Lagrangian diagnostics, such as the finite-time Lyapunov exponent and Lagrangian coherent structures, have become popular tools for analyzing unsteady fluid flows. These diagnostics can help illuminate regions where particles transported by a flow will converge to and diverge from, even in a divergence-free flow. Unfortunately, calculating Lagrangian diagnostics can be time consuming and computationally expensive. Recently, new Eulerian diagnostics have been developed which provide similar insights into the Lagrangian transport properties of fluid flows. These new diagnostics are faster and less expensive to compute than their Lagrangian counterparts. Because Eulerian diagnostics of Lagrangian transport structure are relatively new, there is still much about their connection to Lagrangian diagnostics that is unknown. This paper provides a mathematical bridge between Lagrangian and Eulerian diagnostics. It rigorously explores the mathematical relationship that exists between invariants of the right Cauchy-Green deformation tensor and the Rivlin-Ericksen tensors, primarily the Eulerian rate-of-strain tensor, in the infinitesimal integration time limit. Additionally, this paper develops the infinitesimal-time Lagrangian coherent structures (iLCSs) and demonstrates their efficacy in predicting the Lagrangian transport of particles even in realistic geophysical fluid flows generated by numerical models.
Comments: 42 Pages. This manuscript has been submitted to Nonlinear Dynamics for publication.
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