Thomas Machon

Topological Vorticity Compression in Ideal Fluids

We show how an additional conserved quantity — the Godbillon-Vey Invariant (GV) — arises for three-dimensional ideal fluids whose vorticity field can be related to a codimension-1 foliation. GV is invariant under volume-preserving diffeomorphisms and we interpret it as a topological measure of helical vortex compression. We give examples of fluid flows on S3 where the value of GV is determined by a set of distinguished closed vortex lines. We show that GV being non-zero provides both local and global obstructions to steady flow. Our condition for GV to be defined requires the helicity to vanish and we show how this hierarchical structure fits naturally into the Hamiltonian formulation of ideal fluids, interpreting GV as a ‘restricted’ Casimir invariant.