Helically symmetric extended magnetohydrodynamics: Hamiltonian formulation and equilibrium variational principles

Abstract

Hamiltonian extended magnetohydrodynamics (XMHD) is restricted to respect helical symmetry by reducing the Poisson bracket for 3D dynamics to a helically symmetric one, as an extension of the previous study for translationally symmetric XMHD (D.A. Kaltsas et al, Phys. Plasmas 24, 092504 (2017)). Four families of Casimir invariants are obtained directly from the symmetric Poisson bracket and they are used to construct Energy-Casimir variational principles for deriving generalized XMHD equilibrium equations with arbitrary macroscopic flows. The system is then cast into the form of Grad-Shafranov-Bernoulli equilibrium equations. The axisymmetric and the translationally symmetric formulations can be retrieved as geometric reductions of the helical symmetric one. As special cases, the derivation of the corresponding equilibrium equations for incompressible plasmas is discussed and the helically symmetric equilibrium equations for the Hall MHD system are obtained upon neglecting electron inertia. An example of an incompressible double-Beltrami equilibrium is presented in connection with a magnetic configuration having non-planar helical magnetic axis. †

Hamiltonian kinetic-Hall magnetohydrodynamics with fluid and kinetic ions in the current and pressure coupling schemes

“…The process of deriving these relations has been described several times before, e.g. by Andreussi, Morrison & Pegoraro (2010), Kaltsas, Throumoulopoulos & Morrison (2017), Grasso et al (2017) and Kaltsas, Throumoulopoulos & Morrison (2018). Following the same procedure here, we find…”

“…To obtain a Hamiltonian formulation in the symmetric case we need to translate this field decomposition in a decomposition of the vector functional derivatives in terms of functional derivatives with respect to the scalar potentials χ, ψ, Υ. The process of deriving these relations has been described several times before, e.g., in [50,51,52,53]. Following the same procedure here, we find…”

A class of three-dimensional gyroviscous magnetohydrodynamic models

“…Our model was centred on the introduction of gyroviscosity into the ideal MHD model. However, given that several variants of extended MHD possess Lagrangian and Hamiltonian formulations (Keramidas Charidakos et al 2014; Abdelhamid, Kawazura & Yoshida 2015; Lingam, Morrison & Miloshevich 2015 a ; Lingam, Morrison & Tassi 2015 b ; D'Avignon, Morrison & Lingam 2016; Lingam, Abdelhamid & Hudson 2016 a ; Lingam, Miloshevich & Morrison 2016 b ; Burby 2017; Miloshevich, Lingam & Morrison 2017), it would seem natural to utilize the gyromap and thus formulate the gyroviscous contributions for this class of models; after doing so, their equilibria and stability can be obtained by using the HAP approach along the lines of Andreussi et al (2010, 2012, 2013, 2016), Morrison et al (2014) and Kaltsas, Throumoulopoulos & Morrison (2017, 2018, 2020) where the stability of a variety of equilibria is analysed using Lagrangian, energy–Casimir and dynamically accessibility methods. Likewise, this approach could also be extended to relativistic MHD and XMHD models with HAP formulations (D'Avignon, Morrison & Pegoraro 2015; Grasso et al 2017; Kawazura, Miloshevich & Morrison 2017; Coquinot & Morrison 2020; Ludwig 2020).…”

Energy-Casimir, dynamically accessible, and Lagrangian stability of extended magnetohydrodynamic equilibria

“…The vanishing of the first order variation of the EC functional, i.e., δH C = δ(H − i C i ) = 0 yields the EC equilibrium equations, given by Eqs. (4.25)-(4.31) of [39] with = 0, n = −1 therein, which can be written in a Grad-Shafranov-Bernoulli form (see Eqs. (5.1)-(5.4) in the same reference).…”