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

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. (\cite{wkb}) and Van der Linden & Hood (\cite{vh98}). Qualitatively, the procedure is similar to the one used in these two papers and pioneered by Connor et al. (\cite{CHT}). 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 lc~ ml0+l2/m compared to lc~ ml0+l1 for the unsheared case investigated in Hood et al. (\cite{wkb}). 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 (\cite{vh98}) provide all the tools necessary to perform a complete stability assessment of any one-dimensional cylindrically-symmetric equilibrium model for a coronal loop.