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.
|Effective start/end date||6/15/15 → 10/31/20|
- National Science Foundation: $388,107.00