The investigators model and quantify scalar diffusion in the limit where homogenization theory is invalid. This regime is typical when the scalar in the flow has sources or sinks due to physical or biological forces, such as release or mitigation of contaminants, birth and death processes, predator-prey interactions in biological species, or many others. As a result, the classical assumptions of turbulent isotropy and homogeneity do not apply, and the mathematical challenge is to properly represent enhanced diffusion (from molecular motion) by the associated fluid motion at scales smaller or comparable to the characteristic forcing scale of the flow. The complementary expertise of the investigators enables the development of mathematically based models that are verified with numerical and physical experiments. The investigators study scalar mixing from a variety of angles. Topological measures on the background stirring, based on dynamical systems methods, are used and tested for modeling turbulent diffusion with coherent structures. Additionally, mixing measures associated with sources and sinks are studied with an upper bound approach. Overall, the investigators study how mathematical theory can be used and adapted to explain inhomogeneous, anisotropic diffusion processes observable from physical experiments.
Transport and mixing processes associated with pollutants, such as the Gulf oil spill and radioactive contamination in Japan, and with biological tracers, such as plankton blooms, are greatly affected by stirring patterns in the background flow. The novel theoretical and experimental studies proposed here lay the foundation for improved mathematical models and simulations of scalar mixing for many environmental and geophysical processes; in particular, they can be used to address realistic environmental problems. Other broad impacts include training of a graduate student and a postdoctoral researcher. Moreover, the experiments and visualizations from simulations, along with some analytical and computational tools used in this research, are blended into the investigators' outreach efforts to engage high school students' interest in mathematics and physics. The project is supported by the Division of Chemical, Bioengineering, Environmental, and Transport Systems and by the Division of Mathematical Sciences.
