  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.
