Browsing by Author "Callebaut, D. K."

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On Vladimirov’s approximation for ideal in homogeneous MHD
(2005-08) Callebaut, D. K.; Karugila, G. K.; Khater, A. H.
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.
Powerfull nonlinear plasma waves from moderate first order perturbations
(2005-08) Callebaut, D. K.; Karugila, G. K.
The nonlinear Fourier method of Callebaut consists in concentrating on the family of higher order terms of a single Fourier term of the linearized analysis. Thus we have obtained the higher order terms of plasma perturbations, gravitational ones, etc. In the simplest case of cold plasma this resulted in obtaining an analytical expression for the higher order terms. This allowed to investigate the convergence of the series, which in this case limits the first order amplitude to 1/e of the equilibrium density. For the cases without an analytical expression we developed a numerical-graphical method to obtain the convergence limit. Near this limit the total amplitude of the wave becomes very large. The convergence limit decreases with increasing pressure. Thus a wave with moderate first order amplitude may carry a very large energy due to the higher orders. Moreover, this energy is concentrated in a very narrow interval of the phase interval (0, 2π). This may be relevant in many situations. E.g. in the case of ball lightning a tremendous energy may be accumulated while the glowing is still restricted. The triggering of solar flares or coronal mass ejections may thus be caused. Again, when these eruptions reach the Earth the influence of a first order term may be far too small to cause electric power plants to break down; however, the total of all terms may be much more powerful. Cf. March 1989 when the whole state of Quebec, Canada, was a day without electricity due to a solar storm. This is an alternative mechanism from the one proposed by Callebaut and Tsintsadze based on soliton envelope formation, although there the accent was on the heating of the plasma.