Navier-Stokes Threading on CRT Torus

Visualizing incompressible flow on the modular manifold (p×q torus)

Show Vorticity (ω)
Show Threading Paths
Show Torus Grid
Legend
Threading Paths
Torus Wireframe
Key Insights
  • Triads close on torus: k₁+k₂+k₃≡0 (mod M)
  • Incompressibility locks: ∇·u=0 preserved mod p,q
  • Threading deficit decays: D_j ~ e^(-ν·2³ʲ·t)
  • Entropy bounded: H_j ≤ H₀ + C/(ν·2⁴ʲ)
  • Ω₃ attractor: Flow seeks optimal ratio ≈1.466
  • Full paper.

Watch how vorticity (red) threads along divergence-free cycles without blowing up.