Universality of the Anomalous Enstrophy Dissipation at the Collapse of Three Point Vortices on Euler--Poincaré Models

Abstract

Anomalous enstrophy dissipation of incompressible flows in the inviscid limit is a significant property characterizing two-dimensional turbulence. It indicates that the investigation of non-smooth incompressible and inviscid flows contributes to the theoretical understanding of turbulent phenomena. In the preceding study, a unique global weak solution to the Euler-$\alpha$ equations, which is a regularized Euler equations, for point-vortex initial data is considered, and thereby it has been shown that, as $\alpha \rightarrow 0$, the evolution of three point vortices converges to a self-similar collapsing orbit dissipating the enstrophy in the sense of distributions at the critical time. In the present paper, to elucidate whether or not this singular orbit can be constructed independently on the regularization method, we consider a functional generalization of the Euler-$\alpha$ equations, called the \textit{Euler-Poincar\'{e} models}, in which the incompressible velocity field is dispersively regularized by a smoothing function. We provide a sufficient condition for the existence of the singular orbit, which is applicable to many smoothing functions. As examples, we confirm that the condition is satisfied with the Gaussian regularization and the vortex-blob regularization that are both utilized in the numerical scheme solving the Euler equations. Consequently, the enstrophy dissipation via the collapse of three point vortices is a generic phenomenon that is not specific to the Euler-$\alpha$ equations but universal within the Euler-Poincar\'{e} models.