Canonical Hamiltonian mechanics of Hall magnetohydrodynamics and its limit to ideal magnetohydrodynamics

Abstract

While a microscopic system is usually governed by canonical Hamiltonian mechanics, that of a macroscopic system is often noncanonical, reflecting a degenerate Poisson structure underlying the coarse-grained phase space. Probing into symplectic leaves (local structures in a foliated phase space), we may be able to elucidate the order of transition from micro to macro. The Lagrangian guides our analysis. We formulate canonized Hamiltonian systems of Hall magnetohydrodynamics (HMHD) which have a hierarchized set of canonical variables; the simplest system is the subclass in which the ion vorticity and magnetic field have integral surfaces. Renormalizing the singularity scaled by the reciprocal Hall parameter (as the ion vorticity surfaces and the magnetic surfaces are set to merge), we delineate the singular limit to ideal magnetohydrodynamics (MHD). The formulated canonical equations will be useful in the study of ordered structures and dynamics (with integrable vortex lines) in HMHD and their singular limit to MHD, such as magnetic confinement systems, shocks or vortical dynamics.

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

“…Since the linear stability problem (24) requires two V -variables as initial conditions, we can treat the following two kinds of problems as special cases to the extent that the existence and uniqueness of solutions of Equations (19) and (20) are guaranteed. (ii) By setting t ( ξ(0) η(0)) = t ( ξ 0 ∇ ξ 0 V 0 ), we can obtain a perturbation that satisfies v(0) = 0, i.e., the initial ion and electron velocities are unchanged.…”

Ertel's vorticity theorem and new flux surfaces in multi-fluid plasmas

“…(39) represents the potential energy of the charged plasma particles in the presence of the electric potential U. But viewing U as a Lagrange multiplier to be varied imposes the quasi neutrality condition (37). Likewise, the variation ofL with respect to A yields the pre-Maxwell equation (36), and the variation with respect to P j yields…”

Hamiltonian closures in fluid models for plasmas

“…Hall MHD [63,64], where in the latter reference, a Lagrangian approach in terms of Clebsch variables is proposed.…”

“…[49] for ideal MHD in the barotropic limit. In the limit d e = 0, on the other hand, one retrieves the Hamiltonian structure of Hall MHD [64,81,82]. An action principle derivation of the extended MHD Poisson bracket was provided in Refs.…”