Studying the Transfer of Magnetic Helicity in Solar Active Regions with the Connectivity-based Helicity Flux Density Method

Under ideal conditions, the transfer of magnetic helicity through the photosphere, ${ \mathcal S }$, and into the solar atmosphere is (e.g., Démoulin et al. 2002b; Pariat et al. 2005)

$$\begin{eqnarray}&&\displaystyle \frac{{dH}}{{dt}}={\int }_{{ \mathcal S }}{G}_{\theta }({\boldsymbol{x}})\ d{ \mathcal S }.\end{eqnarray} \tag{ 1 }$$

${G}_{\theta }({\boldsymbol{x}})$ is a surface density of magnetic helicity flux defined as

$$\begin{eqnarray}&&{G}_{\theta }({\boldsymbol{x}})=-\displaystyle \frac{{B}_{n}({\boldsymbol{x}})}{2\pi }{\int }_{{ \mathcal S }^{\prime} }\displaystyle \frac{d\theta ({\boldsymbol{x}}-{\boldsymbol{x}}^{\prime} )}{{dt}}{B}_{n}({\boldsymbol{x}}^{\prime} )\ d{ \mathcal S }^{\prime} ,\end{eqnarray} \tag{ 2 }$$

where

$$\begin{eqnarray}&&\displaystyle \frac{d\theta ({\boldsymbol{x}}-{\boldsymbol{x}}^{\prime} )}{{dt}}=\displaystyle \frac{(({\boldsymbol{x}}-{\boldsymbol{x}}^{\prime} )\times ({\boldsymbol{u}}-{\boldsymbol{u}}^{\prime} )){| }_{n}}{| {\boldsymbol{x}}-{\boldsymbol{x}}^{\prime} {| }^{2}}\end{eqnarray} \tag{ 3 }$$

is the relative rotation rate (or relative angular velocity) between pairs of photospheric magnetic polarities located at ${\boldsymbol{x}}$ and ${\boldsymbol{x}}^{\prime}$ and moving on the photospheric plane ${ \mathcal S }$ with flux transport velocity ${\boldsymbol{u}}={\boldsymbol{u}}({\boldsymbol{x}})$ and ${\boldsymbol{u}}^{\prime} ={\boldsymbol{u}}({\boldsymbol{x}}^{\prime} )$. The flux transport velocity ${\boldsymbol{u}}$ is (Démoulin & Berger 2003)

$$\begin{eqnarray}&&{\boldsymbol{u}}={{\boldsymbol{v}}}_{t}-\displaystyle \frac{{v}_{n}}{{B}_{n}}{{\boldsymbol{B}}}_{t},\end{eqnarray} \tag{ 4 }$$

where subscripts "t" and "n" denote the tangential and normal components of photospheric vector fields, respectively, and ${\boldsymbol{v}}$ and ${\boldsymbol{B}}$ are the photospheric plasma velocity and magnetic fields, respectively.

Equations (1)–(4) show that the total flux of magnetic helicity through the photosphere, ${ \mathcal S }$, can be expressed as the summation of the net rotation of all pairs of photospheric elementary magnetic polarities around each other, weighted by their magnetic flux. It further shows that, at any given time, t, the total flux of magnetic helicity can be solely derived from photospheric quantities, i.e., from a time series of vector magnetograms from which the flux transport velocity can also be derived (Schuck 2008).

The surface density of helicity flux, ${G}_{\theta }$, measures the variation of magnetic helicity in an AR only from the relative motions of its photospheric magnetic polarities. However, we recall that magnetic helicity describes the global linkage of magnetic field lines in the volume. Therefore, what effectively modifies the magnetic helicity of an AR is the relative reorientation of magnetic field lines, i.e., the variation of their mutual helicity, in response to the motions of their photospheric footpoints. Such a global, 3D nature of magnetic helicity and its variation is not taken into account in the definition of ${G}_{\theta }$. As a consequence, ${G}_{\theta }$ has a tendency to misrepresent the local variation of magnetic helicity in ARs by hiding the subtle effects of the mutual helicity variation between magnetic field lines and by overestimating the local helicity flux when an AR is associated with opposite helicity fluxes (e.g., Dalmasse et al. 2014).

Pariat et al. (2005) showed that it is possible to remedy this issue by explicitly rearranging those terms in Equation (1) that are related to the connectivity of elementary magnetic flux tubes. That allows us to express the total flux of magnetic helicity in terms of the sum of the magnetic helicity variation of each individual elementary flux tube of the magnetic field, such that

$$\begin{eqnarray}&&\displaystyle \frac{{dH}}{{dt}}={\int }_{{\rm{\Phi }}}\displaystyle \frac{{{dh}}_{{\rm{\Phi }}}}{{dt}}{| }_{c}\ d{{\rm{\Phi }}}_{c},\end{eqnarray} \tag{ 5 }$$

where ${h}_{{\rm{\Phi }}}$ is the magnetic helicity of the elementary magnetic flux tube, c, and $d{{\rm{\Phi }}}_{c}$ its elementary magnetic flux. ${h}_{{\rm{\Phi }}}$ is the magnetic helicity density and describes how any elementary flux tube is linked to/winded around all other elementary flux tubes (e.g., Berger 1988; Aly 2014; Yeates & Hornig 2014). It can be shown that

$$\begin{eqnarray}&&\displaystyle \frac{{{dh}}_{{\rm{\Phi }}}}{{dt}}{| }_{c}=\displaystyle \frac{{G}_{\theta }({{\boldsymbol{x}}}_{{c}_{+}})}{| {B}_{n}({{\boldsymbol{x}}}_{{c}_{+}})| }+\displaystyle \frac{{G}_{\theta }({{\boldsymbol{x}}}_{{c}_{-}})}{| {B}_{n}({{\boldsymbol{x}}}_{{c}_{-}})| }\end{eqnarray} \tag{ 6 }$$

$$\begin{eqnarray}&&=\,{\dot{{\rm{\Theta }}}}_{B}({{\boldsymbol{x}}}_{{c}_{+}})-{\dot{{\rm{\Theta }}}}_{B}({{\boldsymbol{x}}}_{{c}_{-}}),\end{eqnarray} \tag{ 7 }$$

where ${{\boldsymbol{x}}}_{{c}_{+}}$ (${{\boldsymbol{x}}}_{{c}_{-}}$) is the positive (negative), photospheric, magnetic polarity of the elementary magnetic flux tube c (see Figure 1 of Dalmasse et al. 2014, for a generic sketch of elementary flux tubes and their photospheric connectivity), and

$$\begin{eqnarray}&&{\dot{{\rm{\Theta }}}}_{B}({\boldsymbol{x}})=-\displaystyle \frac{1}{2\pi }{\int }_{{ \mathcal S }^{\prime} }\displaystyle \frac{d\theta ({\boldsymbol{x}}-{\boldsymbol{x}}^{\prime} )}{{dt}}{B}_{n}({\boldsymbol{x}}^{\prime} )\ d{ \mathcal S }^{\prime} .\end{eqnarray} \tag{ 8 }$$

Note that the advantage of using Equation (7) over Equation (6) is to avoid artificial numerical singularities when ${B}_{n}({\boldsymbol{x}})$ is very small.

Then, Pariat et al. (2005) defined a new surface density of helicity flux by redistributing ${{dh}}_{{\rm{\Phi }}}/{dt}$ at each photospheric footpoint of the elementary magnetic flux tube, c, to map the photospheric flux of magnetic helicity

$$\begin{eqnarray}&&{G}_{{\rm{\Phi }}}({{\boldsymbol{x}}}_{{c}_{\pm }})=\displaystyle \frac{1}{2}\displaystyle \frac{{{dh}}_{{\rm{\Phi }}}}{{dt}}{| }_{c}| {B}_{n}({{\boldsymbol{x}}}_{{c}_{\pm }})| .\end{eqnarray} \tag{ 9 }$$

Note that the factor 1/2 in Equation (9) assumes that both photospheric footpoints of an elementary flux tube contribute equally to the variation of its magnetic helicity (a more general case is described in Pariat et al. 2005). Note also that the calculations of ${{dh}}_{{\rm{\Phi }}}/{dt}$ and ${G}_{{\rm{\Phi }}}$ require one significant additional piece of information as compared with ${G}_{\theta }$, i.e., the connectivity of the magnetic field.

Finally, we stress here that both ${G}_{\theta }$ and ${G}_{{\rm{\Phi }}}$ are instantaneous estimations of injected helicity at a particular location on the photosphere, and not the total helicity content in the associated field line.

Dalmasse et al. (2014) introduced a method to compute the connectivity-based helicity flux density proxy, ${G}_{{\rm{\Phi }}}$. Using various analytical case studies and numerical MHD simulations, they showed that ${G}_{{\rm{\Phi }}}$ properly tracks the site(s) of magnetic helicity variations.

Their method is based on field-line integration to derive the magnetic connectivity required to compute ${G}_{{\rm{\Phi }}}$. For observational studies, routine magnetic field measurements are mostly realized at the photospheric level (see Section 1). We thus perform force-free field (FFF) extrapolations to obtain the coronal magnetic field of the studied AR (see Section 4) and compute the photospheric distribution of magnetic helicity flux from Equations (7) and (9).

Each pixel of the photospheric vector magnetogram is identified as the cross section of an elementary magnetic flux tube with the photosphere. Each of these elementary flux tubes is associated with one magnetic field line that is integrated to obtain the connectivity. For any closed magnetic field line, we thus obtain a pair (${{\boldsymbol{x}}}_{{c}_{+}};{{\boldsymbol{x}}}_{{c}_{-}}$) of photospheric footpoints, where ${{\boldsymbol{x}}}_{{c}_{+}}$ is the positive magnetic polarity of the elementary flux tube and ${{\boldsymbol{x}}}_{{c}_{-}}$ is its negative magnetic polarity at the photosphere (z = 0). We then introduce a slight modification to the method proposed in Dalmasse et al. (2014). Instead of computing ${G}_{{\rm{\Phi }}}$ from Equation (6), which can lead to artificial numerical singularities when ${B}_{n}({\boldsymbol{x}})$ is very small, we compute ${{dh}}_{{\rm{\Phi }}}/{dt}$ and ${G}_{{\rm{\Phi }}}$ from Equations (7) and (9). Hence, once the conjugate footpoint of a field line is found, we compute $({\dot{{\rm{\Theta }}}}_{B}({{\boldsymbol{x}}}_{{c}_{+}})$; ${\dot{{\rm{\Theta }}}}_{B}({{\boldsymbol{x}}}_{{c}_{-}}))$ using bilinear interpolation. Finally, field lines for which the conjugate footpoint reaches the top or lateral boundaries are treated as open field lines, and both ${{dh}}_{{\rm{\Phi }}}/{dt}$ and ${G}_{{\rm{\Phi }}}$ are simply set to zero. This is a second modification to the method of Dalmasse et al. (2014) justified by the fact that, in this paper, we focus on comparing ${{dh}}_{{\rm{\Phi }}}/{dt}$ and ${G}_{{\rm{\Phi }}}$ computed using different magnetic field extrapolations, i.e., two quantities that are only defined for closed field lines.

In the connectivity-based helicity flux density approach, half the total helicity flux computed from ${G}_{\theta }$ over the ensemble of positive magnetic polarities is redistributed over the ensemble of negative magnetic polarities, and vice versa. For this redistribution to be perfect, the total magnetic flux summed over the magnetic polarities where ${G}_{{\rm{\Phi }}}$ is computed must vanish. If, for instance, the magnetic flux from the negative magnetic polarities is smaller than that of the positive ones, then a fraction of the total helicity flux computed from ${G}_{\theta }$ over the positive magnetic polarities is not redistributed in the negative ones. This means that part of the total helicity flux computed with ${G}_{\theta }$ for the entire magnetic configuration is missing from that computed with ${G}_{{\rm{\Phi }}}$. We thus introduce a first metric to validate ${G}_{{\rm{\Phi }}}$ maps, i.e., the percentage of magnetic flux imbalance, ${\tau }_{{{\rm{\Phi }}}_{\mathrm{imb}.}}$, for the closed magnetic flux where ${G}_{{\rm{\Phi }}}$ is computed:

$$\begin{eqnarray}&&{\tau }_{{{\rm{\Phi }}}_{\mathrm{imb}.}}=\displaystyle \frac{{\int }_{{{ \mathcal S }}_{\mathrm{cl}.}}{B}_{n}({\boldsymbol{x}})\ d{ \mathcal S }}{\min \left({\int }_{{{ \mathcal S }}_{\mathrm{cl}.}({B}_{n}\gt 0)}{B}_{n}({\boldsymbol{x}})\ d{ \mathcal S };\left|{\int }_{{{ \mathcal S }}_{\mathrm{cl}.}({B}_{n}\lt 0)}{B}_{n}({\boldsymbol{x}})\ d{ \mathcal S }\right|\right)},\end{eqnarray} \tag{ 10 }$$

where ${{ \mathcal S }}_{{{\rm{\Phi }}}_{\mathrm{cl}.}}$ is the part of the photosphere associated with the closed magnetic flux, ${{\rm{\Phi }}}_{\mathrm{cl}.}$.

By definition, the fluxes measured by ${G}_{{\rm{\Phi }}}$ are simply a redistribution of the fluxes measured by ${G}_{\theta }$ at both footpoints of each elementary magnetic flux tube in the closed magnetic field. Thus, the intensity of the magnetic helicity flux in one pixel can be different for ${G}_{{\rm{\Phi }}}$ and ${G}_{\theta }$. However, the total flux of magnetic helicity integrated over the closed magnetic flux from ${G}_{{\rm{\Phi }}}$ must be the same as that integrated from ${G}_{\theta }$ because Equation (5) is equal to Equation (1) for the closed magnetic field. The second metric we defined for the validation of ${G}_{{\rm{\Phi }}}$ maps is thus

$$\begin{eqnarray}&&{C}_{{{ \mathcal S }}_{{{\rm{\Phi }}}_{\mathrm{cl}.}}}=\displaystyle \frac{{\int }_{{{ \mathcal S }}_{{{\rm{\Phi }}}_{\mathrm{cl}.}}}({G}_{{\rm{\Phi }}}({\boldsymbol{x}})-{G}_{\theta }({\boldsymbol{x}}))\ d{ \mathcal S }}{{\int }_{{{ \mathcal S }}_{{{\rm{\Phi }}}_{\mathrm{cl}.}}}{G}_{\theta }({\boldsymbol{x}})\ d{ \mathcal S }}.\end{eqnarray} \tag{ 11 }$$

In theory, ${C}_{{{ \mathcal S }}_{{{\rm{\Phi }}}_{\mathrm{cl}.}}}$ should be strictly equal to 0. In practice, however, its value depends on departures of ${\tau }_{{{\rm{\Phi }}}_{\mathrm{imb}.}}$ from 0. Accuracy and validation of the ${G}_{{\rm{\Phi }}}$ map require that both ${\tau }_{{{\rm{\Phi }}}_{\mathrm{imb}.}}$ and ${C}_{{{ \mathcal S }}_{{{\rm{\Phi }}}_{\mathrm{cl}.}}}$ be close to 0.

Finally, we define two quantification indices to compare the signal intensity between ${G}_{{\rm{\Phi }}}$ and ${G}_{\theta }$ within the closed magnetic field:

$$\begin{eqnarray}&&{C}_{+}=\displaystyle \frac{{\int }_{{{ \mathcal S }}_{\mathrm{cl}.}({G}_{\theta }\gt 0)}{G}_{\theta }({\boldsymbol{x}})\ d{ \mathcal S }}{{\int }_{{{ \mathcal S }}_{\mathrm{cl}.}({G}_{{\rm{\Phi }}}\gt 0)}{G}_{{\rm{\Phi }}}({\boldsymbol{x}})\ d{ \mathcal S }},\end{eqnarray} \tag{ 12 }$$

$$\begin{eqnarray}&&{C}_{-}=\displaystyle \frac{{\int }_{{{ \mathcal S }}_{\mathrm{cl}.}({G}_{\theta }\lt 0)}{G}_{\theta }({\boldsymbol{x}})\ d{ \mathcal S }}{{\int }_{{{ \mathcal S }}_{\mathrm{cl}.}({G}_{{\rm{\Phi }}}\lt 0)}{G}_{{\rm{\Phi }}}({\boldsymbol{x}})\ d{ \mathcal S }}.\end{eqnarray} \tag{ 13 }$$

${C}_{+}$ and ${C}_{-}$ respectively compare the total positive and total negative magnetic helicity fluxes derived from ${G}_{\theta }$ with those derived from ${G}_{{\rm{\Phi }}}$ in the closed magnetic flux. Pariat et al. (2005) and Dalmasse et al. (2014) showed that the helicity flux density proxy ${G}_{\theta }$ hides the true local helicity flux and tends to exhibit larger and spurious helicity flux intensities as compared with the ${G}_{{\rm{\Phi }}}$ proxy when opposite helicity fluxes are present in an AR. ${C}_{+}$ and ${C}_{-}$ allow us to quantify the intensity of the spurious signals in ${G}_{\theta }$, thus providing an idea of the global improvement of the ${G}_{{\rm{\Phi }}}$ maps relative to the ${G}_{\theta }$ ones. In particular, large departures of ${C}_{\pm }$ from 1 are indicative of strong spurious signals in ${G}_{\theta }$.

We test the robustness of the connectivity-based helicity flux density method against different magnetic field extrapolation models of the internally complex AR 11158. This AR appeared on the solar disk at the heliographic coordinates S19 E42 on 2011 February 10. The AR was the result of fast and strong magnetic flux emergence that produced two large-scale bipoles, a northern and a southern one, in close proximity (see Figure 1; e.g., Schrijver et al. 2011). The complex, quadrupolar magnetic field of this AR produced several C-/M-/X-class flares and CMEs during its on-disk passage (e.g., Toriumi et al. 2014). A large fraction of this flaring activity was associated with the collision between the negative magnetic polarity of the northern bipole, NN, and the positive magnetic polarity of the southern bipole, SP, which led to a strong and continuous shearing of their polarity inversion line. More details on the configuration, evolution, and flaring activity of the AR can be found in, e.g., Sun et al. (2012), Jiang et al. (2012), Vemareddy et al. (2012a), and Inoue et al. (2013).

The coronal models of AR 11158 are computed using vector magnetograms taken by the Helioseismic and Magnetic Imager (HMI, e.g., Schou et al. 2012) on board the Solar Dynamics Observatory (SDO; e.g., Pesnell et al. 2012). SDO/HMI provides full-disk vector magnetograms of the Sun with a pixel size of $0\buildrel{\prime\prime}\over{.} 5$. For the purposes of this paper, we reuse part of the data from Dalmasse et al. (2013), who had applied the connectivity-based approach to AR 11158 with an NLFFF extrapolation without addressing the possible dependency of the results on the choice of extrapolation method. In particular, vector magnetograms at 06:22 UT and 06:34 UT on 2011 February 14 from the HMI-SHARP data series, HARP number 377 (Hoeksema et al. 2014), are reused.

These two vector magnetograms are used to derive the photospheric flux transport velocity field with the differential affine velocity estimator for vector magnetograms (DAVE4VM; Schuck 2008), using a window size of 19 pixels as suggested by Liu et al. (2013). The computed flux transport velocity field effectively represents an instantaneous flux transport velocity field at ∼06:28 UT. The two vector magnetograms taken at 06:22 UT and 06:34 UT are averaged to construct an instantaneous vector magnetogram associated with the flux transport velocity field at ∼06:28 UT. The constructed vector magnetic field is then used both to compute ${G}_{\theta }$ and as the photospheric boundary condition for the different FFF extrapolations presented in Section 4. The vector magnetic field and flux transport velocity field at ∼06:28 UT are shown in Figure 1.

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As part of testing the reliability of the connectivity-based helicity flux density method, we also wish to evaluate uncertainties in ${G}_{\theta }$ and ${G}_{{\rm{\Phi }}}$ maps caused by those in the photospheric magnetic field. In particular, our error analysis includes both the effect of photon noise ($\approx 10$ G) and several sources of systematic errors. Hoeksema et al. (2014) recently performed an extensive analysis of the uncertainties in the measurements of the magnetic field strength by SDO/HMI. In particular, they focused on the uncertainty analysis for NOAA 11158. Among several sources of systematic errors, including those related to the inversion code, they found that the dominant contribution of uncertainty in SDO/HMI magnetic field measurements is coming from the daily variation of the radial velocity of the spacecraft along its geosynchronous orbit. Their analysis concludes that the typical uncertainty in SDO/HMI measurements of the magnetic field strength is about 100 G. The latter can be easily checked from the estimated field strength error map of NOAA 11158 provided by the SDO/HMI pipeline and that can be downloaded from JSOC.⁵ We also want to compare these errors with those related to the choice of magnetic field extrapolation used to compute ${G}_{{\rm{\Phi }}}$.

For that purpose, we conduct a Monte Carlo experiment as proposed by Liu & Schuck (2012) for error estimation of the total helicity flux. Random noise with a Gaussian distribution having a width (σ) of 100 G is added to all three components of the magnetic field for both vector magnetograms taken at 06:22 UT and 06:34 UT. The value of 100 G is the $\approx 1\sigma$ uncertainty in the total magnetic field strength from HMI data estimated by Hoeksema et al. (2014) and further reported in Bobra et al. (2014). The uncertainty is then propagated through the chain of helicity flux density calculations to the flux transport velocity field at the photosphere derived from DAVE4VM, to the corresponding vector magnetogram at ∼06:28 UT, and finally to the ${G}_{\theta }$ and ${G}_{{\rm{\Phi }}}$ maps.

Although they did not perform a full parametric analysis, Wiegelmann et al. (2006) showed that extrapolations with the optimization method were not significantly affected by modest noise in the photospheric vector magnetogram. They found that the preprocessing of the photospheric data toward a more force-free state (see, e.g., Section 4.2) strongly helps in that matter. This may be expected considering that the random noise on the photospheric vector magnetic field acts as a source of non-force-free signals on high spatial frequencies while the preprocessing filters such signals out. Even if FFF reconstructions would likely be differently affected by noise in the photospheric data, its global effect on the extrapolations should be limited by the preprocessing stage. On the other hand, FFF extrapolations are more likely to be affected by more global effects, including, but not limited to, large-scale non-force-free regions not suppressed by the preprocessing, magnetic flux imbalance, and lateral boundary conditions. Then, as far as ${G}_{{\rm{\Phi }}}$ is concerned, the effect of random noise on the extrapolation results is to introduce uncertainties in the connectivity of the magnetic field. In this regard, we believe that the choice of extrapolation method and preprocessing level is more fundamental and has a stronger impact on the magnetic connectivity and hence on ${G}_{{\rm{\Phi }}}$. For these reasons, we decided not to propagate the noise to the FFF extrapolations.

The noise propagation experiment is repeated 100 times, producing 100 noise-added ${G}_{\theta }$ and ${G}_{{\rm{\Phi }}}$ maps for all three FFF extrapolations considered in this paper. For each scalar quantity, ${ \mathcal F }$, the $1\sigma$ estimated error, ${\sigma }_{{ \mathcal F }}$, is computed as the mean, over all n pixels of the map, of the rms of the N = 100 noise-added ${ \mathcal F }$-maps, ${{ \mathcal F }}_{{\rm{n}}.{\rm{a}}.}^{i}({{\boldsymbol{x}}}_{j})$, compared with the no-noise ${ \mathcal F }$-map, ${{ \mathcal F }}_{{\rm{n}}.{\rm{n}}.}({{\boldsymbol{x}}}_{j})$:

$$\begin{eqnarray}&&{\sigma }_{{ \mathcal F }}=\displaystyle \frac{1}{n}\displaystyle \sum _{j=1}^{n}{\left(\displaystyle \frac{1}{N}\displaystyle \sum _{i=1}^{N}{({{ \mathcal F }}_{{\rm{n}}.{\rm{a}}.}^{i}({{\boldsymbol{x}}}_{j})-{{ \mathcal F }}_{{\rm{n}}.{\rm{n}}.}({{\boldsymbol{x}}}_{j}))}^{2}\right)}^{\tfrac{1}{2}}.\end{eqnarray} \tag{ 14 }$$

The current-free magnetic field is the minimal-energy possible state for the given distribution of magnetic field at the boundaries of the considered volume. In addition, the potential field is often used as an initial state of numerical methods that build the more complex NLFFF models (see review by, e.g., Wiegelmann & Sakurai 2012). It is therefore an important candidate to consider for testing the connectivity-based helicity flux density method, despite its limitation in reproducing nonlinear features of the coronal field.

The potential field can be directly computed using the potential theory and the reflection principle to solve the Laplace equation for the scalar potential in terms of the flux through the photospheric boundary (Schmidt 1964). However, in order to speed up the calculation, such a method is actually used only to compute the scalar potential on all six boundaries of the considered volume. The magnetic scalar potential in the volume is then computed solving the Laplace equation, subjected to the obtained Dirichlet boundary conditions, using a fast Helmholtz solver from the Intel Mathematical Kernel Library. For the required vertical component of the field on the lower boundary, the same vertical component as for the NLFFF extrapolation described in Section 4.2 is used. The potential field extrapolation is referred to as POT in the following, and selected field lines for this extrapolation are shown in the left panel of Figure 2.

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NLFFF extrapolations are models of the coronal magnetic field that assume the corona to be static on the timescale of interest and to be dominated by magnetic forces that are distributed in such a way that the resulting Lorentz force is everywhere vanishing. Such assumptions are supposed to be valid in the entire volume of interest, boundaries included. The magnetofrictional relaxation method implements numerical relaxation and multigrid techniques to solve the corresponding equations (see Valori et al. 2007, 2010; DeRosa et al. 2015, for more details).

The remapped and disambiguated vector magnetograms from the HMI-SHARP data series were interpolated to $1^{\prime\prime}$ resolution and averaged to construct the vector magnetogram at 06:28 UT (see Section 3.1) used for the NLFFF extrapolation. Vector magnetograms are inferred from spectropolarimetric measurements taken at photospheric heights, where the plasma is non-force-free. Therefore, in order to use the vector magnetogram as a boundary condition for the NLFFF extrapolation code, the forces acting on the magnetogram need to be reduced (preprocessing). To this purpose we use the method of Fuhrmann et al. (2007, 2011). In this application, only the horizontal components of the field are preprocessed, yielding a reduction of the forces from 0.035 to 0.002 in the nondimensional units used in Metcalf et al. (2008). Since smoothing is not necessarily facilitating the extrapolation (Valori et al. 2013), no smoothing was applied. The resulting magnetogram was then extrapolated using the magnetofrictional code into a volume of about 208 × 202 × 145 Mm³.

The resulting extrapolated field has solenoidal errors that can be quantified using the formula by Valori et al. (2013) into 9% of the total magnetic energy. The fraction of the total current that is perpendicular to the field is ${\sigma }_{J}=0.48$ (Wheatland et al. 2000), a rather high value that is not uncommon for extrapolation of HMI vector magnetograms with the magnetofrictional method (see Valori et al. 2012, for an application to the Hinode/SP magnetogram with a much lower ${\sigma }_{J}$). Selected field lines of the NLFFF obtained in this way, and referred to as NLFFF-R in the following, are shown in the middle panel of Figure 2.

For the second NLFFF model considered in this paper we use the weighted optimization method (Wiegelmann 2004), which is an implementation and modification of the original optimization algorithm of Wheatland et al. (2000). The optimization method minimizes an integrated joint measure, which comprises the normalized Lorentz force, the magnetic field divergence, and treatment of the measurement errors, over the computational domain (for more details, see Wiegelmann & Inhester 2010; Wiegelmann et al. 2012).

To perform the extrapolation, the vector magnetogram (Figure 1) is first rebinned to $1^{\prime\prime}$ per pixel and preprocessed toward the force-free condition using the method of Wiegelmann et al. (2006). The extrapolation is finally performed on a uniform grid of $256\times 256\times 200$ points covering ∼185 × 185 ×144 Mm³. We find a solenoidal error of 1% of the total magnetic energy and a fraction of total current perpendicular to the magnetic field ${\sigma }_{J}=0.20$. These values are lower than for the NLFFF-R model (Section 4.2), which is due to both different preprocessing and extrapolation methods and different strategies.

In the following, the NLFFF model built with the optimization method is referred to as NLFFF-O. A set of selected magnetic field lines is shown in the right panel of Figure 2.

Table 1 presents the validation metrics for the ${G}_{{\rm{\Phi }}}$ maps computed from the three FFF extrapolation models described in Section 4. For all three ${G}_{{\rm{\Phi }}}$ maps, we obtain ${\tau }_{{{\rm{\Phi }}}_{\mathrm{imb}.}}$ below 1% and ${C}_{{{ \mathcal S }}_{{{\rm{\Phi }}}_{\mathrm{cl}.}}}$ below 5%. This allows us to verify that the magnetic flux over which the connectivity-based helicity flux density is computed is very well balanced and that the calculation of ${{dh}}_{{\rm{\Phi }}}/{dt}$ and ${G}_{{\rm{\Phi }}}$ well preserves the total helicity flux in that region in the closed magnetic flux. Together, these numbers enable us to confirm the accuracy of the ${G}_{{\rm{\Phi }}}$ and ${{dh}}_{{\rm{\Phi }}}/{dt}$ calculations discussed in the following.

Table 1.  Metrics for Validation and Quantification of ${G}_{{\rm{\Phi }}}$ Maps

FFF Model	${\tau }_{{{\rm{\Phi }}}_{\mathrm{imb}.}}$	${C}_{{{ \mathcal S }}_{{{\rm{\Phi }}}_{\mathrm{cl}.}}}$	${C}_{+}$	${C}_{-}$
POT	1.2 × 10−3	4.2 × 10−4	1.32	1.71
NLFFF-R	−7.6 × 10−3	4.7 × 10−2	1.21	1.67
NLFFF-O	2.6 × 10−3	3.2 × 10−2	1.23	1.57

Note. The table presents the validation and quantification metrics for the maps displayed in Figure 3. All metrics are dimensionless ratios defined by Equations (10)–(13).

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Figure 3 presents the surface density of helicity flux from the purely photospheric proxy, ${G}_{\theta }$, and the connectivity-based proxy, ${G}_{{\rm{\Phi }}}$, computed from all three FFF extrapolations. The mean of the absolute value of the helicity flux signal in most of the AR (i.e., $| {B}_{z}| \geqslant 100\,{\rm{G}}$) is 2.8 × 10⁶ Wb² m⁻² s⁻¹ for the ${G}_{\theta }$ map and 1.7 × 10⁶ Wb² m⁻² s⁻¹ for the ${G}_{{\rm{\Phi }}}$ maps, while a large fraction of the maps are associated with local helicity fluxes of 10⁷ Wb² m⁻² s⁻¹. As shown in Table 2, the errors for ${G}_{\theta }$ and ${G}_{{\rm{\Phi }}}$ estimated from our Monte Carlo experiment are 3.7 × 10⁵ Wb² m⁻² s⁻¹ for ${G}_{\theta }$ and lower than 3 × 10⁵ Wb² m⁻² s⁻¹ for ${G}_{{\rm{\Phi }}}$. The signal intensity of the surface density maps in Figure 3 is thus well above the estimated noise level.

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Table 2.  Error Estimations from Monte Carlo Experiment

	${G}_{\theta }$	${G}_{{\rm{\Phi }}}$(POT)	${G}_{{\rm{\Phi }}}$(NLFFF-R)	${G}_{{\rm{\Phi }}}$(NLFFF-O)
σ	3.7	2.6	2.2	3.0

Note. The errors are in units of 10⁵ Wb² m⁻² s⁻¹, i.e., ∼10–100 times smaller than the typical values displayed by the ${G}_{\theta }$ and ${G}_{{\rm{\Phi }}}$ maps from Figure 3. See Section 3.2 for a description of error calculations.

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Figure 3 shows that the largest differences in helicity flux density maps are between ${G}_{\theta }$ and the three ${G}_{{\rm{\Phi }}}$ maps. In particular, the strongest differences are associated with magnetic flux systems that connect footpoints of opposite ${G}_{\theta }$ signs, i.e., footpoints of NN connected to NP, footpoints of SN connected to SP, and footpoints of SN connected to NP. On the other hand, the ${G}_{\theta }$ map is relatively similar to the three ${G}_{{\rm{\Phi }}}$ maps for the flux system connecting NN to SP, because magnetic field lines are connecting footpoints with similar values of ${G}_{\theta }({\boldsymbol{x}})/| {B}_{n}({\boldsymbol{x}})|$. This is consistent with the work of Pariat et al. (2005) and Dalmasse et al. (2014), who showed that ${G}_{\theta }$ hides the true helicity flux signal when simultaneous opposite helicity fluxes are present in a magnetic configuration. This effect is inherent to the definition of ${G}_{\theta }$ that does not acknowledge the fact that the variation of magnetic helicity in an elementary magnetic flux tube comes from the motions of its two photospheric footpoints with respect to the other elementary flux tubes of the entire magnetic configuration. As a result, the comparison of the total positive helicity flux and total negative helicity flux from ${G}_{\theta }$ in the closed magnetic flux with the same quantities computed for each one of the ${G}_{{\rm{\Phi }}}$ maps leads to values of ${C}_{+}\gt 1.2$ and ${C}_{-}\gt 1.5$ (see Table 1). Such values indicate that ${G}_{\theta }$ is affected by moderate spurious positive signals and rather high spurious negative helicity fluxes that are corrected for by the use of ${G}_{{\rm{\Phi }}}$ (see Section 2.3). Figure 3 thus highlights the fact that the redistribution of helicity flux operated by ${G}_{{\rm{\Phi }}}$ is crucial to the photospheric mapping of helicity flux in ARs, regardless of the coronal magnetic field modeling.

All three ${G}_{{\rm{\Phi }}}$ maps in Figure 3 are in very good qualitative agreement, showing (i) a negative helicity flux in SN, (ii) a strong positive helicity flux in NN and SP, and (iii) strong positive and negative helicity fluxes with a strongly marked interface in NP. The ${G}_{{\rm{\Phi }}}$ maps from the two NLFFF models are very similar. Setting aside the white signal in SN, which is due to open field lines where ${G}_{{\rm{\Phi }}}$ is not computed, and focusing on the common areas where all three extrapolations have closed magnetic flux, we find that the helicity flux density map derived from the potential field is also similar to the ${G}_{{\rm{\Phi }}}$ maps derived from the two NLFFF models. The most noticeable difference is in the southeast part of NN, which does not show any significant helicity flux for ${G}_{{\rm{\Phi }}}$(POT), contrary to both ${G}_{{\rm{\Phi }}}$(NLFFF-R) and ${G}_{{\rm{\Phi }}}$(NLFFF-O).

Finally, the good qualitative agreement between the connectivity-based helicity flux density calculations from the three FFF extrapolations is further emphasized by the 3D representation of ${{dh}}_{{\rm{\Phi }}}/{dt}$ in Figure 4. They all show that the inner part of the AR is dominated by strong positive helicity fluxes and is embedded within a magnetic field region dominated by strong negative helicity fluxes. The core results of Dalmasse et al. (2013) are thus confirmed with a weak dependence on the extrapolation method.

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We now focus on the quantitative comparison of the helicity flux density calculations obtained from the three FFF models. We restrict our analysis to the pixels of the ${G}_{{\rm{\Phi }}}$ maps that are above the noise level (different for each map) estimated from the Monte Carlo experiment. The error levels are given in Table 2.

Figure 5 displays the three maps of sign agreement. The regions where our analysis can be carried out (white and black) are mostly associated with the strong magnetic field of AR 11158, i.e., where $| {B}_{z}| \gtrsim 500$ G. These regions correspond to the area where most of the helicity flux (at least 88% of the total unsigned helicity flux) computed with ${G}_{{\rm{\Phi }}}$ is coming from. This is because the helicity flux intensity outside these regions is below the noise level of the respective ${G}_{{\rm{\Phi }}}$ maps. Despite the presence of some relatively small areas of disagreement (black patches), we find that these regions are dominated by agreement (white signal) over the sign of helicity flux derived using different FFF extrapolations. In particular, the percentage of surface area for which pairs of ${G}_{{\rm{\Phi }}}$ maps agree is always larger than $85 \%$, which translates into more than $\approx 84 \%$ in terms of magnetic flux. Therefore, the local sign of helicity flux computed from the connectivity-based helicity flux density method is very robust to the different FFF models and assumptions used for calculations.

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Figure 6 displays the linear correlation plots of ${G}_{{\rm{\Phi }}}$ values in each pixel, for different pairs of FFF extrapolations. For comparisons between ${G}_{{\rm{\Phi }}}$ from the potential field model and one of the two NLFFFs, the points are colored according to the strength of the photospheric electric current density, $| {j}_{z}|$, from the different preprocessed boundary employed in the NLFFF model under consideration. For the plot comparing ${G}_{{\rm{\Phi }}}$ from the two NLFFF extrapolations, the points are colored according to ${(| {j}_{z}(\mathrm{NLFFF} \mbox{-} {\rm{R}})| \cdot | {j}_{z}(\mathrm{NLFFF} \mbox{-} {\rm{O}})| )}^{1/2}$. Such a color coding was introduced in order to investigate the dependency of the scatter on the electric current density of magnetic field lines where ${G}_{{\rm{\Phi }}}$ is computed.

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In each scatter plot, we find that the spatial distribution of points exhibits a clear ellipsoidal shape aligned along the y = x diagonal line. Each one of these distributions displays a moderate dispersion. The three standard deviations computed from each scatter plot of Figure 6 are ≤2.6 × 10⁶ Wb² m⁻² s⁻¹, which is 5–10 times smaller than most of the signal in the four main magnetic polarities. These standard deviations are $\approx 10$ times larger than the ${G}_{{\rm{\Phi }}}$ errors estimated from the Monte Carlo experiment (Table 2). As anticipated, it means that, despite the substantial agreement on the sign of the injected helicity between different extrapolations, a significant uncertainty in the connectivity-based calculations is coming from the choice of extrapolation model used to derive the field line connectivity.

Figure 6 further shows that the ellipsoidal pattern of the distribution of points is present independently of the electric current density, i.e., of the color of the points. We also notice that the scattering of points away from the y = x line presents some relatively weak dependence on the electric current density of the field lines used for the connectivity-based calculations. In particular, we find a dispersion in the range ≈[1.5, 2.0] × 10⁶ Wb² m⁻² s⁻¹ for black to green points and ≈[2.6, 4.1] × 10⁶ Wb² m⁻² s⁻¹ for green to red points (finer details are provided in Table 3). In addition, the number of pixels with average to strong electric current density (i.e., green to red points) is sensibly the same as the number of pixels with very weak to average electric current density (i.e., black to green points). This implies that the correlation coefficients, displayed in Figure 6 and discussed below, are not dominated by the differences in ${G}_{{\rm{\Phi }}}$ values from nearly potential magnetic field lines.

Table 3.  Scatter Plot Dispersion versus Electric Current Density

jz	]0, 0.2]	]0.2, 3.1]	]3.1, 15.6]	]15.6, 50.0]
${\sigma }_{{G}_{{\rm{\Phi }}}}$	[1.5, 1.8]	[1.8, 2.0]	[2.6, 3.6]	[3.0, 4.1]

Note. The interval for the standard deviation, ${\sigma }_{{G}_{{\rm{\Phi }}}}$, is derived from the values obtained for the three scatter plots of Figure 6. The electric current density is in units of mA m⁻², and the dispersion is in units of 10⁶ Wb² m⁻² s⁻¹. From left to right, the four ranges of electric current density correspond to $]\mathrm{black};\mathrm{dark}\,\mathrm{blue}]$, $]\mathrm{dark}\,\mathrm{blue};\mathrm{green}]$, $]\mathrm{green};\mathrm{yellow}]$, and $]\mathrm{yellow};\mathrm{red}]$ of the color scale in Figure 6.

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For all three scatter plots, we find that the Pearson, c_P, and Spearman, c_S, correlation coefficients are such that ${c}_{P}\geqslant 0.75$ and ${c}_{S}\geqslant 0.72$. We checked that these values are statistically significant by conducting a null hypothesis test (details and results of this test are provided in the Appendix). We therefore conclude that the calculations of the connectivity-based helicity flux density derived from different FFF extrapolations are highly correlated and consistent with each other.

For further comparison, we compute the vector correlation metric, ${C}_{\mathrm{vec}}$, comparing the three 3D magnetic field extrapolations as defined by Equation (28) of Schrijver et al. (2006). For that purpose, the POT and NLFFF-R 3D magnetic fields are interpolated on the same grid as the NLFFF-O (whose extrapolation domain is common to all three models) using trilinear interpolation. We find ${C}_{\mathrm{vec}}({{\boldsymbol{B}}}_{\mathrm{NLFFF}-{\rm{R}}},{{\boldsymbol{B}}}_{\mathrm{POT}})=0.84$, ${C}_{\mathrm{vec}}({{\boldsymbol{B}}}_{\mathrm{NLFFF}-{\rm{O}}},{{\boldsymbol{B}}}_{\mathrm{POT}})=0.90$, and ${C}_{\mathrm{vec}}({{\boldsymbol{B}}}_{\mathrm{NLFFF}-{\rm{O}}},{{\boldsymbol{B}}}_{\mathrm{NLFFF}-{\rm{R}}})\,=0.90$. Such values for the vector correlation metric of the magnetic fields are very high even though the 3D distributions of the magnetic field lines are relatively different when comparing the three extrapolations, as inferred from Figure 2. We thus find that the vector correlation metric for the magnetic fields is higher than the Pearson and Spearman correlation coefficients found when comparing the helicity flux density calculations.