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 = 0 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.

The Godbillon–Vey invariant as a restricted Casimir of three-dimensional ideal fluids*

“…In the finite-dimensional rattleback case, perturbation of the system around the singular manifold leads to interesting dynamical properties [17]. Our own analysis of the Godbillon-Vey invariant elsewhere also suggests a strong connection to dynamics; GV provides a global and local obstruction to steady flow and can be used to estimate the rate of change of vorticity [8]. With that in mind, we suggest that flows with GV = 0 (or perturbations thereof) may prove particularly interesting from a dynamical perspective.…”

“…hence dη = β ∧ γ, for some one-form γ. GV can be thought of as helical compression of vortex lines [8], with η the local direction of vorticity compression.…”

“…foliations [12,13]; in ideal fluids it measures topological helical compression of vortex lines [8]. We show how GV fits naturally into the Lie-Poisson Hamiltonian formulation of ideal fluids [1] as a 'restricted Casimir' invariant.…”

“…Any Casimir that can be written as a regular integral invariant of all vorticity fields is equivalent to helicity [5][6][7]; accordingly, higher-order regular integral invariants can only be defined for special subclasses of vorticity fields. Here we study the Godbillon-Vey invariant (GV) which can be associated to a vorticity field tangent to a codimension-1 foliation [8][9][10][11]. GV originates in the theory of Original content from this work may be used under the terms of the Creative Commons Attribution 4.0 licence.…”