The linear stability of line-tied coronal magnetic fields

The problem of the linear stability of cylindrically symmetric force-free magnetic equilibria is addressed. The aim is to quantify the severity of ideal MHD disturbances on a variety of line-tied coronal equilibria. Accordingly methods are developed which determine, at high spatial resolution, the dominant eigenmodes of the problem. We first introduce a family of equilibrium force-free fields parameterized by the local dimensionless twist T = rν which includes the well-known Gold-Hoyle field in the case ν = 0. The stability problem is then formulated, and the numerical techniques required for its solution are derived. As a preliminary test of our methods we present convergence results for the critical length of the Gold-Hoyle field, obtaining the result lc = 2.46π (dimensionless units). Next, the severity of the instability is quantified. The kink instability (m = 1) is found to be weak even for long lengths (l ≳ 4lc), generating dimensionless force eigenvalues less than O(10-2). The Gold-Hoyle field is shown to be the most unstable member of the family for large l, yet is the most susceptible to line-tying and so possesses the longest critical length. Preliminary investigations for higher m have not yielded instabilities for the moderate tube lengths of interest. The significance of these results to flare theories cannot be known definitively without results from finite-amplitude studies. Our results show, however, that the power release over the linear phase is at least three orders of magnitude below dimensional expectation. This implies that unstable coronal flux tubes cannot be expected to collapse violently on the MHD time scale