A Prospective New Diagnostic Technique for Distinguishing Eruptive and Noneruptive Active Regions

The magnetofrictional simulation describes the magnetic field evolution in a Cartesian 3D domain by considering the simultaneous stressing and relaxation of the coronal magnetic field. The stressing of the field is due to the evolution of the magnetic flux distribution at the lower photospheric boundary determined from a time series of magnetogram observations. The relaxation occurs from specifying the velocity to be proportional to the Lorentz force in the 3D domain. Full details of the NLFFF model can be found in Mackay et al. (2011). In this model, the 3D domain is a Cartesian box where the solar surface is placed at the lower z boundary. The horizontal directions, x and y, extend over a sufficient region of the solar surface to fully contain the active region. In this study, the time series of NLFFF configurations is constructed assuming closed boundaries at the four sides of the 3D box (no normal magnetic field), while the magnetic field can have a normal component across the top boundary. The bottom boundary, which represents the solar surface, is forced to have an evolving and balanced magnetic flux.

The model uses a zero-β approximation where this provides an accurate representation of the evolution of the coronal magnetic field over timescales longer than the Alfv́en crossing time. The initial coronal magnetic field for each active region is assumed to be a potential field, and at later times, the evolution of the magnetic field at the lower boundary (derived from observed line-of-sight magnetograms) leads to the injection and buildup of electric currents in the corona. Thus, the coronal magnetic field evolves to a new NLFFF configuration. It is the evolution of the magnetic flux at the lower boundary that is key to the buildup of magnetic forces in the domain during this quasi-static evolution. The relaxation of the magnetic configuration is tuned to match the relaxation times in the solar corona.

Occasionally, during the quasi-static evolution, the model cannot converge to a new NLFFF equilibrium, due to the buildup of large flux ropes. This usually occurs in conjunction with the liftoff of a magnetic flux rope in the model, where magnetic reconnection below the flux rope leads to a strong outward magnetic tension (Mackay & van Ballegooijen 2006b). At this point, the NLFFF model is no longer appropriate and full MHD is required to describe the correct dynamics (Pagano et al. 2013b).

In order to develop a technique to identify active regions in which magnetic flux rope ejections occur, we consider a number of active regions that have previously been analyzed in detail. In five of these active regions, observable signatures of eruptions have been clearly identified and the other three show no such signatures. Yardley et al. (2018a) provides an overview of what observable signatures can be interpreted as the occurrence of an eruption in an active region. Table 1 shows the main properties of the active regions selected for this study, and we refer to them as eruptive or noneruptive active regions as appropriate (Rodkin et al. 2017; James et al. 2018; Yardley et al. 2018a, 2018b; S.L. Yardley et al. 2019, in preparation). To identify eruptions, we focus mostly on dynamic signatures found within coronal images that indicate a rapid plasma displacement or ejection. These signatures include coronal dimmings, filament eruptions, the disappearance of coronal loops, or postflare magnetic field rearrangement (Yardley et al. 2018a). While CMEs can be linked to solar flares, both phenomena can occur without the other (Gopalswamy 2004). Due to this, we do not use GOES data, which are more related to burst heating or energetic particles compared to a large-scale displacement or rearrangement in the coronal field. For the present study, we favored active regions isolated from large concentrations of magnetic flux in order to simplify the analysis. Each of the active regions is observed to undergo a qualitatively different evolution over the time period studied. Some of them (e.g., AR 11504 and AR 11561) show clear indications of magnetic flux emergence, while others do not.

For each of the active regions, we simulate the evolution of the 3D coronal field over the time period given in Table 1, using the modeling approach discussed in Mackay et al. (2011) and the follow-up works of Mackay et al. (2011), Gibb et al. (2014), Rodkin et al. (2017), Yardley et al. (2018b), and S.L. Yardley et al. (2019, in preparation). For the present study, we do not analyze the evolution of the coronal field in detail, but we focus on defining a metric based on the evolution of the magnetic field configuration and the vertical component of the Lorentz force, LF_z, to identify eruptive active regions.

Figure 1 presents a typical example of the output of our model. The left-hand side panel shows the magnetic field distribution at the solar surface along with magnetic field lines from the model overplotted. The right-hand side shows the associated vertical component of the Lorentz force at the same surface. Initial studies show that simply using these 2D maps at the lower boundary cannot distinguish eruptive from noneruptive active regions.

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We first adopt a proxy for the formation of magnetic flux ropes previously applied in Rodkin et al. (2017) and Pagano et al. (2018). This approach allows us to track magnetic flux ropes using the function

$$\begin{eqnarray}&&{\rm{\Omega }}=\sqrt{{{\rm{\Omega }}}_{x}^{2}+{{\rm{\Omega }}}_{y}^{2}+{{\rm{\Omega }}}_{z}^{2}},\end{eqnarray} \tag{ 2 }$$

where

$$\begin{eqnarray}&&{{\rm{\Omega }}}_{x}=\displaystyle \frac{\left|B\times {\rm{\nabla }}{B}_{x}\right|}{\left|{\rm{\nabla }}{B}_{x}\right|},\end{eqnarray} \tag{ 3 }$$

$$\begin{eqnarray}&&{{\rm{\Omega }}}_{y}=\displaystyle \frac{\left|B\times {\rm{\nabla }}{B}_{y}\right|}{\left|{\rm{\nabla }}{B}_{y}\right|},\end{eqnarray} \tag{ 4 }$$

$$\begin{eqnarray}&&{{\rm{\Omega }}}_{z}=\displaystyle \frac{\left|B\times {\rm{\nabla }}{B}_{z}\right|}{\left|{\rm{\nabla }}{B}_{z}\right|}.\end{eqnarray} \tag{ 5 }$$

The function Ω depends on the twist of the magnetic field and its strength. For example, Ω_z peaks at PILs where gradients of B_z are perpendicular to the direction of ${\boldsymbol{B}}$. Such a function is useful for identifying where flux ropes have formed or are about to form. To produce time-dependent 2D maps that represent the location on the surface where flux ropes exist in the corona, we consider ${\omega }^{* }\left(x,y,t\right)$, the integral of Ω along the z direction,

$$\begin{eqnarray}&&{\omega }^{* }\left(x,y,t\right)={\int }_{z=0}^{z={z}_{\max }}{\rm{\Omega }}\left(x,y,z,t\right){dz},\end{eqnarray} \tag{ 6 }$$

where z = 0 is the lower boundary of the computational box and z = z_max is the upper boundary. Figure 2 shows a map of the function ${\omega }^{* }\left(x,y,t\right)$ normalized to its maximum value for the eruptive active region AR 11561. Typically, we find that ${\omega }^{* }\left(x,y,t\right)$ has significantly larger values at a few locations located across the active region, but it also has nonzero values over a large portion of the active region.

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Finally, we need to obtain a normalized distribution, $\omega \left(x,y,t\right)$, of the quantity ${\omega }^{* }\left(x,y,t\right)$ that can be used for the derivation of the metric ζ. Thus, we exclude a frame of 16 pixels near the x and y boundaries to avoid boundary effects, and we renormalize the functions ω* to be between the values of 0 and 1:

$$\begin{eqnarray}&&\omega \left(x,y,t\right)=\displaystyle \frac{{\omega }^{* }\left(x,y,t\right)-\min ({\omega }^{* }\left(x,y,t^{\prime} \leqslant t\right))}{\max ({\omega }^{* }\left(x,y,t^{\prime} \leqslant t\right))-\min ({\omega }^{* }\left(x,y,t^{\prime} \leqslant t\right))}.\end{eqnarray} \tag{ 7 }$$

It should be noted that this normalization is time dependent, as at time t the function is normalized with respect to the maximum and minimum values for t' ≤ t. This means that the values of $\omega \left(x,y,t\right)$ at time t are not affected by the evolution of the function ${\omega }^{* }\left(x,y,t\right)$ after time t.

Next, we focus on the z-component of the Lorentz force because a large value indicates which active regions favor the ejection of magnetic flux ropes. In each of the simulations, the z-component of the Lorentz force at the lower boundary shows a complex distribution (Figure 1). However, the photospheric Lorentz force represents an incomplete description of the forces that are acting as the equilibrium of magnetic structures results from an interplay of forces exerted at different heights in the atmosphere. Therefore, we compute the integral of the vertical component of the Lorentz force along the z direction (I_LFZ),

$$\begin{eqnarray}&&{I}_{\mathrm{LF}Z}\left(x,y,t\right)={\int }_{z=0}^{z={z}_{\max }}{{LF}}_{z}\left(x,y,z,t\right){dz}.\end{eqnarray} \tag{ 8 }$$

Figure 3 shows the distribution of I_LFZ at the time of the observed eruption for AR 11561. Typically, we find that the function I_LFZ exhibits highly localized positive and negative values. The lower atmosphere below an arbitrary height of 5 Mm usually contributes more than 75% to I_LFZ. However, there are many extended spatial locations where this contribution is smaller and more than 25% of the Lorentz force integral originates from heights greater than 5 Mm.

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To consider the vertical forces acting on a magnetic flux rope, we compute the average of the function ${I}_{\mathrm{LF}Z}\left(x,y,t\right)$ over a moving circular mask ${C}_{0.7\,\mathrm{Mm}}^{({x}_{c},{y}_{c})}$ of radius 0.7 Mm centered in (x_c, y_c), which we define as

$$\begin{eqnarray}&&{\mu }^{* }\left(x,y,t\right)=\displaystyle \frac{{\int }_{{C}_{0.7\,\mathrm{Mm}}^{({x}_{c},{y}_{c})}}^{}{I}_{\mathrm{LF}Z}\left(x^{\prime} ,y^{\prime} ,t\right){dx}^{\prime} {dy}^{\prime} }{\pi {0.7}^{2}}.\end{eqnarray} \tag{ 9 }$$

The value of ${\mu }^{* }\left(x,y,t\right)$ may vary in time at a given location and can change sign within the same polarities. We also find that close to or beneath magnetic flux ropes, both positive or negative values of ${\mu }^{* }\left(x,y,t\right)$ can occur. Positive values of ${\mu }^{* }\left(x,y,t\right)$ at a specific location do not necessarily suggest an ejection has occurred, as it is possible that an outward-directed Lorentz force is balanced by the restoring force of the overlying magnetic field. However, on average, an outwardly directed Lorentz force is a necessary condition for a flux rope ejection. A map of ${\mu }^{* }\left(x,y,t\right)$ is not shown as it is very similar to Figure 3.

For this quantity, we also derive $\mu \left(x,y,t\right)$ from ${\mu }^{* }\left(x,y,t\right)$,

$$\begin{eqnarray}&&\mu \left(x,y,t\right)=\displaystyle \frac{{\mu }^{* }\left(x,y,t\right)-\min ({\mu }^{* }\left(x,y,t^{\prime} \leqslant t\right))}{\max ({\mu }^{* }\left(x,y,t^{\prime} \leqslant t\right))-\min ({\mu }^{* }\left(x,y,t^{\prime} \leqslant t\right))},\end{eqnarray} \tag{ 10 }$$

as explained in Section 3.1.

For the final quantity in the construction of the metric, we are interested in identifying locations where the overlying magnetic field does not balance new positive forces generated at the lower boundary during the evolution. Therefore, we compute the mean quadratic departure from the average Lorentz force, which is computed using the same circular mask as Equation (9),

$$\begin{eqnarray}&&{\sigma }^{* }\left(x,y,t\right)=\displaystyle \frac{{\int }_{{C}_{0.7\,\mathrm{Mm}}^{({x}_{c},{y}_{c})}}^{}\sqrt{{\left[{I}_{\mathrm{LF}Z}\left(x^{\prime} ,y^{\prime} ,t\right)-\mu \left(x,y,t\right)\right]}^{2}}{dx}^{\prime} {dy}^{\prime} }{\pi {0.7}^{2}}.\end{eqnarray} \tag{ 11 }$$

The quantity ${\sigma }^{* }\left(x,y,t\right)$ is a measure of how heterogeneous, ${I}_{\mathrm{LF}Z}\left(x,y,t\right)$, is the area under investigation. We find that there may or may not be a simple correlation between the distributions of ${\sigma }^{* }\left(x,y,t\right)$ and ${\mu }^{* }\left(x,y,t\right)$. However, there are spatial locations where both functions have high values. At these locations, the integral of the Lorentz force is positive and heterogeneous, indicating that within these locations, the Lorentz force is significantly higher and lower than its mean value. Figure 4 shows a map of the function ${\sigma }^{* }\left(x,y,t\right)$ for the eruptive active region AR 11561 at the time of the eruption. We find that only a few elongated structures of high ${\sigma }^{* }\left(x,y,t\right)$ are present in the domain, whereas in most of the active region, ${\sigma }^{* }\left(x,y,t\right)$ remains rather low compared to its maximum value. There is one particular structure, which is highlighted by the green square in Figure 4, that shows a large value of ${\sigma }^{* }\left(x,y,t\right)$.

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Finally, we also apply a normalization to ${\sigma }^{* }\left(x,y,t\right)$ to derive $\sigma \left(x,y,t\right)$, defined as

$$\begin{eqnarray}&&\sigma \left(x,y,t\right)=\displaystyle \frac{{\sigma }^{* }\left(x,y,t\right)-\min ({\sigma }^{* }\left(x,y,t^{\prime} \leqslant t\right))}{\max ({\sigma }^{* }\left(x,y,t^{\prime} \leqslant t\right))-\min ({\sigma }^{* }\left(x,y,t^{\prime} \leqslant t\right))}.\end{eqnarray} \tag{ 12 }$$

The normalization of $\omega \left(x,y,t\right)$, $\mu \left(x,y,t\right)$, and $\sigma \left(x,y,t\right)$ allows for the comparison of the eruption metric ζ between different active regions. Moreover, the normalized quantities plateau their values when the nonnormalized functions increase in time.

By comparing Figures 2–4, it is apparent that for eruptive active regions, the spatial locations over which ω shows higher values include the corresponding locations where either μ or σ shows high values. We also find that each individual function can have a value close to 1; however, this rarely happens simultaneously for all three functions. The same conclusions on the spatial distribution of $\omega \left(x,y,t\right)$, $\mu \left(x,y,t\right)$, and $\sigma \left(x,y,t\right)$ can be drawn for the active regions where no eruptions are found. As anticipated, the newly introduced eruption metric ζ (Equation (1)) combines the information from ω, μ, and σ; is bounded between 0 and 1; and is the product of three normalized quantities that are functions of space and time. For consistency in notation, we define $\zeta \left(x,y,t\right)$ as

$$\begin{eqnarray}&&\zeta \left(x,y,t\right)=\omega \left(x,y,t\right)\mu \left(x,y,t\right)\sigma \left(x,y,t\right).\end{eqnarray} \tag{ 13 }$$

Figure 5 shows the maps of $\zeta \left(x,y,t\right)$ for the five eruptive active regions in this study, at the time of the observed eruption. Over the majority of the domain, we find that the value of $\zeta \left(x,y,t\right)$ is generally close to 0, except at a few locations where it takes a value close to 0.1. The black dashed circles identify the origin of the eruptions as observed by previous studies (Rodkin et al. 2017; James et al. 2018; Yardley et al. 2018a), while the blue triangles identify the maximum value of $\zeta \left(x,y,t\right)$ at different times in the active region evolution. We find that the location of the maximum value of $\zeta \left(x,y,t\right)$ usually matches the location of the eruption in the observations. The match is particularly good for active regions where the eruption was due to a filament eruption (AR 11680, AR 11261, AR 11504), for example, the eruption of a large filament that was associated with the internal PIL of AR 11680. The path of the filament matches the location of high $\zeta \left(x,y,t\right)$ values in the magnetofrictional simulation of the active region. In four out of five cases, the eruption occurs at the location of maximum $\zeta \left(x,y,t\right)$; however, this is not the case for AR 11561. For this active region, there is strong flux emergence during which the two magnetic polarities diverge. During this divergence, the location of maximum $\zeta \left(x,y,t\right)$ moves with one polarity. The eruption does not occur at the location of maximum $\zeta \left(x,y,t\right)$; however, the value of $\zeta \left(x,y,t\right)$ is still high at the location of the observed eruption. Figure 6 shows maps of $\zeta \left(x,y,t\right)$ for the three active regions where no eruption was reported. For each case, $\zeta \left(x,y,t\right)$ is shown at the time when it reaches its maximum value. The values of $\zeta \left(x,y,t\right)$ are in general lower and more localized for the noneruptive active regions compared to those found for the eruptive active regions.

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To distinguish eruptive from noneruptive active regions, we carry out a twofold process. First of all, we compute a spatial average of the eruption metric $\zeta \left(x,y,t\right)$ over a square of size 5.8 Mm to remove local effects. Next, we consider the maximum of the spatial average, and we consider the evolution of this maximum obtained as a function of time, ${\zeta }_{\max }\left(t\right)$. Figure 7 shows the resulting evolution of ${\zeta }_{\max }\left(t\right)$ for each of the active regions considered here. The red curves represent eruptive active regions, with the red asterisks indicating the time of eruption as seen in the observations, and the blue curves represent noneruptive active regions. We find that, for all simulations, there is an increase in ${\zeta }_{\max }\left(t\right)$ at the start of the evolution, which is due to the injection of electric currents from boundary motions. This leads to the magnetic field configuration departing from its initial potential state. We estimate that it takes around 35 magnetograms (between 40 and 55 hr, depending on the cadence of the magnetograms) for the magnetofrictional simulation to lose memory of the initial potential configuration, and we consider this the ramp-up phase of the magnetofrictional simulations. After this ramp-up phase, the system tends to converge to a value of ${\zeta }_{\max }\left(t\right)$ that is significantly different for eruptive active regions compared to the noneruptive ones. Eruptive active regions tend to converge to values between 0.03 and 0.05, while noneruptive active regions usually converge to values around 0.02. However, the evolution of ${\zeta }_{\max }\left(t\right)$ still fluctuates after the initial ramp-up phase. Some active regions (AR 11261, AR 11504, AR 11437) show an instantaneous decline of ${\zeta }_{\max }\left(t\right)$ post-eruption. This does not occur for AR 11561 as the magnetogram series ceases post-eruption. AR 11680 shows the highest value of ${\zeta }_{\max }\left(t\right)$ for all of the simulations and for an extended period of time.

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The present analysis does not provide a unique way to link high values of ${\zeta }_{\max }\left(t\right)$ to the likelihood of an eruption, due to two main features. The first one is that noneruptive active regions show values of ${\zeta }_{\max }\left(t\right)$ instantaneously higher than those of eruptive active regions, and vice versa, close to the ramp-up phase. The second one is that for the eruptive active regions, the time of the eruption does not always coincide with the time when ${\zeta }_{\max }\left(t\right)$ is at its maximum.

However, the values of ${\zeta }_{\max }\left(t\right)$ in eruptive active regions are generally higher than in noneruptive active regions. Therefore, this property can be used to define a new metric that distinguishes between eruptive and noneruptive active regions. For that reason, we consider $\bar{\zeta }$, the time average of ${\zeta }_{\max }\left(t\right)$ between the end of the ramp-up phase and the final magnetogram for each active region. For some active regions, the final magnetogram is after the observed eruption time, but this has a marginal effect on the time average. The time average of ${\zeta }_{\max }\left(t\right)$ is a single number and can therefore be directly associated with the likelihood that an active region will produce an eruption. Figure 8 shows the value of $\bar{\zeta }$ for each active region. We find that eruptive active regions in our sample show significantly higher values of $\bar{\zeta }$ and are separated from noneruptive active regions. For purely operational purposes, we compute a threshold of $\bar{{\zeta }_{\mathrm{th}}}=0.028$ as the average between the maximum value of $\bar{\zeta }$ among the noneruptive active regions and the minimum value of $\bar{\zeta }$ among the eruptive active regions (dashed horizontal line in Figure 8). It should be noted that the scattering in $\bar{\zeta }$ within the populations of eruptive and noneruptive active regions is larger than the minimum difference in $\bar{\zeta }$ between the two populations. As a consequence, it remains a possibility that if a larger sample of active regions is considered, then the two populations would partially overlap. This will be considered in future studies along with improvements in the metric.

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The present study shows that it is possible to identify a metric and a threshold value based only on the magnetic field configuration that discerns between eruptive and noneruptive active regions. Of course, this claim is based on a limited sample of eight active regions that we have analyzed, but an important aspect is that the present analysis makes no specific assumption of the triggering mechanism of the eruption or the processes that lead to the accumulation and release of free magnetic energy. If future larger sample studies show that these results are robust, the present modeling and analysis technique has allowed us to derive a threshold $\bar{\zeta }$ that represents the likelihood of an eruption occurring in an active region from a series of magnetogram measurements coupled with an NLFFF evolutionary model. If the robustness is shown, then the technique will have significant operational capacity.

To evolve the surface and coronal fields beyond t₀, we project the evolution of the magnetograms from t = t₀ to the time of the final magnetogram in the observed time sequence t = t_f. The method of projection is now described. Let ${{\boldsymbol{A}}}_{\mathrm{pt}}=({A}_{x},{A}_{y})$ be the vector potential at the photospheric boundary that can be integrated from B_z. The observed magnetograms (B_z) and derived vector potential, ${{\boldsymbol{A}}}_{\mathrm{pt}}$, are used to drive the evolution of the coronal field until time t₀. We assume that the electric field at the lower boundary remains constant from t₀ to t_f:

$$\begin{eqnarray}&&\displaystyle \frac{\partial {{\boldsymbol{A}}}_{\mathrm{pt}}\left(t\gt {t}_{0}\right)}{\partial t}=\displaystyle \frac{{{\boldsymbol{A}}}_{\mathrm{pt}}\left({t}_{0}\right)-{{\boldsymbol{A}}}_{\mathrm{pt}}\left({t}_{0}-{\rm{\Delta }}t\right)}{{\rm{\Delta }}t}={{\boldsymbol{E}}}_{\mathrm{pt}},\end{eqnarray} \tag{ 14 }$$

where ${{\boldsymbol{E}}}_{\mathrm{pt}}$ is the electric field deduced from the last two observed magnetograms used in the simulation at t = t₀ and t = t₀ − Δt, respectively. The time cadence of the magnetograms is either Δt = 96 or Δt = 60 minutes. This means that the vector potential at the lower boundary after t₀ is constructed by using

$$\begin{eqnarray}&&{{\boldsymbol{A}}}_{\mathrm{pt}}\left(t\gt {t}_{0}\right)={{\boldsymbol{A}}}_{\mathrm{pt}}\left({t}_{0}\right)+\left(t-{t}_{0}\right){{\boldsymbol{E}}}_{\mathrm{pt}}.\end{eqnarray} \tag{ 15 }$$

While ${{\boldsymbol{E}}}_{\mathrm{pt}}$ remains constant in time, it varies spatially from one pixel to the next. This is equivalent to observing an active region until a given time, t = t₀, after which we project its future evolution by using the most recent information available from the magnetogram time series. In this way, we have a hybrid simulation based on both observed and projected magnetograms. We can perform the same analysis as in Section 3 to find the evolution of ${\zeta }_{\max }\left(t\right)$ and the value of $\bar{\zeta }$ associated with the hybrid simulation.

To understand how the evolution of ${\zeta }_{\max }\left(t\right)$ is affected by the projection of the magnetograms, we vary the time at which we stop using the observed magnetograms as the lower boundary conditions of the simulation. To simplify the comparison, we choose to terminate all simulations at t = t_f. The simulations with t₀ = t_f are described in Section 2, where all available observational data are used to reproduce the coronal evolution of the active region. For each active region, we run 19 simulations with projected magnetograms, i.e., varying from t₀ = t_f − 19Δt to t₀ = t_f − Δt.

Figure 9 shows maps of the surface magnetic field B_z for the final state (t = t_f) of the simulations where t₀ = t_f (all observations are used), t₀ = t_f − 5Δt, and t₀ = t_f − 10Δt for AR 11561. We find that the simulation with the projected magnetograms from t₀ = t_f − 5Δt reproduces the majority of the magnetic field features (loops and surface magnetic field distribution) found in the final magnetic configuration of the full data simulation (t₀ = t_f). In contrast, the projected simulations from t₀ = t_f − 10Δt are visibly different from the full observed data simulation. However, these differences are mostly found along the boundaries of the flux concentrations and within weaker magnetic field locations. Due to this, they do not significantly affect the overall connectivity of the coronal field of the active region.

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For the purpose of this work, we are interested in studying the evolution of our eruption metric ζ when we have replaced observed magnetograms with projected ones. These numerical experiments are useful in understanding how much the evolution of our metric depends on the long-term evolution of the active region. Figure 10 shows the time evolution of ${\zeta }_{\max }\left(t\right)$ for the simulations concerning the five eruptive active regions. Unlike Figure 7, we do not normalize the values of $\omega \left(x,y,t\right)$, $\mu \left(x,y,t\right)$, and $\sigma \left(x,y,t\right)$ in order to show how the evolution differs when varying the time at which we switch between observed and projected magnetograms. In Figure 10, the red curve represents the simulation where t₀ = t_f, and the green and blue curves correspond to simulations using projected magnetograms. We find that the evolution of ${\zeta }_{\max }\left(t\right)$ can differ significantly depending on the length of projection. The closer the time t₀ is to t_f, the closer the evolution of ${\zeta }_{\max }\left(t\right)$ is to the simulation where t₀ = t_f, i.e., the cases with the least amount of projection lead to the smallest differences compared to the full observational data case. In most of the cases, when we introduce projected magnetograms, we obtain larger values of ${\zeta }_{\max }\left(t\right)$, as our projection technique is equivalent to a persistent electric field in each magnetogram pixel. In contrast, we expect that the magnetic field variations at the lower boundary are not always constant over an extended period of time. For this reason, the evolution of ${\zeta }_{\max }\left(t\right)$ shows its most significant departures when flux emergence occurs in the projected magnetograms, as these events are assumed to persist over the full projection time. AR 11680 has the longest time series of observed magnetograms where projected magnetograms are used only after around 90 observed magnetograms (∼6 days). However, the evolution of ${\zeta }_{\max }\left(t\right)$ is the least scattered. This emphasizes the importance of having continuous long-lasting data sets, where the persistence of information can be maintained in the coronal field. In contrast, AR 11561 and AR 11504 have the shortest observed magnetogram sequence before projected magnetograms are used, and the corresponding evolution of ${\zeta }_{\max }\left(t\right)$ is highly scattered. In Figure 10, the green curves show the simulations associated with t₀ = t_f − 5Δt and t₀ = t_f − 10Δt. For AR 11561, this is the ${\zeta }_{\max }\left(t\right)$ evolution associated with the images in the center and right columns of Figure 9. It is remarkable that in spite of the differences between the panels in Figure 9, the evolution of ${\zeta }_{\max }\left(t\right)$ does not significantly change when t₀ = t_f − 10Δt. This is true for most of the active regions with the exception of AR 11437. AR 11437 shows observational signatures of an eruption about 10 hr before the end of the magnetogram series, which corresponds to a phase of increasing ${\zeta }_{\max }\left(t\right)$. In light of this, ${\zeta }_{\max }\left(t\right)$ decreases after the eruption because the system has released energy leading to a decrease in the Lorentz force and the magnetic field complexity. The simulation fails to describe this evolution when projected magnetograms are used as the lower boundary conditions.

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Similar conclusions can be drawn from Figure 11, which shows the same evolution of ${\zeta }_{\max }\left(t\right)$ for the noneruptive active regions. In simulations which use projected magnetograms, the evolution of ${\zeta }_{\max }\left(t\right)$ tends to deviate from the reference simulation following the slope at the time when projected magnetograms are introduced. The evolution of ${\zeta }_{\max }\left(t\right)$ can differ substantially when projected magnetograms are introduced; however, the simulation results diverge less from the reference simulation as t₀ approaches t_f.

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Another goal of applying a projected evolution is to identify, in advance, active regions that will erupt. In Section 3, we concluded that the parameter $\bar{\zeta }$ (the time average of ${\zeta }_{\max }\left(t\right)$) best discriminates eruptive from noneruptive active regions. Therefore, we compare the value of $\bar{\zeta }$ obtained for the simulations using only observed magnetograms with the ones using projected magnetograms over different projection timescales. Figure 12 shows the value of $\bar{\zeta }$ for each simulation using projected magnetograms (blue asterisks) in comparison with the simulation with only observed magnetograms (red asterisk) as a function of t₀ for the eruptive active regions. We use green asterisks to signify $\bar{\zeta }$ for the simulations with t₀ = t_f − 5Δt and t₀ = t_f − 10Δt. We find that in all active regions where we use projected magnetograms, the simulations converge to the true value of $\bar{\zeta }$, as t₀ approaches t_f. It is clear that the value of $\bar{\zeta }$ for most of the active regions is accurately reproduced by the predictive simulations when t₀ ≥ t_f − 10Δt, whereas for AR 11261 and AR 11437, it happens only when t₀ ≥ t_f − 5Δt.

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If we compare the value of $\bar{\zeta }$ to the threshold value $\bar{{\zeta }_{\mathrm{th}}}$, we find that for the majority of t₀, $\bar{\zeta }$ remains larger than $\bar{{\zeta }_{\mathrm{th}}}$, indicating the possible occurrence of an eruption. For some cases, the value of $\bar{\zeta }$ oscillates about the threshold, although it always exhibits significant time periods where it is above the threshold. For active regions AR 11561, AR 11261, and AR 11504, the predictions of $\bar{\zeta }$ result in higher $\bar{\zeta }$ compared to the simulation where t₀ = t_f. This occurs when flux emergence is ongoing in the active region at the time we switch from observed to projected magnetograms. This is due to the simple projection technique applied, which leads to a continuous increase of magnetic flux and magnetic stress. When magnetic flux is not emerging, the values of $\bar{\zeta }$ can be predicted more accurately in advance. To improve the accuracy of this approach significantly, a projection technique that mitigates the effect of magnetic flux emergence over long periods of time must be developed.

Figure 13 shows the same plot for the noneruptive active regions, confirming that the final value of $\bar{\zeta }$ can be estimated several time steps before the final magnetogram. In general, the entire set of predictive simulations shows a behavior consistent with the simulation using only observed magnetograms, where the value of $\bar{\zeta }$ remains lower than the threshold value. Again, we find some values of t₀ where the predicted value of $\bar{\zeta }$ is significantly different from the simulation at t₀ = t_f, but they are rather isolated or occur over the longest predictive timescales, where the use of observational information is limited.

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Figures 12 and 13 show that the parameter $\bar{\zeta }$ can consistently identify eruptive from noneruptive active regions. With some limitations, this is also true when we replace observed magnetograms with projected ones. This result emphasizes the potential of the technique in (i) selecting which active regions are most or least likely to erupt or (ii) comparing the likelihood of eruptions between two active regions. In order to further assess the robustness of this approach, in the next section, we investigate the role of additional random noise in simulations during the timeframe of the projected magnetograms.

To test the robustness of this modeling technique, we run additional simulations where projected magnetograms are perturbed with a component of random noise. The purpose of these numerical experiments is to investigate how the predicted value of $\bar{\zeta }$ is affected by random perturbations.

We test the effect of random noise on two active regions from our set, AR 11561 (eruptive) and AR 11813 (noneruptive). As described in Equation (14), the quantity that drives the variation of the lower boundary is the electric field. First, we compute the two components of the surface electric field, ${E}_{x}\left(x,y\right)$ and ${E}_{y}\left(x,y\right)$ (i.e., the right-hand side in Equation (14)) from the final two observed magnetograms. In the previous simulations presented in Section 4.2, ${E}_{x}\left(x,y\right)$ and ${E}_{y}\left(x,y\right)$ were kept constant in time over the projected evolution. In contrast, to introduce a randomly varying electric field, we next compute the mean value of ${E}_{x}\left(x,y\right)$ and ${E}_{y}\left(x,y\right)$, and the standard deviation σ_Ex and σ_Ey of the electric field. These values are computed over the whole computational domain, where we note that the strong field regions only occupy a small subset of the domain. Finally, a random noise component is added to ${E}_{x}\left(x,y\right)$ and ${E}_{y}\left(x,y\right)$ at each pixel where the random component is varied at the magnetogram acquisition cadence (i.e., 96 or 60 minutes). It should be noted that the values of σ_Ex and σ_Ey are approximately two orders of magnitude smaller than the electric field values in the active region center. This is a consequence of computing these values over the full domain, where the strong field regions are localized at the center of the domain. In the simulations with noise, the noise component varies in time around the values of ${E}_{x}\left(x,y\right)$ and ${E}_{y}\left(x,y\right)$, thus its contribution averaged over time is asymptotically zero. The purpose of this exercise is therefore to test the robustness of the metric against localized fluctuations of the electric field and its temporal variations. We run three simulations where the amplitude of the noise component is ${E}_{\mathrm{noise}}^{64}=64{\sigma }_{E}$, ${E}_{\mathrm{noise}}^{256}=256{\sigma }_{E}$, and ${E}_{\mathrm{noise}}^{512}=512{\sigma }_{E}$ (where we replace σ_Ex and σ_Ey with σ_E for simplicity of notation). Large values of the noise relative to the standard deviation are required as when the standard deviation is computed over the full domain it is very small.

We present the results for the simulations with t₀ = t_f − 5Δt and t₀ = t_f − 10Δt for both of the active regions. Figure 14 shows the final distribution of B_z for the three simulations with ${E}_{\mathrm{noise}}^{64}=64{\sigma }_{E}$, ${E}_{\mathrm{noise}}^{256}=256{\sigma }_{E}$, and ${E}_{\mathrm{noise}}^{512}=512{\sigma }_{E}$ for t₀ = t_f − 10Δt for AR 11561. The true observed magnetograms can be seen in Figure 1 for comparison. We find that most magnetic structures are not significantly affected by the noise. The noise only becomes visible at the single pixel level when ${E}_{\mathrm{noise}}={E}_{\mathrm{noise}}^{512}$. We find similar results for AR 11813 (not shown here).

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Figure 15 shows the evolution of ${\zeta }_{\max }\left(t\right)$ for the two active regions. The red curve represents the evolution of ${\zeta }_{\max }\left(t\right)$ when t₀ = t_f (i.e., no noise and only observed magnetograms are used). The green curves represent the evolution when noise is added. This is compared to the blue curves, which represent the corresponding unperturbed simulations using projected magnetograms that deviate from the simulation with t₀ = t_f at the same t₀. We find that the simulations with noise do not largely depart from the associated simulations without noise. Using projected magnetograms in the simulations compared with observed magnetograms has a more prominent effect in departing from the evolution of ${\zeta }_{\max }\left(t\right)$. When we introduce a noise component to the projected magnetogram simulations, there is a marginal effect on the evolution of ${\zeta }_{\max }\left(t\right)$ for AR 11561 whereas there is no significant effect for AR 11813. Moreover, these minor differences are smoothed out when we focus on the value of $\bar{\zeta }$ (Figure 16), as we find that the predictions with noise are similar to the associated prediction without noise. Thus, the distinction between eruptive and noneruptive active regions is still maintained. It is clear that the deviations due to noise are smaller than the variations in the estimation of $\bar{\zeta }$ due to the use of projected magnetograms. This occurs even with large amplitudes of noise with respect to the value of σ_E. This happens for two main reasons. On one hand, the standard deviation of the electric field from its mean is two orders of magnitude smaller than the electric field acting on the active regions and therefore its small-scale fluctuations can only marginally affect the evolution of the magnetic field of the active region. On the other hand, as the time average of the electric field associated with the noise components tends to zero, it does not significantly affect the values of our eruption metric. This analysis shows that our projection technique remains largely insensitive to the introduction of noise and, thus, the underlying mechanisms we are investigating and predicting have a physical nature and are effective in distinguishing eruptive from noneruptive active regions.

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