Barriers to advective transport are computed from the velocity field obtained from a data assimilating primitive equation model of the Gulf of Mexico. The ocean model (CUPOM) used in this study has 24 sigma levels and a horizontal resolution of 1/12 degree. Along-track TOPEX/Poseidon and ERS-2 altimeter data were assimilated as pseudo XBTs using a simple OI-based data assimilation scheme and the near-real-time sea surface temperature was inferred from multichannel infrared imagery (MCSST). The surface was forced by the Navy Operational Global Atmospheric Prediction System (NOGAPS) winds and the model was integrated on a 1/12 degree grid for the summer of 1998. Hyperbolic regions in the near surface flow (50m) are identified with both fixed-time stagnation points and relative dispersion. From these regions, dynamically significant material curves (finite-time invariant manifolds) are constructed, which form Lagrangian boundaries for coherent structures such as eddies and the Loop Current. With this technique, eddy entrainment/detrainment, inter-eddy advection, Loop Current ring formation, and ring cleavage are revealed. Several examples are discussed, and a comparison with drifter data is made.