Enstrophy variations in the collapsing process of point vortices

TL;DR

This paper explores how enstrophy dissipates during the collapse of point vortices in 2D flows, using filtered Euler equations to demonstrate dissipation in systems with four and five vortices.

Contribution

It extends previous work by numerically showing enstrophy dissipation during vortex collapse for systems with more than three vortices.

Findings

Enstrophy dissipation occurs during vortex collapse in four and five vortex systems.

Filtered-point-vortex solutions converge to collapsing orbits and dissipate enstrophy.

Numerical evidence supports the role of vortex collapse in enstrophy dissipation.

Abstract

We investigate enstrophy variations by collapse of point vortices in an inviscid flow and, in particular, focus on the enstrophy dissipation that is a significant property characterizing 2D turbulent flows. Point vortex is an ideal vortex whose vorticity is concentrated on a point and the dynamics of point vortices on an inviscid flow is described by the point-vortex system. The point-vortex system has self-similar collapsing solutions, which are expected to cause the anomalous enstrophy dissipation, but this collapsing process of point vortices cannot be described by the 2D Euler equations. In this study, we consider point-vortex solutions of the 2D filtered Euler equations, which are a regularized model of the 2D Euler equations, and the filtered-point-vortex system describing the dynamics of them. The preceding studies have proven that there exist solutions to the three filtered-point-vortex system such that they converge to self-similar collapsing orbits of the point-vortex system and dissipate the enstrophy at the event of collapse in the zero limit of a filter scale. In this study, we numerically show that the enstrophy dissipation by the collapse of point vortices could occur for the four and five vortex problems.