  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.
