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Astron. Astrophys. 346, 303-312 (1999)

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A complete coronal loop stability analysis in ideal magnetohydrodynamics

II. Force-free cylindrical equilibria

R.A.M. VanderLinden *  1 and A.W. Hood 2

1 Centre for Plasma Astrophysics, Katholieke Universiteit Leuven, Celestijnenlaan 200B, B-3001 Leuven, Belgium (Ronald.Vanderlinden@wis.kuleuven.ac.be)
2 School of Mathematical and Computational Sciences, University of St. Andrews, St. Andrews KY16 9SS, Scotland (alan@dcs.st-andrews.ac.uk)

Received 22 December 1998 / Accepted 29 March 1999


A WKB method to determine approximations to the critical length for the onset of ideal MHD instabilities with high poloidal mode numbers m in one-dimensional force-free cylindrical models of line-tied coronal loops is presented, extending the work of Hood et al. (1994) and Van der Linden & Hood (1998). Qualitatively, the procedure is similar to the one used in these two papers and pioneered by Connor et al. (1979). It is found, however, that the scalings for sheared force-free equilibria are different from those in the other cases, so that significant modifications to the method are necessary.

The WKB method developed only requires solving a simple ordinary differential equation rather than the original set of complicated two-dimensional partial differential equations. For all force-free sheared equilibria we find that for large m the marginal stability length behaves like [FORMULA] compared to [FORMULA] for the unsheared case investigated in Hood et al. (1994). Thus, it appears that in the force-free (or nearly force-free) case the [FORMULA] mode is always the first to become unstable.

The WKB results are complemented with numerical solutions of the full equations and for sufficiently large values of the wave number m excellent agreement is found. The combination of the results and methods described in this paper, together with those in Van der Linden & Hood (1998) provide all the tools necessary to perform a complete stability assessment of any one-dimensional cylindrically-symmetric equilibrium model for a coronal loop.

Key words: Magnetohydrodynamics (MHD) – plasmas – Sun: corona – Sun: magnetic fields

* Postdoctoral Research Fellow of the Flemish Fund for Scientific Research

Send offprint requests to: R.A.M. Van der Linden

© European Southern Observatory (ESO) 1999

Online publication: May 6, 1999