##  Chapter 1 Observations of Magnetic Helicity Proxies in Solar Photosphere:
Helicity with Solar Cycles

Shangbin Yang1,2 Haiqing Xu1 Xiao Yang1 Jie Chen1 Jihong Liu1,3

Hongqi Zhang*1

###### Abstract

Abstract Observations of magnetic helicity transportation through the solar
photosphere reflect the interaction of turbulent plasma movements and magnetic
fields in the solar dynamo process. In this chapter, we have reviewed the
research process of magnetic helicity inferred from the observed solar
magnetic fields in the photosphere and also the solar morphological
configurations with solar cycles. After introducing some achievements in the
study of magnetic helicity, some key points would like to be summarized.

The magnetic (current) helicity in the solar surface layer presents a
statistical distribution similar to that of the sunspot butterfly diagram, but
its maximum value is delayed from the extreme value of the sunspot butterfly
diagram and corresponds in the phase with the statistical eruption of solar
flares. During the spatial transport of magnetic (current) helicity from the
interior of the sun into the interplanetary space at the time-space scale of
the solar cycle, it shows the statistical distribution and the fluctuation
with the hemispheric sign rule. These show that the current helicity and
magnetic helicity transport calculation methods are complementary to each
other.

We also notice that the study of the inherent relationship between magnetic
helicity and the solar cycle still depends on the observed accuracy of the
solar magnetic field.

Keywords: Magnetic field, Helicity, Solar cycles

###  1.1 Introduction

The study of solar magnetic helicity with the solar cycles is an interesting
topic. The solar activity cycle is first found from the observation results of
sunspots, which have 11-year cycles (Schwabe, 1843). Observation of solar
magnetic fields with magnetographs has brought human cognition for
understanding solar activities to a new stage (Hale, 1908).

A large number of observations show that the solar active regions are the
place where the magnetic field energy on the solar surface is most
concentrated (cf. Fisher et al., 2000; Schmieder et al., 2000; Toriumi et al.,
2020). Accompanied by flares or coronal mass ejections, the energy of the
magnetic field is transported from the solar subatmosphere into the
interplanetary space - strong solar storms (cf. Wang & Xu, 2002; Aschwanden,
2005). It is found that the intense solar eruptive phenomena are often
connected with the strong magnetic helicity (cf. Low, 1996; Bothmer & Schwenn,
1998; Yurchyshyn et al., 2001; Nindos & Zhang, 2002; Lynchet al., 2005; Török
& Kliem, 2005).

Due to the opacity of the solar atmosphere, that is, even using the method of
helioseismology to diagnose the structure inside the sun, it is still little
known about the situation of the magnetic fields inside of the sun in detail
(Parker, 1979; Fan, 2009). The magnetic field distribution on the surface of
the sun brings important information about the formation and transport of
magnetic field energy with helicity in the subatmophere.

People try to explore the mechanism of the formation of the magnetic field
inside of the sun through the theory of the solar dynamo, to explain the
statistical phenomenon that the solar activity presents an 11-year cycle, and
so on (cf. Parker, 1955; Moffatt, 1978; Krause & Rädler, 1980; Zeldovich et
al., 1983). The usual theories believe that the chilarity (helicity) of the
magnetic field is also one of the important pieces of evidence for the
formation of the magnetic field inside the sun, and believe that the helicity
tends to show opposite signs in both solar hemispheres in the process of solar
dynamos (cf. Seehafer, 1994). It is normally believed that the observed
magnetic helicity in the solar surface brings the message of the
$\alpha$-effect in the deep solar convection zone from the point of view of
the mean-field $\alpha\omega$ dynamo model (cf. Brandenburg & Subramanian,
2005) or also the flux-transport model (cf. Choudhuri et al., 2004).

The statistical analysis of the observed helical magnetic field in the solar
atmosphere, such as in the solar active regions, provides a window for
studying the possible formation mechanism of the solar active cycles with the
solar dynamos.

In the following, we present the current helicity in active regions, the
injection of the magnetic helicity with solar cycles inferred from observed
photospheric magnetic fields, and the relationship with solar eruptive
activities. We also discuss some questions from the observations of solar
magnetic helicity.

###  1.2 Magnetic Helicity from Observations

####  1.2.1 Early Morphological Observations of helical Patterns in Solar
atmosphere

The chiral patterns of active regions on the Sun were firstly found by Hale
(1908); Richardson (1941) who found that most of the sunspots (about $80\%$)
with discernible whirls had phenomenological counterclockwise (clockwise)
rotations of the fibril patterns in the northern (southern) hemisphere from
the observations. It has been further confirmed and statistically presented on
the distribution of helical sunspots in both hemispheres in 1970-1982 by Ding
et al. (1987), which is covered about a solar cycle, as shown in Figure 1.1.
Interestingly, only about 66% of sunspots follow the helical sign rule.

Figure 1.1: Distribution of spiral patterns of active regions in solar
southern (a) and northern (b) hemispheres. The ordinates (spirals/year) are
the statistical number of twisted sunspots per year. The solid (dashed) lines
indicate the twist of sunspots in right (left) handedness. After Ding et al.
(1987)

The observations of magnetic fields in the solar photosphere using solar
magnetographs or Stokes polarimeters are a general method to diagnose the
twisted pattern of the field in sunspot regions. As an example, Figure 1.2
shows the photospheric vector and chromospheric longitudinal magnetic field in
a delta active region (Zhang, 2019, 2020). It is found that the horizontal
twisting component of the photospheric vector magnetic field of the active
region is consistent with the features of the H$\beta$ chromospheric magnetic
field and also the chromospheric morphological features. These confirm that
the morphological pattern of spiral spot features as before observed by Hale
(1908); Richardson (1941); Ding et al. (1987) reflects the configuration of
the magnetic fields in the atmosphere (such as marked by $f$ in Figure 1.2).

Moreover, the evidence on the hemispheric sign rule of the helical features in
the chromosphere was also indicated by Martin et al. (1994). They found
morphologically that the majority of quiescent filaments were
dextral/sinistral in the northern/southern hemisphere. Rust (1994) pointed out
the relationship amount the patterns of chirality in sunspots, active regions,
filaments, and interplanetary magnetic clouds. These imply that the helical
patterns on the Sun relate to the different scale magnetic fields, due to the
dominant controls of the magnetic fields in the sunspots of active regions,
filaments, clouds, and so on. This means that the helical features extend from
the photosphere into the high atmosphere and the interplanetary space. It is
noticed that the low sensitivity of chromospheric spectral lines to magnetic
field observations, non-local thermal dynamic equilibrium, and disturbance of
photospheric blended spectral lines (such as in the wing of H$\beta$ line)
affect the measurement of the solar chromospheric magnetic field, and make the
difficulty to obtain the chromospheric vector magnetic field (Zhang, 2019,
2020). The reversal patch $r$ in the umbra of chromospheric H$\beta$
magnetogram in Figure 1.2 is caused by the disturbance of photospheric blended
spectral lines in the wing of the H$\beta$ line.

Figure 1.2: Active region NOAA 6619 observed on May 10, 1991. Top left:
Photospheric filtergram. Bottom left: Photospheric vector magnetogram. Top
right: H$\beta$ filtergram. Bottom right: longitudinal magnetogram in
Chromosphere from H$\beta$. Black (white) is negative (positive) polarity in
the magnetograms. After Zhang (2019).

####  1.2.2 Magnetic Helicity from Solar Vector Magnetic Fields

The magnetic helicity is the volume integral (Woltjer, 1958a, b)

$H_{m}=\int_{V}{\bf A}\cdot\nabla\times{\bf A}d^{3}x,$ (1.1)

where $\bf A$ is the magnetic vector potential and is not an immediately
observed quantity and does not satisfy the requirement of gauge invariance. To
simplify calculations as employ the Coulomb gauge (Moffatt, 1969; Arnold,
1974; Berger & Field, 1984)

${\bf A(x)}=-\frac{1}{4\pi}\int d^{3}x^{\prime}\frac{\bf r}{r^{3}}\times{\bf
B(x^{\prime})},$ (1.2)

which gives

$H_{m}=-\frac{1}{4\pi}\int d^{3}x\int d^{3}x^{\prime}{\bf
B(x)}\cdot\left[\frac{\bf r}{r^{3}}\times{\bf B(x^{\prime})}\right],$ (1.3)

with the relationship ${\bf B}=\nabla\times{\bf A}$ (Low, 2015).

The current helicity $H_{c}$ is defined as

$H_{c}=\int_{V}{\bf B}\cdot\nabla\times{\bf B}d^{3}x,$ (1.4)

where $\bf B$ is the magnetic field and the current helicity density can be
difined

$\displaystyle h_{c}$ $\displaystyle={\bf B}\cdot\nabla\times{\bf B}=({\bf
B}\cdot\nabla\times{\bf B})_{z}+({\bf B}\cdot\nabla\times{\bf B})_{t},$ (1.5)

the subscript $z$ and $t$ mark the longitudinal and transverse components of
current helicity respectively. The current helicity density $h_{cz}=({\bf
B}\cdot\nabla\times{\bf B})_{z}$ is observable in the lower solar atmosphere
by observed photospheric vector magnetogam in Figure 1.3, while $({\bf
B}\cdot\nabla\times{\bf B})_{t}$ is difficult because it is impossible to
achieve the construction of the real transverse component of electric current
through the observations of the solar magnetic fields.

Figure 1.3:  The active region NOAA 6619 was taken at Huairou Solar Observing
Station on May 11, 1991, at 03:26 UT. Left: the photospheric vector
magnetogram. Right: the electric current helicity density $h_{cz}=({\bf
B}\cdot\nabla\times{\bf B})_{z}$ (contours) overlapped by the filtergram of
this active region; the black (white) contours correspond to positive
(negative) values of 0.01, 0.05, 0.1, 0.2, 0.5 G2m-1, respectively.  Figure
1.4: Sigmoidal field lines wrapping around a flux rope in the active region
12158, which has a negative helicity (reverse S, left‐hand twist) on 10
September 2014, 2 hr before a flare. The left panel shows an observation in
AIA 94 Å; the right panel shows sigmoidal magnetic field lines using an NLFFF
extrapolation based on the Grad‐Rubin method (Gilchrist & Wheatland, 2014),
overlying a longitudinal component of photospheric magnetic field saturated at
2000 gauss. After Schmieder & Aulanier (2018).

The chirality of the magnetic structures can be analyzed morphologically due
to the magnetic freezing effect of the plasma in the solar atmosphere. It
reflects the spatial characteristics of the magnetic field helicity. This also
expands the distribution and range of magnetic helicity in the solar
atmosphere that we can analyze from an observable perspective. Figure 1.4
shows the extrapolated nonlinear force free (NLFF) magnetic field calculated
from the photospheric magnetic field in an active region to fit AIA 94Å flare
loops (Zhao, 2016; Schmieder & Aulanier, 2018), which has a negative helicity.

From the above discussion, it is found that the distribution of the magnetic
(current) helicity in the solar atmosphere and its evolution can be analyzed
from different perspectives (the magnetic vector potential, magnetic field,
and morphology). The key parameter is the magnetic field, especially from the
observations of the photospheric magnetic field.

###  1.3 Electric Current Helicity from Observations

####  1.3.1 Helicity from solar vector magnetic fields

The early statstical research on the magnetic (current) helicity from the
observations of solar magnetic fields was firstly presented by Seehafer
(1990), and a series of subsequent studies by Pevtsov, Canfield, and Metcalf
(1995); Abramenko et al. (1996); Bao & Zhang (1998); Hagino & Sakurai (2005);
Xu et al. (2007), etc. The observable current helicity density averaged over
an active region is

$\overline{h_{{\rm c}z}}=\overline{({\bf
B}\cdot{\bf{{\bf{\nabla}}\times\,}}{\bf B})_{z}},$ (1.6)

which can be derived from the photospheric vector magnetograms (see Figure
1.3). In the approximation of local homogeneity and isotropy, the value of
$h_{{\rm c}z}$ can be used to analyze the total current helicity $h_{\rm c}$
(Xu et al., 2015). We need to notice that $h_{{\rm c}z}=1/3h_{\rm c}$ is an
estimated value, due to the incompleteness for inversion of the solar magnetic
field through the solar polarized spectral light (Zhang, 2019, 2020).

Another proxy of the magnetic field is twist, i.e., the force-free factor
$\alpha={\bf B}\cdot(\nabla\times{\bf B})/B^{2}$, or an equivalent of the
quantity averaged over an active region is the ratio

$\alpha_{\rm av}=\overline{(\nabla\times{\bf B})_{z}/B_{z}}$ (1.7)

calculated from the observed vector magnetograms.

Pevtsov et al. (2005) reported the result of a study of magnetic helicity in
solar active regions in 1980-2000 (cycles 21-23) using $\alpha_{\rm av}$ as
the proxy for current helicity. They did not see consistency between different
instruments in regards to years of disobeying the hemispheric helicity rule.
They found that the present data sets do not allow for making statistically
significant inferences about the possible cyclic variation of the hemispheric
helicity rule. Xu et al. (2007) compared a series of vector magnetograms of
the same active regions observed by different solar vector magnetographs. It
is found that a similar tendency of calculated current helicities from the
magnetograms was observed at the different observatories. Although there exist
slight differences among them.

Figure 1.5:  Time variation of the mean slope of linear fit calculated from
the latitudinal profile of helicity for each year over 1992–2005 (left) and
1997–2000 (right). The solid (dashed) lines in the left represent HR (MTK)
data, and in the right Mees (MTK) data, respectively. The upper panel is
d$h_{\rm c}$/$d{\theta}$ and the lower is d${\alpha}$ff/$d{\theta}$. Error
bars represent one sigma uncertainty of slope of the linear fit. After Xu et
al. (2007)

The seemingly contradictory results of the helicity mentioned above may mainly
come from the observations of the solar magnetic field, and the processing and
analysis of the data. The diagnostic of solar magnetic fields based on these
magnetically sensitive lines is accompanied by the approximations of the
theoretical analysis, which probably brings the uncertainty of inversion of
spectro-polarimetric observations to obtain magnetic fields. It is important
to prevent or avoid the influence of these errors and try to obtain relatively
accurate measurements of solar magnetic fields if the defects of the different
magnetographs can be ignored. We probably can believe that more reasons
probably bring the observed errors at the vector magnetographs from different
observatories (Ai, 1989; Ronan et al., 1992; Wang et al., 1992; Sakurai et
al., 1995; Sakurai, 2001; Zhang et al., 2003) until now, as we wish to analyze
in detail probably.

####  1.3.2 Magnetic helicity with solar cycles

The helicity evolution with the solar cycle is a notable topic. The mean
distribution of photospheric current helicity density of active regions in the
solar surface by $\overline{h_{{\rm c}z}}=\overline{({\bf
B}\cdot{\bf{{\bf{\nabla}}\times\,}}{\bf B})_{z}}$ in Eq. (1.6) and also
$\alpha_{{\rm av}}=\overline{(\nabla\times{\bf B})_{z}/B_{z}}$ in Eq. (1.7) in
solar cycle 22 was statistically presented by Bao & Zhang (1998) and Zhang &
Bao (1998, 1999), which are inferred from a series of photospheric vector
magnetograms recorded at Huairou Solar Observing Station.

Figure 1.6: Butterfly diagram of the mean current helicity density $h_{{\rm
c}z}$. The sizes of the circles relate to the grades: 0, 1, 3, 5, 7
($10^{-3}G^{2}m^{-1}$) of current helicity density. The white (black) circles
mark the negative (positive) sign of helicity. The dashed-dotted line presents
the distributed average value of current helicity and the dashed line does the
mean value of the imbalance of current helicity after the data smooth.
Vertical coordinates represent the sine of latitude, and horizontal
coordinates are expressed in years. From Zhang & Bao (1998)

Figure 1.6 shows the distribution of the current helicity of active regions in
the form of the Butterfly diagram observed at Huairou. It is found that
besides most of the helicity of active regions following the hemispheric sign
rule, some super $\delta$ active regions have been marked in the figure. Some
high density of current helicity regions (such as NOAA 5395, 6659, and 6619)
occurred at the high latitudes, while NOAA 7321 and 7440 occurred at the low
ones. The notable is that the active region NOAA 6891 shows the reversal sign
of helicity.

Figure 1.7:  Top: the distribution of the averaged twist $\alpha_{\rm ff}$;
and bottom: current helicity $h_{{\rm c}z}$ of solar sunspots in the 22nd and
23rd solar cycles. Superimposed, the underlying grayed “butterfly diagram”
shows how sunspot density varies with latitude over the solar cycle. The open
(closed) circles indicate the positive (negative) sign.

After Zhang et al. (2010b)

A subsequent study was focused on the distribution of the average helical
characteristics of the magnetic field in solar active regions from 1988 to
2005 in the form of butterfly diagrams (latitude-time), which data was
observed at Huairou Solar Observing Station, National Astronomical
Observatories of China (Zhang et al., 2010b). It covers most of the solar
cycle 22 and 23 in Figure 1.7 after removing magneto-optical effects in the
measurements of the magnetic field (Su & Zhang, 2004; Gao et al., 2008).

The maximum values of the mean helicity density of active regions at the solar
surface tend to occur near the edges of the sunspots butterfly diagram. It is
roughly consistent with Figure 1.6, where some super active regions are
marked. The reversal sign of the mean current helicity relative to the
hemispheric sign rule occurs at the beginning of the butterfly wing in 1997 in
Figure 1.7 (also see Bao et al., 2000). It is probably interesting to compare
with the traditional solar dynamo theory (such as Parker, 1955).

Kuzanyan et al. (2003); Zhang et al. (2006) identified the longitudinal
migration of active regions and the corresponding current helicity with their
rotation rates and compared with the internal differential rotation law in the
solar convection zone inferred from helioseismology. They deduced the
distribution of the contributed helicity over depth.

Moreover, Zhang (2006) reported that both $\alpha$ factor and current helicity
in the strong fields present a sign opposite to the weak ones, by analyzing
large simples of photospheric vector magnetic fields of active regions in 1997
- 2004 observed at the Huairou Solar Observing Station. A similar case with
the Spectro-polarimeter (SP) onboard the Hinode satellite was presented by Hao
& Zhang (2011), who showed that the helicity changes the sign from the inner
umbra to the outer penumbra. The question is how to express the possible
reason and mechanism, which relates to the opposite signs of the current
helicity between the umbra and penumbra in the same sunspots or the strong and
weak fields in the same active regions.

Following Matthaeus et al. (1982), it is possible to determine the helicity
spectrum of magnetic fields in the solar active regions from the spectral
correlation tensor if we assume the local statistical isotropy exists. The
Fourier transform of the two-point correlation tensor, $\langle B_{i}({\bf
x},t)B_{j}({\bf x}+{\bf\xi},t)\rangle$ can be written as (cf. Moffatt, 1978)

$\left\langle\hat{B}_{i}({\bf k},t)\hat{B}_{j}^{*}\\!({\bf
k}^{\prime},t)\right\rangle=\Gamma_{ij}({\bf k},t)\delta_{2}({\bf k}-{\bf
k}^{\prime}),$ (1.8)

where $\hat{B}_{i}({\bf k},t)=\int B_{i}({\bf x},t)\,e^{i{\bf k}\cdot{\bf
x}}d^{2}x$ is the 2D Fourier transform, the subscript $i$ and $j$ relate to
the vector magnetic field components, the asterisk denotes complex
conjugation, and ensemble averaging is replaced by averaging over concentric
annuli in wavevector space (Zhang, Brandenburg & Sokoloff, 2016). As $\bf k$
defines the direction in $\Gamma_{ij}$, and that $k_{i}\hat{B}_{i}=0$, the
only possible structure of $\Gamma_{ij}({\bf k},t)$ takes the form (cf.
Moffatt, 1978)

$\Gamma_{ij}({\bf k},t)=\frac{2E_{M}(k,t)}{4\pi
k}(\delta_{ij}-\hat{k}_{i}\hat{k}_{j})+\frac{iH_{M}(k,t)}{4\pi
k}\varepsilon_{ijk}k_{k},$ (1.9)

where $\hat{k}_{i}=k_{i}/k$ is the component of the unit vector of $\bf{k}$,
$k=|\bf{k}|$ is the modulus with $k^{2}=k_{x}^{2}+k_{y}^{2}$, and $E_{M}(k,t)$
and $H_{M}(k,t)$ are the magnetic energy and magnetic helicity spectra,
respectively.

Figure 1.8: (a) $2E_{M}(k)$ (solid line) and $k|H_{M}(k)|$ (dotted line) for
NOAA 11158 at 23:59:54UT on 13 February 2011. Positive (negative) values of
$H_{M}(k)$ are indicated by open (closed) symbols, respectively.
$2E_{M}^{(v)}(k)$ (dotted) and $2E_{M}^{(h)}(k)$ (dash-dotted) are shown for
comparison. (b) Same as the upper panel, but the magnetic helicity is averaged
over broad logarithmically spaced wavenumber bins. After Zhang, Brandenburg &
Sokoloff (2014)

As an example, Figure 1.8 shows the spectrum of magnetic helicity and energy
in an active region, which data is observed by HMI onboard the SDO satellite.
The lower panel of Figure 1.8 shows the sign of magnetic helicity at different
scales, after averaging the spectrum overbroad, logarithmically spaced
wavenumber bins.

Figure 1.9:  Top: The distribution of average scale exponent $\alpha$ of
current helicity ($\alpha_{c}$ solid line) and magnetic energy ($\alpha_{E}$
dashed line) with the time inferred by 6629 vector magnetograms of solar
active regions in 1988-2005. The dotted line is sunspot numbers. Bottom: The
distribution of average scale exponent $\alpha$ of magnetic helicity
($\alpha_{kH}$ dotted line), current helicity ($\alpha_{c}$ solid line), and
magnetic energy ($\alpha_{E}$ dashed line) with the latitude. $\alpha_{kH}$
relates to $k|H_{M}(k)|$. The error bars are 0.3$\sigma$. After Zhang,
Brandenburg & Sokoloff (2016)

Figure 1.9 presents the variation of the slopes of the spectrums of magnetic
energy and helicities of active regions with solar cycles (Zhang, Brandenburg
& Sokoloff, 2016). The magnetic spectrums have been taken in the form of
$k^{-\alpha}$. The statistical correlation coefficient between sunspot numbers
and slopes $\alpha$ of the magnetic energy is 0.827, and that between sunspot
numbers and current helicity is 0.730. The correlation coefficient between
sunspot numbers and slopes of current helicity changes to 0.831, as one takes
the sunspot numbers to one year delay. It is consistent with the observed
result in Figure 1.7, in which the maximum value of mean current helicity of
solar sunspots is delayed than that of sunspot numbers. Similar evidence is
that the complex magnetic configuration of active regions tends to occur in
the decaying phase of the solar cycle (Guo et al., 2010).

Another is that one cannot find the significant variation of the mean slopes
$\alpha$ of the spectrums of magnetic energy and helicities of active regions
with the latitudes after a long-term average in Figure 1.9. It reflects that
the mean scale distribution of the magnetic field of solar active regions
doesn’t show a significant latitudinal tendency.

A similar study for the active cycle is presented by Gosain & Brandenburg
(2019) after computation of the magnetic helicity and energy spectra from
magnetic patches on the solar surface using data observed by the Hinode
satellite in 2006-2017.

Similar studies of the large-scale current helicity are based on the
calculation of the longitudinal full-disk magnetograms, which are obtained by
the Michelson Doppler Imager (MDI) as well as the Kitt Peak Vacuum Telescope
and the Synoptic Long-term Investigations of the Sun (SOLIS) by Pevtsov &
Latushko (2000); Wang & Zhang (2010); Pipin & Pevtsov (2014). The large-scale
magnetic fields reflect a clear and consistent current helicity pattern that
follows the established hemispheric rule. Pipin et al. (2019) have presented
the mean helicity density of the non-axisymmetric magnetic field of the Sun
and separated it into the mean large- and small-scale components of magnetic
helicity density, which display the hemispheric helicity rule of opposite
signs at the beginning of cycle 24.

###  1.4 Magnetic Helicity Injection in Solar Surface

####  1.4.1 Magnetic helicity injection inferred from moving magnetic
structures

Magnetic helicity in Eq. (1.1) is not a directly observable quantity in the
solar atmosphere. The general definition of magnetic helicity does not satisfy
the requirement of gauge invariance, and the concept of relative magnetic
helicity has been introduced (Berger & Field, 1984). The injected magnetic
helicity from the solar subatmosphere is a quantity to reflect its relative
variation. It can be inferred by the motions of footpoints of the magnetic
fields in the solar surface. The change of magnetic helicity (Chae, 2001)

$\frac{dH_{m}}{dt}=-2\oint_{S}({\bf A}_{p}\cdot{\bf
V}_{t})B_{n}ds+2\oint_{S}({\bf A}_{p}\cdot{\bf B}_{t})V_{n}ds,$ (1.10)

where ${\bf A}_{p}$ is the vector potential of the potential field, and the
subscript $t$ and $n$ mark the transverse and vertical components of magnetic
field and velocity field, respectively. The first integral on the right-hand
side of eq. (1.10) is the contribution from the twisting or shearing motions
of footpoints of the magnetic fields on the solar surface, while the second
one is that from the emergences of magnetic fluxes from the sub-photosphere
(Antiochos, 1987; Kusano et al., 2002).

Figure 1.10: Active region NOAA 10488 on Oct 26-31, 2003. Left: Gray-scale
maps of $G\equiv-2(\mathbf{u}\cdot\mathbf{A}_{p})B_{z}$ of one-hour averages.
The white and black colors show the positive and negative signs, respectively.
Right top: (a) Time profile of the longitudinal component of magnetic field
flux. (b) Time profile of the injected rate of magnetic helicity by horizontal
motions. (c) The accumulated helicity $\Delta$H(t) calculated from
${\mathrm{d}}H/{\mathrm{d}}t$. Right bottom: Ratio of the accumulated coronal
helicity to the square of the magnetic flux. After Liu & Zhang (2006)

According to the analysis of Démoulin & Berger (2003), the injected magnetic
helicity can be calculated as (Chae, 2001)

$\frac{dH_{m}}{dt}=-2\oint_{S}({\bf A}_{p}\cdot{\bf U})B_{n}ds,$ (1.11)

where

${\bf U}={\bf V}_{t}-\frac{V_{n}}{B_{n}}{\bf B}_{t},$ (1.12)

and ${\bf A}_{p}$ is magnetic vector potential of the potential field, $t$ and
$n$ mark the horizontal and vertical components of the magnetic and velocity
field, respectively. This implies that one can include the most contribution
of injective helicity from the horizontal motion of magnetic footpoints in the
solar surface.

Figures 1.10 shows an example of the injection of magnetic helicity in a newly
emerging active region by Eq. (1.12). Both polarities of the magnetic flux
increase very quickly in Figure 1.10a. Figures 1.10b and 1.10c display the
temporal variation of ${\mathrm{d}}H_{m}/{\mathrm{d}}t$ and accumulated
helicity $\Delta$$H(t)$, separately. The former represents the injective rate
of magnetic helicity with the horizontal motions of magnetic fields in the
solar surface, and the latter represents the total accumulated quantity of
helicity.

####  1.4.2 Magnetic helicity injection with solar cycles

Berger & Ruzmaikin (2000) evaluated the surface integral using solar
magnetogram data and differential rotation curves, and they proposed the
helicity generation in the solar interior by differential rotation to produce
the correct sign compared to observations of coronal structures. in Figure
1.11. They estimated that the net helicity flows into each hemisphere over
this cycle was approximately $4\times 10^{46}Mx^{2}$. Georgoulis et al. (2009)
estimated a maximum helicity injection of $6.6\times 10^{45}Mx^{2}$ for solar
cycle 23. Hawkes & Berger (2018) applied to data sets covering a total of 60
years to estimate the magnetic helicity as a predictor of the solar cycles,
which is an extension following Berger & Ruzmaikin (2000).

Figure 1.11: Net transfer of helicity into the southern corona and wind
$dH_{CS}/dt=\dot{H}(V_{S}\textrightarrow C_{S})$ (predominantly positive
curve), and into the northern corona and wind
$dH_{CN}/dt=\dot{H}(V_{N}\textrightarrow C_{N})$ (predominantly negative
curve). The units are $10^{40}Mx^{2}/day$. From Berger & Ruzmaikin (2000)

According to Eq. (1.11), the relationship between the mean magnetic helicity
density $\overline{h}_{m}$ and the mean current helicity density
$\overline{h}_{cz}$ can be obtained

$\overline{h}_{m}=-\frac{2}{S_{c}L_{c}}\int_{T_{c}}\oint_{\partial S}({\bf
U}\cdot{\bf A}_{p})B_{n}dsdt\sim k{\overline{\bf A\cdot B}}\sim
L^{2}_{c}\overline{h}_{cz},$ (1.13)

where $T_{c}$ is the typical relaxation time, $S_{c}$ and $L_{c}$ are the
typical horizontal and vertical spatial scale of emerging magnetic flux in the
solar atmosphere before the transport of helicity into the interplanetary
space, and $k$ is a a correlation coefficient (about the order of one).

In principle, it is difficult to obtain $\overline{h}_{m}$ directly from
observation to compare with the current helicity $\overline{h}_{cz}$ in the
active regions (in Eq. (1.6)). The current helicity $\overline{h}_{cz}$ is
inferred from the photospheric vector magnetograms presented above, while Eq.
(1.11) can be used to calculate the injection of magnetic helicity from the
subphotosphere. The statistical relationship between magnetic and current
helicity with solar cycles is probably important to understanding the
distribution of magnetic chirality in the solar atmosphere and its
transformation from the subatmosphere with solar cycles.

Figure 1.12:  Injective magnetic helicity flux from the northern (solid line)
and the southern (dotted line) hemisphere in 1996-2009. The shadow areas mark
the period without the relevant helicity calculation from the magnetograms.
From Zhang & Yang (2013)

For analysis of the injection of global magnetic helicity in the solar surface
with the solar cycle, the MDI full-disk magnetograms had been used by Yang et
al. (2012) by tracking the MDI synoptic charts and Zhang & Yang (2013) by
tracking the full-disk 96 minutes magnetograms. The injection of global
magnetic helicity in the solar surface can also be calculated or estimated by
Eqs. (1.11) and (1.12) from the full disk magnetograms as the projective
effects have neglected nearby the limb of the solar disk. Figure 1.12 shows
the mean injection of magnetic helicity flux in 1996 - 2009 in each solar
rotation for analyzing its long-term evolution in the solar surface. It can be
found that besides the significantly fluctuated injection of magnetic helicity
in the northern and southern hemispheres, we also can find the hemispheric
sign rule for the injective magnetic helicity (Zhang & Yang, 2013). It is
consistent with the result obtained by photospheric vector magnetograms in the
active regions in Figure 1.7, due to the negligible contribution from the
quiet Sun (Welsch & Longcope, 2003). The rough consistency can probably be
estimated by Eq. (1.13), even though there may be a complex internal
relationship between them. This may give us enlightenment from the
observations, the total injective helicity in a solar cycle tends to be zero,
but the fluctuation is significant.

Figure 1.13: Top: The net injective helicity contributed from both
hemispheres. The dotted line marks the net injective helicity after the smooth
of 48 solar rotations. Bottom: Total magnetic helicity flux (thin solid line)
and sunspot numbers (thin dotted line) in 1996-2009, while the black thick
solid (dot-dashed) line shows the total helicity flux (sunspot numbers) after
smoothed significantly. From Zhang & Yang (2013)

It is found that the extreme value of negative helicity flux occur in
1997-2002, after the smooth of net magnetic helicity flux in Figure 1.13. It
provides a rough estimation only, the extreme value is about $-2\times
10^{36}Mx^{2}s^{-1}$. It also depicts the injective rate of the total absolute
value of magnetic helicity flux inferred from the solar surface and the
sunspot numbers in 1996-2009. The injective rate of the total absolute value
of helicity flux is obtained from the absolute value of positive and negative
ones. The mean value of the absolute injective rate of total helicity is
$2.40\times 10^{37}Mx^{2}s^{-1}$ in the calculated solar disk in 1996-2009.
This provides a basic estimation of total injective magnetic helicity flux in
about $5.0\times 10^{46}Mx^{2}$ in the 23rd solar cycle from both hemispheres,
and it is a similar order as estimated by Berger & Ruzmaikin (2000) and also
Yang & Zhang (2012). It may provide a minimum value if the contribution of
small-scale helicity flux and the projective effects of the magnetic field in
the solar surface has been estimated in the solar cycle 23.

It is also noticed that the total magnetic helicity flux tends to delay than
the total sunspot numbers, as comparing both smoothed ones in Figure 1.13. The
maximum of sunspot numbers occurs in 2001, while that of current helicity in
2002 after the smooth. This is consistent with that the maximum of butterfly
diagram of calculated current helicity of solar sunspots delays than that of
sunspot numbers in Figure 1.7. It also means that the sunspot numbers do not
reflect the relevant handedness of magnetic fields generated in the solar
atmosphere completely.

###  1.5 Magnetic Helicity as an Index in the High Solar Atmosphere

When we study the upward transport of magnetic helicity from the solar
photosphere, the helical characteristics of the coronal magnetic field can
give us important instructions. Under the existing observational conditions,
the chirality study of the coronal loops is of great significance.

####  1.5.1 Hemispheric Distribution of Helical Coronal Soft X-Ray Loops

To analyze the magnetic chirality in the solar corona, Figure 1.14 shows the
statistical results of 753 large-scale soft X-ray loops in 1991 - 2001
observed by the Yohkoh satellite. As the unidentified loops are ignored, one
can find that the portion of the systems which are in accord with the
hemispheric rule is $77.3\%$ ($81.5\%$) in the northern (southern) hemisphere.
It is roughly consistent with the hemispheric sign rule of the magnetic
helicity in active regions.

Figure 1.14: The hemispheric handed rule trend of the proportion of soft X-ray
loops in the southern and northern hemispheres. From Zhang et al. (2010a)
Figure 1.15: The mean latitudinal distribution of soft X-ray loops with left
and right-handedness. $\sigma$-error bars are shown by vertical lines. From
Zhang et al. (2010a)

Figure 1.14 also depicts the imbalance of chiral soft X-ray loops in both
hemispheres. The reverse magnetic helicity with a relatively high tendency
occurred in 1991, 1992, and 1995 in the northern hemisphere, while it is
insignificant in the southern hemisphere.

Figure 1.15 shows the statistical distribution of the helical soft X-ray loops
with the latitudes. The mean latitude of soft X-ray loops migrates toward the
equator with the solar cycle in the form of a butterfly diagram. It is similar
to the distributed form of the sunspots. Due to very few soft X-ray loops in
1991, 1995, and 1996 in our statistics, the deviation from the butterfly
diagram in these years can be noted. Besides most of the large-scale loops
following the helicity hemispheric sign rule, the statistical distribution of
the reverse helical soft X-ray loops relative to the sign rule can be found in
Figure 1.15 also.

Figure 1.16:  Left: Correlation of $\alpha_{best}$ of the active region pairs
connected by transequatorial loops. Right: Relationship of $\rho_{h}$ of the
active region pairs connected by TLs. X axis presents the helicity values of
active regions in the southern hemisphere, Y axis shows the values in the
northern hemisphere. Error bars (when present) correspond to $1\sigma$ of the
mean helicity values from multiple magnetograms of the same active region.
Points without error bars correspond to active regions represented by a single
magnetogram. After Chen, Bao & Zhang (2007)

Moreover, from the time scale of solar cycles, the trans-equatorial X-ray
loops connected the solar active regions from both hemispheres and the
relevant current helicity of these pair regions have been statistically
provided by Chen, Bao & Zhang (2007). Figure 1.16 exhibits the helicity
correlation of the 43 pairs of active regions which are connected by
transequatorial loops (Chen, Bao & Zhang, 2007). For the helicity parameter
$\alpha_{best}$ ($\sim\overline{(\nabla\times{\bf B})_{z}/B_{z}}$ in Eq.
(1.7)), 22 pairs (51%) of active regions show the same helicity patterns and
21 pairs (49%) of active regions own the opposite chirality. For the parameter
$\rho_{h}$ ($\displaystyle={\sum{h_{c}(i,j)}}/{\sum{|h_{c}(i,j)|}}$), 26 pairs
(60%) of active regions show the same signs and 17 pairs (40%) the opposite
signs. The results of both proxies exhibit that the active region pairs
connected by transequatorial loops do not necessarily contain the same
chirality. It reflects the large-scale poloidal magnetic field in the high
solar atmosphere. Whether these loops contain more information about the
global transformation of magnetic helicity with solar cycles is worthy of
further study.

####  1.5.2 Helicity with Solar Flare Cycles

The relationship between magnetic helicity and the activity of the flare-
coronal mass ejections is a notable topic. It relates to the transformation of
magnetic helicity from the solar atmosphere into the interplanetary space (cf.
Rust, 2002; Yang et al., 2012; Wang & Zhang, 2015; Kim et al., 2017; Park et
al., 2021).

Figure 1.17: Yearly mean values of $\overline{|h_{cz}|}$ and $|\alpha_{av}|$
of AR samples during 1988-2008. Diamonds represents the yearly mean values of
the samples that produced flares with $\textrm{FI}\geq 10.0$ in the following
24 hours (flare-productive samples). Dots represents the yearly mean values of
the samples that did not produce flares with $\textrm{FI}\geq 10.0$ in the
following 24 hours (flare-quiet samples). The monthly mean sunspot relative
numbers during the same period are also overlapped (dashed line). From Yang et
al. (2012)

For statistical studying the photospheric magnetic nonpotentiality in solar
active regions and its relationship with associated flares, Yang et al. (2012)
selected 2173 photospheric vector magnetograms from 1106 active regions (ARs)
observed by the Solar Magnetic Field Telescope at Huairou Solar Observing
Station in 1988 – 2008, which covers most of the 22nd and 23rd solar cycles. A
series of parameters have been calculated, which include the mean absolute
current helicity density, absolute averaged twist force-free parameter, mean
free magnetic energy, and other non-potential parameters of each vector
magnetogram.

Yang et al. (2012) selected the active samples (i.e., flare-productive ARs)
are defined as the ARs with the equivalent flare strength greater than a
typical value (such as M1.0 flare here) within the same subsequent time window
(24 hours). The rest of them belong to the quiet samples (i.e., flare-quiet
ARs).

In Figure 1.17, the mean current helicity $\overline{|h_{cz}|}$ of the flare-
productive active regions shows statistically higher values than those for
flare-quiet ones in the solar maximum. However, the twist factor
$|\alpha_{av}|$ of active regions shows an insignificant difference between
the quiet and the active samples, in comparison with $\overline{|h_{cz}|}$.
However, it is noticed that the mean helicity density $\overline{|h_{cz}|}$ of
the flare-productive active regions shows two delayed peaks relative to the
sunspot maximum in 1992 and 2005. A similar case of $|\alpha_{av}|$ can also
be found in 2005, respectively. It is consistent with the results calculated
from the magnetic helicity in Figures 1.9 and 1.13. It probably reflects that
the formation on the peaks of the flare-productive active regions with strong
magnetic helicity tends to be statistically delayed than that of sunspot
numbers.

###  1.6  Questions on Solar Dynamos with Observation of Magnetic Helicity

It is noticed that exploring the variation of magnetic helicity with the solar
cycles from different perspectives is of great significance to understanding
the law of formation of the solar internal magnetic field, which relate to the
emergence of twisted magnetic flux in active regions (c.f. Longcope et al.,
1998; Fisher et al., 2000; Kleeorin et al., 2020; Kuzanyan et al., 2020).

In comparing with observations of magnetic helicity, Choudhuri (2003) argued
that an extremely important question is whether, from a solar dynamo
perspective, is it merely a statistical fluctuation of current helicity, or is
there some systematic aspect in it, whether certain current helicity is
preferential at certain latitudes and time appears. Our paper’s more extensive
data analysis provides a chance to settle this question. On theoretical
grounds, they expected that the flux tubes may have corresponding current
helicity systematically in certain latitudes at certain times. They suggested
that flux tubes at the bottom of the convection zone are advected equatorward
by the equatorward meridional flow there, whereas the poloidal field at the
surface is advected poleward by the poleward meridional flow there. A
systematic study of the observed signs of current helicity in different
latitudes in different phases of the solar cycle should throw important light
on the nature of the solar dynamo. The extended numerical simulation for the
solar dynamo with helicity has been presented by Choudhuri et al. (2004).

A similar presentation is that Xu et al. (2009) employed a very simple Parker
dynamo model and direct generalizations of the latitude time distribution of
the helicity obey a polarity rule that is similar to Hales polarity rule.
There is a stable phase shift between the helicity and the toroidal magnetic
field in the dynamo model. Their dynamo models are more sophisticated than
Parker’s simplest model which could predict phase shifts and butterfly
diagrams. It is natural to suppose that a description of the evolution of the
helicity in terms of a differential equation could considerably modify the
phase shift between the helicity and the toroidal magnetic field to compare
with the observed variations in the magnetic field and helicity with solar
cycles (Bao et al., 2000; Zhang et al., 2010b).

It is defined by the evolution of the averaged small-scale magnetic helicity,
$\overline{\chi}=\overline{\mathbf{a}\cdot\mathbf{b}}$ ($\bf{a}$ and is
$\bf{b}$ fluctuating part of the magnetic vector potential and magnetic field
respectively). For the isotropic turbulence (Moffatt, 1978), the current
helicity is related with magnetic helicity, $h_{c}={\overline{{\bf
b}\cdot{\nabla\times{\bf b}}}\sim\overline{\chi}}/\ell^{2}$. The evolution
equation for $\overline{\chi}$ can be obtained from the equations which govern
the evolution of $\mathbf{a}$ and $\mathbf{b}$, it reads as follows (Kleeorin
& Rogachevskii, 1999):

$\displaystyle\frac{\partial\overline{\chi}}{\partial t}$ $\displaystyle=$
$\displaystyle-2\left(\boldsymbol{\mathcal{E}}\cdot\overline{\bf{B}}\right)-\frac{\overline{\chi}}{R_{m}\tau_{c}}-\boldsymbol{\nabla}\cdot\boldsymbol{\boldsymbol{\mathcal{F}}}-\eta\overline{\mathbf{B}}\cdot\mathbf{\overline{J}},$
(1.14)

where
$\boldsymbol{\mathcal{E}}=\alpha_{0}\overline{\mathbf{B}}+\left(\mathbf{\overline{V}}^{(p)}\times\overline{\mathbf{B}}\right)-\eta_{T}\left(\nabla\times\overline{\mathbf{B}}\right)$,
and $\overline{\mathbf{B}}$, $\overline{\mathbf{J}}$ and
$\overline{\bf{V}}^{(p)}$ are mean part of the magnetic field, current and
velocity field respectively, the equipartition value
${\displaystyle\overline{B}_{\rm
eq}\sim\sqrt{4\pi\overline{\rho}\overline{u^{2}}}}$, $R_{m}$ is magnetic
Reynolds number, and the helicity fluxes
$\boldsymbol{\boldsymbol{\mathcal{F}}}=\mathbf{\overline{a\times
u}}\times\mathbf{B}-\mathbf{\overline{a\times(u\times
b)}}-\overline{\mathbf{b}\phi}$, $\phi$ is an arbitrary scalar function which
is related with a gauge of the vector potential (see Kleeorin et al., 1995;
Kleeorin & Rogachevskii, 1999 and references therein). The subsurface kinetic
helicity in solar active regions has been inferred from the helioseismology
(e.g. Gao et al., 2009, 2012; Komm & Gosain, 2019), which tends with the
opposite sign relative to the current helicity statistically (e.g. Gao et al.,
2009, 2012) (to the hemispheric current helicity sign rule).

The relatively small time scale fluctuation or oscillation of mean magnetic
helicity in the solar cycles has been performed in Figures 1.7, 1.12, and
1.13. Some of the theoretical explanations for the evolution of magnetic
helicity in eq. (1.14) in the framework of mean-field dynamo can be found
(Kleeorin et al., 2003; Zhang et al., 2006; Zhang et al., 2012; Yang et al.,
2020) in comparing with the observations. These reflect that the generation of
the magnetic field is a relatively complex process in the convection zone, and
one probably cannot use a simple model to explain the twisting process of the
magnetic field inside of the Sun. It also provides us an important opportunity
to study the solar cycle variation of the magnetic helicity from the
perspective of the characteristics of magnetic turbulence based on the solar
magnetic fields (Hoyng, 1993; Zhang, 2012).

The importance of the magnetic helicity with the kinetic helicity and the
solar dynamo has been noticed (cf. Pouquet et al., 1975; Kleeorin & Ruzmaikin,
1982; Brandenburg & Subramanian, 2005; Rädler & Rheinhardt, 2007). The
$\alpha$ effect is produced by the kinetic helicity and also current helicity,
and it is
${\displaystyle\alpha_{0}=-\frac{\tau_{c}}{3}\left(\overline{{\mathbf{u}}\cdot{\nabla\times}{\mathbf{u}}}-\frac{\overline{{\bf
b}\cdot{\nabla\times{\bf b}}}}{4\pi\overline{\rho}}\right)}$. The latter
effect is interpreted as a resistance of magnetic fields against a twist by
helical motions (Vainshtein & Cattaneo, 1992; Rüdiger & Kitchatinov, 1993).
This effect has been introduced a concept as called “catastrophic quenching”
of the $\alpha$ effect related to the generated large-scale magnetic field. It
was found that

${\displaystyle\alpha_{0}\left(\overline{B}\right)=\frac{\alpha_{0}\left(0\right)+\eta
R_{m}\overline{\bf
B}\cdot\overline{\mathbf{J}}}{1+R_{m}\left(\overline{B}/\overline{B}_{\rm
eq}\right)^{2}}}.$

In the case of $R_{m}\gg 1$, the $\alpha$ effect is quickly saturated for the
large-scale magnetic field if the strength is much below the equipartition
value $\displaystyle\overline{B}_{\rm eq}$ (Ossendrijver et al., 2001).

Furthermore, we may not simply relate many questions in the solar dynamo
mechanism (c.f. Moffatt & Dormy, 2019) from the observational perspective of
magnetic helicity, but a statistically observed magnetic helicity with the
solar cycle provides constraints for this study. A further question is to what
extent we can really diagnose the mechanism inside the sun through the
observed transmission and evolution of magnetic helicity in the solar
atmosphere.

###  1.7 Discussions

In this chapter, we have presented some relationships between magnetic
helicity and solar cycles. We have introduced the magnetic helicity and
current helicity and their possible relationship from the solar observations.
For analyzing the relationship between the magnetic helicity with solar
cycles, we introduce the butterfly diagram of the mean current helicity
density inferred from the vector magnetic fields of active regions and the
relationship with the sunspot butterfly diagram in Figure 1.6 and 1.7, and
soft X-ray loops in Figure 1.15. Another notable topic is the relationship
between the magnetic (current) helicity and solar flare-coronal mass ejections
with solar cycles. It has presented in Figure 1.17. We also have presented the
injection of magnetic helicity with solar cycles inferred from full-disk
magnetograms from the space and ground bases, which provides an aspect of the
magnetic helicity that differs from the current helicity calculated by the
photospheric vector magnetic field in solar active regions. The message which
we infer from the magnetic helicity with solar cycles is as follows:

  1. 1.

The twist and helicity butterfly diagrams of active regions show anti-
symmetric relative to the solar equator. It confirms the hemispheric rule of
magnetic helicity of active regions in 11-year cycles (such as Ding et al.,
1987; Pevtsov, Canfield, and Metcalf, 1995; Bao & Zhang, 1998; Kleeorin et
al., 2003; Zhang & Yang, 2013; Liu et al., 2022). It actually reveals the new
veil on the dynamo process of the magnetic field inside of the Sun.

  2. 2.

The study of current helicity in the solar surface in eq. (1.6) and injective
magnetic helicity through the solar surface in eq. (1.11) reflect different
aspects of the twisted magnetic field with handedness. The basic relationship
between them can also be estimated in the simple form in eq. (1.13), so one
can find a similar tendency in that the magnetic and current helicity also
shows a temporal delay of the maximum of helicities relative to that of
sunspot number in the solar cycles in Figures 1.9 and 1.13.

  3. 3.

The coronal soft X-ray loops reflect the extension of the photospheric
magnetic field. It is found that the handedness of most of the loops tends to
follow the hemispheric rule. The interesting question is how the twisted
magnetic field extends from the photosphere into the corona within the solar
cycles. A similar question is also for the quiescent and active filaments in
the solar atmosphere (Martin et al., 1994; Ouyang et al., 2017), and also how
the magnetic helicity spreads out from the active region, and whether there
are other possible causes.

  4. 4.

The variation of magnetic helicity with solar flare activity was reported by
Bao et al. (1999). We have presented some results on the statistical
correlation between them with solar cycles. However, the complexity between
them is not completely clear, because more magnetic parameters need to be
optimally analyzed for the solar flare-eruptive phenomena (Yang et al., 2012).
It is also noticed that the statistical magnetic (current) helicity related to
flares also shows a temporal delay than sunspot number with solar cycles in
Figure 1.17. Moreover, the relationship between solar flare-coronal mass
ejections with magnetic helicity is still a notable question (Georgoulis et
al., 2009; Wang & Zhang, 2015; Kim et al., 2017; Park et al., 2021). This
involves the basic issues of non-potential magnetic field and the relationship
with magnetic reconnection, which is beyond the scope of discussion here.

  5. 5.

Some basic problems related to observing and studying solar magnetic (current)
helicity with the solar cycles. They involve the measurement and analysis
accuracy of magnetic helicity and the completeness of data. It can be found
from Eq. (1.6) that it is impossible to obtain the complete current helicity
value in the solar active regions, and it is also difficult to obtain complete
data of vector magnetic field in all of the solar active regions on the solar
surface. In the calculation of magnetic helicity transfer in the solar surface
(such as Eq. (1.11)), there are also problems in the measurement accuracy of
magnetic field and velocity field, and it is difficult to cover the whole
surface of the Sun. If we also consider the observation error of the
instruments, it is naturally easy to understand that there are many
uncertainties in the research, and the differences in the observation and
calculation results of different authors (cf. Ai, 1989; Ronan et al., 1992;
Wang et al., 1992; Sakurai et al., 1995; Sakurai, 2001; Zhang et al., 2003;
Pevtsov et al., 2005; Xu et al., 2012). There are still some open questions
until now.

###  1.8 Acknowledgements

The authors would like to thank the Huairou Solar Observing Station, National
Astronomical Observatories, Chinese Academy of Sciences, and the Solar &
Heliospheric Observatory (SOHO) for providing the observational data. The
authors would like especially to thank Profs. D. Sokoloff, A. Brandenburg, T.
Sakurai, S. Bao, Drs. Y. Gao, K. Kuzanyan, and V. Pipin for their kindly
discussions in the study of the magnetic helicity with the solar observations.
This study is supported by grants from the National Natural Science Foundation
(NNSF) of China under the project grant 11673033, 11427803, 11427901,
12073040, 12073041, 12003051, 11973056, and other grants at Huairou Solar
Observing Station, National Astronomical Observatories, Chinese Academy of
Sciences.S. Y. acknowledges support by grants 11427901, 10921303, 11673033,
U1731113, 11611530679, and 11573037 of the National Natural Science Foundation
of China and grants no. XDB09040200, XDA04061002, XDA15010700 of the Strategic
Priority Research Program of the Chinese Academy of Sciences and the Youth
Innovation Promotion Association of CAS (2019059).

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