Extreme Vortex States and the Growth of Enstrophy in 3D Incompressible Flows

TL;DR

This paper characterizes extreme vortex states in 3D incompressible flows that maximize enstrophy growth, constructs these states numerically, and analyzes their implications for turbulence and singularity formation.

Contribution

It provides an analytic and numerical framework for identifying vortex states that maximize enstrophy growth, revealing their properties and limitations in finite-time evolution.

Findings

Extreme vortex states maximize enstrophy growth rate.

These states saturate fundamental growth bounds.

No evidence of finite-time singularity formation.

Abstract

In this investigation we study extreme vortex states defined as incompressible velocity fields with prescribed enstrophy $\mathcal{E}_0$ which maximize the instantaneous rate of growth of enstrophy $d\mathcal{E}/dt$. We provide {an analytic} characterization of these extreme vortex states in the limit of vanishing enstrophy $\mathcal{E}_0$ and, in particular, show that the Taylor-Green vortex is in fact a local maximizer of $d\mathcal{E} / dt$ {in this limit}. For finite values of enstrophy, the extreme vortex states are computed numerically by solving a constrained variational optimization problem using a suitable gradient method. In combination with a continuation approach, this allows us to construct an entire family of maximizing vortex states parameterized by their enstrophy. We also confirm the findings of the seminal study by Lu & Doering (2008) that these extreme vortex states saturate (up to a numerical prefactor) the fundamental bound $d\mathcal{E} / dt < C \, \mathcal{E}^3$, for some constant $C > 0$. The time evolution corresponding to these extreme vortex states leads to a larger growth of enstrophy than the growth achieved by any of the commonly used initial conditions with the same enstrophy $\mathcal{E}_0$. However, based on several different diagnostics, there is no evidence of any tendency towards singularity formation in finite time. Finally, we discuss possible physical reasons why the initially large growth of enstrophy is not sustained for longer times.