A COMPARISON OF TWO METHODS FOR SOLAR NONLINEAR [620530]

A COMPARISON OF TWO METHODS FOR SOLAR NONLINEAR
FORCE-FREE FIELD EXTRAPOLATION
LILIANA DUMITRU
Astronomical Institute of Romanian Academy
Str. Cutitul de Argint 5, 040557 Bucharest, Romania
Email: [anonimizat]
Abstract. In this study we construct the 3D magnetic field of active region NOAA
12673 using two nonlinear force-free field (NLFFF) methods. The event chosen for
the study was the X 9.3 class flare on September 6, 2017, which was the largest flare
of solar cycle 24. We also analyzed the evolution of the free-force field parameter,
called the alpha ( ) parameter. We use the vector magnetogram from Helioseismic and
Magnetic Imager (HMI) instruments on board the Solar Dynamics Observatory (SDO)
in the full disk and Spaceweather HMI Active Region Patch (SHARP) format, for 2
hours before and after X 9.3 class flare.
Key words : Sun – solar cycle – active region – solar flare – extrapolation – force free-
field.
1. INTRODUCTION
Coronal magnetic field modeling is a very important element in the analysis of
solar activity. Over time it is a different method for a face at this time and technical
care to capture the most attention of solar researchers are free force field models
(NLFFF). Various authors need such numerical methods for constructing magnetic
coronal bearing lines, by extrapolating photosphere vector fields (Wheatland et al.
(2000), R ´egnier et al. (2002), Lee (2002), Wiegelmann (2004), Wheatland (2004),
Valori et al. (2005), Wiegelmann et al. (2006), Wheatland (2006), Wheatland and
Regnier (2009), Tadesse et al. (2009), Amari and Aly (2010), Wheatland and Leka
(2011), Wiegelmann et al. (2012), Jiang and Feng (2012), Valori (2016)).
For the use of NLFFF models we need data that specifies the force field. There-
fore, the data used are vector magnetograms obtained from photosphere observa-
tions and from these through magnetohydrodynamic simulations the coronal mag-
netic field is extrapolated.
NLFFF models have been tested on various types of data obtained from space
missions such as magnetograms obtained by Michelson Doppler Imager (MDI) in-
strument on board of Solar and Heliospheric Observatory (SOHO) spacecraft (Scher-
reret al. , 1995), Solar Optical Telescope (SOT) on board of Hinode (Tsuneta et al.
(2008), Suematsu et al. (2008)) and in recent years data Helioseismic and Magnetic
Romanian Astron. J. , V ol. 29, No. 2, p. 1–9, Bucharest, 2019

2 Liliana DUMITRU 2
Imager (HMI) instruments on board the Solar Dynamics Observatory (SDO) (Scher-
reret al. (2012), Schou et al. (2012)) but also from data obtained from terrestrial
observatories such as vector magnetograms from the Vector Spectromagnetograph
(VSM) of the Synoptic Optical Long-term Investigation of the Sun (SOLIS)– U.S.
National Solar Observatory (NSO) project on Kitt Peak (Jones et al. (2002), Keller
et al. (2003)) and the vector magnetogram data observed at the Solar Flare Telescope
(SFT) from the Mitaka Solar Observatory – Tokyo National Astronomical Observa-
tory of Japan (Sakurai et al. , 1995), Solar flare telescope at Mitaka (Hanaoka, 2013)
or vector magnetograms from Solar Magnetic Field Telescope (SMFT) from Huairou
Solar Observatory Station of Beijing Astronomical Observatory (Liu et al. , 2019).
In order to obtain an accurate representation of the coronal magnetic field, we
must take into account the fact that the data are obtained from observations, therefore
they may be subject to errors in the acquisition process. That is why an important role
in using this data is resolving the 180-degree ambiguity in the transversal components
of photospheric data, and this can be done very well with the help of Minimum
Energy method (Metcalf, 1994). As well, an important role is the resolution and the
cadence of the acquired data.
One type of data that try to solve these problems is the magnetograms obtained
by Helioseismic and Magnetic Images (HMI) on board the Solar Dynamics Obser-
vatory (SDO). These are available starting May 1, 2010 and the basic data rate is 135
seconds, but to reduce noise, magnetograms are combined and data is obtained to
every 720 seconds.
The cod (HARPs) automatically identifies and tracks patches of active regions.
Each identified area will receive a HARP number. The HARP numbering is similar
to the NOAA active regions notation without any correspondence between them.
There are two types of HARP: definitive and near real time. The most used ones
are the definitive ones but they are available after the active region has decayed or has
rotated from the visible face of the solar disk. The advantage of the definitive HARPs
is that they cover the same heliographic area throughout their entire lives (Turmon
et al. , 2014), but the disadvantage is that the data is available after the active region
completely rotates on the visible solar disk.
After identifying the HARP number of an active region we can obtain the data
SHARP (Bobra et al. , 2014). These data are remapped to cylindrical equal area
(CEA) heliographic coordinates centered on the HARP center point (Hoeksema et
al., 2014). HMI Times are given in International Atomic Time (TAI), which is with
37 seconds ahead of Universal Time Coordinated (UTC) in 2019 ( https://www.
timeanddate.com/time/international-atomic-time.html ).
In this paper we using two NLFFF methods for calculate the force-free pa-
rameter and 3D coronal magnetic fields. The first method, of Lee, uses the code
described in his master thesis (Lee, 2002) and has been tested for various active

3 A comparison of two methods for solar nonlinear force-free field extrapolation 3
regions (Lee (2002), Lee et al. (2004), Dumitrache et al. (2012), Dumitru (2011),
Dumitru (2013)) and the second method is the method of Alissandrakis (1981) used
in the code of Valori (2016).
We applied the two codes, to the active region NOAA 12673 using two different
types of data. This region was highly analyzed because it was noted for having
produced the largest solar flare for solar cycle 24 (Yang et al. (2017), Mitra et al.
(2018), Yan et al. (2018), Hou et al. (2018), Inoue et al. (2018), Liu et al. (2018),
Thalmann et al. (2019)). Moraitis et al. (2019), determined the magnetic helicity
using two NLFFF extrapolation methods: optimization method Wiegelmann (2004)
and Wiegelmann et al. (2012)) method. Also Moraitis et al. (2019)) and Vemareddy
et al. (2019) have shown that the active region NOAA 12673 does not respect the
helicity rule.
2. DATA ANALYSIS TECHNIQUES
In this paper, we used two NLFFF codes: Lee’s code (Lee, 2002) and Valori’s
code (Valori, 2016), for calculate the force-free parameter ( ) and 3D coronal mag-
netic fields. We used HMI respective HMI – SHARP data, for a period of 2 hours
before and after the X 9.3 class flare, on September 6, 2017.
Lee’s algorithm (Lee, 2002) involves mathematical modeling of magnetic field
lines using the dipole method. From the full disk HMI’s, the areas comprising the
active region to be studied are extracted (the magnetogram selection will be 512
x 512). The x,ylocations of the dipoles are determined by the local minimum and
maximum method known as Powell’s method. On the selected area the magnetic field
lines will be generated using the magnetic flux density components (Bx;By;Bz)
derived from the magnetic dipole equations. The coordinates of the density of the
magnetic flux will be given by the formulas:
Bx=B03xz
r5(1)
By=B03yz
r5(2)
Bz=B013z2
r2
r3; (3)
where x,yandzare the Cartesian coordinates of the position vector for the
field lines, ris the magnitude of the position vector and is given by the relation:
r=p
x2+y2+z2andB0is the constant of magnetic force ( B0measures the power
of the magnetic field). The parameter was calculated using the relation:

4 Liliana DUMITRU 4
=1
Bz
(@By
@x@Bx
@y): (4)
Thus, for each pixel of the magnetogram, a value of was obtained. From the
matrix of its values, obtained for each magnetogram, the global was calculated
(Hagino and Sakurai, 2004) using the formula:
g= (1
N)X
: (5)
The second NLFFF method used, is based on the method of Alissandrakis
(1981) implemented in the IDL of Valori (2016), seeking to find the solutions of
the equations of the free force field:
r !B = !B; (6)
r
!B = 0 (7)
From the SHARP data ( Bp,Br,Bt), we obtain the magnetic field components
making the transformations:
Bx=Bp(axial ) (8)
By=Bt(tangent ¸ial) (9)
Bz=Br(radial) (10)
For the calculation of parameter , the current density will be defined by the
formula:
Jz= (r !B)z: (11)
So
(x;y;z = 0) = (Jz
Bz) = (r !B)z
Bz) =1
Bz
(@By
@x@Bx
@y);z= 0: (12)
It is thus obtained as in the case of Lee’s method (Lee, 2002), the matrix of his
values from which with the same formula as in the first method we determined his
global values for each magnetogram for which we applied the code.
3. RESULTS
The two NLFFF methods used in this paper were applied to the active region
NOAA 12673. Solar active region NOAA 12673 produced the largest eruption of the

5 A comparison of two methods for solar nonlinear force-free field extrapolation 5
solar cycle 24, even though it occurred in the descending phase of the solar cycle.
Besides the fact that it produced the largest eruption of the current solar cycle, the
active region was noted for its eruptive activity. She produced a number of 4 – X
class flare, 27 – M class flare and 46 – C class flare, which shows us that she was very
productive for this phase of the cycle.
Active region NOAA 12673 appeared on the southern hemisphere of the solar
disk, on August 30, 2017, having a simple magnetic configuration. Its coordinates at
the time of occurrence were (S08E50).
Although in the early days this region did not foresee anything special, it de-
veloped rapidly so that in just a few days it came to have a complex configuration
(Yang et al. (2017), Yan et al. (2018)). On September 3, it produced the first eruption
of class C 1.1 and on September 6, after the first part of the morning it produced
an eruption of class X 2.2, at 11:53 UT started the eruption of class X 9.3, with the
maximum at 12:02 UT and ended at 12:10 UT, lasting 17 minutes. At this moment
the active region having the coordinates (S09W34).
This event is associated with a halo Coronal Mass Ejection (CME) detected by
C2 LASCO chronograph at 12:30 UT and C3 at 12:30 UT.
The two NLFFF codes used in this paper, Lee’s code and Valori’s code, use
HMI respective SHARP data, for a period of 120 minutes before and after the X 9.3
class flare, from 6 September 2017. Applying the two NLFFF codes described above
we obtained the values of the global .
The values obtained with both codes and the times at which they were calcu-
lated are shown in the table from figure 1.
Fig. 1 – Values of the global obtained with Lee’s and Valori’s code
Using values of global calculated, we construct the graphs represented in
figure 2. Above is the graph of global values obtained with Lee’s code and below
is the graph of global values obtained with Valori’s code.
The time of the beginning of the eruption is very close to T4 in both graphs
and the end of the eruption is marked by T6. The maximum of the eruption is at
T5. So, in both graphs the eruption is in the interval marked by the red dotted lines,
meaning between T4 and T6. It is noted that in this interval, his global values have
a downward trend. However, there are more variations of the global parameter, so
it cannot be established that there is any rule between the values of the parameter
determined and the eruptive phenomenon.

6 Liliana DUMITRU 6
Fig. 2 – Values of the global obtained with Lee’s code – up and with the Valori’s code – down
The values for the global , obtained for each magnetogram analyzed with
the Valori’s code, are negative, which means that the values of the magnetic helicity
will be negative. This shows that the active region NOAA 12673 does not comply
with the helicity rule. The same result was obtained by Moraitis et al. (2019) and
Vemareddy et al. (2019).
Fig. 3 – 3D extrapolation of the coronal magnetic field obtained with the Valori’s code at left and with
Lee’s code at right

7 A comparison of two methods for solar nonlinear force-free field extrapolation 7
The 3D representations of the coronal magnetic fields, obtained with the two
codes are represented in figure 3. On the right side we have the extrapolated coronal
magnetic field, obtained with the Valori code, superimposed on the magnetogram
at 12:00 and on the right side, obtained with Lee’s code, for the magnetogram at
11:59, these being those close to the peak moment of the X 9.3 class flare, that is,
12:02 UT. In both cases openings of magnetic field lines during the eruption can
be observed. Therefore the NLFFF models used are robust and can mathematically
model the magnetic field free of forces, but inconsistent solar data can lead to results
depending of the NLFFF model chosen.
4. CONCLUSIONS
The study of the magnetic field in the solar corona and the chromosphere re-
quires to use the methods of extrapolation of the photosphere magnetic field. There-
fore, in the present work, we applied to the active region NOAA 12673 two different
nonlinear force-free field (NLFF) codes, to extrapolate the coronal magnetic fields.
Following the analyzes, we were able to establish a concordance between observa-
tions and the mathematical results obtained by magnetohydrodynamic simulations,
for obtaining the coronal magnetic field lines.
I also found that there are differences between the results obtained with the
two methods. Apart from the fact that there are different working methods, another
cause of obtaining different results may be that the data used is different: HMI type
(hmi.M 45s) respectively SHARP (hmi.sharp cea720s) . Thalmann et al. (2013) per-
formed a comparison of force-free coronal magnetic field using vector field from
Hinode and SDO. The very different results were justified by these by the difference
of the data but also by the sensitivity of the instruments and the temporal evolu-
tion of the photosphere magnetic field. The NLFFF models used are robust and can
mathematically model the magnetic field free of forces with a high accuracy, but
inconsistent solar data can lead to results depending on the NLFFF model chosen.
Therefore, in order to obtain the best results is necessary to test the extrapola-
tion methods on different phenomena and solar active regions.
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