Mesochronic classification of trajectories in incompressible 3D vector fields over finite times

TL;DR

This paper introduces a mesochronic classification method for trajectories in 3D incompressible flows, extending finite-time dynamical analysis tools to better understand volume deformation and rotation.

Contribution

It develops a new mesochronic velocity-based approach to classify 3D flow trajectories, extending 2D analysis methods to three dimensions with practical computational techniques.

Findings

Mesochronic velocity characterizes deformation and rotation in 3D flows.

The method extends Okubo--Weiss--Chong analysis to finite times.

Numerical tests on ABC flow validate the approach.

Abstract

The mesochronic velocity is the average of the velocity field along trajectories generated by the same velocity field over a time interval of finite duration. In this paper we classify initial conditions of trajectories evolving in incompressible vector fields according to the character of motion of material around the trajectory. In particular, we provide calculations that can be used to determine the number of expanding directions and the presence of rotation from the characteristic polynomial of the Jacobian matrix of mesochronic velocity. In doing so, we show that (a) the mesochronic velocity can be used to characterize dynamical deformation of three-dimensional volumes, (b) the resulting mesochronic analysis is a finite-time extension of the Okubo--Weiss--Chong analysis of incompressible velocity fields, (c) the two-dimensional mesochronic analysis from Mezic et al. \emph{A New Mixing Diagnostic and Gulf Oil Spill Movement}, Science 330, (2010), 486-489, extends to three-dimensional state spaces. Theoretical considerations are further supported by numerical computations performed for a dynamical system arising in fluid mechanics, the unsteady Arnold--Beltrami--Childress (ABC) flow.