On Vladimirov’s approximation for ideal in homogeneous MHD
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Date
2005-08
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Abstract
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.
Description
Journal article
Keywords
Inhomogeneous MHD, MHD, Vladimirov’s approximation