Robust FEM-based extraction of finite-time coherent sets using scattered, sparse, and incomplete trajectories
This work addresses the challenge of identifying coherent structures in geophysical flows (e.g., ocean eddies, atmospheric vortices) from real-world sparse and incomplete data, providing practical tools for dynamical analysis.
The paper develops three FEM-based numerical methods to extract finite-time coherent sets from scattered, sparse, and incomplete trajectory data, introducing a new dynamic isoperimetric problem with Dirichlet boundary conditions. The methods enable rapid and reliable identification of coherent sets in aperiodic flows, even with limited observational data.
Transport and mixing properties of aperiodic flows are crucial to a dynamical analysis of the flow, and often have to be carried out with limited information. Finite-time coherent sets are regions of the flow that minimally mix with the remainder of the flow domain over the finite period of time considered. In the purely advective setting this is equivalent to identifying sets whose boundary interfaces remain small throughout their finite-time evolution. Finite-time coherent sets thus provide a skeleton of distinct regions around which more turbulent flow occurs. They manifest in geophysical systems in the forms of e.g.\ ocean eddies, ocean gyres, and atmospheric vortices. In real-world settings, often observational data is scattered and sparse, which makes the difficult problem of coherent set identification and tracking even more challenging. We develop three FEM-based numerical methods to efficiently approximate the dynamic Laplace operator, and introduce a new dynamic isoperimetric problem using Dirichlet boundary conditions. Using these FEM-based methods we rapidly and reliably extract finite-time coherent sets from models or scattered, possibly sparse, and possibly incomplete observed data.