The kinetic origin of the fluid helicity—A symmetry in the kinetic phase space

Journal Article · 01 February 2022 · Journal of Mathematical Physics

DOI: https://doi.org/10.1063/5.0050948 · OSTI ID:1978937

Yoshida, Zensho  ^[1]; Morrison, Philip J.  ^[2]

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1. University of Tokyo, Chiba (Japan); Univ. of Tokyo (Japan)
2. University of Texas, Austin, TX (United States)

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 (the so-called epi-two-dimensional flows), the helicity symmetry is written as ∂γ = 0 for a coordinate γ of the configuration space.

• Research Organization: University of Texas, Austin, TX (United States)
• Sponsoring Organization: USDOE Office of Science (SC), Fusion Energy Sciences (FES); National Science Foundation (NSF); Japan Society for the Promotion of Science (JSPS); Alexander von Humboldt Foundation
• Grant/Contract Number: FG02-04ER54742
• OSTI ID: 1978937
• Journal Information: Journal of Mathematical Physics, Journal Name: Journal of Mathematical Physics Journal Issue: 2 Vol. 63; ISSN 0022-2488
• Publisher: American Institute of Physics (AIP)Copyright Statement
• Country of Publication: United States
• Language: English