# The Sun’s small-scale magnetic elements in Solar Cycle 23

C. L. Jin, J. X. Wang, and Q. Song  Key Laboratory of Solar Activity of
Chinese Academy of Sciences
National Astronomical Observatories, Chinese Academy of Sciences, Beijing
100012, China<EMAIL_ADDRESS><EMAIL_ADDRESS>H. Zhao  National Tsing
Hua University, Hsinchu, Taiwan<EMAIL_ADDRESS>
###### Abstract

With the unique database from Michelson Doppler Imager aboard the Solar and
Heliospheric Observatory in an interval embodying solar cycle 23, the cyclic
behavior of solar small-scale magnetic elements is studied. More than 13
million small-scale magnetic elements are selected, and the following results
are unclosed. (1) The quiet regions dominated the Sun’s magnetic flux for
about 8 years in the 12.25 year duration of Cycle 23. They contributed (0.94 –
1.44) $\times 10^{23}$ Mx flux to the Sun from the solar minimum to maximum.
The monthly average magnetic flux of the quiet regions is 1.12 times that of
active regions in the cycle. (2) The ratio of quiet region flux to that of the
total Sun equally characterizes the course of a solar cycle. The 6-month
running-average flux ratio of quiet region had been larger than 90.0% for 28
continuous months from July 2007 to October 2009, which characterizes very
well the grand solar minima of Cycles 23-24. (3) From the small to large end
of the flux spectrum, the variations of numbers and total flux of the network
elements show no-correlation, anti-correlation, and correlation with sunspots,
respectively. The anti-correlated elements, covering the flux of (2.9 -
32.0)$\times 10^{18}$ Mx, occupies 77.2% of total element number and 37.4% of
quiet Sun flux. These results provide insight into reason for anti-correlated
variations of small-scale magnetic activity during the solar cycle.

Sun: magnetic fields — Sun: photosphere — Sun: sunspots

##  1 Introduction

No any other astrophysical process but solar cycle leaves massive footprints
on human’s living environment. This eleven-year cycle was discovered by a
Germany pharmacist Schwabe (1843) from the number changes of solar sunspots. A
primary understanding on the solar cycle has been established based on the
theories and simulations of a mean-field magnetohydrodynamic (MHD) dynamo
(Charbonneau 2005). However, new observations are continuously challenging our
understanding by the myriad of new and seemingly conflicting observations. A
more severe challenge comes from observations of small-scale magnetic elements
(see de Wijn et al. 2009). Therefore, it is still a difficult task to explore
the physics of solar cycle.

Since the 1960s it has been observed that small-scale magnetic fields outside
of sunspots are everywhere on the Sun (Sheeley 1966, 1967; Harvey 1971). The
stronger magnetic elements at the boundaries of supergranulation cells are
network elements, while the smaller and weaker elements within the
supergranulation cells are intra-network (IN) elements (Livingston & Harvey
1975; Smithson 1975). Similar to emerging flux regions (EFRs) in sunspot
groups (or active regions) (Zirin 1972), small-scale emerging bipoles named
ephemeral (active) regions (ERs) were described by Harvey and Martin (1973).
They account for the formation of network elements in addition to the debris
from decaying sunspots. It was noticed that the flux emerging rate in ER
exceeds that in sunspots by two orders of magnitude (Zirin 1987). Moreover,
flux generation rate of IN elements exceeds that of ERs by another two orders
of magnitude. Further smaller magnetic fibrils are believed to be mostly
unresolved by present telescopes, yet their aggregation is the dominant
mechanism by which IN and network elements appear (Lamb et al. 2008). A
substantial amount of solar magnetic flux is probably still hidden (Trujillo
et al. 2004).

As soon as the small-scale magnetic elements were identified, great efforts
have been made to understand how they change during a solar cycle and if they
are correlated with sunspots. Diverse observations are reported, igniting
discussions and debates in the literature. The observations are made either
directly from the magnetic measurements, or indirectly from proxies of small-
scale magnetic flux, e.g., the G-band and CaII K bright points and coronal
X-ray bright points. Key revelations are listed below.

  1. (1)

No cyclic variations: CaII K emission in solar quiet regions (White &
Livingston 1981); modern X-ray bright points observations (Sattarov et al.
2002; Hara & Nakakubo 2003); magnetic flux of networks (Labonte & Howard
1982); flux spectrum and total flux of network elements with flux $\leq
2.0\times 10^{19}$ Mx (Hagenaar et al. 2003); Stokes $\frac{Q}{I}$ profile
(Trujillo et al. 2004).

  2. (2)

Anti-correlation of small-scale fields with sunspot cycle: number of network
bright points in very quiet regions (Muller & Roudier 1984, 1994); HeI 10830 Å
dark points in the higher chromosphere (Harvey 1985); early X-ray bright
points observations (Davis et al. 1977; Davis 1983; Golub et al. 1979); Weak
changes of emergence frequency of ERs with flux less than (3-5)$\times
10^{19}$ Mx (Hagenaar et al. 2003).

  3. (3)

Correlation with sunspot cycle: more ERs appeared during active solar
condition (Harvey & Harvey 1974; Harvey 1989); the number (or magnetic flux)
of network structures (Foukal et al. 1991; Meunier 2003); flux distribution
and total flux of network concentrations with flux $(2.0-3.3)\times 10^{19}$
Mx (Hagenaar et al. 2003).

The observations listed above are related to some fundamental, but not yet
resolved questions in solar physics: the origin, dynamics and active role in
Sun’s global processes of solar small-scale magnetic elements, as well as the
controlling physics of solar activity cycle. However, discrepancy among
different authors is not yet understood, implying problems either in the
observations or on the physics used to interpret the observations. A few
aspects make things even more complicated.

First, for the observations of the proxies of small-scale magnetic elements,
the connections between the magnetic elements and their proxies are not well
quantified, and the underlined physics is not known exactly. There seems to be
not a one-to-one correspondence between network elements and network bright
points (Zhao et al. 2009). In other words, the widely-adopted paradigm of
“magnetic bright points” is still questionable. Moreover, the early revelation
about the magnetic properties of coronal X-ray bright points (Golub et al.
1977), needs to be revisited and updated with state-of-the-art observations.

Secondly, quite many reports listed above went back to early solar
observations, which makes us difficult to evaluate the quality of the
observations. We are confused by rather poor resolution, calibration and
consistency in sensitivity in early magnetic measurements. As an example, the
early Mont. Wilson magnetograph observations (Labonte & Howard 1982) were with
a resolution of $\geq$ 12.5-17.5 arcsec, and the calibration was not
consistent time to time. We are simply not able to say anything confidently
about their conclusions. Additionally, early X-ray bright point measurements,
which suffered from low cadence, purported to show a decrease in the number of
X-ray bright points with the solar cycle. More recent higher cadence
observations have called into question whether this effect is real. It reminds
to be seen whether other observations of variation with the solar cycle also
need to be reinterpreted. New observations with careful and thorough data
reduction and interpretation are crucially required.

Thirdly, even for recent observations, sometimes, the different algorithm and
logic in data analysis make us hard to judge the results too. An interesting
example comes from the analysis of full-disk magnetograms of the Michelson
Doppler Imager aboard the Solar and Heliospheric Observatory (MDI/SOHO)
(Scherrer et al. 1995). By adopting the different detection algorithm and
approaches, Meunier (2003) revealed correlation of the network element number
(or flux) with sunspots; in contrast, Hagenaar et al. (2003) declared some
weak anti-correlated emergence rate of ERs and an independence of the total
absolute flux for smaller network concentrations. This discrepancy should be
clarified with new analysis.

To clarify the problem and to close the debates are an essential task in
understanding the solar cycle phenomena. Fortunately, now MDI/SOHO is
providing a unique database - the full-disk magnetograms over more than 13
years, covering the complete 23rd Solar Cycle. The 13.5 year 5-min average
full disk magnetograms are used in the current study. However, the poor
temporal resolution makes the identity of ERs questionable and the sensitivity
of the full-disk magnetograms rules out the possibility to resolve the IN
elements. Therefore, what we have identified in this study is basically the
network magnetic elements.

In this paper, we aim at learning the cyclic variations of quiet Sun’s
magnetic flux and small-scale magnetic elements. To use the full-disc MDI
magnetograms with the temporal coverage of entire Cycle 23 comes from an
awareness of the intermittency of solar cyclic behavior in both the temporal
and spatial domains. By the intermittency to select the magnetograms of a
short interval, e.g., 10-30 hours, in a month for each year, at the ‘supposed’
different cycle phases. would not guarantee a grape of the key characteristics
of a solar cycle. From our understanding, to choose the database that cover
the entire cycle 23 is of overwhelming importance. The database for the
current study is unique in the sense that it is the only space-borne magnetic
measurements of the full Sun, for which the consistency in sensitivity and
resolution persisted for a cycle-long interval. As we are interested in the
global behavior of small-scale magnetic elements, sampling network elements in
a cycle-long temporal domain and in all different flux ranges (or strengths)
are more important than selecting a few high cadence sequences interruptedly.
Moreover, the magnetic elements with different flux (or size) may have
different origins and characteristics, therefore we group all the network
magnetic elements into different categories in accordance with their magnetic
flux.

In section 2, we describe the observations, the technique of calibration, the
evaluation of noise level of the magnetograms, the separation of active
regions and the quiet Sun, and the selection of network elements. In section
3, we present the results of cyclic behavior of quiet region magnetic flux and
small-scale magnetic elements. In section 4, we make the comparison with
previous studies, and consider the possibilities on how to understand the
anti-correlated network magnetic elements with sunspots. In section 5, we draw
the conclusions.

##  2 Observations and methods

The MDI instrument aboard SOHO spacecraft provides the full-disk magnetogram
with a pixel size of 2”. In order to obtain a low noise level, only those
5-min average magnetograms are selected in the study. We extract one observed
full-disk magnetogram per day, and thusly we totally select 3764 magnetograms
from 1996 September to 2010 February, which include the complete 23rd solar
cycle. In order to further reduce the noise level, we apply a boxcar smoothing
function to each magnetogram by a width of 6”$\times$6”. There are two groups
of authors who first pointed out the under-estimation of magnetic flux by
earlier MDI full-disk magnetogram calibration (Berger et al. 2003; Wang et al.
2003). All the magnetograms used in this study are that retrieved after
recalibration of December 2008. For a better understanding about the cyclic
behavior of solar minima of Cycles 22 and 23, we extend the MDI data base by
adding Kitt Peak full-disk magnetograms from August 1996 back to January 1994.
The data merging is made based on a least-square fitting of the mean flux
density of Kitt Peak magnetograms to that of MDI magnetograms for the common
interval of 1996.

We estimate the noise level of these smoothed 5-min average magnetograms
according to the method described by Hagenaar (2001) and Hagenaar et al.
(2003). Based on these magnetograms, we analyze their histograms of magnetic
flux density. The core of the distribution function is fitted by a Gaussian
function $F(x)=F_{max}exp(-x^{2}/2\sigma^{2})$, where the width $\sigma$ of
the Gaussian function, about 6 Mx/cm2, is defined as the noise level.

We assume that the observed line of sight magnetic flux density is a
projection of the intrinsic flux density normal to the solar surface, so the
magnetic flux density for each pixel is corrected(see Hagenaar 2001 and
Hagenaar et al. 2003) as $B_{cal}=B_{obs}(\alpha)/\cos(\alpha)$. The angle
$\alpha$ of each pixel is defined by $\sin(\alpha)=\sqrt{x^{2}+y^{2}}/R$ Where
x and Y are the pixel position referring to the disk center, at which x and y
is equal to 0, and the R is the solar disk radius. After the correlation, the
magnetogram shows the magnetic flux density normal to solar surface.

After the angle is greater 60 degrees, there are less and less magnetic
signals due to the lower magnetic sensitivity and spatial resolution of
MDI/SOHO magnetograms, and the magnetic noise level would increase according
to the magnetic correction 1/cos($\alpha$). Therefore, we only analyze these
pixels with angle $\alpha$ less than 60 degree, i.e., the region included by
the black circle in the left panels of Fig. 1. The flux density of the pixel
with 60o $\leq$ $\alpha$ $\leq$ 90o is set to zero.

For each smoothed and corrected full-disk magnetogram of MDI/SOHO, we apply a
magnetic flux density of 15 Mx/cm2 as a threshold to define the active regions
and their surroundings, and then create a mask for each magnetogram. These
masks include many small clusters and isolated pixels, so only the islands
with area larger 9$\times$9 pixels are defined as the active regions (Hagenaar
et al. 2003). Considering the active regions close to the edge of 60 degree,
in order to avoid missing them in the automatic procedure, we always search
the active regions in the solar disk with angle $\alpha$ less than 70 degree
first, as that shown in the left panels of Fig. 1. Thusly, the islands with
area less than 81 pixels within 60 degree disk are still defined as the active
regions if they have more than 81 pixels searched within 70 degree disk.

Two magnetograms within the 70$\deg$ from disk center at approximately the
solar maximum and minimum phases, respectively, are displayed in the left
panels of Fig. 1. On these retrieved magnetograms the selected ARs are masked
by red curves. The criterion of selecting ARs appears to work well from a
visually examination for the given cases. In the right panels two selected
sub-windows of the magnetograms are shown with contours outlining the network
elements which are selected by a procedure of automatic feature selection. The
yellow and green contours outline the selected network elements that are
belong to the components of correlated and anti-correlated with sunspots in
the solar cycle, respectively (see Section 3.2).

##  3 Results

###  3.1 Cyclic variations of magnetic flux of solar quiet regions

In order to compare the cyclic variations of magnetic flux of active regions
with that of quiet regions, we calculate their magnetic flux, respectively,
which is shown in the left panel of Fig. 2. At the same time, the area ratio
of quiet regions is also computed, which is shown by purple ‘+’ symbols in the
right panel of Fig. 2. It is found that the quiet Sun contributed
$(0.94-1.44)\times 10^{23}$ Mx flux from approximately the solar minimum to
maximum in Cycle 23. The fractional area of quiet regions always exceeds 80%
in the entire solar cycle 23, and decreased from the cycle minimum to maximum
by a factor of 1.2, although their total flux increased by a factor of 1.53;
as a comparison, the active region flux increased by several orders of
magnitude. The measurements confirm the global behavior of the quiet Sun
fields (see Meunier 2003 and Hagenaar et al. 2003). During the 12.25 years of
Cycle 23, from October 1996 to December 2008 (see http://www.ips.gov.au), the
quiet Sun dominated the Sun’s magnetic flux for 7.92 years. The monthly
average magnetic flux of quiet Sun is 1.12 times that of active regions. The
magnetic fields on the quiet Sun, indeed, are a fundamental component of the
Sun’s activity cycle which maintains the Sun’s magnetic energy and Poynting
flux at a certain level.

It is interesting to notice that the ratio of the quiet Sun’s magnetic flux to
solar total flux (referring to as the flux occupation by the quiet Sun)
equally characterizes the course of a solar cycle, like sunspots. The
occupation of 6-month running-average magnetic flux by the quiet Sun is shown
by purple cross symbols in the right panel of Fig. 2. The active region flux
shown in the left panel answers for the variation of sunspot cycle very well.
However, for the quiet regions, the maximum occupation of magnetic flux marks
the minima of solar cycles. For instance, in our data set, the maximum flux
occupation of quiet Sun, which was 96.0%, first happened in October of 1996 at
the beginning of Cycle 23\. The later maxima happened from July 2008 to August
2009. In December of 2008, the beginning of Cycle 24, the maximum occupation
reached 99.3%. The 6-month running-average fractional flux of quiet Sun had
been larger than 90.0% for 28 continuous months (from July 2007 to October
2009), which characterizes the grand solar minima of Cycles 23-24. Staying at
such a low activity level there were 25 months, for which the total AR flux
was less than $10^{22}$ Mx. However, during the minima of Cycles 22-23 for
only intermittent 7 months, i.e., from December 1995 to April 1996 and from
December 1996 to January 1997, we had witnessed the fraction larger than
90.0%. The distinction between two solar minima are so severe, which can be
seen very clearly in Fig. 2.

###  3.2 Cyclic variations of network magnetic elements

After excluding the active regions, we apply the magnetic noise, i.e., 6
Mx/cm2 as a threshold to create a mask for each quiet magnetogram, and define
these magnetic concentrations with more than 10 pixels in size as network
magnetic elements (Hagenaar et al. 2003). More than 13 million network
elements have been identified for the interval from September 1996 to February
2010. The probability distribution function (PDF) of these magnetic elements
in the studied interval is shown in Fig. 3 as the average flux distribution.
From the figure, it can be found that the distribution of magnetic flux of
network magnetic elements mainly concentrate at the flux of 1019 Mx. This peak
distribution is consistent with that found for multiple MDI full-disk datasets
by parnel et al. (2009) (see their Fig. 5)

For an exclusive examination, we divide all the magnetic elements into 96 sub-
groups according to the flux per element. In this way, a statistical sample is
created, covering the range of magnetic flux per element from the smallest
observable network flux of $1.5\times 10^{18}$ Mx for the current data set to
an upper limit of $3.8\times 10^{20}$ Mx. The monthly variation of magnetic
elements for each sub-group is calculated and examined in term of number
density and absolute total flux in the interval from October 1996 to February
2010, embodying the entire Cycle 23. The influence of the area changes of the
quiet Sun on both quantities has been removed. There are 0.3% of network
elements (or clusters) with flux larger than the upper limit, which were
fragments of decayed sunspots and not included in the sample. By following
Hagenaar et al. (2003) tiny flux pieces with less than 10 pixels are not
considered in the study. As a whole the total flux of these tiny flux pieces
showed a small variation in the scope of $(3.5-4.0)\times 10^{22}$ Mx during
the cycle.

The correlation coefficients between the cyclic variation of numbers of
network elements and sunspots are calculated for each sub-group of network
elements and shown in Fig. 4. They are the linear Pearson correlation
coefficients of two vectors for each sub-group elements. Denote the element
number in sub-group $i$ as $N_{i}$ and the sunspot number $N_{s}$, then the
correlation coefficient between $N_{i}$ and $N_{s}$ will be

$\rho(N_{i},N_{s})=Covariance(N_{i},N_{s})/(Variance{N_{i}}\times
Variance{N_{s}})^{\frac{1}{2}}$ (1)

The confidence level about the correlation can be found in some basic
statistics handbook by taking account of how big was of the sample. As the
sample size for each sub-group elements is 162, which is quite large. If the
coefficient is higher than 0.256, then the failure probability of the linear
correlation would already be $<0.001$.

From the small to the large flux spectrum, there appears a remarkable 3-fold
correlation scheme between the network elements and the sunspots: basically
no-correlation, anti-correlation and correlation. This behavior is held for
both the element number and total flux. Either the anti-correlation or the
correlation has been observed at very high confidence level. The majority of
the correlations show a failure probability $\leq 0.001$. Between the anti-
correlation and correlation, there is a narrow range of magnetic flux per
element of (3.2 - 4.3)$\times 10^{19}$ Mx. Network elements falling in this
flux range show a transition from anti-correlation to correlation with sunspot
cycle (see the narrow shaded column in the middle of Fig. 4.)

The dependence of the correlation coefficient on the element flux hints the
possibility that network elements at different segments of the flux spectrum
may present different physical origins and different cyclic behavior
accordingly. For an detailed examination of cyclic behavior of network
elements, we group all the network elements into 4 categories which show,
respectively, no-correlation, anti-correlation, transition from anti-
correlation to correlation, and correlation with sunspot cycle. For each
category, its flux range, percentage in number and in total flux, as well as
the correlation coefficient with sunspots are listed in Table 1. We, then,
discriminate the cyclic variation of magnetic elements in accordance to the
flux range listed in the table. The detailed cyclic variations of each
category network elements are shown in Fig. 5.

Approximate 77.2% of the magnetic elements, covering the flux range of
(2.9-32.0)$\times 10^{18}$ Mx show anti-correlation with the sunspot cycle.
This anti-correlated component contributes 37.4% of network flux during Cycle
23. Transition from anti-correlation to correlation takes place between (3.20
- 4.27) $\times 10^{19}$ Mx. The correlated component elements have magnetic
flux larger than 4.27$\times 10^{19}$ Mx. They occupy approximately 15.7% in
number but 53.5% of total flux of network elements. In the flux range of
(1.5-2.9)$\times 10^{18}$ Mx, network elements show randomly independent
variation with the sunspot cycle. From this data set, they occupy less than
0.6% of network elements and have neglectable total flux. With the poor
sensitivity in flux measurements at the smallest end of the flux spectrum, it
could not be excluded that the non-correlation component manifested some
random noises in flux measurement. More serious efforts with higher resolution
and sensitivity data are necessary to clarify the cyclic behavior of smallest
observable magnetic elements.

The number changes of the network elements in the flux range of (2.9 \-
32.0)$\times 10^{18}$ Mx show obviously anti-phase correlation with sunspot
cycle, so do the changes of their total unsigned flux. However, the cyclic
minimum of this anti-correlation component is not exactly coincided with the
reversed profile of the maximum of the sunspot cycle, implying complexity in
causing the anti-correlation. Meanwhile, the flux changes of the magnetic
elements with flux larger than 4.3$\times 10^{19}$ Mx show remarkable in-phase
correlation with sunspot cycle. The same is true that the profiles of the
maximum of network elements and that of sunspots are not corresponding one
another exactly. There is a 5-7 month delay of their cyclic maximum related to
that of the sunspot cycle. This seems to be related to the characteristics
dispersal time of active region fields.

To further explore the cyclic variation of magnetic elements, we obtain the
PDFs of yearly network magnetic elements according to the magnetic flux, and
compute the differential PDFs, i.e., the difference between the yearly PDFs
and average PDF (see Fig.3). Here, the differential probability distribution
function is abbreviated as DPDF. We plot the DPDFs, and show the variation for
magnetic flux spectrum from 1996 to 2010 in Fig. 6. From the figure, we
confirm the 3-fold scenario of cyclic variations of network elements.

From the solar minimum to solar maximum (see the first column of the figure),
the distribution of magnetic elements in the flux range about (3-30)$\times
10^{18}$ Mx gradually decrease, which shows the anti-correlation variation
with the sunspot cycle; while the distribution of magnetic elements of flux
larger than about 4$\times 10^{19}$ shows the correlation variation with the
sunspot cycle and reaches the peak in the years 2000, 2001 and 2002.
Furthermore, the distribution of magnetic elements with flux of $\sim$
3$\times 10^{19}$ and less than 3$\times 10^{18}$ Mx shows almost no
variation. The distribution of magnetic elements correlated with the sunspot
cycle reaches the smallest values in the years 2007, 2008 and 2009, which are
the solar minima of cycle 23-24; while the distribution of magnetic elements
anti-correlated with the sunspot cycle shows outstanding peak during this long
interval (see the third column of the figure). The distribution characterizes
the long duration of the solar minima of Cycles 23-24. It is noticed that the
distributions in some of the ascending and declining phases (see that in 1998
and 2005) are, more or less, represent the average distribution of small-scale
magnetic elements shown in Fig.3.

##  4 Discussion

With the unique space-borne observations which comprised a complete solar
cycle, we have revealed a 3-fold correlation scheme of the Sun’s small-scale
magnetic elements with sunspot cycle, and identified an anti-correlation
component of network elements that dominates the element population. Before
coming up with conclusion and discussion on the physics, a comparison with
previous studies that adopted the similar approaches and with that same space-
borne MDI observations (see Section 2) is necessary.

Hagenaar et al.(2003) selected high cadence magnetograms of 6 time-sequences,
each of which covered 10-30 hours in a month from 1996 to 2000. These authors
found that the component of network elements with flux $\geq 30\times 10^{18}$
Mx varied in phase with the sunspot cycle. The magnetogram calibration they
adopted had under-estimated the flux density by a factor of about 1.6 (Bergers
et al. 2003; Wang et al. 2003). With the renewed calibration, this component
would consist of magnetic elements with flux $\geq 4.8\times 10^{19}$. This is
a component in our analysis that changes in phase with sunspots. Meunier
(2003) chose strong magnetic elements with threshold flux density of 25 G and
40 G, respectively, and found naturally a correlation of number and flux of
network elements with sunspots.

When the element flux $\leq 20\times 10^{18}$ Mx (i.e., 32$\times 10^{18}$ Mx
in renewed calibration), Hagenaar et al.(2003) declared that both the flux
spectrum of quiet network elements and the total flux changed a little with
the cycle phase. These authors used the interrupted data in the 6 years of the
ascending cycle phase, they would not be able to guarantee a grasp of the real
trend of the cyclic modulation. We tested their results by using the same
6-month 5-m magnetograms, and found a weak change, but anti-phased with the
cycle phase, in both numbers and flux for network elements in this flux range.
In fact, Hagenaar et al.(2003) reported that the number density of network
concentrations on the quiet Sun decreased by less than 20% from 1997 to 2000,
consistent with our approaches. They also suggested an even anti-correlated
changes in flux emergence rate in this low flux range. The revelation of a
remarkable anti-correlation component of network elements with a broad flux
range from several times of $10^{18}$ Mx to 3 times of $10^{19}Mx$ is likely
to be the true nature of small-scale solar magnetism and inspiring new
considerations of the Sun’s magnetism.

Exploration of the magnetic nature of the Sun’s small-scale activity went back
to earlier solar studies. A few pioneer studies stand still as reliable
references in solar physics. Mehltretter (1974) identified that the network
bright points represented magnetic flux concentration with field strength of
1000-2000 G, each of them had a mean flux of 4.7$\times 10^{17}$ Mx. Later,
the work was extended by Muller & Roudier (1984, 1994). They deduced an
average flux of 2.5$\times 10^{17}$ Mx for an network bright points. The flux
range suggested by these authors for the network bright points changing anti-
phased with sunspot cycle, is out of reach by current data base. Muller &
Roudier (1984) also identified the correlation between the network bright
points in the photosphere and coronal XBPs. For the latter, Golub et al.(1977)
carefully studied their magnetic properties, and found the average total flux
associated with a typical XBP was 2.0$\times 10^{19}$ Mx. The magnetic
measurements were obtained at Kitt Peak with a fine scan of 2.5 arcsec
resolution element and $\sim$2G noise level. They are reasonably reliable to
quantify the magnetic flux of an XBP. This typical flux, even with some
uncertainty, e.g., 50% or larger, is still falling in the flux range of the
anti-correlated component of network elements discovered by this study. We
tentatively suggest that the anti-correlated component of magnetic elements
are responsible for the small-scale activity, e.g., the coronal X-ray bright
points. Updated efforts to quantify the magnetic properties of the so-called
magnetic bright points are crucial to final resolve the long-lasting puzzle of
the anti-phase behavior of the Sun’s small-scale activity in a solar cycle.

Observationally, small-scale network elements come from several sources:
fragmentation of active regions, flux emergence in the form of ephemeral
regions, coalescence of intranetwork flux, and products of dynamic interaction
among different sources of magnetic flux. The 3-fold relationship between
network elements and sunspot cycle has immediate implication on the Sun’s
magnetism. As demonstrated by state-of-the-art simulations (see Vögler &
Schüssler 2007), the magnetic elements at the smallest end of the flux
spectrum, either resolved or un-resolved, manifest a local turbulent dynamo
which operates in the near-photosphere and is independent to the sunspot
cycle. On the other hand, at the larger flux end, the magnetic elements are
likely to be the debris of decayed sunspots. They follow, of course, the solar
cycle.

The key issue here is how to understand the majority of magnetic elements
which are anti-correlated with sunspots in the solar cycle. They are not
likely the debris of decayed sunspots, but probably created by turbulent local
dynamo action that, however, is globally affected or controlled by the sunspot
field from the mean-field MHD dynamo. A few possibilities now are being
considered.

First, during the more active times of the Sun, the smaller magnetic elements
created by the turbulent dynamo have more opportunity to encounter sunspots
and their fragments. The same-polarity encountering results in a merging of
those elements to the flux related to sunspots. Whereas, the opposite polarity
encountering causes flux cancelations with the net results of lost smaller
elements and a diffusion of sunspot flux. What accompanied the sunspot flux
diffusion is the reduced smaller elements with the turbulent origin. This
accounts for the anti-correlated magnetic component possibly. By this kind of
interaction magnetic flux from turbulent dynamo actively takes part in the
operation of the solar cycle, helping with more efficient magnetic diffusion.
To quantify this mechanism, studies of dynamic interaction between small-scale
magnetic elements and active regions fields are crucially required.

Secondly, it is also possible that at the solar maximum, the stronger magnetic
field from sunspots tends to suppress the Sun’s global convection in some
measure. As a result, the local dynamo has been abated somehow, and the
network elements created by turbulence are reduced in number and total flux.
This seems to suggest that the turbulent dynamo is, in fact, global but not
local. Unfortunately, so far there have been no definite observations about
the changes in the global solar convection during the sunspot cycle.

Another possibility is that the anti-correlated component represents the
recycling of parts of the previously diffused or submerged magnetic flux from
the mean-field dynamo (Parker 1987). The diffusion of magnetic flux from
sunspots to the deep convection zone requires 5-7 years (Jiang et al. 2007).
Parts of the diffused or submerged flux serves as the seed field for the
globally turbulent dynamo. Its production is naturally out of phase with
sunspots in the solar cycle, and brings up the magnetic elements that anti-
phased with sunspots.

In a recent literature, Thomas and Weiss (2008) proposed a picture of the
solar Dynamo on three scales (one large and two small), which, according to
the above authors, were only loosely coupled to each other. It is not clear if
some unknown interplay of different scale dynamos may result in the
complicated behavior of the Sun’s small-scale fields. If we adopt the common
vision that the smaller magnetic elements are created by a local turbulent
dynamo, then the local turbulent dynamo on a certain scale must have closely
correlated to the global mean-field dynamo. The global dynamo either provides
seed flux or modifies the condition for this ‘global’ turbulent dynamo. At the
smallest end, the dynamo is likely to be more ‘local’. The turbulent dynamo,
either global or purely local, brings a tremendous amount of turbulent flux to
the Sun that continuously interacts with the products of the mean-field
dynamo. The interaction seems to not only help with the operation of the
global dynamo, but also power the ceaseless small-scale magnetic activity and
maintain the Sun’s Poynting flux to Earth and interplanetary space.

##  5 Conclusions

With the unique database from MDI/SOHO in the interval from September 1996 to
February 2010, which embodies the entire Solar Cycle 23, we analyze the cyclic
variations of quiet Sun’s magnetic flux and Sun’s small-scale magnetic
elements.

The quiet regions contributed $(0.94-1.44)\times 10^{23}$ Mx flux from
approximately the solar minimum to maximum in Cycle 23. The fractional area of
quiet regions decreased from the cycle minimum to maximum by a factor of 1.2,
but their total flux increased by a factor of 1.53. The quiet regions dominate
Sun’s magnetic flux over 60% duration of the cycle. Furthermore, the ratio of
the quiet region magnetic flux to the Sun’s total flux can be used to describe
the course of solar cycle, just as sunspots. The maximum flux occupation of
quiet regions marks the minima of solar cycle. The flux occupation on the
quiet Sun had been larger than 90% for 28 continuous months from July 2007 to
October 2009, which seems to equally characterize the grand minima of Cycles
23 and 24.

With increasing magnetic flux per element the number and total flux of the
Sun’s small-scale magnetic elements follow no-correlation, anti-correlation
and correlation changes with sunspots. The anti-correlated component, covering
the flux range of (2.9 - 32.0)$\times 10^{18}$ Mx, occupies 77.2% of total
elements and 37.4% of flux on the quiet Sun. However, the stronger magnetic
elements with flux larger than 4.3$\times 10^{19}$ Mx dominate the quiet Sun
magnetic flux and follow closely the sunspot cycle.

The definitively identified anti-correlated component of the small-scale
magnetic elements seems to offer an interpretation on the puzzling
observations of anti-correlation variation of many types of small-scale
activity with the solar cycle, e.g., the network bright points, HeI 10830 Å
dark points and coronal X-ray bright points.

It is speculated that the anti-correlated small-scale magnetic elements are
products of some local turbulent dynamo or dynamos that is modulated to be
anti-phased with the global mean-field dynamo.

The authors are grateful to Dean-Yi Chou, Sara Martin and Jie Jiang for their
valuable suggestions and discussions. We appreciate the instructive advice and
valuable suggestions of the anonymous referee, by which the paper has been
significantly improved. The work is supported by the National Natural Science
Foundation of China (10873020, 11003024, 40974112, 40731056, 10973019,
40890161, 10921303, 11025315), and the National Basic Research Program of
China (G2011CB811403).

## References

  * (1) Berger, T. E. & Lites, B. W. 2003, Sol. Phys., 213, 213
  * (2) Charbonneau, P. 2005, Living Reviews in Solar Physics, 2, no.2
  * (3) Davis, J. M. 1983, Sol. Phys., 88, 337
  * (4) Davis, J. M., Golub, L., & Krieger, A. S. 1977, ApJ, 214, L141.
  * (5) de Wijn, A. G., Stenflo, J. O., Solanki, S. K., & Tsuneta, S. 2009, Space Sci. Rev., 144, 275
  * (6) Foukal, P., Harvey, K., & Hill, F. 1991, ApJ, 383, L89
  * (7) Golub, L., Krieger, A. S., Harvey, J. W., Vaiana, G. S. 1977, Sol. Phys., 53, 111
  * (8) Golub, L., Davis, J. M. & Krieger, A. S. 1979, ApJ, 229, L145
  * (9) Hagenaar, H. J. 2001, ApJ, 555, 448
  * (10) Hagenaar, H. J., Schrijver, C. J., & Title, A. M. 2003, ApJ, 584, 1107
  * (11) Hara, H., & Nakakubo, K. 2003, ApJ, 589, 1062
  * (12) Harvey, K. 1985, Aust. J. Phys., 38, 875
  * (13) Harvey, K. 1989, Bull. American Astron. Soc., 21, 839
  * (14) Harvey, K., & Harvey, J. 1974, Bull. American Astron. Soc., 6, 288
  * (15) Harvey, K., & Martin, S. F. 1973, Sol. Phys., 32, 389
  * (16) Harvey, J. 1971, Publ. Astron. Soc. Pacific, 83, 539
  * (17) Jiang, J., Chatterjee, P., & Choudhuri, A. R. 2007, MNRAS, 381, 1527
  * (18) Labonte, B. J., & Howard, R. 1982, Sol. Phys., 80, 15
  * (19) Lamb, D. A., et al. 2008, ApJ, 674, 520
  * (20) Livingston, W. C., & Harvey, J. 1975, Bull. American Astron. Soc., 7, 346
  * (21) Mehltretter, J. P. 1974, Sol. Phys., 38, 43
  * (22) Meunier, N. 2003, A&A, 405, 1107
  * (23) Muller, R., & Roudier, T. 1984, Sol. Phys., 94, 33
  * (24) Muller, R., & Roudier, T. 1994, Sol. Phys., 152, 131
  * (25) Nakakubo, K., & Hara, H. 2000, Adv. Space Res., 25, 1905
  * (26) Parker, E. N. 1987, Sol. Phys., 110, 11
  * (27) Parnell, C. E., DeForest, C. E., Hagenaar, H. J., Johnston, B. A., Welsch, B. T. 2009, ApJ, 698, 75
  * (28) Sattarov, I., Pevtsov, A. A., Hojaev, A. S., & Sherdonov, C. T. 2002, ApJ, 564, 1042
  * (29) Scherrer, P. R. et al. 1995, Sol. Phys., 162, 129
  * (30) Schwabe, M. 1843, AN, 20, 283
  * (31) Sheeley, N. R. Jr. 1966, ApJ, 144, 723
  * (32) Sheeley, N. R. Jr. 1967, Sol. Phys., 1, 171
  * (33) Smithson, R. C. 1975, Bull. American Astron. Soc., 7, 346
  * (34) Thomas, J. H., & Weiss, N. O. 2008, Sunspots and Starspots, Cambridge Univ. Press
  * (35) Trujillo Bueno J., Shchukina, J. N., & Asensio Ramos, A. 2004, Nature, 430, 326
  * (36) Vögler, A., & Schüssler, M. 2007, A&A, 465, L43
  * (37) Wang, J., Zhou, G., Wang, Y., Song, L. 2003, Sol. Phys., 216, 143
  * (38) Webb, D. F., Martin, S. F., Moses, D., Harvey, J. W. 1993, Sol. Phys., 144, 15
  * (39) White, O. R., & Livingston, W. C. 1981, ApJ, 249, 798
  * (40) Zhao, M., Wang, J. X., Jin, C. L., & Zhou, G. P. 2009, RAA, 9, 933
  * (41) Zirin, H. 1972, Sol. Phys., 22, 34
  * (42) Zirin, H. 1987, Sol. Phys., 110, 101

Table 1: Cyclic variation of the NT elements with different flux range Category | Flux (in Mx) | Number ratio | Flux (ratio) | Cor.
---|---|---|---|---
No-correlation | (1.5–2.9)$\times 10^{18}$ | 0.58% | 6.48$\times 10^{21}$ (0.05%) | -0.04
Anti-correlation | (2.9–32.0)$\times 10^{18}$ | 77.19% | 4.72$\times 10^{24}$ (37.40%) | -0.45
Transition | (3.20–4.27)$\times 10^{19}$ | 6.59% | 1.15$\times 10^{24}$ (9.08%) | -0.03
correlation | (4.27–38.01)$\times 10^{19}$ | 15.65% | 6.74$\times 10^{24}$ (53.46%) | 0.82
Figure 1: Left panels: Two retrieved MDI 5-minute full-disk magnetograms
within 70 $\deg$ from disk center, at approximately the solar maximum and
minimum, respectively. Using the threshold on 15 Mxcm-2 to define the edge of
active region, the islands with area larger 9$\times$9 pixels, i.e., the
regions contoured by red line, is defined as active regions. The purple circle
displays the location $\alpha$=70 $\deg$, and the black circle displays the
location $\alpha$=60 $\deg$. The gray scale saturates at $\pm$50 Mxcm-2. Right
panels: enlarged images for the windows framed in the MDI magnetograms, on
which network elements falling in the flux ranges of (2.9-32.0)$\times$ 1018
Mx and (4.3-38.0)$\times$ 1019 Mx are outlines by green and yellow curves,
respectively (see Section 3.2). Figure 2: The left panel is the flux
variations of ARs (cross symbols in black) and quiet Sun (‘+’ symbols in
purple) in an interval including the entire 23rd Solar Cycle. The red curve
represents the sunspot number changes in the cycle. The shaded columns are the
statistical results based on the Kitt Peak full-disk magnetograms. The
magnetic flux for quiet regions rises from 0.94$\times 10^{23}$ Mx in December
1995 to 1.44$\times 10^{23}$ in May 2002, increases by a factor of 1.53. The
fractional quiet Sun area is shown by purple ‘+’ symbols, in the right panel.
It decreases by a factor of 1.2 from the solar minima to maximum. The ratio of
quiet Sun flux to the total Sun’s flux, the flux occupation of the quiet Sun,
is shown by purple cross symbols in the right panel. The quiet Sun flux has
dominated the Sun’s magnetic flux for 7.92 years in the 12.5 year Cycle 23.
Figure 3: The probability distribution function of element magnetic flux for
all the selected network elements during the interval from September 1996 to
February 2010, i.e., average PDF of the quiet Sun’s small-scale magnetic
elements. The peak distribution at $10^{19}$ Mx is consistent with that found
for multiple MDI full-disk datasets by Parnell et al. (2009).  Figure 4:
Correlation coefficients between the sunspot number and network element number
of each of the 96 sub-group elements which are reconstructed according to the
flux per element. There appears a 3-fold correlation scheme between the
network elements and the sunspot cycle: basically no-correlation, anti-
correlation and correlation. At the low end of flux spectrum, there are very
small correction coefficients. With the increasing flux per element, the
correlation coefficients reach approximately to -0.58, then they become
positive and reach as high as 0.92 after a very narrow transition in the flux
range of (3.20-4.27)$\times 10^{19}$ Mx. The color bar represents the
confidence level. Figure 5: Cyclic variations of network element number (right
panel) and flux (left panel) of 4 categories of network elements shown in
Table 1, which represents the 3-fold correlation scheme of network elements
with the sunspot cycle. The green ‘+’ is referring anti-correlation component
elements, while the purple ‘+’ is for the in-phase correlation component
elements. Black and blue dotted lines are elements which have no correlation
or shown transition from anti-correlation to correlation with the solar cycle.
Figure 6: The differential probability distribution function (DPDF), i.e., the
difference between the PDFs of yearly network magnetic elements and average
PDF shown in Fig.3.