Characteristics of Low-latitude Coronal Holes near the Maximum of Solar Cycle 24

Two species of structures appear dark in EUV images: coronal holes and filaments. Coronal holes are the lowest density structures in the Sun's corona at comparably low coronal temperatures of about 1 MK. Therefore, they appear as dark structures in EUV images, but are not visible in Hα images. In contrast, filaments have a high density, a chromospheric temperature of about 10⁴ K, extending from the Sun's chromosphere to the lower corona. Filaments appear dark in both EUV and Hα images.

To create our data set, we first extracted all dark structures in EUV images for the three-year period under study from 2011 January 01 to 2013 December 31 with a cadence of one image per day. Then we visually classify and exclude filaments within the dark structures by comparison with Hα images. The data set is restricted by the availability of EUV images, Hα images and magnetograms, and covers 98% of the time range under study.

We extracted the dark structures in the EUV images by an intensity-based thresholding algorithm based on Rotter et al. (2012) and Reiss et al. (2016). The threshold was set to 0.35 times the median of the AIA-193 intensity of the solar disk. The resulting full-disk binary maps of dark structures were post-processed by morphological operators (erosion and dilation), whereby a median function with a kernel of 9 pixels was applied. This morphological operation adds small bright islands within coronal holes to the area of coronal holes and removes very small dark structures, which are most probably neither coronal holes nor filaments, from the full-disk binary map (Rotter et al. 2012). In order to obtain the single structure binary maps, the full-disk binary maps were segmented. Two structures were recognized as individual structures as soon as they are separated by at least one pixel. Finally, the EUV images and magnetograms of the single structures were obtained by cutting the full-disk EUV images and magnetograms according to the single structure binary maps identified in the EUV.

For further analysis, we restricted the single structure data set to structures with an area of more than 1 × 10¹⁰ km² corresponding to a minimum diameter of 100 Mm, and with an angular distance of the structure's center of mass to the intersection of the solar equator and central meridian of less than 30° in order to reduce the effects of projection in the further analysis. Thus, we arrive at a resulting data set of 343 structures. Figure 1 shows an example of the segmentation procedure and resulting coronal holes.

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The remaining single structures identified in EUV were then divided into coronal holes and filaments by visual comparison with the Hα images. Since filament channels are sometimes visible in EUV before the corresponding channel is visible in Hα, i.e., before the channel is filled with dense cool plasma, we inspected the Hα images within a time range of ±2 days. If the full structure was visible in the Hα images within a time range of ±2 days, the structure was labelled as a filament. If only part of the structure was visible in Hα, the structure was excluded from further analysis. If the structure was not visible in Hα, but had an elongated shape and a balanced magnetic flux, like we expect from filaments, the structure was also excluded from further analysis. All structures remaining were labelled as coronal holes.

In total, our data set includes 288 records of dark structures that were identified as coronal holes, 49 dark structures, which were identified as filaments, and 6 dark structures, which could not be uniquely assigned, were excluded from further analysis.

We analyzed the distribution of characteristic properties in the coronal hole data set, i.e., the areas, latitudes of the center of mass, the mean EUV intensities, the mean value and skewness of the distribution of the radial magnetic field in the photosphere, and the magnetic flux.

We denote a pixel within a coronal hole as i, its projected area as ${A}_{i,\mathrm{proj}}$, its AIA-193 intensity as ${I}_{193,i}$, its heliographic latitude as φ_i, its line-of-sight magnetic field density as ${B}_{i,\mathrm{los}}$, and the total number of pixels within the coronal hole as n.

First, in order to avoid systematic errors due to the different latitudes of the coronal holes, we presume the magnetic field to be radial and correct each pixel for projection effects:

$$\begin{eqnarray}&&{A}_{i}=\displaystyle \frac{{A}_{i,\mathrm{proj}}}{\cos {\alpha }_{i}},\end{eqnarray} \tag{ 1 }$$

$$\begin{eqnarray}&&{B}_{i}=\displaystyle \frac{{B}_{i,\mathrm{los}}}{\cos {\alpha }_{i}},\end{eqnarray} \tag{ 2 }$$

where α_i is the heliographic angular distance of the pixel to the center of the solar disk as seen from SDO.

Then, for each coronal hole, we calculated the projection-corrected area A, the latitude of its center of mass φ, and the mean EUV intensity in the 193 Å filtergrams ${\bar{I}}_{193}$.

Furthermore, we calculated the projection-corrected unbalanced magnetic flux Φ, the total unsigned magnetic flux Φ_t, and the percentaged unbalanced magnetic flux Φ_pu, by

$$\begin{eqnarray}&&{\rm{\Phi }}=\displaystyle \sum _{i}\,{B}_{i}{A}_{i},\end{eqnarray} \tag{ 3 }$$

$$\begin{eqnarray}&&{{\rm{\Phi }}}_{{\rm{t}}}=\displaystyle \sum _{i}\,| {B}_{i}{A}_{i}| ,\end{eqnarray} \tag{ 4 }$$

$$\begin{eqnarray}&&{{\rm{\Phi }}}_{\mathrm{pu}}=\displaystyle \frac{{\rm{\Phi }}}{{{\rm{\Phi }}}_{{\rm{t}}}}.\end{eqnarray} \tag{ 5 }$$

The unbalanced magnetic flux gives the magnetic flux, which is not closed within the coronal hole. Under the standard assumption that only magnetic flux near the border of the coronal hole closes with neighboring regions, the unbalanced magnetic flux is a reasonable estimate of the open magnetic flux. The percentaged unbalanced magnetic flux gives the percentage of magnetic flux, which is not closed within the coronal hole.

Finally, we calculated the projection-corrected mean magnetic field density $\bar{B}$ and the projection-corrected mean unsigned magnetic field density ${\bar{B}}_{\mathrm{us}}$ in the photosphere below the coronal hole,

$$\begin{eqnarray}&&\bar{B}=\displaystyle \frac{{\rm{\Phi }}}{A},\end{eqnarray} \tag{ 6 }$$

$$\begin{eqnarray}&&{\bar{B}}_{\mathrm{us}}=\displaystyle \frac{{{\rm{\Phi }}}_{{\rm{t}}}}{A},\end{eqnarray} \tag{ 7 }$$

and the skewness Skew(B) of the magnetic field distribution. In our context, the skewness is a measure of the distinctness of a dominant polarity.

Whereas in Section 3.2 we analyze the distribution of characteristic properties of coronal holes, here we analyze the distribution of AIA-193 and magnetogram pixels within the coronal holes. To do so, we calculate for each coronal hole a normalized histogram. Then, we stack the normalized histograms of all coronal holes, whereby all histograms are weighted equally, i.e., small coronal holes get the same weight as large coronal holes. The projection of the stacked histograms gives us a median representation of the histograms and a 1σ and a 2σ range in which the histograms lie, i.e., a probability distribution of the histograms. This approach provides us with a characterization of the average pixel distribution within all coronal holes under study.

First, we analyzed the distribution of the EUV intensity within coronal holes. For each coronal hole, we created a histogram of the pixels of the AIA-193 intensity I_193,i, and normalized the histograms by their total number of pixels to obtain for each coronal hole the percentage of pixels belonging to each intensity bin. Then we calculated separately from all coronal holes the median value and the 1σ and 2σ ranges of the percentages of pixels for each intensity bin.

Next, we analyzed the EUV intensities along the major axis of coronal holes. First, we calculated the direction of the major axis by using a least-square fit on the longitude–latitude tupels of all pixels of the coronal hole. We calculated the length of all cross-sections parallel to the derived direction to get the longest cross-section, which we denote as the major axis. We define the width of the coronal hole as the number of pixels along the longest cut perpendicular to the major axis. The AIA-193 intensity along the major axis was then derived by averaging over ±1% of the width of the coronal hole perpendicular to major axis. For each coronal hole, we derived the AIA-193 intensity versus the position along the normalized major axis. We then calculated from all coronal holes the representative median AIA-193 intensity and the 1σ and 2σ ranges for the normalized positions along the major axis.

Furthermore, we analyzed the distribution of the magnetic field densities within coronal holes. Since the dominant polarity of coronal holes can be positive or negative, we transform the magnetic field densities B_i within a coronal hole so that positive values always belong to the dominant polarity of the coronal hole:

$$\begin{eqnarray}{B}_{+}=\left\{\begin{array}{ll}{B}_{i} & \mathrm{for}\ \bar{B}\geqslant 0\\ -{B}_{i} & \mathrm{for}\ \bar{B}\lt 0\end{array}\right..\end{eqnarray} \tag{ 8 }$$

For each coronal hole, we created a histogram of B₊.

Next, we analyzed the distinctiveness of the dominant polarity at various magnetic field density levels within coronal holes. We define the magnetic flux arising from a given signed magnetic field density level l as

$$\begin{eqnarray}&&{{\rm{\Phi }}}_{\mathrm{lv}}(l)=\displaystyle \sum _{| -{B}_{+}+l| \lt 1\,{\rm{G}}}{B}_{+}{A}_{i},\end{eqnarray} \tag{ 9 }$$

where $| -{B}_{+}+l| \lt 1\,{\rm{G}}$ corresponds to the binning of the magnetic field density values B₊. Then, we denote the fraction of the magnetic flux arising from a given magnetic field density level l, which has the same polarity as the overall coronal hole as the polarity dominance Φ_pd(l) of the magnetic field density level l,

$$\begin{eqnarray}&&{{\rm{\Phi }}}_{\mathrm{pd}}(l)=\displaystyle \frac{| {{\rm{\Phi }}}_{\mathrm{lv}}(l)| }{| {{\rm{\Phi }}}_{\mathrm{lv}}(l)| +| {{\rm{\Phi }}}_{\mathrm{lv}}(-l)| }.\end{eqnarray} \tag{ 10 }$$

A polarity dominance of Φ_pd(l) = 0.5 means that the magnetic flux arising from the magnetic field density level l is balanced, whereas a polarity dominance of Φ_pd(l) = 1 (Φ_pd(l) = 0) corresponds to a unipolar magnetic flux arising from the magnetic field density level l with the same (opposite) polarity as the overall coronal hole.

Finally, we analyzed the fraction of the unbalanced magnetic flux Φ of a coronal hole that arises from all pixels above a given threshold magnetic field density b, Φ_fu(b):

$$\begin{eqnarray}&&{{\rm{\Phi }}}_{\mathrm{fu}}(b)=\displaystyle \frac{{\displaystyle \sum }_{| {B}_{+}| \geqslant b}\,{B}_{+}{A}_{i}}{{\rm{\Phi }}}.\end{eqnarray} \tag{ 11 }$$

We investigated the characteristics of magnetic flux tubes within each coronal hole. We extracted the magnetic flux tubes by a thresholding technique on the magnetograms of the coronal holes at different global thresholds of thr = i · 10 G, $i\in \{1,2,...,8\}$. First, all pixels in the magnetogram with $| {B}_{i}| \lt \mathrm{thr}$ were set to Not a Number. Each island remaining within the magnetograms is defined as a magnetic flux tube. The resulting magnetogram was segmented, whereby two flux tubes were identified as individual flux tubes as soon as they were separated by at least one pixel. Note that the flux tubes extracted and their parameters derived depend on the magnetic extraction threshold, thr.

For each flux tube extracted at a threshold thr, we calculated the following parameters: its projection-corrected area ${A}_{\circ ,\mathrm{thr}}$, its projection-corrected mean magnetic field density ${\bar{B}}_{\circ ,\mathrm{thr}}$, and its projection-corrected magnetic flux ${{\rm{\Phi }}}_{\circ ,\mathrm{thr}}$ (see Section 3.2).

For each coronal hole, we counted the number of flux tubes ${\tilde{N}}_{\circ ,\mathrm{thr}}$. We calculated the total area ${\tilde{A}}_{\circ ,\mathrm{thr}}$ of the coronal hole that is covered by flux tubes, their summed unbalanced magnetic flux ${\tilde{{\rm{\Phi }}}}_{\circ ,\mathrm{thr}}$, their summed total unsigned magnetic flux ${\tilde{{\rm{\Phi }}}}_{\circ ,t,\mathrm{thr}}$, and their averaged magnetic field density ${\tilde{B}}_{\circ ,\mathrm{thr}}$, by summing (averaging) over all pixels that belong to magnetic flux tubes extracted at a given threshold thr:

$$\begin{eqnarray}&&{\tilde{A}}_{\circ ,\mathrm{thr}}=\displaystyle \sum _{| {B}_{i}| \geqslant \mathrm{thr}}{A}_{i}\end{eqnarray} \tag{ 12 }$$

$$\begin{eqnarray}&&{\tilde{{\rm{\Phi }}}}_{\circ ,\mathrm{thr}}=\displaystyle \sum _{| {B}_{i}| \geqslant \mathrm{thr}}{B}_{i}{A}_{i}\end{eqnarray} \tag{ 13 }$$

$$\begin{eqnarray}&&{\tilde{{\boldsymbol{\Phi }}}}_{\circ ,t,\mathrm{thr}}=\displaystyle \sum _{| {B}_{i}| \geqslant \mathrm{thr}}| {{\boldsymbol{B}}}_{i}{{\boldsymbol{A}}}_{i}| ,\end{eqnarray} \tag{ 14 }$$

$$\begin{eqnarray}&&{\tilde{B}}_{\circ ,\mathrm{thr}}=\displaystyle \frac{1}{{\tilde{A}}_{\circ ,\mathrm{thr}}}\displaystyle \sum _{| {B}_{i}| \geqslant \mathrm{thr}}{B}_{i}{A}_{i}.\end{eqnarray} \tag{ 15 }$$

In addition, we calculated the mean magnetic field density of the coronal hole without the flux tubes, which we denote as "quiet" coronal hole regions:

$$\begin{eqnarray}&&{\bar{B}}_{\mathrm{qu},\mathrm{thr}}=\displaystyle \frac{1}{(A-{\tilde{A}}_{\circ ,\mathrm{thr}})}\displaystyle \sum _{| {B}_{i}| \lt \mathrm{thr}}{B}_{i}{A}_{i},\end{eqnarray} \tag{ 16 }$$

where A is the area of the coronal hole.

Finally, for each coronal hole, we estimated the fraction of the magnetic flux arising from all flux tubes, which is unbalanced, i.e., the percentaged unbalanced magnetic flux of the flux tubes

$$\begin{eqnarray}&&{\tilde{{\rm{\Phi }}}}_{\mathrm{pu},\mathrm{thr}}=\displaystyle \frac{{\tilde{{\rm{\Phi }}}}_{\circ ,\mathrm{thr}}}{{\tilde{{\rm{\Phi }}}}_{\circ ,t,\mathrm{thr}}}.\end{eqnarray} \tag{ 17 }$$

Note that under the condition of similar absolute unbalanced magnetic fluxes $| {{\rm{\Phi }}}_{\circ ,\mathrm{thr}}|$ of the flux tubes, ${\tilde{{\rm{\Phi }}}}_{\mathrm{pu},\mathrm{thr}}$ is also an estimate of the number of flux tubes that are open and on the number of flux tubes that are closed loops.

An overview of all properties defined is given in Table 1.

Table 1.  Overview of the Parameters Defined in Sections 3.2–3.4

Properties of coronal holes:
A	$={\sum }_{i}{A}_{i}$	Area
φ	$=\tfrac{1}{A}{\sum }_{i}{\varphi }_{i}{A}_{i}$	Latitude of center of mass
I193	$=\tfrac{1}{n}{\sum }_{i}{I}_{193,i}$	Mean AIA-193 intensity
Φ	$={\sum }_{i}{A}_{i}{B}_{i}$	Unbalanced magnetic flux
Φpu	$=\tfrac{{\sum }_{i}{A}_{i}{B}_{i}}{{\sum }_{i}| {A}_{i}{B}_{i}| }$	percentaged unbalanced magnetic flux
$\bar{B}$	$=\tfrac{{\rm{\Phi }}}{A}$	Mean magnetic field density
${\bar{B}}_{\mathrm{us}}$	$\tfrac{{\sum }_{i}| {A}_{i}{B}_{i}| }{A}$	Mean unsigned magnetic field density
Pixel distribution within coronal holes:
B+	$=\left\{\begin{array}{ll}{B}_{i} & \mathrm{for}\ \bar{B}\geqslant 0\\ -{B}_{i} & \mathrm{for}\ \bar{B}\lt 0\end{array}\right.$	Magnetic field density of a pixel, projected so that positive values belong to the dominant polarity of the coronal hole
Φpd(l)	$=\tfrac{{\sum }_{l-1{\rm{G}}\lt {B}_{+}\lt l+1{\rm{G}}}{B}_{+}{A}_{i}}{{\sum }_{l-1{\rm{G}}\lt | {B}_{+}| \lt l+1{\rm{G}}}| {B}_{+}{A}_{i}| }$	Polarity dominance of the pixels at the given magnetic field density level l
Φfu(b)	$=\tfrac{1}{{\rm{\Phi }}}{\sum }_{| {B}_{+}| \geqslant b}{B}_{+}{A}_{i}$	Fraction of the unbalanced magnetic flux that arises from pixels with B+ ≥ b
Magnetic flux tubes extracted at a threshold, thr:
${A}_{\circ ,\mathrm{thr}}$	$={\sum }_{f}{A}_{i}$	Area of a single flux tube
${{\rm{\Phi }}}_{\circ ,\mathrm{thr}}$	$={\sum }_{f}{B}_{i}{A}_{i}$	Magnetic flux of a single flux tube
${\bar{B}}_{\circ ,\mathrm{thr}}$	$=\tfrac{{{\rm{\Phi }}}_{\circ ,\mathrm{thr}}}{{A}_{\circ ,\mathrm{thr}}}$	Mean magnetic field density of a single flux tube
${\tilde{N}}_{\circ ,\mathrm{thr}}$		Number of flux tubes of a coronal hole
${\tilde{A}}_{\circ ,\mathrm{thr}}$	$={\sum }_{| {B}_{i}| \geqslant \mathrm{thr}}{A}_{i}$	Total area covered by all flux tubes of a coronal hole
${\tilde{{\rm{\Phi }}}}_{\circ ,\mathrm{thr}}$	$={\sum }_{| {B}_{i}| \geqslant \mathrm{thr}}{B}_{i}{A}_{i}$	Unbalanced magnetic flux derived from all flux tubes of a coronal hole
${\tilde{{\rm{\Phi }}}}_{\mathrm{pu},\mathrm{thr}}$	$=\tfrac{{\sum }_{| {B}_{i}| \geqslant \mathrm{thr}}{B}_{i}{A}_{i}}{{\sum }_{| {B}_{i}| \geqslant \mathrm{thr}}| {B}_{i}{A}_{i}| }$	Percentaged unbalanced magnetic flux derived from all flux tubes of a coronal hole
${\tilde{B}}_{\circ ,\mathrm{thr}}$	$=\tfrac{1}{{\tilde{A}}_{\circ ,\mathrm{thr}}}\sum _{| {B}_{i}| \geqslant \mathrm{thr}}\,{B}_{i}{A}_{i}$	Average magnetic field density derived from all flux tubes of a coronal hole
${\bar{B}}_{\mathrm{qu},{thr}}$	$=\tfrac{1}{A-{\tilde{A}}_{\circ ,\mathrm{thr}}}\sum _{| {B}_{i}| \lt {thr}}\,{B}_{i}{A}_{i}$	Mean magnetic field density of the coronal hole region without the flux tubes

Note. The subscript i denotes a pixel of a coronal hole, f a pixel of a flux tube, A_i its projection-corrected area, B_i its projection-corrected mean magnetic field density, ${I}_{193,i}$ its mean AIA-193 intensity, φ_i its latitude, and n the total number of pixels within the coronal hole.

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Figure 2 gives an overview of the data set of the low-latitude coronal holes, i.e., the distribution of the areas, latitudes, and polarities of all coronal holes under study.

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Figure 2(a) shows the distribution of the deprojected areas A of the coronal holes. The median area is 2.39 · 10¹⁰ km². Only 18% of all coronal holes under study had areas of $A\gt 5\cdot {10}^{10}\,{\mathrm{km}}^{2}$. The largest coronal hole detected had an area of 1.4 · 10¹¹ km², i.e., covering 2.25% of the visible solar hemisphere.

Figure 2(b) shows the heliographic latitudes φ of the center of mass of coronal holes versus their areas A. In the time range under study, large coronal holes appeared predominantly on the northern hemisphere, whereas small and medium sized coronal holes appeared at both hemispheres.

Figure 2(c) shows the latitudes φ of the center of mass of coronal holes versus their percentaged unbalanced magnetic flux Φ_pu, which corresponds to their magnetic polarity. In the time range under study, i.e., the rising phase of solar cycle 24 up to solar maximum, coronal holes appeared with both polarities in each of the two hemispheres at almost all times. This can be explained by the reversal of the magnetic polarity of polar coronal holes during the time range under study. Coronal holes with a small percentaged unbalanced magnetic flux, i.e., a weakly pronounced polarity, only appeared near the solar equator, with a medium percentaged unbalanced magnetic flux at all latitudes, and with a high percentaged unbalanced magnetic flux only at some distance to the solar equator. Note that this plot is not symmetric to the solar equator, but to ≈5° south.

Figure 3 shows the distribution of the mean AIA-193 intensities ${\bar{I}}_{193}$ of all coronal holes under study. The mean AIA-193 intensity is 36 ± 6 DNs. Only 3% of all coronal holes had mean intensities of ${\bar{I}}_{193}\gt 50\,\mathrm{DNs};$ the brightest coronal hole had a mean intensity of 60 DNs. For comparison, the most common pixel intensity of the solar disk in all the AIA-193 Å filtergrams in our data set, i.e., the quiet-Sun intensity, is in the range of 109 ± 16 DNs, and the median Sun intensity is 150 ± 17 DNs.

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Figures 4(a) and (b) show the distribution of the unbalanced magnetic flux Φ and the percentaged unbalanced magnetic flux Φ_pu of coronal holes. The absolute unbalanced magnetic flux $| {\rm{\Phi }}|$ ranges from 2.2 · 10¹⁹ to 4.7 · 10²¹ Mx with a mean value of 9.9 ± 9.4 · 10²⁰ Mx. The percentaged unbalanced magnetic flux Φ_ro shows two broad peaks at −0.46 ± 0.15 and 0.52 ± 0.17, implying that, on average, ≈49% of the magnetic flux piercing through the photosphere is not closed within the coronal hole. 3% of all coronal holes had absolute values of the percentaged unbalanced magnetic flux $| {{\rm{\Phi }}}_{\mathrm{pu}}| \gt 0.75$, and 2% of all coronal holes had $| {{\rm{\Phi }}}_{\mathrm{pu}}| \lt 0.10$.

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Figures 4(c) and (d) show the distribution of the mean magnetic field densities $\bar{B}$, and the skewness skew(B) of the magnetic field distributions of coronal holes. The mean value of $| \bar{B}|$ is 2.97 ± 1.55 G. The mean value of $| \mathrm{skew}(B)|$ is 6.6 ± 1.6, indicating that, in general, coronal holes have a distinct dominant polarity.

Figures 5(a), (b), and (c) show the mean magnetic field densities $\bar{B}$ of the coronal holes versus their areas A, the mean AIA-193 intensities ${\bar{I}}_{193}$ versus the areas A, and the mean AIA-193 intensities ${\bar{I}}_{193}$ versus the mean magnetic field densities $\bar{B}$. These three parameters do not correlate. However, we note that there is some difference between small and large coronal holes, which is discussed in Section 4.5.

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Figure 5(d) shows the unbalanced magnetic flux Φ of coronal holes versus their areas A. The absolute values of the unbalanced magnetic flux correlate with the areas with a Pearson correlation coefficient of 0.87, the dependency can be expressed as

$$\begin{eqnarray}&&| {\rm{\Phi }}| [\mathrm{Mx}]=-4.26\cdot {10}^{19}+3.45\cdot {10}^{10}\cdot A\ [{\mathrm{km}}^{2}]\end{eqnarray} \tag{ 18 }$$

with a root-mean-square error (RMSE) of 4.79 · 10²⁰ Mx. However, 66% of the coronal holes are rather small and have an absolute value of unbalanced magnetic flux of less than 1 · 10²¹ Mx. Therefore, this relationship is not suited to predict the unbalanced magnetic flux of a coronal hole, but only to estimate the magnitude of the unbalanced magnetic flux.

Figure 6(a) shows the mean unsigned magnetic flux B_us of coronal holes versus the absolute values of their mean magnetic field densities. A distinct, linear correlation is apparent, with a Pearson's correlation coefficient of 0.976. The relationship can be expressed as

$$\begin{eqnarray}&&{B}_{\mathrm{us}}[{\rm{G}}]=3.17+0.86| \bar{B}| ,\end{eqnarray} \tag{ 19 }$$

with an RMSE of 0.30 G.

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Figures 6(b) and (c) show the percentaged unbalanced magnetic flux of coronal holes versus the absolute values of the mean magnetic field densities and versus the mean unsigned magnetic field densities. Distinct relationships are also apparent here, with Spearman's rank correlation coefficients of 0.983 and 0.920. The relationships can be expressed as

$$\begin{eqnarray}&&| {{\rm{\Phi }}}_{\mathrm{pu}}| =\displaystyle \frac{| \bar{B}| \ [{\rm{G}}]}{3.17+0.86| \bar{B}| \ [{\rm{G}}]},\end{eqnarray} \tag{ 20 }$$

$$\begin{eqnarray}&&| {{\rm{\Phi }}}_{\mathrm{pu}}| =\displaystyle \frac{-3.69+1.17\ {\bar{B}}_{\mathrm{us}}\ [{\rm{G}}]}{{\bar{B}}_{\mathrm{us}}\ [{\rm{G}}]},\end{eqnarray} \tag{ 21 }$$

with RMSEs of 0.025, respectively, 0.064. Since the percentaged unbalanced magnetic flux ${{\rm{\Phi }}}_{\mathrm{pu}}$ can be approximated by the ratio of the mean magnetic field density to the mean unsigned magnetic field density $\bar{B}/{\bar{B}}_{\mathrm{us}}$, Equations (20) and (21) are directly related to Equation (19). Note that Equation (19) implies that the mean magnetic field density of a coronal hole strongly depends on its mean unsigned magnetic field density, and Equation (20) implies that the amount of unbalanced magnetic flux relative to the total unsigned flux inside a coronal hole can be predicted purely by the mean magnetic field density!

Figures 5(a), (b), and (d) show differences in the distribution of characteristic properties of small coronal holes, which we define as the 115 coronal holes with areas $A\lt 2\cdot {10}^{10}\,{\mathrm{km}}^{2}$, and the 12 large coronal holes with $A\gt 1\cdot {10}^{11}\,{\mathrm{km}}^{2}$.

Small coronal holes have a wide range of mean AIA-193 intensities ${\bar{I}}_{193}$ ranging from 24 to 60 DNs (Figure 5(b)). The mean magnetic field densities $| \bar{B}|$ range from 0.2 to 8.7 G (Figure 5(a)). The absolute values of the percentaged unbalanced magnetic flux $| {{\rm{\Phi }}}_{\mathrm{pu}}|$ takes values of 0.06–0.81, i.e., the magnetic flux piercing through the photosphere can be balanced well and can also be almost unipolar. The absolute unbalanced magnetic flux $| {\rm{\Phi }}|$ is comparably small, ranging from 2.2 · 10¹⁹ to 1.3 · 10²¹ Mx (Figure 5(d)). Note that we cannot state if the small coronal holes with an almost balanced magnetic flux are coronal holes at their birth/end or if they are artefacts.

In contrast, large coronal holes always appear as dark structures with ${\bar{I}}_{193}\lt 37\,\mathrm{DNs}$ (Figure 5(b)). They have neither very small nor very large magnetic field densities, $| \bar{B}|$ ranges from 2.3 to 4.3 G (Figure 5(a)). The absolute values of the percentaged unbalanced magnetic flux $| {{\rm{\Phi }}}_{\mathrm{pu}}|$ ranges from 0.40 to 0.64, and they always have an absolute unbalanced magnetic flux of $| {\rm{\Phi }}| \gt 2.5\cdot {10}^{21}\,\mathrm{Mx}$. This findings show that the characteristics of large coronal holes are much more constrained than the characteristics of small coronal holes.

Figure 7(a) shows the stacked AIA-193 intensity histograms of coronal holes. The solid line gives the median AIA-193 intensity histogram, the dashed line gives the 1σ range, i.e., the range that contains 68% of all AIA-193 intensity histograms. The AIA-193 intensity is distributed broadly inside coronal holes, ranging from about 15 to 70 DNs; the mode is at 27 DNs. 1% of all pixels have intensities of ${I}_{193,i}\lt 16\,\mathrm{DNs}$, 10% have intensities of ${I}_{193,i}\lt 20\,\mathrm{DNs}$, 10% have intensities of ${I}_{193,i}\gt 50\,\mathrm{DNs}$, and 1% have intensities of ${I}_{193,i}\gt 65\,\mathrm{DNs}$.

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Figure 7(b) shows the stacked AIA-193 intensity distributions along the major axis of the coronal holes. For comparison, the horizontal stripes give the distribution of the median AIA-193 intensities of the entire solar disks (150 ± 17 DNs), which is also an estimate of the median AIA-193 intensities of quiet-Sun regions. At the border of coronal holes, the median representation of the intensities steeply decreases from the quiet-Sun intensity to a low intensity of about 40 DNs. After the steep decrease, the median AIA-193 intensity stays flat, i.e., coronal holes generally do not reveal an intensity minimum at their center. We further investigated the AIA-193 intensity distribution by inspecting the AIA-193 intensity distribution along the cross-section for each coronal hole separately. We found a wide variety on intensity distributions: from almost homogeneous intensity distributions without distinct intensity minimum, intensity distributions with large fluctuations, which most probably arise from coronal bright points located inside of coronal holes, intensity distributions with one and more distinct intensity minima at different locations at the cross-section, intensity distributions that rise sharply at the border of coronal holes, to intensity distributions that rise slowly and steadily at the borders. Thus, Figure 7(b) represents only the statistical cross-section of the AIA-193 intensity as derived by the superposed epoch analysis, but does not hold as a prototype cross-section of coronal holes.

Figure 8(a) shows the stacked histograms of the magnetic field density of coronal holes. Positive values of B₊ represent pixels with the same polarity as the dominant polarity of the coronal hole, negative values of B₊ pixels with the opposite polarity. The imbalance of the magnetic flux within coronal holes is clearly visible.

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Figure 8(b) shows the stacked polarity dominance at given magnetic field density levels l, Φ_pd(l) for all coronal holes. The polarity dominance gives the fraction of the magnetic flux arising from pixels with magnetic field density l, which has the same polarity as the overall coronal hole. Therefore, a value of Φ_pd(l) = 0.5 means that the magnetic flux arising from pixels with a magnetic field density l is balanced; a value of Φ_pd(l) = 1 (Φ_pd(l) = 0) means that the magnetic flux arising from the magnetic level l is unipolar and has the same (opposite) polarity as the overall coronal hole.

The median representation of the polarity dominance at a given magnetic level l is monotonically increasing with the magnetic level l; at a magnetic level of 1 G, 53% of the magnetic flux has the same polarity as the overall coronal hole, at a magnetic level of 10 G, 70% of the magnetic flux, and at a magnetic level of 30 G, 92% of the magnetic flux. The 2σ range shows that even at low magnetic field density levels the polarity dominance is usually >0.5. This indicates that all magnetic levels of magnetic field density predominantly have the same polarity as the overall coronal hole. The magnetic flux arising from pixels with a weak magnetic field density is only slightly unbalanced toward the dominant polarity of the coronal hole, whereas the imbalance strongly increases with increasing magnetic field density, reaching a value of Φ_pd(30) = 0.92 at a magnetic field density level of 30 G.

Figure 9(a) shows the stacked cumulative histograms of the absolute values of the magnetic field density $| {B}_{+}|$ of coronal holes. In the median representation, 90% of all pixels within a coronal hole have an absolute magnetic field density $| {B}_{+}| \lt 11\,{\rm{G}}$, 9% have an absolute magnetic field density $| {B}_{+}|$ of 11–56 G, and 1% have an absolute magnetic field density $| {B}_{+}| \gt 56\,{\rm{G}}$.

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Figure 9(b) shows the stacked plots of the fraction of the unbalanced magnetic flux Φ_fu(b), which arises from pixels above a given threshold magnetic field density b versus b. In the median representation, 81% of the unbalanced magnetic flux arises from pixels with an absolute magnetic field density $| {B}_{+}| \gt 10\,{\rm{G}}$, 54% from pixels with $| {B}_{+}| \gt 30\,{\rm{G}}$, and 38% from pixels with $| {B}_{+}| \gt 50\,{\rm{G}}$. Therefore, most of the unbalanced magnetic flux of coronal holes arises from a small percentage of its area. This aspect is studied in further detail in the following section.

In total, we extracted 26,409 flux tubes at 10 G, 8850 flux tubes at 30 G, and 4941 flux tubes at 50 G.

Figure 11(a) shows the histogram of the mean magnetic field densities ${\bar{B}}_{\circ ,\mathrm{thr}}$ of all flux tubes. Positive values correspond to flux tubes that have the same polarity as the dominant polarity of its coronal hole. At thresholds of 10 G, the mean magnetic field density is 18 ± 20 G, at thresholds of 30 G 58 ± 29 G, and at thresholds of 50 G 91 ± 27 G. The positive values of the mean magnetic field densities, especially at the higher extraction thresholds, show that most flux tubes have the same dominant polarity as their coronal hole. The lower values of mean magnetic field densities at lower extraction thresholds arises from the higher number of weak magnetic flux tubes and the larger area per flux tube.

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Figure 11(b) shows the distribution of the magnetic flux ${{\rm{\Phi }}}_{\circ ,\mathrm{thr}}$ of all flux tubes. Most of the flux tubes have a small absolute magnetic flux: 75% have $| {{\rm{\Phi }}}_{\circ ,10}| \lt 1.0\cdot {10}^{19}\,\mathrm{Mx}$ at a flux-tube threshold of 10 G, 75% have $| {{\rm{\Phi }}}_{\circ ,30}| \lt 2.2\cdot {10}^{19}\,\mathrm{Mx}$ at a threshold of 30 G, and 75% have $| {{\rm{\Phi }}}_{\circ ,50}| \lt 2.6\cdot {10}^{19}\,\mathrm{Mx}$ at a threshold of 50 G. However, there is also a small number of flux tubes with a large magnetic flux, with values as high as 5.3 · 10²⁰ Mx at a threshold of 10 G, 2.6 · 10²⁰ Mx at a threshold of 30 G, and 1.6 · 10²⁰ Mx at a threshold of 50 G.

Figure 11(c) shows the absolute mean magnetic field densities $| {\bar{B}}_{\circ ,50}|$ of all flux tubes extracted at 50 G versus the flux-tube areas ${A}_{\circ ,50}$. 55% of all flux tubes have a small area of ${A}_{\circ ,50}\lt 2.5\cdot {10}^{7}\,{\mathrm{km}}^{2}$ and a small mean magnetic field density of $| {\bar{B}}_{\circ ,50}| \lt 100\,{\rm{G}}$. Flux tubes with a large area always have a medium to high mean magnetic field density. However, flux tubes with a very high mean magnetic field density usually have a small to medium area.

We further investigated the relationship between the absolute mean magnetic field density of flux tubes and their area extracted at 50 G for each coronal hole individually. Figure 11(d) shows the scatter plot of the absolute mean magnetic field densities $| {\bar{B}}_{\circ ,50}|$ of flux tubes of a typical medium sized coronal hole versus their areas ${A}_{\circ ,50}$. At medium size and large size coronal holes, a wide-spread relationship between the absolute mean magnetic field densities and the areas is often visible, with larger areas at higher mean magnetic field densities. This relationship can be explained by magnetic pressure: higher mean magnetic field densities are related to higher magnetic pressures within the flux tubes, and thus to statistically larger cross-sections. At small coronal holes, there are usually not enough flux tubes to make a meaningful statement on the correlation.

Figure 12(a) shows the stacked plots of the number of flux tubes per coronal hole area, ${\tilde{N}}_{\circ ,\mathrm{thr}}/A$, versus the extraction threshold thr. Coronal holes have, on average, 28 ± 3 flux tubes extracted at 10 G per 10¹⁰ km², 9 ± 4 flux tubes extracted at 30 G per 10¹⁰ km², and 5 ± 3 flux tubes extracted at 50 G per 10¹⁰ km².

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In this section, we presume that magnetic flux tubes with the same polarity as the dominant polarity of the coronal hole do not close with quiet regions inside of the coronal hole, but only with flux tubes of opposite polarity inside of the coronal hole. This assumption is supported by the fact that coronal holes usually have the same dominant polarity at all magnetic scales (see Figure 8(b)), and that, therefore, the quiet coronal hole regions, on average, also have the same dominant polarity as the flux tubes with dominant polarity. In this case, the percentaged unbalanced magnetic flux of the flux tubes ${{\rm{\Phi }}}_{\mathrm{pu},\mathrm{thr}}$ gives a lower estimate on the fraction of the magnetic flux arising from flux tubes, which is unbalanced within the coronal hole boundaries.

Figure 12(b) shows the stacked plots of the percentaged unbalanced magnetic flux ${\tilde{{\rm{\Phi }}}}_{\mathrm{pu},\mathrm{thr}}$ of the flux tubes per coronal hole versus the extraction threshold. The median line shows that for 50% of all coronal holes at least 85% of the magnetic flux arising from flux tubes is unbalanced at an extraction threshold of 10 G, at least 98% is unbalanced at an extraction threshold of 30 G, and the complete magnetic flux is unbalanced at an extraction threshold of 50 G. The bottom 1σ line shows that for 82% of all coronal holes, at least 72% of the magnetic flux arising from flux tubes is unbalanced at an extraction threshold of 10 G, at least 93% at 30 G, and all the complete magnetic flux is unbalanced at 50 G. This means that, in general, most of the magnetic flux tubes in coronal holes, and, in particular, the strong magnetic flux tubes, are open.⁵

Note that Figure 9 can also be interpreted in the context of flux tubes. Figure 9(a) is proportional to the fraction of the coronal hole area that is covered by flux tubes extracted at a threshold thr = b, ${\tilde{A}}_{\circ ,\mathrm{thr}}/A$. Figure 9(b) then shows the fraction of magnetic flux to the unbalanced magnetic flux that arises from flux tubes, ${\tilde{{\rm{\Phi }}}}_{\circ ,\mathrm{thr}}/{\rm{\Phi }}$. The flux tubes extracted at thresholds of 10 G cover 10% of the area of coronal holes, and carry 81% of the unbalanced magnetic flux. If we rise the threshold to 30 G, the flux tubes cover 2.5% of the area of coronal holes, and carry 54% of the unbalanced magnetic flux. At extraction thresholds of 50 G, the flux tubes cover 1.2% of the area of coronal holes, and carry 38% of the unbalanced magnetic flux.

This means that most of the unbalanced magnetic flux of coronal holes arise from very localized regions, i.e., magnetic flux tubes, covering only a small fraction of the total coronal hole area.

Figure 13(a) shows the number of flux tubes extracted at 50 G ${\tilde{N}}_{\circ ,50}$ per coronal hole versus the area A of coronal holes. The number of flux tubes depends linearly, but with a high scatter, on the area. Figure 13(b) shows the number of flux tubes per coronal hole normalized on the area of the coronal hole ${\tilde{N}}_{\circ ,50}/A$ versus the absolute values of the mean magnetic field density $| \bar{B}|$ of the coronal hole. The number of flux tubes per area depends linearly on the mean magnetic field density, with about 1.5 flux tubes per 10¹⁰ km² at $| \bar{B}| =1\,{\rm{G}}$ and 9 flux tubes per 10¹⁰ km² at $| \bar{B}| =5\,{\rm{G}}$. The total relationship can be expressed as

$$\begin{eqnarray}&&{N}_{\circ ,50}=(-0.12+1.79| \bar{B}| [G])A[{10}^{10}\,{\mathrm{km}}^{2}],\end{eqnarray} \tag{ 22 }$$

with an RMSE of 2.92 flux tubes.

In contrast, the number of flux tubes extracted at 10 G ${\tilde{N}}_{\circ ,10}$ depends mainly on the coronal hole area A (Figure 13(c)), and only slightly on the mean magnetic field density B of the coronal hole (Figure 13(d)), with

$$\begin{eqnarray}&&{N}_{\circ ,10}=(25+0.77| \bar{B}| [G])A[{10}^{10}\,{\mathrm{km}}^{2}].\end{eqnarray} \tag{ 23 }$$

The RMSE is 10.5 flux tubes.

Altogether, the general number of flux tubes, i.e., extracted at a low extraction threshold of 10 G, depends mainly on the area of the coronal hole, whereas the number of strong magnetic flux tubes, i.e., extracted at 50 G, is also related to the mean magnetic field density of the coronal hole.

Figure 14(a) shows the mean magnetic field density of the quiet coronal hole regions ${\bar{B}}_{\mathrm{qu},50}$, i.e., the coronal hole without the flux tubes extracted at 50 G, versus the averaged magnetic field density of the flux tubes extracted at 50 G ${\tilde{B}}_{\circ ,50}$. The mean magnetic field density of the quiet coronal hole correlates well with the averaged magnetic field density of the flux tubes, with a Spearman's rank correlation coefficient of 0.92. However, this correlation could be artificially created by our flux-tube extraction algorithm in the case in which we only extracted the cores of the flux tubes and the outer layers of flux tubes are contained within our quiet coronal hole regions. Therefore, we cross-checked this relation at low extraction thresholds of 10 G (Figure 14(d)) and found that the correlation still holds at a Spearman's rank correlation coefficient of 0.96. This finding shows that strong flux tubes within the coronal hole result in a distinctly increased mean magnetic field density of the quiet coronal hole region. It also implies that the magnetic field of the flux tubes and of the quiet coronal hole area are coupled.

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Figure 14(b) shows the averaged magnetic field density derived from all flux tubes extracted at 50 G within a coronal hole ${\tilde{B}}_{\circ ,50}$ versus the mean magnetic field density of the coronal hole $\bar{B}$. The averaged magnetic field density of the flux tubes extracted at 50 G correlates strongly with the mean magnetic field density of the overall coronal hole, with a Spearman's rank correlation coefficient of 0.93. By decreasing the flux-tube extraction threshold to 10 G, the correlation coefficient even increases to 0.99 (Figure 14(e)). The relationships can be expressed as

$$\begin{eqnarray}&&{\tilde{B}}_{\circ ,50}[{\rm{G}}]=\displaystyle \frac{115\ \bar{B}[{\rm{G}}]}{0.7\,{\rm{G}}+| \bar{B}| [{\rm{G}}]},\end{eqnarray} \tag{ 24 }$$

$$\begin{eqnarray}&&{\tilde{B}}_{\circ ,10}[{\rm{G}}]=\displaystyle \frac{58\ \bar{B}[{\rm{G}}]}{3.9\,{\rm{G}}+| \bar{B}| [{\rm{G}}]},\end{eqnarray} \tag{ 25 }$$

with an RMSE of 10.7 G and a mean absolute percentage error (MAPE) of 8.6% for flux tubes extracted at 50 G and an RMSE of 1.6 G and an MAPE of 6.8% for flux tubes extracted at 10 G. Note that these correlations result from the correlation between the average magnetic field density of flux tubes and the mean magnetic field density of the quiet coronal hole regions.

Figure 14(c) shows the mean area per flux tube extracted at 50 G derived from all flux tubes within a coronal hole ${\tilde{A}}_{\circ ,50}/{\tilde{N}}_{\circ ,50}$ versus the mean magnetic field density of the coronal hole $| \bar{B}|$. The mean area per flux tube extracted at 50 G correlates well with the mean magnetic field density of the coronal hole, with a Pearsons correlation coefficient of 0.64. By decreasing the flux-tube extraction threshold to 10 G, the correlation coefficient increases to 0.93 (Figure 14(f)). The relationships can be expressed as

$$\begin{eqnarray}&&\displaystyle \frac{{A}_{\circ ,50}}{{N}_{\circ ,50}}[{\mathrm{km}}^{2}]=1.28\cdot {10}^{7}+2.13\cdot {10}^{6}| \bar{B}| [{\rm{G}}],\end{eqnarray} \tag{ 26 }$$

$$\begin{eqnarray}&&\displaystyle \frac{{A}_{\circ ,10}}{{N}_{\circ ,10}}[{\mathrm{km}}^{2}]=1.58\cdot {10}^{7}+6.08\cdot {10}^{6}| \bar{B}| [{\rm{G}}],\end{eqnarray} \tag{ 27 }$$

with an RMSE of 3.8 · 10⁶ km² and an MAPE of 15% for flux tubes extracted at 50 G and an RMSE of 3.6 · 10⁶ km² and an MAPE of 8% for flux tubes extracted at 10 G. These relationships can simply be explained by magnetic pressure: a high mean magnetic field density of the coronal hole is related to high magnetic field densities in the flux tubes (see Figures 14(b) and (e)). A high mean magnetic field density of flux tubes results in a high magnetic pressure within the flux tube, and therefore in a large cross-section of the flux tube under the condition of comparably constant total pressure of the surrounding of the flux tube.

Altogether, the mean area per flux tube correlates well with the mean magnetic field density of the coronal hole. Furthermore, the mean magnetic field density of the coronal hole, the mean magnetic field density derived from the quiet coronal hole regions, and the averaged magnetic field density derived from all flux tubes within a coronal hole strongly correlate with each other. At lower extraction thresholds, i.e., by taking weaker flux tubes into account, the goodness of the relationships even enhance: the correlation coefficients increase and the MAPE decreases.

Figure 15(a) shows the unbalanced magnetic flux derived from all flux tubes extracted at 50 G within a coronal hole ${\tilde{{\rm{\Phi }}}}_{\circ ,50}$ versus the unbalanced magnetic flux Φ of the coronal hole. The unbalanced magnetic flux of the flux tubes correlates with the unbalanced magnetic flux of the complete coronal hole, with a Spearmans rank correlation coefficient of 0.99. By decreasing the flux-tube extraction threshold to 10 G, the Spearmans rank correlation coefficient stays at 0.99. The relationships can be expressed as

$$\begin{eqnarray}&&{\tilde{{\rm{\Phi }}}}_{\circ ,50}[\mathrm{Mx}]=0.42\ {\rm{\Phi }}[\mathrm{Mx}],\end{eqnarray} \tag{ 28 }$$

$$\begin{eqnarray}&&{\tilde{{\rm{\Phi }}}}_{\circ ,10}[\mathrm{Mx}]=0.84\ {\rm{\Phi }}[\mathrm{Mx}],\end{eqnarray} \tag{ 29 }$$

with an RMSE of 8.9 · 10¹⁹ Mx and an MAPE of 79% for flux tubes extracted at 50 G, and an RMSE of 4.7 · 10¹⁹ Mx and an MAPE of 7% for flux tubes extracted at 10 G. This means that about 42% (84%) of the total unbalanced magnetic flux of coronal holes arises from flux tubes extracted at 50 G (10 G). Note that these values slightly differ from the values derived in Section 6.4 (38% for flux tubes extracted at 50 G, and 81% for flux tubes extracted at 10 G), since the least-square fit, which was used to derive Equations (28) and (29), privileges coronal holes with high magnetic fluxes. The high MAPE of Equation (28) arises from the large amount of small coronal holes in our data set, which have small magnetic fluxes and are therefore not described well by Equation (28). The MAPE of Equation (29) is significantly decreased in comparison to Equation (28), since the flux tubes extracted at 10 G already hold most of the unbalanced magnetic flux of the overall coronal hole.

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Figure 15(a) shows the contribution of the flux tubes extracted at 50 G on the unbalanced magnetic flux of the coronal hole, ${\tilde{{\rm{\Phi }}}}_{\circ ,50}/{\rm{\Phi }}$, versus the absolute value of the average magnetic field density of the flux tubes $| {\tilde{B}}_{\circ ,50}|$. The proportion of the magnetic flux arising from the flux tubes on the unbalanced magnetic flux of the coronal hole depends on the averaged magnetic field density of the flux tubes: at a high averaged magnetic field density of all flux tubes within a coronal hole about 60% of the unbalanced magnetic flux arises from flux tubes, at a low averaged magnetic field density only about to 20%; the Spearman's rank correlation coefficient is 0.63. By decreasing the flux-tube extraction threshold to 10 G, the amount of unbalanced magnetic flux arising from the flux tubes increases to about 70%–87% dependent on the averaged magnetic field density of the flux tubes; the Spearman's rank correlation coefficient is 0.62. Taking these correlation into account, Equations (28) and (29) improve to

$$\begin{eqnarray}&&{\tilde{{\rm{\Phi }}}}_{\circ ,50}[\mathrm{Mx}]=(-0.12+0.0055| {\tilde{B}}_{\circ ,50}| [{\rm{G}}]){\rm{\Phi }}[\mathrm{Mx}],\end{eqnarray} \tag{ 30 }$$

$$\begin{eqnarray}&&{\tilde{{\rm{\Phi }}}}_{\circ ,10}[\mathrm{Mx}]=(0.72+0.0041| {\tilde{B}}_{\circ ,10}| [{\rm{G}}]){\rm{\Phi }}[\mathrm{Mx}],\end{eqnarray} \tag{ 31 }$$

with an RMSE of 8.4 · 10¹⁹ Mx and an MAPE of 30% for flux tubes extracted at 50 G, and an RMSE of 3.2 · 10¹⁹ Mx and an MAPE of 4.5% for flux tubes extracted at 10 G.

In summary, all the relationships in this section show that the magnetic characteristics of coronal holes are set by its magnetic flux tubes. A high mean magnetic field density of flux tubes results in a large mean area per flux tube, and thus to a high magnetic flux per flux tube and to high amount of unbalanced magnetic flux arising from flux tubes. In addition, a high magnetic field density of flux tubes is related to a high mean magnetic field density of the quiet coronal hole regions, and therefore to high mean magnetic field density of the overall coronal hole.