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

# Designing a New Coronal Magnetic Field Energy Diagnostic

Marcel F. Corchado-Albelo¹, Kévin Dalmasse², Sarah Gibson³, Yuhong Fan³, and Anna Malanushenko³

¹Department of Astrophysical and Planetary Sciences, University of Colorado Boulder, 2000 Colorado Avenue, Boulder, CO 80305, USA;
marcel.corchado@colorado.edu

²IRAP, Université de Toulouse, CNRS, CNES, UPS, F-31028 Toulouse, France

³National Center for Atmospheric Research, USA

Received 2019 August 14; revised 2020 October 10; accepted 2020 October 15; published 2021 January 21

## Abstract

In the solar corona, the free energy, i.e., the excess in magnetic energy over a ground-state potential field, forms the reservoir of energy that can be released during solar flares and coronal mass ejections. Such free energy provides a measure of the magnetic field nonpotentiality. Recent theoretical and observational studies indicate that the presence of nonpotential magnetic fields is imprinted into the structures of infrared, off-limb, coronal polarization. In this paper, we investigate the possibility of exploiting such observations for mapping and studying the accumulation and release of coronal free magnetic energy, with the goal of developing a new tool for identifying “hot spots” of coronal free energy such as those associated with twisted and/or sheared coronal magnetic fields. We applied forward modeling of infrared coronal polarimetry to three-dimensional models of nonpotential and potential magnetic fields. From these we defined a quantitative diagnostic of nonpotentiality that in the future could be calculated from a comparison of infrared, off-limb, coronal polarization observations from, e.g., the Coronal Multi-channel Polarimeter or the Daniel K. Inouye Solar Telescope, and the corresponding polarization signal forward-modeled from a potential field extrapolated from photospheric magnetograms. We considered the relative diagnostic potential of linear and circular polarization, and the sensitivities of these diagnostics to coronal density distributions and assumed boundary conditions of the potential field. Our work confirms the capacity of polarization measurements for diagnosing nonpotentiality and free energy in the solar corona.

Unified Astronomy Thesaurus concepts: Solar magnetic fields (1503); Infrared astronomy (786); Solar corona (1483); Solar flares (1496)

Supporting material: animations

## 1. Introduction

Observation of structures in the solar corona, e.g., coronal loops and filaments, suggest that the plasma traces out the magnetic field lines (e.g., Nakagawa & Raadu 1972; Aulanier et al. 1998; Malanushenko et al. 2009). These plasma observations are not direct diagnostics of magnetic field strength, however. For this, spectropolarimetric measurements are required. Most routine and available magnetic field data are from the solar photosphere and chromosphere (see the review by, e.g., Lagg et al. 2017, and references therein). These magnetic field data are used as a surface boundary condition on three-dimensional (3D) reconstruction models to determine an approximate 3D solution of the magnetic field in the solar coronal volume (e.g., Démoulin et al. 1997; Mikić et al. 1999; Amari et al. 2006; Inoue et al. 2011; Feng et al. 2012; Malanushenko et al. 2012; Zhu et al. 2013).

3D magnetic reconstruction models often assume a magnetic “force-free” field condition, $j \times B = 0$ (e.g., the Lorentz force vanishes), which can be rewritten as $\nabla \times B = \alpha B$ (see reviews by, e.g., Wiegelmann & Sakurai 2012; Wiegelmann et al. 2014, and references therein). The potential and minimum energy field is the state for which the force-free parameter $\alpha = 0$ and is equivalent to a current-free state of the corona (Woltjer 1958). Any nonpotential field (for example, a force-free field with $\alpha \neq 0$) carries more magnetic energy than the potential field for the same boundary conditions.

Energy exceeding potential energy is known as the free (magnetic) energy (e.g., Régnier & Priest 2007; Shibata & Magara 2011; Gilchrist et al. 2012). The amount of free energy

plays a key role for the production of energetic solar events, such as solar flares and coronal mass ejections (CMEs). Indeed, the free energy is the magnetic energy reservoir available for flaring activity (e.g., Low & Smith 1993; Fan 2010; Valori et al. 2013). Space weather forecasters focus on highly energized events, such as solar flares and CMEs, because of the hazard that this type of event could have on the space/terrestrial environment (e.g., Carrington 1859). Space weather forecast models often make use of observations of magnetic footpoints of coronal structures (e.g., sunspots) and characterize their complexity and nonpotentiality (e.g., Georgoulis & Rust 2007; Falconer et al. 2011; Bobra & Couvidat 2015; Guennou et al. 2017; Kontogiannis et al. 2017; Leka et al. 2018). However, it is also important to consider the free energy stored in the entire coronal volume (e.g., Falconer et al. 2006; Su et al. 2014).

Several 3D force-free reconstruction methods have been developed, differing by the equations solved and/or their treatment of the measured photospheric and/or chromospheric magnetic field used as a boundary condition (e.g., van Ballegooijen 2004; Wheatland 2007; Valori et al. 2010; Wiegelmann & Inhester 2010; Contopoulos et al. 2011; Yeates 2014). These methods are prone to a variety of ambiguities, however, that motivate the use of measurements in the corona itself as further constraints on models (e.g., Metcalf et al. 2006; Leka et al. 2009; Crouch 2015).

Recent theoretical (e.g., Judge et al. 2006; Kramar et al. 2013; Rachmeler et al. 2013) and observational (e.g., Bak-Steślicka et al. 2013; Rachmeler et al. 2014; Gibson et al. 2017) studies showed that the presence of nonpotential magnetic fields is
imprinted into infrared, off-limb, coronal polarization. Since the corona is optically thin and less bright than the photosphere, determining coronal magnetic fields from polarization measurements is challenging, but progress has recently been made in developing frameworks for the inversion of such data (e.g., Kramar et al. 2014; Plowman 2014; Dalmasse et al. 2016, 2019). This, taken with the advent of daily off-limb measurements in the infrared Fe XIII lines by the Coronal Multichannel Polarimeter (CoMP; Tomczyk et al. 2008), and expected future observations from large telescopes such as the Daniel K. Inouye Solar Telescope (DKIST; Keil et al. 2011) and the proposed Coronal Solar Magnetism Observatory Large Coronagraph (COSMO-LC; Tomczyk et al. 2016) make it timely to develop metrics that will help track electric currents and free magnetic energy for space weather forecast applications.

In this paper we introduce a novel way to isolate and quantify “hot spots” of nonpotentiality in coronal magnetic fields using polarization measurements that can be obtained by the CoMP and other polarimeters. In Section 2 we describe the polarimetric data and how they relate to different magnetic field observables. Section 3 presents a model in which a twisted magnetic flux rope emerges and evolves into a CME, along with corresponding potential field configurations. Section 4 describes how we quantify nonpotentiality by comparing the synthetic images from energized versus potential models, both represented as spatial distributions and as a singled-value index for nonpotential energy. We also discuss free magnetic energy and how it relates to the nonpotentiality of each polarimetric observable, and investigate sensitivities to the choice of potential field model and to assumed plasma distribution along the line of sight (LOS). Finally, in Section 5, we present our conclusions.

## 2. Polarimetric Data

The polarimetric measurements that we use to develop the diagnostic of nonpotentiality are taken in the Fe XIII (10747 Å) emission line. At each point in the image, the Stokes polarimetric data is expressed as $[I, Q, U, V]$. The first quantity, *I* is total intensity in the line, and is sensitive to the density and temperature of the coronal plasma. The second and third quantities together define the linear polarization vector, which may also be defined in terms of its magnitude $L = \sqrt{U^2 + Q^2}$ and azimuthal direction $Az = \frac{1}{2} \tan^{-1}(U/Q)$. For the Fe XIII line, the linear polarization signal is dominated by the saturated Hanle effect (e.g., House 1974; Casini & Judge 1999; Lin et al. 2004). This means that the linear polarization provides information about the direction of the magnetic field vector, *B*, but not its strength. Finally, *V* is the net circularly polarized light, and is sensitive to the strength of the LOS component of the magnetic field. Because the corona is optically thin, all of these measurements are intensity-weighted averages along the LOS (see Rachmeler et al. 2012 for further discussions).

CoMP obtains daily measurements of *I*, *Q*, and *U* in the Fe XIII (10747 Å) coronal emission line, and so provides information about the coronal magnetic field direction and topology (see, e.g., Rachmeler et al. 2013, 2014). Because it is a relatively small telescope (20 cm) it does not possess sufficient light-gathering capacity to obtain images of *V* (circularly polarized light is substantially fainter than linearly polarized light in this emission line). Future telescopes such as the DKIST and COSMO-LC would be large enough to provide

observations of *V*, adding information about LOS magnetic field strength. Therefore, for the purposes of this study we consider both circularly and linearly polarized light, expressed as $[L/I, Az, V/I]$. We use fractions of circularly and linearly polarized light relative to total intensity in order to minimize the effects of density and temperature distribution along the LOS (to be discussed further in Section 4.3).

## 3. Magnetic Models

A particular feature frequently seen by CoMP in linearly polarized light, which resembles a rabbit head (also called a lagomorph), provides evidence of the presence of magnetic flux ropes in coronal cavities (Bak-Steślicka et al. 2013). This is of particular interest because the continued emergence of such flux ropes in the solar atmosphere can build up the free magnetic energy, which leads to coronal eruptions (Fan & Gibson 2007; Savcheva et al. 2016). We therefore have chosen a flux rope model as our test case for identifying regions of nonpotentiality, an important first step in modeling CME source regions.

### 3.1. Emerging Flux Rope Simulation

Figure 1 shows representative magnetic field lines from a simulation of a twisted flux rope, which emerges into the closed magnetic field part of a helmet streamer. The flux rope emergence increases the magnetic pressure below the streamer, which in turn pushes the cusp of the helmet streamer higher up in the corona. Eventually, the flux rope reaches a coronal height beyond which it erupts (Fan 2017). This simulation solves the full equations of magnetohydrodynamics to yield the temporal evolution of the magnetic fields and associated plasma density and temperature, resulting in the formation of a prominence and its subsequent eruption. As the twisted flux rope emerges (blue and green field lines) the magnetic energy increases, reaching a maximum energy at time step 89 (which corresponds to *t* = 8.82 hr). We will use the time steps of the rope's emergence to represent a sequence of increasingly energized coronal magnetic fields.

### 3.2. Potential Field Models

The potential field model, as discussed in Section 1, represents the current-free magnetic field configuration for a given distribution of magnetic flux at the domain boundaries. To calculate the potential field, we use the following boundary conditions. For the sides and the bottom of the computational volume, we use the values taken from the simulation. We then explore two choices for the top boundary that result in the following two potential field configurations within the simulation cube domain: (1) $B_p$—the potential field that further matches the radial magnetic flux distribution at the upper boundary of the simulation box, and (2) $B_{pfss}$— a potential field whose upper “source surface” boundary (where magnetic fields are purely radial) is set to the height ($r_{ss}$) of the helmet streamer cusp above the flux rope and for which the magnetic field above the source surface is computed from the magnetic flux conservation (i.e., $B(r > r_{ss}) = B_r(r > r_{ss}) \hat{f} = B(r_{ss}) r_{ss}^2 / r^2$). Note that $B_p$ and $B_{pfss}$ are computed for each time step of the simulation and evolve due to changes of the simulation boundaries, and, for $B_{pfss}$, also due to changes of the cusp height ($r_{ss}$).

The first of these potential fields, $B_p$, is of interest because it represents the true, minimum energy for the boundary conditions
<figure><img src="image_2.png" /><figcaption>**Figure 1.** Field lines showing a twisted flux rope (shown by sample blue and green field lines) emerging into the closed magnetic field part of a helmet streamer (shown by red field lines) as it begins to erupt. Each panel shows the magnetic field from the simulation at a different time step: (a) Time step 54 ($t = 5.35$ hr). (b) Time step 89 ($t = 8.82$ hr). (c) Time step 149 ($t = 14.77$ hr). The maximum magnetic energy is obtained at time step 89; once the eruption begins, magnetic energy decreases as it is transferred to kinetic energy. From Fan (2017). An animation of the emergence is available. The animation covers the first 149 time steps ($t = 0$ to 14.77 hr).</figcaption></figure>

(An animation of this figure is available.)

<figure><img src="image_3.png" /><figcaption>**Figure 2.** Field lines for potential field configurations (top) exact $B_p$, and (bottom) evolving source surface $B_{pfss}$, corresponding to the Fan (2017) simulation. The red, blue, and green lines represent the potential field lines corresponding to the same three time steps as in Figure 1. (a) Time step 54 ($t = 5.35$ hr). (b) Time step 89 ($t = 8.82$ hr). (c) Time step 149 ($t = 14.77$ hr). Note that since magnetic flux emergence through the lower boundary is stopped at time step 89, the potential field is unchanged thereafter. An animation of the temporal evolution of both potential fields is available. The animation has the exact $B_p$ on the left and the evolving source surface $B_{pfss}$ on the right. The animated version covers the first 149 time steps ($t = 0$ to 14.77 hr).</figcaption></figure>

(An animation of this figure is available.)

on the simulation box. The second, $B_{pfss}$, will have higher energy associated with the open magnetic flux passing through the upper boundary—this is associated with the solar wind

opening the magnetic field. We will discuss these two potential fields in more detail in Section 4. Figure 2 shows examples of both $B_p$ and $B_{pfss}$ for the same three time steps as Figure 1.
<figure><img src="image_4.png" /><figcaption>**Figure 3.** Nonpotentiality diagnostics of the energized flux rope simulation, $B_{sim}$, at time step 89, computed from the fraction of linear polarization, $L/I$, using the $B_p$ potential field. (a) $L/I$ map for the flux rope simulation (see Section 3.1). (b) $L/I$ map for the potential field, $B_p$ (see Section 3.2). (c) Map of the proxy of 2D-density of nonpotentiality in a logarithmic scale, $s^{L/I}$ (see Equation (2)). (d) $\tilde{z}_{np}$ (see Equation (7)). Green lines represent the POS direction of the magnetic field. Solid yellow line indicates the surface of the Sun. An animation showing the temporal evolution the nonpotentiality diagnostics is available. The animation covers the first 89 time steps.</figcaption></figure>

(An animation of this figure is available.)

<figure><img src="image_5.png" /><figcaption>**Figure 4.** Same as Figure 3, but using the $B_{pfss}$ potential field as the reference field. Dashed yellow line shows the cusp height in the simulation and source surface for $B_{pfss}$. An animation showing the temporal evolution of this figure is available. The animation covers the first 89 time steps.</figcaption></figure>

(An animation of this figure is available.)

<figure><img src="image_6.png" /><figcaption>**Figure 5.** Same as Figure 3, but for the linear polarization direction (or azimuth), $Az$, using the $B_p$ potential field as the reference field. An animation showing the temporal evolution of this figure is available. The animation covers the first 89 time steps.</figcaption></figure>

(An animation of this figure is available.)

Solving for potential magnetic fields in the force-free field approximation does not provide us with plasma parameters (density and temperature). We thus need to separately establish a description of the plasma density and temperature that are needed to calculate the synthetic Stokes observations (see Section 4.2). However, as we will use ratios of linearly/

circularly polarized intensity versus total intensity, the plasma distribution primarily enters in as a weighting function along the LOS. For this reason, we define our baseline density/temperature distribution associated with the potential field to be identical to that of the Fan (2017) simulation. We note that this is not in force balance with the potential magnetic fields, and in
<figure><img src="image_7.png" /><figcaption>**Figure 6.** Same as Figure 5, but using the $B_{pfss}$ potential field as the reference field. An animation showing the temporal evolution of this figure is available. The animation covers the first 89 time steps.</figcaption></figure>

(An animation of this figure is available.)

<figure><img src="image_8.png" /><figcaption>**Figure 7.** Same as Figure 3, but for the fraction of circular polarization, $V/I$, using the $B_p$ potential field as the reference field. An animation showing the temporal evolution of this figure is available. The animation covers the first 89 time steps.</figcaption></figure>

(An animation of this figure is available.)

<figure><img src="image_9.png" /><figcaption>**Figure 8.** Same as Figure 7, but using the $B_{pfss}$ potential field as the reference field. An animation showing the temporal evolution of this figure is available. The animation covers the first 89 time steps.</figcaption></figure>

(An animation of this figure is available.)

Section 4.3 we will explore how using a spherically symmetric density/temperature distribution in hydrostatic balance would affect our results.

# 4. Quantifying Nonpotentiality

## 4.1. Definitions

Potential magnetic fields can be readily constructed from photospheric magnetograms and are often used in analyses of the global solar magnetic field (e.g., Schatten et al. 1969; Wang et al. 2000; Wang & Sheeley 2002; Cook et al. 2009;

Schrijver et al. 2013) or solar wind modeling (e.g., Levine et al. 1977; Wang & Sheeley 1990; Arge & Pizzo 2000; Cohen et al. 2007; Woolsey & Cranmer 2014; Peleikis et al. 2016). Forward modeling can then be used to produce synthetic observables, such as coronal polarization data, from these potential fields and a plasma (density and temperature) model. In principle, it is thus possible to quantify the nonpotentiality in the coronal polarization data by comparing it to the corresponding synthetic data generated from the potential field. In particular, we are interested in developing a proxy that could map the regions that store nonpotential magnetic energy in the corona.
<caption>**Table 1**</caption>

Correlations between 2D Maps of *s* and $\tilde{\epsilon}_{np}$ for Various Observables for Time Step 89

<table><thead><tr><th>Quantities Being Compared</th><th>Reference Field</th><th>Correlations (Pearson, Spearman)</th></tr></thead><tbody><tr><td>$s^{L/1}$; $\tilde{\epsilon}_{np}$</td><td>$\mathbf{B}_p$</td><td>-0.47, -0.52</td></tr><tr><td>$s^{L/1}$; $\tilde{\epsilon}_{np}$</td><td>$\mathbf{B}_{pfss}$</td><td>-0.07, 0.08</td></tr><tr><td>$s^{L/1}$; $\tilde{\epsilon}_{np}$—masked</td><td>$\mathbf{B}_p$</td><td>-0.45, -0.48</td></tr><tr><td>$s^{L/1}$; $\tilde{\epsilon}_{np}$—masked</td><td>$\mathbf{B}_{pfss}$</td><td>-0.07, 0.17</td></tr><tr><td>$s^{Az}$; $\tilde{\epsilon}_{np}$</td><td>$\mathbf{B}_p$</td><td>0.25, 0.32</td></tr><tr><td>$s^{Az}$; $\tilde{\epsilon}_{np}$</td><td>$\mathbf{B}_{pfss}$</td><td>0.35, 0.33</td></tr><tr><td>$s^{Az}$; $\tilde{\epsilon}_{np}$—masked</td><td>$\mathbf{B}_p$</td><td>0.25, 0.44</td></tr><tr><td>$s^{Az}$; $\tilde{\epsilon}_{np}$—masked</td><td>$\mathbf{B}_{pfss}$</td><td>0.35, 0.53</td></tr><tr><td>$s^{V/1}$; $\tilde{\epsilon}_{np}$</td><td>$\mathbf{B}_p$</td><td>0.79, 0.72</td></tr><tr><td>$s^{V/1}$; $\tilde{\epsilon}_{np}$</td><td>$\mathbf{B}_{pfss}$</td><td>0.79, 0.39</td></tr><tr><td>$s^{V/1}$; $\tilde{\epsilon}_{np}$—masked</td><td>$\mathbf{B}_p$</td><td>0.80, 0.83</td></tr><tr><td>$s^{V/1}$; $\tilde{\epsilon}_{np}$—masked</td><td>$\mathbf{B}_{pfss}$</td><td>0.81, 0.52</td></tr></tbody></table>

**Note.** “Masked” refers to correlations limited to regions below the simulation cusp height.

First, if $\rho$ is the position in the plane of sky (POS) for synthetic data, and $\mathbf{r}$ is the vector position in the solar atmosphere, then we can express a Stokes-related image as $\mathcal{D}(\rho, \mathbf{B}(\mathbf{r}), n(\mathbf{r}), T(\mathbf{r}))$ (i.e., any of $L/1$, $Az$, or $V/1$), where $\mathbf{B}(\mathbf{r})$ is the vector magnetic field, $n(\mathbf{r})$ is the plasma density, and $T(\mathbf{r})$ is the plasma temperature. We then define the corresponding map of nonpotentiality as

$$s^D(\rho, \mathbf{B}, n_B, T_B, \mathbf{B}_{\text{pot}}, n_{B_{\text{pot}}}, T_{B_{\text{pot}}}) = (\mathcal{D}(\rho, \mathbf{B}, n_B, T_B) - \mathcal{D}(\rho, \mathbf{B}_{\text{pot}}, n_{B_{\text{pot}}}, T_{B_{\text{pot}}}))^2, \quad (1)$$

$\mathbf{B}_{\text{pot}}$ is a reference potential field with regard to which the nonpotentiality is computed, and $n_B$ (respectively $n_{B_{\text{pot}}}$) and $T_B$ (respectively $T_{B_{\text{pot}}}$) are the plasma density and temperature model associated with the magnetic field, $\mathbf{B}$ (respectively $\mathbf{B}_{\text{pot}}$). In other words, $s^D$ images are maps of the squared difference between the energized (e.g., $\mathbf{B} = \mathbf{B}_{\text{sim}}$) and potential (e.g., $\mathbf{B}_{\text{pot}} = \mathbf{B}_p$ or $\mathbf{B}_{\text{pot}} = \mathbf{B}_{pfss}$) fields for each observable. For tractability and because we want to emphasize the dependency on the magnetic field, we however use the simplified notation

$$s_i^D(\mathbf{B}, \mathbf{B}_{\text{pot}}) = (\mathcal{D}_i(\mathbf{B}) - \mathcal{D}_i(\mathbf{B}_{\text{pot}}))^2, \quad (2)$$

where $i$ is the $i$th pixel of the Stokes-related image. Because it is a square of differences rather than a difference of squares, this metric is sensitive to change in sign between the energized and the potential reference fields. This is important because it allows for consideration of the orientation of the energized field with respect to the potential field in evaluating nonpotentiality. Also we note that while now being hidden in Equation (2), one should bear in mind that $s^D$ explicitly depends on the plasma (density and temperature) models used with $\mathbf{B}$ and $\mathbf{B}_{\text{pot}}$.

The $s^D$-maps are made for diagnosing the degree and spatial location of nonpotentiality from each Stokes observable. These maps essentially show a 2D-density of nonpotentiality. In order to distill this information into a single number, we define a nonpotentiality index, $\text{NPI}^D = \text{NPI}^D(\mathbf{B}, \mathbf{B}_{\text{pot}})$, as the surface

integral of $s^D$ in the POS, i.e.,

$$\text{NPI}^D(\mathbf{B}, \mathbf{B}_{\text{pot}}) = \int_{\text{POS}} s^D(\mathbf{B}, \mathbf{B}_{\text{pot}}) d\mathcal{S}, \\ \approx \sum_{i=1}^{n} \mathcal{A} \cdot s_i^D(\mathbf{B}, \mathbf{B}_{\text{pot}}), \quad (3)$$

where $n$ is the number of pixels in the $s^D$-map, and $\mathcal{A}$ is the area of a pixel.

We test the capabilities of our nonpotentiality proxies with a fully known, energized, 3D magnetic field from which we can readily compute the potential field, and then the nonpotential magnetic energy and its 3D-density. The nonpotential magnetic energy, $E_{np}(\mathbf{B}, \mathbf{B}_{\text{pot}})$, is defined as

$$E_{np}(\mathbf{B}, \mathbf{B}_{\text{pot}}) = \int_{\mathcal{V}} \epsilon_{np}(\mathbf{B}, \mathbf{B}_{\text{pot}}) d\mathcal{V}, \quad (4)$$

where

$$\epsilon_{np}(\mathbf{B}, \mathbf{B}_{\text{pot}}) = \frac{\mathbf{B}^2 - \mathbf{B}_{\text{pot}}^2}{8\pi}, \quad (5)$$

quantifies the volume distribution of the nonpotential magnetic energy. It should be noted that, for the particular case in which $\mathbf{B}_{\text{pot}}$ is the potential field sharing the same distribution of magnetic flux as $\mathbf{B}$ on the boundaries of $\mathcal{V}$ (i.e., $\mathbf{B}_{\text{pot}} = \mathbf{B}_p$ for the models considered in Section 3), Equation (4) leads to the classical definition of free magnetic energy (e.g., Valori et al. 2013) and represents energy available both for flaring and for opening up the field to the solar wind.

By decomposing the volume integral of Equation (4) in terms of a 1D-integral along the LOS and a 2D-integral along the POS, we obtain

$$E_{np}(\mathbf{B}, \mathbf{B}_{\text{pot}}) = \int_{\text{POS}} \tilde{\epsilon}_{np}(\mathbf{B}, \mathbf{B}_{\text{pot}}) d\mathcal{S}, \\ \approx \sum_{i=1}^{n} \mathcal{A} \cdot \tilde{\epsilon}_{np,i}(\mathbf{B}, \mathbf{B}_{\text{pot}}), \quad (6)$$

where

$$\tilde{\epsilon}_{np} = \int_{\text{LOS}} \epsilon_{np} dl, \quad (7)$$

is therefore the column density of $\epsilon_{np}$. The ability of our proxies to map and quantify the nonpotentiality of the coronal magnetic field from coronal polarization data is then tested by comparing $s^D$-maps with $\tilde{\epsilon}_{np}$-maps (see Section 4.2), and $\text{NPI}^D$ with $E_{np}$ (see Section 4.3).

## 4.2. 2D Maps

FORWARD is a SolarSoft IDL package that, given a model of coronal magnetic fields, densities, and temperatures, synthesizes a variety of observables including $[I, Q, U, V]$ in the Fe XIII line (Gibson et al. 2016). We first used it to create these observables from the energized twisted flux rope simulation, focusing on a field of view covering the region in which the flux rope is emerging (e.g., Figure 3–8, first column). Next, we applied it to the two potential field models considered in this paper (see $\mathbf{B}_p$ and $\mathbf{B}_{pfss}$ in Section 3.2) assuming the same plasma density and temperature as the simulation, as discussed above (e.g., Figures 3–8, second column). Finally, we compute our proxies of nonpotential energy from
<figure><img src="image_10.png" /><figcaption>**Figure 9.** NPI of the energized flux rope simulation, $B_{sim}$ (see Section 3.1), when using $B_p$ as the reference potential field (see Section 3.2). Polarimetric observables $(L/I, A_z, \text{and } V/I)$ possess the same density and temperature distributions. Left column panels represent NPI evolution vs. time step. Right panels show NPI vs. nonpotential magnetic energy (in units of erg).</figcaption></figure>

Equations (2) (third column) and (3), and their true physical counterpart from Equations (7) and (6) (fourth column).

Since we have set the plasma density and temperature to be the same for the flux rope and the potential field models, the differences between the polarization signals are due to differences in their magnetic configurations. The $s^{L/I}$ maps for time step 89 shown in Figures 3-4 illustrate differences between the synthetic Stokes images that are due to the presence of the flux rope. In particular, the presence of a lagomorph (although faint) is apparent for both $s^{L/I}(B_{sim}, B_p)$ and $s^{L/I}(B_{sim}, B_{pfss})$ (third column—c), within the closed field. This structure introduces a depletion of $s^{L/I}$ in the very region where both the $\tilde{\epsilon}_{np}(B_{sim}, B_p)$ and $\tilde{\epsilon}_{np}(B_{sim}, B_{pfss})$ maps (fourth column—d) are bright, due to nonpotential energy associated with the twisted flux rope. Moreover, the $s^{L/I}(B_{sim}, B_p)$ is generally bright above the flux rope while the $\tilde{\epsilon}_{np}$ is not. Consequently, the Pearson and Spearman correlation coefficients between the 2D images of $s^{L/I}(B_{sim}, B_p)$ and $\tilde{\epsilon}_{np}(B_{sim}, B_p)$ are negative indicating that there is a moderate anticorrelation between the values of both quantities and their ranking (see Table 1). For $B_{pfss}$, there is not as extreme a difference between $s^{L/I}$ and $\tilde{\epsilon}_{np}$ for much of the area above the simulation cusp height (dashed curved line in Figure 4), because the source surface attempts to approximate how the solar wind opens the magnetic field. When we calculate the Pearson and Spearman correlation coefficients between $s^{L/I}(B_{sim}, B_{pfss})$ and $\tilde{\epsilon}_{np}(B_{sim}, B_{pfss})$ for the PFSS case, we get

values close to zero. We note that when restricting our analysis to the region below the cusp, the Spearman correlation coefficient between $s^{L/I}(B_{sim}, B_{pfss})$ and $\tilde{\epsilon}_{np}(B_{sim}, B_{pfss})$ increases slightly, indicating that there may be some weak correlation between the ranking of the values of $s^{L/I}(B_{sim}, B_{pfss})$ and $\tilde{\epsilon}_{np}(B_{sim}, B_{pfss})$ for the PFSS case (the Pearson correlation coefficient remains close to zero).

Figure 5 shows the $s^{Az}$ maps for time step 89 for $B_p$. We find that the $s^{Az}(B_{sim}, B_p)$ signal associated with the flux rope is relatively large in the region of strong $\tilde{\epsilon}_{np}$, and the Pearson and Spearman correlation coefficients between them indicate a mild correlation between these 2D quantities at this time step (see Table 1). However, as the animation associated with Figure 5 illustrates and as we will discuss below in Section 4.3, the time evolution of $s^{Az}(B_{sim}, B_p)$ versus $\tilde{\epsilon}_{np}(B_{sim}, B_p)$ is rather complex. In particular, for earlier time steps Figure 5 shows $s^{Az}(B_{sim}, B_p)$ relatively low values in the region of the flux rope, introducing an anticorrelation with $\tilde{\epsilon}_{np}$. The situation is simpler for the PFSS reference field. $s^{Az}(B_{sim}, B_{pfss})$ shown in Figure 6 is also bright in the region of the flux rope and thus has a weak correlation to $\tilde{\epsilon}_{np}$ (see Table 1), but some weak correlation is maintained for all but the earliest stages of the flux rope's emergence as we will discuss in Section 4.3.

Finally in Figures 7 and 8 (Column 3), $s^{V/I}$ shows a very strong, white, saturated signal confined to the region associated with the axial magnetic field of the flux rope, hence, below the streamer cusp. We note that this is true for both $s^{V/I}(B_{sim}, B_p)$
<figure><img src="image_11.png" /><figcaption>Figure 10. Same as Figure 9, but using $B_{pfss}$ as the reference potential field (see Section 3.2).</figcaption></figure>

<table><caption>**Table 2**
Correlations between 1D Nonpotentiality Index NPI<sup>D</sup> and Nonpotential Magnetic Energy $E_{np}$ for Various Observables and Reference Potential Fields</caption><thead><tr><th>Quantities Being Compared</th><th>Reference Field</th><th>Correlations (Pearson, Spearman)</th></tr></thead><tbody><tr><td>NPI<sup>L</sup>/<sup>1</sup>; E<sub>np</sub></td><td><b>B</b><sub>p</sub></td><td>0.96, 1.0</td></tr><tr><td>NPI<sup>L</sup>/<sup>1</sup>; E<sub>np</sub></td><td><b>B</b><sub>pfss</sub></td><td>0.91, 0.91</td></tr><tr><td>NPI<sup>Az</sup>; E<sub>np</sub></td><td><b>B</b><sub>p</sub></td><td>-0.80, -0.67</td></tr><tr><td>NPI<sup>Az</sup>; E<sub>np</sub></td><td><b>B</b><sub>pfss</sub></td><td>0.90, 0.75</td></tr><tr><td>NPI<sup>V</sup>/<sup>1</sup>; E<sub>np</sub></td><td><b>B</b><sub>p</sub></td><td>0.99, 1.00</td></tr><tr><td>NPI<sup>V</sup>/<sup>1</sup>; E<sub>np</sub></td><td><b>B</b><sub>pfss</sub></td><td>0.99, 1.0</td></tr></tbody></table>

**Note.** In all cases, contributions to NPI<sup>D</sup> and $E_{np}$ are only included from below the cusp height.

and $s^{V/I}(\mathbf{B}_{sim}, \mathbf{B}_{pfss})$, i.e., regardless of the potential field used to apply our diagnostics of nonpotentiality. Comparing the $s^{V/I}$ and $\tilde{\epsilon}_{np}$ maps, we find that our 2D-density of nonpotentiality defined from $V/I$ provides a very good qualitative map of the coronal nonpotentiality. This is quantitatively confirmed by the Pearson and Spearman correlation coefficients. In particular for $s^{V/I}(\mathbf{B}_{sim}, \mathbf{B}_{p})$ versus $\tilde{\epsilon}_{np}(\mathbf{B}_{sim}, \mathbf{B}_{p})$, the Pearson and Spearman coefficients indicate strong correlation between the values of both quantities and their ranking (see Table 1). The values of $s^{V/I}(\mathbf{B}_{sim}, \mathbf{B}_{pfss})$ and $\tilde{\epsilon}_{np}(\mathbf{B}_{sim}, \mathbf{B}_{pfss})$ are also strongly correlated, although we note that their ranking correlation is not as high.

The fact that $s^{V/I}$ is such a good proxy of nonpotentiality is due to the following reasons. When we observe a coronal cavity, the LOS is approximately along the flux rope/ prominence/neutral line. Since the circular polarization in the Fe XIII line is produced by the Zeeman effect, $V$ is sensitive to the LOS magnetic field (Casini & Judge 1999). $V/I$ is thus a direct measure of the shear (or axial field) of the flux rope associated with the cavity (see also Fan et al. 2018), and hence, of the nonpotentiality of the flux rope magnetic field. In this very specific geometry, the potential field contribution to $V/I$ is in fact essentially zero, and the correlation to free energy could thus be obtained directly from a measurement of the circular polarization $V/I$ without requiring the calculation of a potential field. However, our analysis may be extended to the more general scenarios where there may be significant nonzero potential field oriented along the LOS. Our approach also provides a means for quantifying nonpotentiality from linear polarization measurements.

### 4.3. Nonpotentiality Index

In order to focus on the nonpotentiality of the flux rope, we compute NPI<sup>D</sup> (Equation (3)) and $E_{np}$ (Equation (6)) only for the region located below the cusp height. Figures 9–10 show time series (left column) of NPI<sup>L</sup>/<sup>1</sup>, NPI<sup>Az</sup>, and NPI<sup>V</sup>/<sup>1</sup>, for the two potential field options, as well as plotting these quantities versus $E_{np}$. NPI<sup>L</sup>/<sup>1</sup>, NPI<sup>Az</sup>, and NPI<sup>V</sup>/<sup>1</sup> all show sensitivity to the flux rope emergence after the initial stages of emergence.

For both choices of potential field, NPI<sup>L</sup>/<sup>1</sup> increases slowly until approximately time step 60, at which point it increases
<figure><img src="image_12.png" /><figcaption>**Figure 11.** Left three columns: time evolution (time steps 30, 54, and 89, top to bottom) of the maps of 2D-density of nonpotentiality computed from $L/I$, $Az$, and $V/I$, using the $B_p$ model as the reference potential field (see Section 3.2). Far right column: column density of nonpotential magnetic energy (see Equation (7)). Note that a hydrostatic plasma density/temperature distribution (Gibson et al. 1999) is assumed for the Stokes parameters calculated for the $B_p$ model. An animation showing the temporal evolution of this figure is available. The animation covers the first 89 time steps.</figcaption></figure>

(An animation of this figure is available.)

more rapidly. Although the spatial maps $s^{L/I}$ and $\tilde{\epsilon}_{np}$ are anticorrelated, as we demonstrated in Section 4.2, the rising flux rope nevertheless increases the integrated brightness for both with increasing time steps (see animations associated with Figures 3–4). As a consequence the Pearson and Spearman correlation coefficient values (see Table 2) indicate a very strong correlation between $\text{NPI}^{L/I}(\mathbf{B}_{\text{sim}}, \mathbf{B}_p)$ and $E_{\text{np}}(\mathbf{B}_{\text{sim}}, \mathbf{B}_p)$. Similarly, we find a very good correlation between $\text{NPI}^{L/I}(\mathbf{B}_{\text{sim}}, \mathbf{B}_{\text{pfss}})$ and $E_{\text{np}}(\mathbf{B}_{\text{sim}}, \mathbf{B}_{\text{pfss}})$.

When using the Stokes azimuth, we find very different relationships between $\text{NPI}^{Az}$ and $E_{\text{np}}$ depending on which potential field is used as the reference field. This is likely due to the complex time evolution of $s^{Az}(\mathbf{B}_{\text{sim}}, \mathbf{B}_p)$ versus $\tilde{\epsilon}_{np}(\mathbf{B}_{\text{sim}}, \mathbf{B}_p)$ that we have already mentioned in Section 4.2. In

particular, we find a weak-to-moderate anticorrelation between $\text{NPI}^{Az}(\mathbf{B}_{\text{sim}}, \mathbf{B}_p)$ and $E_{\text{np}}(\mathbf{B}_{\text{sim}}, \mathbf{B}_p)$ compared to a very strong correlation between $\text{NPI}^{Az}(\mathbf{B}_{\text{sim}}, \mathbf{B}_{\text{pfss}})$ and $E_{\text{np}}(\mathbf{B}_{\text{sim}}, \mathbf{B}_{\text{pfss}})$ (see Table 2 and Figures 9–10).

Finally, $\text{NPI}^{V/I}$ shows a sharp increase at about time step 35 and then flattens out at time step 89, which is when the rope has fully emerged. $\text{NPI}^{V/I}$ is clearly the quantity that best correlates with $E_{\text{np}}$ regardless of reference field chosen (see Table 2). We note that the calculation of the $\text{NPI}^{V/I}$ was done scaling the $V/I$ data points as described in (Equation (3)) from Fan et al. 2018). Each signal of $V/I$ was scaled to the $B_{\text{LOS}}$ as follows, $B_{\text{LOS}} = \frac{\sqrt{\pi}}{2} \frac{\Gamma V}{k I}$, where $\Gamma = 0.1$ nm corresponds to the line width from the Fe XIII emissions, and $k = 8.1 \times 10^{-6}$ nm G$^{-1}$
<figure><img src="image_13.png" /><figcaption>**Figure 12.** Same as Figure 9, but using a hydrostatic plasma density/temperature distribution (Gibson et al. 1999) to compute the Stokes parameters calculated for the $B_p$ model.</figcaption></figure>

the Zeeman sensitivity of the line. When calculating our NPI$^{V/I}$ we use the square of this coefficient because it diagnoses magnetic energy, which scales to $B^2$. We note that the magnitude of NPI$^{V/I}$ is roughly one-quarter of the nonpotential energy $E_{np}$, which is consistent with the findings of Fan et al. (2018) who determined that LOS weighting resulted in approximately a factor of 2 decrease in the $B_{los}$ predicted from $V/I$.

In all of the calculations so far, we have used the same plasma distribution in the potential field Stokes calculations as the simulation at the same time step. In order to test the sensitivity of our analysis to this assumption, we have performed all the same calculations assuming a hydrostatic plasma (Gibson et al. 1999) for the potential field instead (see Figures 11–14). The differences are minimal (especially below the source surface/cusp height), as seen by comparing the third column of Figures 3, 5, and 7 to the third row of Figure 11 using the $B_p$ potential field, and in the same way comparing Figures 4, 6, and 8 to the third row of Figure 13 for $B_{pfss}$. We see that when comparing the NPIs for the different plasma models seen in Figures 9 and 12, and 10 and 14, only the NPI$^{L/I}$ and NPI$^{Az}$ are sensitive to the change in plasma model. Although the values of NPI$^{L/I}$ show an order of magnitude difference, the correlation of NPI$^{L/I}$ and $E_{np}$ is essentially unchanged. NPI$^{Az}$ show some differences but do not significantly affect our interpretation of the degree of correlation. NPI$^{V/I}$ shows the same results independent of plasma model,

both for the quantitative value of the index and the correlation coefficients (see Table 3).

As a final note, in order to test the sensitivity of our results to the orientation of the flux rope along the LOS, we ran two cases, one with a 15° axial tilt, and one with a 30° axial tilt, and found our results largely unchanged (in both cases we chose $B_{pfss}$ as our reference field). We find that although the Spearman correlations between the 2D-density of nonpotentiality $s^D$ versus $\tilde{e}_{np}$ do generally decrease as the flux rope is tilted away from the LOS (compare Tables 1 and 4), the 1D correlation between the time evolution of NPI$^D$ versus nonpotential energy $E_{np}$ remains strong, regardless of the polarization component (i.e., circular or linear) used in defining the NPI (compare Tables 2 and 4).

## 5. Conclusions

Off-limb coronal polarimetry, such as routinely obtained by the CoMP in the Fe XIII line, is a powerful tool for probing the solar coronal magnetic field. Despite recent advances (e.g., Kramar et al. 2014; Plowman 2014; Dalmasse et al. 2019), inverting such coronal polarization measurements into 3D vector magnetic fields remains a real challenge and it will likely take time before near real time inversions of off-limb coronal polarimetry become available. On the other hand, several theoretical (e.g., Judge et al. 2006; Kramar et al. 2013; Rachmeler et al. 2013) and observational (e.g., Bak-Steślicka et al. 2013; Rachmeler et al. 2014; Gibson et al. 2017) studies have demonstrated that current-carrying (a.k.a., nonpotential)
<figure><img src="image_14.png" /><figcaption>**Figure 13.** Same as Figure 11, but using the *B*<sub>pfss</sub> model as the reference potential field (see Section 3.2). An animation showing the temporal evolution of this figure is available. The animation covers the first 89 time steps.

*(An animation of this figure is available.)*</figcaption></figure>

magnetic fields leave very specific polarization signatures that are distinguishable from those of potential fields. In principle, it means that such data could be used to locate nonpotential coronal magnetic fields, such as those associated with coronal cavities, which are density-depleted regions often seen in extreme ultraviolet (e.g., in the 193 Å channel of the Atmospheric Imaging Assembly on board the Solar Dynamics Observatory; Lemen et al. 2012; Bak-Steślicka et al. 2013). Coronal cavities are likely associated with 3D twisted and/or sheared magnetic fields (e.g., Low 1997; Aulanier & Demoulin 1998) preceding quiet-Sun CMEs and are thus natural reservoirs of free (magnetic) energy that should also be monitored in the context of space weather forecasts.

In this paper, we introduced and tested three new proxies to track nonpotential coronal magnetic fields and their free energy from IR coronal polarimetric observations of coronal cavities. Our proxies rely on combining potential field solutions, which can be readily extrapolated from photospheric vector-magneto-grams observed on-disk and then propagated to the limb (either through rigid rotation or using photospheric flux transport models; e.g., Yeates et al. 2007), with forward modeling of IR coronal polarimetry (Gibson et al. 2016). By subtracting the potential field signal we obtain a physically sensible measure relative to a meaningful, quantifiable baseline for general analyses of nonpotentiality using coronal polarimetric measurements. Nonpotentiality maps are then defined by computing the
<figure><img src="image_15.png" /><figcaption>Figure 14. Same as Figure 12, but for the Stokes parameters calculated for the *B*<sub>pfss</sub> model.</figcaption></figure>

<table><caption>**Table 3**
Same as Table 2, but with Hydrostatic Density Model for the Potential Fields</caption><thead><tr><th>Quantities Being Compared</th><th>Reference Field</th><th>Correlations (Pearson, Spearman)</th></tr></thead><tbody><tr><td>NPI<sup>L</sup>/<sup>L</sup>; E<sub>np</sub></td><td><b>B</b><sub>p</sub></td><td>0.96, 1.0</td></tr><tr><td>NPI<sup>L</sup>/<sup>L</sup>; E<sub>np</sub></td><td><b>B</b><sub>pfss</sub></td><td>0.95, 1.0</td></tr><tr><td>NPI<sup>Az</sup>; E<sub>np</sub></td><td><b>B</b><sub>p</sub></td><td>-0.83, -0.69</td></tr><tr><td>NPI<sup>Az</sup>; E<sub>np</sub></td><td><b>B</b><sub>pfss</sub></td><td>0.36, 0.16</td></tr><tr><td>NPI<sup>V</sup>/<sup>V</sup>; E<sub>np</sub></td><td><b>B</b><sub>p</sub></td><td>0.99, 1.0</td></tr><tr><td>NPI<sup>V</sup>/<sup>V</sup>; E<sub>np</sub></td><td><b>B</b><sub>pfss</sub></td><td>0.99, 1.0</td></tr></tbody></table>

squared differences between the real polarization measurements and the corresponding synthetic data generated from the potential field. While the image of squared differences provides a 2D map of nonpotentiality, its surface integral defines a nonpotentiality index (NPI). In particular, we proposed three nonpotentiality proxies that differ by the data type used to define them, i.e., the fraction of linear polarization (L/I), the direction of linear polarization (Az), or the fraction of circular polarization (V/I).

We applied our nonpotentiality diagnostics to the synthetic data generated from a simulation of the emergence of a twisted flux rope below a helmet streamer (Fan 2017). Such a choice was made to test the capabilities of our proxies in a case for which we could directly compare them with the actual free energy value and its true 3D structure. In particular, our goal was to demonstrate that there is a correlation between the free energy of the system and the nonpotentiality proxies defined

<table><caption>**Table 4**
Consideration of Different Orientations of the Simulation Flux Rope</caption><thead><tr><th>Quantities Being Compared</th><th>Axial Tilt</th><th>Correlations (Pearson, Spearman)</th></tr></thead><tbody><tr><td>s<sup>L</sup>/<sup>L</sup>; ε<sub>np</sub></td><td>15°</td><td>-0.23, 0.04</td></tr><tr><td>NPI<sup>L</sup>/<sup>L</sup>; E<sub>np</sub></td><td>15°</td><td>0.90, 0.86</td></tr><tr><td>s<sup>Az</sup>; ε<sub>np</sub></td><td>15°</td><td>0.37, 0.39</td></tr><tr><td>NPI<sup>Az</sup>; E<sub>np</sub></td><td>15°</td><td>0.95, 0.92</td></tr><tr><td>s<sup>V</sup>/<sup>V</sup>; ε<sub>np</sub></td><td>15°</td><td>0.66, 0.38</td></tr><tr><td>NPI<sup>V</sup>/<sup>V</sup>; E<sub>np</sub></td><td>15°</td><td>0.99, 0.99</td></tr><tr><td>s<sup>L</sup>/<sup>L</sup>; ε<sub>np</sub></td><td>30°</td><td>-0.32, -0.28</td></tr><tr><td>NPI<sup>L</sup>/<sup>L</sup>; E<sub>np</sub></td><td>30°</td><td>0.88, 0.81</td></tr><tr><td>s<sup>Az</sup>; ε<sub>np</sub></td><td>30°</td><td>0.27, 0.14</td></tr><tr><td>NPI<sup>Az</sup>; E<sub>np</sub></td><td>30°</td><td>0.95, 0.98</td></tr><tr><td>s<sup>V</sup>/<sup>V</sup>; ε<sub>np</sub></td><td>30°</td><td>0.54, 0.64</td></tr><tr><td>NPI<sup>V</sup>/<sup>V</sup>; E<sub>np</sub></td><td>30°</td><td>0.97, 0.98</td></tr></tbody></table>

**Note.** The $s^D$; $ε_{np}$ correlations are evaluated for time step 89. All cases use $B_{pfss}$ as reference field and the simulation density, and are masked to exclude contributions from above the simulation cusp height.

from IR coronal polarimetry. We explored two choices of potential fields: (1) the potential field, $B_p$, that matches the magnetic flux distribution at all six boundaries of the simulation box and that must be computed to derive the free energy of the magnetic field from the MHD simulation, and (2) a potential field, $B_{pfss}$, with upper source surface boundary set to heights corresponding to the height of the cusp of the helmet streamer below which the flux rope emerged. $B_{pfss}$ was chosen to provide a solution that could identify the free energy
associated with the flux rope/cavity system while removing the
contribution from the open magnetic field caused by the solar
wind pressure.

We found that the nonpotentiality determined from the
circular polarization is the best proxy of those we tried for
mapping the distribution of off-limb, coronal nonpotential
energy and for tracking its accumulation, regardless of the
potential field used as a reference field (i.e., **B**<sub>p</sub> or **B**<sub>pfss</sub>). In
particular, we obtained high Pearson and Spearman correlation
coefficients between the 2D-density map of nonpotentiality
defined from *V*/*I* for time step 89 and the equivalent map of ē<sub>np</sub>
(see Table 1). Similarly, we obtained extremely high correla-
tion coefficients between the NPI defined from *V*/*I* and the
nonpotential magnetic energy (see Table 2).

The nonpotentiality diagnostics determined from the linear
polarization observables (L/I and Az) showed more complex
results. While the maps of 2D-density of nonpotentiality
derived from L/I and Az both displayed sensitivity to the
presence of nonpotentiality, they failed to meaningfully map to
the coronal nonpotential energy, regardless of whether Bp or
Bpfss was used as a reference field. In particular, we found that
stronger (respectively weaker) signals in the maps of 2D-
density of nonpotentiality were not systematically associated
with stronger (respectively weaker) nonpotential energies. This
is likely due to the fact that the linear polarization is not directly
sensitive, nor related, to the magnetic field strength, as opposed
to the circular polarization that is linearly related to the LOS
magnetic field (e.g., Fan et al. 2018).

Interestingly, despite this lack of correlation between the 2D-
density of nonpotentiality from L/I and ē<sub>np</sub>, we found a strong
correlation between the time variation of the corresponding
NPI (i.e., the integral of the 2D-density of nonpotentiality) and
the coronal free energy, leading to Pearson correlation
coefficients larger than 0.8 when the reference potential field
used is either B̄<sub>p</sub> or B<sub>pfss</sub> (see Table 2). This result means that
the NPI derived from L/I does provide capability for
monitoring the accumulation of free energy in, e.g., coronal
cavities.

This work therefore opens new perspectives for exploiting
IR, off-limb, coronal polarimetry to diagnose the nonpotenti-
ality and free energy in the solar corona. Among the proposed
proxies, our nonpotentiality diagnostics defined from V/I will
be a valuable, original tool for mapping the structure of coronal
free energy and monitor its accumulation in coronal cavities, in
particular for space weather applications. In a future work, we
will further investigate the possibility of using time series of the
V/I-related 2D-density of nonpotentiality to build up synoptic
volumes of coronal free energy that could be used to provide
volume information on the coronal current-carrying systems
that power CMEs. Such data would be further useful for, e.g.,
constraining nonlinear force-free field extrapolations of the
solar coronal magnetic field, as well as for studying the
evolution of these 3D current-carrying systems through the
solar cycle. However, we note that all these applications will
require larger-aperture telescopes, such as the DKIST or the
COSMO-LC, to accurately measure the circular polarization.
Meanwhile, the nonpotentiality index derived from L/I will
provide a practical alternative to study the build up of coronal
free energy prior to an eruption, with smaller-aperture
telescopes (e.g., CoMP) that are only sensitive enough to
measure the linear polarization signal. This will be the subject
of a future observational study.

We thank the anonymous referee for a careful consideration
of the manuscript and helpful comments. M.C.A. gratefully
acknowledges the support of the Significant Opportunities in
Atmospheric Research and Science (SOARS) program. K.D. is
supported by the French Centre National d'Études Spatiales.
This material is based upon work supported by the National
Science Foundation (NSF) under grant No. AGS-1641177
(SOARS), as well as by the National Center for Atmospheric
Research, which is a major facility sponsored by the NSF under
Cooperative Agreement No. 1852977. This work was further
supported by the Air Force Office of Scientific Research grant
FA9550-15-1-0030. Any opinions, findings, and conclusions or
recommendations expressed in this material are those of the
author(s) and do not necessarily reflect the views of the
National Science Foundation.

ORCID iDs

Marcel F. Corchado-Albelo https://orcid.org/0000-0003-1597-0184

Kévin Dalmasse https://orcid.org/0000-0001-8929-4006

Sarah Gibson https://orcid.org/0000-0001-9831-2640

Yuhong Fan https://orcid.org/0000-0003-1027-0795

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