Where and How Does a Decay-index Profile Become Saddle-like?

A dipole field can be written as

$$\begin{eqnarray}&&{\boldsymbol{B}}({\boldsymbol{r}})=\displaystyle \frac{3({\boldsymbol{m}}\cdot {\boldsymbol{r}}){\boldsymbol{r}}}{{r}^{5}}-\displaystyle \frac{{\boldsymbol{m}}}{{r}^{3}},\end{eqnarray} \tag{ 1 }$$

where m is the dipole moment and r the displacement vector. The plane at a height h₀ above the dipole moment, where the bipolar flux distribution (Figure 1(a)) shares similarities with a pair of sunspots with opposite polarities, is taken as a pseudophotosphere. In other words, the horizontal dipole m = m e _x is placed at (0, 0, − h₀) in the Cartesian coordinate system. Right above the PIL, which is along the y-axis, the magnetic field can be written as

$$\begin{eqnarray}&&{{\boldsymbol{B}}}_{\mathrm{PIL}}({\boldsymbol{r}})=-\displaystyle \frac{m}{{r}^{3}}\,{{\boldsymbol{e}}}_{x}=-\displaystyle \frac{m}{{\left[{D}^{2}+{\left(h+{h}_{0}\right)}^{2}\right]}^{\tfrac{3}{2}}}\,{{\boldsymbol{e}}}_{x},\end{eqnarray} \tag{ 2 }$$

where h is the height above the photosphere and D is the linear distance from a point on the PIL to the origin. The decay index right above the PIL is then given as follows:

$$\begin{eqnarray}&&{n}_{\mathrm{PIL}}=\displaystyle \frac{3h(h+{h}_{0})}{{D}^{2}+{\left(h+{h}_{0}\right)}^{2}}.\end{eqnarray} \tag{ 3 }$$

The critical height h_cr is defined as the height where n_PIL = 1.5. In a dipole field,

$$\begin{eqnarray}&&{h}_{\mathrm{cr}}={\left({h}_{0}^{2}+{D}^{2}\right)}^{\tfrac{1}{2}}.\end{eqnarray} \tag{ 4 }$$

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We define the centroid distance d_c as the linear distance between the centroids of opposite polarities in the pseudophotosphere. The flux-weighted centroid of the positive polarity, e.g., is (x_c , y_c ) = (∫xB_z dA/∫B_z dA, ∫yB_z dA/∫B_z dA), where B_z > 0. We obtain d_c in terms of the depth of the dipole h₀, i.e.,

$$\begin{eqnarray}&&{d}_{c}=\pi {h}_{0}.\end{eqnarray} \tag{ 5 }$$

Parameter h_cr can then be expressed in terms of d_c to facilitate the comparison with observations,

$$\begin{eqnarray}&&{h}_{\mathrm{cr}}={\left(\displaystyle \frac{{d}_{c}^{2}}{{\pi }^{2}}+{D}^{2}\right)}^{\displaystyle \frac{1}{2}}.\end{eqnarray} \tag{ 6 }$$

Taking the average critical height ${\bar{h}}_{\mathrm{cr}}$ over the PIL from D = − 0.7d_c to 0.7d_c , we found

$$\begin{eqnarray}&&{\bar{h}}_{\mathrm{cr}}\simeq {d}_{c}/2.\end{eqnarray} \tag{ 7 }$$

From Equation (3) one can see that the decay index n increases monotonically with the increasing height above the photosphere, which leaves no room for a saddle-shaped n(h) profile. Hence, below we investigate the decay index in quadrupole fields.

In quadrupole fields, it becomes difficult to derive h_cr analytically. Hence, similar to Sun et al. (2013) and Liu et al. (2016b), we placed two horizontal dipoles at the same depth below a Cartesian computation domain (−255.5 < x, y < 255.5, 0 < z < 511). They yield together a potential field above the z = 0 plane, which is again taken as the photosphere. Note that in the 2D model of Filippov (2018) two dipoles are placed at different depths to generate the background field with two different spatial scales above the surface. Here, for simplicity, m ₁'s flux distribution at the z = 0 plane, termed "BP1" hereafter, is maintained constant, and m ₁ is always oriented in the x-direction. We adjust m ₂'s flux distribution at the z = 0 plane, termed "BP2" hereafter, to produce different quadrupole configurations. The centroid distance d_c of each individual dipole is the same as in Equation (5) and taken as the unit length in the following analysis.

Two types of saddle-like profiles n_sdl(h) can be distinguished, based on whether the local maximum value at the saddle top n_t is above n_cr and the local minimum value at the saddle bottom n_b is below n_cr. Saddle-like profiles satisfying n_b < n_cr < n_t (e.g., Figure 1(g)) only occur above the inter-bipole PIL, i.e., the external PIL of each individual bipole but internal PIL of the quadrupole. This type of saddle-like profile crosses the n_cr = 1.5 line at least three times, which is denoted by ${n}_{\mathrm{sdl}}^{{\rm{m}}}(h)$ hereafter. Other saddle shapes satisfying either n_b > n_cr or n_t < n_cr (e.g., Figures 1(h) and (i)) are found above the intra-bipole PIL, mostly located close to the center of each individual bipole. They cross the n_cr = 1.5 line only once, which is denoted by ${n}_{\mathrm{sdl}}^{{\rm{s}}}(h)$ hereafter.

3.2.1. Parameters Regulating the Occurrence of n_sdl(h)

3.2.2. Occurrence Rate of n_sdl(h) above PIL

To quantify the probability of encountering saddle-like n(h) profiles above the PIL in quadrupole fields, we define the "saddle occurrence rate" as follows:

$$\begin{eqnarray}&&{R}_{\mathrm{sdl}}=\displaystyle \frac{{L}_{\mathrm{sdl}}}{L},\end{eqnarray} \tag{ 8 }$$

where L is the length of the PIL from the center of BP1 to that of BP2, and L_sdl denotes the length of the PIL segments above which n_sdl(h) appears. Similarly, we may obtain the occurrence rate of ${n}_{\mathrm{sdl}}^{{\rm{m}}}(h)$ among saddle-shaped profiles, i.e.,

$$\begin{eqnarray}&&{R}_{\mathrm{sdl}}^{{\rm{m}}}=\displaystyle \frac{{L}_{\mathrm{sdl}}^{{\rm{m}}}}{{L}_{\mathrm{sdl}}}.\end{eqnarray} \tag{ 9 }$$

We set the height of the computational domain by the displacement of m ₂ relative to m ₁ in each individual quadrupole configuration. The result has been checked against those obtained in larger computational domains, but we found no qualitative difference.

First, two bipoles are set to be parallel to each other and have equal magnitude, but m ₂ is displaced with respect to m ₁ by Δ_x and Δ_y in the X- and Y-direction, respectively. The distributions of R_sdl and ${R}_{\mathrm{sdl}}^{{\rm{m}}}$ in this Δ_x –Δ_y space are indicated by colors (Figures 2(a) and (b)). One can see that Δ_y /Δ_x is an important parameter in regulating the occurrence rate of saddle-shaped n(h) above the PIL, i.e., R_sdl becomes significant (>0.4) when Δ_y /Δ_x < 1.1 and ${R}_{\mathrm{sdl}}^{{\rm{m}}}$ starts to appear (>0.1) when Δ_y /Δ_x < 0.8.

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Next, as mentioned above, the angle θ between m ₁ and m ₂ can influence the generation of saddle-shaped n(h). We set up a quadrupole configuration with an initial Δ_x = 0 (e.g., Figure 1(e)) to ensure the absence of n_sdl(h), and then we change m ₂'s orientation to provide different θ. In Figure 2(c) the saddle occurrence rate R_sdl increases with θ, when θ exceeds a threshold value (∼90°), but soon becomes saturated when θ ∼ [130°, 145°], and thence decreases gradually. At θ = 180°, however, R_sdl suddenly drops to nearly 0, because this symmetric configuration possesses a single null point above the center. Only through the null point does the decay-index profile become extremely saddle-like (see the yellow curve in Figure 1(g)). Apparently, the threshold value depends on other parameters of the quadrupole configuration;, e.g., the appearance of ${n}_{\mathrm{sdl}}^{{\rm{m}}}(h)$ requires a higher θ threshold than ${n}_{\mathrm{sdl}}^{{\rm{s}}}(h)$.

Finally, we keep m ₁ constant and weaken m ₂ continuously to emulate asymmetric quadrupolar regions with unbalanced magnetic flux between two pairs of sunspots (Figure 1(d)). We divide PIL into two segments, namely, PIL1, which is close to BP1, and PIL2, which is close to BP2, based on the distance to the flux-weighted center of each pair of sunspots. The total saddle generation rate R_sdl above the entire PIL (black), R_sdl,1 above PIL1 (blue), and R_sdl,2 above PIL2 (red) are shown in Figure 2(d). One can see that saddle-shaped n(h) tends to appear above the PIL belonging to the weaker sunspot pair.

Here we investigate a few selected ARs that have simple configurations, typically composed of either one or two pairs of major sunspots with opposite polarities, so that they can be emulated by either one or two dipoles. We further compare the decay index of the AR background fields that are approximated by a potential field with those emulated by dipoles.

3.3.1. Bipolar Active Regions

Three ARs, NOAA AR 12192, 11166, and 11428, which have two major sunspots of opposite polarities, are shown in Figure 3. The centroid distance d_c is again given as the linear distance between the flux-weighted centroids of opposite polarities, and the midpoint is taken as the AR center (yellow triangles in Figures 3(a)–(c)). The color-coded decay index n as a function of height above the PIL is plotted along the PIL (Figures 3(d)–(f)), whose origin is designated as the closest point to the AR center (yellow circles in Figures 3(a)–(c)). The decay index n increases monotonically with increasing height above almost all segments of PIL. We took the average of h_cr over the PIL points whose arc length from the PIL origin along the PIL ranges from − 0.7d_c to 0.7d_c (thick curves in Figures 3(a)–(c)) and found that this average value, ${\bar{h}}_{\mathrm{cr}}$, is approximately d_c /2, consistent with Equation (7).

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To further compare with the dipole model, we show h_cr as a function of D, the linear distance from a PIL point to the AR center (Figures 3(g)–(i)). Parameter h_cr obtained for PIL segments to the north (south) of the PIL origin is denoted in blue (red). That h_cr increases with the linear distance from the AR center suggests that magnetic structures located on the outskirts of a bipolar AR generally experience a stronger confinement imposed by the background field than those close to the AR center. An overall consistency between h_cr obtained from the potential field approximation and those from the dipole field is demonstrated by Figures 3(g)–(i), in which the dashed curves are given by Equation (4), with h₀ replaced by the critical height at the PIL origin obtained in the potential field.

3.3.2. Quadrupolar Active Regions

Here the coronal background fields of two exemplary ARs that consist of two major pairs of sunspots are compared with those idealized by a quadrupole field consisting of a pair of dipoles.

AR 11158 (Figure 4(a)) can be emulated by a pair of dipoles that differ in both location and orientation, as well as slightly in intensity (Figure 4(b)). The n(h) profiles plotted along the PIL (similar to Figures 3(d)–(f)) again demonstrate an overall similarity between the background field approximated by the potential field and that by a pair of dipoles. We also plot n(h) profiles at three specific locations in Figures 4(g1)–(g3). Both n(h) profiles at Location 1 and Location 3 can be categorized as ${n}_{\mathrm{sdl}}^{{\rm{s}}}(h)$, which crosses the n = 1.5 line only once, but those at Location 2 can be categorized as ${n}_{\mathrm{sdl}}^{{\rm{m}}}(h)$, which crosses the n = 1.5 line at least three times. A reversal in the transverse field component B _t and a strongly enhanced squashing factor Q (Titov et al. 2002) are found above Location 2 at similar heights in both the potential field and the quadrupole field (red and blue curves, respectively, in Figures 4(h2 and i2)), which will be discussed in Section 4.3.

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Figure 5 shows a quadrupolar region composed of two bipolar ARs, NOAA AR 11504 and 11505. To emulate this quadrupolar field, we adopted two dipoles that have similar orientation but are relatively displaced only in the direction perpendicular to the dipole orientation. The n(h) profiles show a similar trend, i.e., increasing monotonically with height, over the major PIL.

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In the dipole field, we found that the critical height averaged over above the major PIL in the core region [−0.7d_c , 0.7d_c ] is approximately half of the centroid distance d_c (Equation (7)). This is consistent with the statistics of two-ribbon flares by Wang et al. (2017, their Equation (1)). Similarly, Kliem et al. (2014) found that h_cr is slightly below the half-distance between two magnetic charges serving as the external field for a flux rope. These results generally imply that the bipolar ARs that have a large centroid distance between opposite polarities also have a large critical height, therefore providing strong confinement to underlying nonpotential structures. This is reminiscent of the extremely large AR 12192, which produced 32 M-class and six X-class flares but only one CME (Liu et al. 2016a). Equation (6) further implies that h_cr increases with increasing distance from the center of a bipolar region. However, this appears to contradict some observations, in which confined flares are located closer to the flux-weighted magnetic center than eruptive ones in the same AR (e.g., Wang & Zhang 2007; Cheng et al. 2011; Baumgartner et al. 2018). We note that the ARs in these studies are often far more complex than bipolar regions that we tried to emulate here with a dipole, and also the overlying field of an eruptive structure located in the AR periphery must be modulated by the magnetic field of neighboring ARs (e.g., Figure 5), which is seldom taken into account in previous studies.

Placing two dipoles at the same depth below a plane taken as the photosphere, we found two types of saddle-shaped n(h) profiles in quadrupole fields: one crosses the n = 1.5 line multiple times, and the other crosses the same line only once. The former, ${n}_{\mathrm{sdl}}^{{\rm{m}}}(h)$, is found only above the inter-bipole PIL between two bipoles, while the latter, ${n}_{\mathrm{sdl}}^{{\rm{s}}}(h)$, is close to and inside each bipole, i.e., mainly above the intra-bipole PIL.

We investigated three parameters regulating the generation of saddle-shaped n(h), namely, the relative location, orientation, and magnitude of the two dipoles.

• 1.  
  For two parallel dipoles, the rate of n_sdl(h) along the PIL becomes significant (R_sdl > 0.4) when their relative displacement along the dipole direction is larger than that perpendicular to the dipole direction; ${n}_{\mathrm{sdl}}^{{\rm{m}}}(h)$ appears when the former displacement is significantly larger than the latter.
• 2.  
  n_sdl(h) appears more frequently above the PIL with larger angle between the two dipoles.
• 3.  
  n_sdl(h) tends to appear above the PIL segments closer to the bipole that is weaker in terms of unsigned flux.

The saddle-shaped profile n_sdl(h), especially ${n}_{\mathrm{sdl}}^{{\rm{m}}}(h)$, provides a zone of stability for an eruptive structure when it rises to the height corresponding to the bottom of the saddle where n < n_cr. It appears to be the case in some confined flares (e.g., Liu et al. 2018a) or two-step eruptions (e.g., Gosain et al. 2016). However, it is also found that eruptions tend to occur at the "external PIL" of bipolar ARs (Yardley et al. 2018), which suggests that remote ARs might also play a role, or at the "collisional PIL" between two pairs of sunspots within quadrupolar ARs (Chintzoglou et al. 2019). In both occasions, a saddle-like profile is likely present, from our experience of quadrupole fields (Section 3.2).

Further investigating the characteristics of magnetic field, we found that in quadrupole fields the saddle-like decay-index profile is associated with a strong variation of magnetic connectivity; it is hence also associated with a significant rotation of the transverse field component B _t , whose magnitude ${B}_{t}=\sqrt{{B}_{x}^{2}+{B}_{y}^{2}}$, with respect to the photospheric PIL (see also Filippov 2020). In the example shown in Figure 6, two dipoles are relatively displaced but have the same orientation, strength, and depth (Figure 6(a)). Two quasi-separatrix layers (QSLs; Titov et al. 2002) can be seen as the isosurfaces of logarithmic squashing factor, ${\mathrm{log}}_{10}Q=3.5$. Their footprints at the pseudophotosphere are shown as the blue and red lines in the map of signed ${\mathrm{log}}_{10}Q$ (Figure 6(b)). Such two QSLs intersecting each other are collectively known as a hyperbolic flux tube (HFT), whose intersections continuously transform from one end to the other (Titov et al. 2002), and in the center the intersection exhibits a typical X-shaped morphology (Figure 6(c)). In the cross section above the inter-bipole PIL (marked by the thick curve in Figures 6(a) and (b), where ${n}_{\mathrm{sdl}}^{{\rm{m}}}(h)$ is present; Figure 6(d)), one can see that with increasing height the decay index changes from over n_cr = 1.5 (reddish) to below it (blueish) near where the two QSLs intersect to give an X shape. Near the X shape, magnetic field lines are roughly parallel to the PIL (magenta), but below and above, they become increasingly perpendicular to the PIL (yellowish and greenish, respectively). Right above the center of the PIL (Figures 6(e)–(g)), one can see that the peak of ${\mathrm{log}}_{10}Q$ corresponds to where n decreases most rapidly and to where the field lines rotate with respect to the PIL (shaded bar).

For the background field of the representative AR 11158 as approximated by both potential field and two dipoles (Figure 4), we see similar height variation of various parameters, i.e., a reversal of the angle φ between B _t and PIL is associated with strongly enhanced Q, along the PIL segment between sunspot pairs, where ${n}_{\mathrm{sdl}}^{{\rm{m}}}(h)$ is present (e.g., Location 2 in Figure 4 and panels (g2, h2, j2)). Along the PIL segments within the sunspot pairs (e.g., Location 3 in Figure 4), where only ${n}_{\mathrm{sdl}}^{{\rm{s}}}(h)$ is present, field lines rotate within a much smaller range without an orientation reversal with respect to PIL, despite the enhanced Q. An exception is found at Location 1, where n(h) is not exactly saddle-like in the potential field extrapolation, despite that the quadrupole field gives an untypical ${n}_{\mathrm{sdl}}^{{\rm{s}}}(h)$ profile. Location 1 is located at the intra-bipole PIL of the sunspot pair in the northwest of AR 11158. Compared with its southeastern counterpart, this pair is quite sprawled and its opposite polarities are highly sheared, which may contribute to the poor performance of the dipole in this region. Overall, unlike the relatively gradual rotation of the field lines, the drastic changes in magnetic connectivity as measured by the mapping of field-line footpoints, i.e., the squashing factor Q, can obviously serve as a more sensitive indicator for the presence of a saddle-shaped decay-index profile.

Imagine that initially a low-lying MFR aligned along the PIL is strapped by the downward Lorentz force between the axial current J _a of the MFR and the transverse component B _t of the overlying field, i.e., $F={J}_{a}\,{B}_{t}\,\sin \varphi$, where φ is the angle between J _a and B _t and $\sin \varphi \lt 0$ in the low atmosphere. One may argue that the Lorentz force would turn upward, if the MFR somehow rose into the region where the B _t orientation is reversed with respect to the PIL (Figure 6(f)). However, without magnetic reconnection to open up the overlying field, the rising MFR would just push the overlying field upward.

An HFT is a preferential site for magnetic reconnection (Titov et al. 2002, 2003). When a rising MFR impacts the overlying field, the supposed reconnection between them would reduce the overlying flux and therefore weaken the constraining force, which facilitates the MFR's eruption, similar to the role of the null-point reconnection in the breakout model (Antiochos et al. 1999). On the other hand, the reconnection would shed magnetic twist from the MFR (e.g., Xue et al. 2016; Wyper et al. 2017; Huang et al. 2018; Yang et al. 2019), causing a redistribution of magnetic helicity over the larger system, and even the (partial) disintegration of the MFR (e.g., Liu et al. 2018b; Yan et al. 2020; Chen et al. 2021), which hampers the MFR's eruption.

To summarize, our study corroborates that the critical height is roughly half of the centroid distance between opposite polarities in a dipole field and clarifies where a saddle-like decay-index profile can be found in a quadrupole field. With some caution, these results can be applied to bipolar and quadrupolar ARs, respectively. Our study also reveals that saddle-like profiles may be associated with magnetic skeletons, such as a null point (see Figures 1(b) and (g)) or an HFT (Figure 6), which is anticipated to serve as a "double-edged sword" in eruptive processes in the case of magnetic reconnection with the eruptive structure taking place at such skeletons: reconnection may open up the overlying field, facilitating the eruption, but at the same time shed the flux of the eruptive structure, causing its disintegration or even failed eruption. Thus, the zone of stability at the saddle may indeed provide some constraint for the eruptive structure. This may explain the mixing results obtained in observational studies, as mentioned in the beginning of Section 4.3.