On the kinetic theory origin of fluid helicity

Helicity,  a topological measure of the winding and linking of vortex lines,  is preserved by ideal fluid dynamics.   In the Hamiltonian description,  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 is another Casimir invariant, whose invariance guarantees the mass conservation. In a kinetic description (e.g. the Vlasov equation),  the helicity is no longer an invariant (although the total mass/particle number remains one in the Vlasov Poisson algebra).  Thus,  some "kinetic effect"  violates the constancy of the helicity. To see how the helicity constraint emerges or submerges, we examine the fluid reduction of the Vlasov system; the fluid system is a "sub-algebra" of the kinetic  Vlasov system. In the Vlasov system, the helicity can be conserved, if a special helicity symmetry condition holds -- breaking  helicity symmetry induces a change in the helicity. We delineate the geometrical meaning of  helicity symmetry, and show for a special class of flows how to explicitly write the symmetry.  Poster based on arXiv:2103.03990v1.