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

TL;DR

This paper introduces a novel formalism combining geometric and network methods to analyze three-dimensional fluid transport between two layers, focusing on particle dynamics in flows with a dominant direction.

Contribution

It develops a two-layer map framework and introduces the Finite Depth Lyapunov Exponent, integrating geometric and probabilistic network approaches for fluid transport analysis.

Findings

The formalism effectively characterizes particle transport in complex flows.

Numerical results demonstrate the approach on a modified ABC flow.

Relationships between geometric and network measures are established.

Abstract

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.