<figure><img src="image_1.png" /></figure>

# Magnetofrictional Modeling of an Erupting Pseudostreamer

Nishu Karna<fnref n="1" />, Antonia Savcheva<fnref n="1" />,<fnref n="2" />, Sarah Gibson<fnref n="3" />, Svetlin Tassev<fnref n="1" />, Katharine K. Reeves<fnref n="1" />, Edward E. DeLuca<fnref n="1" />, and Kévin Dalmasse<fnref n="4" />

Received 2019 July 5; revised 2021 March 24; accepted 2021 March 25; published 2021 May 24

## Abstract

In this study, we present the magnetic configuration of an erupting pseudostreamer observed on 2015 April 19, on the southwest limb of the Sun, with a prominence cavity embedded inside. The eruption resulted in a partial halo coronal mass ejection. The prominence eruption begins with a slow rise and then evolves to a fast-rise phase. We analyze this erupting pseudostreamer using the flux-rope insertion method and magnetofrictional relaxation to establish a sequence of plausible out-of-equilibrium magnetic configurations. This approach allows the direct incorporation of observations of structures seen in the corona (filament and cavity) to appropriately model the pseudostreamer based on SDO/HMI line-of-sight photospheric magnetograms. We also perform a topological analysis in order to determine the location of quasiseparatrix layers (QSLs) in the models, producing Q-maps to examine how the QSL locations progress in the higher iterations. We found that the axial flux in our best-fit unstable model was a factor of 20 times higher than we found in our marginally stable case. We computed the average magnetic field strength of the prominence and found that the unstable model exhibits twice the average field strength of the stable model. The eruption height from our modeling matches very well with the prominence eruption height measured from the AIA observation. The Q-maps derived from the model reproduce structures observed in LASCO/C2. Thus, the modeling and topological analysis results are fully consistent with the observed morphological features, implying that we have captured the large magnetic structure of the erupting filament in our magnetofrictional simulation.

Unified Astronomy Thesaurus concepts: Solar corona (1483)

## 1. Introduction

Coronal mass ejections (CMEs), the largest-scale eruptive phenomenon in our solar system, are one of the key drivers of our space weather. CMEs can eject enormous amounts of plasma, impacting interplanetary and planetary environments and causing geomagnetic storms. Various triggering mechanisms have been proposed either conceptually or through magnetohydrodynamics (MHD) analysis in interpretations of eruption initiations, including tether cutting (e.g., Moore et al. 2001; Kusano et al. 2004, 2012; Joshi et al. 2015; Zhang et al. 2015), magnetic breakout (Antiochos et al. 1999; Wyper et al. 2017), emerging flux (e.g., Louis et al. 2015), kink instability (e.g., Hood & Priest 1979, 1980), torus instability (e.g., Török et al. 2004; Kliem & Török 2006; Fan & Gibson 2007), and double arc instability (Ishiguro & Kusano 2017).

CMEs are often associated with the eruption of filaments embedded in the lobes of pseudostreamers. Pseudostreamers are also known as unipolar streamers, separating same polarity coronal holes and consisting of twin loop arcades without a current sheet in the outer corona (Hundhausen 1972; Zhao & Webb 2003; Wang et al. 2007; Karna et al. 2019). Pseudostreamer configurations are prone to producing both single (Wang et al. 2009; Zuccarello et al. 2012) and sympathetic twin filament eruptions (Schrijver & Title 2011; Török et al. 2011; Titov et al. 2012, 2017; Lynch & Edmondson 2013; Panasenco et al. 2013). Various numerical simulations have modeled pseudostreamer eruptions. Török et al. (2011) presented an idealized three-dimensional (3D) MHD simulation of a sympathetic eruption of twin filaments in a pseudostreamer observed on 2010 August 1. The authors inserted two flux ropes into both lobes within a

pseudostreamer and one flux rope located next to the pseudostreamer, demonstrating that the eruptions were initiated by reconnection at a separator high in the corona. The topology of the same event was also analyzed with the large-scale background coronal field using potential field source surface (PFSS) models (e.g., Titov et al. 2012, 2017). Panasenco et al. (2013) presented a PFSS extrapolation of a pseudostreamer observed on 2007 May 16 embedding a polar crown filament under the southern lobe of the pseudostreamer. The polar crown filament erupted nonradially and rolled northward in the direction of the pseudostreamer null line. From several other observations and modelings, it has been shown that a pseudostreamer topological structure provides a favorable environment for preconditioning sympathetic or sequential eruptions (e.g., Török et al. 2011; Titov et al. 2012, 2017; Lynch & Edmondson 2013; Panasenco et al. 2013).

The flux-rope insertion method developed by van Ballegooijen (2004) has been used successfully to model eruptions. Kliem et al. (2013) performed an MHD simulation using nonlinear force-free field (NLFFF) models from Su et al. (2011) as initial conditions. The authors showed that the flux rope in the unstable model experiences a full eruption in the MHD simulation. Further, the authors showed that the evolution of the currents and connectivities in the MHD simulation were very similar to what the magnetofrictional simulation produces. Moreover, Savcheva et al. (2015, 2016) used the flux-rope insertion method to study the evolution of erupting sigmoid regions, producing unstable models by adding more axial flux to the best-fit stable model so that the flux-rope stable model is pushed over the stability limit determined by balancing the constraining flux and the flux rope

<footnote n="1">¹Harvard & Smithsonian Center for Astrophysics, 60 Garden St., Cambridge, MA 02138, USA; nishu.karna@cfa.harvard.edu</footnote>

<footnote n="2">²Institute of Astronomy and National Astronomical Observatory, Bulgarian Academy of Sciences, 72 Tsarigradsko Chaussee Blvd., 1784 Sofia, Bulgaria</footnote>

<footnote n="3">³High Altitude Observatory, 3300 Center Green Dr., Boulder, CO, USA</footnote>

<footnote n="4">⁴IRAP, Université de Toulouse, CNRS, CNES, UPS, F-31028 Toulouse, France</footnote>
<figure><img src="image_2.png" /><figcaption>**Figure 1.** Snapshot images of the pseudostreamer as seen on the southwest limb on 2015 April 18, including the pseudostreamer embedding filament cavity, observed in (a) SDO/AIA 193 Å (19:30 UT, 2015 April 18) and (b) the best-fit NLFFF stable model of the pseudostreamer for the same date.</figcaption></figure>

flux. The authors then derived quasiseparatrix layer (QSL) maps from those models. The magnetic field solutions at different steps were compared with the flare ribbon evolution as observed by SDO/AIA. The authors showed a good match between QSL maps taken at low heights in the corona and chromosphere with the flare ribbons locations and extent at the beginning (Savcheva et al. 2015) and at different times during the progression (Savcheva et al. 2016) of the flare. These results imply that the magnetofrictional simulation is sufficient to look at the evolution of several flare features with less computational power and time. These past works give us confidence that the unstable solutions found by the magnetofrictional relaxation technique are suitable representations of the pseudostreamer eruption morphology, if not the dynamics.

In this paper, we study a filament eruption observed on 2015 April 19 which resulted in a partial halo CME with a round front and prominence core intersected by a streamer as seen in the Solar and Heliospheric Observatory's Large Angle and Spectrometric Coronagraph (SOHO/LASCO, Brueckner et al. 1995) C2 coronograph. The prominence was embedded in the northern lobe of the pseudostreamer. The pseudostreamer was observed from 2015 April 17 to 19 on the southwest limb in the AIA (Lemen et al. 2012) on board SDO (Pesnell et al. 2012; Figure 1(a)), however, a prominence and an associated cavity were evident only in the northern lobe. Observations from HMI (Hoeksema et al. 2014) on board SDO showed that the pseudostreamer was bounded by two negative polarity magnetic fields and was centered above a positive polarity field.

This pseudostreamer was the subject of recent papers (Gibson et al. 2017; Karna et al. 2019). Gibson et al. (2017) found that a potential field extrapolation of the photospheric field (assuming no currents) resulted in a magnetic null height that was too low and a magnetic flux expansion that was too small in comparison to linear polarization measurements of the pseudostreamer.

Karna et al. (2019), hereafter Article I, modeled the same pseudostreamer observed on April 18 using the flux-rope insertion method (i.e., incorporating a current channel at the location of the filament). The authors found that the null height for synthetic data produced from our NLFFF flux-rope inserted

models match infrared (IR) linear polarization from Coronal Multichannel Polarimeter (CoMP, Tomczyk et al. 2008) observations better than synthetic data produced from a PFSS model.

Matching the NLFFF stable model with the pseudostreamer observation on April 18 motivated the current work to extend our grid of models into the unstable range, finding realizable 3D nonlinear magnetic field configurations that are consistent with the observations. We know that the MHD dynamics associated with the eruption cannot be captured, however, this work demonstrates that by starting with a validated NLFFF configuration and rendering it unstable (albeit in a nonphysical way), we can obtain 3D field configurations that are consistent with the observations. Our approach is designed to capture the balance (or lack of balance) of magnetic forces for a given field configuration, although it lacks the physical description required to capture the resulting hydrodynamics and any feedback from the hydrodynamics on the magnetic field. One of the important contributions of this paper is the analysis that is enabled by the 3D representations of the field configurations we find. The solar observations are used to select the most appropriate regions to study and the topology tools are used for analysis.

The advantage of using the flux insertion NLFFF model is that: (i) it uses observed line-of-sight (LOS) photospheric magnetic flux as a boundary condition and further constrains the magnetic field in the corona by comparing with observed coronal features, (ii) models with different values of the axial and poloidal fluxes and height and length of the flux rope can be constructed, (iii) the critical values of flux rope close to eruption can be determined, and (iv) this process is much more economic in numerical resources.

The paper is organized as follows. In Section 2, we present extreme ultraviolet (EUV) and white light observations describing the coronal structures. In Section 3, we present a brief description of the initial magnetic field configuration, the flux-rope insertion method, the magnetofrictional relaxation method, the magnetic model, and the description of the QSL calculation. In Section 4, we present our observation, modeling, and topological analysis results. We conclude with a summary of our findings and discussion in Section 5.
<figure><img src="image_3.png" /><figcaption>**Figure 2.** SDO/AIA 193 Å observations of the prominence eruption on 2015 April 19 from 18:20 UT to 20:36 UT. The regions of interest are marked with arrows. Brightenings are pointed out by the blue arrows. Measurements are taken along the white dashed-dotted lines.</figcaption></figure>

## 2. Observations

SDO/AIA records solar images in seven EUV channels with a resolution of 0.6' px⁻¹ and a 12 s cadence. We used the 193 Å band to study the pseudostreamer evolution because the fine magnetic structures in the corona are clearly defined in this passband. To further enhance fine structure, we used the Solarsoft AIA radial filter routine.<fnref n="5" /> The AIA radial filter sums a number of sequential images from the same channel and divides the coronal component into rings. Each ring is then scaled based on its radius and associated pixel brightnesses relative to its neighbors. The disk is pulled from the central image in the sum to create the final output, e.g., Figure 2.

Figure 2 panels show observations of the targeted prominence within the pseudostreamer configuration prior to, during, and after eruption. The erupting prominence is embedded within the northern lobe of the pseudostreamer in the southern hemisphere on 2015 April 19 as observed in AIA 193 Å. This figure shows that the prominence is composed of a series of vertical threads prior to the eruption (indicated by the red arrow in the top left panel). The pseudostreamer was first observed on 2015 April 17 on the southwest limb with a dark cavity and a prominence only in the northern lobe and stayed visible for two days until April 19. The prominence begins to slowly rise

around 2015 April 19 16:02 UT at S44W and extends to its maximum height below the null point at 2015 April 19 18:20 UT (top left panel) before it starts rising. The cavity inflates a lot around this time. The prominence starts to rise quickly around 18:40 UT (top right panel) and propagates nonradially toward the null point. The cavity reforms quickly around 19:00 UT as indicated by the yellow arrow (middle left panel) and grows bigger in size. The erupted prominence leaves the AIA field of view completely around 2015 April 19 20:36 UT (bottom right panel). However, the pseudostreamer stays at the same location. We measured the height of the erupting prominence by fitting the motion of the top of the prominence (white dashed-dotted line in the top right and bottom left panels). We then measured the velocity of the erupting prominence by creating a stack height-time plot along the white dashed-dotted lines in Figure 2 from 2015 April 19 16:50 UT to 2015 April 19 18:30 UT with 10 minutes cadence. Figure 3 is the velocity height plot of the prominence eruption seen in AIA 193 Å. The black asterisks are the observed velocity and the blue line is the fit to the velocity. The maximum attained velocity within the AIA field of view was less than 21 km s⁻¹.

In association with this prominence eruption event, SOHO observed a partial halo CME (around 15° width). Figure 4 shows the running difference image observations of the prominence eruption and CME propagation as seen in LASCO/C2 and AIA 193 Å on 2015 April 19 from 18:48 UT to 22:00 UT with 24 minutes cadence. The sharp narrow streamer entered the (2–6 $R_{\odot}$) LASCO/C2 field of view starting around 19:12 UT (indicated by the yellow arrow in the top middle panel), about 32 minutes after prominence liftoff observed in the AIA image (top right panel). However, the CME front indicated by the blue arrow became visible only around 21:12 UT (bottom left panel), after an hour and half of prominence eruption. The red arrows point to the eruption region observed in the AIA 193 Å passband and the yellow and blue arrows indicate the region of interest in LASCO/C2. The CME had a velocity of 672 km s⁻¹, according to the SOHO/LASCO CME catalog.<fnref n="6" />

Figure 5 is the MLSO/COSMO K-coronagraph (K-Cor) observations of the prominence eruption to CME propagation on 2015 April 19 from 18:45 UT to 20:48 UT with a 30-minute cadence. The erupting prominence enters the K-Cor field of view around 18:45 UT as a small white sharp structure indicated by the red arrow in the top left panel. Eventually, around 19:48 UT, the prominence gets faint (indicated by the red arrow) but a streamer with open field lines becomes visible (indicated by the yellow arrow) in the middle left panel. The regions of interest are marked with red and yellow arrows.

## 3. Modeling and Topological Analysis

### 3.1. Flux-rope Insertion Method

The key to understanding the physical processes in the solar atmosphere is the solar magnetic field. Unfortunately, most routine full-disk vector magnetic field measurements at high spatio-temporal cadence are only available for the photosphere where the magnetic field is not force free. Therefore, we rely on magnetic field models constrained by the photospheric magnetic field measurements (Mackay & Yeates 2012). We

<footnote n="5">⁵ http://www.lmsal.com/solarsoft/</footnote>

<footnote n="6">⁶ https://cdaw.gsfc.nasa.gov/CME_list/</footnote>
<figure><img src="image_4.png" /><figcaption>Figure 3. Velocity calculated using SDO AIA 193 Å. The maximum propagation speed of prominence is 18 ± 3 km s⁻¹.</figcaption></figure>

extrapolate the photospheric measurements into the corona under the assumption that the field is force free, i.e., $j \times B = 0$, where $B(r)$ is the magnetic field and $j(r)$ is the electric current density, because in the corona the plasma pressure is very small compared to the magnetic pressure, i.e., the corona is a low-$\beta$ environment.

Our eruptive magnetic field models are constructed using the flux-rope insertion method. This flux rope insertion method has been successfully used to study different coronal structures: active regions with filaments by Bobra et al. (2008), Su et al. (2011), and Asgari-Targhi & van Ballegooijen (2012); the evolution of soft X-ray sigmoids by Savcheva & van Ballegooijen (2009) and Savcheva et al. (2012a, 2012b, 2012c 2015, 2016); a quiescent prominence by Su & van Ballegooijen (2012); and to study the magnetic structure of the pseudostreamer by Karna et al. (2019). Here we briefly describe the method: we use the Coronal Modeling System (CMS) software package to construct NLFFF models of the solar corona based on photospheric magnetograms. First of all, a wedge-shaped domain covering a large area surrounding the region of interest is selected. The magnetic field $B(r)$ in the domain is expressed in terms of the vector potential ($A$, where $B = \nabla \times A$) so that the solenoidal condition $\nabla \cdot B = 0$ is automatically satisfied. The LOS magnetograms obtained from SOHO/MDI or SDO/HMI are used because in the quiet Sun areas, the LOS components of the photospheric fields are measured with much greater precision than the transverse fields and are not subject to the 180 ambiguity problem (Bobra et al. 2008). The LOS magnetograms serve as the lower boundary condition and the “source surface” is positioned at the top boundary where the magnetic field is assumed to be radial and open. We also use an HMI/MDI synoptic map to compute a global potential field, which provides the side boundary conditions. The next step is to specify a flux rope path on the solar surface and its axial ($\phi_{axi}$) and poloidal ($F_{pol}$) fluxes. The $\phi_{axi}$ of the flux rope is represented by a thin tube that runs horizontally along the filament path and the $F_{pol}$ is represented by a set of inserted

closed field lines that wrap around the tube. The flux-rope path is manually selected following the path of observed filaments in EUV images. The two ends of the flux rope path are anchored at the photosphere via two vertical sections. We then modify the flux distribution by subtracting the fluxes equal to $F_{pol}$ at the two ends. The reason to do so is to ensure that after flux rope insertion the original flux distribution is recovered.

Next, we compute the vector potential $A(r)$ of the potential field in a domain based on the modified magnetic map. The potential field computation method is described in Appendix B of van Ballegooijen et al. (2000). The gauge of $A$ is chosen such that $A_r(r, \theta, \phi) = 0$. Our next step is to create a field-free cavity along the flux-rope path. This is achieved by setting $A = (R_\odot/r)A(r, \theta, \phi)$ above the selected path in the domain. As $A = 0$, the horizontal components of the magnetic field vanish in the cavity ($B_\theta = B_\phi = 0$). This has the effect of pushing up the horizontal fields to the top of the cavity, leaving only the vertical fields associated with flux located at its base which are along the polarity inversion line where vertical fields are weak, creating a nearly field-free cavity (van Ballegooijen et al. 2000; Bobra et al. 2008).

Then, the flux rope is inserted into the cavity as a combination of axial and poloidal flux. A grid of models is computed with different combinations of $\phi_{axi}$ and $F_{pol}$. These models are not in force-free equilibrium. So, they are relaxed to a force-free state using magnetofrictional relaxation with some amount of hyperdiffusion. The magnetofrictional method uses the induction equation to evolve the coronal field, expressed in terms of the vector potential according to the following equation (described in Bobra et al. 2008, Appendix A)

$$ \frac{\partial A}{\partial t} = v \times B - \eta_i \nabla \times B + \frac{B}{B^2} \nabla \cdot (\eta_4 B^2 \nabla \alpha), \quad (1) $$

where, $v$ is the plasma velocity, $\eta_i$ is the ordinary diffusion constant, $\eta_4$ is the hyperdiffusion constant (Bhattacharjee & Hameiri 1986; Boozer 1986), and $\alpha$ is a magnetic torsion
<figure><img src="image_5.png" /><figcaption>Figure 4. Running difference image observations of the prominence eruption to CME propagation seen in LASCO and AIA 193 Å on 2015 April 19 from 18:48 UT to 21:48 UT with a 24-minute cadence.</figcaption></figure>

parameter defined by $\alpha \equiv j \cdot \frac{B}{B^2}$, where the current $j = \nabla \times B$. The magnetofrictional velocity $v$ is given by

$$v = (fj - v_1 \hat{r} \times B) \times \frac{B}{B^2} \quad (2)$$

where $f$ is the magnetofriction coefficient and $v_1$ is an additional upward velocity that simulates the effects of magnetic buoyancy and pressure gradients in the photosphere.

The second term in the Equation (1) describes the ordinary diffusion. The third term in Equation (1) describes hyperdiffusion which is used to conserve magnetic helicity and smooth out any numerical artifacts in the electric current distribution. The vector potential in the corona is evolved according to the magnetic induction equation with a plasma velocity that is proportional to the Lorentz force. In the coronal volume of the magnetic field reconstruction, magnetofrictional velocities must vanish in order to produce a stable relaxed model (van Ballegooijen 2004).

This approach allows us to isolate the topologically distinct features of the pseudostreamer, determine critical values of fluxes beyond which structures do not form, examine their role in the definition of the stability boundary, and constrain them with direct comparison to observations. While it is theoretically possible to do a similar comparison with full MHD models, the computational parameter space is so large as to make it impractical.

In this paper, we follow the approach of Savcheva et al. (2016). Starting from a stable NLFFF model of the pseudostreamer, we produce a set of other models with flux ropes possessing increasingly large axial and poloidal fluxes to obtain solutions in which the magnetofrictional code can no longer relax to a stable equilibrium. We used the same procedure as in Article I for our model set up, described here briefly: we combined four full-disk HMI photospheric LOS magnetograms taken on 2015 April 13–16 (each at 17:01 UT, 24 hr cadence). The northern filament barbs appeared left-bearing in AIA absorption, so we designed the inserted
<figure><img src="image_6.png" /><figcaption>**Figure 5.** K-Cor observations of the prominence eruption to CME propagation on 2015 April 19 from 18:45 UT to 20:48 UT with a 30-minute cadence.</figcaption></figure>

flux rope similarly to have sinistral orientation and thus be a positive helicity right-handed flux rope, and we positioned the flux rope center at 14 Mm height. The domain extends from the photosphere to 2.8 $R_⊙$ source surface from the Sun center.

We started with the best-fit NLFFF model parameters ($4 \times 10^{19}$ Mx, $8 \times 10^9$ Mx cm$^{-1}$) of the pseudostreamer achieved in Article I shown in Figure 1(b). The best-fit stable model was selected by comparing the filament's location and height, cavity size, and overlying closed and open magnetic field lines to AIA images. We then used different combinations of axial and poloidal fluxes (higher than that NLFFF), ranging from $0.05-5 \times 10^{21}$ Mx in axial and $1-10 \times 10^{10}$ Mx cm$^{-1}$ in poloidal fluxes, respectively. For these axial and poloidal flux combinations, the magnetofrictional code no longer converges to an equilibrium solution. The flux rope rises higher as the solutions are iterated.

For all the iterations (time-steps) we use only hyperdiffusion and no ordinary diffusion. For the first 100 steps we use a value of $7.03 \times 10^{20}$ km$^4$ s$^{-1}$ for the hyperdiffusion coefficient. For the next 10,000 iterations we scale down the hyperdiffusion coefficient from $2.34 \times 10^{20}$ km$^4$ s$^{-1}$ to $7.03 \times 10^{19}$ km$^4$ s$^{-1}$ to $2.34 \times 10^{19}$ km$^4$ s$^{-1}$. After 10,000 iterations we use the final value of $2.34 \times 10^{19}$ km$^4$ s$^{-1}$ for the coefficient. We relaxed our models up to 27,000 iterations. This relaxation scheme stops the flux rope from diffusing too much during the relaxation procedure (Su et al. 2011).

## 3.2. Magnetic Field Topology

We used squashing factor ($Q$, Titov 2007) maps to analyze the eruptive pseudostreamer magnetic structure. The QSLs separate the volume into magnetically disparate regions connected by bundles of field lines and are used as a proxy to determine where strong electric current sheets can form during the evolution of a magnetic field having a high Lundquist number in the corona (Aulanier et al. 2005). The study of QSL magnetic reconnection has attracted much attention in recent years (e.g., Guo et al. 2012; Janvier et al. 2014; Zhao et al. 2014; Dalmasse et al. 2015; Mandrini et al. 2015; Savcheva et al. 2015). The position of QSLs has been well connected with the location of flares and CMEs (Savcheva et al. 2015 and references therein), coronal and chromospheric heating (Wang et al. 2000), the source region of slow solar wind (Titov et al. 2011), and the magnetic reconnection at QSLs such as the origin of the active region outflows (Baker et al. 2009; Mandrini et al. 2015), active regions bordering coronal holes contributing to a slow solar wind (van Driel-Gesztelyi et al. 2012), and magnetic null points and separators associated with the pseudostreamers (Titov et al. 2012).

We performed the QSL calculation in the same way as described in Article I. We used the Tassev & Savcheva (2017) open source code (QSL squasher) to compute $Q$ for the entire volume of our unstable model for each iteration (until 210,000 iteration) and studied the evolution of the QSL from one iteration to the other. We used the same starting grid size as that of our unstable models for the QSL calculations.

## 4. Results

From our different combinations of axial and poloidal fluxes, we found that the models with axial fluxes more than $4 \times 10^{19}$ Mx were unstable, as the flux ropes have more axial flux than can be contained. Out of 33 unstable models, the best eruptive model was selected by comparing to the location and height of the filament, overlying closed and open magnetic field lines in AIA images, and the CME propagation in the SOHO/LASCO C2 images.

We found that the model with an axial flux of $8 \times 10^{20}$ Mx and a poloidal flux of $3 \times 10^{10}$ Mx cm$^{-1}$ is the best unstable model. The axial flux in this unstable model is much higher than the marginally stable case we found in our Article I, i.e., $4 \times 10^{19}$ Mx, however, the magnetic structure generated from the relatively high poloidal flux matches the observed untwisting motion during the filament eruption. In order to verify the model's flux parameters, we roughly estimated the time ($T$) it takes to bring the flux rope's axial flux from its pre-eruptive value ($4 \times 10^{19}$ Mx) to its eruptive one ($8 \times 10^{20}$ Mx) using $T = (\text{difference in the axial flux})/(f \times L \times B \times \text{Vphot})$, assuming that flux cancellation is happening all along the polarity inversion line. In this case, $L$ is the length of flux rope path (223 Mm from Article I), Vphot is the typical photospheric plasma velocity ($0.1$ km s$^{-1}$), $B$ is the magnetic flux contained in 1 pixel, and $f$ is the filling factor. We found that the time to eruption is (i) at most $T = 9.46$ hr for $f = 1/B = 100$ G (using $B$ typical of evolved active regions), (ii) $T = 26.29$ hr for $f = 1/B = 36$ G ($B$ is computed from the unstable model), (iii) $T = 473.34$ hr for $f = 0.1/B = 20$ G (using a $B$ that is typical of a high latitude filament). Hence, spanning different values of $f$ and $B$ will provide different times. Case (iii) would be comparable with the lifetime of a high latitude filament. Cases
<figure><img src="image_7.png" /><figcaption>**Figure 6.** 3D magnetic field evolution of an erupting pseudostreamer at 10,000, 90,000, 130,000, and 150,000 iterations. Unrelaxed reconnecting field lines around the cusp are indicated by the gray shaded rectangular box in the bottom right panel and the reconnection region of the field lines around the null points are enclosed by the turquoise shaded rectangular box. Notice the untwisting magnetic field lines away from the pseudostreamer.</figcaption></figure>

(i) and (ii) both result in a reasonable time for our case because the stable NLFFF model (Article I) was optimized to fit coronal data taken at around 23 hr prior to the eruption time (2015-04-19T18:20). However, in case (i) the magnetic field is unrealistically high for a polar crown filament, and in both cases, the filling factor is likely to be less than one. This result leads to the conclusion that flux is canceled via a different mechanism than through random photospheric motions prior to the eruption. Models indicate that before an eruption, the flux cancellation accelerates below the flux rope, building up more flux into the flux rope accompanied by a rise of the structure (van Ballegooijen & Martens 1989). The slow rise of the filament observed in our study prior to the eruption could be an indication that the flux cancellation rate is increasing in the photosphere below our flux rope during the pre-eruption phase.

Pre-eruption flux cancellation rates similar to our value of $7.6 \times 10^{20}$ Mx over 23 hr have been found in prior studies. For example, Yardley et al. (2019) studied an intermediate filament and presented two models when the filament was in equilibrium. The authors found that the total flux canceled as calculated from the HMI observations was $3.2 \times 10^{20}$ Mx over a period of 33 hr. This amount of flux is comparable to the difference in axial flux between two NLFFF models in their work, suggesting that flux cancellation builds flux into the filament’s magnetic structure prior to an eruption. These observations suggest that our calculated increase of $7.6 \times 10^{20}$ Mx over 23 hr is a reasonable result.

<figure><img src="image_8.png" /><figcaption>**Figure 7.** 3D magnetic model at 140,000 iterations plotted over the LASCO/C2 image at 21:24 UT. The red and yellow arrows indicate the streamer and the CME leading edge. Magnetic field lines of the model end at $2.8R_⊙$.</figcaption></figure>

Figure 6 is a 3D view of the erupting pseudostreamer at 10,000 (top left), 90,000 (top right), 130,000 (bottom left), and 150,000 (bottom right) iterations. We interpret the sequence of magnetofrictional steps as an evolution of the CME propagation. Magnetic field lines of the model end at $2.8R_⊙$. We note
<figure><img src="image_9.png" /><figcaption>**Figure 8.** Free energy of the unstable model from 10,000 to 290,000 iterations.</figcaption></figure>

that as the relaxation progresses, the connectivities of the field lines change per the evolution of the unstable flux rope. The untwisted magnetic field lines are away from the pseudostreamer. In the top right panel of Figure 6, the first two open field lines of the flux rope can be observed. However, at higher iterations, the field lines are not relaxed. This is illustrated by the turquoise shaded rectangular box which encloses the reconnecting field lines around the null region and the gray box around the cusp in the bottom panels of Figure 6.

We found that the input axial flux remained unchanged but the input poloidal flux changed significantly during the magnetofrictional evolution. The input poloidal flux ($3 \times 10^{10}$ Mx cm$^{-1}$) rapidly reduced until 30,000 iterations ($4.2 \times 10^{09}$ Mx cm$^{-1}$) and reduced more slowly later on (e.g., $2 \times 10^{09}$ Mx cm$^{-1}$ by 10,000 iterations) during the magnetofrictional relaxation process. This nonuniform change in the poloidal flux was due to the fact that the diffusion in the magnetofrictional evolution does not proceed uniformly.

The unstable model had $6.7 \times 10^{20}$ Mx total poloidal flux taking the initial input poloidal flux into account and the ratio of axial to poloidal flux was 0.6. The twist angle of the flux rope was $3.34 \pi$ estimated using $\phi = 2\pi F_{\text{pol}}L/\Phi_{\text{axi}}$ (Savcheva & van Ballegooijen 2009) exceeding the critical twist ($2.5\pi$) derived by Hood & Priest (1981). L = 223 Mm is the length of the inserted flux rope from Article I.

We also calculated the line-of-sight average magnetic field of the inserted flux rope determined from the dipped grid points as done in Mackay et al. (2020). Using CMS, we calculated magnetic dips extending up to $1.086 R_{\odot}$ (60,000 km) along the polarity inversion line. This is the height around which the prominence erupted. The average magnetic field for the stable model from Article I was 4.4 G with minimum and maximum values of 0.03 and 19 G. The unstable model is more than two times larger than the stable case. The average B for the unstable model is 9.8 G with minimum and maximum values of 0.1 and 36 G. We are unaware of any magnetic observations of the prominence we are modeling; such measurements are not as numerous and routine as photospheric and chromospheric magnetic field measurements. Moreover, these measurements are also limited to individual prominences and not full off-limb disks like CoMP is doing in infrared. However, Mackay et al. (2020) measured the magnetic strength of prominence observed on 2013 September 24, using THEMIS/MTR observations.

Most of the magnetic field values of this prominence were in the range 5–15 G and some were as high as 18 G, in line with our model values.

Figure 7 shows the unstable model at 140,000 iterations plotted over the AIA 193 Å and LASCO/C2 images taken at 21:24 UT. There is an overlap between $2R_{\odot}$ and $2.8R_{\odot}$ in the model and LASCO/C2 image. We find that the location of the streamer, the linear bright feature, coincides well with the open field lines indicated by the red arrow. Moreover, twisted field lines indicated by the yellow arrow match with the CME bright leading edge as seen in LASCO/C2. However, the two bright regions enclosed by the two yellow rectangular boxes are missing in our model as well as in the K-cor image in Figure 5. CMEs are optically thin and the nature of their scattering allows us to see along the LOS with equal radiance, which makes it difficult to distinguish between plasma that is part of the CME (including its compression region) and plasma that is elsewhere along the LOS. In addition, the orientation of the CME feature structure changes with observing location (Howard 2015). The CME bubble in our unstable model appears narrower than the one observed by LASCO/C2, but the major features of the eruption—the leading edge, the streamer position from the LASCO/C2 image, and the cavity location from the AIA image—are well captured in our model.

Figure 8 gives a summary of free energies for the unstable model from iteration 10,000 to 290,000. The magnetic free energy is the difference between magnetic energies of the force-free field and potential field energies, where the potential field satisfies the same boundary conditions at each iteration. The free energy computation method is described in Bobra et al. (2008), Appendix B. The free energy keeps decreasing with increase in the iteration number. The bulk of the energy (83%–89%) decreases due to frictional dissipation where energy is taken out of the field as the friction term acts or through energy transport through the top boundary. Some amount of energy (11%–17%) decreases as field lines reconnect. We note that during the magnetofrictional process, the frictional dissipation goes on decreasing with each iteration and the field line reconnection goes on increasing. Between 10,000 to 290,000 iterations, free energy decreased from $1.9 \times 10^{31}$ to $1.17 \times 10^{31}$. A significant decline in free energy is noted until 100,000 iterations, which then slows down.
<figure><img src="image_10.png" /><figcaption>**Figure 9.** Top: height-time plot of erupted prominence on 2015 April 19, measured using SDO AIA 193 Å. The blue arrow and red horizontal dashed lines indicate the critical height at which the prominence erupted (58 Mm) with an error of ±2 Mm. Bottom: decay index plot from the model. The decay index *n* is calculated from the potential field model. The peak of the curve indicates *n* = 1.5, which corresponds to the critical value for the torus instability at the height of 56 Mm ± 5 Mm indicated by a vertical red dashed line.</figcaption></figure>

The top panel of Figure 9 displays the mean height-time plot of the prominence eruption seen in AIA 193 Å. We measured the height of the erupting prominence several times by fitting the top of the prominence (e.g., white dashed curve in Figure 2 top right and bottom left panels), which allowed us to produce a mean and standard deviation of the likely position of the erupting prominence height. The standard deviation represents the uncertainty in the measurement. The bottom panel of Figure 9 displays the decay index plot from the potential field model. The decay index ($n$)=(-dlnB/dlnz) measures the rate at which the overlying magnetic field falls off as computed from the potential field model. The decay index of the overlying magnetic field reaches about 1.5 at 56 ± 5 Mm. This corresponds to the height at which the prominence erupted as

observed in AIA 193 Å (Figure 9, top panel), i.e., around 58 Mm with an error of ±2 Mm. The eruption height of our model matches quite well with the AIA observation.

Figure 10 shows the Log₁₀Q map (vertical 2D slice of a 3D QSL map) for two different iterations: 10,000 (left) and 90,000 (right). Q is highest in the red regions and lowest in the blue. A well defined pseudostreamer structural skeleton with two separatrix surfaces, vertical and dome-like, can be seen in Figure 10 (left panel). In the left panel, the pseudostreamer configuration consists of an embedding cavity (red arrows), the fan (black arrows), the spine field line (white arrows), and the 2D-projected magnetic null points (green arrows). The maximum Q values are observed at the core and edge of the cavity, the fan, the spine field line, and the 2D-projected magnetic null points, which are the most probable sites for reconnection. In the
<figure><img src="image_11.png" /><figcaption>**Figure 10.** Log₁₀Q maps at 10,000 (left) and 90,000 (right) iterations of the erupting pseudostreamer in a spherical domain expanded to 2.8R⊙. Red, black, white, and green arrows in the left panel point out the cavity, the fan, the spine field lines, and the 2D-projected magnetic null points. The purple arrow in the right panel points out the transferred high Q region in the higher iteration.</figcaption></figure>

<figure><img src="image_12.png" /><figcaption>**Figure 11.** Log₁₀Q maps plotted over LASCO/C2 with AIA 193 images. 10,000 iterations over 19:00 UT (top left), 140,000 iterations over 21:12 UT (top right), 130,000 iterations over 21:24 UT (bottom left), and 150,000 iterations over 21:36 UT (bottom right) in a spherical domain expanded to 2.8R⊙. The bright linear feature indicated by the red arrow is a pseudostreamer. The yellow arrow indicates the leading edge of the CME.</figcaption></figure>

later iteration (90,000) the high Q is transferred to the blob region (purple arrow, Figure 10, right panel).

Next, we compared 2D Log₁₀Q maps for several different iterations with LASCO/C2 with AIA 193 observations taken at different times shown in Figure 11. Q-maps for an erupting pseudostreamer at iterations: 10,000, 130,000, 140,000, and 150,000 are plotted over LASCO/C2 with AIA 193 images at: 19:00 UT (top left), 21:12 UT (top right), 21:24 UT (bottom left), and 21:36 UT (bottom right), respectively. The Q map

expands until 2.8R⊙. The bright features in the LASCO/C2 image are the regions of high plasma density. There are several small features in the Q map that do not have corresponding features in the coronagraph image and vice versa, however, the major CME features observed in LASCO/C2 coincide well with the Q map. In the top left panel, the prominence has already erupted but has not reached the LASCO/C2 field of view. In the top right panel, the high Q value indicated by the red arrow coincides well with the streamer, a bright linear
feature associated with the pseudostreamer LASCO/C2. The other high Q value pointed by the yellow arrow is the CME leading edge. In the bottom left panel, we find both high Q value streamer (indicated by the red arrow) and the CME blob (indicated by the yellow arrow) coincides well with the LASCO/C2 observation. In the bottom right panel, the streamer location in the Q map overlaps well in the LASCO/ C2. However, the CME leading edge outgrows the Q map. Moreover, the eruption extends from the south side of the Q map. If the Q map grid had been extended more to the south then we might have captured the southward extended portion of the CME. From our Q map we find that the CME leading edge had high Q values compared to the streamer. The evolution and location of the QSLs are strongly correlated with the high current buildup regions, which are the probable sites of reconnection. Overall there is a good match between the motion of the QSLs and the CME features. Thus, the CME topology is more or less well captured in the Q map.

**5. Conclusion and Discussion**

In this study, starting with a well constrained magnetic field model of a pseudostreamer from Article I, we have followed the evolution of an unstable flux rope and compared the evolving topological structures with observations. The consistency of the model results with clearly observed morphological features suggests that we have captured the large-scale magnetic structure of the erupting filament. This gives us confidence that our analysis of the pseudostreamer eruption onset is the result of the processes described below.

1. There are three phases of the eruption (a) a slow rise of the prominence until it reaches critical height, (b) the prominence detaching from the limb and propagating toward the null point making a curve with maximum speed of 21 km s⁻¹, (c) reconnection at the null point, after which the CME propagates with a speed about 672 km s⁻¹ in LASCO/C2.

2. Brightenings are observed in AIA 193 Å (Figure 2, blue arrows) as the detached prominence makes a curve (nonradial propagation) and reconnects at the null point. These brightenings may be an indication of heating due to energy released from reconnection around the null point and below the pseudostreamer separatrix.

3. A comparison of several of our unstable models with SDO and SOHO observations motivated the use of a strong twisted flux rope model. The axial flux in our best-fit unstable model is a factor of 20 times higher than the threshold of stability. However, we find that if there is too much axial and poloidal flux in the inserted flux rope, then a pseudostreamer with a cavity structure does not form.

4. The average field strength within the prominence for the unstable model (9.8 G) is twice that of the stable model (4.4 G) from Article I.

5. The decay index of the overlying potential magnetic field computed from the unstable model reaches about 1.5 at 56 ± 5 Mm which matches quite well with the prominence eruption height (58 ± 2 Mm) in the AIA observation.

6. High Q values are observed at the cavity core and edge, the fan, spine field lines, and 2D-projected magnetic null points and later transferred to the CME blob region in

higher iterations (Figure 10). QSLs gives a better approximation of the evolution of the eruption, suggesting that the CME topology is well captured in the Q map.

The slow rise phase of a filament observation lasting up to several hours before it erupts is consistent with the filament eruption observations of Sterling et al. (2007). The prominence progressively elevates and reaches the critical height at which the decay index *n* of the overlying potential magnetic field reaches about 1.5, after which the prominence detaches from the photosphere.

We find that the prominence enters the regime of torus instability at the onset of the eruption. McCauley et al. (2015) studied 106 limb filament eruptions and found that the decay index at the onset of the fast-rise phase spanned values from 0.7–2. The strongest change in the filament height–time plot happened at *n* = 1.5 (Figure 9) which is very close to the value of *n* = 1.4 ± 0.1 computed by Zuccarello et al. (2016) at the apex of the flux rope axis. The erupting prominence propagated toward the null point. This event is strongly supportive of the idea that pseudostreamer configurations are favorable for eruptive prominences reconnecting at a null point. The curved (nonradial) propagation of the erupting prominence toward the null point is due to two reasons. (1) Nearby coronal holes deflect the erupting plasma and embedded fields and (2) CME and filaments motion are guided toward the null point (i.e., weaker field regions at different heights in the overlying configuration; Panasenco et al. 2013). Panasenco et al. (2013) also demonstrated these results using both low and high latitude filament eruptions and associated CMEs and found that the large-scale forces acting on the CME were asymmetric due to the presence of the coronal hole. As a consequence, the CME propagates nonradially away from the coronal hole. The cavity evolves from a circular to a narrow oval structure and is nonradially ejected in the direction of the X-point.

The authors thank the two anonymous reviewers whose constructive comments helped improve this manuscript. The authors thank Rosa Niembro Hernández for the helpful discussion. The AIA and HMI data are courtesy of NASA/SDO and the AIA and HMI Science Investigation Teams. MLSO is operated by the High Altitude Observatory, a division of the National Center for Atmospheric Research (NCAR). This material is based upon work supported by the NCAR, which is a major facility sponsored by the National Science Foundation under Cooperative Agreement No. 1852977. This work was supported by the Air Force Office of Scientific Research grant FA9550-15-1-0030 and in part by NASA Heliophysics Supporting Research 80NSSC19K0860. K.D. gratefully acknowledges the support of the French Centre National d'Études Spatiales. A.S. acknowledges the support of NASA Grand Challenge grant 80NSSC17K0016 and NASA Heliophysics Supporting Research grant 80NSSC19K0860.

**ORCID iDs**

Nishu Karna https://orcid.org/0000-0002-1314-4690
Antonia Savcheva https://orcid.org/0000-0002-5598-046X
Sarah Gibson https://orcid.org/0000-0001-9831-2640
Katharine K. Reeves https://orcid.org/0000-0002-6903-6832
Edward E. DeLuca https://orcid.org/0000-0001-7416-2895
Kévin Dalmasse https://orcid.org/0000-0001-8929-4006
References

Antiochos, S. K., DeVore, C. R., & Klimchuk, J. A. 1999, ApJ, 510, 485

Asgari-Targhi, M., & van Ballegooijen, A. A. 2012, ApJ, 746, 81

Aulanier, G., Pariat, E., & Démoulin, P. 2005, A&A, 444, 961

Baker, D., van Driel-Gesztelyi, L., Mandrini, C. H., Démoulin, P., & Murray, M. J. 2009, ApJ, 705, 926

Bhattacharjee, A., & Hameiri, E. 1986, PhRvL, 57, 206

Bobra, M. G., van Ballegooijen, A. A., & DeLuca, E. E. 2008, ApJ, 672, 1209

Boozer, A. H. 1986, JPIPh, 35, 133

Brueckner, G. E., Howard, R. A., Koomen, M. J., et al. 1995, SoPh, 162, 357

Dalmasse, K., Chandra, R., Schmieder, B., & Aulanier, G. 2015, A&A, 574, A37

Fan, Y., & Gibson, S. E. 2007, ApJ, 668, 1232

Gibson, S. E., Dalmasse, K., Rachmeler, L. A., et al. 2017, ApJL, 840, L13

Guo, Y., Ding, M. D., Schmieder, B., Démoulin, P., & Li, H. 2012, ApJ, 746, 17

Hoeksema, J. T., Liu, Y., Hayashi, K., et al. 2014, SoPh, 289, 3483

Hood, A. W., & Priest, E. R. 1979, SoPh, 64, 303

Hood, A. W., & Priest, E. R. 1980, SoPh, 66, 113

Hood, A. W., & Priest, E. R. 1981, GApFD, 17, 297

Howard, T. A. 2015, JSWSC, 5, A22

Hundhausen, A. J. 1972, Coronal Expansion and Solar Wind (Berlin: Springer)

Ishiguro, N., & Kusano, K. 2017, ApJ, 843, 101

Janvier, M., Aulanier, G., Bommier, V., et al. 2014, ApJ, 788, 60

Joshi, N. C., Liu, C., Sun, X., et al. 2015, ApJ, 812, 50

Karna, N., Savcheva, A., Dalmasse, K., et al. 2019, ApJ, 883, 74

Kliem, B., Su, Y. N., van Ballegooijen, A. A., & DeLuca, E. E. 2013, ApJ, 779, 129

Kliem, B., & Török, T. 2006, PhRvL, 96, 255002

Kusano, K., Bamba, Y., Yamamoto, T. T., et al. 2012, ApJ, 760, 31

Kusano, K., Maeshiro, T., Yokoyama, T., & Sakurai, T. 2004, ApJ, 610, 537

Lemen, J. R., Title, A. M., Akin, D. J., et al. 2012, SoPh, 275, 17

Louis, R. E., Kliem, B., Ravindra, B., & Chintzoglou, G. 2015, SoPh, 290, 3641

Lynch, B. J., & Edmondson, J. K. 2013, ApJ, 764, 87

Mackay, D., & Yeates, A. 2012, LRSP, 9, 6

Mackay, D. H., Schmieder, B., López Ariste, A., & Su, Y. 2020, A&A, 637, A3

Mandrini, C. H., Baker, D., Démoulin, P., et al. 2015, ApJ, 809, 73

McCauley, P. I., Su, Y. N., Schanche, N., et al. 2015, SoPh, 290, 1703

Moore, R. L., Sterling, A. C., Hudson, H. S., & Lemen, J. R. 2001, ApJ, 552, 833

Panasenco, O., Martin, S. F., Velli, M., & Vourlidas, A. 2013, SoPh, 287, 391

Pesnell, W. D., Thompson, B. J., & Chamberlin, P. C. 2012, SoPh, 275, 3

Savcheva, A., Green, L., van Ballegooijen, A., & DeLuca, E. 2012a, ApJ, 759, 105

Savcheva, A., Pariat, E., McKillop, S., et al. 2015, ApJ, 810, 96

Savcheva, A., Pariat, E., McKillop, S., et al. 2016, ApJ, 817, 43

Savcheva, A., Pariat, E., van Ballegooijen, A., Aulanier, G., & DeLuca, E. 2012b, ApJ, 750, 15

Savcheva, A., & van Ballegooijen, A. 2009, ApJ, 703, 1766

Savcheva, A., Van Ballegooijen, A., & DeLuca, E. 2012c, ApJ, 744, SBD12

Schrijver, C. J., & Title, A. M. 2011, JGRA, 116, A04108

Sterling, A. C., Harra, L. K., & Moore, R. L. 2007, ApJ, 669, 1359

Su, Y., Surges, V., van Ballegooijen, A., DeLuca, E., & Golub, L. 2011, ApJ, 734, 53

Su, Y., & van Ballegooijen, A. 2012, ApJ, 757, 168

Tassev, S., & Savcheva, A. 2017, ApJ, 840, 89

Titov, V. S. 2007, ApJ, 660, 863

Titov, V. S., Mikić, Z., Linker, J. A., Lionello, R., & Antiochos, S. K. 2011, ApJ, 731, 111

Titov, V. S., Mikic, Z., Török, T., Linker, J. A., & Panasenco, O. 2012, ApJ, 759, 70

Titov, V. S., Mikić, Z., Török, T., Linker, J. A., & Panasenco, O. 2017, ApJ, 845, 141

Tomczyk, S., Card, G. L., Darnell, T., et al. 2008, SoPh, 247, 411

Török, T., Kliem, B., & Titov, V. S. 2004, A&A, 413, L27

Török, T., Panasenco, O., Titov, V. S., et al. 2011, ApJL, 739, L63

van Ballegooijen, A. A. 2004, ApJ, 612, 519

van Ballegooijen, A. A., & Martens, P. C. H. 1989, ApJ, 343, 971

van Ballegooijen, A. A., Priest, E. R., & Mackay, D. H. 2000, ApJ, 539, 983

van Driel-Gesztelyi, L., Culhane, J. L., Baker, D., et al. 2012, SoPh, 281, 237

Wang, H., Yan, Y., Sakurai, T., & Zhang, M. 2000, SoPh, 197, 263

Wang, Y.-M., Muglach, K., & Kliem, B. 2009, ApJ, 699, 133

Wang, Y.-M., Sheeley, N. R., Jr., & Rich, N. B. 2007, ApJ, 658, 1340

Wyper, P. F., Antiochos, S. K., & DeVore, C. R. 2017, Natur, 544, 452

Yardley, S. L., Savcheva, A., Green, L. M., et al. 2019, ApJ, 887, 240

Zhang, Q. M., Ning, Z. J., Guo, Y., et al. 2015, ApJ, 805, 4

Zhao, J., Li, H., Pariat, E., et al. 2014, ApJ, 787, 88

Zhao, X. P., & Webb, D. F. 2003, JGRA, 108, 1234

Zuccarello, F. P., Aulanier, G., & Gilchrist, S. A. 2016, ApJL, 821, L23

Zuccarello, F. P., Bemporad, A., Jacobs, C., et al. 2012, ApJ, 744, 66