Bjerknes Vortex Rings
Two coaxial vortex rings of equal circulation perform the leapfrog: the rear ring accelerates into the front, which is slowed and expanded, until they pass through each other — then roles reverse indefinitely. V. F. K. Bjerknes studied these interactions in the 1890s as an analogy for atom-atom forces. Watch rings leapfrog, collide head-on, or orbit each other in tilted configurations.
vz = Γ/(4πR) · [ln(8R/a) − ¼] (thin-ring self-induced velocity)
Click on the canvas to spawn a new vortex ring at that position. The coloured glow represents each ring’s induced velocity field.
Vortex ring dynamics
A vortex ring in an ideal (inviscid, incompressible) fluid is topologically stable. The ring propagates along its axis with a self-induced velocity proportional to its circulation Γ and inversely proportional to its radius R. When two coaxial rings travel in the same direction, the mutual induction is asymmetric: the rear ring sees the front ring as a source that accelerates and contracts it, while the front ring is decelerated and expanded. This drives the rear ring through the front, reversing their roles — and the cycle repeats indefinitely.
The key dimensionless parameter is the ratio of ring radii. If R&sub2;/R&sub1; is close to 1, leapfrogging is stable and periodic. If the ratio is too large or the rings too close, the interaction becomes chaotic.
Bjerknes forces
Vilhelm Bjerknes (1897) noted that two pulsating or oscillating bubbles in a fluid attract each other when pulsating in phase (like two rings with the same circulation sense) and repel when out of phase. He proposed this as a model for electromagnetic forces between atoms, anticipating the idea that forces could arise from wave-mechanical interactions in a medium.
Head-on collision
When two equal rings travel in opposite directions along the same axis, they approach, expand, and ultimately reconnect. In a real viscous fluid this produces a complex splashing pattern; in the inviscid limit the rings expand to infinite radius at finite time (a topological singularity). The simulation handles this by reversing the interaction once the rings become very large.