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Point Vortex Sandbox

Learning goal

Convince yourself that one sentence — each vortex is advected by the field of all the others, and by none of its own — generates the entire zoo: orbiting pairs, translating pairs, wall glide, leapfrogging rings and Hamiltonian chaos.

The integrator is RK4 on the exact point-vortex equations. The faint dots are passive tracers — they are advected by the flow but exert no influence, so they show you the velocity field without changing it.

Things to try

  1. Co-rotating pair. The two orbit their midpoint forever. Check the readout: the measured period matches \(T = 2\pi^2d^2/\Gamma\) to better than a hundredth of a percent. Two like-signed vortices never merge in this model — real ones do, because they have finite cores that deform, which is the elementary coarsening step of 2-D turbulence.

  2. Counter-rotating pair. Now they translate in a straight line at \(V = \Gamma/2\pi d\), with the separation preserved exactly. This is a wingtip pair head-on, and the cross-section of a smoke ring — it is why smoke rings move. Nothing pushes them; they induce their own motion.

  3. Vortex near a wall. The faint circle below the line is the image vortex, of opposite sign, which enforces no penetration. The real vortex glides along the wall at \(\Gamma/4\pi h\) — and note that its height never changes, because the image is always directly below. Same method of images as electrostatics.

  4. Leapfrogging. Two counter-rotating pairs in line. The rear pair is squeezed by the front pair's field, narrows, speeds up (since \(V \propto 1/d\)), overtakes through the middle, then widens and slows while the other catches up. They alternate indefinitely. This is the 2-D cross-section of two vortex rings chasing each other, and it is genuinely one of the prettiest results in classical fluid mechanics.

  5. Three vortices. Watch the trail. Three point vortices are integrable in general, but the motion is already intricate; nudge to four and the system is chaotic — one of the first recognised examples of Hamiltonian chaos.

  6. Watch the invariants. The Hamiltonian and the linear impulse hold to about one part in \(10^{13}\) over long runs. They are not enforced by the integrator; they are conserved because the equations have that structure and RK4 respects it well over these timescales. A drifting invariant would mean a bug.

What this model cannot do

Point vortices are singular: infinite velocity at the core, infinite kinetic energy, no viscosity. Consequently they never merge, never diffuse, and never dissipate. Real vortices with finite cores do all three. The model is exact for the motion of well-separated vortices and badly wrong once cores approach — a limitation worth keeping in view when a simulation shows two vortices orbiting at tiny separation at enormous speed.

Point vortices · Rankine vortex · Helmholtz theorems · Wingtip vortices · Conformal mapping & images