Kink instability of flux ropes anchored at one end and free at the other

The kink instability of a magnetized plasma column (flux rope) is a fundamental process observed in laboratory and in natural plasmas. Previous theoretical, experimental, and observational work has focused either on the case of periodic (infinite) ropes (relevant to toroidal systems) or on finite ropes with both ends tied to a specified boundary (relevant to coronal ropes tied at the photosphere). However, in the Sun's corona and in astrophysical systems there is an abundant presence of finite flux ropes tied at one end but free at the other. Motivated by recent experiments conducted on the RSX device (Furno et al., 2006) and by recent theoretical work (Ryutov et al., 2006), the present paper investigates by simulation the linear and nonlinear evolution of free-ended flux ropes. The approach is based on comparing the classic case of a periodic flux rope with the case of a rope tied at one end and free at the other. In the linear phase, periodic and free ropes behave radically differently. A simulation analysis of the linear phase confirms the experimental and phenomenological findings relative to an increased instability of a free rope: the new stability limit is shown to be just half of the classic limit for periodic ropes. In the nonlinear phase, reconnection is observed to be a fundamental enabler to reach the eventual steady state. The mechanism for saturation of a flux rope is investigated and compared with the classic theory (the so-called bubble state model) by Rosenbluth et al. (1976). A remarkable agreement is found for the classic periodic case. The case of a free rope is again very different. The saturated state is observed to present a outwardly spiraling configuration with the displacement of the plasma column increasing progressively and monotonically from the tied end to the free end. The maximum displacement is observed at the free end where it is consistent with the displacement observed in a periodic rope. The key distinction is that in a periodic rope the same displacement is observed throughout the whole rope to form a helix with constant radius.