的一个目标是从数学上证明这一预测现象的合理性。

The objective of this research is to study, from a mathematical perspective, physical properties of heterogeneous media. Heterogeneous media are ubiquitous. Some examples are ocean flows, atmospheric turbulence, oil-bearing sands, and biological tissues. Understanding of properties of such media is relevant to aspects of virtually every branch of science and engineering (e.g., materials science, chemical engineering, geophysics, medical imaging, fluid dynamics). One of the components of this project concerns adaptive optics. Adaptive optics is a method to improve accuracy of astronomical telescopes, laser communication systems, and other optical devices. The main source of distortion in these devices comes from atmospheric turbulence. The aim of this part of the project is to understand the mechanism of this distortion. Another component of this research concerns changes in salinity or chlorofluorocarbon on the surface of the ocean. Oceanic vortices may dramatically change mixing rates of various chemical compounds. The aim of this element in project is to estimate these rates in simpler mathematical models to illuminate the mechanisms present in the full problem.Numerical simulation of wave propagation in heterogeneous media is still beyond reach of modern computers. In practice it leads to use of various simplified approximations. The derivation and justification of these approximations is delicate. Many difficulties of such derivation are already found in a simpler setup of particle propagation in heterogeneous media. One of the goals of this research is to develop tools and a better understanding of approximate models for particle propagation in heterogeneous media. This project concentrates on transport of particles in slowly decorrelating media, because it was observed recently that propagation in slowly decorrelating media has properties with no analogs in rapidly decorrelating media. One of the outcomes of this project will be a rigorous justification of the Random Travel Time Approximation - a method used in adaptive optics. The second project concentrates on anomalous diffusion of tracer particles in a strong array of opposing vortices. Anomalously fast transport of particles in presence of strong vortices was observed by oceanographers in 1980s. A goal of this project is to justify mathematically this predicted phenomenon.