  We consider transport in a fluid flow of arbitrary complexity but with a
dominant flow direction. This is the situation encountered, for example, when
analyzing the dynamics of sufficiently small particles immersed in a turbulent
fluid and vertically sinking because of their weight. We develop a formalism
characterizing the dynamics of particles released from one layer of fluid and
arriving in a second one after traveling along the dominant direction. The main
ingredient in our study is the definition of a two-layer map that describes the
Lagrangian transport between both layers. We combine geometric approaches and
probabilistic network descriptions to analyze the two-layer map. From the
geometric point of view, we express the properties of lines, surfaces and
densities transported by the flow in terms of singular values related to
Lyapunov exponents, and define a new quantifier, the Finite Depth Lyapunov
Exponent. Within the network approach, degrees and an entropy are introduced to
characterize transport. We also provide relationships between both
methodologies. The formalism is illustrated with numerical results for a
modification of the ABC flow, a model commonly studied to characterize
three-dimensional chaotic advection.
