A complete solar coronal stabilitiy analysis in ideal magnetohydrodynamics: I non-force-free cylindrical equilibria

A procedure is introduced to perform a complete ideal MHD stability analysis of one-dimensional cylindrical equilibrium models for coronal loops, including the important effect of line-tying. The stability is completely determined by calculating the critical (marginally stable) length for the onset of ideal MHD instabilities for every azimuthal wave number m. The analysis consists of the combination of a WKB method to determine the critical length of intermediate to high (infinite) values of m with a numerical code (using bicubic finite elements) for the low to intermediate values of m. As before it is found that for large enough m the critical length can be expressed as l(c) = l(0) +l(1)/m. It is also demonstrated that in general either the m = 1 or the m --> infinity mode has the shortest critical length, the farmer being the first to become unstable for nearly force-free magnetic fields, the latter for strongly non-force-free fields. Therefore, a stability analysis of these two modes will normally suffice, with perhaps a need for some more numerical calculations near the point where the modes cross over. The combination of these two tools provides a complete stability assessment.