A moment model for vortex interactions of the two-dimensional Euler equations. Part 1. Computational validation of a Hamiltonian elliptical representation

Abstract

We consider the evolution of finite uniform-vorticity regions in an unbounded in viscid fluid. We perform a perturbation analysis based on the assumption that the regions are remote from each other and nearly circular. Thereby we obtain a self-consistent infinite system of ordinary differential equations governing the physical-space moments of the individual regions. Truncation yields an Nth-order moment model. Special attention is given to the second-order model where each region is assumed elliptical. The equations of motion conserve local area, global centroid, total angular impulse and global excess energy, and the system can be written in canonical Hamiltonian form. Computational comparison with solutions to the contour-dynamical representation of the Euler equations shows that the model is useful and accurate. Because of the internal degrees of freedom, namely aspect ratio and orientation, two like-signed vorticity regions collapse if they are near each other. Although the model becomes invalid during a collapse, we find a striking similarity with the merger process of the Euler equations.