TY - JOUR
T1 - Conservation laws and evolution schemes in geodesic, hydrodynamic, and magnetohydrodynamic flows
AU - Markakis, Charalampos
AU - Uryu, Koji
AU - Gourgoulhon, Eric
AU - Nicolas, Jean Philippe
AU - Andersson, Nils
AU - Pouri, Athina
AU - Witzany, Vojtěch
N1 - Funding Information:
We thank Brandon Carter for pointing out the second line of Eq. (3.29) and a similar remark in the analogous equation for a fluid Killing tensor. We thank Theocharis Apostolatos, Jacob Bekenstein, Brandon Carter, Greg Comer, John Friedman, Roland Haas, David Hilditch, Darryl Holm, David Kaplan, Alan Kostelecky and Panagiotis Stavrinos for very fruitful discussions and comments. This work was supported by JSPS Grant-in-Aid for Scientific Research(C) 20540275, MEXT Grant-in-Aid for Scientific Research on Innovative Area 20105004, the Greek State Scholarships Foundation (IKY), NSF Grant No. PHY1001515, DFG grant SFB/Transregio 7 “Gravitational Wave Astronomy,” STFC Grant No. PP/E001025/1 and ANR Grant No. 06-2-134423 Méthodes mathématiques pour la relativité générale . K. U. and E. G. acknowledge support from a JSPS Invitation Fellowship for Research in Japan (short-term) and the invitation program of foreign researchers at the Paris observatory. C. M. and J. P. N. thank the Paris Observatory for hospitality during the course of this work.
Publisher Copyright:
© 2017 American Physical Society.
PY - 2017/9/13
Y1 - 2017/9/13
N2 - 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.
AB - 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.
UR - http://www.scopus.com/inward/record.url?scp=85031717851&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=85031717851&partnerID=8YFLogxK
U2 - 10.1103/PhysRevD.96.064019
DO - 10.1103/PhysRevD.96.064019
M3 - Article
AN - SCOPUS:85031717851
SN - 2470-0010
VL - 96
JO - Physical Review D
JF - Physical Review D
IS - 6
M1 - 064019
ER -