Burning Invariant Manifolds: The Geometry of Front Propagation in Advection-Reaction-Diffusion Dynamics

该奖项的目标是发展平流-反应-扩散动力学中反应前沿传播的几何理论。特别是，工作将集中于将众所周知的不变流形和拉格朗日相干结构的结构--对被动平流有价值--推广到平流介质中的锋面传播的情况。最近，燃烧不变流形(BIM)的存在从实验和理论上证明了这种方法的可行性。BIM将被动平流中不变流形的概念推广到锋面传播的情形。(这里，燃烧泛指任何类型的正面传播。)BIM为锋面演变创造了局部障碍，类似于它们的对流同行。因此，该奖项的一个中心目标是在BIM的基础上制定一个全面的正面传播框架。将具体考虑波瓣动力学技术、反应前沿的锁模、拓扑动力学、三维流体流动和时间非周期流体流动(利用有限时间Lyapunov指数和拉格朗日相干结构)。此外，BIMS还将应用于混沌流动中反应前沿的控制。该奖项下的理论工作将受益于与巴克内尔大学汤姆·所罗门教授的实验小组的密切合作。许多物理过程将流体运动与分离两种不同介质的锋面的传播或生长结合在一起。重要的例子包括气流中的火焰锋面、微流控装置中的化学反应、洋流中的浮游生物水华，甚至是生物种群流动中的流行病。这项研究的结果将提高对这类过程的几何理解。一个潜在的应用将是研究沿海水域入侵浮游生物水华的生长。此外，通过更好地了解前沿传播的几何约束，人们可以设计出更好的控制机制。例如，从这项工作中得出的机制可以用于控制芯片实验室应用中的化学反应。通过研究生、本科生和博士后的研究经验，这项工作还将有助于培养下一代科学家，并有助于培育加州大学默塞德分校(UC Merced)新成立的校区。加州大学默塞德分校于2005年开业，已经对加州大中央山谷的经济和教育产生了重大影响，大中央山谷是该州的农业中心。

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.