Quantifying the tangling of trajectories using the topological entropy.

S Candelaresi,G Hornig,D I Pontin

doi:10.1063/1.5000812

Abstract

We present a simple method to efficiently compute a lower limit of the topological entropy and its spatial distribution for two-dimensional mappings. These mappings could represent either two-dimensional time-periodic fluid flows or three-dimensional magnetic fields, which are periodic in one direction. This method is based on measuring the length of a material line in the flow. Depending on the nature of the flow, the fluid can be mixed very efficiently which causes the line to stretch. Here, we study a method that adaptively increases the resolution at locations along the line where folds lead to a high curvature. This reduces the computational cost greatly which allows us to study unprecedented parameter regimes. We demonstrate how this efficient implementation allows the computation of the variation of the finite-time topological entropy in the mapping. This measure quantifies spatial variations of the braiding efficiency, important in many practical applications.

Highlights

We discuss the concept of the topological entropy1 as a measure of mixing of fluid particles in a two-dimensional flow, or field line tangling in a three-dimensional vector field
These mappings could represent either two-dimensional time-periodic fluid flows or three-dimensional magnetic fields, which are periodic in one direction
We study a method that adaptively increases the resolution at locations along the line where folds lead to a high curvature. This reduces the computational cost greatly which allows us to study unprecedented parameter regimes. We demonstrate how this efficient implementation allows the computation of the variation of the finite-time topological entropy in the mapping

Summary

Introduction

We discuss the concept of the topological entropy as a measure of mixing of fluid particles in a two-dimensional flow, or field line tangling in a three-dimensional vector field. The notion of topological entropy was developed in the context of dynamical systems, and has been used in the field of fluid dynamics to understand fluid mixing.. The notion of topological entropy was developed in the context of dynamical systems, and has been used in the field of fluid dynamics to understand fluid mixing.2–4 Such fluid mixing can be the stirring of a substance (solid or liquid) with applications in engineering and production. In plasma physics, for magnetic fields in tokamaks and spheromaks, the tangling of magnetic field lines is crucial for transport processes in the plasma, the concept of topological entropy has rarely been used in this context To make that interpretation complete, we need to assume that the field is static and periodic in the direction of the field

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