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%%%%%%%%%%%%%%%%%%% TITLE PAGE %%%%%%%%%%%%%%%%%%%

% Title of the paper, and the short title which is used in the headers.
% Keep the title short and informative.
\title[Injection of magnetic helicity in solar cycle 24]{Injection of magnetic helicity in solar cycle 24 from the observations of magnetic fields}

\author[H. Zhang, S. Yang, K. Kuzanyan, A. Brandenburg, H. Xu and D. Sokoloff]
{ H. Zhang$^{1}$\thanks{E-mail: hzhang@bao.ac.cn},
S. Yang$^{1}$, K. Kuzanyan$^{1,2}$, A. Brandenburg$^{3,4}$, H. Xu$^{1}$, \& D. Sokoloff$^{2,5}$
 \newauthor { }
\\$^{1}$National Astronomical Observatories, Chinese Academy of Sciences, Beijing 100012, China,
\\$^{2}$IZMIRAN, Russian Academy of Sciences, Troitsk, Moscow 142190, Russia
\\$^{3}$Nordita, KTH Royal Institute of Technology and Stockholm University, Roslagstullsbacken 23, SE-10691 Stockholm, Sweden
\\$^4$Department of Astronomy, AlbaNova University Center, Stockholm University, SE-10691 Stockholm, Sweden
\\$^{5}$Department of Physics, Moscow University, 119992 Moscow, Russia
}




% These dates will be filled out by the publisher
\date{Accepted XXX. Received YYY; in original form ZZZ}

% Enter the current year, for the copyright statements etc.
\pubyear{2015}

% Don't change these lines
\begin{document}
\label{firstpage}
\pagerange{\pageref{firstpage}--\pageref{lastpage}}
\maketitle

% Abstract of the paper
\begin{abstract}

The injection of magnetic helicity with solar cycle 24th has been calculate based on the analysis of  a series of magnetic synoptic charts.
%The injective magnetic helicity  with solar cycles is contributed from the magnetic fields in active regions mainly.
%AB: changed like so:
During the cycle, the injective magnetic helicity is mainly contributed by the magnetic field in active regions.
It is hard to find the helicity near the polar regions due to the eruption of the helicity into the interplanetary space from the solar surface.
The mean values of the injective magnetic helicity flux is consistent with the hemispheric sign rule of current helicity of solar active regions, i.e.\ the negative (positive) sign tends to be in the northern (southern) hemisphere.
Significant fluctuations of the injected magnetic helicity from the
%subatmosphere probably reflects the complex process of twisted magnetic
%AB: subatmosphere -> subsurface layers
subsurface layers probably reflect the complex process of how the twist
%field inside of the convection, which bring out by magnetic fields
%AB: "twisted magnetic field ... bring out by magnetic fields" was unclear
%AB: Is the following still what you mean?
from the convection zone ejects magnetic fields
%characterized by a series of  active regions with different temporal
%AB: modified
through a series of active regions on different temporal
%and spatial scales in the solar surface.
%AB: modified
and spatial scales at the solar surface.
\end{abstract}

% Select between one and six entries from the list of approved keywords.
% Don't make up new ones.
\begin{keywords}
Sun -- Photosphere -- Magnetic field
\end{keywords}


%%%%%%%%%%%%%%%%% BODY OF PAPER %%%%%%%%%%%%%%%%%%

\section{Introduction}

Helicity is an important quantity that reflects the topology of
the field, such as linkage, twist, and writhe of the field lines
%\citep{Woltjer58a,Woltjer58b,Taylor86} and also an integral measure of
%AB: new sentence
\citep{Woltjer58a,Woltjer58b,Taylor86}. It is an integral measure of
the topological properties of the field in a closed volume $V$:
\begin{equation}
\label{ }
H_f =\int{\bf F}\cdot\nabla \times {\bf F}dV,
\end{equation}
where $\bf F$ can be the magnetic vector potential {\bf A}, the magnetic
%field {\bf B}, and the velocity field {\bf V}, which is
%AB: and -> or, inserted "assumed to be"
field {\bf B}, or the velocity field {\bf V}, which is assumed to be
confined to the volume $V$.

%In the solar atmosphere, the magnetic field and velocity field are
In the solar atmosphere, the magnetic and velocity fields are
observable quantities with vector magnetographs, while the magnetic
potential {\bf A} and the electric current density {\bf J}
%$\left(=\nabla\times{\bf B}/\mu_0\right)$ are the derivative quantities
%AB: the derivative ->  derived
$\left(=\nabla\times{\bf B}/\mu_0\right)$ are derived quantities
%under some assumptions \citep[cf.][]{Zhang00,Zhang19, ZhangHQ03} in the sense of astrophysics.
%AB: not clear what is meant by "in the sense of astrophysics". Could we omit it?
under some assumptions \citep[cf.][]{Zhang00,Zhang19, ZhangHQ03}.
%Computation of helicity on the Sun requires knowledge of vector field in
%AB: inserted "the full"
The computation of helicity in the Sun requires the knowledge of the full vector field in
%a 3D region, but observations of the magnetic fields are usually taken in
%AB: region -> volume
a 3D volume, but observations of the magnetic fields are usually taken in
a shallow layer of the solar atmosphere (typically, in the photosphere).
Due to the limitation of the observations of vector magnetic fields
in a single layer of the solar atmosphere, it is difficult to construct
%the real distribution of the whole components of the electric current
%AB: the whole -> all, electric current -> current density
the real distribution of all components of the current density
in the solar photosphere \citep{Xu15}.
%This means that we still cannot get the total components of electric
%AB: the total -> all, omitted "electric", J is just "current density"
This means that we still cannot get all components of the
%current density, even if in the lower solar photosphere, and to infer
%AB: can one write like so?
current density, not even in the lower solar photosphere.
%the spatial distribution of the current.
%AB: I think this is a bit redundant, so I omitted it.
The similar case is also for the magnetic potential {\bf A}.
This means that the completeness of the helicity has been lost from the
calculations by means of the observations of vector magnetograms.

%Even through, the magnetic helicity density $h_m={\bf A}\cdot\nabla\times {\bf A}$,
%AB: Even through -> Nevertheless
Nevertheless, the magnetic helicity density $h_{\rm m}={\bf A}\cdot\nabla\times {\bf A}$,
current helicity density $h_c={\bf B}\cdot\nabla\times {\bf
B}$, and velocity helicity density $h_v={\bf V}\cdot\nabla\times {\bf V}$
are still important detectable quantities.
For example, one can calculate a part of the current helicity density
$h_c=3({\bf B})_z\cdot(\nabla\times {\bf B})_z$ inferred from observed
photospheric vector magnetograms under the isotropic assumption
\citep{Xu15} and the injection rate of magnetic helicity from the solar
%sub-atmosphere \citep{cha01} based on the photospheric magnetograms.
%AB: sub-atmosphere -> subsurface regions
subsurface regions \citep{cha01} based on the photospheric magnetograms.
%These mean the real analysis of the helicities is started from the
%AB: slight changes; still ok like so?
This implies that the analysis of helicities involves
observable quantities in the solar atmosphere only.
In addition, it is normally believed that the complex distribution
of magnetic (current) helicity density in the solar active
%regions relates with the trigger of the powerful flare-CMEs
%AB: slight changes; still ok like so?
regions relates with the trigger of powerful flares and CMEs
\citep[cf.][]{Bao99,liu06,Zhangy08}.

The helical topology of magnetic fields in active regions was firstly
observed from the handedness of sunspot penumbral configurations by \cite{Hale08}
%and statistically with hemispheres by \cite{Ding87}.
%AB: modified
and statistically with respect to hemispheres by \cite{Ding87}.
%The helicity with hemispheric rule was subsequently analyzed by
%AB: modified
The hemispheric sign rule of helicity was subsequently analyzed by
\cite{Seehafer90}, who found that the current helicity density and
twist in solar active regions follows the hemispheric helicity rule with
predominantly negative values in the northern hemisphere and positive
values in the southern hemisphere.
%It is noticed that the reverse of magnetic helicity  sign in the
It is noted that a violation of the magnetic helicity sign rule in the
%beginning of each solar cycle was firstly presented by \cite{bao00} based
beginning of each solar cycle was discovered by \cite{bao00} based
on observations of a series of vector magnetograms at Huairou Solar
Observing Station and a reversal of the helicity sign rule with
time by \cite{Hagino05} from the observations at National Astronomical
Observatory of Japan.
%The Distribution of current helicity of active regions in 22nd and 23rd
%solar cycle was presented by \cite{zetal10}, the injection of magnetic
%AB: 22nd and 23rd solar cycle -> solar cycles 22 and 23
The distribution of current helicity of active regions in solar cycles
22 and 23 was presented by \cite{zetal10}, the injection of magnetic
%AB.
helicity by \cite{YangSB12} and \cite{Zhang13}, and also the magnetic
%helicity by \cite{pipin19} in 24th solar cycle based on the vector field
%AB: similarly
helicity by \cite{pipin19} for solar cycle 24 based on the vector field
synoptic maps.

The importance of the magnetic helicity for the solar dynamo has been
noticed \citep[cf.][]{pouquet-al:1975,kleruz82,bra-sub:05}  after the
development of the solar dynamo theory with the turbulence of velocity
field in the solar convection zone (i.e., the early stage of the mean
field dynamos).
%It is noticed that \cite{Choudhuri04} explained the hemispheric helicity
%could reverse sign at the beginning of each cycle by means of the
%flux transport dynamo model after the observations by \cite{bao00}.
%AB: reordered
By means of a flux transport dynamo model, \cite{Choudhuri04} explained
that the hemispheric helicity could reverse sign at the beginning of
each cycle, as seen in the observations of \cite{bao00}.
%AB.
A similar work has been proposed based on the analytical solution of
the mean field dynamo model by \cite{Xu09}, who found that a reversed
sign of mean hemispheric helicity can also occur in the decaying
stage of a cycle.
While the more complex distribution of magnetic helicity with the solar
cycle can be found from the butterfly diagrams of mean current helicity
of solar active regions in solar cycles 22 and 23 \citep{zetal10},
and also confirmed {from} the injection of global magnetic helicity from
the solar surface by \cite{YangSB12} and \cite{Zhang13}.

Section~2 presents the injection of magnetic helicity during solar cycle
24 inferred by the large-scale magnetic fields. Section~3 describes the
magnetic helicity and the relationship with evolution of magnetic field
in the individual active regions. Section~4 discusses our findings.

\section{Injection of magnetic helicity in solar cycle 24}

It is generally believed that the magnetic fields and the corresponding
helicity in the solar surface are important parameters to understanding
%the possible solar dynamo process in the solar convection zone,
%AB: minor edits in this line
possible effects on the solar dynamo process in the solar convection zone
where the $\alpha$ effect is one of the key parameters in this
process \citep[e.g.][]{par55,Steenbeck66,Zeldovich83}.
The magnetic (current) helicity is an important ingredient in the $\alpha$
%component \citep[e.g.][]{bra-sub:05}.
%AB: minor edits in this line
effect; see \cite{bra-sub:05} for a review.

The collective effect of the active regions on the contribution of
%magnetic (current) helicity with solar cycles have been studied from
%AB: minor edits in this line
magnetic (current) helicity with the solar cycle has been studied from
observations \citep[such as][]{zetal10,Yangx12,ZhangBS2016}
%and also with the solar dynamo models \citep[such as,]
%[]{ketal03,Choudhuri04,Xu09,pietal13mn}.
%AB: minor edits in this line
and also with solar dynamo models \citep[see, e.g.,]
[for example]{ketal03,Choudhuri04,Xu09,pietal13mn}.

In the following, we study the injection of magnetic helicity during solar
cycle 24, using the method of \cite{YangSB12} and \cite{Zhang13}.
We also study the distribution of the magnetic helicity flux in the
butterfly diagram.

\subsection{Method}

The transfer of magnetic helicity is accompanied with the dynamo
process within the Sun.
Corresponding  discussions have been provided by \cite{kleruz82}
and \cite{ketal95}.
The variation of the mean value of the fluctuating small-scale magnetic helicity,
$h_{\rm m}=\langle{\bf a\cdot b}\rangle$, can be written in the form
\begin{equation}
\label{eq:heliflux3}
\begin{aligned}
\frac{\partial h_{\rm m}}{\partial t}=&-2\langle({\bf u}\times {\bf b})\cdot {\bf B}\rangle-2\eta_{m}\langle({\bf b}\cdot(\nabla\times {\bf b})\rangle\\
&-\langle(\nabla\cdot[{\bf a}\times ({\bf u\times b})]\rangle,
\end{aligned}
\end{equation}
where {\bf b}, {\bf u}, and {\bf B} are the fluctuating magnetic
field, the fluctuating velocity field, and the mean magnetic field,
respectively.
%The effective electric field $\bf E_{eff}\equiv\langle({\bf u}\times
%AB: electric field -> electromotive force
The effective electromotive force $\bf E_{eff}\equiv\langle({\bf u}\times
{\bf b})\rangle=\alpha {\bf B}\!-\!\eta(\nabla\times {\bf B})$
is give by \cite{M78,Parker1979CMF,Zeldovich83,bra-sub:05}, so
equation~(\ref{eq:heliflux3}) becomes
\begin{equation}
\label{eq:heliflux4}
\begin{aligned}
\frac{\partial h_{\rm m}}{\partial t}=&2[\eta{\bf B}\cdot(\nabla\times {\bf B})-\alpha{\bf B}^2- \eta\langle({\bf b}\cdot(\nabla\times {\bf b})\rangle]-\nabla\cdot {\bf F}_{\rm hel},
\end{aligned}
\end{equation}
where $\alpha=\alpha_v+\alpha_{\rm m}$ and $\eta=\eta_{\rm m}+\eta_T$.
The integrals of the magnetic helicity in the northern and southern
hemispheres are, even if the total magnetic helicity is normally
conserved, can be written
\begin{equation}
\label{eq:heliflux}
\begin{aligned}
   \frac{d H_{\rm m}}{\partial t}=&\int_V2[\eta{\bf B}\cdot(\nabla\times {\bf B})-\alpha{\bf B}^2- \eta\langle({\bf b}\cdot(\nabla\times {\bf b})\rangle]dv\\
%   &+\oiint_S(h_m{\bf V}-{\bf F}_{hel})\cdot d{\bf s},
   &-\oiint_S{\bf F}_{\rm hel}\cdot d{\bf s},
\end{aligned}
\end{equation}
where the helicity flux ${\bf F}_{\rm hel}= \bf \langle a\cdot b\rangle
V-\langle(V\cdot a)b\rangle   +\langle a\times u\rangle \times
B+\langle a\times(u\times b)\rangle \cdots$, and $\bf a$ is the
fluctuating magnetic potential and $\bf V$ is the mean velocity field
\citep{kle-rog99,bra-sub:05}.
The first term in the right-hand side of equation~(\ref{eq:heliflux}) relates
to the generation of magnetic helicity field inside the domain $V$
(such as the Sun) and the second to the injection of helicity flux from
%the surface $S$ in the perspective of the solar dynamo.
%AB: perspective -> exterior (or what do you mean?)
the surface $S$ in the exterior of the solar dynamo.

%Moreover, the injective magnetic helicity from the  surface containing
%AB: starting a paragraph with Moreover isn't so good.
The injective magnetic helicity from the surface containing
%the twisted or linkage magnetic fields can be written in the form
%AB: minor changes
twist or linkage of magnetic fields can be written in the form
\citep{BergerField84,Demoulin03}
\begin{equation}
\label{eq:heliflux1}
\begin{aligned}
F_{\rm m}&=\frac{dH_{\rm m}}{dt}=\frac{d}{dt}\int_V h_{\rm m}\, dv\\
&=-2\oiint_S[({\bf V}_t\cdot {\bf A}_p){\bf B}_n -({\bf
A}_p\cdot {\bf B}_t){\bf V}_n]\cdot d{\bf s},
\end{aligned}
\end{equation}
where the magnetic field ${\bf B}$ and the velocity field ${\bf V}$ are
observed quantities in the solar atmosphere, and the boundary value
of the magnetic vector potential ${\bf A}_p$ of the potential of the reference field reference field can be inferred
from the vertical component of the magnetic field $B_n$. 


where the magnetic field ${\bf B}$ and the velocity field ${\bf V}$
%are observed quantities in the solar atmosphere, and the magnetic vector
%potential ${\bf A}_p$ of the potential reference field can be inferred
%from the vertical component of the magnetic field $B_n$.
%AB: minor reordering
are observed quantities in the solar atmosphere, and the boundary value of
the magnetic vector potential of the potential reference field, ${\bf A}_p$,
can be inferred from the vertical component of the magnetic field $B_n$.
The subscripts $n$ and $t$ indicate the normal and transverse component,
respectively.

\begin{figure*}
\begin{center}
 	\includegraphics[width=165mm]{butterfly_helicityac}	
 \end{center}
 \vspace{5mm}
    \caption{ a) Butterfly diagram of magnetic fields in solar cycle 24th. The white (black) in positive (negative) polarity. b) The distribution of the injective rate of magnetic helicity in the both hemispheres. The white (black) shows positive (negative) sign. c) The  injective rate of magnetic helicity in the both hemispheres after the smooth. The red (blue) contours show the relative injective rates of 1,  5, 20, 50 $\times 10^{32}$(Mx)$^2/s$ in the positive (negative) signs.
    }
    \label{fig:butterfly1}
\end{figure*}

The first term in the right hand side of equation~(\ref{eq:heliflux1})
provides the contribution from the horizontal motion of footpoints of
%magnetic field in the solar surface, while the second one does that
%vertical motion of the magnetic flux from the integral surface.
%AB: slightly modified
magnetic field at the solar surface, while the second one reflects the
contribution from the vertical motion of magnetic flux at the surfaces
of the integral.

Both equations~(\ref{eq:heliflux4}) and (\ref{eq:heliflux1}) have been used
to analyze the evolution of magnetic helicity from different perspectives.
%The formal equation has been used in the study of the magnetic helicity
%AB: formal -> former
The former equation has been used in the study of the magnetic helicity
with the dynamo process, and latter on the injective flux of magnetic
helicity from the enveloping surface, such as the solar surface.
%The consistent sense of both would be suitable for the analysis of the
%evolution of the large-scale magnetic fields with solar cycles.
%AB: can we also say this like so?
Agreement between the two methods can be taken as an indication of the
validity of the analysis concerning the evolution of large-scale magnetic
fields over the solar cycle.

%The vector velocity field in the photosphere also can
%AB: can we also say this like so?
The three components of the velocity field in the photosphere can also
be derived from the Differential Affine Velocity Estimator for Vector
Magnetograms \citep[DAVE4VM;][]{Schuck08}.
As one neglects the second term in the right hand side of
%Eq.~(\ref{eq:heliflux1}), it caused the errors of about 10\% in the
%AB: how about this
equation~(\ref{eq:heliflux1}), there is an error of about 10\% in the
%calculation of the injected magnetic helicity as comparing with the method
%AB: comparing -> compared
calculation of the injected magnetic helicity as compared with the method
DAVE4VM, as pointed out by \cite{Liu12} from their calculated results.

%According to the analysis of \cite{Demoulin03}, one can determine that
%AB: can determine -> finds
According to the analysis of \cite{Demoulin03}, one finds that
\begin{equation}
\label{eq:heliflux2}
\frac{dH_{\rm m}}{dt}=-2\oiint_S({\bf U}\cdot {\bf A}_p){\bf B}_n\cdot d{\bf s},
\end{equation}
where
\begin{equation}
\label{ }
{\bf U}={\bf V}_t-\frac{V_n}{B_n}{\bf B}_t.\nonumber
\end{equation}
%This means that the influence of the second term in eq. (\ref{eq:heliflux1}) has been neglected because only the line of sight magnetograms have been used.
%AB: means -> implies
This implies that the influence of the second term in
equation~(\ref{eq:heliflux1}) has been neglected because only the line
of sight magnetograms have been used.
It has been used in our following calculation, as the one ignores the influence of some errors discussed above.

It is noticed that equation~(\ref{eq:heliflux2}) with the local correlation
tracking (LCT) method \citep{cha01} for calculating the horizontal
velocity field of magnetic features has been used to analyze the
%large-scale injective magnetic helicity in solar surface  in solar cycle 23
%AB: in -> at the, in solar cycle 23 -> of cycle~23
large-scale injective magnetic helicity at the solar surface of cycle~23
by means of  MDI magnetic synoptic charts \citep{YangSB12}  and the MDI
96min full-disk magnetograms \citep{Zhang13}.
%It also shows the similar tendency on the large scale reversal sign
%AB: hyphen, added "of the"
It also shows the similar tendency on the large-scale reversal of the sign
distributions of magnetic helicity with solar cycles in the northern
and southern hemispheres as compared with the calculation of the mean
current helicity of active regions by \cite{zetal10}.

%In the following, we will study the injective helicity in solar cycle
%AB: omit "will"
In the following, we study the injective helicity in solar cycle
%24 and its possible latitudinal distribution with solar cycles.
%AB: solar cycles -> time
24 and its possible latitudinal distribution with time.

\subsection{ Distribution of injective magnetic helicity in solar cycle 24}

In this study, 122 HMI magnetic synoptic charts in the period
between 2010-June-02 and  2019-March-26 have been used in the
calculation to obtain the longitudinal butterfly diagrams in
Fig.~\ref{fig:butterfly1}a.
It includes the most part of the magnetic fields in solar cycle 24.
It is noticed that  most of the magnetic field  migrates toward the
equator, and also some to the polar regions \citep{Wangy89} along the
meridional flow in the solar convection zone \citep{Zhao13}.

For analyzing the transfer of the magnetic helicity contributed from
the  magnetic fields, Fig.~\ref{fig:butterfly1}b shows the butterfly
diagram of mean injective magnetic helicity inferred by a seres of HMI
synoptic charts by means of equation~(\ref{eq:heliflux2}).
It shows the mean values of injective helicity with the latitude at
various solar Carrington rotation cycles.
This result reflects the contribution of large temporal and
spatial scale magnetic fields to the magnetic helicity in the
magnetic synoptic charts. This means that the results reflected in
Figs~\ref{fig:butterfly1}a and \ref{fig:butterfly1}b show similar
temporal and spatial scales.

It is pointed out by  \cite{ZhangBS2016} that around the solar
maximum the magnetic energy and helicity spectra are steeper,
emphasizing the large-scale field.
This tendency can also be found in Fig.~\ref{fig:butterfly1}, where
%the large-scale structures contributed by the structure elements of magnetic field and helicity with the same signs tend in the maximum of the solar cycle.
%AB: could one also say like so?
the large-scale structures are the result of smaller elements of magnetic field and helicity of the same sign tend during the maximum of the solar cycle.

From Fig.~\ref{fig:butterfly1}, it is found that the injected magnetic
helicity at the solar surface shows a trace to the equator with the
migration of the large scale magnetic field and also shows a tendency
toward the northern and southern poles, but it is hard to find the
helicity in the polar regions.
It reflects that most of the helicity has been erupted with the evolution
of magnetic fields of active regions and cannot reach the solar polar
regions.
Figure~\ref{fig:butterfly1}b shows the injected magnetic helicity with
the variation of magnetic features corresponding to temporal and spatial
scales calculated from a series of synoptic magnetic charts.

Figure~\ref{fig:butterfly1}c shows the smoothed injective helicity
with solar cycle 24 for displaying the mean contribution of magnetic
helicity. It is easy to see that the relatively strong large-scale
%positive injective helicity patterns occur near solar rotation~2155
%AB: solar rotation~2155 -> Carrington rotation (CR) 2155
%in the southern hemisphere and that near 2175 in the northern hemisphere,
%and negative one occurs between the solar rotation 2115 and 2135 in the
%AB: added CR
positive injective helicity patterns occur near Carrington rotation (CR)
2155 in the southern hemisphere and that near CR~2175 in the northern
hemisphere, and negative one occurs between CRs~2115 and 2135 in the
northern hemisphere etc.
%
%AB: join paragraphs
%
%It also needs to be noted that the accuracy on the calculation of
%AB: noted -> emphasized, on -> of
It also needs to be emphasized that the accuracy of the calculation of
the magnetic helicity  in high latitudes is low due to the influence of
projective effects on the observed magnetograms and the corresponding
synoptic charts.

A similar pattern of the butterfly diagram of mean current helicity of
%active regions has been presented by \cite{zetal10} inferred from
%AB: inferred from -> using
active regions has been presented by \cite{zetal10} using
%the observed 6205 vector magnetograms of 984 solar active regions at
%AB: swap
the 6205 observed vector magnetograms of 984 solar active regions at
Huairou Solar Observing Station (most of the large solar active regions
in solar cycle 22 and 23 have been included).
From a theoretical point of view, consistence between equations~(\ref{eq:heliflux1})
and (\ref{eq:heliflux4}) can be found, because the variation of mean
magnetic helicity density ${\partial h_{\rm m}}/{\partial t}$ in
equation~(\ref{eq:heliflux4}) relates to the current helicity and also the
$\alpha$ factor of the solar dynamo statistically.
This means that the transfer of magnetic helicity (twisted magnetic
flux loops) brings the current helicity from the subsurface to the
solar surface, and the latter can be detectable from the photospheric
vector magnetograms.
The correlation between the change of magnetic and current helicity
in individual solar active regions has been discussed by \cite{liu06}
and \cite{Zhangy08}.


\begin{figure}
\begin{center}
	\includegraphics[width=50mm]{butterfly_helicityc}
 \end{center}
 \vspace{5mm}
    \caption{ The mean injective rate of magnetic helicity with the latitude in solar cycle 24th. The error bars show the relative deviation of the mean values only. The red line marks the distribution of injective magnetic helicity after the smooth.}
    \label{fig:butterfly4}
\end{figure}

Figure~\ref{fig:butterfly4} shows the mean injective rate  of magnetic
helicity with latitude.
It is found that the dominant sign is negative in the northern hemisphere
and positive in the southern  hemisphere.
This tendency can also be found from the smoothed result in
Fig.~\ref{fig:butterfly1}c.
It is consistent with the hemispheric sign rule of magnetic helicity
\citep{Seehafer90} and also the result for the solar cycle 23 by
\cite{YangSB12} and \cite{Zhang13}, and for the solar cycle 24 by
\cite{pipin19}.
Some slight difference relative to the results of \cite{pipin19} for
%solar cycle 24 is probably caused by the different methods.
%AB: inserted "the use of"
solar cycle 24 is probably caused by the use of different methods.
%The result of \cite{pipin19} mainly provides remained magnetic (current)
%AB: "remained" is unclear; how about this:
The result of \cite{pipin19} provide additional magnetic (current)
helicity in the solar atmosphere.

Figure~\ref{fig:butterfly3} shows the mean injective rate of magnetic
helicity in the northern and southern hemispheres with solar cycle 24
based on the calculation of a series of HMI synoptic magnetic charts.
The mean values in the northern and southern hemispheres are
$-83.7\,{\rm Mx}^2,{\rm s}^{-1}$ and $66.6\,{\rm Mx}^2\,{\rm s}^{-1}$,
respectively.
It provides a rough estimation on the distribution of the injective
magnetic helicity in the solar atmosphere.
The total flux of injective magnetic helicity flux in this calculation is
on the order of $10^{44}\rm Mx^2$ in the whole solar cycle 24, while it
is lower than $5\times10^{46}\,{\rm Mx}^2$, as estimated by \cite{Zhang13},
because this calculation is based on the magnetic synoptic charts, some
short temporal and spatial magnetic flux and its contribution for the
helicity flux have been ignored.

The difference on the estimation of the total magnetic helicity from
the solar surface between that by \cite{Zhang13} with the MDI 96 minute
full-disk magnetograms and our calculation with the magnetic synoptic
charts is of the order of $5\times10^2$.
The difference on the temporal intervals of the data series of the
magnetic field for the calculation of the injected helicity by us and that
by \cite{Zhang13} is a order of $4\times10^2$.
This means that a certain mount of injected helicity has been lost in
our calculation.
%This also means that \cite{Zhang13} provide a low value of the injective
%AB: can we say like so?
This also means that the results of \cite{Zhang13} provide a lower estimate
%of the injective helicity from the solar surface with the solar cycles,
%AB: with -> during
of the injective helicity from the solar surface during the solar cycles,
because the contribution from short time scales of less than 96~minutes
has been neglected.

\begin{figure}
\begin{center}
	\includegraphics[width=50mm]{butterfly_helicityb}
 \end{center}
 \vspace{5mm}
    \caption{ The injective rate of magnetic helicity in the northern (red) and southern (black) hemisphere along the latitude in solar cycle 24th. }
    \label{fig:butterfly3}
\end{figure}

Figure~\ref{fig:longiheli} shows the distribution of the mean magnetic
field with longitude in the northern and southern hemispheres in solar
cycle 24 and the corresponding injective rate of magnetic helicity
inferred from the magnetic synoptic charts, which are contributed by
the magnetic fields of a series of active regions.
In Figs \ref{fig:longiheli}a and d, it is easy to see the slanted
arrangement of a series of magnetic field structures like scratches in
both hemispheres caused by the differential rotation of the Sun.
One can also see a large-scale pattern of the injective helicity after the
data are smoothed in Fig.~\ref{fig:longiheli}c and \ref{fig:longiheli}f.
It is consistent with the idea that the dominant sign of the injective
helicity shows a negative sign in the northern hemisphere and a positive
one in the southern hemisphere.
It is also true of Fig.~\ref{fig:butterfly1}c and \ref{fig:butterfly4}.
The relatively strong large-scale negative injective helicity patterns
%occurs near solar rotation 2135 in the northern hemisphere and positive
%ones  near the solar rotation 2155 in the southern hemisphere.
occurs near CR~2135 in the northern hemisphere and positive
ones near CR~2155 in the southern hemisphere.
This is consistent with the result in Fig.~\ref{fig:butterfly1}c.

\begin{figure*}
\begin{center}
	\includegraphics[width=165mm]{longitudefly_helicityc}
 \end{center}
\vspace{5mm}
\caption{Left: mean magnetic field with the longitude in the northern
(a) and southern (d) hemisphere in solar cycle 24.
Middle: the corresponding injective rate of magnetic helicity in the
both hemispheres (b and e).
Right: injective rate of magnetic helicity in the both hemispheres
(c and f) after smoothing.
Red (blue) contours show the injective rates of 1, 5, 10, and
$20\times 10^{32}\,{\rm Mx}^2\,{\rm s}^{-1}$ in positive (negative) signs.}
\label{fig:longiheli}
\end{figure*}

%\section{Magnetic helicity with individual active regions}
%AB: with -> from
\section{Magnetic helicity from individual active regions}

Although the injection of magnetic helicity in the solar atmosphere
has been presented from photospheric vector magnetograms \citep[such
as][]{Zhang01, liu06}, the comparison between the averaged magnetic
helicity and the individual contributions from active regions is also
useful for understanding the contribution of magnetic helicity from the
solar surface.
%
%AB: the above paragraph is only one sentence, so maybe better
%AB: merge with the rest:
%
To analyze the magnetic helicity in the lower solar atmosphere, we
introduce a sample of active regions such as active region NOAA12673
and the subsequent relative active regions.
Active region NOAA12673 was observed in September 2017 by the
\textit{Heliospheric Magnetic Imager} (HMI) and \textit{Atmospheric
Imaging Assembly} (AIA) on board the \textit{Solar Dynamics Observatory}
(SDO).
This active region was a typical fast developing $\delta$ region with
4 X-class and 27 M-class flares during its passage across the visible
solar disk \citep[cf.][]{Yang17,Sun17,Romano19}.

\begin{figure}
\begin{center}
	\includegraphics[width=40mm]{eme12673b.eps}
	\includegraphics[width=40mm]{Fluxi.eps}
 \end{center}
    \caption{ Left: The interaction between the existed magnetic fields in active region NOAA 12673 (the green line with arrow) and the reversely emerging magnetic flux (red one) from deep convection zone in the top of view on September 4, 2017. Size is $125''\times 125''$. Right: The relationship between the magnetic field between active region NOAA 12673 and 12674 in the large scale field of view. Size is $750''\times 750''$. The top is the north and left is east.}
    \label{Fig:topo2}
\end{figure}

\begin{figure*}
\begin{center}
	\includegraphics[width=160mm]{synoptic2017_4dif_vels2}
 \end{center}
\vspace{5mm}
    \caption{ Left: Evolution of large scale magnetic fields of active region NOAA 12673 and the relationship with active regions NOAA 12670, 12682, and 12685 in the synoptic charts of different Carrington rotations 2193-2196. The blue dashed lines with arrows mark the regular direction of magnetic fields in the convection zone according to the Hale polarity low of magnetic fields. The contour levels (black/white) of the smoothed magnetic fields are $\pm$5, 20, 50, 100, and 200 G.
Right: Injective rate $G=-({\bf V}_t\cdot {\bf A}_p)B_n$ (white (black) shows the positive (negative) sign), and the horizontal velocity (arrows) inferred by LCT.  }
    \label{Fig:syncharts}
\end{figure*}

The magnetic field in active region NOAA 12673 and the possible topology
of overlapping magnetic lines of force  in the top of view are shown in
Fig.~\ref{Fig:topo2}.
The newly emerging magnetic flux is marked by red arrowed lines and the
existing magnetic field by green ones.
Comparing with the evolution of the large-scale magnetic field, it is
found that the new magnetic flux emerges near the photospheric inversion
line of the active region to form highly sheared magnetic field in the
solar atmosphere.
This probably reflects the interaction of two different systems of
magnetic lines of force, i.e., the new reversed magnetic lines formed
in the deeper layer of the sub-atmosphere occurred in the developed
active region to reverse the distribution of the magnetic field in the
photosphere from Hale's rule \citep{Getling19}.
%This means the local evolution of the magnetic fields in the solar surface
%probably controls by the magnetic fields formed in the different layers
%of the convection zone.
%AB: slightly reordered
This probably means that the local evolution of the magnetic field at
the solar surface is controlled by the magnetic field formed in different
layers of the convection zone.

The statistical analysis of the emergence of magnetic flux with twisted
magnetic loops from the subsurface was proposed by \cite{Longcope98}.
Although it is  favorable to compare with observations in both mean
value and statistical dispersion, it is also very important to track
the evolution process of magnetic flux  with helicity in individual
active regions.

Figure~\ref{Fig:syncharts} shows a series of synoptic charts of the
%magnetic field at different solar Carrington rotation cycles and the corresponding time difference.
%AB: why rotation cycles? Can't we just say:
magnetic field at different CRs and the corresponding time difference.
It provides an opportunity to analyze the large-scale magnetic fields
for the detection of a possible evolution of magnetic fields from the
subsurface.
This is a sequence of solar active regions NOAA 12670, 12673, 12682,
and 12685.
It is found that the leading and following polarities of the large-scale
%magnetic bipoles change in time before and after the active region NOAA 12673.
%AB: should one say "emergence of", like so?
magnetic bipoles change in time before and after the emergence of
active region NOAA 12673.
This provides evidence for a reversed variation of the observed
large-scale magnetic bipole of active regions due to the emergence of
negative magnetic helicity flux from the deeper solar convection zone.
This is similar for the highly sheared magnetic field occurred in the
active region NOAA12681 in the southern hemisphere with the injection
of magnetic helicity.

Figure~\ref{Fig:syncharts} also shows the injective rate $G\!=\!-({\bf
V}_t\cdot {\bf A}_p)B_n$ calculated from the synoptic magnetic charts
and the corresponding horizontal velocity ${\bf V}_t$ by LCT after
some smoothing.
The horizontal velocity arrows due to the evolution of the large-scale
magnetic fields in the solar surface are inferred from LCT.
%The consistency in the direction of the velocity field with evolved
%AB: is |evolving direction" ok?
The consistency in the direction of the velocity field with the evolving
direction of the large-scale magnetic fields can normally be found,
%such as that of the active region NOAA 12670-12673-12682.
%AB: such as that of -> for example, use commas
for example in the active regions NOAA 12670, 12673, and 12682.
It is also noticed that the contribution of the helicity in the quiet
Sun is negligible.

It is found that a major contribution of the magnetic helicity
$G\!=\!-({\bf V}_t\cdot {\bf A}_p)B_n$ comes from the active regions, even
if this only provides the variation of magnetic fields on the synoptic
scale, while the relatively small temporal scale ones have been ignored.
%However, comparing with whole solar cycles, it is also can be
%AB: modified
However, comparing with all solar cycles, it can also be
presented as the contribution from the fluctuated magnetic fields.
%
%AB: merge paragraphs (second one too short)
%
From the evolution of the large-scale magnetic fields in
Fig.~\ref{Fig:syncharts}, we suggest that these reflect the local
exchange between the poloidal and toroidal fields emerging from in
the subsurface.

\begin{figure*}
\begin{center}
%\hspace{25mm}	
\includegraphics[width=80mm]{AutoCorrelationTime_64Lat.eps}
\includegraphics[width=80mm]{AutoCorrelationTime_79Lat.eps}
 \end{center}
\vspace{0.05mm}
\caption{Auto correlation function distribution of the different
latitude along the time direction of  CR2193 (labeled by two straight
white horizontal solid lines in the figure \ref{Fig:syncharts}a.)
Left: relative to active region, the low line and right: the quiet Sun,
the top line in figure \ref{Fig:syncharts}a.
}\label{fig:autocorrelation}
\end{figure*}

In Fig.~\ref{Fig:syncharts} it is seen that the contribution of the
horizontal velocity with the injective magnetic helicity occurs in local
areas of active regions.
This occupy only a relative small amount in the magnetic charts due
to the smaller contribution from the quiet Sun.

Since we use the LCT method to obtain the horizontal velocity, there
may be a loss of some magnetic helicity signal.
Here we use the auto-correlation function to evaluate the
lost.
As an example, Fig.~\ref{fig:autocorrelation} shows the auto-correlation
function for different latitudes along the time direction of CR~2193 in
Fig.~\ref{Fig:syncharts}a.
%It is found that the distribution function is much sharper at the region
%AB: distribution -> auto-correlation
It is found that the auto-correlation function is much sharper at the region
without sunspots.
While using the LCT method, this may cause a low correlation
and underestimates the movements at small scales.
Thus, we obtain a relatively small amplitude of magnetic helicity
fluxes compared with previous results, such as those of \cite{YangSB12}
and \cite{Zhang13}.
However, this could still reflect helicity flux characteristic at the
specific spatial resolution and temporal cadence.
This is also consistent with the local contribution of injective helicity
from the large-scale field distributed in Fig.~\ref{Fig:syncharts}a.

\section{Explanation of injective magnetic helicity with solar cycles}

As one follows the theoretical analysis of the solar dynamo and
comparing with the observations, the magnetic field can be separated
to the mean field component and fluctuation field one. The formal
one relates to the mean field dynamo  with large scale helicity
\citep{ketal03,2012ApJ...751...47Z}, while second one to the fluctuated
helicity, probably with some exchange of the signs of the helicity in
the solar surface. The emergence of magnetic field brings the magnetic
helicity from the solar convection zone into the solar surface. It is
usually believed that in this process the total magnetic helicity tends
to be conserved in the approximation of large Reynolds number, and the
fluctuation contribution of the different scale magnetic helicity is
neglected in many studies of the  solar cycle.

It is noticed that the obvious fluctuations of injective helicity
occurred at the surface comparing on solar cycle scales; see, for example,
Figs~\ref{fig:butterfly1}b, \ref{fig:longiheli}b, \ref{fig:longiheli}e,
%and \ref{Fig:syncharts}, as the finding by \cite{Zhang13}.
%AB: should one insert "well as"?
and \ref{Fig:syncharts}, as well as the findings by \cite{Zhang13}.
This reflects the generation of magnetic fields in the convection zone
is a complex process, i.e., the helical magnetic field probably formed at
the different layers in the deep solar convection zone and with different
helical tendencies probably.

\begin{figure*}
\begin{center}
%\hspace{25mm}	
\includegraphics[width=150mm]{dynamo201905}
 \end{center}
\vspace{5mm}
\caption{The schematics on the relationship between the magnetic field
and injective helicity with the solar cycle. Left: large scale magnetic
field with the solar cycle. The solid (dotted) lines show the positive
(negative) polarity. Middle: the large-scale magnetic field overlapped
by the mean values of large scale injective magnetic helicity, the
blue is negative and green is positive sign.
Right: the large-scale magnetic field overlapped by the small-scale
dispersed helicity from fluctuating magnetic fields to morphologically
compare Figs~\ref{fig:butterfly1}b and c.
}\label{fig:butterfly5}
\end{figure*}

%It is noticed that amount of magnetic flux has been injected from
%AB: is this change ok?
It is interesting that the amount of magnetic flux injected from
the convection zone into the solar atmosphere during the solar cycle
%as provided above by means of the eq.~(\ref{eq:heliflux2}), and
%AB: is this change ok?
is provided by means of equation~(\ref{eq:heliflux2}), and
%this is roughly relevant with the description of the second term
%AB: is this change ok?
this is roughly compatible with the description of the second term
$\displaystyle-\oiint_S{\bf F}_{\rm hel}\cdot d{\bf s}$ in the right-hand
side of equation~(\ref{eq:heliflux1}).

For analyzing the distribution of magnetic helicity with the cycle, the
simplest case of a solar dynamo by \cite{par55} has been used to compare
with the behavior of an $\alpha\omega$ dynamo wave near a given latitude
$\theta$, and the equations assume the form \citep{ketal03}
\begin{equation}
\begin{aligned}
\frac{\partial B}{\partial t}+ V\frac{\partial B}{\partial
\theta}&=D\frac{\partial A}{\partial \theta}+\lambda \frac{\partial^2
B}{\partial\theta^2 },\\
\frac{\partial A}{\partial t}+ V\frac{\partial A}{\partial
\theta}&=\alpha B+\lambda \frac{\partial^2 A}{\partial \theta^2},\label{eq:dyna161}
\end{aligned}
\end{equation}
where $V$ is the large-scale flow, $D$ is the dynamo number ($D\sim\Omega(\theta,r)$), where is the angular velocity and $r$ is in the direction of the solar radius) and $\lambda$ is the turbulent diffusivity. $D(\Omega)$ and $\alpha$
are topologically important parameters relative to the stretch and twist of the magnetic field in the solar dynamo model. The solar differential rotation is related to the quantity $D(\Omega)$.
 The spectral analysis of $\alpha$ effect has been presented in the theory of solar dynamo \citep[such as][]{M78,Parker1979CMF,Zeldovich83,bra-sub:05}. The contribution of the different typical scales of the velocity and magnetic fields is a notable question.

As one follows the Figs~\ref{fig:butterfly1}-\ref{Fig:syncharts} and the
variation of the magnetic field with the depth, we can assume the form
of a traveling wave in equations~(\ref{eq:dyna161}) in the form  of the
Fourier components \cite[such as][]{Zeldovich83}
\begin{equation}
\label{ }
\begin{aligned}
B&=\sum_m b_m\cdot \exp(\gamma_m t + i\kappa_m\theta+\varphi_{m})\\
A&=\sum_n a_n\cdot \exp(\gamma_n t + i\kappa_n\theta+\varphi_{n}),
\end{aligned}
\end{equation}
where $B$ and $A$ are composed of different time and spatial scale components $b_m$ and  $a_n$, respectively.%, {\color{magenta} as we can believe that the typical space-time scale and field are relative concepts in the  magnetohydrodynamics and mathematics \citep{M78}.} %This means that the magnetic field has been combined by some components with different temporal and spatial scales.

As the phase difference between $\varphi_m$ and $\varphi_n$ and the interaction of the different temporal and spatial scales of magnetic fields and their potentials are negligible (i.e. $\varphi_m=\varphi_n$, $\gamma_m=\gamma_n$ and $\kappa_m=\kappa_n$),
the magnetic helicity density can be written in the form \citep[see][]{Xu09,Popova10}
\begin{equation}
\label{}
\begin{aligned}
H_{\rm m}= A\cdot B%=&\left(\sum_n a_n\cdot \exp(\gamma_n t + i\kappa_n\theta+\varphi_{n}) \right)\\
%&\left(\sum_m b_m\cdot \exp(\gamma_m t + i\kappa_m\theta+\varphi_{m}) \right)\\
=&\sum_n a_nb_n\cdot \exp(2\gamma_n t +2i\kappa_n\theta)+....,%\\
%&=(\frac{\alpha}{\kappa |D|})^{\frac{1}{2}}\cdot \exp(2\gamma t+2i\kappa\theta+\frac{i\pi}{4}).
\end{aligned}\end{equation}
where the magnetic helicity consists with different temporal and spatial scale helicity components.
Thus the flux change of magnetic helicity can be written in the form
\begin{equation}
\label{eq:maghelich}
\begin{aligned}
\frac{\partial H_{\rm m}}{\partial t}= \sum_n a_nb_n2\gamma_n\cdot \exp(2\gamma_n t +2i\kappa_n\theta)+....%\\
%&=(\frac{\alpha}{\kappa |D|})^{\frac{1}{2}}\cdot \exp(2\gamma t+2i\kappa\theta+\frac{i\pi}{4}).
\end{aligned}
\end{equation}
and it can be written, after some calculation  \citep{Zhang12}  in the form 
\begin{equation}
\label{ }
\frac{\partial H_{\rm m}}{\partial t}=\sum_n2\gamma_n\left(\frac{\alpha_n}{\kappa_n |D_n|}\right)^{{1}/{2}}\exp\left(2\gamma_n t +2i\kappa_n\theta+\frac{i\pi}{4}\right).
\end{equation}

Figure~\ref{fig:butterfly5} shows the schematics of the distribution of
magnetic field and injective helicity with solar cycles.
The mean large scale magnetic helicity tends to show a negative
(positive) sign in the northern (southern) hemisphere, while the reverse
of large-scale magnetic helicity in the decaying phase of the solar cycles
was analytically demonstrated by \cite[][]{Xu09,Popova10,Zhang12}.
For analyzing in more detail, the small-scale fluctuating components of
magnetic helicity have been overlaid with the large-scale ones.
It is roughly consistent with the distribution of the observed injective
helicity in the form of butterfly diagrams in Fig.~\ref{fig:butterfly1}
and consistent with the butter-diagram of the observed current helicity of
active regions by \cite{zetal10} (i.e.\ ${d H_{\rm m}}/{d t}\sim a_nb_n$
in equation~(\ref{eq:maghelich})).
This also probably can be used to reflect the formation and transformation
between the large scale toroidal and poloidal fields inside of the Sun.

\section{Discussions and conclusions}

In this paper, the collective effect of injective magnetic helicity
during solar cycles  and individual samples of magnetic helicity with
the evolution of magnetic field in active regions have been presented.
%
%AB: join this short sentence with rest
%
To analyze the helicity formation in the solar subsurface, the
collective effect of the injective magnetic helicity with the magnetic
fields of solar cycle 24 has been presented based on the analysis of
a series of magnetic synoptic charts.
The sign distribution of injective magnetic helicity shows the dispersed
form contributed from individual local regions (such as active regions)
with solar cycles, and it is hard to find the helicity in the polar
regions, due to the eruption of magnetic field from the sub-atmosphere
into the interplanetary space.

For presenting the evolution of magnetic helicity with solar cycles, a
morphological analysis of the helicity patterns with butterfly diagram
has been proposed, which are composed of different scale fluctuating
components, such as characterized by solar active regions.
This probably reflects the complex process of twisted magnetic field
inside of the convection due to the interaction the Coriolis force and
meridional circulation on the formation of magnetic lines of force.

We have analyzed the evolution of the magnetic helicity in active
region NOAA 12673 and the relationship with the sequence of solar active
regions NOAA 12670, 12682, 12685 in different solar Carrington rotation
cycles. The reversal magnetic helicity occurs in the solar surface with
emergence of newly emergence of magnetic flux in the active region.
This probably reflects the interaction the existed magnetic field of the
active regions with the new magnetic flux formed in the deep convection
zone with the injection of the magnetic helicity, with the  exchange of
the local poloidal and toroidal fields.

We have formally tried to reproduce the helicity distribution
in the solar surface in Fig.~\ref{fig:butterfly5} by means of
equation~(\ref{eq:maghelich}).
There is still the related fundamental question on the generation of
magnetic fields of the different scales with velocity fields in the
deep convection zone, while the strong magnetic flux loops are mainly
characterized  by  the solar active regions.

Although the calculation of the injective magnetic helicity flux with the
synoptic magnetic charts has been calculated based on the corresponding
temporal and spatial scales of magnetic fields, the detailed analysis of
the contribution of the evolution of the magnetic field with the solar
cycle still needs to be  based on the observed full disk magnetograms.
It is noticed that as the different temporal and spatial scale observed
solar magnetic fields have been used in the analysis, the statistical
results probably has some difference, while the trend probably is
the same.

\section*{Acknowledgements}

The authors would like to thank the %referee for his/her comments for improving the manuscript, and
staff at the \textit{Solar Dynamics Observatory} (SDO) for providing
the observational data.
The authors would like to thank Dr.\ V.\ Pipin for useful discussions.
This study is supported by grants from the National Natural Science
Foundation (NNSF) of China under the project grant 11673033, 11427803,
11427901, {11703042, 11911530089} and other grant at Huairou Solar
Observing Station, National Astronomical Observatories, Chinese Academy
of Sciences.
This work was supported in part through the Swedish Research Council,
grant 2019-04234.

%\newpage
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{\color{red}

\appendix

\section{DETAILED DESCRIPTION OF
THE SOLUTION OF EQUATION (3)  }

We can assume the form of a traveling wave for the discrete components of magnetic fields and their potentials in equations (1) and (2) as
\begin{align}
a_n&=a_{n0}\exp(\gamma_n t + i\kappa_n\theta)\\
b_n&=b_{n0}\exp(\gamma_n t + i\kappa_n\theta)
\end{align}
Under the assumption of the mirror dissymmetry of magnetic helicity $h_{\rm m}=AB=\sum a_nb_n$ with solar cycle, the similar simple solution and the corresponding picture of magnetic helicity are presented by \cite{Xu09}. The solution of the discrete components in the equations (\ref{eq:dyna162}) and (\ref{eq:dyna161}) gives the mirror asymmetrical component of magnetic helicity relative to the equator with opposite sign. Equations (\ref{eq:dyna162}) and (\ref{eq:dyna161}) can be written in the form as one neglects the subscripts
\begin{equation} %\label{eq:1}
%\left\{
 \left\{\begin{aligned}
\gamma a+i\kappa Va&=\alpha b-\lambda\kappa^2a\\
\gamma b+i\kappa Vb&=i\kappa Da-\lambda\kappa^2b.
                          \end{aligned}\right.
%                          \right.
                          \end{equation}
The dispersion relation can be written
 \addtocounter{equation}{0}
\begin{equation}
(\gamma+\lambda\kappa^2+i\kappa V)^2=i\kappa D\alpha,
\end{equation}
thus
\begin{equation}
\gamma=-\lambda\kappa^2-i\kappa V\pm\surd \overline{i\kappa
D\alpha}.
\end{equation}
As $D>0$ , we have
\begin{equation}
\gamma=-\lambda\kappa^2-i\kappa V\pm(\kappa
|D|\alpha)^{{1}/{2}}\frac{1+i}{\surd \overline{2}},
\end{equation}
and thus as $Re ~\gamma>0$
\begin{equation}
\gamma=-\lambda\kappa^2-i\kappa V+(\kappa
|D|\alpha)^{{1}/{2}}\frac{1+i}{\surd \overline{2}}.
\end{equation}
For $D<0$ , we have
\begin{equation}
\gamma=-\lambda\kappa^2-i\kappa V\pm(\kappa
|D|\alpha)^{{1}/{2}}\frac{-1+i}{\surd \overline{2}},
\end{equation}
and thus as $Re ~\gamma>0$
\begin{equation}
\gamma=-\lambda\kappa^2-i\kappa V+(\kappa
|D|\alpha)^{{1}/{2}}\frac{1-i}{\surd \overline{2}}.
\end{equation}
%{\color{blue}
{ For maximum  $Re ~\gamma$, then
\begin{equation} -\lambda\kappa^2+\frac{1}{2^{{1}/{2}}}(\kappa|D|\alpha)^{{1}/{2}}=Max,
\end{equation}
and
\begin{equation}
\frac{d\gamma}{d\kappa}=-2\lambda\kappa+\frac{1}{2^{{3}/{2}}}\kappa^{-{{1}/{2}}}(|D|\alpha)^{{1}/{2}}=0.
\end{equation}
We find
\begin{equation}
\kappa=\frac{(|D|\alpha)^{{{1}/{3}}}}{2^{{5}/{3}}\lambda^{{2}/{3}}}.
\end{equation}

\begin{equation}
\begin{aligned}
\gamma&=-\lambda\kappa^2-i\kappa V+(\kappa|D|\alpha)^{{1}/{2}}\frac{1-i}{\surd \overline{2}}\\
&=-\lambda\left[\frac{(|D|\alpha)^{{{1}/{3}}}}{2^{{5}/{3}}\lambda^{{2}/{3}}}\right]^2-i\frac{(|D|\alpha)^{{{1}/{3}}}}{2^{{5}/{3}}\lambda^{{2}/{3}}}V+\left[\frac{(|D|\alpha)^{{{1}/{3}}}}{2^{{5}/{3}}\lambda^{{2}/{3}}}|D|\alpha\right]^{{1}/{2}}\frac{1-i}{\surd \overline{2}}               \\
&=-\frac{(|D|\alpha)^{{{2}/{3}}}}{2^{{10}/{3}}\lambda^{4/3}}-i\frac{(|D|\alpha)^{{{1}/{3}}}}{2^{{4/3}}\lambda^{2/3}}V+\frac{(|D|\alpha)^{{{2}/{3}}}}{2^{{4}/{3}}\lambda^{1/3}}(1-i).
\end{aligned}\end{equation}
As set $\lambda=1$, then
\begin{equation}
\begin{aligned}
\gamma&=-\frac{(|D|\alpha)^{{{2}/{3}}}}{2^{{10}/{3}}}+\frac{(|D|\alpha)^{{{2}/{3}}}}{2^{{4}/{3}}}  -i\left[\frac{(|D|\alpha)^{{{1}/{3}}}}{2^{{5}/{3}}}V+\frac{(|D|\alpha)^{{{2}/{3}}}}{2^{{4}/{3}}}   \right]\\
&=-\frac{3(|D|\alpha)^{{{2}/{3}}}}{2^{{10}/{3}}} -i\left[\frac{(|D|\alpha)^{{{1}/{3}}}}{2^{{5}/{3}}}V+\frac{(|D|\alpha)^{{{2}/{3}}}}{2^{{4}/{3}}}   \right].
\end{aligned}\end{equation}
}

As set $b=1$, then
\begin{equation}
a=\frac{\alpha}{\gamma+\lambda\kappa^2+i\kappa V}.
\end{equation}

As an aggregation of the discrete components of magnetic helicity density $a_nb_n$ has been considered, it can be written in the form
\begin{equation}
\begin{aligned}
A\cdot B&=\sum_na_nb_n\exp(2\gamma_n t +2i\kappa_n\theta)\\
&=\sum_n\frac{\alpha_n}{\gamma_n+\lambda_n\kappa_n^2+i\kappa_n V}\exp(2\gamma_n t +2i\kappa_n\theta)\\
&=\sum_n\frac{2^{{1}/{2}}\alpha_n}{(1-i)(\kappa_n
|D_n|\alpha)^{{1}/{2}}}\exp(2\gamma_n t +2i\kappa_n\theta)\\
&=\sum_n\frac{\alpha_n^{{1}/{2}}}{2^{{1}/{2}}(\kappa_n
|D_n|)^{{1}/{2}}}(1+i)\exp(2\gamma_n t +2i\kappa_n\theta)\\
&=\sum_n\left(\frac{\alpha_n}{\kappa_n |D_n|}\right)^{{1}/{2}}\exp\left(2\gamma_n t
+2i\kappa_n\theta+\frac{i\pi}{4}\right),
\end{aligned}\end{equation}
and the change of the magnetic helicity can be written in the form
\begin{equation}
\label{ }
\frac{\partial}{\partial t}(A\cdot B)=\sum_n2\gamma_n\left(\frac{\alpha_n}{\kappa_n |D_n|}\right)^{{1}/{2}}\exp\left(2\gamma_n t
+2i\kappa_n\theta+\frac{i\pi}{4}\right).
\end{equation}

}



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