Hamiltonian magnetohydrodynamics: Helically symmetric formulation, Casimir invariants, and equilibrium variational principles

Abstract

The noncanonical Hamiltonian formulation of magnetohydrodynamics (MHD) is used to construct variational principles for symmetric equilibrium configurations of magnetized plasma including flow. In particular, helical symmetry is considered and results on axial and translational symmetries are retrieved as special cases of the helical configurations. The symmetry condition, which allows the description in terms of a magnetic flux function, is exploited to deduce a symmetric form of the noncanonical Poisson bracket of MHD. Casimir invariants are then obtained directly from the Poisson bracket. Equilibria are obtained from an energy-Casimir principle and reduced forms of this variational principle are obtained by the elimination of algebraic constraints. * Electronic address: t.andreussi@alta-space.com †

Hamiltonian magnetohydrodynamics: Lagrangian, Eulerian, and dynamically accessible stability—Theory

“…The usual equations of motion for Z follow from either of the expressions in terms of q or Q (see [2]). The notation ∂/∂t will be used to denote differentiation of Eulerian quantities at fixed x.…”

“…Using the notation of Ref. [2] the set of Eulerian variables for MHD is denoted by Z := (ρ, v, s, B), or alternatively Z := (ρ, M := ρv, σ := ρs, B), with the map from the Lagrangian variables (q, π) to Eulerian variables Z given…”

“…Whereas in Refs. [1,2] the subject matter concerned the construction and origin of variational principles for equilibria, here we present the comprehensive approach to stability of magnetohydrodynamics (MHD) equilibria that is a direct consequence of the Hamiltonian nature of this system. The presentation organizes scattered approaches into the cohesive Hamiltonian framework, which will be seen to be useful for obtaining, interpreting, and comparing stability results Ultimately, the stability results we consider, which are a consequence of the Hamiltonian form, have their origin in two energy theorems of mechanics: Lagrange's theorem and Dirichlet's theorem (see Ref.…”

“…[2] and its predecessor Ref. [1], we explore further ramifications of the Hamiltonian nature of ideal magnetohydrodynamics (MHD).…”

“…A main goal of this paper is to explore the consequences for stability for such different ways of enforcing constraints. The results of [1,2] will be used in Secs. II, III, and IV to construct three kinds of energy principles for stability of both static and stationary MHD equilibria.…”