The kinetic origin of the fluid helicity -- a symmetry in the kinetic phase space

TL;DR

This paper explores the kinetic origins of fluid helicity, revealing how helicity conservation depends on symmetry conditions in the kinetic phase space and how it can be violated in a kinetic description.

Contribution

It demonstrates the geometric and symmetry-based conditions under which helicity is conserved or broken in kinetic and fluid models.

Findings

Helicity is conserved under specific symmetry conditions in the Vlasov system.

Breaking helicity symmetry leads to changes in helicity in kinetic descriptions.

Helicity symmetry relates to a coordinate condition in special flow classes.

Abstract

Helicity, a topological degree that measures the winding and linking of vortex lines, is preserved by ideal (barotropic) fluid dynamics. In the context of the Hamiltonian description, the helicity is a Casimir invariant characterizing a foliation of the associated Poisson manifold. Casimir invariants are special invariants that depend on the Poisson bracket, not on the particular choice of the Hamiltonian. The total mass (or particle number) is another Casimir invariant, whose invariance guarantees the mass (particle) conservation (independent of any specific choice of the Hamiltonian). In a kinetic description (e.g. that of the Vlasov equation), the helicity is no longer an invariant (although the total mass remains a Casimir of the Vlasov's Poisson algebra). The implication is that some "kinetic effect" can violate the constancy of the helicity. To elucidate how the helicity constraint emerges or submerges, we examine the fluid reduction of the Vlasov system; the fluid (macroscopic) system is a "sub-algebra" of the kinetic (microscopic) Vlasov system. In the Vlasov system, the helicity can be conserved, if a special helicity symmetry condition holds. To put it another way, breaking helicity symmetry induces a change in the helicity. We delineate the geometrical meaning of helicity symmetry, and show that, for a special class of flows (so-called epi-2 dimensional flows), the helicity symmetry is written as $\partial_\gamma =0$ for a coordinate $\gamma$ of the configuration space.