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首页> 外文期刊>Physics of plasmas >Hamiltonian canonical formulation of Hall-magnetohydrodynamics: Toward an application to weak turbulence theory
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Hamiltonian canonical formulation of Hall-magnetohydrodynamics: Toward an application to weak turbulence theory

机译：霍尔磁流体力学的哈密顿量正则表述：对弱湍流理论的应用

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The different levels of description of fluid media [e.g., magnetohydrodynamics (MHD), Hall-magnetohydrodynamics, bi-fluid,...] are commonly known under the form of Newtonian systems of equations. Nevertheless, this form proves to be ill-suited to derive a fully analytical weak turbulence theory of these media, due to the well-known complexity of the calculations implied. For such studies, therefore, a more appropriate mathematical frame needs to be found and this is shown to be the Hamiltonian formalism, even though it can often appear difficult to handle. The goal of this paper is to look for Hamiltonian formulations for the different levels of the fluid description of a plasma using the variational principle. Starting from the bi-fluid system, it is shown that such a formulation can be obtained by combining the Lagrangians already used for describing: (i) the motion of a charged particle in an electromagnetic field; (ii) the evolution of an electromagnetic field in presence of sources; (iii) the motion of a neutral fluid (Clebsch variables). The equivalence of the obtained description in terms of the generalized-Clebsch variables to the familiar Newtonian formulation is discussed. It is shown that each solution of the Hamiltonian system is also a solution for the Newtonian one, but that the converse is not true. The origin and the implication of this restriction are discussed. Reducing the Hamiltonian formulation obtained for the bi-fluid system to lower orders of the fluid approximations is then shown to be mandatory when one tries to obtain analytical results for linear waves and nonlinear wave-wave couplings. It is shown that this goal can be reached in two steps. The first one leads to a "reduced bi-fluid" system, which is identical to the bi-fluid one when the displacement current is neglected but the electron inertia is still working. The number of linear modes then goes down from six to three. The second step, leading to the Hall-MHD system, consists in neglecting the electron mass. It is demonstrated that the only four generalized Clebsch variables are sufficient to describe the full Hall-MHD dynamics. Some future applications of such a powerful formalism are outlined. (C) 2003 American Institute of Physics. [References: 29]
机译：在牛顿方程组的形式下，流体介质的不同描述水平[例如，磁流体动力学（MHD），霍尔磁流体力学，双流体等]是众所周知的。然而，由于所隐含的众所周知的计算复杂性，这种形式被证明不适合得出这些介质的完全分析性弱湍流理论。因此，对于此类研究，需要找到一个更合适的数学框架，这被证明是汉密尔顿式的形式主义，尽管通常看起来很难处理。本文的目的是使用变分原理为不同水平的血浆流体描述寻找哈密顿公式。从双流体系统开始，表明可以通过组合已经用于描述的拉格朗日方程获得这样的公式：（i）带电粒子在电磁场中的运动; （ii）在有辐射源的情况下电磁场的演变; （iii）中性流体的运动（Clebsch变量）。讨论了根据广义克莱布斯变量获得的描述与熟悉的牛顿公式的等价关系。结果表明，哈密顿系统的每个解也是牛顿解的一个解，但是反之则不成立。讨论了此限制的起源和含义。当人们试图获得线性波和非线性波-波耦合的分析结果时，必须将为双流体系统获得的哈密顿公式简化为较低的流体近似值是强制性的。结果表明，可以通过两个步骤实现此目标。第一个导致“减少双流体”系统，该系统与当忽略位移电流但电子惯性仍在起作用时的双流体系统相同。线性模式的数量从六个减少到三个。通往霍尔MHD系统的第二步是忽略电子质量。结果表明，仅有的四个广义Clebsch变量足以描述完整的Hall-MHD动力学。概述了这种强大的形式主义在未来的一些应用。 （C）2003美国物理研究所。 [参考：29]

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