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Criteria for Flux Rope Eruption: Non Equilibrium versus Torus Instability

The coronal magnetic configuration of an active region typically evolves quietly during few days before becoming suddenly eruptive and launching a coronal mass ejection (CME). The precise origin of the eruption is still debated. Among several mechanisms, it has been proposed that a loss of equilibrium, or an ideal magneto-hydrodynamic (MHD) instability such as the torus instability, could be responsible for the sudden eruptivity. Distinct approaches have also been formulated for limit cases having circular or translation symmetry. We revisit the previous theoretical approaches, setting them in the same analytical framework. The coronal field results from the contribution of a non-neutralized current channel added to a background magnetic field, which in our model is the potential field generated by two photospheric flux concentrations. The evolution on short Alfvenic time scale is governed by ideal MHD. We show analytically first that the loss of equilibrium and the stability analysis are two different views of the same physical mechanism. Second, we identify that the same physics is involved in the instability of circular and straight current channels. Indeed, they are just two particular limiting case of more general current paths. A global instability of the magnetic configuration is present when the current channel is located at a coronal height, h, large enough so that the decay index of the potential field, (d ln |Bp|) / (d ln h) is larger than a critical value. At the limit of very thin current channels, previous analysis found a critical decay index of 1.5 and 1 for circular and straight current channels, respectively. However, with current channels being deformable and as thick as expected in the corona, we show that this critical index has similar values for circular and straight current channels, typically in the range [1.1,1.3].