Mixing by Resonances in Multi-Scale Systems

本课题的目标是发展多尺度系统中存在有限惯性、随机波动和分子扩散等现象的共振混合理论。所构建的理论将提供一种有效而准确的方法来计算给定系统中混合域的大小，估计混合的速率、彻底性和均匀性，并描述混沌海洋中相干结构的外观。共振表面上的过程理论将与这些表面之间输运的绝热模型相结合，以发展具有共振系统中输运的统计长期描述。数值模拟和研究来自微流体实验室和空间等离子体实验的实验数据，以确定引起粒子输运的共振机制，并描述附加随机过程对输运稳定性的影响。如果成功，这项工作将导致将经常脱节的方法和技术整合到广泛的流、离散图和具有共振的哈密顿系统的一般输运理论中。其中一个主要组成部分将是一种新的技术，用于量化混合的不同方面，以及在各种多尺度系统中存在不确定性和波动时定量特征的持久性。所获得的结果将直接应用于科学和工程中的广泛问题，例如微流体或其他低雷诺数（微尺度）流，彗星和小行星在太阳系中的传输，凝聚态物质中激发模式之间的能量交换，以及带电粒子在地球磁层或磁约束聚变装置的各种电磁场中的运动。

The goal of this project is to develop the theory of mixing by resonances in multi-scale systems in the presence of such phenomena as finite inertia, random fluctuations, and molecular diffusion. The constructed theory will provide an effective and accurate method to compute the size of the mixing domain in a given system, to estimate the rate, thoroughness, and uniformity of mixing, and to describe the appearance of coherent structures in a chaotic sea. The theory of processes on resonance surfaces will be combined with the adiabatic model of transport between those surfaces to develop a statistical long-term description of transport in systems with resonances. Numerical simulations and study of experimental data from microfluidics laboratory and space plasma experiments will be performed to identify the resonance mechanisms causing the transport of particles and describe the impact of additional stochastic processes on the stability of transport.If successful, this work will result in the integration of often disconnected methods and techniques into a general transport theory for a wide class of flows, discrete maps, and Hamiltonian systems with resonances. One of the main components will be a novel technique to quantify different aspects of mixing and the persistence of quantitative characteristics in the presence of uncertainties and fluctuations in various multi-scale systems. The obtained results will have direct applications to a wide range of problems in science and engineering, such as microfluidics or other low Reynolds-number (microscale) flows, the transport of comets and asteroids through the solar system, energy exchange between excitation modes in condensed matter, and motion of charged particles in various electromagnetic fields of the Earth's magnetosphere or magnetic confinement fusion devices.