# Why do 2D hydrodynamic vortices merge?

Two point-like vortices with the same vorticity in 2D perfect fluid can rotate forever. Why do finite-sized vortices merge as soon as they touch? Why in general 2D fluid vortices of the same sign want to merge? Is there any analogy between vortex merging and droplet merging, i.e. is there something similar to surface tension for vortices?

• You are right. I have deleted my incorrect answer. Although you have mentioned perfect fluid in your question, the merging video is for viscous fluid (Reynolds number = 100). The counter-rotating vortex video also seems to be for a viscous fluid because overall kinetic energy seems to be going down with time.
– Deep
Apr 7, 2021 at 5:04
• yes, the videos are for low Reynolds numbers. Some papers say they see merging for finite vortices for perfect fluid too though. It is hard to get an understanding why it happens though. Apr 8, 2021 at 5:50
• Onsager explained this behaviour in "Statistical hydrodynamics", 1949, see the review journals.aps.org/rmp/abstract/10.1103/RevModPhys.78.87 . Vortices of the same sign tend to cluster in 2D and vortices of opposite sign tend to "unbind" due to a sort of "inverse cascade". This is related to the fact that the 2 coordinates of the vortex are conjugated variables in the phase space, leading to the possibility of having negative temperature states. Related question: physics.stackexchange.com/q/383159/226902 Apr 28, 2022 at 17:14

From the abstract of "The physical mechanism of symmetric vortex merger: A new viewpoint":

Their 'results suggest that a complete merging process is as follows: the vortices deform first due to the mutually induced straining; the deformation results in elliptical vortices and an angle between the major axis of each elliptical vortex and the line joining the two vortices, which in turn cause an attraction of fluid particles from one vortex to the other; sheetlike vortex structures are thus formed; and finally the velocity field induced by these sheetlike structures readily pushes two vortex cores together.'

It also states that 'when the flow is viscous, the separation between vortices reduces and the mutual attraction increases with time by diffusion. As the mutual attraction dominates over the self-induced rotation, sheetlike structures are formed gradually and merger eventually occurs. The onset time of merger is thus found to depend not only on the initial separation but also on the Reynolds number. The former determines when the mutual attraction will become dominant and the latter controls the speed at which sheetlike structures grow.'

• Thank you for the answer. Unfortunately, this explanation rises more questions than it answers. Why do vortices induce elliptical deformation onto each other? Why does this, in turn, causes fluid particles to go from one vortex to the other? What are these sheet-like structures and why they "readily push to vortex cores together"? Would the vortices merge if the vorticity fields did not overlap? Apr 9, 2021 at 7:22