Linear and nonlinear stability criteria for compressible MHD flows in a gravitational field

Moawad, S. M.

doi:10.1017/s0022377813000627

Cited by 14 publications

(14 citation statements)

References 71 publications

(42 reference statements)

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“…[39] the authors have used a form of Noether's theorem in an action principle setting of MHD to compare Lagrangian and dynamically accessible perturbations, while in Ref. [40] the author considers a case of energy-Casimir stability that is in the same vein as that of our Sec. III.…”

Section: Discussionmentioning

confidence: 99%

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

2013

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Stability conditions of magnetized plasma flows are obtained by exploiting the Hamiltonian structure of the magnetohydrodynamics (MHD) equations and, in particular, by using three kinds of energy principles. First, the Lagrangian variable energy principle is described and sufficient stability conditions are presented. Next, plasma flows are described in terms of Eulerian variables and the noncanonical Hamiltonian formulation of MHD is exploited. For symmetric equilibria, the energy-Casimir principle is expanded to second order and sufficient conditions for stability to symmetric perturbation are obtained. Then, dynamically accessible variations, i.e. variations that explicitly preserve invariants of the system, are introduced and the respective energy principle is considered. General criteria for stability are obtained, along with comparisons between the three different approaches.

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Section: Discussionmentioning

confidence: 99%

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

2013

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“…(2008), Andreussi et al. (2010, 2012), Moawad (2013), Morrison, Lingam & Acevedo (2014) and Kaltsas et al. (2017).…”

Section: Casimir Invariants and Equilibrium Variational Principle Witmentioning

confidence: 97%

“…With the helically symmetric Casimirs at hand, we can build the EC variational principle to obtain equilibrium conditions. For analogous utilizations of this methodology for symmetric or 2D plasmas the reader is referred to Holm et al (1985); Almaguer et al (1988); Andreussi & Pegoraro (2008); Tassi et al (2008); Andreussi et al (2010Andreussi et al ( , 2012; Moawad (2013); Morrison et al (2014); Kaltsas et al (2017). As mentioned in Sec.…”

Section: Equilibrium Variational Principle With Helical Symmetrymentioning

confidence: 99%

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Helically symmetric extended magnetohydrodynamics: Hamiltonian formulation and equilibrium variational principles

2018

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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. †

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“…Detailed consequences of the original noncanonical Hamiltonian structure of Morrison and Greene [12], were explored in a series of papers [14][15][16][17][18], including various variational principles for equilibria and their use in ascertaining stability via energy principles that incorporate different constraints. Given that XMHD is a Hamiltonian theory and that the investigations of [14][15][16][17][18] are generic to Hamiltonian theories, all of the considerations of these and other works can be worked out for XMHD. This is the main motivation for conducting this study in the framework of noncanonical Hamiltonian mechanics, i.e.…”

Section: Introductionmentioning

confidence: 99%

Translationally symmetric extended MHD via Hamiltonian reduction: Energy-Casimir equilibria

2017

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The Hamiltonian structure of ideal translationally symmetric extended MHD (XMHD) is obtained by employing a method of Hamiltonian reduction on the three-dimensional noncanonical Poisson bracket of XMHD. The existence of the continuous spatial translation symmetry allows the introduction of Clebsch-like forms for the magnetic and velocity fields. Upon employing the chain rule for functional derivatives, the 3D Poisson bracket is reduced to its symmetric counterpart. The sets of symmetric Hall, Inertial, and extended MHD Casimir invariants are identified, and used to obtain energy-Casimir variational principles for generalized XMHD equilibrium equations with arbitrary macroscopic flows. The obtained set of generalized equations is cast into Grad-Shafranov-Bernoulli (GSB) type, and special cases are investigated: static plasmas, equilibria with longitudinal flows only, and Hall MHD equilibria, where the electron inertia is neglected. The barotropic Hall MHD equilibrium equations are derived as a limiting case of the XMHD GSB system, and a numerically computed equilibrium configuration is presented that shows the separation of ion-flow from electromagnetic surfaces.

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Linear and nonlinear stability criteria for compressible MHD flows in a gravitational field

Cited by 14 publications

References 71 publications

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

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

Helically symmetric extended magnetohydrodynamics: Hamiltonian formulation and equilibrium variational principles

Translationally symmetric extended MHD via Hamiltonian reduction: Energy-Casimir equilibria

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