Efficient ridge tracking algorithms for computing Lagrangian coherent structures in fluid
Lagrangian coherent structures (LCS) are recently defined structures used to analyze transport in dynamical systems with general time dependence. LCS techniques have seen increasing use over the past decade, but several factors have limited their application to highly complex and three-dimensional flows. In this dissertation, I study the computation of LCS in the context of fluid dynamics applications. The primary examples used here are axisymmetric simulations of swimming jellyfish, a three-dimensional ocean current simulation, a three-dimensional hurricane simulation, and various test cases and analytically defined flows. All these flows involve complicated dynamics and fluid transport that can be analyzed using LCS to reveal the flow structures and underlying transport behavior.
The main contribution of this dissertation is the development and application of a class of efficient algorithms for computing LCS in a given velocity field. Large computational time has been a major hurdle to the widespread adoption of LCS techniques, especially in three dimensions. The ridge tracking algorithms presented here take advantage of the definition of LCS as codimension-one manifolds by avoiding computations in parts of the domain away from the LCS surfaces. By detecting and tracking LCS through the domain of interest, the computational order is reduced from O(1=δxn) to O(1=δxn-1) in n-dimensional problems. In three dimensions, this algorithm is used to compute the LCS in a warm-core ring in the Gulf of Mexico and a hurricane simulation, revealing a new type of LCS structure in the boundary layers of these geophysical vortices. The transport of these structures is analyzed and found to enhance the potential for diffusive mixing in these flow regions through the generation of small length scales.