  We develop a methodology to identify finite-time Lagrangian structures from
data and models using an extension of the Koopman operator-theoretic methods
developed for velocity fields with simple (periodic, quasi-periodic)
time-dependence. To achieve this, the notion of the Finite Time Ergodic (FiTER)
partition is developed and rigorously justified. In combination with a
clustering-based approach, the methodology enables identification of the
temporal evolution of Lagrangian structures in a classic, benchmark,
oceanographic transport problem, namely the cross-stream flux induced by the
interaction of a meso- scale Gulf Stream Ring eddy with the main jet. We focus
on a single mixing event driven by the interaction between an energetic cold
core ring (a cyclone), the strong jet, and a number of smaller scale cyclones
and anticyclones. The new methodology enab les reconstruction of Lagrangian
structures in three dimensions and analysis of their time-evolution.
