On Maximum Enstrophy Dissipation in 2D Navier-Stokes Flows in the Limit of Vanishing Viscosity

TL;DR

This paper investigates how enstrophy dissipation in 2D Navier-Stokes flows behaves as viscosity approaches zero, using optimization and numerical methods to identify flow configurations that maximize dissipation.

Contribution

It introduces an optimization framework to identify initial conditions that maximize enstrophy dissipation, revealing multiple mechanisms and confirming the sharpness of theoretical bounds.

Findings

Multiple local maximizers with distinct amplification mechanisms

Maximum enstrophy dissipation aligns with theoretical estimates

Numerical solutions show the dependence on viscosity is sharp

Abstract

We consider enstrophy dissipation in two-dimensional (2D) Navier-Stokes flows and focus on how this quantity behaves in thelimit of vanishing viscosity. After recalling a number of a priori estimates providing lower and upper bounds on this quantity, we state an optimization problem aimed at probing the sharpness of these estimates as functions of viscosity. More precisely, solutions of this problem are the initial conditions with fixed palinstrophy and possessing the property that the resulting 2D Navier-Stokes flows locally maximize the enstrophy dissipation over a given time window. This problem is solved numerically with an adjoint-based gradient ascent method and solutions obtained for a broad range of viscosities and lengths of the time window reveal the presence of multiple branches of local maximizers, each associated with a distinct mechanism for the amplification of palinstrophy. The dependence of the maximum enstrophy dissipation on viscosity is shown to be in quantitative agreement with the estimate due to Ciampa, Crippa & Spirito (2021), demonstrating the sharpness of this bound.