Differential-geometrical approach to the dynamics of dissipationless incompressible Hall magnetohydrodynamics: II. Geodesic formulation and Riemannian curvature analysis of hydrodynamic and magnetohydrodynamic stabilities

Abstract

Abstract. In this study, the dynamics of a dissipationless incompressible Hall magnetohydrodynamic (HMHD) medium are formulated as geodesics on a direct product of two volume-preserving diffeomorphism groups. Formulations are given for the geodesic and Jacobi equations based on a linear connection with physically desirable properties, which agrees with the Levi-Civita connection. Derivations of the explicit normal-mode expressions for the Riemannian metric, Levi-Civita connection, and related formulae and equations are also provided using the generalized Elsässer variables (GEVs). Examinations of the stabilities of the hydrodynamic (HD, α = 0) and magnetohydrodynamic (MHD, α → 0) motions and the O(α) Hall-term effect in terms of the Jacobi equation and the Riemannian sectional curvature tensor are presented, where α represents the Hall-term strength parameter. It is very interesting that the sectional curvatures of the MHD and HMHD systems between two GEV modes were found to take both the positive (stable) and negative (unstable) values, while that of the HD system between two complex helical waves was observed to be negative definite. Moreover, for the MHD case, negative sectional curvatures were found to occur only when mode interaction was "local," i.e., the wavenumber moduli of the main flow (say p) and perturbation (say k) were relatively close to each other. However, in the nonlocal limit (k p or k p), the sectional curvatures were always positive. This result leads to the conjecture that the MHD interactions mainly excite wavy or non-growing motions; however, some local interactions cause dynamical instability that leads to chaotic or turbulent plasma motions. Additionally, it was found that the tendencies of the O(α) effects are opposite between the ion cyclotron and whistler modes. Comparison with the energy-Casimir method is also discussed using a remarkable constant of motion which relates the Riemannian curvature to the second variation of the Hamiltonian.

A Note on Stability Analysis by the Second Variations of Lagrangian and Hamiltonian for Ideal Incompressible Plasmas

“…The expressions formally agree with those found in the Section 5.3 of Ref. [4], where the results were limited to the HMHD system and the mode expansion by the generalized Elsässer variables. According to our consideration made in Ref.…”

“…As aforementioned in Ref. [4] in terms of sectional curvature analysis, the second term can be positive or negative.…”

“…is used to derive the second order perturbation of the velocity field, where [ * , * ] is the Lie bracket of Lie algebra, 4 which satisfies antisymmetry and the Jacobi identity:…”

“…Differential geometrical approaches to the dynamics of continuous media have revealed general and fundamental mathematical structure [1,2]. Regarding plasma physics, the dynamics of dissipationless incompressible magnetohydrodynamic (MHD) and Hall magnetohydrodynamic (HMHD) media have been commonly described as geodesics on some appropriate Lie groups [3,4]. These dynamical systems are defined by the pair of the appropriate Riemannian metric (or inner product) and the Lie bracket of basic variables.…”

Particle-relabeling symmetry, generalized vorticity, and normal-mode expansion of ideal incompressible fluids and plasmas in three-dimensional space

“…Group theoretic stability anlyses of homogeneous, isotropic MHD and HMHD turbulence conjecture that unstable interactions are limited to "local" interactions in wavenumber space, while non-local interactions are stable and have a "wavy" nature. 15 Investigations related to the helicity conservation laws have a very long history and contain a wide variety of topics, 16 for example, their relationship to the topological nature of flow fields 17 and their influence on the dynamics of the plasma equilibrium state 18 or turbulent phenomena. 19 Of the numerous studies that have been conducted, helicity conservation, in particular, has been discussed from the viewpoint of relabeling symmetry by Salmon.…”