Maximum Palinstrophy Growth in 2D Incompressible Flows

TL;DR

This paper investigates vortex structures that maximize palinstrophy growth in 2D incompressible flows, revealing optimal configurations and their implications for understanding potential singularities in fluid dynamics.

Contribution

The study systematically identifies vortex structures that maximize palinstrophy growth using variational optimization, demonstrating sharpness of theoretical bounds and analyzing structures across different regimes.

Findings

Optimal vortex structures are identified and classified.

Some families of solutions saturate theoretical growth estimates.

Optimal initial states can reach the maximum palinstrophy growth in finite time.

Abstract

In this study we investigate vortex structures which lead to the maximum possible growth of palinstrophy in two-dimensional incompressible flows on a periodic domain. The issue of palinstrophy growth is related to a broader research program focusing on extreme amplification of vorticity-related quantities which may signal singularity formation in different flow models. Such extreme vortex flows are found systematically via numerical solution of suitable variational optimization problems. We identify several families of maximizing solutions parameterized by their palinstrophy, palinstrophy and energy and palinstrophy and enstrophy. Evidence is shown that some of these families saturate estimates for the instantaneous rate of growth of palinstrophy obtained using rigorous methods of mathematical analysis, thereby demonstrating that this analysis is in fact sharp. In the limit of small palinstrophies the optimal vortex structures are found analytically, whereas for large palinstrophies they exhibit a self-similar multipolar structure. It is also shown that the time evolution obtained using the instantaneously optimal states with fixed energy and palinstrophy as the initial data saturates the upper bound for the maximum growth of palinstrophy in finite time. Possible implications of this finding for the questions concerning extreme behavior of flows are discussed.