Reduced Models for Transport of Particles in Heterogeneous Media

这个项目的总体目标是研究异构环境中的物理系统。异构环境无处不在。一些例子是海洋流动，大气湍流，含油砂和生物组织。了解这种介质的整体性质与科学和工程的几乎每个分支的方面有关;特别是材料科学、化学工程、物理学、医学成像和流体动力学。这项研究的一个项目涉及海洋表面盐度或氯氟烃的变化。海洋涡旋可以显著改变各种化合物的混合速率。该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。 这项研究的另一个项目涉及光在大气中的传播。这些装置中的主要失真源来自大气湍流。本项目的目的是了解这种扭曲的机制。这项工作进行了混乱环境中的粒子传播和流体中的对流扩散过程的数学研究。第一个项目集中在长期相关性的影响。这是随机介质中传播理论的一个新方向，并且还没有开发出在缓慢去相关介质中传播的近似模型。目的是解释长程相关性如何导致各种现象的时间尺度分离。第二个项目集中在不可压缩流的标准和分数扩散的相互作用。所提出的工作的智力挑战是理解这种相互作用的不可压缩流与大振幅。该奖项反映了NSF的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

The overall objective of this project is to study physical systems in heterogeneous environments. Heterogeneous environments are ubiquitous. Some examples are ocean flows, atmospheric turbulence, oil-bearing sands, and biological tissues. Understanding overall properties of such media is relevant to aspects of virtually every branch of science and engineering; especially materials science, chemical engineering, geophysics, medical imaging and fluid dynamics. One of the projects 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 project is to estimate these rates in simpler mathematical models to illuminate the mechanisms present in the full problem. Another project of this research concerns light propagation through the atmosphere. The main source of distortion in these devices comes from atmospheric turbulence. The aim of this project is to understand the mechanism of this distortion.This work carries out mathematical studies of particle propagation in cluttered environments and of convection-diffusion processes in fluids. The first project concentrates on the effect of long-range correlations. This is a novel direction in the theory of propagation in random media, and approximate models of propagation in slowly decorrelating media are not yet developed. The goal is to explain how long-range correlations lead to time-scale separation of various phenomena. The second project concentrates on the interaction of the standard and the fractional diffusion with incompressible flows. The intellectual challenge of the proposed work is to understand this interaction for incompressible flows with large amplitude. This is particularly relevant to understanding diffusion in turbulent flows.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.