Three-phase Evolution of a Coronal Hole. II. The Magnetic Field

The HMI/SDO instrument acquires velocity, magnetic field, and spectral line measurements of the solar photosphere from narrowband filtergrams of six wavelengths centered on the spectral line of Fe i (6173 Å). Sequences are obtained every 45 s or 135 s with an image size of 4096 × 4096 pixel and an angular resolution of 05 (Couvidat et al. 2016). For this study LoS magnetograms were used for the magnetic field measurements. The 720 s LoS data were used due to the lower photon noise of ∼3 G measured near the center of the solar disk and because of the higher signal-to-noise ratio at lower field strengths in comparison to vector magnetograms.

To minimize projection effects, LoS magnetic field data were analyzed for each rotation of the CH around the time of the central meridian passage (CMP) of the CH's center of mass (CoM). This covers a spatial range of ±10° longitude from the central meridian and is related to a time window of roughly ±18 hr around the time of the CMP of the CoM (assuming that the magnetic field does not change substantially over that period). For each rotation during that time window, data are taken at a 1 hr cadence and downloaded from the Joint Science Operations Center. This results in a data set of ∼25–30 images per CMP. We calculate each parameter from each image and then average over each set to reduce noise and short-term variations. In total, this gives 10 data points for studying the magnetic field evolution of the CH.

Basic data reduction was applied using the SolarSoft Suite of the Interactive Data Language (SSW-IDL). The images were prepped to level 1.5 and bad images (e.g., saturation or high noise) were removed.

Using LoS magnetic field data, we assumed a radial magnetic field, which was corrected by applying a pixelwise correction:

$$\begin{eqnarray}&&{B}_{i,{\rm{corr}}}=\displaystyle \frac{{B}_{i}}{\cos ({\alpha }_{i})},\end{eqnarray} \tag{ 1 }$$

with B_i,corr being the corrected value of each pixel of the magnetic field map, B_i the uncorrected one, and α_i the respective angular distance from the center of the solar disk.

To analyze the magnetic field underlying the CH, we apply CH masks extracted from EUV images to the co-registered photospheric magnetic field maps. The CH boundaries were extracted from EUV 193 Å images from the Atmospheric Imaging Assembly (AIA, Lemen et al. 2012) by applying an intensity-based threshold method (35% of the median intensity of the solar disk; for more details see paper I).

The projection-corrected area was calculated from the binary CH maps under the assumption of a spherical Sun:

$$\begin{eqnarray}&&{A}_{i,{\rm{corr}}}=\displaystyle \frac{{A}_{i}}{\cos ({\alpha }_{i})},\end{eqnarray} \tag{ 2 }$$

with A_i being the area per pixel and α_i the angle toward the center of the solar disk. From this we were able to calculate the size of the CH by adding up the corrected areas for all CH pixels:

$$\begin{eqnarray}&&A=\displaystyle \sum _{i}^{N}{A}_{i,{\rm{corr}}}.\end{eqnarray} \tag{ 3 }$$

The area is given in square kilometers.

In a similar way to that for the magnetic field, we average the calculated area in 1 hr cadence over a time window of ±18 hr around the time of the CMP of the CoM (see Section 2.5 in paper I). The error bars shown in Figures 3, 6–8, and 10 are the 1σ standard deviations from the mean values.

The parameters of the global CH magnetic field, as extracted from the EUV mask, include the magnetic field strength, the magnetic flux, and the magnetic field distribution with its moments (mean, variance, skewness, and kurtosis).

The mean (signed) magnetic field strength $\bar{B}$, in gauss [G] was calculated as

$$\begin{eqnarray}&&\bar{B}=\displaystyle \frac{1}{N}\displaystyle \sum _{i}^{N}{B}_{i,{\rm{corr}}},\end{eqnarray} \tag{ 4 }$$

where i represents the ith pixel of the CH and N is the total number of pixels within the CH. Correspondingly, the unsigned magnetic field strength is calculated as

$$\begin{eqnarray}&&{\bar{B}}_{{\rm{us}}}=\displaystyle \frac{1}{N}\displaystyle \sum _{i}^{N}| {B}_{i,{\rm{corr}}}| .\end{eqnarray} \tag{ 5 }$$

From the magnetic field strength and the area of the structure we can derive the magnetic flux, Φ. The magnetic flux is given in maxwell [Mx] and can be divided into signed (or open) and unsigned (or total) flux. The signed magnetic flux Φ_s is calculated as the net flux through the respective area:

$$\begin{eqnarray}&&{{\rm{\Phi }}}_{{\rm{s}}}=\displaystyle \sum _{i}^{N}({B}_{i,{\rm{corr}}}\,{A}_{i,{\rm{corr}}}),\end{eqnarray} \tag{ 6 }$$

with B_i and A_i the magnetic field strength and area for each pixel respectively. The unsigned magnetic flux Φ_us is calculated as the total flux through the respective area:

$$\begin{eqnarray}&&{{\rm{\Phi }}}_{{\rm{us}}}=\displaystyle \sum _{i}^{N}(| {B}_{i,{\rm{corr}}}| \,{A}_{i,{\rm{corr}}}).\end{eqnarray} \tag{ 7 }$$

From the signed and unsigned magnetic flux we can define the magnetic flux balance R_Φ of a CH as

$$\begin{eqnarray}&&{R}_{{\rm{\Phi }}}=\displaystyle \frac{| {{\rm{\Phi }}}_{{\rm{s}}}| }{{{\rm{\Phi }}}_{\mathrm{us}}}.\end{eqnarray} \tag{ 8 }$$

The signed flux divided by the unsigned flux of a CH can be seen as a measure of the flux that is not balanced within the CH, and this provides a measure of the percentage of open flux.

To analyze the fine structure within the CH that is composed of FTs, i.e., small-scale unipolar structures, we use a simple threshold-based method that is applied to the magnetograms with a threshold value of ±20 G. All structures with pixels above this threshold and containing at least three pixels were extracted as FTs. The value of ±20 G was chosen because structures extracted with this and with higher thresholds show unipolar properties, whereas structures that have been extracted with a lower threshold may also contain pixels with opposite polarity (see also Hofmeister et al. 2017). The value for the average unipolarity for all FTs extracted is ${0.99}_{-0.02}^{+0.01}$, which gives support for the choice of the threshold of 20 G. The unipolarity (or magnetic flux balance) is calculated using Equation (8) as described in Section 2.3; for values around 1, it indicates that all pixels of the structure (i.e., the FT) are of the same polarity.

The total number of extracted FTs is called the FT ensemble. This ensemble is then sorted into three categories based on the absolute value of the mean magnetic field strength of each FT structure $| {\bar{B}}_{\mathrm{FT}}|$, which is calculated as

$$\begin{eqnarray}&&| {\bar{B}}_{{\rm{FT}}}| =\displaystyle \frac{1}{N}\displaystyle \sum _{i}^{N}| {B}_{i,{\rm{corr}}}| ,\end{eqnarray} \tag{ 9 }$$

where i represents the ith pixel of the FT and N is the total number of pixels that make up the FT.

FTs in the first category have the lowest magnetic field strengths and are called weak FTs. If the absolute value $| {\bar{B}}_{\mathrm{FT}}|$ of an FT is between 20 and 35 G it is placed into the pool of weak FTs. The next category are the medium FTs, which have an absolute value of the mean magnetic field strength between 35 and 50 G. All FTs that have a mean magnetic field strength exceeding ±50 G are called strong FTs. Figure 1 shows three snapshots of the CH magnetic field (one of each phase) with the FTs highlighted. The strong FTs are shown in cyan, the medium FTs in orange, and the weak FTs in magenta. The FTs seem to be aligned along the magnetic network (Gabriel 1976; Dowdy et al. 1986), and a change in the relative abundance of the FTs during the CH evolution can be seen.

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The area of each FT is calculated using

$$\begin{eqnarray}&&{A}_{{\rm{FT}}}=\displaystyle \sum _{i}^{N}{A}_{i,{\rm{corr}}},\end{eqnarray} \tag{ 10 }$$

with A_i,corr being the area of each FT pixel corrected for a spherical Sun (Equation (2)) and N the number of pixels of the FT. The flux for each FT was calculated using Equations (6) and (7). By summing over all FTs of one category in one image, we can calculate the FT property of the total CH, e.g., Φ_FT,s is the signed flux coming from all (weak, medium, or strong) FTs within the CH.

Table 1.  Overview of Parameters Defined in Section 2

Parameter	Definitiona,b	Description
A	$={\sum }_{i}{A}_{i}$	Area
$\bar{B}$	$=\tfrac{1}{N}{\sum }_{i}{B}_{i}$	Signed mean magnetic field strength
${\bar{B}}_{\mathrm{us}}$	$=\tfrac{1}{N}{\sum }_{i}| {B}_{i}|$	Unsigned mean magnetic field strength
Φs	$={\sum }_{i}({B}_{i}\,{A}_{i})$	Signed magnetic flux
Φus	$={\sum }_{i}(| {B}_{i}| \,{A}_{i})$	Unsigned magnetic flux.
RΦ	$=| \tfrac{{{\rm{\Phi }}}_{{\rm{s}}}}{{{\rm{\Phi }}}_{\mathrm{us}}}|$	Flux balance, ratio of signed to unsigned magnetic flux
rΦ	$=\tfrac{| {{\rm{\Phi }}}_{\mathrm{FT}}| }{{{\rm{\Phi }}}_{\mathrm{CH}}}$	Flux ratio, ratio of flux from FTs to the CH flux
rA	$=| \tfrac{{A}_{\mathrm{FT}}}{{A}_{\mathrm{CH}}}|$	Area ratio, ratio of area of FTs to the CH area

Notes.

^aNote that A_i and B_i represent the corresponding corrected versions A_i,corr and B_i,corr. ^bProperties of coronal hole and flux tube are denoted with the subscripts CH and FT respectively.

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Lastly we can define the FT proportions—the area proportion and the flux proportion. These ratios show how much a certain category of FTs contributes to the respective parameter of the total CH in terms of flux or area:

$$\begin{eqnarray}{r}_{{\rm{\Phi }}} & = & | \displaystyle \frac{{{\rm{\Phi }}}_{\mathrm{FT}}}{{{\rm{\Phi }}}_{\mathrm{CH}}}| \\ {r}_{{\rm{A}}} & = & \displaystyle \frac{{A}_{\mathrm{FT}}}{{A}_{\mathrm{CH}}}.\end{eqnarray} \tag{ 11 }$$

The correlation between the various extracted parameters is calculated using the Pearson correlation coefficient and the Spearman correlation coefficient. To consider the significance of the relation for a low number of data points, we apply a bootstrapping algorithm (Efron & Tibshirani 1993; Efron 1979) to the data set with over 10⁶ repetitions (replacement for each repetition is taken from the initial set, with each data point in this subset coming from a Gaussian distribution of itself plus its standard deviation). The correlation coefficients are calculated from the derived subsets and given as mean values of all repetitions. The confidence intervals (CIs) for the correlation coefficients (90%, 95%, 99%) were calculated using the respective quantiles. An overview of the parameters defined in this section is given in Table 1. A summary of all results and statistical parameters is given in Table 2 in the Appendix.

In the following, we take up the result of paper I where we show the three-phase evolution of the CH as derived from the evolution of its area (and other parameters such as the intensity and the in situ peak velocity of the associated HSS of the solar wind). Figure 3(a) shows the evolution of the CH area as observed from Earth together with the evolution of the mean magnetic field strength of the CH over its lifetime. From the CH area (dashed black line) we clearly obtain the three-phase evolutionary pattern. In the growing phase, lasting from 2012 February 4 until 2012 May 13, we see a first peak in the deprojected area at ∼6 · 10¹⁰ km² followed by a fast decline to ∼2 · 10¹⁰ km² and a growth until a maximum is reached. The maximum phase settles at an area of ∼9 · 10¹⁰ km² around 2012 June 3 and lasts about one month. In the decaying phase the area drops first sharply then moderately to ∼1 · 10¹⁰ km² until the CH can no longer be observed in 2012 October. The error bars represent the 1σ deviation from the averaging as described in Section 2. The three phases are marked by the color bar at the bottom of Figure 3(a).

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For comparison, we also plot in Figure 3(a) the evolution of the signed magnetic field strength (blue line), representing a measure of the open field within the CH, and the unsigned field strength (red line), measuring the absolute field strength. From this we see a synchronized evolutionary behavior that seems to be linked to the evolution of the CH area; nevertheless some differences are obtained. The mean field strength varies between −1 and −5 G, hence the predominant negative polarity does not change over the CH lifetime. The profile shows one early peak (−2.7 G) around 2012 April 9, which does not match the CH evolution, and a main peak (−4.4 G) around 2012 June 3 that coincides with the peak in the area. During the maximum phase the signed mean magnetic field strength declines to −3.8 G before dropping significantly at the start of the decaying phase. In the decaying phase the value drops below −1 G. The unsigned mean magnetic field strength shows a similar behavior to the signed field strength. It reveals an early peak (6.7 G) during the growing phase of the CH area, as well a clear maximum (8.6 G) and a decrease to 5.6 G that both match the CH evolution.

Figure 3(b) explores the relation of the signed field strength to the area. We find a linear correlation with a Pearson correlation coefficient of c = −0.82 with a 95% CI of [−0.36, −0.97]. The linear regression fit for this CH can be expressed as

$$\begin{eqnarray}&&A=(-1.21\pm 0.33)+(-2.46\pm 0.16)\bar{B}.\end{eqnarray} \tag{ 12 }$$

The area A is given in 10¹⁰ km² and the mean magnetic field strength $\bar{B}$ is given in gauss. A similar significant relation can be found when comparing the unsigned mean magnetic field strength to the CH area (Figure 3(c)), where we find a linear correlation with a Pearson correlation coefficient of c = 0.83 in a 95% CI of [0.38, 0.97]. The linear regression fit can be described as

$$\begin{eqnarray}&&A=(-14.22\pm 1.25)+(2.88\pm 0.19)\bar{{B}_{{\rm{us}}}}.\end{eqnarray} \tag{ 13 }$$

Figure 4 shows the normalized magnetic field pixel distribution for five time steps, each corresponding to one data point in Figure 3, representing different stages in the evolution of the CH. The first two lines (green, dark green) mark the growing phase around 2012 March 13 and May 6. The red line is the magnetic field distribution during the maximum around 2012 June 3. The blue lines (dark blue, blue) represent the decaying phase around 2012 July 26 and September 18. The distribution follows a Lorentzian-like profile that is shifted to the dominant polarity of the CH (between −0.25 and −0.5 G). By comparing the distributions of the different evolutionary stages we find a broadening from the growing to the maximum phase. We find an increase in the density of pixels with higher field strengths (flanks of the distribution) in the maximum phase compared to the growing and decaying phases.

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Figure 5 shows the second, third, and fourth moments of the magnetic field distribution, corresponding to the standard deviation (square root of the variance), the skewness (a measure of the lopsidedness of a distribution), and the kurtosis (a measure of the heaviness of the tail of the distribution). Interestingly, we find a similar behavior in all the profiles: a clear peak in the maximum phase followed by a steep drop to low values in the decaying phase. This suggests a significant change in the magnetic field distribution in the maximum phase, which coincides with the maximum in the area. The asterisks represent the last data point (around 2012 October 14), which we excluded as an outlier in calculating the moments of the distribution, because of large uncertainties in the extraction and deprojection of the magnetic field with a small area. The outlier value arises from a few erroneously detected CH pixels, which cover strong fields. Due to the small CH area at this time, these erroneous pixels greatly alter the calculated moments of the distribution.

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Figure 6 shows the CH's signed flux Φ_s, unsigned flux Φ_us, and the flux balance (Φ_s/Φ_us). The unsigned flux has a negative polarity, but for visualization purposes in Figure 6 the absolute value is plotted. The fluxes show the same trend as the area and the mean magnetic field strength, with growing, maximum, and decaying phases. The signed flux (red) peaks at a maximum of 4 · 10²¹ Mx and reaches down to 1 · 10²¹ Mx during the CH "formation" and decay phase. The unsigned flux (blue) peaks at the same time as the signed flux with 8 · 10²¹ Mx and ranges down to 1.5 · 10²¹ Mx. The flux balance also peaks during the maximum phase with a flux balance of 0.5, and it ranges down to 0.2 during the early and late phases. The variation within the CH area correlates with the variation of magnetic flux and clearly shows the three-phase evolution.

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Figure 7(a) shows the evolution of the average number of FTs in the CH separately for the three different FT categories, namely weak (magenta), medium (orange), and strong FTs (cyan). The number of strong FTs ranges from 7 to over 160, the number of medium FTs from 12 to 230, and the number of weak FTs from 40 to 870. Although weak FTs appear most frequently during the whole evolution, the strong FTs have the highest area proportion, i.e., the area of all FTs of one category in comparison to the total area of the CH (Figure 7(b)). During the maximum phase, strong FTs cover up to 3.9% of the total CH area. During growing and decaying phases this number is up to a factor of 3 lower. Variations in the area proportion of the other FT categories (weak, medium) are barely given, although their number does vary slightly following the three-phase evolution. Their area proportion averages around (1.0 ± 0.3)%. Figure 1 illustrates the evolution of the FT number and area proportion in three snapshots (one of each phase). Note that these findings also imply that strong FTs make the highest contribution to the total CH flux, because their field strength and their area proportion are the largest. This indicates that strong FTs play a major role in the evolution of a CH.

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The number of FTs and the total area they cover show the three-phase evolution, therefore a relation would seem reasonable. Figure 8 shows the correlation between FT number and CH area using different colors for the different FT categories (weak FTs: magenta; medium FTs: orange; strong FTs: cyan). For weak FTs, we find a Pearson correlation coefficient of c = 0.89 with a 95% CI of [0.62, 0.99]. Although this reveals a high correlation, we note that the linear fit does not cover three data points (including error bars), making the correlation less significant. The medium FTs, however, have a more significant correlation. The Pearson correlation coefficient is c = 0.96 with a 95% CI of [0.85, 0.99], and the linear fit can be described by

$$\begin{eqnarray}&&N=(-2.5\pm 8.1)+(26.8\pm 1.6)A\,[{10}^{10}\,{{\rm{km}}}^{2}].\end{eqnarray} \tag{ 14 }$$

For the strong FTs we see the strongest correlation with a Pearson correlation coefficient of c = 0.96 and a 95% CI of [0.89, 0.99], and the linear fit can be expressed as

$$\begin{eqnarray}&&N=(-7.1\pm 4.7)+(17.4\pm 0.9)A\,[{10}^{10}\,{{\rm{km}}}^{2}].\end{eqnarray} \tag{ 15 }$$

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Figure 9 shows the magnetic field distribution of FTs per area of different stages in the CH evolution. The color indices match Figure 4: the first two lines (green, dark green) represent the growing phase around 2012 March 13 and May 6. The red line is the magnetic field distribution during the maximum around 2012 June 3. The blue lines (dark blue, blue) represent the decaying phase around 2012 July 26 and September 18. From this we find clear differences between the distributions of the three phases. Besides the obvious and expected asymmetry of the distributions, due to one dominant polarity, the peak of the distribution in the growing phase is slightly lower than in the following phases. Toward the maximum phase the dominant (negative) polarity of the distribution rises, especially the flanks (i.e., at high field strengths), whereas the flank for FTs of the non-dominant polarity declines. We find a significant rise in the FTs below −70 G, which shows that strong FTs cause the main difference between the distributions. The decaying phase is marked by a decline of the dominant polarity flank and an increase of the non-dominant polarity flank. The number of weaker FTs does not change significantly (see also Figure 7).

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Figure 10 shows how much flux the FTs supply to the total signed flux of the CH, which may act as a measure of the open flux of the CH. We assume that flux from FTs of opposite polarity cancels within the CH. We find that a large fraction of the signed flux of the CH comes from FTs (70%–80%), which occupy less than 7% of the CH's area. The major part of the flux comes from strong FTs (48%–71%). In the evolution of FT flux proportion, the three-phase evolution is emphasized. We see the three distinctive phases as clearly as in the area, especially in the strong FTs. For strong FTs we have a constant flux proportion of 60% in the growing phase, which rises to ∼70% in the maximum phase and drops to around 54% in the decaying phase. It is also interesting to note that during the maximum phase the rise in contribution of the strong FTs is accompanied by a drop in the contribution of the medium FTs. Other than that, the contribution of the medium FTs (6%–14%) and the weak FTs (2.5%–4.5%) plays only a minor role in comparison to the strong FTs.

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