WKB estimates for the onset of ideal MHD instabilities in solar coronal loops

A Wentzel-Kramer-Brillouin (WKB) approach, based on the method of Connor, Hastie, and Taylor (1979), is used to obtain simple estimates of the critical conditions for the onset of ideal magnetohydrodynamic (MHD) instabilities in line-tied solar coronal loops. The method is illustrated for the constant twist, Gold-Hoyle (1960) field, and the critical conditions are compared with previous and new numerical results. For the force-free case, the WKB estimate for the critical loop length reduces to (2 * pi * m) + (pi * square root of 2). For the sufficiently non-force-free case the critical length can be expressed in the form (l0 + l1)/m. The results confirm the findings of De Bruyne and Hood (1992) that for force-free fields the m = 1 mode is the first mode to become unstable but for the sufficiently strong non-force-free case this reverses with the m approaches infinity mode being excited first.