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

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