of extreme vortex states maximizing the instantaneous rate of enstrophy production in the limit of vanishing enstrophy. In addition, based on these findings, we identify the set of initial data, parameterized by its energy and enstrophy, for which one can a priori guarantee global-in-time existence of smooth solutions. This result therefore offers a physically appealing interpretation of an “abstract” mathematical theorem concerning global existence of classical solutions corresponding to “small” initial data [@lady69]. We also emphasize that, in order to establish a direct link with the results of the mathematical analysis discussed below, in our investigation we follow a rather different strategy than in most of the studies referenced above. While these earlier studies relied on data from a relatively small number of simulations performed at a high (at the given time) resolution, in the present investigation we explore a broad range of cases, each of which is however computed at a more moderate resolution (or, equivalently, Reynolds number). With such an approach to the use of available computational resources, we are able to reveal trends resulting from the variation of parameters which otherwise would be hard to detect. Systematic computations conducted in this way thus allow us to probe the sharpness of the mathematical analysis relevant to the problem.

The question of regularity of solution to the Navier-Stokes system is [usually addressed using “energy” methods which rely on finding upper bounds (with respect to time) on certain quantities of interest, typically taken as suitable Sobolev norms of the solution. A key intermediate step is obtaining bounds on the rate of growth of the quantity of interest, a problem which can be studied with ODE methods.]{} While [for the Navier-Stokes system]{} different norms of the velocity gradient or vorticity can be used to study the regularity of solutions, the use of enstrophy $\E$ (see equation ) is privileged by the well-known result of @ft89, where it 