# Topological Chaos in a Three-Dimensional Spherical Fluid Vortex

## TL;DR

This paper extends topological analysis techniques to three-dimensional fluid systems, enabling detailed characterization of chaos, mixing mechanisms, and fractal structures in a spherical vortex, which was previously challenging.

## Contribution

It develops a 3D extension of homotopic lobe dynamics, allowing topological chaos analysis in realistic 3D fluid flows, overcoming previous limitations.

## Key findings

- Built a symbolic dynamics model from 3D chaotic scattering data.
- Predicted self-similar fractal structures in scattering data.
- Identified two distinct mixing mechanisms in 3D fluid flow.

## Abstract

In chaotic deterministic systems, seemingly stochastic behavior is generated by relatively simple, though hidden, organizing rules and structures. Prominent among the tools used to characterize this complexity in 1D and 2D systems are techniques which exploit the topology of dynamically invariant structures. However, the path to extending many such topological techniques to three dimensions is filled with roadblocks that prevent their application to a wider variety of physical systems. Here, we overcome these roadblocks and successfully analyze a realistic model of 3D fluid advection, by extending the homotopic lobe dynamics (HLD) technique, previously developed for 2D area-preserving dynamics, to 3D volume-preserving dynamics. We start with numerically-generated finite-time chaotic-scattering data for particles entrained in a spherical fluid vortex, and use this data to build a symbolic representation of the dynamics. We then use this symbolic representation to explain and predict the self-similar fractal structure of the scattering data, to compute bounds on the topological entropy, a fundamental measure of mixing, and to discover two different mixing mechanisms, which stretch 2D material surfaces and 1D material curves in distinct ways.