The 3D scene shows vortex lines moving in the (x,y) plane but never draws the fluid's actual velocity field or checks the one property that defines a quantized vortex in the first place. This 2D companion computes both directly. Every vortex still obeys the real Hall–Vinen equation of motion — Biot–Savart induction from every other vortex, an opposite-sign image vortex enforcing the no-flow-through-wall boundary condition, and mutual friction with the co-rotating normal fluid — but now the resulting velocity field v(x,y) is evaluated on a grid and rendered as a live heatmap, and a draggable loop measures the discretized line integral around itself every frame:
v(x,y) = Σᵢ κ/(2π rᵢ²) ẑ×r̂ᵢ (Biot–Savart, real vortex i + its wall image)
Γ = ∮ v · dl (trapezoidal sum over 720 points on the loop)
Because each point vortex carries exactly one quantum of circulation κ = h/m₄, the residue-theorem-like result from 2D potential flow says the loop integral can only depend on how many real vortex cores it encloses — not on their exact positions, and not on the (equal-and-opposite) image charges sitting outside the container, since those never cross the loop:
Γ = n · κ (n = number of vortices strictly inside the loop)
Drag the loop across a vortex and Γ/κ jumps by exactly 1 the instant the core crosses the boundary — the discrete, topological signature of quantized circulation, verified here numerically frame by frame rather than assumed.
- Ω slider — sets the target vortex density via Feynman's rule nv = 2Ω/κ; changing it re-nucleates the array.
- T slider — sets the mutual-friction strength that crystallizes the array into a triangular lattice, and the two-fluid superfluid fraction shown for reference.
- Velocity-field heatmap — brighter regions show faster local superfluid flow; note how the field diverges (and reverses circulation sense) at each vortex core and cancels far from any core.
- Loop radius / drag — move the measurement loop to enclose more or fewer vortices and watch Γ/κ track the enclosed count exactly, with the small residual error coming only from the finite vortex-core softening used to avoid a literal 1/r singularity.
This is the same real-world quantization that lets neutron scattering and rotating-frame NMR experiments count vortices in liquid helium-4 indirectly — by measuring circulation, not by seeing the (nanometre-thin) cores themselves.