Dong Wang^{1,2}, Rui Liu^{1}, Yuming Wang^{1}, Kai Liu^{1}, Jun Chen^{1}, Jiajia Liu^{1}, Zhenjun Zhou^{1}, & Min Zhang^{2}

^{1 }CAS Key Laboratory of Geospace Environment, Department of Geophysics and Planetary Sciences, University of Science and Technology of China, Hefei 230026, China

^{2 }Department of Mathematics and Physics, Anhui Jianzhu University, Hefei 230601, China

Flares and coronal mass ejections (CMEs) are two intimately related phenomena on the Sun, yet a significant fraction of flares are not associated with CMEs, therefore being referred to as *confined* flares, whereas the complement are known as *eruptive* flares. The torus instability comes into play when the external field above a flux rope decreases sufficiently rapidly with increasing height^{1}. Quantified by the decay index *n* = – *d*ln*B*_{ex} / *d*ln*h* (ref. 2), the threshold value of the instability *n*_{crit} is found in the range of 1-2 by various models and numerical simulations. Typically *n*_{crit} = 1.5 is taken as a yardstick number.

Here we carried out a statistical investigation to evaluate to what extent the decay index affects solar eruptions^{3}, which has significant implications for space weather forecasting. We studied the background field of 60 two-ribbon, M-class-and-above flares during 2011-2015, located within ~45 degree from the disk center. The working assumption is that a magnetic flux rope is present during a two-ribbon flare, no matter wether the rope is preexistent or formed during the eruption. Due to the difficulty in decoupling *B*_{ex} from the flux-rope field, the background field is approximated by a potential field extrapolated from the *B _{z}* component of vector magnetograms provided by the Helioseismic and Magnetic Imager. The transverse component of the potential field,

*B*, is employed to approximate the external field component orthogonal to the axial current of the flux rope, since potential field is almost orthogonal to polarity inversion line (PIL), along which a flux rope in equilibrium typically resides. To quantify the onset point of the torus instability, we calculated the decay index (

_{t}*n*≃ –

*d*ln

*B*/

_{t}*d*ln

*h*) above the PIL segment in between the two flare ribbons, and then the critical height

*h*

_{crit}corresponding to the theoretical threshold (

*n*

_{crit}= 1.5).

The distribution of *h*_{crit} peaks at 20-30 Mm (left panel of Fig. 1), but for confined flares *h*_{crit} significantly spreads to higher heights than eruptive flares. The average *h*_{crit} is 36.3 ± 17.4 Mm for the 35 eruptive flares, and 53.6 ± 21.3 Mm for the 25 confined flares. Moreover, *h*_{crit} is roughly proportional to the flux centroid distance *d* of active regions (right panel of Fig. 1), i.e., *h*_{crit} ≃ *d* / 2, which may serve as a rule of thumb for the scale of *h*_{crit}.

Figure 1 |Distribution ofh_{crit}(left) and its relation to the centroid distancedof active regions (right). In the right panel, plus and diamond symbols denote dipolar (D) and multipolar (M) magnetic field, respectively. Eruptive (E) and confined (C) events are shown in blue and red, respectively. Notion ‘sl’ indicates the slope given by linear fitting and ‘cc’ the correlation coefficient with the confidence interval denoted in the brackets.h_{crit}=d/ 2 is marked by the dotted line.

Two distinct types of *n*(*h*) profiles emerge: 1) *n* increases monotonically as the height increases in 30 of 35 (86%) eruptive flares and in 21 of 25 (84%) confined flares; and 2) the rest 9 events, all originating from multipolar magnetic field, exhibit a saddle-like *n*(*h*) profile, with a local minimum *n _{b}* at a height higher than

*h*

_{crit}(e.g., left panel of Fig. 2).

*n*of the 5 eruptive flares (black; right panel Fig. 2) is generally larger than that of the 4 confined flares (red).

_{b}

Figure 2 |Saddle-liken(h) profile. Left panel shows an exemplaryn(h) profile from the confined flare on 2014 February 4.nand_{b}h_{crit}are marked. Right panel shows the distribution ofnfor 5 eruptive (black) and 4 confined (red) flares._{b}

Thus, this investigation confirms that the decay index profile of the background field plays an important role in deciding whether a two-ribbon flare would lead up to a CME. In particular, the saddle-like profile present in some active regions may provide an additional confinement effect on eruptions.

### References

[1] Török, T., & Kliem, B., 2005, *ApJL*, **630**, L97

[2] Kliem, B., & Török, T., 2006, *PRL*, **96**, 255002

[3] Wang, D., Liu, R., Wang, Y., et al., 2017, *ApJL*, **843**, L9