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 ons et of ideal MHD instabilities with high poloidal mode numbers m in one-dime nsional force-free cylindrical models of line-tied coronal loops is present ed, extending the work of Hood et al. (1994) and Van der Linden & Hood (199 8). Qualitatively, the procedure is similar to the one used in these two pa pers 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 differenti al equation rather than the original set of complicated two-dimensional par tial differential equations. For all force-free sheared equilibria we find that for large m the marginal stability length behaves like l(c) approximat e to ml(0) + l(2)/m compared to l(c) approximate to ml(0) + l(1) for the un sheared case investigated in Hood et al. (1994). Thus, it appears that in t he force-free (or nearly force-free) case the m = 1 mode is always the firs t to become unstable. The WKB results are complemented with numerical solutions of the full equat ions and for sufficiently large values of the wave number m excellent agree ment 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-dimen sional cylindrically-symmetric equilibrium model for a coronal loop.