Numerical and Theoretical Developments for Coherent Structures

Abstract

The field of nonautonomous dynamical systems has undergone an exceptional amount of growth over the last few decades, with the geometric viewpoint leading to the development of Lagrangian and objective Eulerian coherent structures. This theory extends notions developed for autonomous systems to those that may have arbitrary dependence on time and whose flow data may not be known or available for all time. The structures obtained from this theory are material surfaces that have an exceptional effect on the contents of a flow and are key for understanding transport. These structures organize the contents of a flow and lead to a qualitative description of the fate of material under the influence of a flow. While these methods have been adopted in geophysical and engineering applications, the theory is still relatively new. This dissertation aims to contribute to the maturation of this theory and its numerical implementations by providing an efficient and user-friendly software package for these tools, demonstrating the usefulness of these methods on a large-scale transport application, further extending the theory in a specific context, and presenting a new numerical framework for computing LCS from their variational theory.