We study the flow obtained from a 3-layer, eddy-resolving quasi-geostrophic circulation model subject to an applied wind stress curl. For this model we will consider transport between the northern and southern ``gyres'' separated by a jet. We will focus on the importance of invariant manifolds in forming geometric structures that govern transport. By ``govern'', we mean they can be used to compute Lagrangian transport quantities, such as the flux across the jet. We will consider periodic, quasi-periodic, and chaotic velocity fields, and thus assess the effectiveness of dynamical systems techniques in flows with progressively more spatio-temporal complexity. The numerical methods necessary to implement the dynamical systems techniques and the significance of invariant manifolds as signatures of specific ``events'', such as rings pinching off from a meandering jet, will also be discussed.