The kinetic origin of the fluid helicity—A symmetry in the kinetic phase space
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 thatmore »
- Authors:
-
[1];
[2]
- University of Tokyo, Chiba (Japan)
- University of Texas, Austin, TX (United States)
- Publication Date:
- Research Org.:
- Univ. of Texas, Austin, TX (United States)
- Sponsoring Org.:
- 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
- OSTI Identifier:
- 1978937
- Alternate Identifier(s):
- OSTI ID: 1843308
- Grant/Contract Number:
- FG02-04ER54742; 1440140; 17H01177; FG02-04ER-54742
- Resource Type:
- Accepted Manuscript
- Journal Name:
- Journal of Mathematical Physics
- Additional Journal Information:
- Journal Volume: 63; Journal Issue: 2; Journal ID: ISSN 0022-2488
- Publisher:
- American Institute of Physics (AIP)
- Country of Publication:
- United States
- Language:
- English
- Subject:
- 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; Hamiltonian mechanics; Vlasov equation; differentiable manifold; lie algebras; gauge symmetry; gauge group; fluid systems; fluid dynamics
Citation Formats
Yoshida, Zensho, and Morrison, Philip J. The kinetic origin of the fluid helicity—A symmetry in the kinetic phase space. United States: N. p., 2022.
Web. doi:10.1063/5.0050948.
Yoshida, Zensho, & Morrison, Philip J. The kinetic origin of the fluid helicity—A symmetry in the kinetic phase space. United States. https://doi.org/10.1063/5.0050948
Yoshida, Zensho, and Morrison, Philip J. Wed .
"The kinetic origin of the fluid helicity—A symmetry in the kinetic phase space". United States. https://doi.org/10.1063/5.0050948. https://www.osti.gov/servlets/purl/1978937.
@article{osti_1978937,
title = {The kinetic origin of the fluid helicity—A symmetry in the kinetic phase space},
author = {Yoshida, Zensho and Morrison, Philip J.},
abstractNote = {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.},
doi = {10.1063/5.0050948},
journal = {Journal of Mathematical Physics},
number = 2,
volume = 63,
place = {United States},
year = {Wed Feb 02 00:00:00 EST 2022},
month = {Wed Feb 02 00:00:00 EST 2022}
}
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Works referenced in this record:
A hierarchy of noncanonical Hamiltonian systems: circulation laws in an extended phase space
journal, May 2014
- Yoshida, Z.; J. Morrison, P.
- Fluid Dynamics Research, Vol. 46, Issue 3
Reduction of symplectic manifolds with symmetry
journal, February 1974
- Marsden, Jerrold; Weinstein, Alan
- Reports on Mathematical Physics, Vol. 5, Issue 1
Phase transitions and flux-loop metastable states in rotating turbulence
journal, October 2020
- Clark Di Leoni, P.; Alexakis, A.; Biferale, L.
- Physical Review Fluids, Vol. 5, Issue 10
Epi-Two-Dimensional Fluid Flow: A New Topological Paradigm for Dimensionality
journal, December 2017
- Yoshida, Z.; Morrison, P. J.
- Physical Review Letters, Vol. 119, Issue 24
Plasma in a monopole background does not have a twisted Poisson structure
journal, November 2019
- Lainz, Manuel; Sardón, Cristina; Weinstein, Alan
- Physical Review D, Vol. 100, Issue 10
Poisson brackets for fluids and plasmas
conference, January 1982
- Morrison, Philip J.
- AIP Conference Proceedings Volume 88
The local structure of Poisson manifolds
journal, January 1983
- Weinstein, Alan
- Journal of Differential Geometry, Vol. 18, Issue 3
Gauge-independent canonical formulation of relativistic plasma theory
journal, December 1984
- Bialynicki-Birula, Iwo; Hubbard, John C.; Turski, Lukasz A.
- Physica A: Statistical Mechanics and its Applications, Vol. 128, Issue 3
The Maxwell-Vlasov equations as a continuous hamiltonian system
journal, December 1980
- Morrison, Philip J.
- Physics Letters A, Vol. 80, Issue 5-6
Clebsch parameterization: Basic properties and remarks on its applications
journal, November 2009
- Yoshida, Z.
- Journal of Mathematical Physics, Vol. 50, Issue 11
The Hamiltonian description of incompressible fluid ellipsoids
journal, August 2009
- Morrison, P. J.; Lebovitz, Norman R.; Biello, Joseph A.
- Annals of Physics, Vol. 324, Issue 8
A Theorem on Force-Free Magnetic Fields
journal, June 1958
- Woltjer, L.
- Proceedings of the National Academy of Sciences, Vol. 44, Issue 6
Complete measurement of helicity and its dynamics in vortex tubes
journal, August 2017
- Scheeler, Martin W.; van Rees, Wim M.; Kedia, Hridesh
- Science, Vol. 357, Issue 6350
Adiabatic charged-particle motion
journal, January 1963
- Northrop, Theodore G.
- Reviews of Geophysics, Vol. 1, Issue 3
Moreau's hydrodynamic helicity and the life of vortex knots and links
journal, March 2018
- Irvine, William T. M.
- Comptes Rendus Mécanique, Vol. 346, Issue 3
Closure theory for the split energy-helicity cascades in homogeneous isotropic homochiral turbulence
journal, October 2017
- Briard, Antoine; Biferale, Luca; Gomez, Thomas
- Physical Review Fluids, Vol. 2, Issue 10
Hamiltonian moment reduction for describing vortices in shear
journal, August 1997
- Meacham, S. P.; Morrison, P. J.; Flierl, G. R.
- Physics of Fluids, Vol. 9, Issue 8
The Hamiltonian structure of the Maxwell-Vlasov equations
journal, March 1982
- Marsden, Jerrold E.; Weinstein, Alan
- Physica D: Nonlinear Phenomena, Vol. 4, Issue 3