The objective of this award centers on the development of a geometric theory of reaction front propagation in advection-reaction-diffusion dynamics. In particular, work will focus on extending the well-known constructs of invariant manifolds and Lagrangian coherent structures - valuable for passive advection - to the case of front propagation in advecting media. The viability of such an approach has recently been experimentally and theoretically demonstrated by the existence of burning invariant manifolds (BIMs). BIMs generalize the concept of invariant manifolds in passive advection to the case of front propagation. (Here, burning refers generally to any kind of front propagation.) BIMs create local barriers to front evolution, analogous to their advective counterparts. A central objective of this award is thus to develop a comprehensive framework for front propagation based on the BIMs. Specific consideration will be given to the technique of lobe dynamics, to mode-locking of reaction fronts, to topological dynamics, to three-dimensional fluid flows, and to time-aperiodic fluid flows (utilizing finite-time Lyapunov exponents and Lagrangian coherent structures). Furthermore, BIMs will be applied to the control of reaction fronts in chaotic flows. The theoretical work under this award will benefit from a close collaboration with the experimental group of Prof. Tom Solomon at Bucknell University.
Many physical processes combine fluid motion with the propagation, or growth, of a front separating two different media. Important examples include flame fronts in an airflow, chemical reactions in microfluidic devices, plankton blooms in ocean currents, or even epidemics in moving biological populations. The results of this research will improve the geometric understanding of a broad class of such processes. A potential application would be to study the growth of invasive plankton blooms in coastal waters. Furthermore, by better understanding geometric constraints on front propagation, one can engineer better control mechanisms. For example, mechanisms derived from this work could be applied to control chemical reactions in lab-on-a-chip applications. Through graduate, undergraduate, and postdoctoral research experiences, this work will also help train the next generation of scientists and help to nurture the newly established campus of the University of California at Merced (UC Merced). UC Merced was opened in 2005 and has already made a significant impact on the economy and education in California's Great Central Valley, the agricultural heart of the state.
