Linear and nonlinear stability analysis for two-dimensional ideal magnetohydrodynamics with incompressible flows

Khater, A. H.; Moawad, S. M.; Callebaut, D. K.

doi:10.1063/1.1828464

Cited by 9 publications

(9 citation statements)

References 46 publications

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“…The EC method evolved out of early plasma work [3] and there is now considerable prior application to plasma physics problems, as can be found in Refs. [15][16][17][30][31][32].…”

Section: B Crmhd Equilibria and Energy Stabilitymentioning

confidence: 99%

Stability of compressible reduced magnetohydrodynamic equilibria—Analogy with magnetorotational instability

2013

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Stability analyses for equilibria of the compressible reduced magnetohydrodynamics (CRMHD) model are carried out by means of the Energy-Casimir (EC) method. Stability results are compared with those obtained for ideal magnetohydrodynamics (MHD) from the classical δW criterion. An identification of the terms in the second variation of the free energy functional for CRMHD with those of δW is made: two destabilizing effects present for CRMHD turn out to correspond to the kink and interchange instabilities in usual MHD, while the stabilizing roles of field line bending and compressibility are also identified in the reduced model. Also, using the EC method, stability conditions in the presence of toroidal flow are obtained. A formal analogy between CRMHD and a reduced incompressible model for magnetized rotating disks, due to Julien and Knobloch [EAS Pub. Series, 21, 81 (2006)], is discovered. In light of this analogy, energy stability analysis shows that the condition for magnetorotational instability (MRI) for the latter model, corresponds to the condition for interchange instability in CRMHD, with the Coriolis term and shear velocity playing the roles of the curvature term and pressure gradient, respectively. Using the EC method, stability conditions for the rotating disk model, for a large class of equilibria with possible non-uniform magnetic fields, are obtained. In particular, this shows it is possible for the MRI system to undergo, in addition to the MRI, another instability that is analogous to the kink instability. For vanishing magnetic field, the Rayleigh hydrodynamical stability condition is recovered.

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“…The EC method evolved out of early plasma work [3] and there is now considerable prior application to plasma physics problems, as can be found in Refs. [15][16][17][30][31][32].…”

Section: B Crmhd Equilibria and Energy Stabilitymentioning

confidence: 99%

Stability of compressible reduced magnetohydrodynamic equilibria—Analogy with magnetorotational instability

2013

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“…In previous papers [46][47][48] we investigated the equilibrium and stability for incompressible cases of ideal MHD plasmas.…”

Section: Introductionmentioning

confidence: 99%

Equilibrium properties, variational principles, and linear stability for steady-state, two-dimensional, ideal gravitating plasma of a barotropic compressible flow

Moawad

2012

Can. J. Phys.

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The equilibrium and stability properties of a planar gravitating plasma in the theory of ideal magnetohydrodynamic flows are investigated. The flow is taken to be barotropic. Variational principles for the steady state equations of aligned magnetic field and plasma flow are formulated. A simple and efficient way is proposed for investigating the stability properties of this state. Sufficient conditions for linear stability of the steady-state flow are obtained. , 47.65.-d, 95.30.Qd, 95.30.Lz, 95.30.Sf Résumé : Nous étudions, en théorie magnétohydrodynamique idéale, les propriétés d'équilibre et de stabilité d'un plasma plan sous champ gravitationnel. Nous considérons un écoulement baratrope. Nous formulons les principes variationnels pour les équations d'état stationnaire du champ magnétique aligné et du flot de plasma. Nous proposons une façon simple et efficace pour étudier les propriétés de stabilité de cet état. Nous obtenons les conditions qui sont suffisantes pour la stabilité linéaire du flot stationnaire. [Traduit par la Rédaction]

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“…Formal stability of fluids and plasmas has been established by several authors who employed some aspects of Arnold's method. 47,[63][64][65][66][67] Equilibrium states with mass flow are becoming increasingly interesting in magnetic fusion research. 47,[63][64][65][66][67] Equilibrium states with mass flow are becoming increasingly interesting in magnetic fusion research.…”

Section: Introductionmentioning

confidence: 99%

Nonlinear periodic solutions for isothermal magnetostatic atmospheres

2008

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Magnetohydrodynamic equilibria for a plasma in a gravitational field are investigated analytically. For equilibria with one ignorable spatial coordinate, the equations reduce to a single nonlinear elliptic equation for the magnetic potential A, known as the Grad-Shafranov equation. Specifying the arbitrary functions in the latter equation, one obtains three types of nonlinear elliptic equations ͑a Liouville equation, a sinh Poisson equation, and a generalization of those with a sum of exponentials͒. Analytical solutions are obtained using the tanh method; this is elaborated in the Appendix. The solutions are adequate to describe an isothermal atmosphere in a uniform gravitational field showing parallel filaments of diffuse, magnetized plasma suspended horizontally in equilibrium.

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Linear and nonlinear stability analysis for two-dimensional ideal magnetohydrodynamics with incompressible flows

Cited by 9 publications

References 46 publications

Stability of compressible reduced magnetohydrodynamic equilibria—Analogy with magnetorotational instability

Stability of compressible reduced magnetohydrodynamic equilibria—Analogy with magnetorotational instability

Equilibrium properties, variational principles, and linear stability for steady-state, two-dimensional, ideal gravitating plasma of a barotropic compressible flow

Nonlinear periodic solutions for isothermal magnetostatic atmospheres

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