Two like-signed vorticity regions can pair or merge into one vortex. This phenomenon occurs if the original two vortices are sufficiently close together, that is, if the distance between the vorticity centroids is smaller than a certain critical merger distance, which depends on the initial shape of the vortex distributions. Our conclusions are based on an analytical/numerical study, which presents the first quantitative description of the cause and mechanism behind the restricted process of symmetric vortex merger. We use two complementary models to investigate the merger of identical vorticity regions. The first, based on a recently introduced low-order physical-space moment model of the two-dimensional Euler equations, is a Hamiltonian system of ordinary differential equations for the evolution of the centroid position, aspect ratio and orientation of each region. By imposing symmetry this system is made integrable and we obtain a necessary and sufficient condition for merger. This condition involves only the initial conditions and the conserved quantities. The second model is a high-resolution pseudospectral algorithm governing weakly dissipative flow in a box with periodic boundary conditions. When the results obtained by both methods are juxtaposed, we obtain a detailed kinematic insight into the merger process. When the moment model is generalized to include a weak Newtonian viscosity, we find a ‘metastable’ state with a lifetime depending on the dissipation timescale. This state attracts all initial configurations that do not merge on a convective timescale. Eventually, convective merger occurs and the state disappears. Furthermore, spectral simulations show that initial conditions with a centroid separation slightly larger than the critical merger distance initially cause a rapid approach towards this metastable state.

We consider the evolution of an isolated elliptical vortex in a weakly dissipative fluid. It is shown computationally that a spatially smooth vortex relaxes inviscidly towards axisymmetry on a circulation timescale as the result of filament generation. Heuristically, we derive a simple geometrical formula relating the rate of change of the aspect ratio of a particular vorticity contour to its orientation relative to the streamlines (where the orientation is defined through second-order moments). Computational evidence obtained with diagnostic algorithms validates the formula. By considering streamlines in a corotating frame and applying the new formula, we obtain a detailed kinematic understanding of the vortex's decay to its final state through a primary and a secondary breaking. The circulation transported into the filaments although a small fraction of the total, breaks the symmetry and is the chief cause of axisymmetrization.

The detailed mechanisms in vortex cross-linking are unveiled by adequately resolved, direct numerical simulation of two viscous vortex tubes. There are three characteristic phases: (i) inviscid induction followed by core flattening and stretching; (ii) bridging of the two vortices by accumulation of annihilated and then cross-linked vortex lines; and (iii) threading of the remnants of the initial vortex pair in between the two bridges as they pull apart. These phases and the role of threading—along with bridging—in the mixing and the enstrophy cascade are explained, and it is shown that the mechanism is insensitive to asymmetries.

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.