The Godbillon-Vey Invariant as Topological Vorticity Compression and Obstruction to Steady Flow in Ideal Fluids

Abstract

If the vorticity field of an ideal fluid is tangent to a foliation, additional conservation laws arise. For a class of zero-helicity vorticity fields the Godbillon-Vey (GV) invariant of foliations is defined and is shown to be an invariant purely of the vorticity, becoming a higher-order helicity-type invariant of the flow. GV non-zero gives both a global topological obstruction to steady flow and, in a particular form, a local obstruction. GV is interpreted as helical compression and stretching of vortex lines. Examples are given where the value of GV is determined by a set of distinguished closed vortex lines.

arXiv comment14 pages, 2 figures. V2: Changes to discussion of local conservation law, non-unique $\theta$

References in corpus (12)

• Helicity is the only integral invariant of volume-preserving transformations
• Local and Nonlocal Advected Invariants and Helicities in Magnetohydrodynamics and Gas Dynamics I: Lie Dragging Approach
• Helicity is the only invariant of incompressible flows whose derivative is continuous in $C^1$-topology
• The Godbillon-Vey Invariant as a Restricted Casimir of Three-dimensional Ideal Fluids
• The third order helicity of magnetic fields via link maps
• A new cohomological formula for helicity in ${\mathbb{R}}^{2k+1}$ reveals the effect of a diffeomorphism on helicity
• Pontryagin invariants and integral formulas for Milnor's triple linking number
• A note on the stationary Euler equations of hydrodynamics
• On volume-preserving vector fields and finite type invariants of knots
• The third order helicity of magnetic fields via link maps II
• Godbillon-Vey Helicity and Magnetic Helicity in Magnetohydrodynamics
• Generalized Gauss maps and integrals for three-component links: toward higher helicities for magnetic fields and fluid flows, Part 2

Cited by in corpus (1)

• The Godbillon-Vey Invariant as a Restricted Casimir of Three-dimensional Ideal Fluids