Nonlinear stability of axisymmetric ideal magnetohydrodynamic flows

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

doi:10.1017/s0022377803002356

Cited by 7 publications

(12 citation statements)

References 5 publications

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“…In fact some of the present authors (A.H.K and D.K.C) have used Vladimirov's approximation with success in several papers [5][6][7]. Hence a critical analysis may be useful.…”

Section: Introductionmentioning

confidence: 86%

“…Hence ρ is treated there as not varying in the inertia term. However, in the studies at hand [1][2][3][4][5][6][7], the variation in ρ is a variation in space in the configuration, besides possible variations due to thermal effects or to a perturbation (which moreover should be negligible for incompressible matter). Hence we are not dealing with a proper Boussinesq approximation, but with a different kind, although there is some similarity.…”

Section: Use Of Reduced Quantitiesmentioning

confidence: 99%

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On Vladimirov's Approximation for Ideal Inhomogeneous MHD

Callebaut¹,

Karugila²,

Khater³

2005

PIERS Online

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Vladimirov and Vladimirov and Moffat have considered configurations in ideal magnetohydrodynamics, i.e. inviscid and perfectly conducting. The matter is considered as incompressible. However, the density is allowed to vary slowly. They base the following approximation on this slow variation: they omit the mass density in front of the total derivative of the velocity in the equation of motion. Normally the mass density should appear in front of Du. This is a tremendous simplification which allows them to obtain various interesting results concerning the stability of the configurations. However, in such a kind of approximation the results might be only crude. However, in many applications the results are OK, because crucial in those papers is the vanishing of ∇ρ × ∇ϕ. Often both gradients are parallel and the results obtained by Vladimirovs approximation are nevertheless valid, e.g. in the application to inhomogeneous gas clouds and protostars. Moreover for small density gradients and/or nearly parallel gradients the approximation is fair. We even suggest an approximation which may be more correct and avoids the term ∇ρ × ∇ϕ. Hence for linear perturbations and stability analyses the results may turn out to be acceptable. However, for nonlinear stability a more extended analysis is required.

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“…In fact some of the present authors (A.H.K and D.K.C) have used Vladimirov's approximation with success in several papers [5][6][7]. Hence a critical analysis may be useful.…”

Section: Introductionmentioning

confidence: 86%

Section: Use Of Reduced Quantitiesmentioning

confidence: 99%

On Vladimirov's Approximation for Ideal Inhomogeneous MHD

Callebaut¹,

Karugila²,

Khater³

2005

PIERS Online

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“…For cylindrical and axisymmetric geometries, a work developed by Vladimirov et al [46][47][48] for describing the stability properties of ideal MHD plasmas is applied to study the nonlinear stability of a wide class of incompressible MHD states. 49,50 Previously, the stability of ideal incompressible MHD flows in the plane with constant density was investigated by Holm et al 8 An approach to the study of the equilibrium and stability of ideal MHD flows was introduced by Vladimirov et al 46,47 They dealt only with fields having two planar components, 47 and consequently obtained stability conditions equivalent to those obtained by Holm et al 8 Here we investigate the problem in the presence of nonplanar components of both the velocity and magnetic field. We use a principle of minimum constrained energy to derive the equilibrium equations for this class of MHD flows.…”

Section: Introductionmentioning

confidence: 94%

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

Khater

Moawad

Callebaut³

2005

Physics of Plasmas

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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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Nonlinear stability of axisymmetric ideal magnetohydrodynamic flows

Cited by 7 publications

References 5 publications

On Vladimirov's Approximation for Ideal Inhomogeneous MHD

On Vladimirov's Approximation for Ideal Inhomogeneous MHD

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

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

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