# Magnetic and Velocity Field Topology in Active Regions of Descending Phase
of the Solar Cycle 23

R. A. Maurya  Department of Physics, National Institute of Technology Calicut,
Calicut-673601, India.  A. Ambastha  Udaipur Solar Observatory, Physical
Research Laboratory, Udaipur-313001, India.

###### keywords:

Helioseismology: observations, Helicity: Current, Magnetic; Active Regions:
Magnetic Fields, Velocity Field; Solar Cycle: Observations

figuret

##  1 Introduction

sec:intro

Solar active regions are three-dimensional magnetic structures extending from
the interior below the photosphere to the coronal heights. They consist of
magnetic flux tubes that emerge from the convection zone and extend into the
solar atmosphere (Hood, Archontis, and MacTaggart, 2011, and references
therein). Solar interior near the photosphere is an excellent conductor where
frozen-in-field condition may satisfy (Priest, 2014). Over a period of time,
near the photosphere, plasma flows may evolve with magnetic field lines, and
may swirl around them. A flux tube emerging from the convection region is
prevented from being destroyed by the hydrodynamics vortex behind it when
twisting effects are present (Fan et al., 1998). Therefore, we can expect
topological association between the sub-photospheric flows and photospheric
magnetic fields.

Mathematically, the topology of a vector field $\mathbf{X}$ is represented by
the parameter “helicity”. It is defined by the volume integral of the scalar
product of the vector field and its curl, $\mathbf{Y}=\nabla\times\mathbf{X}$,

$H=\int_{V}\mathbf{X}\cdot\mathbf{Y}\>{\rm d}V.\ilabel{eq:hel-def}$ (1)

The helicity is a fundamental topological quantity in laboratory as well as in
astrophysical plasma. In general, helicity (Equation eq:hel-def) describes
what kind of handedness is preferable for the field $\mathbf{X}$, left or
right hand screw. If $\nabla\times\mathbf{X}$ is clockwise when viewed from
ahead of the body, then helicity is positive; and if counter-clockwise, it is
negative. The helicity defined in Equation eq:hel-def is a global parameter
over the entire volume $V$.

The magnetic helicity of a divergence-free field $\mathbf{B}$ within a volume
$V$, bounded by a surface $S$, is defined by (Elsasser, 1956),

$H_{\rm m}=\int_{\rm V}\mathbf{A}\cdot\mathbf{B}\>{\rm d}V\ilabel{eq:mag-hel-
def}$ (2)

where, the vector potential $\mathbf{A}$ satisfies,
$\mathbf{B}=\nabla\times\mathbf{A}$. However, Equation eq:mag-hel-def is gauge
invariant (independent of $\phi$ where
$\mathbf{A^{\prime}}=\mathbf{A}+\nabla\phi$) when the magnetic field is fully
contained inside the volume $V$. The helicity $H_{{\rm m}}$ is a measure of
linking, knotting and twisting of field lines (Moffatt, 1969; Berger and
Field, 1984). Note that there is a subtle mathematical difference between the
general definition of the helicity $H$ (Equation eq:hel-def) and the magnetic
helicity $H_{{\rm m}}$ (Equation eq:mag-hel-def). In the definition of the
helicity $H$ of the field $\mathbf{X}$, we integrate the scalar product of the
field $\mathbf{X}$ with its curl, while for the helicity $H_{{\rm m}}$ of
magnetic field $\mathbf{B}$, we integrate the scalar product of vector
potential $\mathbf{A}$ of magnetic field $\mathbf{B}$ (and NOT $\mathbf{B}$
itself) with its curl. This makes difficult to compute the magnetic helicity,
as the vector potential $\mathbf{A}$ is gauge invariant and the full 3D
magnetic fields are not observable. Therefore, force-free parameter
$\alpha_{f}$ given in the equation,

$\nabla\times\mathbf{B}=\alpha_{f}\mathbf{B},\ilabel{eq:alpha}$ (3)

is taken as a proxy for the magnetic helicity by several researchers (Pevtsov
et al., 2008, and references therein). The topology of flow fields can be
given by the kinetic helicity (Moffatt and Tsinober, 1992; Moffatt, 1969). It
is defined as the volume integral of the scalar product of velocity
($\mathbf{u}$) and vorticity $\omega$ ,

$H_{{\rm k}}=\int_{{\rm V}}\mathbf{u}\cdot\boldmath{\omega}\>{\rm
d}V\ilabel{eq:kin-hel-def}$ (4)

where, vorticity, $\omega=\nabla\times\mathbf{u}$, is another important
topological parameter to describe the flow which represents circulation per
unit area or twist of fluid fields. It is a pseudo-vector (or axial vector)
that transforms like a vector under a proper rotation, but in three dimensions
again an additional sign flip occurs under an improper rotation such as a
reflection. Note that $H_{{\rm k}}$ retains the same mathematical form for the
flow field $\mathbf{u}$ as of $H$ for the vector field $\mathbf{X}$. It
represents extent to which corkscrew-like motion occurs. If a parcel of fluid
is moving and rotating about an axis parallel to the direction of motion, it
will be positive. In the absence of complete velocity field observations, we
compute the kinetic helicity density,

$h_{k}=\mathbf{u}\cdot\omega=\mathbf{u}\cdot\nabla\times\mathbf{u},\ilabel{eq:kin-
hel-den}$ (5)

as a proxy for the swirls in the sub-photospheric flows. The kinetic helicity
density, $h_{k}$, is a scalar quantity and it represents the component of
vorticity along the direction of flow. In magneto-hydrodynamics, the kinetic
helicity of flow field has similar mathematical form as that of the “current
helicity”,

$H_{{\rm c}}=\int\mathbf{J}\cdot\mathbf{B}\>{\rm d}V\ilabel{eq:cur-hel-def}$
(6)

where, $\mathbf{J}$ is current density which is related with magnetic field by
formula, $\mathbf{J}=\frac{1}{\mu_{0}}\nabla\times\mathbf{B}$, and
$\mu_{0}=4\pi\times 10^{-7}\,{Hm^{-1}}$ is the permeability of free space. The
current helicity quantifies the location of twist and sheared non-potential
structures in a magnetic field of ARs. It defines the degree of twist and
linkage of electric currents. Since full 3D magnetic fields are not observed,
we consider the current helicity density as follow,

$h_{\rm c}=(\nabla\times\mathbf{B})\cdot\mathbf{B}.\ilabel{eq:cur-hel-den}$
(7)

It is evident from Equation eq:mag-hel-def and eq:cur-hel-def that only
current helicity density can be estimated from the vector magnetograms under
the assumption of non-linear force-free field, and the calculation of magnetic
helicity density can only be made under the more restrictive assumption of
constant $\alpha_{f}$ of force-free field. However, the force-free parameter
$\alpha_{f}$ has the same sign as that of magnetic field under the assumption
of linear force field (Hagyard and Pevtsov, 1999). They are related by the
formula, $h_{\rm c}=\alpha_{f}B_{z}^{2}$ (Seehafer, 1990), which implies that
for a given AR, the current helicity has same sign as that of the magnetic
helicity. However, in a recent study, Russell et al. (2019) conjectured that
current and magnetic helicities having the same sign is not true in general.

In the solar chromosphere, magnetic structures such as H
${}_{\alpha}\leavevmode\nobreak\ $vortices, filament chirality, etc., manifest
the existence of helicity in ARs. The topology in all the above features
demonstrate a consistent pattern relating a large scale ordering of solar
magnetic fields. From a study of chromospheric H
${}_{\alpha}\leavevmode\nobreak\ $images of 51 sunspots from 1901 to 1944,
Hale (1927) first reported the preference of hemispheric trend in the H
${}_{\alpha}\leavevmode\nobreak\ $vortices. Later, Richardson (1941) included
some more sunspots and reported $71\%$ $(66\%)$ with left-handed (right-
handed) vortices in the northern (southern) hemisphere. The hemispheric sign
rule (HSR) has also been found in the chirality of chromospheric filaments
(e.g., Martin, Bilimoria, and Tracadas, 1994; Pevtsov, Balasubramaniam, and
Rogers, 2003; Lim and Chae, 2009).

Vector magnetic field observations of solar ARs have revealed that on average
solar ARs have a small but statistically significant mean twist and they show
hemispheric biases. The magnetic helicity (Pevtsov, Canfield, and Metcalf,
1995; Longcope, Fisher, and Pevtsov, 1998; Bao, Ai, and Zhang, 2000; LaBonte,
Georgoulis, and Rust, 2007; Yang, Zhang, and Büchner, 2009; Zhang and Yang,
2013; Liu, Hoeksema, and Sun, 2014) and the current helicity (Seehafer, 1990;
Bao and Zhang, 1998; Abramenko, Wang, and Yurchishin, 1996; Pevtsov, Canfield,
and Latushko, 2001; Hagino and Sakurai, 2005; Gosain et al., 2013, among
others) of ARs have been found to follow HSR. However, opposite HSR in the
global force free parameter $\alpha_{f}$ has been reported earlier during the
beginning (Bao, Ai, and Zhang, 2000) and decay phases (Tiwari,
Venkatakrishnan, and Sankarasubramanian, 2009; Pipin et al., 2019) of solar
cycles. From an analysis of several ARs, Zhang (2006) pointed out that the ARs
with strong magnetic fields show opposite hemispheric trend. Pevtsov et al.
(2008) compared data from four different instruments and concluded that HSR
sign change in some phases of the solar cycle is not supported at high level
of significance. Choudhuri, Chatterjee, and Nandy (2004) have numerically
shown the hemispheric preference of magnetic helicity parameter $\alpha_{f}$.
They also noticed opposite HSR at the beginning of solar cycle. However, there
is also some debate about this rule.

Twist in a magnetic flux tube can be formed due to: (1) Coriolis force acting
on rising flux tube (Holder et al., 2004) and $\Sigma$-effect (Longcope,
Fisher, and Pevtsov, 1998). The Coriolis force tilts one leg of the tube
towards equator and the other away. This deformation generates helical
distortion in the tube’s central axis. (2) Flux tube buffeted by the
turbulence of convection outside the tube during its rise through the
convection zone. The kinetic helicity of the turbulent flows in convection
zone twists the rising flux tube. (3) Background poloidal field (Choudhuri,
2003; Choudhuri, Chatterjee, and Nandy, 2004). The magnetic helicity is
expected to change its sign too because flows drags field lines with it
(Gilman, 1983). Longcope, Fisher, and Pevtsov (1998) showed fluid motion with
right handed coil deform the flux tube with a left handed.

Using helioseismic measurement of sub-photospheric flows, Komm et al. (2007)
have reported cyclonic vorticity (counter clockwise) in the northern
hemisphere. The hemisphere preference has also been found in other studies
(Gao, Zhang, and Zhao, 2009; Maurya, Ambastha, and Reddy, 2011; Seligman,
Petrie, and Komm, 2014; Komm, Gosain, and Pevtsov, 2014). There are a few
studies on the combined topology of sub-photospheric flows and photospheric
magnetic helicities (Gao, Zhang, and Zhao, 2009; Maurya, Ambastha, and Reddy,
2011; Seligman, Petrie, and Komm, 2014; Komm and Gosain, 2015). In some of
these studies, kinetic helicity has been found to show statistically opposite
HSR than that of current helicities (Gao, Zhang, and Zhao, 2009).

There are not enough studies reported on the topology of the combined
photospheric magnetic fields and sub-photospheric velocity fields from the
peak to descending phase of the solar cycle 23. Therefore, it would be
interesting to investigate the topological parameters of sub-photospheric
flows, photospheric magnetic fields and their hemispheric preferences,
associations. With this aim, we explore for any possible relationship between
the magnetic and kinetic helicities using photospheric vector magnetograms and
Dopplergrams observations of a large sample of ARs. Rest of the manuscript is
organized as follow: In Section sec:data-analysis, we describe the selection
of ARs and the observational data used. We discuss the methodology for
computing different helicity parameters. Section sec:result-discn is devoted
to the results and discussion, and finally, we present the summary and
conclusions in Section sec:sum-conc.

##  2 The Active Regions, Data and Analysis

sec:data-analysis

The main observational data used in this study consists of dopplergrams and
vector magnetograms of several ARs of the main to decay phase of the solar
cycle 23. Doppler observations of the full disc Sun for several years are
available from the Global Oscillation Network Group (GONG) while the
corresponding vector magnetic field observations are not available as such
during the aforementioned period. Therefore, for our studies, we have obtained
vector magnetograms from the Marshall Space Flight Center (MSFC). Furthermore,
GONG high resolution Doppler observations are available from 22 July 2001
onward while MSFC vector magnetograms are available only for the periods 19
September 2000 to 25 October 2004. Due to this constraint, we have chosen only
the common sets of data for the period of 22 July 2001 to 25 October 2004. The
other limitations are the availability of daily vector magnetic field data.
There are data gaps of several days. ARs far away from the disc center suffer
from the foreshortening effects (Maurya, Ambastha, and Chae., 2014, and
references therein) and produce spurious results in helioseismic computations.
Hence, we selected only the ARs within the radius of $\pm 40^{{\rm o}}$ in
heliographic longitude and latitude. In total, we short-listed 189 ARs for the
common period of MSFC and GONG observations, where some of these ARs were
observed during multiple rotations. In our sample of 189 ARs, 118 ARs were
found to be located in the southern hemisphere and 71 ARs in the northern
hemisphere. Thus the number of ARs for the southern hemisphere is larger than
that in the northern hemisphere. However, we do not expect our results on HSR
to be statistically affected by the different numbers of ARs in the two
hemispheres.

###  2.1 The Vector Magnetograms and Analysis

sec:mag-fields

The MSFC vector magnetograph facility was assembled in 1973 to support the
Skylab mission, and thereafter went through several improvements (MSFC,
Hagyard et al., 1982; Hagyard and Kineke, 1995; West et al., 2002). The vector
magnetic fields data were taken in the magnetically sensitivity spectral line
Fe i 5250.2 Å using the Zeiss birefringent filter with band pass of 125 mÅ.
Its spatial resolution over the $7^{\prime}\times 5.2^{\prime}$ field-of-view
is $0.64\mbox{${}^{\prime\prime}$}pixel^{-1}$. These observations are taken in
the far wing of the spectral line at -90 mÅ(or -120 mÅ) from the line center,
where the Faraday rotation effect decreases (Ronan et al., 1992). The pixel
resolution of the MSFC magnetograms is 1′′.28 along both x- and y-directions.

The MSFC observations consist of line-of sight magnetic field ($B_{l}$),
transverse component $(B_{t})$ and the azimuth angle $(\phi)$ of $B_{t}$. The
horizontal components $(B_{x},B_{y})$ of the vector magnetic field can be
computed from $B_{t}$ and $\phi$. We resolved the usual $180^{\rm\circ}$
ambiguity in the transverse components from all the vector magnetograms using
the acute angle method although there are several other methods available as
reviewed by Metcalf et al. (2006). Further, to avoid any projection effects,
we transformed the images to the disc center using the method described by
Venkatakrishnan and Gary (1989).

The MSFC vector magnetograms are not aligned in the east-west and north-south
directions which should be corrected for comparing the photospheric magnetic
fields with sub-photospheric flows obtained from the GONG. Active region’s
NOAA number and their central locations are provided in the header of every
vector magnetogram. However the central coordinates are found to be incorrect
in some cases. In order to correct the coordinates and orientation of the MSFC
vector magnetograms, we have used the full disc SOHO/MDI magnetograms for
reference. First of all, based on the location and time provided in the header
of the vectors magnetograms, we estimated corresponding location in the MDI
magnetograms and then after cross-correlating the ARs between MSFC and MDI
magnetograms, we corrected for the locations and orientations of MSFC
observations.

####  2.1.1 Determination of Magnetic and Current Helicities

sec:determine-mag-cur-hel

As discussed in the Section sec:intro, the force-free parameter $\alpha_{f}$
(Equation eq:alpha) can be used as a proxy for the magnetic helicity (Equation
eq:mag-hel-def). But, $\mathbf{B}$ is observed only at a single height, so we
can have its derivatives only along $x-$ and $y-$ directions. Thus, we can
compute only the vertical part of $\alpha_{f}$. Basically, there are two
approaches for calculating it: i) $(\alpha_{f})_{\rm best}^{z}$ which is the
best fit single value for the whole AR in the least squares sense (Pevtsov,
Canfield, and Metcalf, 1995), and ii) $(\alpha_{f})_{av}^{z}$ which is the
average of the vertical component of $\alpha_{f}$ over the entire AR (Bao and
Zhang, 1998; Abramenko, Wang, and Yurchishin, 1996; Hagino and Sakurai, 2004).
Both methods have been found to be in fairly good agreement to each other
(Leka and Skumanich, 1999). For a linear force-free field, the values of
$\alpha_{f}$ for a given position $(x,y)$, can be given by,

$\alpha_{f}^{z}(x,y)=\frac{\sum J_{z}\cdot B_{z}}{\sum
B_{z}^{2}}=\left[\sum_{x,y}\left(\frac{\partial B_{y}}{\partial
x}-\frac{\partial B_{x}}{\partial y}\right)\cdot
B_{z}\right]/\sum_{x,y}B_{z}^{2}\ilabel{eq:alpha-pix}$ (8)

where, numerical derivatives of magnetic field components can be determined
using the Lagrange’s three point interpolation. While computing the pixel
values of $\alpha_{f}^{z}(x,y)$ in an AR, we selected only those pixels where
the total magnetic field strength were above the quiet regions. To get the net
twist in an AR, we took average over the entire AR,

$(\alpha_{f}^{z})_{\rm
av}=\frac{1}{N}\sum\limits_{x,y}\alpha(x,y)\ilabel{eq:alpha-av}$ (9)

where, $N$ represents the total number of points where $\alpha_{f}^{\rm
z}(x,y)$ are estimated. Along with the magnetic helicity parameter
$(\alpha_{f}^{\rm z})_{\rm av}$, we computed the vertical current helicity,

$(h_{\rm c}^{\rm z})_{\rm av}=\frac{1}{N}\sum_{x,y}J_{z}\cdot
B_{z}=\frac{1}{N}\sum_{x,y}\left(\frac{\partial B_{y}}{\partial
x}-\frac{\partial B_{x}}{\partial y}\right)B_{z}\ilabel{eq:cur-hel-av}$ (10)

We have also computed the net absolute magnetic flux ($B$) for every AR by
adding the fluxes for those positions where helicity is estimated. From the
above expressions for the magnetic helicity parameter $\alpha_{\rm av}^{\rm
z}$ (Equation eq:alpha-av) and current helicity parameter $h_{\rm c}^{\rm z}$
(Equation eq:cur-hel-av), it is evident that both these parameters could have
same sign for a given AR.

###  2.2 Sub-photospheric Flows

sec:sub-surf-flows

The horizontal components ($u_{x}$, $u_{y}$) of the sub-photospheric flow
parameters for all ARs are computed from the Doppler observations using the
ring-diagram analysis (Hill, 1988). The basis of this method is that the
frequency of sound waves inside the solar interior are perturbed by the sub-
photospheric plasma motion,
$\delta\omega_{n,\ell}=\mathbf{k}_{n,\ell}\cdot\mathbf{U}_{n,\ell}$, where,
$\mathbf{k}_{n,\ell}$ is the propagation vector and $\mathbf{U}_{n,\ell}$ is
the flow vector for a given mode ($n,\ell$). Mode parameter
$\mathbf{U}_{n,\ell}$ can be computed by fitting the three dimensional Doppler
power spectra (Haber et al., 2000). It is related with the sub-photospheric
flows, $\mathbf{U}_{n,\ell}=\int
K_{n,\ell}(r)\mathbf{u}(r)dr+\epsilon_{n,\ell}$, where the kernel $K_{n,\ell}$
can be computed based on standard solar model (Christensen-Dalsgaard et al.,
1996), $\mathbf{u}(r)$ is the horizontal velocity at radius $r$, and
$\epsilon_{n,\ell}$ is the error in the mode parameter $U_{n,\ell}$. This is
an inverse problem, Fredholm equation of first kind, which can be solved using
inversion methods such as regularized least squares (Gough, 1985). For more
detail about the ring-diagram analysis one can refer to the review paper by
Antia and Basu (2007).

The sub-photospheric flows are largely affected by the solar differential
rotation and meridional circulations. We computed the residual flows by
removing these large scale trend following the methods as described in Komm et
al. (2004).

Ring-diagram yields only the horizontal components, $u_{x}$ and $u_{y}$, of
the flow, but not the vertical component, $u_{z}$, at any point in the
interior. We estimated the $u_{z}$ from the divergence of horizontal
components, assuming mass conservation (Komm et al., 2004), or steady-state
continuity equation,

$\frac{\partial\rho(z)u_{z}}{\partial
z}+\rho(z)(\nabla\cdot\mathbf{u})_{h}=0,\ilabel{eq:continuity}$ (11)

with horizontal divergence,

$div_{h}\equiv(\nabla\cdot\mathbf{u})_{h}\equiv\frac{\partial u_{x}}{\partial
x}+\frac{\partial u_{y}}{\partial y}.\ilabel{eq:div-h}$ (12)

Note that $z$ is the vertical height increasing with radial distance with the
origin on the surface. The interior is specified by a negative value of $z$.
By supposing $u_{z}$ on the surface, $z=0$, and integrating the continuity
Equation eq:continuity over $z$ from $d$ (negative) to 0, we obtain,

$u_{z}(d)=-\frac{1}{\rho(d)}\int_{0}^{d}\rho(z)(\nabla\cdot\mathbf{u})_{h}dz\ilabel{eq:uz}$
(13)

which has the same form as Equation (5) of Komm et al. (2004). Now, suppose
$(\nabla\cdot\mathbf{u})_{h}>0$ in the height range from $d$ to 0, then it
follows that $u_{z}(d)>0$. This means that a horizontally diverging flow rises
upward, and hence $u_{z}$ and $(\nabla\cdot\mathbf{u})_{h}$ are expected to
have the same sign.

####  2.2.1 Sub-photospheric Topological Parameters

sec:sub-surf-flow-top

Following, Equation eq:cur-hel-av for the current helicity, we computed the
vertical kinetic helicity density as follow,

$h_{\rm k}^{\rm z}=\left<u_{z}\cdot\omega_{z}\right>\ilabel{eq:kin-hel-den-z}$
(14)

where, the vertical vorticity is given by,

$\omega_{z}\equiv curl_{z}=\frac{\partial u_{y}}{\partial x}-\frac{\partial
u_{x}}{\partial y},\ilabel{eq:vert-vort}$ (15)

represents the twist of the flow around the vertical (radial direction). The
parameter $h_{\rm k}^{\rm z}$ is analogous to $h_{\rm c}$ (Equation eq:cur-
hel-den), and we have taken it as a proxy for the kinetic helicity density.

Note that the vertical kinetic helicity density $h_{\rm k}^{z}$ (Equation
eq:kin-hel-den-z) is obtained from the weighted average of $\omega_{z}$ by
$u_{z}$. But one should note that in strong magnetic field regions vertical
flows are suppressed due to the Lorentz force. The horizontal divergence
(Equation eq:div-h) of flows from the sunspot regions to outwards are common.
Therefore, weighted average of $\omega_{z}$ with $div_{h}$ would be a better
estimate for the vertical helicity. Such a parameter can be given by,

$C_{\rm k}^{\rm z}=\left<div_{h}\cdot curl_{z}\right>\ilabel{eq:div-curl-z}$
(16)

The parameter, $C_{\rm k}^{z}$, is expected to have same sign as that of
$h_{\rm k}^{z}$ because $div_{h}$ and $u_{z}$ are expected to have same sign
as explained earlier. This expression for the vertical twist has been used
earlier for studying the kinetic helicity of supergrannular flows (Egorov,
Rüdiger, and Ziegler, 2004) and sub-photospheric flows (Komm et al., 2007).
The validity of above helicity proxy has been confirmed for the depth range
obtained from ring-diagram as numerically established by Miesch et al. (2008).

In order to compute the sub-photospheric flow topological parameters for an
AR, we first computed them over patches of $60\hbox{${}^{\circ}$}\times
60\hbox{${}^{\circ}$}$ around the disc center. Then the parameters associated
with the ARs were interpolated based on their coordinates on the solar disc.

Earlier studies (Komm et al., 2005; Maurya, Ambastha, and Tripathy, 2009;
Maurya and Ambastha, 2010) have shown bipolar structure of sub-photospheric
flows in several ARs. Therefore, two different types of twists are possible in
the underlying flows of ARs. To deal with it, we analysed $h_{k}^{z}$ and
$C_{k}^{z}$ at the two depths of 2.4 Mm and 12.6 Mm. For distribution, the
associated parameters are labeled by the subscripts ‘1’ and ‘2’, e.g.,
$h_{k1}^{z}$ and $h_{k2}^{z}$, respectively.

The sub-photospheric flows of ARs are obtained from the 1664 minute time-
series duration with a cadence of one-minute Doppler observations and thus
correspond to average subsurface flows over a day. Therefore, for comparing
magnetic field topology with sub-photospheric flows of an AR, the magnetic
field topological parameters are also averaged accordingly.

Note that, next section onward, we have used the vertically averaged
quantities only, and therefore, we have dropped the scripts “z” and “av” from
all the topological parameters, e.g., we have used $h_{\rm c}$ instead of
$(h_{\rm c})_{\rm av}^{\rm z}$.

##  3 Results and Discussion

sec:result-discn

Results of our analysis of the 189 ARs are shown in Figures fig:mag-cur-hel-
lat – fig:correl-kin-mag-hel-depth. The hemispheric distribution of
topological parameters are summarized in the Tables tab:hemisp-distb and
tab:top-para-bav.

###  3.1 Photospheric Magnetic Fields

sec:phot-mag-fields

The photospheric magnetic field topology is studied in terms of magnetic
helicity parameter $\alpha_{f}$ (Section sec:mag-hel) and current helicity
$h_{c}$ (Section sec:cur-hel). In the following, we discuss the results
obtained.

####  3.1.1 Magnetic Helicity

sec:mag-hel

The results of our analysis for the magnetic helicity parameter $\alpha_{f}$
are shown in Figures fig:mag-cur-hel-lat – fig:mag-cur-hel-time. Figure
fig:mag-cur-hel-lat(a) shows the latitudinal distribution of magnetic helicity
parameter $\alpha_{f}$ for the 189 ARs. We found $68\%(67\%)$ ARs in the
northern (southern) hemisphere with negative (positive) $\alpha_{f}$. This
result further confirms the earlier reports on HSR for the magnetic helicity.
We notice that $\alpha_{f}$ shows large scatter in both the hemispheres.
Therefore, the following questions arise: Is the hemispheric sign
statistically significant? How do $\alpha_{f}$ values vary with latitude?

Figure 1.: (a) Latitudinal distribution of the force-free parameter
$\alpha_{f}$, where solid lines represent the linear least square fitting
between the parameter $\alpha_{f}$ and latitude, while the dashed curves
correspond to 95$\%$ confidence intervals. (b) Probability distribution
function of the parameter $\alpha_{f}$ for the northern hemisphere (solid &
red) and southern hemisphere (dotted & blue). Bottom row is similar to the top
row but for the current helicity $h_{c}$.fig:mag-cur-hel-lat

To address the above questions, we fitted a linear regression model through
the parameters $\alpha_{f}$. In Figure fig:mag-cur-hel-lat(a), the solid line
shows the linear regression line while dashed curves correspond to the $95\%$
confidence ($\sim 2\sigma$) interval. The confidence interval $95\%$ contains
the best-fit regression line. The fitted line has a slope, $-3.4\times
10^{-10}m^{-1}degree^{-1}$, such that the magnitude of $\alpha_{f}$ increases
with latitude. However, linear regression is not fully satisfactory because we
do not have any known reason for the straight line to be the appropriate fit.
But at the same time, we can not justify higher order polynomials. The average
values of $\alpha_{f}$ in northern (southern) hemisphere is found to be
$-4.7(+7.6)\times 10^{-9}m^{-1}$.

To examine the statistical significance of hemispheric trend, we calculated
probability distribution function (PDF) for $\alpha_{f}$ which is shown in
Figure fig:mag-cur-hel-lat(b). The peak of density function for northern
(southern) hemisphere lies at $-5.4(+3.4)\times 10^{-8}m^{-1}$. These results
confirm the hemispheric trend of the estimated $\alpha_{f}$ values for our set
of ARs. The FWHM of PDF of $\alpha_{f}$ for northern (southern) hemispheric
points is $2.3(2.6)\times 10^{-8}m^{-1}$. That is, $\alpha_{f}$ of the
southern hemisphere is spread slightly more than that of the northern
hemisphere.

Figure 2.: (a) The helicity parameter $\alpha_{f}$ as a function of absolute
magnetic flux of ARs. Solid (dashed) line below (above) $\alpha_{f}=0$
represents the linear least square fitting through northern (southern) data
points. The slopes of the fitted lines are mentioned within the rectangular
boxes. (b) Hemispheric distribution of ARs based on $\alpha_{f}$ values as a
function of magnetic field. The dashed (solid) lines represent the linear
regression for the northern (southern) ARs. Corresponding equations are given
in the rectangular boxes. Bottom row is similar to the top row but for the
current helicity $h_{c}$.fig:mag-hel-absB

Next we address whether ARs of different magnetic field strengths have similar
twist in their field lines. For this, we plotted the magnetic field parameter
$\alpha_{f}$ as a function of absolute magnetic flux ($B$) in Figure fig:mag-
hel-absB(a). In this plot, we have only shown the data points with
$|\alpha_{f}|\geq 0.5\times 10^{-8}{\rm m^{-1}}$ for clarity. A linear
regression is performed through positive and negative values of $\alpha_{f}$
of ARs. The values of slope of the regression is found to be $+5.6(-5.8)\times
10^{-11}m^{-1}G^{-1}$ for positive (negative) $\alpha_{f}$. This result shows
that the ARs with stronger magnetic fields have larger twist irrespective of
ARs’s hemispheric locations.

Figure fig:mag-hel-absB(b) shows magnetic field related HSR in the helicity
parameter $\alpha_{f}$. For this, we ran a boxcar magnetic flux of size 50 G,
and counted ARs following the HSR as a function of $B$. Linear regressions
through the data are shown with dashed lines with a slope $0.06(0.10)$. This
implies that the HSR becomes stronger with magnetic fields strength of ARs.

Figure 3.: (a) Sunspot number as a function of time for reference. The
complete sunspot cycle 23 is shown in the inset, where vertically dashed lines
mark the time period of ARs which are analyzed in this paper. (b) Latitudinal
distribution of magnetic helicity parameter $\alpha_{f}$ as a function of
time, where sizes of opened(filled) circles represent magnitudes of negative
(positive) values of the parameter $\alpha_{f}$, and dashed lines represent
the linear regression lines through ARs latitude. (c) Similar to the panel (b)
but for the current helicity parameter $h_{c}$. (d) Hemispheric distribution
of ARs as a function of time for northern (solid) and southern (dotted)
hemispheres.fig:mag-cur-hel-time

Next, we explore the time dependent hemispheric trend of the magnetic helicity
parameter $\alpha_{f}$ (Figure fig:mag-cur-hel-time). Figure fig:mag-cur-hel-
time(a) shows the monthly averaged sunspot number as a function of time; in
the period where ARs are taken for this paper. A complete solar cycle 23 is
also shown in the inset of the top panel where vertically dotted lines mark
the time window over which ARs are taken for this analysis. This period
corresponds to the peak to the descending phase of the solar cycle 23.

Magnetic helicity parameter $\alpha_{f}$ for the northern and southern
hemisphere is plotted in Figure fig:mag-cur-hel-time(b). The symbols with
opened(filled) circles represent negative(positive) values of the magnetic
helicity parameter $\alpha_{f}$. This panel not only shows the predominantly
hemispheric distribution of the parameter $\alpha_{f}$ but also part of the
butterfly-diagrams for sunspots which indicates that the sunspots appear at
larger latitude in the beginning of solar cycle and as time passes they appear
at lower latitude as shown by the linear regression lines (dashed) through ARs
latitudes.

Parameter (P) | N/S | HSR | HSR | Total | Slope | Average | PDF | PDF
---|---|---|---|---|---|---|---|---
|  | (%) | (no.) | (no.) | $(\times 10^{-2}deg^{-1})$ |  | Max. | FWHM
$\alpha_{f}$ | N | 68 | 48 | 71 | -3.4 | -0.5 | -0.5 | 2.3
$(m^{-1})$ | S | 67 | 79 | 118 |  | 0.8 | 0.3 | 2.6
$h_{c}$ | N | 68 | 48 | 71 | -2.0 | -0.3 | -0.4 | 1.3
$(G^{2}m^{-1})$ | S | 68 | 80 | 118 |  | 0.4 | 0.2 | 1.5
$h_{k1}$ | N | 65 | 46 | 71 | -1.8 | -0.3 | -0.2 | 2.0
$(ms^{-2})$ | S | 56 | 66 | 118 |  | 0.2 | 0.1 | 1.6
$h_{k2}$ | N | 69 | 49 | 71 | -12.4 | -1.9 | -0.7 | 8.1
$(ms^{-2})$ | S | 67 | 79 | 118 |  | 1.7 | 1.1 | 9.8
$C_{k1}$ | N | 65 | 46 | 71 | -2.4 | -0.4 | -0.3 | 2.2
$(s^{-2})$ | S | 55 | 65 | 118 |  | 0.2 | 0.0 | 1.9
$C_{k2}$ | N | 65 | 46 | 71 | -2.5 | -0.4 | -0.2 | 1.8
$(s^{-2})$ | S | 70 | 83 | 118 |  | 0.3 | 0.4 | 2.2
Table 1.: Himispheric distribution of topolgical parameters. For the actual values of the parameters listed in the columns 6-9, the multiplication factors, $10^{-8},10^{-3},10^{-9}$ and $10^{-15}$ should be applied for $\alpha_{f}$, $h_{c}$, $h_{k}$ and $C_{k}$, respectively.tab:hemisp-distb Parameter | +ve P | -ve P | HSR-N | HSR-S
---|---|---|---|---
(P) | ($G^{-1}$) | ($\times 10G^{-1}$)
$\alpha_{f}(m^{-1})$ | 5.6 | -5.8 | 0.6 | 1.0
$h_{c}(G^{2}m^{-1})$ | 5.1 | -5.8 | 0.6 | 1.0
$h_{k1}(ms^{-2})$ | 0.1 | -0.1 | 1.9 | 1.1
$h_{k2}(ms^{-2})$ | 1.4 | -1.3 | -1.1 | -1.2
$C_{k1}(s^{-2})$ | 0.8 | -3.0 | 0.9 | 1.2
$C_{k2}(s^{-2})$ | 1.6 | -2.6 | -0.7 | -0.9
Table 2.: Slopes of the distributions of the topological parameters (columns
$2^{nd}$ and $3^{rd}$) and their hemispheric distributions (columns $4^{rth}$
and $5^{th}$) as a function of $B$. For the actual values of the parameters
listed in the columns 1-2, the multiplication factors,
$10^{-8},10^{-3},10^{-9}$ and $10^{-15}$ should be applied for $\alpha_{f}$,
$h_{c}$, $h_{k}$ and $C_{k}$, respectively.tab:top-para-bav

These distributions show a clear indication of the hemispheric distribution of
the magnetic helicity in both the northern and southern hemispheres. Around
the end of year 2002, there are ARs with slightly large magnetic helicity.
During the end phase of the solar cycle, the magnetic helicity reduces, as is
obvious from the smaller circles. Overall, there seems to be equator-ward
propagation of the magnetic helicity parameter $\alpha_{f}$ similar to the
sunspots of the solar cycle. Such results have also been reported in another
study by Zhang et al. (2010) in their data obtained from other sources.

In order to get the time dependent hemisphere distribution of the parameter
$\alpha_{f}$, we ran a boxcar of width 90 days and counted the numbers of ARs
following the HSR in both the hemispheres. Figure fig:mag-cur-hel-time(d)
shows the boxcar averaged hemispheric distribution of the parameter
$\alpha_{f}$. This plot shows that the ARs on average follow HSR for the
magnetic helicity parameter $\alpha_{f}$ except at peak and end phase of the
cycle for northern and southern ARs, respectively. The opposite hemispheric
trend during the decay phase of this solar cycle has been found in earlier
studies (e.g., Bao, Ai, and Zhang, 2000; Tiwari, Venkatakrishnan, and
Sankarasubramanian, 2009; Pipin et al., 2019).

####  3.1.2 Current Helicity

sec:cur-hel

Next, we analysed the current helicities $h_{c}$ of all the 189 ARs similar to
the magnetic helicity parameter $\alpha_{f}$ (see Section sec:mag-hel). The
results of our analysis are shown in Figures fig:mag-cur-hel-lat – fig:mag-
cur-hel-time. Figure fig:mag-cur-hel-lat(c) shows the latitudinal distribution
of the vertical current helicity $h_{c}$. There are 68% ARs in both northern
and southern hemispheres with negative and positive current helicities
$h_{c}$, respectively. The small deviation in $h_{c}$ from $\alpha_{f}$ for
the southern ARs is due to averaging of $h_{c}$ values for ARs that are
observed multiple times. The values of $h_{c}$ show a similar hemispheric
trend as the force-free parameter $\alpha_{f}$ (Figure fig:mag-cur-hel-lata).
The slope of linear least square fit through $h_{c}$ of ARs is found to be
$-2.0\times 10^{-10}Gm^{-1}degree^{-1}$. This hemispheric trend of ARs based
on $h_{c}$ confirms the earlier reports (Abramenko, Wang, and Yurchishin,
1996, and others).

The probability distribution function for $h_{c}$ is shown in Figure fig:mag-
cur-hel-lat(d). This shows hemispheric trend for $h_{c}$ as the PDFs for
northern and southern hemispheric ARs peak at $-4.1\times 10^{-4}G^{2}m^{-1}$
and $+1.8\times 10^{-4}G^{2}m^{-1}$, respectively. The FWHM of PDF for
northern hemisphere is $1.3\times 10^{-3}{\rm G}^{2}\,{\rm m}^{-1}$ and for
the southern hemisphere $1.5\times 10^{-3}{\rm G}^{2}\,{\rm m}^{-1}$. The PDF
profiles of the parameter $h_{c}$ and the parameter $\alpha_{f}$ are look-
alike representing similar distributions of two topological parameters in ARs.

Figure fig:mag-hel-absB(c) shows the variation of current helicity $h_{c}$ as
a function of absolute magnetic flux ($B$). It is found that $h_{c}$ increases
on increasing the magnetic flux of ARs similar to the magnetic helicity
parameter $\alpha_{f}$ (Figure fig:mag-hel-absBa). The slope of the linear
least square fitting is found to be $5.1(-5.8)\times 10^{-6}Gm^{-1}$ for the
positive (negative) $h_{c}$ values of ARs. This suggests that the ARs with
stronger magnetic fields may have more current helicity than typical ARs.
Figure fig:mag-hel-absB(d) shows the hemispheric distribution of the current
helicity $h_{c}$ as a function of $B$, similar to Figure fig:mag-hel-absB(b)
for the helicity parameter $\alpha_{f}$. It shows that the HSR becomes
stronger with the magnetic field strength $B$ as the slopes for
northern(southern) ARs is found to be $0.06(0.10)$.

We have also analyzed the solar cyclic variation in the current helicity
$h_{c}$. Figure fig:mag-cur-hel-time(c) shows the hemispheric distribution of
the current helicity parameter $h_{c}$ as a function of time. This plot
further confirms the equator-ward propagation of helicity $h_{c}$ patterns.
Since $h_{c}$ has same sign as $\alpha_{f}$, it shows similar time dependent
distribution as shown in Figure fig:mag-cur-hel-time(d) for the parameter
$\alpha_{f}$.

Figure 4.: Similar to Figure fig:mag-cur-hel-lat but for the kinetic helicity
density $h_{k}$ (first two columns) and divergence-curl $C_{k}$ (last two
columns) at two different depths 2.4 Mm (top row) and 12.6 Mm (bottom
row).fig:kin-hel-lat-pdf

###  3.2 Sub-photospheric Flow Fields

sbsec:subsurf-flow

Helicities in the sub-photospheric velocity fields in our sample of 189 ARs
are studied by computing the kinetic helicity density $h_{k}$ (Equation
eq:kin-hel-den-z) and the divergence-curl $C_{k}$ (Equation eq:div-curl-z). In
Figures fig:kin-hel-lat-pdf – fig:kin-hel-time-d2, we explain our results
based on these parameters.

####  3.2.1 Kinetic Helicity Density

sec:kin-hel

Figure fig:kin-hel-lat-pdf(a-d) shows the latitudinal distribution of the
vertical kinetic helicity $h_{k}$ for depths 2.4 Mm (a) and 12.6 Mm (c). There
is a clear hemispheric preference in the parameter $h_{k}$ as there are
$65\%(56\%)$ ARs at depth of 2.4 Mm and $69\%(67\%)$ ARs at depth of 12.6 Mm
in the northern (southern) hemisphere having negative(positive) values. The
linear regression through ARs shows slopes of $-1.8\times
10^{-10}s^{-2}degree^{-1}$ for the depth of 2.4 Mm and $-1.2\times
10^{-9}s^{-2}degree^{-1}$ for the depths of 12.6 Mm. These results show that
the kinetic helicity have same hemispheric preference as that of magnetic
helicity $\alpha_{f}$ and current helicity $h_{c}$. The larger values of
$h_{k}$ for deeper layers indicate that the parameter $h_{k}$ have better
hemispheric preference at larger depths.

Figure 5.: (a) Hemispheric probability density function for the sub-
photospheric flow parameter $h_{k}$ for several depths. The PDF is scaled down
for deeper layers. Thick curves correspond to depths of 2.4 Mm and 12.6 Mm (b)
The peak point of the PDFs (in panel a) as a function of depth. Buttom row is
similar to the top row but for the parameter $C_{k}$.fig:kin-hel-depth-pdf

To further confirm the hemispheric preference of kinetic helicity $h_{k}$, we
have computed the PDF for the parameter $h_{k}$ of northern and southern ARs.
Figures fig:kin-hel-lat-pdf(b) and fig:kin-hel-lat-pdf(d) shows the PDF of
$h_{k}$ for the depths 2.4 Mm and 12.6 Mm. The PDF for $h_{k}$ peaked at
$-0.2(+0.1)\times 10^{-9}ms^{-2}$ for northern (southern) hemispheric ARs for
the depth 2.4 Mm. Similarly, the PDF for $h_{k}$ at the depth 12.6 Mm peaked
at $-0.7(+1.1)\times 10^{-9}ms^{-2}$ which further confirms the HSR for
parameter $h_{k}$.

In order to check depth dependent HSR for $h_{k}$, we computed the PDFs for
all the available depths and plotted in Figure fig:kin-hel-depth-pdf(a). We
have also estimated the $h_{k}$ associated with peaks of PDFs and displayed in
Figure fig:kin-hel-depth-pdf(b). We find that the peak position shifted
towards larger $h_{k}$ values at larger depths for both hemispheric ARs. The
southern ARs show larger $h_{k}$ values than northern ARs at deeper layers
($>6$ Mm) but at shallower depths ($<6$ Mm) northern ARs have larger $h_{k}$
values than southern ARs. This could be due to stronger swirls in the sub-
photospheric flows at deeper layers than shallower.

Figure 6.: Variation in the slopes of the distribution of kinetic helicity
$h_{k}$ (a) and divergence-curl $C_{k}$(b) vs $B$ as a function of depth for
$h_{k}>0$ (solid and red) and $h_{k}<0$ (dotted and blue).fig:kin-hel-slope-b

We have also analysed the variations in the kinetic helicity as a function of
magnetic flux $B$. For this, we computed the slope of the distribution of
$h_{k}$ vs $B$ as a function of depth (Figure fig:kin-hel-slope-ba). We found
that the magnitude of the slope enhances considerably at larger depths
$>3\,Mm$ for both positive and negative helicities. For example, the slope for
the depth 2.4 Mm is found to be $+1.9(+1.1)\times 10^{-10}ms^{-2}$ for
positive(negative) $h_{k}$ values while it increased to $+1.4(-1.3)\times
10^{-9}ms^{-2}G^{-1}$ at the depth of 12.6 Mm. The positive (negative) values
of slope for the parameter $h_{k}>0$($h_{k}<0$) represent the linear relation
between the helicity parameter $h_{k}$ and $B$. Similar result has been found
for the magnetic helicity (Figure fig:mag-hel-absBb) and the current helicity
(Figure fig:mag-hel-absBd).

Figure 7.: (Top row) Hemispheric distribution of sub-photospheric flow
parameters $h_{k}$ for northern (a) and southern (b) hemisphere, as a function
of depth and magnetic flux $B$. (Bottom row) Similar to the top row but for
the parameter $C_{k}$.fig:kin-hel-hemishp-map

Next, we find the hemispheric distribution of the parameter $h_{k}$ as a
function of depth and magnetic flux. For this we counted the number of ARs
following the HSR by running a magnetic flux window of width 50 G for all
depths. These results are shown in Figure fig:kin-hel-hemishp-map(top row) for
northern (left panel) and southern (right panel) hemispheres. This clearly
shows that the HSR, statistically, becomes stronger at larger depths. Also, as
the magnetic flux $B$ increases the hemispheric biases of ARs shifted towards
larger depths. However, in ARs with smaller fluxes show opposite HSR trend in
both hemispheres. Similar preference is also discernible at shallower layers
in ARs having stronger fluxes. The increase in hemispheric preference of with
magnetic flux has also been found for the photospheric magnetic helicity
$\alpha_{f}$ (Figure fig:mag-hel-absBb) as well as for the current helicity
$h_{c}$ (Figure fig:mag-hel-absBd).

Figure fig:kin-hel-time-d1(b) and fig:kin-hel-time-d2(b) show the time
dependent variations in the kinetic helicity parameter $h_{k}$ for the depths
2.4 Mm and 12.6 Mm, respectively. The hemispheric biases are noticeable during
the peak to decay phase of the solar cycle. But, these preferences are
slightly weaker than the magnetic and current helicity parameters. The kinetic
helicity $h_{k}$ is found to be scattered significantly for both depths
throughout the time period considered in the analysis. The magnitude of
kinetic helicity of all ARs for the depth of 12.6 Mm are found to be larger
throughout than that for the depth of 2.4 Mm. This shows largely variant sub-
photospheric twisted flows in the interior of ARs. We could not find clear
signature of equator-ward propagation of ARs for the parameter $h_{k}$ at both
depths.

In the first half year of 2003, the ARs in northern hemisphere deviated
largely from usual HSR for the depth of 2.4 Mm (Figure fig:kin-hel-time-d1d).
Similarly, southern ARs show opposite HSR during November 2002 to January
2003. For the depth of 12.6 Mm (Figure fig:kin-hel-time-d1d), the ARs show
better HSR even during the aforesaid period. Southern ARs show better HSR than
northern ARs from the peak to decay phase of the solar cycle. But, at the end
phase of the solar cycle, some ARs in southern hemisphere were found to have
opposite hemispheric trend.

Figure 8.: Similar to Figure fig:mag-cur-hel-time but for the parameters
$h_{k}$ and $C_{k}$ at depth 2.4 Mm.fig:kin-hel-time-d1

Figure 9.: Similar to Figure fig:kin-hel-time-d2 but for the depth 12.6
Mm.fig:kin-hel-time-d2

####  3.2.2 Divergence-Curls of Sub-photospheric Flows

sbsbsec:kin-hel

Figure fig:kin-hel-lat-pdf(e-h) shows the latitudinal distribution of sub-
photospheric flow parameter $C_{k}$ for depths 2.4 Mm (e) and 12.6 Mm (g). For
the above two depths, the latitudinal distributions of ARs are found to be
$65\%(55\%)$ and $65\%(70\%)$ in the northern (southern) hemispheres having
negative (positive) values of $C_{k}$. We noticed that the hemispheric
distribution of ARs for $C_{k}$ and $h_{k}$ are same for the depth 2.4 Mm but
different for the depth of 12.6 Mm. This is due to averaging of topological
parameters of same ARs observed multiple times. The linear regression through
distribution of the parameter $C_{k}$ shows a slope of $-2.4\times
10^{-17}s^{-2}degree^{-1}$ for the depth of 2.4 Mm and $-2.5\times
10^{-17}s^{-2}degree^{-1}$ for the depth of 12.6 Mm. These results show
hemispheric preferences similar to the magnetic helicity parameter
$\alpha_{f}$(Figure fig:mag-cur-hel-lata), current helicity $h_{c}$(Figure
fig:mag-cur-hel-latc) and the kinetic helicity parameter $h_{k}$ (Figure
fig:kin-hel-lat-pdfa and c).

The probability density function for the sub-photospheric flow parameters
$C_{k}$ is shown in Figures fig:kin-hel-lat-pdf(f) for the shallower depth at
2.4 Mm and fig:kin-hel-lat-pdf(h) for the deeper layer at 12.6 Mm. For the
depth 2.4 Mm the parameter $C_{k}$ peaks at $-2.5(+0.0\times 10^{-16})\,{\rm
s^{-2}}$ and for the depth 12.6 Mm it peaks at $-2.0(+3.6\times
10^{-16})\,{\rm s^{-2}}$ for the northern (southern) hemisphere while average
values of $C_{k}$ were found to be $-4.2(+0.22)\times 10^{-16}s^{-2}$ for the
depth of 2.4 Mm and $-0.43(+0.34)\times 10^{-15}s^{-2}$ for the depth of 12.6
Mm. This further confirms the HSR in $C_{k}$ for the sub-photospheric flows of
ARs.

Figure fig:kin-hel-depth-pdf(c) shows depth dependent PDF for the parameter
$C_{k}$ for northern (red) and southern (blue) hemispheres similar to Figure
fig:kin-hel-depth-pdf (a) for the kinetic helicity $h_{k}$. Figure fig:kin-
hel-depth-pdf(d) represents the variations in peak position of PDF for
northern (red) and southern(blue) hemispheres as function of depth. The
hemispheric distribution for the parameter $C_{k}$ is found to exist for all
depth range 0 – 20 Mm. At depths beyond 2.0 Mm it is significant as evident
from Figure fig:kin-hel-depth-pdf(d). The peaks of PDFs, between depth range 5
– 20 Mm, for both hemispheres are found to be nearly constant at
$-2.0(+3.5)\times 10^{-16}s^{-2}$ for northern (southern) hemisphere while it
shifted towards larger values at larger depth for the parameter $h_{k}$. The
constant distribution of maximum peak of PDF of $C_{k}$ is due to horizontal
divergence. The flows are found to be largely diverging near the photosphere
than in deeper layers. Hence increase in vorticity is compensated by decrease
in divergence and results into constant value for the parameter $C_{k}$.

In order to study the variation in the topological parameter $C_{k}$ as a
function of depth and magnetic fields, we computed the slope of the
distribution, $C_{k}$ vs. $B$. The results are shown in the Figure fig:kin-
hel-slope-b(b) for $C_{k}>0$(solid and red curves) and $C_{k}<0$(dotted and
blue curves). For example, the slope for $C_{k}>0$ ($C_{k}<0$) for the depth
of 2.4 Mm is found to be $+0.78(-3.04)\times 10^{-18}s^{-2}G^{-1}$ while it
changed to $+1.61(-2.63)\times 10^{-18}s^{-2}G^{-1}$ for the depth of 12.6 Mm.
The slopes, on average, are found to increase as we go deeper ($>4\,Mm$).

Similar to the helicity $h_{k}$ (Figure fig:kin-hel-hemishp-map, top row), we
computed the depth and magnetic field dependent hemispheric preference of the
parameter $C_{k}$. This result is shown in Figure fig:kin-hel-hemishp-
map(bottom row). It clearly shows that the hemispheric distribution of $C_{k}$
shifted towards larger depths as the magnetic field increases. Also, it can be
noticed that the HSR became stronger with depth and magnetic flux $B$. Near
the photosphere with stronger flux regions and deeper layers with smaller
fluxes, the opposite hemispheric trends have been found in both the
hemisphere. The hemispheric preference for the parameter $C_{k}$ is very
similar to that of $h_{k}$ (Figure fig:kin-hel-hemishp-map, top row) with same
sign of handedness.

In order to check the solar cyclic variation of the parameter divergence-curl
($C_{k}$), we have plotted it as a function of time for two depths 2.4 Mm
(Figure fig:kin-hel-time-d1c) and 12.6 Mm (Figure fig:kin-hel-time-d1c). We
found that most of ARs shows $C_{k}$ around an absolute mean of $1.0\times
10^{-15}s^{-1}$ for both the depths. We found that the parameter $C_{k}$
ranges between $\pm 4\times 10^{-15}s^{-2}$ for both the depths (Figure
fig:kin-hel-time-d1c and fig:kin-hel-time-d1c) while $h_{k}$ (Figure fig:kin-
hel-time-d1b and fig:kin-hel-time-d1b) have larger ranges for deeper depths.
Similar to the parameter $h_{k}$, we did not find clear evidence of equator-
ward propagation of the parameter $C_{k}$.

###  3.3 Relation between Kinetic and Magnetic Helicities

sbsec:rel-kine-mag-hel

The relation between the surface and sub-surface helicities are shown in
Figures fig:assoc-kin-mag-hel – fig:correl-kin-mag-hel-depth. Figure
fig:assoc-kin-mag-hel(top row) shows the correlation between the force-free
parameter $\alpha_{f}$ and the sub-photospheric twist parameters for the
parameters $h_{k}$ (a and c) and $C_{k}$ (b and d) for two depths 2.4 Mm (a
and b) and 12.6 Mm (c and d). The parameter $\alpha_{f}$ shows correlations
with $C_{k}$ and $h_{k}$ at both depths. The correlation coefficient is
significantly larger for deeper layer than the shallower. The significant
values of the correlation coefficient $CC$ shows relation between the two
topological parameters of the surface magnetic fields and the sub-photospheric
flows.

Figure 10.: (top row) Kinetic helicity parameters $h_{k}$ and $C_{k}$ for two
depths 2.4 Mm (a and b) and 12.6 Mm (c and d), as a function of magnetic
helicity parameter $\alpha_{f}$, where the values of $CC$ represent the
Pearson correlation coefficients between the topology of photospheric magnetic
fields and sub-photospheric flows. Bottom row is similar to the top row but as
a function of current helicity $h_{c}$.fig:assoc-kin-mag-hel

Figure fig:assoc-kin-mag-hel(bottom row) shows correlation between the current
helicity and the twist parameter of sub-photospheric flows similar to Figure
fig:assoc-kin-mag-hel(top row) for the parameter $\alpha_{f}$ and sub-
photospheric flow parameters $h_{k}$ and $C_{k}$. Corresponding correlation
coefficients are given in respective panels. The magnitudes of correlations
imply association between the current helicity and the twist of sub-
photospheric flows of ARs.

In order to check the depth dependent association between topology of
photospheric magnetic fields and sub-photospheric flows, we have computed the
correlation coefficients between magnetic and flows helicities parameters as a
function of depth. The result is shown in Figure fig:correl-kin-mag-hel-depth.

An interesting pattern of $CC$ with depth is that it is not random. It changes
gradually with depths. Very near the photosphere the sub-photospheric flows
and magnetic fields topology show no significant relation as the $CC$ is close
to zero. But the absolute $CC$ gradually increases with depth up to around of
1.5 Mm, then decreases and becomes minimum around a depth of 3.3 Mm, and
thereafter starts increasing. There is a sharp increase in $CC$ between 3.4 –
12 Mm and after that it becomes almost constant. This shows that the
photospheric magnetic field lines and the sub-photospheric flows at larger
depths have similar topological trend. The two peaks in the $CC$-profile
indicates the bi-polar structure in the sub-photospheric flows as reported
earlier (Komm et al., 2005; Maurya, Ambastha, and Tripathy, 2009; Maurya and
Ambastha, 2010).

Figure 11.: Depth dependent correlation coefficients between kinetic helicity
and magnetic helicity parameters: (a) $C_{k}$ & $\alpha_{f}$ (solid line),
$C_{k}$ & $h_{c}$ (dashed line), (b) $h_{k}$ & $\alpha_{f}$ (solid line),
$h_{k}$ & $h_{c}$ (dashed line).fig:correl-kin-mag-hel-depth

Another, interesting pattern of $CC$ is that at larger depths (> 5 Mm) the
magnitude of $CC$ between current helicity and sub-photospheric topology is
larger than the magnitude of $CC$ between force-free parameter $\alpha_{f}$
and sub-photospheric topology parameters. This shows that the current helicity
has association with the sub-photospheric flows topology. This is also evident
from the mean field dynamo (Biskamp, 2003).

The gradual variation of $CC$ between surface and sub-surface helicities could
be due to the relation between the magnetic field and sub-photospheric flows.
But, one should note that we are comparing the topology of magnetic field at
the photosphere and the flows at depths. Since, there is no direct measurement
of magnetic field in sub-photospheric depths, we cannot confidently suggest
their association with flows. But from the above results it suggests that they
may have some relation.

##  4 Summary and Conclusions

sec:sum-conc

Using MSFC vector magnetograms and GONG Dopplergrams observations, we have
analyzed the topological properties of photospheric magnetic fields and sub-
photospheric flow fields of 189 ARs that cover peak to descending phase of the
solar cycle 23. Our analysis confirms previous reports, and show some new
interesting results.

We found clear evidence of hemispheric preferences for the magnetic and
current helicities. The ARs of the northern(southern) hemisphere are found
with dominantly negative (positive) helicities which confirms earlier reports
(Pevtsov et al., 2008, and others). However, many ARs were found to show
opposite hemispheric preferences, especially at beginning and end phase of the
solar cycle as indicated in others studies (Bao, Ai, and Zhang, 2000; Tiwari,
Venkatakrishnan, and Sankarasubramanian, 2009; Pipin et al., 2019). One should
note that in complex magnetic field regions magnetic helicity proxy
$\alpha_{f}$ may deviate from the force-free field, i.e., $\alpha_{f}$ may not
be a true proxy for helicity in complex sunspot regions to correctly reflect
the twistedness of an AR as conjectured by Russell et al. (2019).

We have also found signature of equator-ward propagating pattern in the
magnetic and current helicities similar to sunspots, in both hemispheres,
during descending phase of the solar cycle. Zhang et al. (2010) have also
reported similar results and suggested that the solar dynamo could be helical
in nature. An AR with larger area and stronger fields may have larger
helicities. Therefore such patterns of helicities could also be due to
variations of areas and magnetic fields of sunspots during a solar cycle, from
larger areas with stronger fields regions to smaller areas with weaker field
regions (Watson, Fletcher, and Marshall, 2011).

On comparing the magnetic and current helicity parameters with field strength
of ARs, we found that the magnitude of both helicities increases. Another
interesting pattern was found that the hemispheric preferences become stronger
for both the helicity types, implying that ARs with larger field strengths
show better hemispheric preferences. Hemispheric trend of current helicity
infers that the regular vortex structure of sunspots is caused by forces such
as the Coriolis and differential rotations.

In the sub-photospheric flows, we found similar hemispheric trend for the
topological parameter, kinetic helicity $h_{k}$ and divergence-curl $C_{k}$ at
all depths. Both the flow parameters follow same hemispheric preference as
that of magnetic and current helicity parameters, i.e., $h_{k}$ and $C_{k}$
are found to have dominantly negative (positive) sign in ARs of
northern(southern) hemisphere. This confirms the earlier reports (Maurya,
Ambastha, and Reddy, 2011; Seligman, Petrie, and Komm, 2014; Komm, Gosain, and
Pevtsov, 2014; Komm and Gosain, 2015). But Gao, Zhang, and Zhao (2009) have
found opposite hemispheric preferences in a small sample of ARs. Further, we
could not detect clear signature of equator-ward propagating patterns in
kinetic helicities of ARs as found for the magnetic helicities. Since kinetic
and current helicities are related by mean field dynamo model such patterns
are expected to exist in both the parameters.

The hemispheric preference of the parameter $h_{k}$ could be due to mean field
dynamo $\alpha_{d}$ (Biskamp, 2000, see Equation 5.19). Let us think of the
hemispheric dependence of force-free alpha $\alpha_{f}$, dynamo alpha
$\alpha_{d}$ and kinetic helicity,
$h_{k}=<\mathbf{u}\cdot\nabla\times\mathbf{u}>\approx
3<u_{z}(\nabla\times\mathbf{u})_{z}>$. According to the mean-field dynamo,
$\alpha_{d}$ and $h_{k}$ are strongly related to each other by the relation
$\alpha_{d}=-\frac{1}{3}\tau<\mathbf{u}\cdot\nabla\times\mathbf{u}>=-\frac{1}{3}h_{k}\tau$,
where $\tau$ is the velocity correlation time. Thus $h_{k}$ and $\alpha_{d}$
have opposite signs. In the northern hemisphere, the Coriolis force causes a
flow to be left-helical and $h_{k}<0$. From our analysis, we have found
statistically negative kinetic helicity for ARs which support the mean-field
dynamo theory. Further, considering $h_{k}<0$, we come to have $\alpha_{d}>0$
in the northern hemisphere. This is commonly adopted view in the solar dynamo.
It also seems to be consistent with Joy’s law (Hale et al., 1919). The
condition of $\alpha_{d}>0$ as well as the Joy’s law suggests that the larger-
scale magnetic field comprising bipolar ARs may be right helical. On the other
hand, the force-free alpha $\alpha_{f}$ is empirically found to be negative in
the northern hemisphere. This means that magnetic fields are left-helical in
scales smaller than individual ARs.

The solar differential rotation may be another factor for the hemispheric
trend of the sub-photospheric flows as rotation may introduce left- and right-
handed vorticity in northern and southern hemispheres with magnitude of
$0.01\times 10^{-6}s^{-1}$ to $2.5\times 10^{-6}s^{-1}$ for an AR of sizes 1
Mm to 40 Mm along the meridional direction.

The hemispheric preference for kinetic helicity is found to increase with
depths. The magnitude of peak of kinetic helicity distribution is found to
increases with depth for both hemispheric ARs while divergence-curl is found
to be constant around depths 5 – 20 Mm. The reason for growth in kinetic
helicity with depth is mainly due to increase in vertical velocity while
constant value of the parameter $C_{k}$ is caused by net effects of increase
in swirl and decrease in divergence.

Our analysis shows interesting patterns of hemispheric preferences, of sub-
photospheric flow topology, as a function of magnetic field strength and
depths. The HSR for sub-photospheric flows topology enhances with magnetic
field strength and with depths. Poor hemispheric preferences of flow topology
near the photosphere, may be due to smaller plasma density and diverging
magnetic field lines. This may be due to different scales of magnetic field
lines. This supports the solar dynamo theory which predicts the bi-helical
properties of magnetic fields (Blackman and Brandenburg, 2002; Brandenburg and
Subramanian, 2005).

Further, we analysed the correlation between sub-photospheric flows and
photospheric magnetic fields. The kinetic helicity, near the photosphere, do
not show significant relation with magnetic or current helicities. But we
found that the correlation between them enhances as we go deeper into the
solar interior. The kinetic helicity and vertical divergence-curl show
correlation with magnetic and current helicities at all depths. Further, we
found that the kinetic helicity shows better covariance with magnetic and
current helicities than divergence-curl of flow fields, which supports the
mean field dynamo models. Near-surface flow topology shows poor correlation
with magnetic and current helicities. This suggests that the magnetic fields
and flows may have different topologies. Another reason could be the frozen-in
field conditions in the interior plasma, which may not be satisfied at these
depths and results into weak association of magnetic and flow fields. But
deeper into the interior there is increasingly high density plasma. There the
field lines may be concentrated close to the flux tubes which may lead to
similar topology in both physical quantities.

It should be noted that we have used a single (or a few) vector magnetograms
map corresponding to an AR to derive the photospheric twist while the twist in
sub-photospheric flow was obtained using the data cube for 1664 minutes. This
could be one reason for weak relation between the photospheric magnetic fields
and sub-photospheric flows. But magnetic helicity is likely to be
approximately conserved in solar atmosphere on time scale of AR while passing
the disc (Berger and Field, 1984). It should also be noted that we have used
magnetic and Doppler fields observations obtained from different sources.
These factors may be responsible in affecting our results and in reducing
correlations of the twist parameters. It is expected that these results may
improve in the future with magnetic and Doppler velocity observations obtained
from the same instrument.

##### Acknowledgements

This work utilizes data obtained by the GONG program operated by AURA, Inc.
and managed by the National Solar Observatory under a cooperative agreement
with the National Science Foundation, U.S.A. The vector magnetogram data were
obtained from MSFC. This work also utilizes data from the Solar Oscillations
Investigation/Michelson Doppler Imager (SOI/MDI) on the Solar and Heliospheric
Observatory (SOHO). SOHO is a project of international cooperation between ESA
and NASA. MDI was supported by NASA grants NAG5-8878 and NAG5-10483 to
Stanford University. We are grateful to Prof. J. Chae of Seoul National
University, South Korea for his valuable discussions and suggestions. R.A.M.
thankfully acknowledge the support by the NITC/FRG-2019.

Disclosure of Potential Conflicts of Interest. The authors declare that they
have no conflicts of interest.

## References

  * Abramenko, Wang, and Yurchishin (1996) Abramenko, V.I., Wang, T., Yurchishin, V.B.: 1996, Analysis of electric current helicity in active regions on the basis of vector magnetograms. Sol. Phys. 168, 75\. DOI. ADS.
  * Antia and Basu (2007) Antia, H.M., Basu, S.: 2007, Local helioseismology using ring diagram analysis. Astron. Nachr. 328, 257\. DOI. ADS.
  * Bao and Zhang (1998) Bao, S., Zhang, H.: 1998, Patterns of current helicity for the twenty-second solar cycle. Astrophys. J. Lett. 496, L43. DOI. ADS.
  * Bao, Ai, and Zhang (2000) Bao, S.D., Ai, G.X., Zhang, H.Q.: 2000, The hemispheric sign rule of current helicity during the rising phase of cycle 23. J. Astrophys. Astron. 21, 303\. DOI. ADS.
  * Berger and Field (1984) Berger, M.A., Field, G.B.: 1984, The topological properties of magnetic helicity.  J. Fluid Mech. 147, 133\. DOI. ADS.
  * Biskamp (2000) Biskamp, D.: 2000, Magnetic reconnection in plasmas, Cambridge monographs on plasma physics 3, Cambridge University Press, ISBN 0521582881. ADS.
  * Biskamp (2003) Biskamp, D.: 2003, Magnetohydrodynamic Turbulence, Cambridge University Press, ISBN 0521810116. ADS.
  * Blackman and Brandenburg (2002) Blackman, E.G., Brandenburg, A.: 2002, Dynamic nonlinearity in large-scale dynamos with shear. Astrophys. J. 579(1), 359\. DOI.
  * Brandenburg and Subramanian (2005) Brandenburg, A., Subramanian, K.: 2005, Astrophysical magnetic fields and nonlinear dynamo theory. Phys. Rep. 417(1-4), 1\. DOI.
  * Choudhuri (2003) Choudhuri, A.R.: 2003, On the connection between mean field dynamo theory and flux tubes. Sol. Phys. 215(1), 31\. DOI. ADS.
  * Choudhuri, Chatterjee, and Nandy (2004) Choudhuri, A.R., Chatterjee, P., Nandy, D.: 2004, Helicity of Solar Active Regions from a Dynamo Model. Astrophys. J. Lett. 615(1), L57. DOI. ADS.
  * Christensen-Dalsgaard et al. (1996) Christensen-Dalsgaard, J., Dappen, W., Ajukov, S.V., Anderson, E.R., Antia, H.M., Basu, S., Baturin, V.A., Berthomieu, G., Chaboyer, B., Chitre, S.M., Cox, A.N., Demarque, P., Donatowicz, J., Dziembowski, W.A., Gabriel, M., Gough, D.O., Guenther, D.B., Guzik, J.A., Harvey, J.W., Hill, F., Houdek, G., Iglesias, C.A., Kosovichev, A.G., Leibacher, J.W., Morel, P., Proffitt, C.R., Provost, J., Reiter, J., Rhodes, E.J. Jr., Rogers, F.J., Roxburgh, I.W., Thompson, M.J., Ulrich, R.K.: 1996, The current state of solar modeling. Science 272, 1286\. DOI. ADS.
  * Egorov, Rüdiger, and Ziegler (2004) Egorov, P., Rüdiger, G., Ziegler, U.: 2004, Vorticity and helicity of the solar supergranulation flow-field. Astron. Astrophys. 425, 725\. DOI. ADS.
  * Elsasser (1956) Elsasser, W.M.: 1956, Hydromagnetic dynamo theory. Rev. Mod. Phys. 28(2), 135\. DOI.
  * Fan et al. (1998) Fan, Y., Zweibel, E.G., Linton, M.G., Fisher, G.H.: 1998, The rise of kink-unstable magnetic flux tubes in the solar convection zone. Astrophys. J. 505(1), L59. DOI.
  * Gao, Zhang, and Zhao (2009) Gao, Y., Zhang, H., Zhao, J.: 2009, Comparison of photospheric current helicity and subsurface kinetic helicity. Monthly Not. RAS 394, L79. DOI. ADS.
  * Gilman (1983) Gilman, P.A.: 1983, Dynamically consistent nonlinear dynamos driven by convection in a rotating spherical shell. ii - dynamos with cycles and strong feedbacks. Astrophys. J. Sup. 53, 243\. DOI. ADS.
  * Gosain et al. (2013) Gosain, S., Pevtsov, A.A., Rudenko, G.V., Anfinogentov, S.A.: 2013, First synoptic maps of photospheric vector magnetic field from solis/vsm: Non-radial magnetic fields and hemispheric pattern of helicity. Astrophys. J. 772, 52\. DOI. ADS.
  * Gough (1985) Gough, D.: 1985, Inverting helioseismic data. Sol. Phys. 100, 65\. DOI. ADS.
  * Haber et al. (2000) Haber, D.A., Hindman, B.W., Toomre, J., Bogart, R.S., Thompson, M.J., Hill, F.: 2000, Solar shear flows deduced from helioseismic dense-pack samplings of ring diagrams. Sol. Phys. 192, 335\. ADS.
  * Hagino and Sakurai (2004) Hagino, M., Sakurai, T.: 2004, Latitude variation of helicity in solar active regions. Pub. ASJ 56, 831\. ADS.
  * Hagino and Sakurai (2005) Hagino, M., Sakurai, T.: 2005, Solar-cycle variation of magnetic helicity in active regions. Pub. ASJ 57, 481\. ADS.
  * Hagyard and Pevtsov (1999) Hagyard, M.J., Pevtsov, A.A.: 1999, Studies of solar helicity using vector magnetograms. Sol. Phys. 189(1), 25\. DOI. ADS.
  * Hagyard and Kineke (1995) Hagyard, M.J., Kineke, J.I.: 1995, Improved method for calibrating filter vector magnetographs. Sol. Phys. 158, 11\. DOI. ADS.
  * Hagyard et al. (1982) Hagyard, M.J., Cumings, N.P., West, E.A., Smith, J.E.: 1982, The msfc vector magnetograph. Sol. Phys. 80, 33\. DOI. ADS.
  * Hale (1927) Hale, G.E.: 1927, The fields of force in the atmosphere of the sun. Nature 119, 708\. DOI. ADS.
  * Hale et al. (1919) Hale, G.E., Ellerman, F., Nicholson, S.B., Joy, A.H.: 1919, The magnetic polarity of sun-spots. ApJ 49, 153\. DOI. ADS.
  * Hill (1988) Hill, F.: 1988, Rings and trumpets - three-dimensional power spectra of solar oscillations. Astrophys. J. 333, 996\. DOI. ADS.
  * Holder et al. (2004) Holder, Z.A., Canfield, R.C., McMullen, R.A., Nandy, D., Howard, R.F., Pevtsov, A.A.: 2004, On the tilt and twist of solar active regions. Astrophys. J. 611(2), 1149\. DOI.
  * Hood, Archontis, and MacTaggart (2011) Hood, A.W., Archontis, V., MacTaggart, D.: 2011, 3d MHD flux emergence experiments: Idealised models and coronal interactions. Sol. Phys. 278(1), 3\. DOI.
  * Komm and Gosain (2015) Komm, R., Gosain, S.: 2015, Current and kinetic helicity of long-lived activity complexes. Astrophys. J. 798, 20\. DOI. ADS.
  * Komm, Gosain, and Pevtsov (2014) Komm, R., Gosain, S., Pevtsov, A.A.: 2014, Hemispheric distribution of subsurface kinetic helicity and its variation with magnetic activity. Sol. Phys. 289(7), 2399\. DOI.
  * Komm et al. (2004) Komm, R., Corbard, T., Durney, B.R., González Hernández, I., Hill, F., Howe, R., Toner, C.: 2004, Solar subsurface fluid dynamics descriptors derived from global oscillation network group and michelson doppler imager data. Astrophys. J. 605, 554\. DOI. ADS.
  * Komm et al. (2005) Komm, R., Howe, R., Hill, F., González Hernández, I., Toner, C.: 2005, Kinetic helicity density in solar subsurface layers and flare activity of active regions. Astrophys. J. 630, 1184\. DOI. ADS.
  * Komm et al. (2007) Komm, R., Howe, R., Hill, F., Miesch, M., Haber, D., Hindman, B.: 2007, Divergence and vorticity of solar subsurface flows derived from ring-diagram analysis of mdi and gong data. Astrophys. J. 667, 571\. DOI. ADS.
  * LaBonte, Georgoulis, and Rust (2007) LaBonte, B.J., Georgoulis, M.K., Rust, D.M.: 2007, Survey of magnetic helicity injection in regions producing x-class flares. Astrophys. J. 671(1), 955\. DOI.
  * Leka and Skumanich (1999) Leka, K.D., Skumanich, A.: 1999, On the value of ‘$\alpha$AR’ from vector magnetograph data - I. Methods and Caveats. Sol. Phys. 188(1), 3\. DOI. ADS.
  * Lim and Chae (2009) Lim, E., Chae, J.: 2009, Chirality of intermediate filaments and magnetic helicity of active regions. Astrophys. J. 692, 104\. DOI. ADS.
  * Liu, Hoeksema, and Sun (2014) Liu, Y., Hoeksema, J.T., Sun, X.: 2014, Test of the hemispheric rule of magnetic helicity in the sun using the helioseismic and magnetic imager (hmi) data. Astrophys. J. Lett. 783(1), L1. DOI.
  * Longcope, Fisher, and Pevtsov (1998) Longcope, D.W., Fisher, G.H., Pevtsov, A.A.: 1998, Flux-tube twist resulting from helical turbulence: The sigma-effect. Astrophys. J. 507, 417\. DOI. ADS.
  * Martin, Bilimoria, and Tracadas (1994) Martin, S.F., Bilimoria, R., Tracadas, P.W.: 1994, Magnetic field configurations basic to filament channels and filaments. In: Rutten, R.J., Schrijver, C.J. (eds.) Solar Surface Magnetism, 303\. ADS.
  * Maurya and Ambastha (2010) Maurya, R.A., Ambastha, A.: 2010, Sub-surface meridional flow, vorticity and the life time of solar active regions. Astrophys. J. Lett. 714, L196. DOI. ADS.
  * Maurya, Ambastha, and Chae. (2014) Maurya, R.A., Ambastha, A., Chae., J.: 2014, Activity related variations of high-degree p-mode amplitude, width and energy in solar active regions. Astron. Astrophys. 561, A123. DOI. ADS.
  * Maurya, Ambastha, and Reddy (2011) Maurya, R.A., Ambastha, A., Reddy, V.: 2011, Kinetic and magnetic helicities in solar active regions. J. Phys. Conf. Ser. 271(1), 012003(1. DOI. ADS.
  * Maurya, Ambastha, and Tripathy (2009) Maurya, R.A., Ambastha, A., Tripathy, S.C.: 2009, Variations in p-mode parameters with changing onset time of a large flare. Astrophys. J. Lett. 706, L235. DOI. ADS.
  * Metcalf et al. (2006) Metcalf, T.R., Leka, K.D., Barnes, G., Lites, B.W., Georgoulis, M.K., Pevtsov, A.A., Balasubramaniam, K.S., Gary, G.A., Jing, J., Li, J., Liu, Y., Wang, H.N., Abramenko, V., Yurchyshyn, V., Moon, Y.: 2006, An overview of existing algorithms for resolving the 180 degree ambiguity in vector magnetic fields: Quantitative tests with synthetic data. Sol. Phys. 237, 267\. DOI. ADS.
  * Miesch et al. (2008) Miesch, M.S., Brun, A.S., DeRosa, M.L., Toomre, J.: 2008, Structure and evolution of giant cells in global models of solar convection. Astrophys. J. 673(1), 557\. DOI.
  * Moffatt (1969) Moffatt, H.K.: 1969, The degree of knottedness of tangled vortex lines. J. Fluid. Mech. 35(1), 117\. DOI.
  * Moffatt and Tsinober (1992) Moffatt, H.K., Tsinober, A.: 1992, Helicity in laminar and turbulent flow. Annu. Rev. Fluid Mech. 24, 281\. DOI. ADS.
  * Pevtsov, Balasubramaniam, and Rogers (2003) Pevtsov, A.A., Balasubramaniam, K.S., Rogers, J.W.: 2003, Chirality of chromospheric filaments. Astrophys. J. 595, 500\. DOI. ADS.
  * Pevtsov, Canfield, and Latushko (2001) Pevtsov, A.A., Canfield, R.C., Latushko, S.M.: 2001, Hemispheric helicity trend for solar cycle 23. Astrophys. J. Lett. 549, L261. DOI. ADS.
  * Pevtsov, Canfield, and Metcalf (1995) Pevtsov, A.A., Canfield, R.C., Metcalf, T.R.: 1995, Latitudinal variation of helicity of photospheric magnetic fields. Astrophys. J. Lett. 440, L109. DOI. ADS.
  * Pevtsov et al. (2008) Pevtsov, A.A., Canfield, R.C., Sakurai, T., Hagino, M.: 2008, On the solar cycle variation of the hemispheric helicity rule. Astrophys. J. 677, 719\. DOI. ADS.
  * Pipin et al. (2019) Pipin, V.V., Pevtsov, A.A., Liu, Y., Kosovichev, A.G.: 2019, Evolution of magnetic helicity in solar cycle 24. Astrophys. J. Lett. 877, L36. DOI. ADS.
  * Priest (2014) Priest, E.R.: 2014, A life of fun playing with solar magnetic fields (special historical review). Sol. Phys. DOI. ADS.
  * Richardson (1941) Richardson, R.S.: 1941, The nature of solar hydrogen vortices. Astrophys. J. 93, 24\. DOI. ADS.
  * Ronan et al. (1992) Ronan, R.S., Orrall, F.Q., Mickey, D.L., West, E.A., Hagyard, M.J., Balasubramaniam, K.S.: 1992, A comparison of vector magnetograms from the marshall space flight center and mees solar observatory. Sol. Phys. 138(1), 49\. DOI.
  * Russell et al. (2019) Russell, A.J.B., Demoulin, P., Hornig, G., Pontin, D.I., Candelaresi, S.: 2019, Do current and magnetic helicities have the same sign? Astrophys. J. 884(1), 55\. DOI.
  * Seehafer (1990) Seehafer, N.: 1990, Electric current helicity in the solar atmosphere. Sol. Phys. 125, 219\. DOI. ADS.
  * Seligman, Petrie, and Komm (2014) Seligman, D., Petrie, G.J.D., Komm, R.: 2014, A combined study of photospheric magnetic and current helicities and subsurface kinetic helicities of solar active regions during 2006-2013. Astrophys. J. 795, 113\. DOI. ADS.
  * Tiwari, Venkatakrishnan, and Sankarasubramanian (2009) Tiwari, S.K., Venkatakrishnan, P., Sankarasubramanian, K.: 2009, Global twist of sunspot magnetic fields obtained from high-resolution vector magnetograms. Astrophys. J. Lett. 702, L133. DOI. ADS.
  * Venkatakrishnan and Gary (1989) Venkatakrishnan, P., Gary, G.A.: 1989, Off disk-center potential field calculations using vector magnetograms. Sol. Phys. 120, 235\. DOI. ADS.
  * Watson, Fletcher, and Marshall (2011) Watson, F.T., Fletcher, L., Marshall, S.: 2011, Evolution of sunspot properties during solar cycle 23. A&A 533, A14. DOI. ADS.
  * West et al. (2002) West, E.A., Hagyard, M.J., Gary, G.A., Smith, J., Adams, M., Cloyd, R.A.: 2002, In: Goldstein, D.H., Chenault, D.B., Egan, W.G., Duggin, M.J. (eds.) Development of a new vector magnetograph at Marshall Space Flight Center, Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series 4481, 270\. DOI. ADS.
  * Yang, Zhang, and Büchner (2009) Yang, S., Zhang, H., Büchner, J.: 2009, Magnetic helicity accumulation and tilt angle evolution of newly emerging active regions. Astron. Astrophys. 502, 333\. DOI. ADS.
  * Zhang and Yang (2013) Zhang, H., Yang, S.: 2013, Distribution of magnetic helicity flux with solar cycles. Astrophys. J. 763(2), 105\. DOI.
  * Zhang et al. (2010) Zhang, H., Sakurai, T., Pevtsov, A., Gao, Y., Xu, H., Sokoloff, D.D., Kuzanyan, K.: 2010, A new dynamo pattern revealed by solar helical magnetic fields. MNRAS 402(1), L30. DOI.
  * Zhang (2006) Zhang, M.: 2006, Helicity observations of weak and strong fields. Astrophys. J. 646(1), L85. DOI.