DISTRIBUTION OF ELECTRIC CURRENTS IN SUNSPOTS FROM PHOTOSPHERE TO CORONA

The two sunspots studied here are simple round sunspots; one is identified in the active region (AR) NOAA 11084 (southern hemisphere) and the other one in AR NOAA 11092 (northern hemisphere). Both spots conform to the hemispheric pattern of chirality (Balasubramaniam et al. 2004). These two spots were chosen for our study because (1) they are regular, of round shape with a nearly uniform twist in the azimuthal direction in chromospheric and coronal images; (2) they are isolated and not surrounded by other strong polarities, and therefore of minimal complexity; and (3) they are not involved in flaring activity in the range of times selected for the present analysis.

The contextual pictures (chromospheric, coronal, photospheric continuum, and longitudinal magnetogram) of the two sunspots are displayed in Figure 1. The images shown in this figure were obtained by the instruments Atmospheric Imaging Assembly (AIA; Lemen et al. 2012) and Helioseismic and Magnetic Imager (HMI; Scherrer et al. 2012) on board the Solar Dynamics Observatory (SDO) spacecraft (Pesnell et al. 2012).

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Further, for high-resolution vector magnetic field data of these sunspots we use observations from Solar Optical Telescope (SOT; Kosugi et al. 2007; Tsuneta et al. 2008) on board the Hinode spacecraft. The vector magnetograms are derived by performing Milne–Eddington (M-E) inversion of the spectropolarimetric profiles of the Fe i 630 nm line pair observed by the Spectro-Polarimeter (SP) instrument (Ichimoto et al. 2008) on board Hinode SOT. The data reduction and calibration of the Stokes profiles are performed by using Solarsoft routines (Lites & Ichimoto 2013), and the retrieval of the magnetic field vector under M-E approximation is carried out by the data analysis pipeline at HAO/CSAC (Community Spectropolarimetric Analysis Center of High Altitude Observatory, Boulder). The data are called Hinode level 2 data sets at HAO/CSAC. We estimate the background (quiet-Sun region in the field of view shown in Figure 1) noise in the longitudinal and horizontal component of the magnetic field in these maps to be ≈50 G and ≈120 G, respectively, at 1σ level.

The 180° azimuth ambiguity of the observed magnetic field is easily solved with the acute angle method (Metcalf et al. 2006) as the two sunspots studied here are simple round sunspots. In this method the azimuth solution making acute angle with respect to the potential field azimuth direction is taken as the correct solution. It is known that, except for the highly sheared PILs of complex active regions, the acute-angle method works very well. For a review of azimuth ambiguity resolution algorithms see (Metcalf et al. 2006). The vector magnetograms are then transformed and remapped to the heliographic coordinate system (Venkatakrishnan et al. 1988).

We model the twisted coronal and chromospheric magnetic structures in both ARs assuming the LFFF approximation. To compare the models with observations, we use SDO/AIA data taken in the 171 Å and 304 Å wavelength bands (henceforth AIA 171 and AIA 304). The magnetic field models are obtained using a fast Fourier transform (FFT) method described by Alissandrakis (1981), developed by Démoulin et al. (1997), and used in a number of studies (see, e.g., López Fuentes et al. 2006). As boundary conditions for the model we use longitudinal magnetograms from SDO/HMI corresponding to the closest time to the AIA observations. The above procedure consists of the extrapolation of the observed photospheric field into the corona following the LFFF equation:

$$\begin{equation} \nabla \times {\bf B} = \alpha {\bf B}. \end{equation} \tag{ 1 }$$

In the case of AR 11084 the size of the extrapolation box is 300 Mm in x, y, and z directions, and it is the same for both wavelengths. The grid has 129 points in x and y (horizontal) directions and 81 points in z (vertical) direction. The grid spacing is not uniform. It is 1 Mm at the center of the x–y plane and the base of the box (z = 0) and increases linearly in the vertical and x and y directions. Since the magnetograms have a uniform grid, before the computation the code interpolates the magnetogram pixel values to the above grid points. For AR 11092 it is the same except that the x–y plane size is 350 Mm × 350 Mm.

We obtain magnetic field models with different values of the force-free parameter α and integrate sets of field lines that we compare with the structures observed in AIA images. For the comparison we align the model with the data and compute the mean distance from model field lines to selected observed structures, such as EUV loops or chromospheric whirls, depending on the type of data. Each structure is defined by a series of points selected by eye on contrast-enhanced images. These selected structures are represented by colored plus signs in the top panels of Figures 2 and 3. A large number of field lines is computed in the vicinity of each structure. The distance of each field line, projected on the plane of sky, to the structure is computed. The field line minimizing this distance is selected, then new field lines are computed around it, and the procedure is continued until the minimum distance has no significant change. The last identified field line defines the minimum distance to the selected structure. This procedure is applied to an ensemble of structures (loops, fibrils), and a mean distance is defined. Repeating the above analysis for different α values defines the curves shown at the bottom of Figures 2 and 3. A detailed description of this procedure can be found in Green et al. (2002) and López Fuentes et al. (2006).

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In Figure 2 we show the results of LFFF modeling for NOAA 11084. In the top row we show the SDO/AIA intensity images in AIA 304 and AIA 171 channels. In the middle row we show a set of model field lines for the best-fit model, and in the bottom row we show the mean distance between model field lines and observed AIA structures in NOAA 11084 for different α values. Here we use an adimensional parameter α_ad that is related to α from Equation (1) by α_ad = α10⁸ m/2π, where α is in m⁻¹.

We consider the best-fit model as the one that corresponds to the minimum mean distance. As can be seen in the plot, the best fit is obtained for α_ad ≈ 0.23, corresponding to α_cor ≈ 1.44 × 10⁻⁸ m⁻¹ for AIA 171, while the best fit of the model field lines to the AIA 304 whorls is obtained for α_ad ≈ 0.4, corresponding to α_ch ≈ 2.5 × 10⁻⁸ m⁻¹.

Similarly, in the case of NOAA 11092 we show results of LFFF modeling in Figure 3. In this case we find the best-fit value of α for AIA 171 to be α_ad ≈ −0.15, corresponding to α_cor ≈ −0.94 × 10⁻⁸m⁻¹. For AIA 304 we observe that the fibrils show different amounts of twist in different sectors, and therefore we try to model different regions around the sunspot separately, as shown in the middle left panel of Figure 3. Different sets of fibrils around the sunspot (shown by different colors) are found to correspond to different values of best-fit α_ad. The average value of α_ad corresponds to α_ch ≈ −1.8 × 10⁻⁸ m⁻¹, and α_ch is in the range [ − 2.5, −0.94] × 10⁻⁸ m⁻¹. The high value of α_ch in some parts of the sunspot is in contrast to the coronal AIA 171 loops, which are described by nearby α values.

We conclude that the α values of the magnetic structures depend on height in the solar atmosphere, being larger at chromospheric heights by a factor of two than for coronal loops (Table 1).

The best-fit value of α for the photospheric vector field, α_ph, can be derived by LFFF computing for different values of α and minimizing the difference between the observed transverse component, $\boldsymbol {B}_{\rm tr, obs}$, and the modeled one $\boldsymbol {B}_{\rm tr, lfff}$ (Pevtsov et al. 1995). More precisely, by minimizing the quantity $({1}/{N}) \sum _{i=1}^{N} (\boldsymbol {B}_{\rm tr, obs}-\boldsymbol {B}_{\rm tr, lfff})^{2}$, where i is the summation index for all pixels of a given region, we derive the best-fit value α_ph. We use the longitudinal component of the Hinode vector magnetograms for the LFFF computation. Further, the pixels below the noise level threshold of 120 G for transverse field $\boldsymbol {B}_{\rm tr, obs}$ were excluded in the summation. The best-fit values of α_ph that we derive using this method are ≈2.4 × 10⁻⁸ m⁻¹ for NOAA 11084 and ≈ − 1.0 × 10⁻⁸ m⁻¹ for NOAA 11092. The sign and magnitude of the α_ph corresponds well with the best-fit LFFF model in the chromosphere, considering the fact that these are derived independently from two methods and two magnetographs with different types of measurements.

In the case of NOAA 11092, various chromospheric fibrils show different amounts of α (see left middle panel in Figure 3), and so a global best-fit α_ph ≈ −1.0 × 10⁻⁸ m⁻¹ might be an underestimate. However, the value of |α| in NOAA 11084 is consistently larger than the twist in NOAA 11092 at all three heights. This inference can also be made visually in AIA 171 images in Figure 1, where, on average, the loops in NOAA 11084 appear more twisted than loops in NOAA 11092 (taking into account that the images have the same field-of-view size).

The high-resolution photospheric vector magnetograms obtained from Hinode SOT/SP observations easily resolve the sunspot fine structures such as the penumbral fibrils (e.g., Figure 4). As a result of the fine structures, the maps of the vertical component of the electric current density J_z thus show strong local currents (Figures 5(d) and 6(d)). Such high-resolution maps of electric current density and the corresponding map of α parameter show elongated patterns, corresponding to elongated penumbral fine structure, with alternating positive and negative sign, as shown by Su et al. (2009) and Venkatakrishnan & Tiwari (2009).

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It is currently well established that the penumbral magnetic field has an azimuthal variation of the zenith angle of a few times 10° on the transverse scale of the fibrils, which can be at arcsecond and even subarcsecond scale (Title et al. 1993; Solanki & Montavon 1993; Martínez Pillet 2000). From Ampere's law, such variations of the magnetic field on short scales imply the presence of strong alternating electric currents as present in the J_z maps. Since these currents are present on small scales, it is quite likely that a large part of these +/− currents close across field lines in the photosphere and chromosphere and do not reach higher up in the corona. The penumbral fibrils have also different densities and hence different opacities (e.g., Puschmann et al. 2010); therefore, $\boldsymbol {B}$ is measured at different heights. It implies a variable measured field strength across the fibrils, so again on short scales. This introduces false strong currents (on top of a possible real one owing to a true variation of the field strength) in the J_z maps.

The strong electric currents, with alternating positive and negative sign across the penumbral filaments, mask the more distributed and weaker current pattern owing to the global twist of the sunspot magnetic field. In order to remove this "zebra-pattern," one needs to separate the contributions of currents due to spatial structure across the fibrils and the overall twist of the sunspot magnetic field. Such a decomposition can be performed assuming cylindrical symmetry of the sunspots (Venkatakrishnan & Tiwari 2009). In general, however, the fibrils are not exactly radial, and an offset remains between radial direction and fibril elongation direction. In order to avoid this problem, we perform the decomposition of electric current density j_z by taking the derivative along and perpendicular to the fibril direction as described below.

In order to perform such decomposition, we need to estimate the direction of the fibrils. We first tried to define the fibril direction as the normal to the direction of steepest descent in the high-pass-filtered $|\boldsymbol {B}_{h}|$, as well as the continuum intensity map. However, this led to a lot of noise in the deduced directions, as the computation of the gradient amplifies the fluctuations already seen in the fibril pattern. This procedure is found to be unsatisfactory for current decomposition. We then considered the transverse field direction as an approximation to the fibril orientation. This assumes that the flux tube of each fibril is weakly twisted.

We applied a high-pass filter to enhance the high-frequency component (for, e.g., fibrils) in the image, as shown in the left panel of Figure 4. The fibrils, as seen in the intensity images, are closely aligned with $\boldsymbol {B}_{h}$. Moreover, the observed horizontal magnetic field, $\boldsymbol {B}_{h}$, is well resolved and has a coherent direction between adjacent pixels (Figure 4). This is shown more closely in two zoom-in regions, selected as examples and displayed in the adjacent panels on the right. Therefore, in the following the transverse field direction is selected to perform the decomposition of the electric current density j_z.

Let us define the unit vector $\hat{t}$ along the horizontal field $\boldsymbol {B}_{h}$ and its horizontal normal $\hat{n}$ as

$$\begin{eqnarray} \hat{t}&=& \boldsymbol {B}_{h}/ B_h = \cos \psi \ \hat{x}+ \sin \psi \ \hat{y}\ \, \end{eqnarray} \tag{ 2 }$$

$$\begin{eqnarray} \hat{n}&=& \hat{z}\times \hat{t}= -\sin \psi \ \hat{x}+ \cos \psi \ \hat{y}\protect, \protect \end{eqnarray} \tag{ 3 }$$

where $\hat{x},\hat{y}$ are the orthogonal horizontal directions, $\hat{z}$ is the local vertical, $|\hat{t}|=|\hat{n}|=|\hat{x}|=|\hat{y}|=|\hat{z}|=1$, and ψ is the azimuthal angle between $\hat{x}$ and $\hat{t}$. Then, the vertical current density,

$$\begin{equation} \mu _0\ j_z = (\boldsymbol {\nabla } \times \boldsymbol {B}).\hat{z}= (\boldsymbol {\nabla }_{h} \times (B_{h} \hat{t})).\hat{z}\,, \end{equation} \tag{ 4 }$$

is decomposed as the sum of two contributions computed with spatial derivatives along and orthogonal to $\boldsymbol {B}_{h}$ as

$$\begin{equation} \mu _0\ j_z = \mu _0\ j_{z,t}+ \mu _0\ j_{z,n} \end{equation} \tag{ 5 }$$

with

$$\begin{eqnarray} \mu _0\ j_{z,t}&=& \left(\hat{t}\frac{\partial }{\partial t} \times \boldsymbol {B}_{h}\right). \hat{z}= \boldsymbol {B}_{h}. \boldsymbol {\nabla }\psi\ \ \, \end{eqnarray} \tag{ 6 }$$

$$\begin{eqnarray} \mu _0\ j_{z,n}&=& \left(\hat{n}\frac{\partial }{\partial n} \times \boldsymbol {B}_{h}\right). \hat{z}= -\hat{n}. \boldsymbol {\nabla }B_{h}. \end{eqnarray} \tag{ 7 }$$

The derivation of the right-hand sides of Equations (6) and (7) is given in the Appendix. Equation (6) implies that j_{z, t} is only due to the variation of the azimuth angle, ψ, along the horizontal magnetic field, $\boldsymbol {B}_{h}$, while Equation (7) implies that j_{z, n} is only due to the change of the horizontal field strength orthogonal to $\boldsymbol {B}_{h}$.

The above decomposition, Equation (4), splits the current density into two parts without loss of signal, in contrast to the application of a spatial filter where some signal is lost. However, it is worth remembering that both j_{z, t} and j_{z, n} contain large-scale currents that are expected to flow into the corona. Indeed, j_{z, n} is designed to keep most of the current coming from the fibril structure, but it still contains a part of nonfibril currents. In order to better visualize the distribution of these last currents, one can apply a low-pass filter to j_{z, n}. A similar filtering can be applied to j_{z, t} to further remove the remaining fibril contribution. The interest of the decomposition is to keep the highest possible spatial resolution on the part of j_z, which is less affected by the fibril pattern, i.e., j_{z, t}, rather than applying a strong low-pass filter to j_z. Such filtering is only an optional last step of the proposed method. It allows us to better visualize the weak large-scale current densities.

In Figures 5 and 6 maps of continuum intensity and electric current density are shown. The maps of j_{z, t} and low-pass-filtered j_{z, t} (henceforth $j_{z,t}^{f}$) are shown in the top panels, while the maps of j_z, j_{z, n}, and low-pass-filtered j_{z, n} (henceforth $j_{z,n}^{f}$) are shown in the bottom panels. j_{z, t} and j_{z, n} are computed with the expressions written in Cartesian coordinates (Equations (A11) and (A12)). The electric current density in all the maps is scaled between ± 10 mA m⁻² so that they are directly comparable.

The j_z map of the NOAA 11084 spot has elongated alternate ± currents, and they are dominant over the penumbral region owing to the presence of penumbral fibrils (Figure 5(d)). However, after the decomposition of j_z into j_{z, t} and j_{z, n}, the map of j_{z, t} is almost free from this elongated ± current pattern (Figure 5(b)). This is even clearer in the $j_{z,t}^{f}$ (Figure 5(c)), which has mostly negative values. On the other hand, the map of j_{z, n} (Figure 5(e)) mostly contains the elongated ± current pattern due to the penumbral fibrils. Only weak currents remain on the low-pass-filtered $j_{z,n}^{f}$ map (Figure 5(f)) as quantified in Table 2 by the comparison of $\Sigma |J_{z,n}^f|$ with Σ|J_z|. Thus, in this case the decomposition of j_z succeeds in keeping most of the fibril pattern in j_{z, n} maps even if the twist of $\boldsymbol {B}_{h}$ is weak, as the magnetic field is mostly radial within the sunspot penumbra (Figure 4).

Table 2. The Current Imbalance (CI)

Active Region No.	Jz	Jz, t	Jz, n	$J_{z,t}^f$	$J_{z,n}^f$	Σ|Jz|	Σ|Jz, t|	Σ|Jz, n|	$\Sigma |J_{z,n}^f|$
NOAA 11084	−11%	−34%	9%	−69%	38%	5.3	3.1	4.9	1.2
NOAA 11092	9%	14%	−1%	33%	−1.3%	7.4	5.0	6.9	2.2

Notes. The current imbalance, CI(%)=Σ_ij/Σ_i|j|, is given in Columns 2–6. The summation is done over all the pixels above the noise threshold, as shown in Figures 5 and 6. The total unsigned current (in units of 10¹² A) for the maps is given in Columns 7–10.

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Similarly, in Figure 6 maps of electric current density are shown for the NOAA 11092 spot. Here, again dominant alternating ± currents are present in the penumbral region in the j_z map (Figure 6(d)). After the decomposition, the ± current pattern is mostly present in j_{z, n}, but a part is still present in j_{z, t} (Figure 6(b)). These remaining ± current patterns are mostly removed in the low-pass-filtered $j_{z,t}^{f}$ (Figure 6(c)). The corresponding filtered $j_{z,n}^{f}$ (Figure 6(f)) contains more currents than the previous case (Figure 5(f) and Table 2).

While the decomposition of j_z provides comparable results for both spots, the decomposition has more difficulties in separating the fibril pattern from other currents for the second case. Indeed, the currents are stronger for NOAA 11084 than for NOAA 11092 (compare panels c and f of Figures 5 and 6). This is in agreement with the results of Section 3: both LFFF modeling and visual inspection of coronal and chromospheric structures show that twist is lower in NOAA 11092. It implies that the fibril pattern is even more dominant on the distributed current in case of NOAA 11092 and the j_z decomposition has more difficulties in separating it from other currents (Figure 6).

Further, in NOAA 11084 the dominant current is negative, while in NOAA 11092 the dominant current is positive. This is confirmed by a spatial integration of j_z. This is also true for j_{z, t}; however, for j_{z, n} the dominant current sense is opposite in the two spots, i.e., positive in NOAA 11084 and negative in NOAA 11092. More precisely, we define the current imbalance (CI) as 100 × total(j)/total(|j|). The results are summarized in Table 2. We conclude that the imbalance increases for filtered cases. Moreover, the imbalance is stronger for j_{z, t} and $j_{z,t}^{f}$, and they are coherent with the dominance of j_z.

Since both the sunspots are of negative magnetic polarity, i.e., B_z < 0, this means that the electric currents are flowing dominantly parallel and anti-parallel to the magnetic field in NOAA 11084 and NOAA 11092, respectively. This is in agreement with the chirality of the coronal and chromospheric structures seen in the AIA images for these two sunspots (Figure 1).

The distribution of j_{z, t} in Figures 5(b) and 5(c) has a global negative distribution with a concentrated positive current near the sunspot center. A similar pattern is present in the low-pass-filtered $j_{z,n}^{f}$ (Figure 5(f)). Indeed, the real j_z currents have a contribution in both j_{z, t} and j_{z, n}, so that related patterns are expected, but we cannot exclude also the effect of imperfect decomposition of j_z. The distribution of j_{z, t} in Figures 6(b) and (c) has a comparable distribution, but with a more extended and stronger positive core, while the surrounding annular negative current is less extended and weaker. A comparable pattern, while not complete and more fragmented, also seems to be present in the low-pass-filtered $j_{z,n}^{f}$ (Figure 6(f)). Such a ring-like pattern of oppositely directed electric currents at the umbra-penumbra boundary was also reported in Su et al. (2009) in a different sunspot; however, this was not investigated in detail. Finally, both the central positive current and its surrounding negative annular current pattern could be distinguished from the +/− fibril pattern in both j_z maps of the spots (Figures 5(d) and 6(d)), so that both spots seem to have a similar current pattern in j_z maps in contradiction with the opposite chirality observed in the chromospheric and coronal AIA images (Figure 1)!

Opposite currents are indeed expected to be present in flux ropes within the convective zone, as their magnetic field is confined to a small volume in a mostly field-free plasma (Parker 1996), as follows. If the flux rope has a finite total net current, then Faraday's law implies that its azimuthal field decreases inversely to the distance to the flux-rope axis. This is too slow a decrease, and indeed a twisted flux rope confined to finite volume needs a return electric current around its core that would cancel the core electric current. This return current can have a volume distribution, or it can be a current layer at the flux-rope boundary. However, it is not currently settled how far this return current can go in the solar atmosphere, as it may flow across field lines at and around the photospheric region as the magnetic field expands and fills all the available space (e.g., Longcope & Welsch 2000; Török et al. 2014). An annulus of return currents is also naturally formed when a twisting motion of finite extension is applied to an untwisted magnetic field. However, neutralization is only partial in a bipolar field if a finite magnetic shear is present along the photospheric inversion line between the two magnetic polarities (Aulanier et al. 2005).

The j_z maps of the spots (Figures 5(d) and 6(d)) seem to both indicate the presence of a central positive current surrounded by an annulus of negative current as expected theoretically with a localized twisted flux rope. Since B_z < 0 for both spots, it would imply a negative helicity, in contradiction with a positive chromospheric and coronal helicity for NOAA 11084. The decomposition of j_z allows us to solve this paradox as j_{z, t} maps show only a weak positive central current (Figures 5(b) and 5(c)) while only mixed currents remain in j_{z, n} maps (Figures 5(e) and (f)). The direct current for this spot is in fact negative, and no significant positive return current is detected around it (see Section 4.3). The second spot shows a more expected current pattern (Figure 6). The negative return current is well imaged, as it is strong and around the umbra/penumbra boundary.