Vortices, Maximum Growth and the Problem of Finite-Time Singularity Formation

TL;DR

This paper investigates extreme vortex states in 2D viscous flows that maximize palinstrophy growth, providing insights into vortex dynamics and implications for understanding finite-time singularities in fluid systems.

Contribution

It extends previous work by identifying vortex mechanisms responsible for extreme palinstrophy growth and explores long-term flow behaviors in 2D viscous flows.

Findings

Maximum palinstrophy growth occurs at short times.

Vortex scattering influences long-time flow evolution.

Extreme vortex states saturate mathematical growth estimates.

Abstract

In this work we are interested in extreme vortex states leading to the maximum possible growth of palinstrophy in 2D viscous incompressible flows on periodic domains. This study is a part of a broader research effort motivated by the question about the finite-time singularity formation in the 3D Navier-Stokes system and aims at a systematic identification of the most singular flow behaviors. We extend the results reported in Ayala & Protas (2013) where extreme vortex states were found leading to the growth of palinstrophy, both instantaneously and in finite-time, which saturates the estimates obtained with rigorous methods of mathematical analysis. Here we uncover the vortex dynamics mechanisms responsible for such extreme behavior in time-dependent 2D flows. While the maximum palinstrophy growth is achieved at short times, the corresponding long-time evolution is characterized by some nontrivial features, such as vortex scattering events.