Existence of The Closed Magnetic Field Lines Crossing The Coronal Hole Boundaries

The LOS magnetic fields are converted into radial magnetic fields ($B_{r}$) assuming that the surface fields are radial (Svalgaard et al., 1978; Petrie & Patrikeeva, 2009; Gosain & Pevtsov, 2013). The synoptic maps of the radial fields and AIA 193 Å emissions are constructed using the stripes of $\pm 15^{\circ}$ around the central meridian. The size of the resulting maps is 3600 pixels in longitude ($\phi$) and 1440 pixels in sine-latitude ($\sin\lambda$).

The coronal holes are identified as the dark regions with predominantly unipolar magnetic field. The identification procedure consists of three steps: (1) extraction of dark regions from AIA 193Å synoptic maps; (2) evaluation of the stability of the dark-region boundaries, following the stability test procedure developed by Heinemann et al. (2019); and (3) evaluation of the level of unipolarity of the dark regions.

In the first step, an automated dark-region detecting procedure (Krista & Gallagher, 2009; Huang et al., 2019) is applied to AIA synoptic maps to determine an optimal intensity threshold of the Extreme UltraViolet (EUV) emission, $t_{\rm EUV}$, for each synoptic map. The determined $t_{\rm EUV}$ of each synoptic map over the time period examined in this study is shown in Figure 1(a). The regions that are darker than $t_{\rm EUV}$ and larger than 400 pixels ($\approx 470$ Mm²) are extracted as dark regions.

In the second step, the stability of a dark region is evaluated by the variability of its area (area uncertainty) due to the variation of the intensity threshold (Heinemann et al., 2019). The smaller the area uncertainty, the higher the stability of the dark region. In brief, the stability evaluation is conducted as follows: The intensity threshold is varied from $t_{\rm EUV}-2$ digital number (DN) to $t_{\rm EUV}+2$ (DN), producing five area values ($A_{i}$, $i=1$ to $5$) and an average area, $A$, for each dark region. The area uncertainty $\epsilon_{A}$ of the dark region is subsequently computed as follows:

To determine the stability of the dark-region boundaries, we use a power function $f(A)=\alpha A^{\beta}+\gamma$, where $\alpha$, $\beta$ and $\gamma$ are the fitting parameters, to fit the scatter plot of $\epsilon_{A}$ vs. $A$ (shown in Fig. 1b). The best-fit curve $f_{\rm fit}$ is used as the bench mark to define high, medium and low stability regions: $\epsilon_{A}<f_{\rm fit}$ is high, $f_{\rm fit}<\epsilon_{A}<2f_{\rm fit}$ is medium, and $\epsilon_{A}>2f_{\rm fit}$ is low stability. They correspond to the green, yellow and red regions in Fig. 1b.

The third step is to determine the dark regions that are predominantly unipolar. A region with high level of unipolarity can be observed as a region with the radial-field distribution highly skewed toward one polarity, or with a large average radial field $|\langle B_{r}\rangle|$, or both. Since the radial fields are measured at the photosphere, the dark-region boundaries are projected down to the photosphere by assuming that the 193 Å images represent the corona at a height of 0.02$R_{\odot}$ above the solar surface (Caplan et al., 2016; Huang et al., 2019). The histogram of the skewness of the dark regions is plotted in the upper left panel of Fig. 2. The x axis is $|{\rm Skew}|$, and y axis is the number of the dark regions. The vertical line marks the location of $|{\rm Skew}|=0.5$, which is approximately the turning point of the histogram. It is thus selected as the threshold for the skewness (Huang et al., 2019). To determine the threshold for $|\langle B_{r}\rangle|$, the percentage of the dark regions qualified as CHs is plotted as a function of $|\langle B_{r}\rangle|$ threshold in the lower left panel of Fig. 2. In addition to the result of setting the skewness threshold $|{\rm Skew}|=0.5$ (black line), the profiles of setting $|{\rm Skew}|=0.1$ (blue) and 1.0 (red) are also plotted for comparison. The plot shows that the percentage of the dark regions qualified as CHs approaches a constant of $\approx 70\%$ after 3 Gauss. Therefore, 3 Gauss is chosen as the threshold, and the dark regions with $|\langle B_{r}\rangle|\geq 3$ Gauss or magnetic-field skewness $|{\rm Skew}|\geq 0.5$ are considered as predominantly unipolar, and identified as coronal holes.

Since the boundary of a dark region can be changed by the variation of the intensity threshold, $|\langle B_{r}\rangle|$ and $|{\rm Skew}|$ within the boundary can also be changed. We consider the dark regions that are qualified as coronal holes under all five intensity thresholds as reliable coronal holes. The regions that only satisfy the unipolarity criteria for some intensity thresholds are classified as uncertain coronal holes. The coronal holes identified from the synoptic maps of AIA 193Å and HMI radial fields are referred to as CH₁₉₃ hereinafter.

The global coronal magnetic field is often constructed by the Potential Field Source Surface (PFSS) model (Schatten et al., 1969), which is based on the assumptions that the current is zero in the region of interest and that the tangential magnetic field vanishes at the upper boundary, called “source surface”, leading to strictly radial field lines afterward. Earlier studies (e.g., Schatten et al., 1969) have shown that the current-free assumption is broadly valid in most part of the corona except for the regions with rapidly varying magnetic fields, such as sunspots and streamers.

In this study, the PFSS magnetic fields are computed using the Global Potential and Linear Force-Free Fields code (GLFFF; Jiang & Feng, 2012), which solves the Laplace equation by the relaxation method. The upper boundary, i.e., the source surface, is placed at $2.5R_{\odot}$ from the Sun center (e.g., Wang & Sheeley, 1990; Obridko & Shelting, 1999). The upper boundary condition is that all field lines at the source surface are assumed to be radial. The lower boundary is at the solar surface, and the HMI magnetic field synoptic maps are used as the lower boundary condition. To comply with our computing capacity, the HMI synoptic maps (3600$\times$1400 pixels) are down sampled by Sifting Convolution on the Sphere (Roddy & McEwen, 2021). In brief, we first apply a low-pass Gaussian filter on the spherical harmonic coefficients of the maps and then inverse transform them back to a size of 360 pixels in longitude ($\phi$) and 180 pixels in latitude ($\lambda$). The detail of the down sampling procedure is described in Appendix.

The data of MHD modelled solar corona for the time period studied in this work are from Predictive Science, who computed the magnetic and thermal structures of the corona using their Magnetohydrodynamic Algorithm outside a Sphere (MAS) code, which solves the full set of thermodynamic MHD equations. The plasmas are assumed to be collisional below $10R_{\odot}$ and collisionless beyond $10R_{\odot}$. The energy equation in the code includes the contributions from coronal heating, dissipation and radiative loss.

Using the electron density $n_{e}$ and temperature $T_{e}$ in the MAS data, we compute the synthetic EUV emission maps at 193 Å wavelength by the following equation:

where $n_{e}(z),T_{e}(z)$ are the electron number density and the electron temperature along the line of sight $z$, and $f_{193}(T_{e})$ is the AIA-193 temperature response function. The coronal holes for the MHD model are identified following the same procedure described in Section 3.2: (1) the dark regions are extracted from the synthetic EUV synoptic maps by intensity thresholding, (2) their stability is evaluated by the stability test, and (3) a set of unipolarity criteria is applied to identify the coronal holes. The unipolarity criteria are determined by the same procedure described in Sec. 3.2: The histogram of $|{\rm Skew}|$ is plotted in the upper right panel of Fig. 2. The vertical line in the upper right panel marks the location $|{\rm Skew}|=0.2$, which is where the slope of the histogram shows the sharpest change. The percentage of the qualified CHs as a function of $|\langle B_{r}\rangle|$ is shown in the lower right panel of Fig. 2. The result of setting the skewness threshold to 0.2 is plotted in black, and compared with the results of setting the threshod to 0.1 (blue) and 0.5 (red). The plot shows that the percentage of the qualified CHs approaches a constant of $70\%$ after $|\langle B_{r}\rangle|=2$ G. Therefore, $|\langle B_{r}\rangle|=2$ G is selected as the threshold for $|\langle B_{r}\rangle|$. In short, based on these plots, we determine the unipolarity criteria for the synthetic maps to be $|{\rm Skew}|\geq 0.2$ or $|\langle B_{r}\rangle|\geq 2.0$ Gauss. The CHs identified from the synthetic synoptic maps will be referred to as synthetic MHD CHs or CH_MHD, to be distinguished from CH₁₉₃, the coronal holes identified from the AIA and HMI synoptic maps.

The magnetic field lines are computed by tracing the field lines from the surface upward. The “open” magnetic field lines in a PFSS magnetic field are defined as the field lines reaching the source surface, which is $2.5R_{\odot}$ from the Sun center. To define the OMF lines in the MHD magnetic fields, we computed the distance at which the thermal pressure of the MHD model becomes larger than the tangential magnetic pressure. The result indicates that except for CR2157, “open” MHD field lines can also be defined as the field lines reaching $2.5R_{\odot}$ from the Sun center.

After the field line tracing, the field lines that are rooted in the reliable coronal holes are considered as reliable field lines to be used for the analysis of CH magnetic structures. Other field lines are categorized as uncertain field lines and discarded.

The temporal variations of the optimal intensity thresholds $t_{\rm EUV}$ in AIA-193 synoptic maps and in MHD synthetic EUV maps are plotted as black lines in Fig. 1a and b, respectively. The overplotted red lines are the SSN, and follow the y-axis on the right side. The panels show that the $t_{\rm EUV}$ variations of both AIA and synthetic maps are large before 2016, when the solar activity is higher, and become small with $t_{\rm EUV}$ approaching a constant after 2016, when the Sun becomes quieter.

The scatter plots of $\epsilon_{A}$ vs. $A$ are shown in Fig. 1b and d for the dark regions identified from the AIA-193 synoptic maps and synthetic EUV maps, respectively. The best-fit curves to the scatter plots are plotted as green dashed curves, and the functional forms printed in the titles of corresponding panels. The points corresponding to high, medium and low stabilities are located in green, yellow and red sections, respectively. Among the 3210 dark regions extracted from the AIA-193 synoptic maps, 1894 ($\approx 59.0\%$) regions are categorized as high stability, 1133 ($\approx 35.2\%$) as medium stability, and 183 ($\approx 5.7\%$) as low stability. From the synthetic EUV synoptic maps, 1506 dark regions are identified, of which 924 ($\approx 61.4\%$) are categorized as high stability, 308 ($\approx 20.4\%$) as medium stability, and 274 ($\approx 18.2\%$) as low stability.

The y-axis of the scatter plots show that the area uncertainties for AIA-193 dark regions are all less than 25% of their average areas while $\epsilon_{A}$ for the dark regions extracted from the synthetic EUV maps can be higher than 200%, indicating that the dark regions extracted from the AIA maps are more stable than those from the MHD synthetic maps. The large $\epsilon_{A}$ for the synthetic dark regions is because the synthetic EUV maps are smoother, i.e., the intensity gradient is smaller. As a result, a same change in $t_{\rm EUV}$ from -2 to 2 DN results in larger area change.

Large area uncertainty can lower the reliability of the identified coronal holes. This is because the distribution of the magnetic field within the boundary is more likely to change from predominantly unipolar to non-unipolar, or vice versa, if the enclosed area is changed by a large percentage. Fig. 3 and Fig. 4 show such examples found in the AIA-193 synoptic map of CR2099 and synthetic EUV map of CR2100, respectively. The background images in Fig. 3a and Fig. 4a are the AIA-193 map and synthetic EUV map, respectively. The contours of different colors mark the boundaries of the dark regions determined by different intensity thresholds, as indicated next to the two panel (a). Both panels contain both high-stability dark region(s), of which the variation of intensity threshold leads to a relatively small area variation, and low-stability regions, of which the same threshold variation results in significant boundary changes.

By comparing the background images of Fig. 4a and Fig. 3a, we can see that the synthetic EUV map (Fig. 4a) is smoother than the AIA-193 map (Fig. 3a), and therefore the area uncertainty of the former is larger than that of the latter, as pointed out in the discussion of $\epsilon_{A}$ vs. $A$ scatter plots.

Fig. 3b and Fig. 4b show the coronal holes identified from the dark regions in Fig. 3a and Fig. 4a, respectively. The dark regions that do not qualify as coronal holes under any intensity threshold are not shown. The white and gray areas correspond to the reliable and uncertain coronal holes, and blue and red lines the reliable and uncertain field lines, respectively. The gray region indicated by the white arrow in each figure is an uncertain coronal hole whose average magnetic field and skewness are plotted as a function of intensity threshold in Panel (c). The dashed horizontal line in each Panel (c) marks the threshold of unipolarity ($|\langle B_{r}\rangle|\geq 3$ G or $|{\rm Skew}|\geq 0.5$ for Fig. 3c, and $|\langle B_{r}\rangle|\geq 2$ G or $|{\rm Skew}|\geq 0.2$ for Fig. 4c). The $|\langle B_{r}\rangle|$ and $|{\rm Skew}|$ of the indicated gray regions are plotted as black and red lines, respectively. The x axis is the intensity threshold $t_{\rm EUV}$. $|\langle B_{r}\rangle|$ follows the y axis on the left, and $|{\rm Skew}|$ the y axis on the right.

Fig. 3c indicates that both $|{\rm Skew}|$ and $|\langle B_{r}\rangle|$ become lower than the unipolarity thresholds when $t_{\rm EUV}$ is greater than approximately $67.5$ DN, disqualifying the region as a CH. Fig. 4c indicates that the indicated region in Fig. 4b is disqualified as a CH when $t_{\rm EUV}$ is approximately 18 (17.5 DN $<t_{\rm EUV}<18.4$ DN), when both $|{\rm Skew}|$ and $|\langle B_{r}\rangle|$ are lower than the unipolarity thresholds. Since they are not qualified as coronal holes for all five intensity thresholds, these two regions are categorized as uncertain coronal holes. In contrast, the white regions in panel (b) of Fig. 3 and Fig. 4 satisfy the unipolarity criteria for all five intensity thresholds.

The field lines that are rooted in the reliable CHs are considered as reliable field lines. In total, 81.8% of the field lines in CH₁₉₃ and 61.8% of the field lines in CH_MHD are identified as reliable field lines. All CHs and field lines discussed in the rest of the paper are the reliable CHs and reliable field lines, and “reliable” will be dropped for brevity.

The examination of the field lines reveals that in addition to the well-known open magnetic field lines and closed field lines with both foot points located in the same CH, there are closed field lines that cross the boundary of the CH and end in either a bright region or a different CH. For conciseness, we use “1” and “0” to indicate the field-line foot points located inside and outside a CH, respectively, and abbreviate OPen and CLose as OP and CL. Therefore, OPen field lines will be referred to as OP1, and CLosed field lines with one foot point ending in a non-CH region will be referred to as CL10. To distinguish the CLosed field lines with both foot points rooted in a same CH and those connecting two different CHs, the former is referred to as normal CLosed field lines CL11n, and the latter as abnormal CLosed field lines CL11a.

The results of CR2105 are shown in Fig. 5 as an example. The field lines are computed from the PFSS magnetic fields. The background in all panels is the AIA 193 Å synoptic map projected to the solar surface, assuming that the height of the AIA 193 images is at $0.02R_{\odot}$. The white contours mark the identified CH₁₉₃, and different types of field lines are plotted in different colors in separate panels, as indicated in the panel titles. Although OP1 (panel a) and CL10 (panel b) appear to be the dominant types for this Carrington Rotation, they occupy less than 21% and 28% of the total number of field lines in this Carrington Rotation, respectively. The most numerous type is actually CL11n (panel c). CL11n, while being shorter than the other three field line types, can be found in all CHs, and accounts for nearly 52% of the field lines in CR2105.

Comparing panel (a) and other panels, it can be seen that four CHs (indicated by white arrows) in this CR contain no OP1 and only closed field lines. CHs with such magnetic structures do not intersect with OMF regions, and would not be the source regions for high-speed solar wind streams. A close examination of panel (b) reveals that some brighter ends of CL10 are found in the neighborhoods of active regions and just outside the edges of another CH. CL11a, shown in panel (d), is usually the rarest among the four types. In this Carrington Rotation, they account for only 0.5% of the total number of the field lines. In this specific CR, they are seen to connect two CHs (marked in thicker contours) that are located in different hemispheres. Comparison with panel (b) shows that these two CHs are also connected by several CL10.

The field lines computed from the MHD model of the same Carrington Rotation, CR2105, are plotted in Figure 6 in the same format as Figure 5. The background in all panels is the synthetic EUV map and the white contours are the identified CH_MHD. The synthetic map and CH_MHD are qualitatively similar to, albeit spatially less resolved than, the observed AIA map and CH₁₉₃ in Figure 5.

Figure 6 shows that the coronal holes from the MHD model contain all four types of the field lines: OP1 (panel a) is the most numerous field line type, accounting for $\approx 73\%$ of the total field lines in this CR, and CL11a (panel d) is the rarest type ($\approx 0.2\%$ of the total number). Despite being the most numerous, OP1 does not exist in all CH_MHD. In this Carrington Rotation, there are two CH_MHD without any OP1, as indicated by the arrows. CL11n, which was the most numerous type in Figure 5, constitutes only $\approx 11\%$ of the total field lines in the CH_MHD of CR2105, and does not exist in all CH_MHD. Similar to the CL10 in the PFSS magnetic fields, the brighter ends of CL10 (panel b) in the MHD model are also found in the neighborhoods of active regions and near the boundary of another coronal hole.

As revealed in Sec. 4.2, there are coronal holes without open magnetic field lines. They constitute more than 50% of all CH₁₉₃, and more than 17% of all CH_MHD.

The CHs without OP1 are on average smaller than those with OP1. For CH₁₉₃, the average area of the former is $\approx 0.46\times 10^{4}$ Mm² and is $\approx 3.42\times 10^{4}$ Mm² for the latter. For CH_MHD, the values are $\approx 0.85\times 10^{4}$ Mm² (without OP1) and $\approx 12.79\times 10^{4}$ Mm² (with OP1).

To examine their magnetic fields, the histograms of $|\langle B_{r}\rangle|$ and $|{\rm Skew}|$ of the coronal holes with and without OP1 are plotted in Fig. 7, and the averages are printed in the corresponding panels. The upper four panels are the histograms of $|\langle B_{r}\rangle|$ and the lower four the histograms of $|{\rm Skew}|$. The histograms for CH₁₉₃ and CH_MHD are placed in the left and right panels, respectively.

Compared with the $|\langle B_{r}\rangle|$ histograms of the coronal holes with open magnetic field lines (panel a and e), the histograms of the CHs without OP1 (panel b and f) are more concentrated near lower $|\langle B_{r}\rangle|$ with fewer cases at the high $|\langle B_{r}\rangle|$ tail. This is especially prominent in panel f, with more than 50% of CH_MHD concentrated at small $|\langle B_{r}\rangle|$ ($<1$ G). The high average $|\langle B_{r}\rangle|$ in panel f is caused by a few outliers with very large $|\langle B_{r}\rangle|$ ($>30$ G). For the histograms of $|{\rm Skew}|$, the peaks of the distributions of the CHs with and without OP1 are at similar locations. However, the CHs without OP1 (panel d and h) have fewer cases than the CHs with OP1 (panel c and g) at the tails of high $|{\rm Skew}|$.

In short, these histograms indicate that, for both CH₁₉₃ and CH_MHD, the coronal holes without open magnetic field lines are less likely to have large $|\langle B_{r}\rangle|$ and large skewness. In other words, CHs without OP1 are less unipolar than the CHs with OP1.