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

机译：测地，流体动力学和磁力流体流动中的保护法和演化方案

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摘要

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 mag-netohydrodynamics. 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, Alfven and Bekenstein-Oron, emerge simply as special cases of the Poincare-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.
机译：Carter和Lichnerowicz已经确定了波奇渗透压流体流动是一般的Geodesic和Obey Hamilton的原则。这种变分别方法可以适应中性或充电，导电差，流畅。我们表明，与以前认为的不同之处，这种方法也可以通过Bekenstein-Oron描述理想的Mag-Netohydroynamics描述完全进行磁流体流体。当与杀死载体或张量相关联的NoEther对称存在于测量流中时，它们在动矩中导致运动多项式的常数。我们将这些概念概括为流体动力流动。此外，理想磁流动动力学的Hamiltonian描述允许人们将演化方程铸造成用于在数值一般相对性中具有磁场的旋转或二进制紧凑型对象的双曲形式。在该框架中，氨基氯液的潜在涡流定理是对任何哈密顿系统有效的保护法的特殊情况。此外，保守的流通法则，例如Kelvin，Alfven和Bekenstein-Oron的普遍存在Hamiltonian系统的庞然军车内不变性的特殊情况。我们使用这种方法基于温度依赖的时间参数来获得Kelvin定理的延伸至氨基甲基（非熵）流体。我们进一步扩展了这一结果，以完全或不良导电的雄心磁性制剂。最后，在波衡器壳体中，这种磁铁显示在芬德勒（而不是Riemann）空间中也是测地的。

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  • 来源
    《Physical Review D》 |2017年第8期|064019.1-064019.26|共26页
  • 作者

    Charalampos Markakis; Koji Uryu; Eric Gourgoulhon; Jean-Philippe Nicolas; Nils Andersson; Athina Pouri; Vojtech Witzany;

  • 作者单位

    Mathematical Sciences University of Southampton Southampton SO1 7 1BJ United Kingdom NCSA University of Illinois at Urbana-Champaign Illinois 61801 USA;

    Department of Physics University of the Ryukyus Senbaru Nishihara Okinawa 903-0213 Japan;

    LUTh UMR 8102 du CNRS Observatoire de Paris Universite Paris Diderot F-92190 Meudon France;

    Departement de Mathematiques Universite de Bretagne Occidentale 6 avenue Victor Le Gorgeu 29238 Brest Cedex 3 France;

    Mathematical Sciences University of Southampton Southampton SO1 7 1BJ United Kingdom;

    RCAAM Academy of Athens Soranou Efesiou 4 11527 Athens Greece;

    ZARM Universitaet Bremen Am Fallturm 28359 Bremen Germany;

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