Enstrophy Variations in the Incompressible 2D Euler Flows and $\alpha$ Point Vortex System

Abstract

The dissipation of the enstrophy, which is the $L^2$ norm of the vorticity, in the zero-viscous limit gives rise to the emergence of inertial range in the ensemble average of the energy density spectrum in 2D fluid turbulence. However, it is mathematically known that not only smooth solutions but also weak solutions in $L^p(\mathbb {R}^2)$, $p>2$ to the 2D Euler equations never dissipates the enstrophy [7]. This indicates that weak solutions for initial vorticity distributions belonging to weaker function spaces such as the space of Radon measure on $\mathbb {R}^2$ should be constructed to obtain such singular solutions with the enstrophy dissipation, but no existence result in this function space has not yet been established. We here consider the 2D Euler-$\alpha$ equations, which is a dispersive regularization of the Euler equations with a scaling parameter $\alpha$, for the initial vorticity distributions whose support consists of a set of N points, called $\alpha$-point vortices. We shall construct singular weak solutions to the Euler equations from those of the evolution equations of the $\alpha$ point vortices by taking their $\alpha \rightarrow 0$ limit. We then numerically demonstrate that the self-similar collapse of the $\alpha$ point vortices gives rise to the anomalous enstrophy dissipation in the distributional sense and it is a robust mechanism of the enstrophy dissipation observed for a wide range of initial configurations of $\alpha$ point vortices.