The ideal MHD stability of line-tied coronal loops: A truncated Fourier series approach

The stability behaviour of a line-tied cylindrically symmetric coronal loop is investigated using a general method presented by Velli, Einaudi and Hood. The plasma perturbation in the linearised equation of motion is expanded in a truncated Fourier series in the poloidal and axial directions, and the resulting system of ordinary differential equations for the radial displacement is solved as an eigenvalue problem. The eigenvalue, be it the critical loop length or the growth rate, is found to converge rapidly with the order of the truncation (approximately as N-2, where N is the number of terms in the Fourier series). Results for the non-force-free uniform-twist field of Gold and Hoyle are presented and compared with a previous study based on the energy principle. The instability threshold for the m = 1 kink mode and the m = ∞ balloning mode are found to be of the same order of magnitude when substantial pressure gradients are present. Preliminary results for higher-m modes indicate that they yield instability thresholds comparable to the ballooning threshold for smaller gradients too.