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。异构媒体无处不在。例如洋流、大气湍流、含油砂和生物组织。对这类介质性质的理解几乎与科学和工程的每个分支（例如，材料科学、化学工程、地球物理、医学成像、流体动力学）有关。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。这些装置的主要畸变源来自大气湍流。这部分项目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化碳的变化。海洋涡旋可以极大地改变各种化合物的混合速率。项目中这个元素的目的是用更简单的数学模型来估计这些速率，以阐明整个问题中存在的机制。波在非均质介质中传播的数值模拟仍然是现代计算机无法实现的。在实践中，它导致使用各种简化的近似。这些近似的推导和证明是很微妙的。在非均质介质中粒子传播的一个更简单的设置中，已经发现了这种推导的许多困难。本研究的目标之一是开发工具和更好地理解颗粒在异质介质中传播的近似模型。本项目主要研究粒子在缓慢脱相关介质中的输运，因为最近观察到，在缓慢脱相关介质中的传播具有在快速脱相关介质中没有类似物的特性。这个项目的成果之一将是对随机旅行时间近似的严格证明——一种用于自适应光学的方法。第二个项目集中于示踪粒子在强相反涡旋阵列中的异常扩散。20世纪80年代，海洋学家观察到强涡旋存在时粒子的异常快速输运。这个项目的目标是用数学方法证明这种预测的现象。

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.