Finding Lagrangian Structures via the Application of Braid Theory

The ability to accurately identify regions of mixing in two dimensional systems has applications in ocean and geophysical systems, as well as granular flows. Over the last decade, much work has been put into finding coherent structures in the Lagrangian frame. Most of these methods require field data (e.g. velocity or vorticity fields) which in situations like the ocean are not readily available. However, trajectories of tracers for these systems can accurately be measured and are available for analysis. This indicates that a method which estimates the location of Lagrangian structures based on trajectory data would be highly valuable. The Lagrangian structures we are looking for are defined as regions of mixing surrounded by a transport boundary. These moving regions contain fluid which mixes with other fluid within the boundary but not with fluid outside the region. This self mixing region can move through the full system and can change shape. The transport boundaries in the fluid effectively divide up the fluid into regions of qualitatively different dynamics. What separates these transport boundaries from other material lines in the fluid is that the length of these boundaries stays the same order of magnitude throughout the time evolution. If the material line perimeter stays of the same order of length then there is negligible mixing occurring between fluid inside and outside of the transport boundary. The goal of this project is to develop a method for finding Lagrangian structures from trajectory data. We will use the property of a negligibly growing perimeter as the differentiating characteristic between transport boundaries and other material lines in the fluid. In order to do this, we have developed an algorithm that analyzes the trajectory data and makes predictions of where these structures are located. The novelty of this algorithm is its reliance on the tools of braid theory.
Published in 2010 Program of Study: Swirling and Swimming in Turbulence, ADA544302, p146-167. The original document contains color images