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

Abstract

The formal stability analysis of Eulerian extended MHD (XMHD) equilibria is considered within the noncanonical Hamiltonian framework by means of the energy-Casimir variational principle and dynamically accessible stability method. Specifically, we find explicit sufficient stability conditions for axisymmetric XMHD and Hall MHD (HMHD) equilibria with toroidal flow and for equilibria with arbitrary flows under constrained perturbations. A Lyapunov functional that can potentially provide explicit stability criteria for generic equilibria under dynamically accessible variations is also obtained. Moreover, we examine the Lagrangian stability of the general quasi-neutral twofluid model written in terms of MHD-like variables, by finding the action and the Hamiltonian functionals of the linearized dynamics, working within a mixed Lagrangian-Eulerian framework. Upon neglecting electron mass we derive a HMHD energy principle and in addition the perturbed induction equation arises from Hamilton's equations of motion in view of a consistency condition for the relation between the perturbed magnetic potential and the canonical variables.

Lagrangian and Dirac constraints for the ideal incompressible fluid and magnetohydrodynamics

“…We mention a few. Since the Hamiltonian structure of extended and relativistic MHD are now at hand (Charidakos et al 2014;Abdelhamid, Kawazura & Yoshida 2015;D'Avignon, Morrison & Pegoraro 2015;D'Avignon, Morrison & Lingam 2016;Lingam, Miloshevich & Morrison 2016;Kaltsas, Throumoulopoulos & Morrison 2020) calculations analogous to those presented here can be done for a variety of magnetofluid models. Another valuable class of models that could be studied, ones that are known to have Lagrangian and Hamiltonian structure, are those with various finite-Larmor-radius effects (e.g.…”

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).…”

A class of three-dimensional gyroviscous magnetohydrodynamic models

“…Lastly, our model was centered around 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;Lingam et al 2015b;Abdelhamid et al 2015;Lingam et al 2015aD'Avignon et al 2016;Miloshevich et al 2017;Burby 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 Morrison et al (2014); Kaltsas et al (2017Kaltsas et al ( , 2018Kaltsas et al ( , 2020.…”