Topological vortex identification for two-dimensional turbulent flows in doubly periodic domains

TL;DR

This paper introduces a topological classification method for vortex structures in 2D turbulence, converting complex flow patterns into simple tree representations to analyze their statistical properties.

Contribution

It develops a novel topological classification theory for particle orbits in 2D flows, enabling effective extraction of vortex structures from turbulent flow data.

Findings

Complex vortex structures can be represented by simple trees and sequences.

The method successfully extracts and analyzes vortex statistical properties.

Application to turbulence data demonstrates the approach's effectiveness.

Abstract

The dynamics and statistical properties of two-dimensional (2D) turbulence are often investigated through numerical simulations of incompressible, viscous fluids in doubly periodic domains. A key challenge in 2D turbulence research is accurately identifying and describing statistical properties of its coherent vortex structures within complex flow patterns. This paper addresses this challenge by providing a classification theory for the topological structure of particle orbits generated by instantaneous Hamiltonian flows on the torus $\mathbb{T}^2$, which serves as a mathematical model for 2D incompressible flows. Based on this theory, we show that the global orbit structure of any Hamiltonian flow can be converted into a planar tree, named a partially Cyclically-Ordered rooted Tree (COT), and its corresponding string expression (COT representation). We apply this conversion algorithm to 2D energy and enstrophy cascade turbulence. The results show that the complex topological structure of turbulent flow patterns can be effectively represented by simple trees and sequences of letters, thereby successfully extracting coherent vortex structures and investigating their statistical properties from a topological perspective.