Reduced Models for Transport of Particles in Heterogeneous Media

该项目的总体目标是研究异类环境中的物理系统。异类环境无处不在。一些例子包括海洋流动、大气湍流、含油砂和生物组织。了解这种介质的整体性质几乎涉及到科学和工程的每一个分支；特别是材料科学、化学工程、地球物理、医学成像和流体动力学。这项研究的项目之一是关于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比率，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。这些设备中失真的主要来源是大气湍流。这个项目的目的是为了了解这种扭曲的机制。这项工作开展了粒子在杂波环境中的传播和流体中对流-扩散过程的数学研究。第一个项目集中在长程关联的影响上。这是随机媒质中传播理论的一个新方向，目前还没有发展出慢去相关媒质中传播的近似模型。其目的是解释长程关联如何导致各种现象的时间尺度分离。第二个项目集中于标准扩散和分数扩散与不可压缩流动的相互作用。这项拟议工作的智力挑战是理解这种大振幅不可压缩流动的相互作用。这与理解湍流中的扩散特别相关。这一奖项反映了NSF的法定使命，并通过使用基金会的智力优势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.