Linear stability theory for extended magnetohydrodynamics (XMHD), which incorporates the Hall and electron-inertia effects into MHD, is developed by introducing Lagrangian displacement fields for ions and electrons. For simplicity, incompressible and dissipationless XMHD is assumed in this work, since the present consideration is greatly inspired by an analogy with incompressible ideal fluid. The governing equations for the Lagrangian displacement fields are written as a gyroscopic dynamical system, to which the well-established canonical Hamiltonian theory is ready to apply. This linear perturbation can be further restricted to the isovortical one in the same manner as ideal fluid, which is more generally understood as dynamically accessible perturbation in a constrained Hamiltonian system. In XMHD, there are two isovortical constraints corresponding to the two canonical vorticities for ions and electrons. It is shown that the Frieman–Rotenberg equation for ideal MHD is reproduced by not only neglecting the Hall and electron-inertia effects but also imposing one of the two isovortical constraints. As an application, a new stability condition for static equilibria of inertial MHD is derived.

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关于拉格朗日位移场和等涡动扰动的扩展磁流体动力学线性化动力学系统

通过引入离子和电子的拉格朗日位移场，开发了扩展的磁流体动力学（XMHD）线性稳定性理论，该理论将霍尔和电子惯性效应纳入了MHD。为简单起见，在这项工作中假设使用不可压缩且无耗散的XMHD，因为当前的考虑很大程度上是通过与不可压缩的理想流体进行类比得出的。拉格朗日位移场的控制方程写为陀螺动力学系统，已建立的规范的哈密顿理论已准备好应用到该系统中。可以以与理想流体相同的方式，将该线性摄动进一步限制为等涡动摄动，更通常地，将其理解为受约束的哈密顿系统中的动态可访问的摄动。在XMHD中 有两个等离子约束对应于离子和电子的两个经典涡旋。结果表明，理想MHD的Frieman-Rotenberg方程不仅通过忽略霍尔和电子惯性效应，而且还施加了两个等张约束之一来重现。作为一种应用，推导了惯性MHD静平衡的新稳定条件。