Estimation of the Magnetic Flux Emergence Rate in the Quiet Sun from Sunrise Data

The quiet Sun covers most of the solar surface, in particular at activity minimum, but also plays an important role even during the active phase of the solar cycle. The magnetic field in the quiet Sun is composed of the network (Sheeley 1967), internetwork (IN, Livingston & Harvey 1971, 1975), and the ephemeral regions (Harvey & Martin 1973). For an overview of the small-scale magnetic features, see Solanki (1993), de Wijn et al. (2009), Priest (2014), and Wiegelmann et al. (2014).

The IN features are observed within the supergranular cells and carry hecto-Gauss fields (Solanki et al. 1996, Khomenko et al. 2003, Martínez González et al. 2008b) although kilo-Gauss fields have also been observed in the IN (Lagg et al. 2010, 2016). They evolve as unipolar and bipolar features with typical lifetimes of less than 10 minutes (Zhou et al. 2010; Lamb et al. 2013; Anusha et al. 2017), i.e., they continuously bring new flux to the solar surface, flux that has been either freshly generated, or recycled. They carry fluxes $\leqslant {10}^{18}$ Mx, with the lower limit on the smallest flux decreasing with increasing spatial resolution and polarimetric sensitivity of the observing instruments, although the identification technique also plays an important role.

Ephemeral regions are bipolar magnetic features appearing within the supergranular cells carrying fluxes $\approx {10}^{19}$ Mx (Chae et al. 2001; Hagenaar et al. 2003) and are much longer-lived compared to the IN features, with lifetimes of 3–4.4 hr (Title 2000; Hagenaar 2001). The ephemeral regions also bring new magnetic flux to the solar surface.

The network is more stable, with typical lifetimes of its structure of a few hours to a day, although the individual kG magnetic elements within the network live for a much shorter time, as the entire flux within the network is exchanged within a period of 8–24 hr (Hagenaar et al. 2003; Gošić et al. 2014). The flux in the network is fed by ephemeral regions (Schrijver et al. 1997; Hagenaar 2001) and IN features (Gošić et al. 2014). The network features are found along the supergranular boundaries and carry fields of kG strength with a typical flux of 10¹⁸ Mx (Wang et al. 1995).

The magnetic flux is produced by a dynamo, the location of which is currently the subject of debate, as is whether there is only a single dynamo acting in the Sun (e.g., Stenflo 2012) or whether there is a small-scale dynamo acting in addition to a global dynamo (Petrovay & Szakaly 1993; Cattaneo 1999; Cattaneo & Hughes 2001; Vögler & Schüssler 2007; Schüssler & Vögler 2008; Danilovic et al. 2010; Buehler et al. 2013; Hotta et al. 2015; Karak & Brandenburg 2016). In addition, it is unclear if all the magnetic flux appearing on the Sun is actually new flux produced by a dynamo, or possibly recycled flux transported under the surface to a new location, where it appears again (e.g., Ploner et al. 2001). This may be particularly important at the smallest scales.

An important parameter constraining the production of magnetic flux is the amount of magnetic flux appearing at the solar surface. In particular, the emergence of magnetic flux at very small scales in the quiet Sun provides a probe for a possible small-scale dynamo acting at or not very far below the solar surface. The deep minimum between solar cycles 23 and 24 offered a particularly good chance to study such flux emergence, as the long absence of almost any activity would suggest that most of the emerging flux is newly produced and not transported from decaying active regions to the quiet Sun (although the recycling of some flux from ephemeral regions cannot be ruled out).

The IN quiet Sun displays by far the largest magnetic flux emergence rate (FER). Zirin (1987) pointed out that two orders of magnitude more flux appears in ephemeral regions than in active regions, while the FER in the IN is another two orders of magnitude larger. This result is supported by more recent studies (e.g., Socas-Navarro & Sánchez Almeida 2002; de Wijn et al. 2009; Parnell et al. 2009; Thornton & Parnell 2011). Given the huge emergence rate of the magnetic flux in the IN, it is of prime importance to measure the amount of flux that is brought to the surface by these features.

The current estimates of the FER in the IN vary over a wide range, which include: ${10}^{24}\,\mathrm{Mx}\,{\mathrm{day}}^{-1}$ (Zirin 1987), $3.7\times {10}^{24}\,\mathrm{Mx}\,{\mathrm{day}}^{-1}$ ($120\,\mathrm{Mx}\,{\mathrm{cm}}^{-2}\,{\mathrm{day}}^{-1}$, Gošić et al. 2016) and $3.8\times {10}^{26}\,\mathrm{Mx}\,{\mathrm{day}}^{-1}$ (Zhou et al. 2013). By considering all the magnetic features (small-scale features and active regions), Thornton & Parnell (2011) measure a global FER of $3\times {10}^{25}\,\mathrm{Mx}\,{\mathrm{day}}^{-1}$ ($450\,\mathrm{Mx}\,{\mathrm{cm}}^{-2}\,{\mathrm{day}}^{-1}$), while Thornton (2011) measures $3.9\times {10}^{24}\,\mathrm{Mx}\,{\mathrm{day}}^{-1}$ ($64\,\mathrm{Mx}\,{\mathrm{cm}}^{-2}\,{\mathrm{day}}^{-1}$), whereby almost all of this flux emerged in the form of small IN magnetic features. The FER depends on the observations and the method used to measure it. A detailed comparison of the FERs from different works is presented Section 4.3.

To estimate the FER, Zirin (1987) and Thornton & Parnell (2011) considered features with fluxes $\geqslant {10}^{16}$ Mx, while Zhou et al. (2013) and Gošić et al. (2016) included features with fluxes as low as $6\times {10}^{15}$ Mx and $6.5\times {10}^{15}$ Mx (M. Gošić 2017, private communication), respectively. However, with the launch of the balloon-borne Sunrise observatory in 2009 (Solanki et al. 2010; Barthol et al. 2011; Berkefeld et al. 2011; Gandorfer et al. 2011) carrying the Imaging Magnetograph eXperiment (IMaX, Martínez Pillet et al. 2011), it has now become possible to estimate the FER including the contribution of IN features with fluxes as low as 9 × 10¹⁴ Mx (Anusha et al. 2017, hereafter referred to as LSA17). The IMaX instrument has provided unprecedented high-resolution magnetograms of the quiet Sun observed at 5250 Å. The high resolution is the main reason for the lower limiting flux. A detailed statistical analysis of the IN features observed in Stokes V recorded by Sunrise/IMaX is carried out in LSA17. In the present paper we estimate the FER in the IN region using the same data.

In Section 2 we briefly describe the employed Sunrise data. The IN features bringing flux to the solar surface that are considered in the estimation of FER are outlined in Section 3. The FER from Sunrise are presented, discussed, and compared with previously obtained results in Section 4 while our conclusions are presented in Section 5.

The data used here were obtained during the first science flight of Sunrise described by Solanki et al. (2010). We consider 42 maps of the line-of-sight (LOS) magnetic field, ${B}_{\mathrm{LOS}}$, obtained from sets of images in the four Stokes parameters recorded with the IMaX instrument between 00:36 and 00:59 UT on 2009 June 9 at the solar disk center, with a cadence of 33 s, a spatial resolution of $0\buildrel{\prime\prime}\over{.} 15\mbox{--}0\buildrel{\prime\prime}\over{.} 18$ (plate scale is $0\buildrel{\prime\prime}\over{.} 054$ per pixel), and an effective field of view (FOV) of $43^{\prime\prime} \times 43^{\prime\prime}$ after phase diversity reconstruction. The data were reconstructed with a point-spread function determined by in-flight phase diversity measurements to correct for the low-order aberrations of the telescope (defocus, coma, astigmatism, etc., see Martínez Pillet et al. 2011). The instrumental noise of the reconstructed data was 3 × 10⁻³ in units of continuum intensity. For identification of the features, spectral averaging was done which further reduced the noise to $\sigma =1.5\times {10}^{-3}$. All features with signals above a $2\sigma$ threshold, which corresponds to 12 G, were used (Martínez Pillet et al. 2011).

To measure the flux, we use ${B}_{\mathrm{LOS}}$ determined with the center-of-gravity (COG) method (Rees & Semel 1979; Orozco Suárez et al. 2010; LSA17). The inclination of the IN fields has been under debate, with several studies dedicated to measuring their angular distribution. By analyzing the data from Hinode, Orozco Suárez et al. (2007) and Lites et al. (2008) concluded the IN fields to be predominantly horizontal. However, using the same data set, Ishikawa & Tsuneta (2011) found some of the IN fields to be vertical. Jafarzadeh et al. (2014) arrived at similar conclusions (vertical inclination) by analyzing the magnetic bright points observed from the first flight of Sunrise. Variations in the inclination of the IN fields with heliocentric angle (μ) have been reported by Orozco Suárez & Bellot Rubio (2012) and Borrero & Kobel (2013), Stenflo (2013). Isotropic and quasi-isotropic distribution of the IN field inclinations is favored by Martínez González et al. (2008a), using the Fe i $1.56\,\mu {\rm{m}}$ infrared lines, and by Asensio Ramos (2009) and Asensio Ramos & Martínez González (2014) using Hinode data. More recently, Danilovic et al. (2016) found the distribution of IN field inclination to be quasi-isotropic by applying 2D inversions on Hinode data and comparing them with 3D magnetohydrodynamic simulations. For a detailed review see Borrero et al. (2015).

In the determination of the FER, we use ${B}_{\mathrm{LOS}}$ for consistency and for easier comparison with earlier studies on the FER. Also, the determination of the exact amount of flux carried in horizontal field features is non-trivial and requires estimates of the vertical thickness of these features and the variation of their field strength with height. In addition, if they are loop-like structures (as is suggested by local-dynamo simulations, e.g., Vögler & Schüssler 2007), then there is the danger of counting the flux multiple times if one or both of their footpoints happen to be resolved by the Sunrise data. We avoid this by considering only the vertical component of the magnetic field. It is likely that we miss the flux carried by unresolved magnetic loops by concentrating on Stokes V, but this problem is suffered by all previous studies of FER and should decrease as the spatial resolution of the observations is increased. With the Sunrise I data analyzed here having the highest resolution, we expect them to see more of the flux in the footpoints of the very small-scale loops that appear as horizontal fields in Hinode and Sunrise data (Danilovic et al. 2010).

The small-scale features were identified and tracked using the feature tracking code developed in LSA17. For the sake of completeness we summarize the most relevant results from LSA17 as follows. All the features covering at least 5 pixels were considered with Stokes V being larger than $2\sigma$ in each pixel. To determine the flux per feature, the ${B}_{\mathrm{LOS}}$ averaged over the feature, denoted as $\langle {B}_{\mathrm{LOS}}\rangle$ was used. $\langle {B}_{\mathrm{LOS}}\rangle$ had values up to 200 G, even when the maximum field strength in the core of the feature reached kG values. A total of 50,255 features of both polarities were identified. The sizes of the features varied from 5 to 1585 pixels, corresponding to an area range of $\approx 8\times {10}^{-3}\mbox{--}2.5$ Mm². The tracked features had lifetimes ranging from 0.55 to 13.2 min. The smallest detected flux of a feature was 9 × 10¹⁴ Mx and the largest $2.5\times {10}^{18}$ Mx.

At the time of the flight of Sunrise in 2009 the Sun was extremely quiet, with no signs of activity on the solar disk. Two sample magnetograms at 00:47 UT and 00:58 UT are shown in Figure 1. Most of the features in these maps are part of the IN and in this paper, we determine the rate at which they bring flux to the solar surface.

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The different processes increasing the magnetic flux at the solar surface are schematically represented in Figures 2(a)–(d). In the figure, f_i refers to the flux of the feature at its birth and f_m is the maximum flux that a feature attains over its lifetime. A typical evolution of the flux of a feature born by unipolar or bipolar appearances, and by splitting/merging is shown in Figure 2(e) (top and bottom, respectively). The gain in the flux of a feature after its birth is ${f}_{m}-{f}_{i}$. Magnetic flux at the surface increases through the following processes:

• 1.  
  Unipolar appearance: birth of an isolated feature with no spatial overlap with any of the existing features in the current and/or previous time frame (Figure 2(a)).
• 2.  
  Bipolar appearance: birth of bipolar features, with the two polarities closely spaced, and either appearing simultaneously or separated by a couple of time frames (referred to as time symmetric and asymmetric emergence in LSA17; see also Figure 2(b)).
• 3.  
  Flux gained by features in the course of their lifetime: the gain in the flux of a feature in the course of its lifetime, i.e., the increase in flux between its birth and the time it reaches its maximum flux, before dying in one way or another, either by interacting with another feature, or by disappearing.This gain can take place in features born in different ways, be it by growth, or through the merging or splitting of pre-existing features (Figures 2(c)–(e)).

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Note that the bipolar appearance of magnetic flux is often referred to as "emergence" in earlier papers including LSA17. However, the term "emergence" in FER describes the appearance of new flux at the solar surface from all three processes described above. To avoid confusion, we refer to the emergence of bipolar features as bipolar appearance. Of all the newly born features over the entire time series, 19,056 features were unique (for area ratio 10:1, see Section 4). Among them 48% (8728 features) were unipolar and 2% (365 features) were part of bipolar appearances. Features born by splitting constituted 38% (6718 features), and 12% (2226 features) were born by merging. The remaining 1019 features correspond to those alive in the first frame. A comparison of the rates of birth and death of the features by various processes for different area ratio criteria is given in Table 2 of LSA17.

In the FER estimations, the flux brought by the features born by unipolar and bipolar appearances is the maximum flux that they attain (f_m) over their lifetime. In the case of features born by splitting or merging, the flux gained after birth is taken as the flux brought by them to the surface. This gain is the difference between their flux at birth f_i and the maximum flux they attain f_m, i.e., ${f}_{m}-{f}_{i}$.

Our approach is conservative in the sense that if a feature reaches multiple peaks of flux in the course of its lifetime, as in the example shown in Figure 3, then we consider only the largest one (the flux gained during the first peak in Figure 3), and neglect increases in flux contributing to smaller peaks such as the second and third peaks in Figure 3. Multiple peaks in the flux of a feature (shown in Figure 3) are rarely seen, as most features do not live long enough to display them (see LSA17).

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Changes in the flux of a feature in the course of its lifetime can cause it to seemingly appear and disappear with time if its total flux is close to the threshold set in the study (given by a signal level twice the noise in at least five contiguous pixels). If it disappears and reappears again, then it will be counted twice. This introduces uncertainties in the measurement of the FER. Uncertainties are discussed in Section 4.2.

4.1. Flux Emergence Rate

In this work, we consider the results from the area ratio criterion 10:1 of LSA17. In that paper, the authors devise area ratio criteria (10:1, 5:1, 3:1 and 2:1) to avoid that a feature dies each and every time that a tiny feature breaks off, or merges with it. For example in a splitting event, the largest of the features formed by splitting must have an area less than n times the area of the second largest, under the n:1 area ratio criterion. We have verified that the choice of the area ratio criterion does not drastically alter the estimated FER, with variations being less than 10% for area ratios varying between 10:1 and 2:1.

The instantaneous and maximum fluxes of the features in different processes are given in Tables 1–5 of LSA17. A summary is repeated in Table 1 for convenience, where fluxes are given for features born by the four processes listed in the first column. The instantaneous flux, in the second column, refers to the flux of a feature at its birth (f_i in Figure 2). In the third column is the maximum flux of a feature during its lifetime (f_m in Figure 2). The flux gain in the fourth column is the difference of the second and third columns (${\rm{\Delta }}f={f}_{m}-{f}_{i}$). In the fifth column is the factor by which the flux increases from its birth to its peak (${f}_{m}/{f}_{i}$). The fluxes given in this table are the sum total over all features in the entire time series, for each process.

Table 1.  Instantaneous and Maximum Fluxes of the Features, Integrated over All Features and T Frames, in Different Processes Measured in LSA17

Process	Instantaneous Flux	Maximum Flux	Flux Gain	Factor of Increase
	(fi in Mx)	(fm in Mx)	(${\rm{\Delta }}f={f}_{m}-{f}_{i}$ in Mx)	(${f}_{m}/{f}_{i}$)
Unipolar appearance	$4.69\times {10}^{19}$	$9.69\times {10}^{19}$	$4.99\times {10}^{19}$	2.06
Splitting	$1.76\times {10}^{20}$	$2.12\times {10}^{20}$	$3.60\times {10}^{19}$	1.20
Merging	$1.64\times {10}^{20}$	$1.85\times {10}^{20}$	$2.20\times {10}^{19}$	1.31
Bipolar appearance	$3.85\times {10}^{18}$	$9.53\times {10}^{18}$	$5.67\times {10}^{18}$	2.47

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To compute the FER, we add the fluxes from the various processes described in the previous section. For features born by appearance (unipolar and bipolar), we take their maximum flux (f_m) to be the fresh flux emerging at the surface. For features formed by merging or splitting only the flux increase after birth (${\rm{\Delta }}f$) is considered. From the first two processes alone, the total flux brought to the surface is $1.1\times {10}^{20}$ Mx over an FOV of 43'' × 43'' in 22.5 minutes. This gives an FER of $700\,\mathrm{Mx}\,{\mathrm{cm}}^{-2}\,{\mathrm{day}}^{-1}$. Including the flux gained by split/merged features increases the FER to $1100\,\mathrm{Mx}\,{\mathrm{cm}}^{-2}\,{\mathrm{day}}^{-1}$. Figure 4 shows the contribution from each process to the total FER. The isolated features appearing on the solar surface contribute the largest, nearly 60%. Given that the emerging bipoles contain only 2% of the total observed flux (Table 5 in LSA17), they contribute only about 5.7% to the FER.

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However, the flux brought to the solar surface by features born by splitting or merging after their birth is 5 × 10¹⁹ Mx, which is quite significant and contributes $\approx 35 \%$ to the FER. The contribution to solar surface flux by this process is comparable to the flux brought to the surface by features born by unipolar appearance ($9.7\times {10}^{19}$ Mx) and nearly an order of magnitude higher than that from features born by bipolar appearance ($9.5\times {10}^{18}$ Mx).

Over their lifetimes, the features born by splitting and by merging gain 1.2 times their initial flux (i.e., ${f}_{m}=1.2\times {f}_{i}$). The fluxes gained by features born by appearance relative to their flux at birth is slightly higher ($\approx 2$ times, i.e., ${f}_{m}=2\times {f}_{i}$). This is because the initial magnetic flux of the features born by appearance is quite low. The flux at birth of split or merged features is already quite high because the parent features which undergo splitting or merging are at later stages in their lives (see Figure 2(e)). This is also evident from the fact that the average initial flux per feature of the features born by splitting or merging ($2.9\times {10}^{16}$ Mx and $7.4\times {10}^{16}$ Mx, respectively) is an order of magnitude higher than the average initial flux per feature of the appeared unipolar or bipolar features ($5.4\times {10}^{15}$ Mx, see Table 2 of LSA17).

The fact that the small-scale magnetic features are the dominant source of fresh flux in the quiet photosphere is discussed in several publications (Socas-Navarro & Sánchez Almeida 2002; de Wijn et al. 2009; Parnell et al. 2009; Thornton & Parnell 2011). Our results extend these earlier findings to lower flux per feature values. As shown in Figure 5, over the range ${10}^{15}\mbox{--}{10}^{18}$ Mx, nearly 65% of the detected features carry a flux $\leqslant {10}^{16}$ Mx (left panel). They are also the dominant contributors to the FER (right panel). In this figure, only the features that are born by unipolar and bipolar appearances are considered. Below 2 × 10¹⁵ Mx, we see a drop as we approach the sensitivity limit of the instrument.

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4.1.1. Flux Loss Rate

Flux is lost from the solar surface by disappearance, cancellation of opposite polarity features, and decrease in the flux of the features in the course of their evolution (i.e., the opposite process to the "flux gain" described earlier in Section 3). As seen from Tables 3 and 4 of LSA17, the increase in flux at the solar surface balances the loss of flux, as it obviously must if the total amount of flux is to remain unchanged. To compute the flux loss rate, we take the maximum flux of the features that die by cancellation and by disappearance to be the flux lost by them in the course of their lifetime and during disappearance or cancellation. For the features that die by splitting/merging we take the difference between the maximum flux of the features and the flux at their death as a measure of the flux lost during their lifetimes. By repeating the analyses for the 10:1 area ratio criterion, we find that the flux is lost from the solar surface at a rate of $1150\,\mathrm{Mx}\,{\mathrm{cm}}^{-2}\,{\mathrm{day}}^{-1}$ which corresponds within 4.5% to the obtained FER. This agreement serves as a consistency check of the FER value that we find.

4.2. Uncertainties

4.3. Comparison with Previous Studies

In this paper, we have estimated the FER in the quiet Sun from the IN features using the observations from Sunrise/IMaX recorded during its first science flight in 2009. We have included the contribution from features with fluxes in the range $9\times {10}^{14}\mbox{--}2.5\times {10}^{18}$ Mx, whose evolution was followed directly. By accounting for the three important processes that bring flux to the solar surface: unipolar and bipolar appearances, and flux gained after birth by features born by splitting or merging over their lifetime, we estimate an FER of 1100 $\mathrm{Mx}\,{\mathrm{cm}}^{-2}\,{\mathrm{day}}^{-1}$. The third process is found to contribute significantly to the FER. The smaller features with fluxes ≤10¹⁶ Mx bring most of the flux to the surface. Since our studies include fluxes nearly an order of magnitude smaller than the smallest flux measured from the Hinode data, our FER is also an order of magnitude higher when compared to studies using a similar technique (i.e., Gošić et al. 2016). We compare also with other estimates of the FER in the literature. Thornton & Parnell (2011) obtained a range of values. Those near the lower end of the range (also quoted by Thornton 2011), which are possibly the more reliable ones, are roughly consistent with the results of Gošić et al. (2016). The high FER of $3.8\times {10}^{26}\,\mathrm{Mx}\,{\mathrm{day}}^{-1}$ found by Zhou et al. (2013) is, however, difficult to reconcile with any other study. It is likely so high partly due to the excessively large ${B}_{\mathrm{LOS}}$ of IN fields of 12.4 G used by these authors, which is more than a factor of 4 times larger than the averaged ${B}_{\mathrm{LOS}}$ of 2.8 G that we find. Even the absolute upper limit of the spatially averaged ${B}_{\mathrm{LOS}}$ in our data (including noise) is below the value used by Zhou et al. (2013). We therefore expect that they have overestimated the FER.

There is clearly a need for further investigation, not only to better quantify the reasons for the different results obtained by different techniques. There are also still multiple open questions. What is the cause of the increase and decrease of the flux of a feature during its lifetime? Is this due to interaction with "hidden" flux? Is this hidden flux not visible because it is weak and thus below the noise threshold, or because it is structured at very small scales, i.e., it is below the spatial resolution? How strongly does the "hidden" or missed flux change with changing spatial resolution? The most promising approach to answering these and related questions is to study the flux evolution in a magnetohydrodynamic simulation that includes a working small-scale turbulent dynamo.

We thank F. Kahil for helping with the comparison of COG and inversion results. H.N.S. and L.S.A. acknowledge the financial support from the Alexander von Humboldt foundation. The German contribution to Sunrise and its reflight was funded by the Max Planck Foundation, the Strategic Innovations Fund of the President of the Max Planck Society (MPG), DLR, and private donations by supporting members of the Max Planck Society, which is gratefully acknowledged. The Spanish contribution was funded by the Ministerio de Economía y Competitividad under Projects ESP2013-47349-C6 and ESP2014-56169-C6, partially using European FEDER funds. The HAO contribution was partly funded through NASA grant number NNX13AE95G. This work was partly supported by the BK21 plus program through the National Research Foundation (NRF) funded by the Ministry of Education of Korea.