Critical Height of the Torus Instability in Two-ribbon Solar Flares

We studied the background field for 60 two-ribbon flares of M-and-above classes during 2011-2015. These flares are categorized into two groups, i.e., eruptive and confined flares, based on whether a flare is associated with a coronal mass ejection or not. The background field of source active regions is approximated by a potential field extrapolated from the B-z component of vector magnetograms provided by the Helioseismic and Magnetic Imager. We calculated the decay index n of the background field above the flaring polarity inversion line, and defined a critical height h(crit) corresponding to the theoretical threshold (n(crit) = 1.5) of the torus instability. We found that h(crit) is approximately half of the distance between the centroids of opposite polarities in active regions and that the distribution of h(crit) is bimodal: it is significantly higher for confined flares than for eruptive ones. The decay index increases monotonously with increasing height for 86% (84%) of the eruptive (confined) flares but displays a saddle-like profile for the rest, 14% (16%), which are found exclusively in active regions of multipolar field configuration. Moreover, n at the saddle bottom is significantly smaller in confined flares than that in eruptive ones. These results highlight the critical role of background field in regulating the eruptive behavior of two-ribbon flares.