J. Plasma
J.
Plasma Fusion
Fusion Res.
Res. SERIES,SERIES,Vol.
Vol.6 6(2004)(2004)115–118
000–000

Study of Magnetic Helicity and Magnetohydrodynamic
Relaxation in Solar Flare Processes
KUSANO Kanya1,2, MAESHIRO Tomohiro2, MIIKE Haruka2, YOKOYAMA Takaaki3 and SAKURAI Takashi4
1

Earth Simulator Center, Japan Agency for Marine-Earth Science and Technology, Kanazawa-ku 236-0001, Japan
2
Graduate School of Advanced Sciences of Matter, Hiroshima University, Higashi-Hiroshima 739-8530, Japan
3
Graduate School of Sciences, University of Tokyo, Bunkyo-ku 113-0033, Japan
4
National Astronomical Observatory, Mitaka 181-8588, Japan(Received: 10 December 2003 / Accepted: 26 January 2004)

Abstract
The solar coronal activities, such as solar flares, are believed to arise as a result of photospheric plasma motion,
which supplies magnetic free energy as well as magnetic helicity into the solar corona. In this paper, we investigatedthe physical relationship between magnetic helicity and the coronal activity, based on the three-dimensional numericalsimulations and on the vector magnetograph observations. We found that magnetic helicity flow across the solar
surface has a patchy structure, in which the right-handed helicity and the left-handed helicity were almost
simultaneously injected within an active region. Furthermore, it was also revealed, in the various flare events, that theinitial brightening of TRACE 1,600 Å image was located at a region where magnetic shear is sharply reversed on thephotosphere. Motivated by these observations, we carried out the numerical simulations, in which the magnetic shear
in a magnetic arcade is reversed by a slow foot-point motion. It was demonstrated by the simulations that the resistive
instability growing on the shear inversion layer can lead to the eruption of large-scale plasmoid through the interactionof two different kinds of magnetic reconnections. These results strongly suggest that the solar coronal activity andsolar flares are a consequence of the magnetohydrodynamic relaxation process driven by the cancellation of the
oppositely sheared magnetic fluxes.
Keywords:
solar flare, the Sun, MHD, magnetic reconnection

1. Introduction

discussed based on the vector magnetogram observation andthe observation-based numerical simulations.

For the last decade, several satellite observations by
Yohkoh, SOHO, and TRACE have revealed that the solar
corona is a dynamic plasma system, in which various typesof energetic activity always proceed. Particularly, solar flares
are powerful explosion in the solar corona, powered by thesudden release of the magnetic energy accumulated in thecoronal magnetic field. Although there are many evidencesthat magnetic reconnections play a key role in flare processes,
the triggering mechanism of flares still remains to be solved.
Since magnetic shear and sigmoid structure are often
observed in flare productive active regions by themagnetogram and the soft-X ray observations, respectively,magnetic helicity is widely believed to be an important
quantity for understanding the trigger mechanism of solar
flares. The objective of this paper is to briefly review ourrecent study of magnetic helicity in solar flare processes. In
particular, the relationship between magnetic helicity in thesolar corona and the trigger mechanism of solar flares will be

2. Measurement of magnetic helicity flux
Since the measurable magnetic field in the sun is limited
only on the photospheric and the chromospheric levels, wecannot directly measure magnetic helicity contained in thesolar corona. On the other hand, according to Berger andField’s formula [1], the gauge-invariant relative helicity should
be defined by the volume integral,H = ∫ (A + AP) · (B – BP)dV ,and the flux across a plane S can be described by

Ḣ = 2 ∫ S E × AP · dS ,(1)

where A and AP are vector potentials of magnetic field B andcurrent free field BP , which have common boundary conditionof the normal component on S, respectively. It means that, if
the electric field E on the solar surface can be measured in©2004 by The Japan Society of Plasma
Science and Nuclear Fusion ResearchCorresponding author’s e-mail: kusano@jamstec.go.jp1115

Kusano K. et al., Study of Magnetic Helicity and Magnetohydrodynamic Relaxation in Solar Flare Processes

addition to the magnetic field, we could derive the magnetic
helicity flux into the solar corona from the sun. Recently,
Kusano et al. developed the new methodology, which enables
to derive the electric field from the vector magnetogram
observations [2]. The principle of the new measurement isbased on the induction equation of the normal component on
the solar surface,∂Bn /∂ t = [∇ × (V × B)]n .

the area (labeled with ‘A’), in which magnetic helicity wasmost actively injected. This is a negative result for the simple
model that the over-injection of magnetic helicity can trigger
solar flares, and suggests that some other properties should
be relevant to the onset of flares rather than the amount ofmagnetic helicity.
On the other hand, our measurement of magnetic helicity
revealed also that the structure of helicity injection in active
regions was highly complicated both in time and space, andthat even the sign of helicity flux often changed within asingle active region [2]. Motivated by the results, we recentlyproposed a new hypothesis that annihilation between thepositive and negative magnetic helicities is responsible for the
release of magnetic energy and also for the trigger of solar
flares [5]. Magnetic helicity is a quantity having either
positive or negative sign, which represents the right-handed
linkage or the left-handed linkage of magnetic fluxes,
respectively. Therefore, if the positive and negative helicities
coexist within an active region, magnetic reconnection cancancel magnetic helicity by merging the flux systems ofopposite helicities. Since the reconnected field may relax tothe helicity-free state, it is likely that the free energy
corresponding to the magnetic helicity of mixed signs isefficiently released through the helicity annihilation process.
In fact, we found that the brightening in some solar flares
extended from a point where the magnetic shear is steeply
reversed, based on the correlation analyses between TRACE1,600 Å images and the vector magnetogram observations [6].
Since the bright areas in TRACE 1,600 Å images could
correspond to the foot-points of the field lines, which are
subject to energy liberation, these results strongly support thehelicity annihilation hypothesis.

(2)

Because the magnetogram observation can provide theinformation of the magnetic field vector B as well as thevariation ∂Bn /∂ t, the velocity V can be derived as a solution
from the inverse problem of the induction equation (2).
However, it is not a well-defined problem, because an infinite
number of the solutions is possible. The uncertainty comesfrom the fact that the induction equation (2) cannot determine
the velocity component orthogonal to the vertical plane, whichcontains the magnetic field line. Therefore, the velocity vector
V has to be derived from (2) after determining the orthogonal
velocity, which is tangential to the solar surface. We showedthat it can be obtained by applying the correlation tracking
technique onto the sequential data of magnetograms. The
proposed procedure can always provide the consistent electricfield, even though the derived velocity vector V is not thesame as the real plasma velocity.
Using this new method, Maeshiro et al. [3] found thatthe absolute intensity of the magnetic helicity flux wellcorrelates with the averaged energy flux of the soft-X ray
radiation, based on the data analyses of the seven different
active regions appearing on the photosphere from the year
1997 to 2000. This result indicates that magnetic helicity isindeed responsible for the energetic activity in the solar
corona. However, the detail analyses of each flare event
implied that there were no clear correlation between theamplitude of injected helicity and the onset of flares [4]. For
instance, in Fig. 1, we can see that the solar flare, which wasobserved by Yohkoh HXT, occurred in a region away fromFig. 1

3. Numerical simulation
In order to examine the helicity annihilation hypothesis,
we developed the three-dimensional numerical simulation
model [7]. The simulations were carried out by applying thefoot-point shear motion, which reverses the pre-loaded
magnetic helicity, on the vicinity of magnetic neutral line in
the arcade configuration. As a result, it was clearly
demonstrated that the reversal of magnetic helicity can causea large scale eruption of the magnetic arcade through a seriesof two different kinds of magnetic reconnections. The
physical mechanism is explained as follows: First, theresistive tearing mode instability grows on the helicity
inversion surface, on which the axial magnetic field along themagnetic arcade is reversed. Secondly, the resistive tearing
mode instability drives magnetic reconnection, whichannihilates the positive and negative axial magnetic fluxes.
Thirdly, the flux annihilation is followed by collapse of themagnetic arcade into the reconnection point, and then thevertical current sheet is generated. Fourthly, the second
reconnection starts on the new current sheet, and drives theexplosive up-welling flow, as seen in Fig. 2a. Finally, the total
system reaches a loss-of-equilibrium state, and the wholemagnetic arcade is erupted upward (see Fig. 2b).

A map of active region NOAA 8100 at 04:34:39 UT,Nov. 3, 1997. Gray scale indicates intensity of helicity
flow, and thin contours (white and black) represent themagnetic field component along the line-of-sightobserved by SOHO/MDI. Thick contour is hard-X ray
intensity of solar flare observed by Yohkoh HXT.
Region where magnetic helicity is most intensively
injected is labeled with ‘A’.

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Kusano K. et al., Study of Magnetic Helicity and Magnetohydrodynamic Relaxation in Solar Flare Processes

collapse seen in Fig. 2a. Because two reconnections drive
each other, the eruptive dynamics may start quickly, eventhough the growth rate of the original instability (tearingmode) is much slower than the MHD time-scale.
It was also found from the simulation results that theoriginal instability growing on the helicity inversion layer may
cause the MHD energy relaxation in the embedded region
bounded by the helicity inversion surface in advance of thelarge scale eruption. Figure 3 indicates the vertical distributionof the averaged shear parameter

These processes are understood as a chain-reaction, in
which the resistive magnetohydrodynamic (MHD) instability(tearing mode) brings about a loss-of-equilibrium dynamics(eruption). In this model, the onset of solar flares can beexplained as a result of mutual excitation between twodifferent reconnections, which grows on the helicity inversion
surface and on the current sheet generated by the internal

α = B⋅∇×BB⋅B,

where the bra-ket denotes the average along the magnetic
arcade axis. Here, we can see that 〈α 〉 is greatly decreased in
the lower part (z < 0.1) for the early phase (t = 8.8). It is a
result of the shear reversal due to the foot-point motion
imposed near the magnetic neutral line. After the relaxation
phase (t = 30.5), however, 〈α 〉 in the negative shear region (z< 0.2) is dramatically flattened.
According to the Taylor’s theory, the absolute value ofthe shear parameter in the relaxed state must be smaller than
the smallest eigenvalue of the curl operator [8,9,10]. The
limitation of α arises as a result of transition of the linear
force-free field from the so-called coupled state into the mixed
state in the words of Taylor [8]. In our model, since the sizeof the revered shear region is L = 0.2 in the non-dimensional
unit, and thus the corresponding eigenvalue is approximately
given by 2π/L = 31.4. In fact, 〈α 〉 in the negative helicity
region at t = 30.5 has a value close to α0 as seen in Fig. 3. It
is an evidence of the Taylor-type relaxation. Furthermore, wecan see in Fig. 4 that the magnetic field lines form the inverse
S-shape structure after the energy relaxation.
Sigmoidal structure, which showed the S- or the inverse200

α0−20−40

t=30.5<α> −60−80Fig. 2t=8.8−100

Three-dimensional structure of the magnetic field lines
in the eruptive phase calculated by the numericalsimulation [5]. (a) Sheared loop field lines are collapsed
into the center of arcade, as a result of the annihilationof the axial magnetic field parallel to the magnetic
arcade axis, and they are reconnected. The up-stream
jet region is plotted by the iso-value surface, in which
the up-welling velocity is faster than 10% of thecharacteristic Alfvén speed. The typical flow structure isillustrated by thick arrows. (b) The large scale plasmoid
is ejected upward.

−120−14000.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5z
Fig. 33117

Vertical distribution of the averaged α calculated by thethree-dimensional simulation. Dashed line indicates thesmallest eigenvalue α0 = –31.4 corresponding to size ofthe reversed shear region.

Kusano K. et al., Study of Magnetic Helicity and Magnetohydrodynamic Relaxation in Solar Flare Processes

4. Conclusion

Fig. 4In this paper, based on the vector magnetogram
observations and the numerical simulations, we havedemonstrated that the reversal of magnetic shear in thecoronal magnetic field may cause a large scale eruption of amagnetic arcade and the MHD energy relaxation process. The
physical mechanism of flare onset is explained as a chainreaction, in which the resistive MHD instability (tearingmode) brings about a loss-of-equilibrium dynamics (eruption).
The simulation results are well consistent with theobservations of magnetic field and X-ray images. However, it
is still an open problem how generally is this model
applicable to various types of energetic events. Therefore,
further investigation is required to carefully examine therelationship between the magnetic shear structure and thecoronal activity.

Top views of the magnetic field lines in the inside of
the reversed shear region before (t = 8.8) and after (t =30.5) the MHD energy relaxation.

References[1] M.A. Berger and G.B. Field, J. Fluid Mech. 147, 133(1984).
[2] K. Kusano, T. Maeshiro, T. Yokoyama and T. Sakurai,Astrophys. J. 577, 501 (2002).
[3] T. Maeshiro, K. Kusano, T. Yokoyama and T. Sakurai,Astrophys. J. (2005) in press.
[4] T. Yokoyama, K. Kusano, T. Maeshiro and T. Sakukrai,
Advances in Space Research 32, 1949 (2003).
[5] K. Kusano, T. Maeshiro, T. Yokoyama and T. Sakurai,Advances in Space Research 32, 1917 (2003).
[6] T. Maeshiro, K. Kusano, T. Yokoyama and T. Sakurai,
Astrophys. J. (to be submitted).
[7] K. Kusano, T. Maeshiro, T. Yokoyama and T. Sakurai,
Astrophys. J. 610, 537 (2004).
[8] J.B. Taylor, Revs. Modern Phys. 58, 741 (1986).
[9] K. Kusano, Y. Suzuki and K. Nishikawa, Astrophys. J.
441, 942 (1995).
[10] K. Kusano and K. Nishikawa, Astrophys. J. 461, 415(1996).

S-shapes corresponding to the sign of magnetic helicity, wasoften observed by soft-X ray observation of active regions.
Since sigmoidal structures sometimes appeared prior to solar
flares and coronal mass ejection (CME) events, it was pointed
out that sigmoids could be a precursor of eruption in the solar
corona. The numerical results, that the MHD energy
relaxation forms the inverse S-shape field line preceding thelarge scale eruption, is consistent with the observations. Itimplies that the MHD energy relaxation, which is similar tothe self-sustainment activity of the reversed-field pinch
experiments, could be realized also in the solar corona, andthat the coronal activities, in particular solar flares, areunderstood as a consequence of that.

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