k. kusano et al- the trigger mechanism of solar flares in a coronal arcade with reversed magnetic...

8/3/2019 K. Kusano et al- The Trigger Mechanism of Solar Flares in a Coronal Arcade with Reversed Magnetic Shear

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THE TRIGGER MECHANISM OF SOLAR FLARES IN A CORONAL ARCADEWITH REVERSED MAGNETIC SHEAR

K. Kusano and T. Maeshiro

Graduate School of Advanced Sciences of Matter, Hiroshima University, 1-3-1 Kagamiyama, Higashi-Hiroshima, Hiroshima 739-8530, Japan;

[email protected]

T. Yokoyama

Graduate School of Sciences, University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033, Japan

and

T. Sakurai

National Astronomical Observatory, 2-21-1 Osawa, Mitaka, Tokyo 181-8588, Japan

Receivved 2003 November 27; accepted 2004 April 1

ABSTRACT

We have investigated the possibility that magnetic reconnection between oppositely sheared magnetic loopsworks as a trigger mechanism of solar flares, based on three-dimensional numerical simulations. The simulationswere carried out by applying a slow footpoint motion, which reverses a preloaded magnetic shear, in the vicinity

of the magnetic neutral line. The simulation results clearly indicated that the reversal of magnetic shear can causea large-scale eruption of the magnetic arcade through a series of two different kinds of magnetic reconnections.The first reconnection is initiated by the resistive-tearing mode instability growing on the magnetic shearinversion layer and annihilates the sheared magnetic fluxes, which are oppositely directed along the magneticneutral line. As a result of this, the magnetic arcade collapses into the reconnection point, and a new current sheetis generated above and below the shear inversion layer. The generation of new current sheets is followed byanother magnetic reconnection, which drives the eruption of the sheared magnetic arcade. Mutual excitation ofthe two reconnections may explain the explosive property of the flare onset.

Subject headinggs: MHD Sun: activity Sun: flares Sun: magnetic fields

1. INTRODUCTION

Although magnetic reconnection is widely believed to playa crucial role in energy relaxation in solar flares, the triggermechanism of flares still remains an open problem. Magneticreconnection is a self-sustaining process in which stagnatedplasma flow and current sheet thinning mutually drive eachother, once the process starts. However, the fundamentalquestions, how and why reconnection starts as an explosive process in flare events, are not yet well understood.

In order to solve these questions, we have to explain the physical causality that connects the slow evolution of the photospheric magnetic field and the eruptive dynamics inflares. Theoretically, two possibilities, the loss of stability andthe loss of equilibrium, have been pointed out as the cause. Inthe loss-of-stability model, the flare trigger could be explained

by the destabilization of the magnetohydrodynamic (MHD)mode, for instance the kink instability (Hood & Priest 1979;Rust & Kumar 1996; Gerrard & Hood 2003). However, sincethe growth rate of the conventional MHD instability couldgradually increase as the system slowly comes away from thestability limit, some nonlinear mechanism, which acceleratesthe dynamic timescale, could be necessary in order to explainthe explosive property of flares. On the other hand, the loss-of-equilibrium model may offer a promising prospect for theexplanation of explosiveness (Forbes & Priest 1995). How-ever, it is still unknown whether the coronal field can indeedlose the equilibrium condition before the onset of flares in therealistic boundary condition on the photosphere.

Since magnetic helicity, which is a measure of the linkage

between magnetic fluxes, is a key quantity for stability and

equilibrium in various MHD systems (Brown et al. 1999), itis likely that magnetic helicity is relevant to the triggering

mechanism of solar flares. When the current-free field B0 isadopted as the reference field, the gauge-invariant relativehelicity of magnetic field B can be described by

HR(B)

ZV

(A A0) = (B B0)dV; 1

where B and B0 have a common boundary condition for thenormal component and A and A0 are the vector potentials ofB and B0, respectively (Berger & Field 1984). By definition,HR must vanish when the magnetic field B is relaxed to thecurrent-free field B0. This implies that the relative helicitycontained in the coronal field has to be reduced when a solar

flare liberates the magnetic free energy that is defined as theexcess from the current-free field energy.However, magnetic helicity is a topological invariant in

perfectly conducting media (Woltjer 1958), so it can hardlychange in a high-temperature plasma such as the solar corona(Taylor 1986). Even magnetic reconnection cannot expeditethe decay of magnetic helicity, but can only transfer it fromone flux system to another. Hence, as a trigger mechanism offlares a great deal of attention has been paid to magnetic re-connection, which expels the excess helicity out of the solarcorona by detaching the magnetic field lines from the solarsurface (Mikic et al. 1988; Biskamp & Welter 1989; Mikic &Linker 1994; Kusano et al. 1995; Birn et al. 2000; Amari et al.2003; Roussev et al. 2003). These conventional models were

based on the idea that solar flares should occur after too much537

The Astrophysical Journal, 610:537549, 2004 July 20

# 2004. The American Astronomical Society. All rights reserved. Printed in U.S.A.


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helicity is injected into the coronal field (Priest & Forbes2000).

Recently, Kusano et al. (2002) developed a new method-ology that enables the measurement of magnetic helicity fluxthrough the photosphere. They used an induction equationand local correlation tracking to infer the electric field onthe photosphere from vector magnetograms. This techniqueallows us to accurately evaluate the rate of magnetic helicityinjection due to flux emergence and shearing motions. Usingthis new technique, we have recently analyzed the relationshipbetween helicity injection and the onset of flares that occurredin NOAA Active Region 8100 (Yokoyama et al. 2003). How-ever, a clear correlation between the amplitude of helicityinjection and the onset of flares was not found. This is anegative result for the simple model in which the overinjectionof magnetic helicity can trigger solar flares and suggests thatsome properties beyond the amount of magnetic helicity couldbe relevant to the onset of flares.

On the other hand, our measurement of magnetic helicityalso revealed that the structure of helicity injection in activeregions is highly complicated in both time and space and that

even the sign of the helicity injection often changes within asingle active region ( Kusano et al. 2002; Yokoyama et al.2003). This new finding suggested another possibility for re-ducing magnetic helicity in the coronal plasma. Magnetichelicity is a quantity having either a positive or a negativesign, which represents the right-handed linkage or the left-handed linkage of magnetic fluxes, respectively. This meansthat if positive and negative helicities coexist in a single do-main, magnetic reconnection can cancel magnetic helicity bymerging flux systems of opposite helicities. If it does, thereconnected field may relax toward a helicity-free state, andthus the free energy corresponding to magnetic helicity ofmixed signs should be released.

Magnetic helicity in the solar corona is closely related to

magnetic shear, which is usually defined as the parallel component of a transverse magnetic field along the magneticneutral line on the solar surface. The topological relationship between magnetic helicity and magnetic shear is easily un-derstood in a simple magnetic arcade model that is symmetricalong the magnetic neutral line, as illustrated in Figure 1. Whenthe x-axis is taken along the magnetic neutral line, the mag-netic field can be described by

B Bxex : < ex; 2

where ex is the unit vector directed to the x-axis and is themagnetic flux through the shaded ribbon S1 of a unit length

along the x-coordinate. Since the isosurface of correspondsto the magnetic arcade on which the field vector is tangent, theaxial magnetic flux through the cross section S2,

()

ZS2

Bx dydz; 3

is defined as a function of. In this case, as explained in theAppendix, the magnetic helicity density h for unit flux can bedefined on the surface , and it is given by the sum of the twofluxes,

h() s ; 4

where s d=d reflects the shearing displacement betweenthe field line footpoints along the neutral line. Equations (3)and (4) indicate that the axial magnetic field Bx, which cor-responds to the magnetic shear, is a principal ingredient formagnetic helicity. If the axial field is reversed in the vicinity ofthe magnetic neutral line, the sign of the helicity density h()must also be changed between the central region and the outerregion of the arcade because the signs of and s are deter-mined by Bx .

In fact, our recent analyses of the magnetic helicity injection activity and the sheared magnetic field structure inflare-productive active regions indicated a tendency that thecoexistence of the positive and negative magnetic shear couldbe associated with solar flares (Yokoyama et al. 2003). Kusano

et al. (2003a) also detected in NOAA Active Regions 9026 and9077 that the intensive injection of magnetic helicity of mixedsigns was seen prior to the X-class flare events in the GOESclassification. Furthermore, our preliminary simulation indi-cated that magnetic reconnection indeed takes place on theshear inversion layer, when the arcade feet are twisted into theopposite sense, compared to the preloaded shear (Kusano et al.2003b). These results suggest that shear reversal might be re-lated to the trigger process of solar flares.

The basic idea of reconnection between the reversed-shearfields is also consistent with simulations by Linton et al.(2001), who showed that the collision between counterhelicityflux tubes oriented in opposite directions is the most dynamicof the possible configurations of the loop-loop interaction.

Also, Mok et al. (2001) studied the process in which a pre-existing magnetic loop collides with an emerging loop ofopposite helicity and showed that magnetic reconnection be-tween the counterhelicity loops may explain the energy re-lease, at least in small flares.

The objective of this paper is to examine the hypothesisthat shear reversal in the coronal magnetic field can causethe onset of solar flares. In order to achieve this purpose, westudy the detailed dynamics of the shear reversal process inthe magnetic arcade, based on three-dimensional numericalsimulations.

The rest of this paper is organized as follows. The numer-ical model and the simulation results are described in xx 2 and3, respectively. Based on the simulation results, we propose a

Fig. 1.Schematic illustration of the magnetic arcade system.

KUSANO ET AL.538 Vol. 610


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new trigger model for solar flares in x 4, and important con-clusions and remaining problems are summarized in x 5.

2. NUMERICAL MODEL

The simulation domain spans a rectangular box, 0 < x 3) to thelower mode (m 1). This can be understood as an inversecascade of magnetic energy. Actually, it is seen in Figures 8b(t 12:4) to 8e (t 33:0) that the undulation of the shear in-version surface gradually changes from the short-wavelength

Fig. 8.Variation of the magnetic field Bx (color scale) and the electric current density jJj (red contours) on the x-z plane above the magnetic neutral line in thethree-dimensional simulation.

TRIGGER MECHANISM OF SOLAR FLARES 543No. 1, 2004


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mode to the longer wavelength mode. As a result, the cancel-lation between the positive and negative axial field Bx prefer-entially proceeds in a particular region (x 0:4 $ 0:6), as seenin Figure 8d.

In order to evaluate the reconnection activity, the minimumvalue of the absolute flux across the y-zplane, which is definedby

F(t) minx

ZLz0

ZLy0

Bx(t;x;y;z)j jdydz;

is calculated. If the positive and negative Bx are canceled byreconnection, we may detect this as a notable decrease of F.

In Figure 11 it is seen that the absolute flux of the two-dimensional simulation constantly increases, after the fieldlines are completely reversed at t 5. This means that thefield reversal caused by the photospheric shear motion mayovercome the resistive diffusion. On the other hand, the ab-solute flux of the three-dimensional simulation intermittentlydecreases after t 8. Each stepwise decrease corresponds to

magnetic reconnection, which proceeds on the shear inversionlayer as explained above.3. Eruptive phase.In Figure 5c we can see that a new

phase starts at t 32 in which the kinetic energy

EK(m)

ZLz0

ZLy0

vmj j2

dydz

Fig. 9.Three-dimensional structure of the field lines (a) before (t 8:47) and (b) after (t 8:79) reconnection on the shear inversion layer in the three-dimensional simulation. Converging flows into the reconnection point are illustrated by thick arrows in ( a). Red strings in (b) indicate typical field lines subjected toreconnection. Color shading on the bottom plane represents Bx , the same as in Fig. 2.

Fig. 10.Geometrical relation for the field lines plotted in Fig. 9 b. Theabscissa (z) and the ordinate (s) indicate the z-coordinate, at which the linesintersect the vertical plane (y 0), and the x-coordinate displacement betweenthe two footpoints of each line, respectively. The two open circles correspondto the reconnected field lines ( Fig. 9b, red strings), and the filled circles are for

the reversed-shear field, which is not yet subject to reconnection.

Fig. 11.Evolution of the absolute flux F in the two-dimensional andthree-dimensional simulations. The dotted line indicates the time when the

eruptive phase starts.



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of the m 0 component remarkably increases. This is a sig-nature that the magnetic arcade finally erupts through thenonlinear activity in the relaxation phase.

Figures 12a and 12b represent the three-dimensional struc-ture of the magnetic field lines at t 32:4. We can see thatthe positively sheared field lines (blue lines) contact eachother just above the shear inversion layer. The vertical currentsheet, which is labeled C2 in Figure 8e, is newly generatedbetween these field lines and works as a diffusion region formagnetic reconnection. The vertical magnetic field (Bz) andthe horizontal velocity (Vy) are mainly involved in this new

reconnection process. Magnetic reconnection at the current

sheet C2 creates a weakly sheared field and a highly shearedfield below and above the reconnection point, respectively, asshown by the red lines in Figures 12a and 12b. Therefore, thistype of reconnection plays a role in transferring positivelysheared flux from below to above the reconnection point.

The highly sheared field above the reconnection pointaccelerates the plasma upward by the slingshot effect, as il-lustrated by the upward arrow in Figure 12b, and a high-speedup-welling jet is generated. The green surface of this figurerepresents the region where Vz is larger than 0:1VA. The plasmain this region is accelerated by up to 36% of the local Alfven

speed.

Fig. 12.Three-dimensional structure of the magnetic field lines in the eruptive phase of the three-dimensional simulation. The plot format is the same as inFig. 2, and (a), (c), and (d) are snapshots at t 32:4, 37.2, and 44.7, respectively; (b) is the zoom-in view of the region bounded by the black square in (a). Typical

plasma flows are illustrated by thick arrows. The green surface in (b) represents an isovalue surface, on which Vz 0:1VA.



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On the other hand, the downward flow from the reconnec-tion region C2 collides with the shear inversion layer. The

collision of the flow may further strengthen the original re-connection with the negatively sheared field lines (corre-sponding to gray lines in Fig. 12b) at the current sheet C1 inFigure 8e. Note that both reconnections at C1 and C2 tend toreduce magnetic shear around the shear inversion layer, al-though the first and second reconnections play different roles,namely, the annihilation of the sheared field and the transfer ofthe sheared field, respectively.

Finally, while the up-welling flow lifts up the magneticarcade, the current sheet C2 extends along the x-coordinate,and then the whole arcade is subject to reconnection, whichlaunches a large-scale plasmoid, as seen in Figures 12c12d.Once this eruption starts, the growth of kinetic energy doesnot stop until the calculation is terminated. This could be

an indication of the fact that the system reaches a loss-of-equilibrium state.

4. DISCUSSION: REVERSED-SHEAR FLARE MODEL

In the previous section, it was clearly demonstrated by thenumerical simulations that the reversal of magnetic shear maycause large-scale eruption of the magnetic arcade through aseries of magnetic reconnections. Based on the results, herewe propose a new model for the triggering mechanism of solarflares.

Our model predicts that the flare process should be initiatedat some point on the shear inversion layer where the sign ofmagnetic shear is steeply changed in an active region. Let usconsider a reversed-shear system in which the left-handedaxial field Ba is embedded in the magnetic arcade with theright-handed axial field Ba , as illustrated in Figure 13a. When

Fig. 13.Illustration of the shear annihilation process. (a d) Time sequence seen as a projection onto the cross section of the magnetic arcade. The thin lineswith arrows and the thick lines (C1 and C2) correspond to the magnetic field lines and the current sheets, respectively. The large arrows depict typical plasmaflows.



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the direction of the axial field is steeply switched, an intensivecurrent sheet C1 appears on the shear inversion layer, and itcould be destabilized against the resistive-tearing mode in-stability. The growth of the instability stimulates the verticallystagnating flow and may drive magnetic reconnection betweenBa and B

a .

Here note that the criterion of the tearing-mode instability ismainly decided by the gradient of the current density. There-fore, the local structure of magnetic shear, rather than the totalamount of magnetic helicity, could be more important as atrigger for the process considered here. If the tearing instability grows sufficiently on the shear inversion layer, recon-nection may cancel the right-handed and left-handed magneticfluxes near the reconnection point, as schematically illustratedby the open ellipsoid in Figure 13b. The flux cancellation wasclearly seen in our calculations (Fig. 11).

In the sheared-arcade system, the MHD equilibrium can be retained by a balance between the magnetic pressure ofthe axial field and the tension force of the magnetic loop.Therefore, flux cancellation erodes the equilibrium conditionnear the reconnection point, and finally the magnetic arcade

collapses inward (see Fig. 13c). As a result, the vertical cur-rent sheet C2 is generated, and a second reconnection, whichleads to the arcade eruption, will be triggered (see Fig. 13 d).

This is a sort of the chain reaction in which reconnectioncauses another reconnection. Furthermore, since the down-ward outflow from the second reconnection point (C2) worksas the inflow into the first reconnection point (C1), the tworeconnections mutually excite each other, at least immediatelyafter the second reconnection starts. It is likely that the mutualexcitation of the two reconnections, which appears only in thetransition process from the relaxation phase to the eruptive phase, may cause the impulsive feature in solar flares.

As the eruptive phase proceeds, the reconnection point onthe vertical current sheet (C2) moves upward away from the

shear inversion, and the mutual relation of the two recon-nections should be weakened. However, the second recon-nection between the originally sheared loops (see Fig. 12c)can be self-sustained as the reconnection region extends alongthe arcade axis. Finally, the whole arcade is subject to re-connection, and a detached flux tube (plasmoid) is launchedout of the coronal region (see Fig. 12d). Once the detachedflux rope is formed, the basic mechanism of the plasmoidejection can be explained by conventional loss-of-equilibriumtheory (Priest & Forbes 1990; Forbes & Priest 1995), althoughthe trigger mechanism is different from the previous models.

Note also that, in our model, the flux rope does not nec-essarily need to preexist for the triggering of an eruption,unlike in the previous models (Priest & Forbes 1990; Forbes

& Priest 1995; Chen & Shibata 2000). Furthermore, even the photospheric converging flow into the magnetic neutral line,which was introduced to drive reconnection in the models byBirn et al. (2000), Amari & Luciani (2000), and Amari et al.(2003), is not necessary, because the annihilation of the re-versed shear can spontaneously generate the converging flowinto the shear inversion layer. Theoretically speaking, thedouble-reconnection process can be interpreted as a transitionmechanism in which the resistive instability (tearing mode)gives rise to the loss-of-equilibrium process through fluxcancellation.

According to our model, the preflare phase, which maycorrespond to the phases before the eruption, could be con-trolled by the competition between the shear reversal and the

resistive-tearing mode instability. If the shear reversal takes

place much more slowly than the instability, the coronal fieldcan adjust the change of boundary condition without devel-oping a shear inversion layer, and thus the instability may notgrow sufficiently. Actually, in the numerical simulation thegrowth time of the initial-phase instability is as long as thetypical time of the shear reversal, shear w=V0.

For practical parameters (temperature T 106 K and plasma density n

1015 m3), the growth time of the most

unstable tearing mode tearing (A)0:5 is given by 4 ;

105(3=B)0:5 (s) for a current sheet of width (m) andmagnetic field B (T), where A and are the Alfven transittime and the resistive diffusion time, respectively. If tearing iscomparable to the observed timescale of the helicity injection(104 s; Kusano et al. 2002), is estimated to be 100 to 200 kmfor B 102 to 101 T. This result suggests that the narrowcurrent sheet, which can be observed as a discontinuity of thehorizontal magnetic field by vector magnetographs, is a can-didate region for solar flares, because the estimated width is 1order of magnitude smaller than the resolution of the currentvector magnetographs.

5. SUMMARY

In this paper, using numerical simulations we have dem-onstrated that shear reversal in the coronal magnetic field maycause a large-scale eruption of a magnetic arcade and pro-posed a new model of the solar flare triggering mechanism.The physical process is explained by the following scenario.First, the resistive-tearing mode instability grows on the shearinversion layer if the magnetic shear is steeply reversed.Second, magnetic reconnection driven by the tearing-modeinstability cancels the antiparallel magnetic fluxes along themagnetic neutral line. Third, the flux cancellation is followedby collapse of the magnetic arcade into the reconnection point,and then the vertical current sheet is generated. Fourth, asecond reconnection is triggered on the new current sheet and

drives the up-welling jet. Finally, the arcade system reaches aloss-of-equilibrium state, and the whole magnetic arcadeerupts upward.

These processes are understood as a chain reaction in whichthe resistive MHD instability (tearing mode) brings about loss-of-equilibrium dynamics (eruption). Even though the growthrate of the original instability is much slower than the MHDtimescale, the mutual excitation of the two different kinds ofmagnetic reconnection may well explain the explosive prop-erty of flare onset.

The reversed-shear flare model is still based on the hy-pothesis that there is a steep gradient of magnetic shear nearthe magnetic neutral line. However, the careful analyses of thevector magnetogram indicated that there indeed was a region

where the transverse magnetic field steeply varied in an activeregion (Solanki et al. 2003). Our recent analyses reveal thatthe position of H emission by the flares in NOAA ActiveRegion 8100 corresponded well to the shear inversion line,which was seen as the region where the parallel component ofthe nonpotential magnetic field along the vector potential A0 issharply reversed (Yokoyama et al. 2003). Also, the correlationanalyses of TRACE 1600 8 images (Handy et al. 1999) andvector magnetograms indicate that the initial brightening inthe preflare phase was located along the shear inversion line(H. Miike et al. 2003, private communication). Furthermore,the statistical analyses of vector magnetograms for multipleactive regions suggest that there is a positive correlation between the length of the shear inversion layer and the soft

X-ray activity (T. Maeshiro et al. 2004, in preparation). These



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results strongly support the validity of the reversed-shear flaremodel.

Flare kernels observed in X-ray, UV, and optical wave-lengths are believed to be the footpoints of loops in which theenergy has been released probably through reconnection processes. Tanaka (1987) studied the kernels observed inthe He D3 line and found that in the initial phase of a flare, thefootpoints showed a sheared configuration with respect to themagnetic neutral line and moved in a very complex manner.Hard X-ray footpoints observed by the Yohkoh hard X-raytelescope (Kosugi et al. 1991) were found to separate in an-tiparallel directions with respect to the neutral line in someflares (Sakao et al. 2000). Our new model can account for suchnonstandard motions of the footpoints.

On the other hand, several open problems still remain forthe proof of the new model. For instance, because of 180

ambiguity in the polarization measurements, uncertainty mayexist in the observation of the shear inversion layers. One wayto overcome the problem will be given by continuous obser-vation of the vector magnetic field, which is not interruptedeven by the night period. If we can detect the whole process of

the shear reversal, we may determine the magnetic orientationfrom the temporal continuity. For that, a global network ob-servation of the vector magnetic field should be organized.Also, the vector magnetograph and the Stokes polarimeter,with which the Solar-B satellite (Shimizu 2002) is equipped,will be powerful tools for this purpose.

The theoretical understanding of the transition process fromthe first reconnection (tearing mode) to the second reconnec-tion (eruption) is still insufficient, even though it is clearlydemonstrated in our simulation. It is particularly important to

make clear the physical conditions for triggering the secondreconnection in more realistic situation. This problem is relatedto the question of how we should define magnetic shear as wellas shear reversal in a three-dimensional system without anygeometrical symmetry. In order to obtain definitive answersto these questions, the physical relationship between magneticshear distribution and MHD relaxation should be investigatedin more detail. Furthermore, we also have to develop a moresophisticated theory of three-dimensional reconnection be-cause multiple reconnection is a three-dimensional phenome-non in which the horizontal and vertical current sheets form aninverse-T shaped structure, as seen in Figure 8e.

Finally, we should emphasize the possibility that flare-productive active regions consist of multiple subregions, eachof which are characterized by different magnetic shear. Theannihilation of magnetic shear may arise on the boundary between different subregions. This process is similar to themechanism whereby earthquakes are most frequently pro-duced on the boundary between different plates. Recently,Priest et al. (2002) discussed coronal heating as an analogyof geophysical plate tectonics. Our simulation results imply

that the shear inversion layer could correspond to the plateboundary and that the steep shear gradient on the solar surfacewill be an indicator for predicting the onset of solar flares.

The present work is partially supported by Grants-in-Aidfor Scientific Research from the Japan Society for the Pro-motion of Science and by the REIMEI Research Resourcesof Japan Atomic Energy Research Institute.

APPENDIX

MAGNETIC HELICITY IN A LINEAR ARCADE

The relative helicity (eq. [1]) of the domain Vis equivalent to the difference between two whole space helicities, including theexternal space V0, i.e.,

HR(B) H(B;B0) H(B0;B

0);

where

H(B1;B2)

ZVV0

A = B dV;

B B1 in V;

B2 in V0

;

and A : < B. The reference field B0 is usually given by the current-free field : < B0 0 for convenience. When Vand V0 are

bounded by a plane, if the current-free field : < B00 0 is used as the common extension B0 in V0 and if the Coulomb gauge is

adopted, the reference helicity H(B0;B00) vanishes (Berger & Field 1984). In this case, the relative helicity can be given by

HR(B) H(B;B00);

because HR is independent of the gauge B0.

If the volume integral is carried out along magnetic flux tubes, which contains infinitesimal flux d, then

H(B;B00)

Z ZVV0

A = B dld

jBj

Z ZVV0

A = dld;



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where lis the arc-length vector parallel to B and B00. The equation above indicates that magnetic helicity for unit flux can be definedon field lines by

dH

d

ZA = dl; A1

where : = A 0 both in Vand V0.If and only if the field line orbit forB and B0

0

stays on some restricted region, equation (A1) can work as a local quantity of therelative helicity. For instance, in the linear arcade system illustrated in Figure 1, the extended field line periodically turns around onan isovalue surface . In this case, one-turn integration of equation (A1) along the field line A-B-C in Figure 1,

h

Zs0

Axdx

I(Aydy Azdz) s ;

can be defined as a function of flux , and it corresponds to helicity density for unit flux, where s indicates the shearingdisplacement between the footpoints (A and B) along the magnetic neutral line.

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