A complete coronal loop stability analysis in ideal magnetohydrodynamics: II. Force-free cylindrical equilibrium.

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 l(c) approximate to ml(0) + l(2)/m compared to l(c) approximate to ml(0) + l(1) for the unsheared case investigated in Hood et al. (1994). Thus, it appears that in the force-free (or nearly force-free) case the m = 1 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.