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Conservation laws and evolution schemes in geodesic, hydrodynamic, and magnetohydrodynamic flows

Conservation laws and evolution schemes in geodesic, hydrodynamic, and magnetohydrodynamic flows
Conservation laws and evolution schemes in geodesic, hydrodynamic, and magnetohydrodynamic flows
Carter and Lichnerowicz have established that barotropic fluid flows are conformally geodesic and obey Hamilton's principle. This variational approach can accommodate neutral, or charged and poorly conducting, fluids. We show that, unlike what has been previously thought, this approach can also accommodate perfectly conducting magnetofluids, via the Bekenstein-Oron description of ideal magnetohydrodynamics. When Noether symmetries associated with Killing vectors or tensors are present in geodesic flows, they lead to constants of motion polynomial in the momenta. We generalize these concepts to hydrodynamic flows. Moreover, the Hamiltonian descriptions of ideal magnetohydrodynamics allow one to cast the evolution equations into a hyperbolic form useful for evolving rotating or binary compact objects with magnetic fields in numerical general relativity. In this framework, Ertel's potential vorticity theorem for baroclinic fluids arises as a special case of a conservation law valid for any Hamiltonian system. Moreover, conserved circulation laws, such as those of Kelvin, Alfvén and Bekenstein-Oron, emerge simply as special cases of the Poincaré-Cartan integral invariant of Hamiltonian systems. We use this approach to obtain an extension of Kelvin's theorem to baroclinic (nonisentropic) fluids, based on a temperature-dependent time parameter. We further extend this result to perfectly or poorly conducting baroclinic magnetoflows. Finally, in the barotropic case, such magnetoflows are shown to also be geodesic, albeit in a Finsler (rather than Riemann) space.
2470-0029
Markakis, Charalampos
8c2d83e9-d3f9-45ba-ab26-1e6023f9bbcc
Uryu, Koji
d75ab5b6-4112-4264-8482-eb4cbb69d8d6
Gourgoulhon, Eric
d9fc53cc-7125-483c-b6a6-9c20884f79b7
Nicolas, Jean-Philippe
b464a108-157b-4608-a5a5-96250b5555b0
Andersson, Nils
2dd6d1ee-cefd-478a-b1ac-e6feedafe304
Pouri, Athina
d2ade2be-6a8f-47a0-b42f-7b46e39d564f
Witzany, Vojtěch
357f122e-43a1-48c4-98b2-e22fe2dde9da
Markakis, Charalampos
8c2d83e9-d3f9-45ba-ab26-1e6023f9bbcc
Uryu, Koji
d75ab5b6-4112-4264-8482-eb4cbb69d8d6
Gourgoulhon, Eric
d9fc53cc-7125-483c-b6a6-9c20884f79b7
Nicolas, Jean-Philippe
b464a108-157b-4608-a5a5-96250b5555b0
Andersson, Nils
2dd6d1ee-cefd-478a-b1ac-e6feedafe304
Pouri, Athina
d2ade2be-6a8f-47a0-b42f-7b46e39d564f
Witzany, Vojtěch
357f122e-43a1-48c4-98b2-e22fe2dde9da

Markakis, Charalampos, Uryu, Koji, Gourgoulhon, Eric, Nicolas, Jean-Philippe, Andersson, Nils, Pouri, Athina and Witzany, Vojtěch (2017) Conservation laws and evolution schemes in geodesic, hydrodynamic, and magnetohydrodynamic flows. Physical Review D, 96 (064019), [064019]. (doi:10.1103/PhysRevD.96.064019).

Record type: Article

Abstract

Carter and Lichnerowicz have established that barotropic fluid flows are conformally geodesic and obey Hamilton's principle. This variational approach can accommodate neutral, or charged and poorly conducting, fluids. We show that, unlike what has been previously thought, this approach can also accommodate perfectly conducting magnetofluids, via the Bekenstein-Oron description of ideal magnetohydrodynamics. When Noether symmetries associated with Killing vectors or tensors are present in geodesic flows, they lead to constants of motion polynomial in the momenta. We generalize these concepts to hydrodynamic flows. Moreover, the Hamiltonian descriptions of ideal magnetohydrodynamics allow one to cast the evolution equations into a hyperbolic form useful for evolving rotating or binary compact objects with magnetic fields in numerical general relativity. In this framework, Ertel's potential vorticity theorem for baroclinic fluids arises as a special case of a conservation law valid for any Hamiltonian system. Moreover, conserved circulation laws, such as those of Kelvin, Alfvén and Bekenstein-Oron, emerge simply as special cases of the Poincaré-Cartan integral invariant of Hamiltonian systems. We use this approach to obtain an extension of Kelvin's theorem to baroclinic (nonisentropic) fluids, based on a temperature-dependent time parameter. We further extend this result to perfectly or poorly conducting baroclinic magnetoflows. Finally, in the barotropic case, such magnetoflows are shown to also be geodesic, albeit in a Finsler (rather than Riemann) space.

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Accepted/In Press date: 17 August 2017
e-pub ahead of print date: 13 September 2017

Identifiers

Local EPrints ID: 414886
URI: http://eprints.soton.ac.uk/id/eprint/414886
ISSN: 2470-0029
PURE UUID: e6d4cbed-49b1-46bc-a53d-f1e7b53312fa
ORCID for Nils Andersson: ORCID iD orcid.org/0000-0001-8550-3843

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Date deposited: 13 Oct 2017 16:30
Last modified: 10 Dec 2019 01:51

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Contributors

Author: Charalampos Markakis
Author: Koji Uryu
Author: Eric Gourgoulhon
Author: Jean-Philippe Nicolas
Author: Nils Andersson ORCID iD
Author: Athina Pouri
Author: Vojtěch Witzany

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