Topological Fluid Dynamics: Euler Number & Helicity

H = ∫ v·ω dV — helicity counts vortex linkage. A topological invariant of ideal flow.

Topology
Vortex strength Γ 1.0
Ring radius R 0.8
Animation speed 1.0
Helicity H = ∫ v·ω dV = Σ Γᵢ·Γⱼ·Lk(Cᵢ,Cⱼ) where Lk is the Gauss linking number. Moffatt (1969): helicity is an ideal-MHD/Euler invariant = topological degree of flow. Trefoil vortex: H = Γ²·(writhe + twist). Hopf fibration: every fiber links every other fiber exactly once. Dissipation (reconnection events) changes topology and helicity.