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