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2D Point-Vortex Dynamics

A set of ideal point vortices, each of fixed circulation Γa\Gamma_a, advecting one another through the velocity each induces (the 2D Biot-Savart law v⃗(p⃗)=∑aΓa2π z^×(p⃗−r⃗a)/∣p⃗−r⃗a∣2\vec v(\vec p)=\sum_a \frac{\Gamma_a}{2\pi}\,\hat z\times(\vec p-\vec r_a)/|\vec p-\vec r_a|^2). This is a Hamiltonian system: the total circulation, the linear and angular impulse, and the Kirchhoff-Routh Hamiltonian H=−14π∑a<bΓaΓbln⁡∣r⃗a−r⃗b∣H=-\frac1{4\pi}\sum_{a\lt b}\Gamma_a\Gamma_b\ln|\vec r_a-\vec r_b| are all conserved (Saffman; Aref). Two vortices of equal and opposite circulation a distance dd apart form a dipole that travels in a straight line at exactly v=Γ/2πdv=\Gamma/2\pi d; an equal co-rotating pair spins about its centroid; three or more give the integrable-to-chaotic vortex motion of Aref. Tracer particles ride the induced flow as streaklines. The headline readout is the relative drift of HH, held under 10−310^{-3} by the RK4 integrator.

Figure 1. Interacting 2D point vortices (warm = positive circulation, cool = negative) advecting one another via the Biot-Savart law; tracer particles trace the induced streaklines. A dipole translates at v=Γ/2πdv=\Gamma/2\pi d; co-rotating vortices orbit; three or more are integrable-to-chaotic. Method: RK4 integration of the Hamiltonian point-vortex equations (gate-tested sim.js), Canvas2D; the live readout is the relative drift of the conserved Hamiltonian.
configuration
strength1.00
speed3

WHAT TO TRY

  • Switch the configuration: a dipole (counter-rotating pair) translates in a straight line, a co-rotating pair orbits its centre, and the tripole and quadrupole trace richer Biot-Savart dances. Each is an exact point-vortex solution.
  • Watch the conserved quantities in the rail: total circulation, linear impulse and angular impulse stay fixed while the energy H barely drifts. That tiny H drift is the integrator error, the honest accuracy check.
  • Raise the strength or speed: the vortices induce faster motion on each other, but the conserved invariants hold, because point-vortex dynamics is Hamiltonian.