Network and geometric characterization of three-dimensional fluid transport between two layers

de la Fuente, Rebeca; Drótos, Gábor; Hernández-Garcı́a, Emilio; López, Cristóbal
Submitted (2021)

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.

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