2D Point-Vortex Dynamics

A set of ideal point vortices, each of fixed circulation $\Gamma_a$, advecting one another through the velocity each induces (the 2D Biot-Savart law $\vec v(\vec p)=\sum_a \frac{\Gamma_a}{2\pi}\,\hat z\times(\vec p-\vec r_a)/|\vec p-\vec r_a|^2$). This is a Hamiltonian system: the total circulation, the linear and angular impulse, and the Kirchhoff-Routh Hamiltonian $H=-\frac1{4\pi}\sum_{a\lt b}\Gamma_a\Gamma_b\ln|\vec r_a-\vec r_b|$ are all conserved (Saffman; Aref). Two vortices of equal and opposite circulation a distance $d$ apart form a dipole that travels in a straight line at exactly $v=\Gamma/2\pi d$; 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 $H$, held under $10^{-3}$ by the RK4 integrator.