Experimental and Theoretical Developments in the Application of Lagrangian Coherent Structures to Geophysical Transport
The transport of material in geophysical fluid flows is a problem with important implications for fields as diverse as: agriculture, aviation, human health, disaster response, and weather forecasting. Due to the unsteady nature of geophysical flows, predicting how material will be transported in these systems can often be challenging. Tools from dynamical systems theory can help to improve the prediction of material transport by revealing important transport structures. These transport structures reveal areas of the flow where fluid parcels, and thus material transported by those parcels, are likely to converge or diverge. Typically, these transport structures have been uncovered by the use of Lagrangian diagnostics. Unfortunately, calculating Lagrangian diagnostics can often be time consuming and computationally expensive.
Recently new Eulerian diagnostics have been developed. These diagnostics are faster and less expensive to compute, while still revealing important transport structures in fluid flows. Because Eulerian diagnostics are so new, there is still much about them and their connection to Lagrangian diagnostics that is unknown. This dissertation will fill in some of this gap and provide a mathematical bridge between Lagrangian and Eulerian diagnostics.
This dissertation is composed of three projects. These projects represent theoretical, numerical, and experimental advances in the understanding of Eulerian diagnostics and their relationship to Lagrangian diagnostics. The first project rigorously explores the deep mathematical relationship that exists between Eulerian and Lagrangian diagnostics. It proves that some of the new Eulerian diagnostics are the limit of Lagrangian diagnostics as integration time of the velocity field goes to zero. Using this discovery, a new Eulerian diagnostic, infinitesimal-time Lagrangian coherent structures is developed. The second project develops a methodology for estimating local Eulerian diagnostics from wind velocity data measured by a fixed-wing unmanned aircraft system (UAS) flying in circular arcs. Using a simulation environment, it is shown that the Eulerian diagnostic estimates from UAS measurements approximate the true local Eulerian diagnostics and can predict the passage of Lagrangian diagnostics. The third project applies Eulerian diagnostics to experimental data of atmospheric wind measurements. These are then compared to Eulerian diagnostics as calculated from a numerical weather simulation to look for indications of Lagrangian diagnostics.