Global Organization of Three-Dimensional, Volume-Preserving Flows: Constraints, Degenerate Points, and Lagrangian Structure

TL;DR

This paper investigates how volume-preserving constraints and periodic lines in 3D flows create degenerate points that organize Lagrangian transport, revealing their crucial role in understanding fluid mixing and transport structures.

Contribution

It introduces a method to identify degenerate points in 3D volume-preserving flows and explains their role in organizing transport structures, especially after flow perturbations.

Findings

Degenerate points are identified via the trace of the deformation tensor.

Periodic lines and degenerate points organize all Lagrangian transport.

Flow invariants constrain deformation and structure formation.

Abstract

Global organization of 3-dimensional (3D) Lagrangian chaotic transport is difficult to infer without extensive computation. For 3D time-periodic flows with one invariant we show how constraints on deformation that arise from volume-preservation and periodic lines result in resonant degenerate points that periodically have zero net deformation. These points organize all Lagrangian transport in such flows through coordination of lower-order and higher-order periodic lines and prefigure unique transport structures that arise after perturbation and breaking of the invariant. Degenerate points of periodic lines and the extended 3D structures associated with them are easily identified through the trace of the deformation tensor calculated along periodic lines. These results reveal the importance of degenerate points in understanding transport in one-invariant fluid flows.