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In this paper we develop a method for the estimation of {\bf T}ransport {\bfI}nduced by the {\bf M}ean-{\bf E}ddy interaction (TIME) in two-dimensionalunsteady flows. The method is built on the dynamical systems approach and canbe viewed as a hybrid combination of Lagrangian and Eulerian methods. The(Eulerian) boundaries across which we consider (Lagrangian) transport arekinematically defined by appropriately chosen streamlines of the mean flow. Byevaluating the impact of the mean-eddy interaction on transport, the TIMEmethod can be used as a diagnostic tool for transport processes that occurduring a specified time interval along a specified boundary segment.  We introduce two types of TIME functions: one that quantifies theaccumulation of flow properties and another that measures the displacement ofthe transport geometry. The spatial geometry of transport is described by theso-called pseudo-lobes, and temporal evolution of transport by their dynamics.In the case where the TIME functions are evaluated along a separatrix, thepseudo-lobes have a relationship to the lobes of Lagrangian transport theory.In fact, one of the TIME functions is identical to the Melnikov function thatis used to measure the distance, at leading order in a small parameter, betweenthe two invariant manifolds that define the Lagrangian lobes. We contrast thesimilarities and differences between the TIME and Lagrangian lobe dynamics indetail. An application of the TIME method is carried out for inter-gyretransport in the wind-driven oceanic circulation model and a comparison withthe Lagrangian transport theory is made.

Transport Induced by Mean-Eddy Interaction: I. Theory, and Relation to Lagrangian Lobe Dynamics