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

Abstract

The Lagrangian mechanical consideration of the dynamics of ideal incompressible hydrodynamic, magnetohydrodynamic, and Hall magnetohydrodynamic media, which are formulated as dynamical systems in appropriate Lie groups equipped with Riemannian metrics, leads to the notion of generalized vorticities, as well as generalized coordinates, velocities, and momenta. The action of each system is conserved against the integral path variation in the direction of the generalized vorticity, and this invariance is associated with the particle relabeling symmetry. The generalized vorticities are formulated by the operation of integro-differential operators on the generalized velocities. The eigenfunctions of the operators provide sets of orthogonal functions, and we obtain common mathematical expressions concerning these dynamical systems using the orthogonoal functions. In particular, we find that the product of the Riemannian metric, g im , and the structure constants of the Lie group, C m jk , is given by the product of the eigenvalue of the operator, Λ(i), and a certain totally antisymmetric tensor, T ijk : g iα C α jk = Λ(i)T ijk . Its physical implications, including the weak interaction conjecture of MHD turbulence, are also discussed.

On magnetohydrodynamic gauge field theory

“…Tanehashi and Yoshida (2015), use the known Casimirs for barotropic MHD, and the non-canonical Poisson bracket of Greene (1980, 1982) to uncover gauge symmetries in MHD, by using a Clebsch variable expansion for both u and B. Araki (2016) provides an alternative viewpoint of fluid relabelling symmetries in MHD involving generalized vorticity and normal mode expansions for ideal incompressible fluids and MHD by using integro-differential operators acting on the generalized where the symbol means cyclically permute (b, u, Γ) and then sum. It is interesting to note that ∇ × Q = 0.…”

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

“…The details of the mathematical backgrounds were described in [13,14,25]. In this section, we introduce some mathematical notions and notation which was not explained in the previous studies but will use in the following sections.…”

“…As was discussed in [25], the dissipationless, incompressible HD, MHD, and HMHD systems have common mathematical structures. Thus, the formal derivations of differentialgeometrical properties and equations such as given in the previous two sections are also applicable to the Euler equation case.…”

“…The normal-mode expansion expression of the Levi-Civita connection equation (C.18) and the Riemannian curvature tensor equation ( 27) had its foundation on the formula that the combination of the Riemannian metric and the Lie bracket is equal to the product of the eigenvalue of the helicity-based, particle-relabeling operator and a certain totally antisymmetric tensor, Φ( k)| Φ( p), Φ( q) = λ( k) k p q . As was described in [25], this decomposition form ula is based on the general eigenvalue problem that derives the DBFs under appropriate boundary condition. Thus, the expression of the curvature tensor is applicable to the systems with arbitrary shape, for example, to cylindrical configuration.…”

“…As for the normal-mode analysis, it should be remarked that application of the geometrical method to the dissipationless, incompressible HD, MHD, and HMHD systems has its foundation on the following fact: these three systems were found to have common dynamical system features [25]; that is, each system has its own action-preserving integro-differential operator, and the corresponding eigenfunctions yield formally the same spectral representation as that of the related formulae and equations. In particular, we found that the product of the Riemannian metric g ij with the structure constants C i jk of the Lie group was given by the product of the eigenvalue Λ(i) of the operator with a certain totally antisymmetric tensor T ijk : g iα C α jk = Λ(i)T ijk .…”