T. Amari et al- Coronal Mass Ejection: Initiation, Mass Helicity, and Flux Ropes. II. Turbulent...

8/3/2019 T. Amari et al- Coronal Mass Ejection: Initiation, Mass Helicity, and Flux Ropes. II. Turbulent Diffusion-Driven Evolution

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CORONAL MASS EJECTION: INITIATION, MAGNETIC HELICITY, AND FLUX ROPES. II.TURBULENT DIFFUSIONDRIVEN EVOLUTION

T. Amari, J. F. Luciani, and J. J. Aly1

CNRS,Centre de PhysiqueTheorique de lEcole Polytechnique, F-91128 PalaiseauCedex, France; [email protected]

and

Z. Mikic and J. LinkerScience Applications International Corporation, 10260 Campus Point Drive, San Diego, CA 92121

Received 2002 December 26;accepted2003June 10

ABSTRACT

We consider a three-dimensional bipolar magnetic fieldB, occupying a half-space, which is driven into evo-lution by the slow turbulent diffusion of its normal component on the boundary. The latter is imposed by fix-ing the tangential component of the electric field and leads to flux cancellation. We first present generalanalytical considerations on this problem and then construct a class of explicit solutions in which Bkeepsevolving quasi-statically through a sequence of force-free configurations without exhibiting any catastrophicbehavior. Thus, we report the results of a series of numerical simulations in which Bevolves from differentforce-free states, the electric field on the boundary being imposed to have a vanishing electrostatic part (thelatter condition is not enforced in the analytical model, and thus it is possible a priori for the results of the

two types of calculations to be different). In all the cases, we find that the evolution conserves the magnetichelicity and exhibits two qualitatively different phases. The first one, during which a twisted flux rope is cre-ated, is slow and almost quasi-static, while the second one is associated with a disruption, which is confinedfor a small initial helicity and global for a large initial helicity. Our calculations may be relevant for modelingthe coronal mass ejections that have been observed to occur in the late dispersion phase of an active region.In particular, they may allow us to understand the role played by a twisted flux rope in these events.

Subject headings: MHD stars: coronae stars: flare stars: magnetic fields Sun: coronal mass ejections (CMEs) Sun: flares

On-line material: color figures

1. INTRODUCTION

Large-scale eruptive processes like coronal mass ejections(CMEs) are very interesting phenomena occurring in active

regions of the atmosphere of the Sun throughout their lives.They are observed indeed to be produced in the phase ofappearance and development of such regions, but also,although at a slower rate, during their longer dispersionphase (Demoulin et al. 2002). For a few decades, solar phys-icists have done much to try to understand CMEs, and sev-eral models have been proposed, most of them resting onthe idea that a key process is the change induced in the pho-tospheric conditions by the motions of the dense plasma(see, e.g., the calculations in Cartesian geometry in Aly &Amari 1985; Aly 1990; Amari et al. 1996a, 1997 and thosein spherical geometry in Mikic & Linker 1994; Aly 1995;Amari et al. 1996b; Tokman & Bellan 2002, as well as refer-ences therein). However, it appears that many basic ques-

tions have not yet been fully answered, and much workseems to still be necessary to settle such basic issues as thetriggering mechanism, the energy and helicity budget, andthe possible role of a twisted flux rope.

The aim of this series of papers is to contribute to ourunderstanding of these issues essentially by carrying outthree-dimensional magnetohydrodynamic (MHD) numeri-cal simulations of the evolution of a magnetic field Bgener-ated by different types of drivers imposed on the boundary.In the first paper (Amari et al. 2003, hereafter Paper I), we

have considered the case where Bevolves as a consequenceof slow boundary motions converging toward the inversionline, a situation considered before in two dimensions byPriest & Forbes (2002). We have shown that no magnetichelicity is injected in the configuration during the converg-ing phase, implying that helicity storage needs to be per-formed earlier and that a large-scale disruption eventuallyoccurs, with a flux rope being created by reconnection dur-ing this nonequilibrium process (related results wereobtained in Antiochos, DeVore, & Klimchuk 1999 andPaper I).

In this second paper, we study a different class of drivers.We suppose indeed that the evolution of B is due to aturbulent diffusion of the photospheric flux generated bysmall-scale horizontal plasma motions. In our approach,this diffusion is enforced by imposing a particular form onthe tangential component of the electric field on theboundary, which leads to a well-posed problem. We firstconsider the behavior ofBfrom an analytical point of view.We concentrate in particular on the establishment of generalformulae giving the variation of the relative helicity and thatof the energy of B, and on the derivation of solutionsdescribing the diffusion ofBz on the boundary and the asso-ciated flux cancellation. We also try to adapt the simplemodel introduced in x 4 of Paper I in order to exhibit explicitclasses of diffusion-driven evolution in which B passesquasi-statically through a sequence of force-free configura-tions. Next we consider the problem from the numericalpoint of view. As in Paper I, we start by constructing a seriesof force-free fields with a range of helicity and energy con-tents. In asecond step, we take each of them to be the initial

1 CEA, DSM/DAPNIA, Service dAstrophysique (URA 2052 associeeau CNRS), Centre dEtudes de Saclay, F-91191 Gif sur Yvette Cedex,France.

The Astrophysical Journal, 595:12311250,2003 October 1

# 2003.The American AstronomicalSociety. All rightsreserved.Printedin U.S.A.

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state of an evolutionary process, paying much attention tothe basic questions already quoted: What is the helicity andenergy budget (in particular, are the numerical results com-patible with the analytical ones)? Does a nonequilibriumprocess develop in the system? Is there creation of a fluxrope, either in equilibrium or not? We note that our workhere extends a previous attempt made by Amari et al.(1999b), who were able to show the formation of a twistedflux rope but did not follow its subsequent evolution.

We hope that the results reported here will be useful forunderstanding the reasons why a nonnegligible number ofCMEs are still observed in the phase of dispersion of anactive region into the network (Wang, Sheeley, & Nash1991). It appears indeed quite reasonable to model the long-term dispersion process by turbulent boundary flux diffu-sion, as first proposed by Leighton (1964) and thus abun-dantly discussed by many authors (see, e.g., DeVore,Sheeley, & Boris 1984; Ruzmaikin & Molchanov 1997; vanBallegooijen 1999; Amari et al. 1999b). For this application,the helicity contents of an initial state in our numerics canbe taken to represent the amount of helicity that has beenleft at the end of the more active early phase of the region.Indeed, the field is not expected to fully relax to a potentialstate after each of the CMEs, and thus the high amount ofhelicity that is certainly present initially as a consequence offlux emergence and differential rotation cannot be fully con-sumed. Another question that can be addressed on the basisof our results is the reason why prominences are observed toreform (entirely or partly) during the dispersion of the activeregion. Such a phenomenon may be related indeed tothe presence of a flux rope, which is well known to be afavorable site for cold matter condensation and support.

The paper is organized as follows: In x 2 we describe ourmodel, analyze the diffusion process suffered by Bz on theboundary, and give (by using new general relations derivedin Appendix A) some formulae for the time variations of themagnetic energy and the magnetic relative helicity. In

x3 we

develop our simple analytical quasi-static model, usingsome basic ingredients presented in Appendices B and C.Our numerical simulations are described in x 4 (where wefirst construct the set of initial numerical force-free configu-rations by applying boundary twisting motions to the foot-points of the lines of a dipolar potential field), and theirresults are reported and analyzed in detail in x 5. Finally, theresults are summarized in x 6, where we also discuss some oftheir possible implications for a theory of CMEs.

2. A MODEL OF MHD EVOLUTION DRIVEN BYBOUNDARY FLUX DIFFUSION

This section is devoted to a short description of our modeland to the presentation of a few analytical relations obeyedby the magnetic energy and helicity.

2.1. The Model

The model we consider in this paper is defined by thefollowing conditions:

1. An active region of the solar corona is represented bythe upper half-space fz > 0g, assumed to contain amagnetic field B embedded in a low-density perfectlyconducting slightly viscous plasma.

2. At some initial time t t0, the field B0 Bt0 hasfinite energy and finite magnetic relative helicity, and it is

force-free; i.e., it obeys the equations

D B0 0B0 ; 1D

xB0 0 ; 2with 0 satisfying the well-known constraint B0 x

D0 0.

3. For t ! t0, the field and the plasma evolve accordingto the MHD equations (or some of their approximations).

4. The evolution is driven by imposing the tangentialcomponent Es of the electric field on S, which can beHelmholtz-decomposed according to (see Appendix A)

cEs Ds Ds zz ; 3where

Ds xx@x ^yy@y. Specifically, we set

x;y; t bBzx;y; 0; t ; 4with b > 0 being a constant having the dimension of amagnetic diffusivity, and

x;y; t 0 : 5The latter condition is not assumed, however, in the simpleanalytical model presented in the following section. In thatmodel, the value of results from the form chosen for thefield Bin : it can be computed only a posteriori rather thanbeing imposed a priori.

It is worth stressing that fixing Esx;y; t on S (and theinitial field B0) leads to a well-posed problem. Our calcula-tions are conducted without adding any other conditions onS. In particular, the tangential component Bs ofB is notrequired to satisfy any a priori constraint.

Finally, we note that the form of chosen here differsfrom the one resulting from the condition

D2sx;y; t lBzx;y; 0; t0 ; 6

which was imposed in Amari et al. (2000) (l < 0 is a con-stant), and found therein to lead to a linear variation of Bzon S.

2.2. The Diffusion of Bz on S

Quite generally (see eq. [A16]), the z-component qrs; t Bzrs; 0; t of the field on S(rs xxx y^yy) evolves accordingto

@tq D2s ; 7i.e., it is fully determined by only. Using the form (4) forthe latter quantity leads to

@tq bD2s q on S : 8Therefore, q obeys a two-dimensional diffusion equation.Its solution is given by the well-known formula (see, e.g.,Tikhonov & Samarskii 1963)

qrs; t 14b

ZS

q0r0sejrsr0sj2=4b ds0 ; 9

where we have set q0rs qrs; 0 and t t0. Conse-quently, q exhibits both flux dispersal (q decreasing whileoccupying larger and larger areas) and flux cancellation[the flux t through the region of positive polarity Sdecreasing].

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These phenomena can be discussed in terms of thevelocity

vs bD

sBz

Bz: 10

Using this definition and equation (4) in equation (3), weobtain indeed

cEsvs

Bzzz

Ds ;

11

showing that vs is a flux-preserving horizontal velocity[but remember that such a velocity is not uniquely defined;vs v0s is also flux-preserving if

Ds x Bzv0s 0; see, e.g.,

Boozer 2002]. Clearly, vs transports Bz from regions whereit is intense toward regions where it is weaker, which resultsin a spreading of that quantity. Near the inversion line Iwhere Bz 0, DsBz=Bz is directed toward I, and then vsadvects continuously magnetic flux up to I, where it can-cels. Note that there is no trouble with the divergence of vson I, since vs is not a material velocity. Actually, it is thepresence of this singularity that allows flux to disappear(Boozer 2002). An interesting question emerges at thispoint: What happens to the magnetic lines in when their

two footpoints on SreachI? It is difficult to give a generalanswer to this question, but we can at least note that thereare two limiting processes that are made possible by thedivergence ofvs onI (the actual behavior may be a combi-nation of both): (1) Reconnection occurs on I, leading tothe formation of a flux rope; in that case, the lines are con-served in. (2) The entire lines disappear at I, leading toa decrease of the flux inside.

As a last point, we remark that only the normal compo-nent of the field on Sis constrained to satisfy an equation ofdiffusion. In contrast, the tangential component is notimposed to satisfy any a priori condition, as already notedabove. Rather, it is fully determined by the physics inside .Thus, it is clear that it is not possible to use our calculations

to describe a magnetic evolution driven by photosphericresistive diffusion. To treat correctly this latter situation,one has to replace the plate Sby a layerD of resistive plasmain a given state of motion, for instance, and both Bz and Bsdiffuse in D in addition to being advected (as a result, thelines suffer a resistive slippage through D and some det-wisting in ). Clearly, we need to solve both the transportequation for B in D and the MHD equations in and tomatch their solutions on the separating plane: the problemcan no longer be set in the only domain with appropriateboundary conditions, as in the situation we consider here.

2.3. Physical Interpretation of the Boundary Conditions

We now establish a link between our choice of boundaryconditions and the numerous works that have been effectedafter Leighton (1964) on the turbulent diffusion of the mag-netic flux at the surface of the Sun (e.g., DeVore et al. 1984;Ruzmaikin & Molchanov 1997; van Ballegooijen 1999;Amari et al. 1999b). On the photosphere, horizontalturbu-lent motions of the plasma generate a random wandering ofthe footpoints ofB. As first noted by Leighton (1964), thisleads to a two-dimensional horizontal diffusion of the nor-mal component Bn ofB, a process that is described (at leastto first approximation) by equation (8) with an appropriatevalue ofb. To be more precise, it is not the actual value ofBn that suffers diffusion but rather its average Bn over a scalelarger than the turbulent one. Therefore, there is no contra-

diction between flux conservation for Bn, which is submittedto the horizontal motions of the highly conducting plasma,and flux cancellation for Bn, which results from mixing nearthe inversion line of small-scale patches of positive and neg-ative polarities. Moreover, in contrast to the resistive caseanalyzed above, we do not expect a diffusion ofBs owing tothe horizontal nature of the turbulent motions. If we inter-pret our Bas the large-scale average magnetic field above aturbulent surface, it thus appears that our calculations maybe used to describe the evolution of the coronal fieldresulting from photospheric turbulent diffusion.

We stress the fact, however, that the reader who is eitherunconcerned or uncomfortable with the boundary diffusionprocess can merely consider our calculations as a technicalway for producing a twisted flux rope embedded in anarcade and for studying its possible disruption.

2.4. Evolution of the Magnetic Energy and Relative Helicity

Quite generally (i.e., for arbitrary values of and ), thetime variations of the (finite) energy

W

t

1

8Z

B2 dv

12

and the (finite) relative helicity (Berger & Field 1984)

Ht Z

A xB A xB dv 13

of the field Bare given respectively by (see Appendix A)

_WW 14

ZS

@zBz ds 14

ZS

Jz ds ; 14

_HH 2Z

S

D2s ds 2

ZS

Bz ds : 15

Equation (14), in which Jz denotes the normal component

of the current flowing through S, shows the presence of twocontributions to _WW: one related to (which is the quantitymaking Bz change), and one related to (which is a quantitywithout effect on Bz). On its side, equation (15) showsno explicit dependence on . The latter quantity, how-ever, plays an implicit role, which is hidden in the timedependence ofBz on S(see eq. [7]).

With conditions (4) and (5) being enforced, we thus have

_WW b4

ZS

Bz@zBz ds ; 16_HH 0 : 17

The magnetic relative helicity is thus exactly preserved. This

analytical result is checked to hold true in our numericsbelow. Note that H-conservation is also obtained with theboundary conditions (5)(6) used in Amari et al. (2000).

For the type of problem considered here, it is always par-ticularly important to compare at any time t the energy ofBt to those of the potential field Bt and of the open fieldBt satisfying Bzt Bzt Bzt on S. The latter aregiven by

Wt 1162

ZSS

Bzx;y; 0; tBzx0;y0; 0; tjr r0j dsds

0 ; 18

Wt 1162

ZSS

jBzx;y; 0; tBzx0;y0; 0; tjjr r0j dsds

0 ; 19

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respectively. On general grounds, the energy of the initialfield B0 is expected to satisfy (Aly 1984, 1991; Sturrock1991)

Wt0 Wt0 Wt0 : 20By a standard theorem, we know that the first equalityactually has to be satisfied at any time t. This may not be thecase for the second one, and thus we check carefully whether

it holds true or not for all the values oft in our calculationsbelow.

3. A SIMPLE ANALYTICAL MODEL FORDIFFUSION-DRIVEN EVOLUTION

In this section, we consider the possibility of using thesimple analytical scaling method of Paper I (or rather theslight generalization of it described in Appendix B) for con-structing an explicit example of diffusion-driven evolution.

3.1. Statement of the Problem

Let B0

r

be an arbitrary force-free field given in , and

choose two differentiable positive functions l and of t t0 ! 0 satisfying l0 0 1. Then (AppendixB) the fields

Br; t lB0r 21form a force-free time sequence, and it is natural to addressthe following problem: Is it possible to choose B0, , and lin equation (21) in such a way that (1) the normal compo-nent qrs; t Bzrs; 0; t on S solves the equation ofdiffusion (8) and (2) the evolution of the field Br; t iscompatible with ideal MHD inside ?

3.2. Existence of a Sequence Satisfying the First Condition

We first consider the existence of a sequence satisfyingcondition 1 above. From equation (21), we need to have

qrs; t lq0rs ; 22and to begin with, we have to discuss the existence of solu-tions to the diffusion equation (8) of this form having zeroflux trough S. As shown in Appendix C, such solutions doexist. To construct them, (1) we select a solution q0 to theelliptic equation (C7) on S, where the arbitrary number kis chosen to be large enough (for the explicit solutions wepropose, k! 3), and (2) we set

1

1 1=2

; 23

l 11 k=2

: 24

Once an adequate q0 has been obtained, we build up in a finite energy and finite helicity force-free field B0 such thatB0z q0 on S. As discussed in Paper I, this can be done bysolving the boundary value problem (BVP) in in which wealso fix a not too large value of 0 on S. Eventually, wethus end up with the force-free sequence

Br; t 11 k=2

B0r

1 1=2" #

; 25

whose energy and relative helicity decrease according to

Wt 11 k3=2

Wt0 ; 26

Ht 11 k2Ht0 27

(see eqs. [B6] and [B7]).

3.3. Compatibility of the Sequence with the Frozen-in LawWe now prove that the sequence we have constructed sat-

isfies automatically condition 2 above. For that, we firstcompute the electric field E in associated by Faradayslaw with the time variation ofB, Ebeing thus defined up toan additive gradient. We start from equation (B3), in whichwe assume for later considerations that A0 is the uniquepotential ofB0 satisfying the gauge conditions of Paper I(this implies immediately that A has the same property atany time). Differentiating A with respect to t and usingequations (23)(24), we obtain

cEr; t @A@t

r; t c DVr; t

2k1t k 1A0 r x DA0 jtr

c DVr; t : 28To go on, we impose the ideal condition ExB 0, whichimplies

B0r x DVr; t 2c

k1t B0 x k 1A0 r x DA0gf jtr :

29This relation can be satisfied only if

V

r; t

k

t

V0

t

r

;

30

with V0 satisfying at any point r

B0 xD

V0 2c

B0 x k 1A0 r x DA0 : 31Our problem thus reduces to finding a solution to the latterequation, which is clearly always possible (at least if weassume that the field B0 has no neutral points). We just needto fix arbitrarily the values ofV0 on S, for instance, and tointegrate along the magnetic lines to get its values in all of.As announced above, the electric field can always be chosenin a way compatible with ideal MHD.

From equation (28), we have on S

cEs

rs;0; t

k1

t

&

2k 1A0s r x DA0s c DsV0

'trs;0

;

32and it results at once from our choice of gauge that the termdepending on A0 and that depending on V0 in the right-hand side have to be identified, respectively, with the -termand the-term of the decomposition (3) ofEs. Then we needto have

rs; t kt0trs ; 33cVrs;0; t rs; t kt0trs ; 34



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for some functions 0 and 0. Clearly, is necessarily of theform (4). But condition (5) (which is used as a boundarycondition in our numerics) can never be satisfied by . Wecould choose cV0 0 on S, for instance, but afterintegration of equation (31) along an entire line, we do notend up in general with a zero value ofV0, and thus 0 inthe region of negative polarity. This impossibility is alsoapparent in equation (27), which shows an actual variationof the helicity: this would be in contradiction with thegeneral formula (15) if we had 0 on S.

Finally, we note that we can compute a plasma velocity in by using Ohms law once a solution V0 to equation (31)has been fixed. Of course, this velocity is determined from Eonly up to an arbitrary component along the lines, and wecan use this arbitrariness to impose our constraint vz 0 onS. However, as in the case discussed in x 2.2, this makes thetangential component vs diverge at the inversion line I.There, it is clear that reconnection does not occur [the topol-ogy is conserved owing to the scaling properties of Bt!],and then we are in a situation where the lines disappear in asingular way at I. Of course, this leads to a decrease of theflux and the helicity in .

Our analytical model thus describes a very quiet evolu-tion. We do not even expect an ideal instability to destroythe sequence of equilibria if we have taken a stable initialfield B0 (see the last property listed in Appendix B). Diffu-sion ofBz on S merely leads in to ideal flux transport, inparticular toward the inversion line, where it is made to dis-appear in a singular way. It is interesting to contrast thisbehavior with the more dynamic one exhibited by thenumerical solutions that we present immediately below.

4. NUMERICAL SIMULATIONS OF AN EVOLUTIONDRIVEN BY BOUNDARY FLUX DIFFUSION

In this section, we describe our numerical simulations,

which are done in a cubic box h of size much larger thanthe characteristic scales of the boundary conditions.

4.1. MHD Equations and Values of the Parameters

The full system of the MHD equations, which can betreated by our code using a semi-implicit scheme (Amari,Luciani, & Joly 1999a), is given in nondimensionalized formby

@v

@t v x Dv D B B Dp Dx Dv g ;

35@B

@t

D

v B

D

j

;

36

@@t

Dx v ; 37@p

@t v x Dp p Dx v H ; 38

j D B ; 39D

xB 0 ; 40where , , , and C are, respectively, the mass density,kinematic viscosity, resistivity, and adiabatic index of theplasma.

For the particular situation considered in this paper,these equations are solved in the cube h after discretization

on a nonuniform mesh (141 121 91 nodes). Beforehand,some simplifications are made and the values of the parame-ters are fixed as now indicated. Small values are chosen forthe dissipation coefficients: 102 to 103 for the kine-matic viscosity and 104, 105, and 0 for the resistivity,giving for our mesh resolution a Lundquist number of order104. This allows us to neglect the term H in equation (38).The plasma B2=8p is taken to be either of the order ofthe very small coronal value 103 or to vanish, without anydifference between the results. When making the choice 0, we have to fix a mass density profile and, of course,to neglect the gravity term in equation (35). Here we chooseeither B2, which gives a constant Alfven velocity, or 1. Alternative choices of density profiles (exhibiting forinstance a slower decrease with distance) do not lead to verydifferent results.

4.2. The Set of Initial Force-free Configurations

We now construct a set of force-free magnetic fields withvarious helicity and energy contents. They are obtained asin Paper I (to which we refer the reader for more details) bythe following procedure:

1. At time t 0, we compute a bipolar potentialmagnetic field

B DV 41by solving in h a nonhomogeneous Dirichlet-NeumanBVP for the harmonic scalar potential V. In particular,we impose on the bottom face Sh of the cube the boundarycondition

q0x;y ex2=2xeyyc2=2y eyyc2=2y : 42

As for the values of the parameters, we choose x 2,y 1, and yc 0:8. The computational domain is taken tobe h

20; 20

20; 20

0; 40

. Note that q0 takes

very small values on the boundary of the bottom face.2. Next we introduce a boundary flow by means of the

Helmholtzs decomposition

Bzvs Ds f Dsg zz 43introduced in Paper I. In a first twisting phase extendingfrom t 0 to ts (the unit of time is the transit Alfven timeA), we impose f 0 and g gBz; t, with

@g

@Bz Bz @

@Bz; 44

Bz; t v0RtB2z eB2zB2zm= B2zm : 45The corresponding motion is generated by two symmetricvortices that introduce shear along the neutral line. In thelatter equation, Bzm supSh , v0 102 (thus v0 is smallcompared to the Alfven speed vA 1), and R is a linearramp function used to smoothly switch on or off the velocityfield in 10 units of time. During this entire phase as well asduring the next one, the resistivity is set equal to zero in hand on Sh.

The computations are done for five values of the finaltime ts: ts 2T f0; 50; 100; 200; 400g, with ts 0 beingused as a reference state. Thus we get a set of five fields ofincreasing energy and helicity.

3. In a second phase extending from ts to t0, the boun-dary flow is taken to vanish (i.e., f g 0), and the system



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relaxes to an equilibrium under the action of the viscosity.The duration of this phase is 200, except for the configura-tion reached at ts 400, for which a duration of 400 isneeded. This is due to the fact that the latter field is so highlytwisted that it is near the transition to nonequilibrium(Amari et al. 1996b).

Finally, we end up with five equilibrium states that welabel with the values of ts: Uts Bts ; ts ;pts. They are allobtained from U0 by an ideal MHD mapping, and theirshears range from moderate to high values. They are shownin Figure 1.

4.3. Evolution Driven by the Diffusion of Bz on S

The force-free configurations constructed above are nexttaken in turn as initial states for an evolution of the fieldand the plasma obeying the MHD equations and the par-ticular prescriptions of x 4.1. To force this evolution, weimpose for t ! t0 the tangential electric field on the lowerboundary Sh according to equations (4) and (5). We firsttake b

103 for the value of the diffusion coefficient,

which gives a typical diffusion time of the order of2x= 4000 in the x-direction and 2y= 1000 in the

Fig. 1.Selected field lines of the initial force-free configurations reached after a shearing-twisting phase of duration (a) ts 0, (b) ts 50, (c) ts 100, (d)ts 200,and (e) ts 400,followedby aviscous relaxation phase of duration200. [Seethe electronic edition of theJournal fora color version of this figure.]



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y-direction, this last time being comparable to the durationof our simulations.

Although the evolution ofBz is computed numerically, itis worth noting that it can be also expressed in closed analyt-ical form (in the whole S) by computing the integral on theright-hand side of equation (9). Noting that q0 is the productof a function of x and one of y, as is the kernel in theintegral, we see indeed that q is the product of two simpleintegrals involving Gaussian functions, and we eventuallyget

qx;y; t xyxy e

x2=2x

eyyc2=2y eyyc2=2y ; 46where we have set

2x=y 2x=y 4b : 47(We note for future reference that the same type of calcula-tions would allow us to give explicit solutions of eq. [8] whenq0 is a superposition of an arbitrary number of Gaussianfunctions each having its own center r

kc , intensity B

k0 , and

extension k

x=y, all these parameters being constrained onlyby the condition that the total flux ofq0 vanishes.)

For the solution (46), the total magnetic flux t thread-ing the region of positive polarity S fy > 0g is given by

t Z

Sqrs; t ds 2

ffiffiffi

pxyB0

Zyc=y0

eY2

dY : 48

When y4yc, equation (48) reduces tot 2 ffiffiffip xyB0 yc

y ; 49

i.e., the unsigned flux decreases eventually as 1=2. By usingequations (18)(19), we can also compute the evolution of

the energy Wt of the potential field in associated withthis form ofq, and the evolution of the energy Wt of theopen field. Integrations cannot be carried out explicitly,however, and thus we just note that both quantities decreaseas 3=2 for large enough values of time (b42x=y).

4.4. Search for Neighboring Equilibria

In addition to the diffusion-driven evolution describedabove, we perform tests that allow us to characterize the nonequilibrium degree ofBt. At some selected time tr,we switch off the diffusion process on Sh and let the fieldBtr evolve freely, energy being dissipated by the viscousforces acting on the plasma in which it is embedded. Theneither of two different behaviors is a priori possible:

1. The magnetic energy quickly decreases to a finite limitwhile the kinetic energy goes to zero. In this case, the fieldhas relaxed to a nearby equilibrium and the evolution isalmost quasi-static around tr.

2. Such a fast relaxation does not occur, the kineticenergy increasing at least during some phase. In this case,there is no nearby equilibrium, and two situations mayobtain: (a) After some time, the kinetic energy startsdecreasing and eventually goes to zero: the field reaches anequilibrium, but the latter is not located near our initialBtr. (b) During the entire duration of the computation, thekinetic energy keeps some positive value and does not showany tendency to vanish: there is no available equilibrium for

the field (or there is one, but it is so far away that it is likelyto be physically irrelevant).

5. PRESENTATION AND ANALYSIS OF THENUMERICAL RESULTS

We now describe the main results that we have obtainedin our numerics.

5.1. Magnetic Energy and Helicity of the Solutions

For all the computed cases, i.e., for all ts 2T, the numer-ical solutions of the ideal ( 0) MHD evolution driven bythe boundary diffusion ofBz are characterized at any time tby the following relations (see Figs. 25):

_HHt 0 ; 50

Fig. 2. Variation of the magnetic energy during the various phases:phase of twisting by boundary motions applied to the potential configura-tion from t 0 to t ts, phase of viscous relaxation from t ts to t t0,and phase of evolution driven by flux diffusion on the boundary. The vari-ous cases ts 0, 50, 100, 200, and 400 are represented. Magnetic energy Wis expressed in units ofW (see text). The arrows indicate the three phasesfor the particular case ts 400. Curves shown from bottom to top corre-spond the labels given in the box from top to bottom. [ See the electronicedition of theJournalfor a color version of this figure.]



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_WWt < 0 ; 51_WWt < 0 ; 52_WWt < 0 ; 53

Wt < Wt : 54We note in particular the following:

1. The preservation of the magnetic helicity (Fig. 3), incomplete agreement with the analytical prediction ofx 2.3.

2. The nonconstant rate of decrease of the magneticenergy, except for the case ts 0, where we start from a cur-rent-free configuration. In particular, Wt suffers a charac-teristic change at some time, whose significance is analyzedbelow.

3. The fact that the magnetic energy always stays belowthe energy of the open field (Fig. 5), a property that hasalready been noted (x 2.3) to be certainly valid for anyforce-free field.

4. The increase of the ratio Wt=Wt < 1 (Fig. 5),which also occurs at a nonconstant rate and shows a charac-teristic change at some time (for ts 6 0). It should be noted

Fig. 3. Variation of the magnetic relative helicity during the variousphases (twisting, viscous relaxation, flux diffusion; see Fig. 2). The various

cases ts 0, 50, 100, 200, and 400 are represented. The arrow indicates thepoint at which boundary flux diffusion starts to be applied for the particularcase ts 200. Curves shown from bottom to top correspond to the labelsgiven in the box from bottom to top. [ See the electronic edition of theJournal for a color version of this figure.]

Fig. 4.Variation of the magnetic energy (in units of W) of the openfield having the same distribution ofBz on the bottom boundary as B. Thisvariation is represented here for the three phases of evolution and for thefive choices ofts. Curves shown from right to left correspond to the labelsgiven in the box from bottom to top. [See the electronic edition of theJournal fora color version of this figure.]

Fig. 5.Evolution of the magnetic energy vs. open-field energy duringthe turbulent diffusion phase for (a) ts 0, (b) ts 50, (c) ts 100, (d)ts 200, and (e) ts 40. It appears clearly that the energy Wof the evolv-ing configuration remains bounded by the open-field energy W and that,apart in the current-free case, W decreases faster than W. [See theelectronic edition of the Journal fora color version of this figure.]



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that this variation of Wt=Wt < 1 is not surprising,since Wt depends only on the distribution of Bz on S(and therefore its evolution is fully determined by the boun-dary diffusion) while Wt also depends on the currentspresent in the corona, which increase with time. A similarproperty was reported by Amari et al. (2000) for a linearvariation ofBz on the boundary, although in this later case,the two energies W

t

and W

t

were found to become

equal at some time.

5.2. Existence of a Dynamic Transition

For all ts 2T, except ts 0, there exists a critical timetc > t0 such that

1. For t0 < t < tc, the system evolves in an approxi-mately quasi-static way. Indeed, when the boundary driveris suppressed (test of x 4.4), magnetic energy quicklyapproaches a minimum value while kinetic energy smoothlydecreases to zero (see Fig. 6), which implies that there isalways an accessible equilibrium state in the immediateneighborhood of the actual state.

2. For t > tc, the evolution is dynamic, with the transi-tion quasi-static regime/dynamic regime around t tcbeing characterized by the change noted above in the rate ofvariation of the monotonically decreasing magnetic energyand, correlatively, by a stronger variation of the kineticenergy (see Fig. 7). Moreover, there is a disappearance ofneighboring equilibria, as proven by our test. When switch-ing off the diffusion process on Sh, we observe indeed that

kinetic energy no longer decreases smoothly, but on the con-trary increases while magnetic energy suffers a jump down-ward (see also Fig. 7). The value of tc depends on ts, asdo the nature of the solution after the transition and thepossibility of a relaxation toward a new equilibrium with adifferent topology (see below).

In the case ts 0, the evolution never becomes dynamic.

5.3. Topological Properties of the Solutions

We now consider the topology of the magnetic lines. Forall ts 2T except ts 0, there is a time tfl, ts < t0 < tfl < tc,such that

Fig. 6.Variations of the magnetic energy and the kinetic energy. These variations are represented for the four values of ts 6 0 and for the three phasesof evolution (twisting, relaxation, and flux diffusion) and an additional phase of relaxation during which flux diffusion is switched off. In this fourth phase,starting at the time indicated in the panels, the magnetic energy keeps a constant value while the kinetic energy decreases, indicating the presence of a nearbyequilibrium. [Seethe electronic edition of theJournal for a color version of this figure.]



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1. For t < tfl, the topology of the lines is arcade-like.2. For t > tfl, the topology of the lines is flux ropelike.

To prove the existence of tfl, it is sufficient to show (sincethe initial force-free configuration has an arcade topology)that a magnetic flux rope forms for some t > t

fland stays in

equilibrium. Figure 8 shows that the equilibrium configura-tions obtained after a viscous relaxation has been performed(see Fig. 6) have a flux ropelike topology for all thecomputed cases.

As shown on Figure 9, during the first slow quasi-staticphase (t0 < t < tc) the magnetic shear (defined as the anglebetween the inversion line and the transverse magneticfield) increases continuously up to time tfl, at which thereis a reversal in the sign of By (i.e., By starts pointingtoward the positive polarity) in some subdomain of Sh:this is a signature for the transition from an arcade topol-ogy to a twisted rope one. Actually, the amount of twistfor the various values of ts may be different. For the case

ts 400 and for a relaxation started at t 825, the twistis smaller.

5.4. The Disruption of the Magnetic Field

5.4.1. The Case of Small Enough Initial MagneticHelicity and Energy

For the two cases ts 50 and ts 100, i.e., for small val-ues ofHt0 and Wt0, the evolution leads at time tcne tcto a confined nonequilibrium: the flux rope experiences a dis-ruption but remains confined by the overlaying arcade, asshown in Figure 10. There is no wind or jet beingproduced. The confined character of the disruption is alsoclearly seen quantitatively in Figure 7: after the first changein the slope (energy release) of the magnetic energy (associ-ated with a jump in the kinetic energy), this quantitydecreases smoothly to a minimum while kinetic energydecreases too.

Fig. 7.Same as Fig. 6, but with the late relaxation phase starting at a later time. During that phase, the magnetic energy decreases while the kinetic energyrises strongly. This behavior is associated with the nonexistence of a nearby equilibrium. For ts 50 and 100, the system relaxes to a new equilibrium state in afinite time, while for ts 200 and 400, kinetic energy saturates at a high value in the finite duration of the simulation. [See the electronic edition of the Journal

for a color version of this figure. ]



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The amount of magnetic energy that is dissipated in thisdisruption appears to be limited, being associated only withthe partial reconnection of a part of the flux rope. Eventu-ally, the field relaxes to the equilibrium shown in the rightpanel of Figure 10, in which a flux rope structure, although broken, is seen to be still present.

5.4.2. The Case of Large Enough Initial MagneticHelicity and Energy

For ts

200 and ts

400, i.e., for large enough H

t0and Wt0, the system experiences a global nonequilibrium

process at tgne tc. As shown in Figure 8, there is anincrease of the kinetic energy during the final relaxationphase as in the previous case, but it is not followed by adecrease. This is just the signature of the nonexistence ofequilibria accessible to the field and of an unconfined dis-ruption with upward flows (as also found by Mikic & Linker1994). The flux rope thus rises, inducing a major disruptionof the configuration, and it strongly reconnects with theoverlaying arcades, as shown in Figure 11, with a dissipa-tion rate higher than during the first quasi-static phase.

As shown in Figure 12, large electric currents generatingstronger dissipation and thus plasma heating lead to the

well-known sigmoidal shape associated with an unconfineddisruption (Canfield, Hudson, & McKenzie 1999), which isnot the case for the smaller values ofts. During the eruptivephase, the electric current distribution exhibits an inverse-Sshape in the horizontal plane, and the dissipation appears tobe larger in the two spots corresponding to the nonuniformlocalization of twist along the rising flux rope (see Fig. 13 a,taken at t 1050 for ts 400 and corresponding to the con-figuration shown in Fig. 11c). Moreover, in the central verti-cal plane the dissipation is stronger near the current sheet(see Fig. 13b) created below the plasmoid-like structureshown in Figure 11d as well as below, in the reformingpostdisruptive loops.

Since we have considered only five values of ts in thispaper, we cannot give a very precise estimate of the value tscof ts at which the transition from confined to unconfineddisruption occurs. We can only assert that 100 < tsc 200.

5.5. Comparison of the Two Cases

The fact that the field can live two quite different fates,depending on the initial values of the helicity and energy, isa new result that may be possibly relevant for our under-standing of CMEs and flares, and, as emphasized by the

Fig. 8.Selection of field lines of the configuration obtained at the end of the relaxation phase (following the diffusion phase) specified in Fig. 5 in the four

cases: (a) ts 50, (b) ts 100, (c) ts 200, and (d) ts 400. For all ts, the relaxed state contains a twisted flux rope. [See the electronic edition of the Journalfor a color version of this figure. ]



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referee, it is important to try to discern the underlying physi-cal factors leading to one behavior or the other. We do notyet have a definite answer to this question, but we can makea few points by comparing some features of the twosituations.

Our first point concerns the structure of the initial force-free configurations that are generated by applying some par-ticular flow on the boundary (x 4.2). This flow creates shearnear the inversion line on S and twist on larger scales. For

small values ofts (see Fig. 1), we obtain shear only very closeto the inversion line, while outer field lines suffer a very smallamount of twist: thus the energy is far from the open-fieldenergy W, and the overlaying lines have good confiningproperties. For large values ofts, on the contrary, high shearalong the inversion line is present, while the global configu-ration suffers twist on the large spot scale, which leads to amuch higher value of the ratio W=W and thus to a highertendency to open. At tc, this ratio has grown up to a value ofthe order of 0.8 that characterizes opening behavior, asalready noted for twisting motions by Amari et al. (1996b).

As we have seen before, there is in both cases a flux ropepresent in the preerupting configuration, and it turns out tohave a twist of order 2 or larger, depending on the cases(see Fig. 8). However, we remark that its structure and itspossibility to erupt depend on ts because of the nature of theinitial distribution of twist recalled just above. For low val-ues of ts, only the field lines close to the inversion line canreconnect. Thus we get equilibria with a very low-lying small thickness twisted flux rope in which the field linescan make several turns around some axis, while most ofthe overlaying lines remain potential and confining (as alsoobtained by Amari & Luciani 1999). For large ts, on thecontrary, we have much more magnetic helicity and recon-nection at the footpoints of the lines close to the inversionline (which are highly sheared), which implies the presenceof a flux rope with a larger scale but with a smaller numberof turns. But now the outer lines too are twisted, and furtherreconnection implies a larger amount of current being trans-ferred to the coronal flux rope, which eventually appears tobe favorable to a global disruption.

Finally, we note that, although disruption occurs in bothcases when neighboring equilibria cease to exist, there is stillan accessible equilibrium available to the field in the formercase but not in the latter one. As already explained, per-forming our test for nearby equilibria leads indeed to twoquite different behaviors after the first change in the slope ofthe magnetic energy, associated with a jump in the kineticenergy (Fig. 6). When ts 50 or 100, both the magneticand the kinetic energies eventually decrease, the formerapproaching a minimum, while when ts 200 or 400,kinetic energy no longer decreases.

5.6. Effects of Changing the Value of the DiffusionCoefficient

We have checked that the results obtained above for aflux diffusion coefficient b 103 are actually conservedwhen this coefficient is reduced, which shows the robustnessof our conclusions. Owing to the scaling with b of the evo-lution timescale, we have considered the cases b

5

104

and b 104. As shown on Figure 14, the transitiondescribed above still occurs (later) for the case ts 200. Thisimportant result shows that the sequence of equilibriumdoes not diffuse on an intermediate scale but truly scaleswith b.

6. DISCUSSION

In this paper we have considered a class of three-dimensional model problems describing the evolution of amagnetic field B in a highly conducting half-space , thisevolution being driven by a turbulent anisotropic diffusionof the magnetic flux (i.e., of Bz) on the boundary plane S.

Fig. 9.Transverse component of the magnetic field on the lower boun-dary Sh for the initial potential configuration at t 0 (top), the relaxedequilibrium for ts 400 (middle), and the relaxed equilibrium correspond-ing to the last panel of Fig. 5 (bottom). Note a change in the sign in By alongthe inversion line (as indicated by the color reversal), which indicates thepresence of a twisted flux rope. [See the electronic edition of the Journal for acolor version of this figure.]

1242 AMARI ET AL.


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Fig. 10.Selection of field lines for a few of the configurations through which the field evolves in the case ts 50. Viscous relaxation is started at t 650.The two panels correspond to t 650 (left) and t 1050 (right). As shown in Fig. 6, the field does not relax to a nearby equilibrium, but it reaches at t 1050a new equilibrium after a transition involving reconnection with some of the overlaying lines. Thus, the small disruption remains confined. [See the electronicedition of theJournalfor a color version of this figure.]

Fig. 11.Selection of field lines for a few of the configurations through which the field evolves in the case ts 400 and late viscous relaxation is started att 850. The four panels correspond to (a) t 850, (b) t 950, (c) t 1050, and (d) t 1050. As shown on Fig. 6, the field does not relax at t 850 to anearby equilibrium, but it does experience a global disruption involving opening, reconnection through the overlaying arcade and below, and the formation ofa current sheet, associated witha high dissipation of magnetic energy and a strong increase of kinetic energy. [See the electronic edition of the Journal for a colorversion of this figure.]


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To impose this diffusion, we have first introduced a generalformalism in which the tangential electric field on S isexpressed in terms of two scalar potentials and (Helmholtzs representation), and established general for-mulae giving the time variations of the magnetic energy andthe relative helicity. We have shown that a diffusion of Bz isobtained indeed for an adequate choice of, and we havederived a few explicit solutions for B

zt

on S[in particular,one of our formulae gives the Bzt corresponding to the ini-tial value Bzt0 used in our numerics]. Thus, we haveconstructed a simple analytical model of diffusion-drivenevolution in which the field passes quasi-statically through asequence of force-free configurations. In this model, mag-netic lines initially present inside eventually disappear in asingular way at the inversion line on S, which leads to adecrease of the flux and the helicity in . No disruption isobtained.

In a second part of the paper, we have reported the resultsof detailed numerical simulations of the evolution of B. Inthese numerics, we have constructed a set of force-free fieldshaving different magnetic and helicity contents and used

them as initial conditions for diffusion-driven evolutions inwhich the value of the second potential is taken to vanish.By choosing several values for the coefficient of diffusion bon S, we have found that helicity keeps a constant value(in accordance with the analytical prediction) and thatsave in the case where we start from the potential state (zerohelicity)there exist several critical times associated withcharacteristic changes in the energy, topology, and existenceof a nearby equilibrium. More explicitly, we have found thefollowing: (1) The rate of dissipation of magnetic energybecomes suddenly stronger at some tc. The evolution is thusdivided into two phases, quasi-static and dynamic, duringwhich magnetic energy decreases at different rates. (2) Thetopology changes from an arcade type to a flux rope type atsome tfl < tc; i.e., a twisted flux rope appears spontaneouslyduring the first slow quasi-static phase and stays in equili-brium. Rope formation is associated with a reconnectionprocess occurring at the inversion line on S. (3) Nonequili-brium develops at tc and leads to a confined disruption forsmall initial helicity and to an unconfined major disruptionfor large initial helicity. Moreover, we have shown that, for

Fig. 12.Horizontal cut of the distribution ofJ2, which controls resistive dissipation, for (a) ts 50, (b) ts 100, (c) ts 200, and (d) ts 400. The cutsare taken above the various flux ropes seen in Fig. 7. Sigmoids appear more clearly on the last two panels, which correspond to larger values of helicity andenergy. [Seethe electronic edition of theJournalfor a color version of this figure.]



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all the values of the initial helicity, the energy of the fieldremains always smaller than that of the open field havingthe same distribution ofBz on the boundary plane.

The difference between the behavior of the field in theanalytical model on the one hand and the simulations on theother hand is interesting, since it allows us to understandbetter the underlying physics. Clearly, this difference isrelated to the nature of the conditions occurring in theneighborhood of the inversion line I. In the analyticalmodel, magnetic lines initially in disappear in some wayatI, and there is a continuous matching between the flux ofthe tubes staying in and the boundary flux on S thatdecreases as a consequence of diffusion. On the contrary,lines swallowing at I are made impossible in the numerics(this explains why helicity is conserved), and the tubes thatcan no longer connect to Sbecause there is not enough fluxavailable have no other choice than reconnecting. A similarbehavior was obtained in the two-dimensional calculationsreported by van Ballegooijen (1999).

It is worth comparing the results reported in this paperwith some of our earlier results. In Amari et al. (1999b), itwas shown in a particular case that it is possible to obtain atwisted flux rope in equilibrium. The results here are muchmore completewe have proven that pursuing the evolu-tion may lead to nonequilibriumand moreover, theyprovide a general theoretical background for describingevolution due to photospheric changes. In Amari et al.(2000), we have studied an evolution driven by flux submer-gence through the boundary. It appears that this processcan be described with the help of our Helmholtz decomposi-tion of the tangential electric field on S. In particular, the

Fig. 13.Closer look at (a) the horizontal and (b) the vertical cuts of thedistribution ofJ2 during the global nonequilibrium rising of the flux ropeshown in Figs. 10c and 10d. Panel a shows nonuniform dissipation alongthe sigmoid. Panel b shows that electric current dissipation is largerin a cur-rent sheet developing below the plasmoid structure (appearing in Fig. 10d)and in the reforming postdisruptive loops. [See the electronic edition of theJournal for a color version of this figure.]

Fig. 14.Variation of the magnetic energy along the MHD evolution inthe case ts 200 for three different values of the flux diffusion coefficient onthe boundary. The change in the slope of the graph (magnetic dissipationrate)is stronger for smaller values of the diffusion coefficient. Curves shownfrom right to left correspond to the labels given in the box from bottom totop.[See the electronic edition of theJournalfor a color version of this figure.]



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fact that the helicity keeps a constant value in the latterprocess finds a clear explanation in the framework pre-sented here. We have to note, however, that the timescalesassociated with the processes considered therein and hereare quite different. The former is controlled by the imposedvelocity of emergence (typically 102vA), while the latter ismuch larger, being associated with diffusion (although wehave been able, because of computational time saving needs,to perform simulations with diffusivity b only up to 104).Moreover, these two mechanisms and the one considered inPaper I (where converging motions were applied to theforce-free configurations also considered here) correspondto approaching the footpoints near the inversion line.Unlike in Amari et al. (2000), the magnetic energy of thesolutions remains bounded by the open-field energy.

Let us now examine some implications of our results forthe problem of the origin of the helicity and the mechanismof the initiation of CMEs. The results obtained in this paperare actually relevant for trying to understand the observedpersistence of CMEs in the late phase of dispersion of anactive region. Our results show that this dispersion process(insofar as it can be modeled by boundary flux diffusion, asfirst proposed by Leighton 1964), can trigger eruptive eventsthat may be either confined or unconfined, depending onthe value of the initial helicity. Moreover, it sheds some lighton the question of the necessity or not of the presence of atwisted flux rope in the preerupting configuration. Itappears that such a rope can be formed during the diffusion-driven evolution and stay in equilibrium for a while (see alsoAmari et al. 2000). This is in contrast with the results ofPaper I and of Antiochos et al. (1999; where a quadrupolarconfiguration was studied), since it was found therein thatin an evolution driven by some types of boundary motionsthe flux rope can be produced only by reconnection duringthe CME itself. This rope may be the site of the formationof a prominencethe lines have a shape favorable to masssupport against the Suns gravitational fieldand this couldexplain why prominences reform in between CMEs duringthe active region dispersion phase.

It also appears that the helicity, which keeps a constantvalue through the diffusion-driven evolution, cannot be theonly parameter controlling the triggering of an ejection: the

initial configuration does not erupt, in spite of the fact thatit has the same helicity as the final erupting one. Thus, hav-ing a large enough helicity seems to be a necessary conditionfor an ejection to occur, but not a sufficient one.

Finally, we note that, from the observational point ofview, changes at the photospheric level in both Jz (verticalcomponent of the electric current density) and Bz can bemeasured, as well as twist changes in the coronal configura-tion. For instance, each half-turn twist observed in thetwisted arcade configuration merges to give a flux rope oftwist 2, with the same magnetic helicity. By the same token,the coronal magnetic helicity contents cannot be explainedby this process. Therefore, the amount of magnetic helicityin the pre- and post-CME configurations depends entirelyon that of the initial force-free configuration possiblyinjected by emergence (although we cannot exclude the pos-sibility of having during the dispersion phase some additionof helicity due to differential rotation and to boundarymotions different from the converging ones considered inPaper I, which have been proven to not lead to helicity injec-tion). As already indicated in the conclusion of Paper I, theclass of velocity fields we have taken to construct the initialstate Ut0 (f

0) can lead to an important helicity storage,

but we do not claim that this process actually occurs on theSun since emergence of subphotospheric structures mayalso be responsible for injection of helicity into the corona.To settle this point, we need some observational evidencethat such transverse twisting photospheric velocity fields doactually exist.

We thank an anonymous referee for his/her remarks,which led us to clarify our description of the diffusionprocess on the boundary and to improve the general pre-sentation of the paper. We acknowledge support fromNASAs Sun-Earth Connection Theory Program, NASAsSTEREO/SECCHI Consortium, and the Centre NationaldEtudes Spatiales, which also supported Dr. Amaris vis-its to SAIC in San Diego. The numerical simulations per-formed in this paper have been done on the NEC SX5supercomputer of the Institute IDRIS of the CentreNational de la Recherche Scientifique.

APPENDIX A

NEW FORMULAE FOR ENERGY AND HELICITY EVOLUTION

A1. ELECTRIC FIELD DECOMPOSITION

The tangential component of the electric field on Scan always be decomposed according to

cEs Ds Ds zz : A1(A similar decomposition was introduced in Paper I for the tangential vector vsBz, where v denotes the plasma velocity.) Thetwo potentials x;y; t and x;y; t appearing here are in fact related to the electrostatic potential V and to the tangentialcomponent As of the vector potential on S. Under our gauge conditions (see x 2.3 of Paper I), the latter quantity can beexpressed in the form

As As Ds zz ; A2with the function x;y; t being the solution to the BVP on S:

D2s Bz ; A3lim

r1 0 : A4



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Then, comparing equation (3) with the quite general relation

cEs c DsV @tAs c DsV Ds@t zz ; A5we obtain indeed

cV ; A6 @t : A7

The potentials and can be computed from Es

as follows. From equation (3), we haveD2

s cDs xEs ; A8

D2s czz x

Ds Es : A9

Therefore, imposing and to decrease to zero at infinity, we can write

r c2

ZS

ln jr r0j Ds xEsr0 ds0

c2

ZS

Esr0 x r r0

jr r0j2 ds0 ; A10

r c2

ZS

ln jr r0jzz x Ds Esr0 ds0

c2

ZS

r r0jr r0j2x zz Esr0 ds0 ; A11

where we have used the well-known Greens function for the two-dimensional Laplacian and effected integrations by parts toget the second expressions of both and .

It is also possible to write

Ds xU ; A12 D? xU ; A13

with

Ur c2

ZS

ln jr r0jEsr0 ds0 ; A14D? @y;@x : A15

An important property of is that it fully determines the evolution ofBz on S. Using Faradays law, we have indeed

@tBz czz x Ds Es D2s : A16

A2. EVOLUTION OF THE MAGNETIC ENERGY

The energy

Wt 18

Z

B2 dv A17

of a field Bchanges according to

_WW c4

ZS

Es Bs x zz ds ; A18

the integral on the right-hand side just being the Poynting flux across the boundary S. With the help of the representation (A1)for Es, we obtain

_WW c4

ZS

Ds Bs x zz Ds xBs ds

c4

ZS

Ds Bs x zz Ds Bs x zz Ds x Bs Ds xBs ds

c4

ZS

Ds Bs x zz Ds xBs ds; A19



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where we have effected two integrations by parts and used Stokess and Gausss theorems. Finally, using Amperes law and thedivergence ofBleads to

_WW 14

ZS

@zBz ds 14

ZS

Jz ds ; A20

where Jz is the normal component of the current flowing through S.

A3. EVOLUTION OF THE MAGNETIC RELATIVE HELICITY

Let A and A be vector potentials ofB and B, respectively, which satisfy the gauge restrictions of Paper I. Then themagnetic relative helicity ofBis defined by

Ht Z

A xB A xB dv ; A21

and its time variation is given by

_HHt 2cZ

S

Es xD

s ds ; A22

where is the quantity introduced in equation (A2) (Berger & Field 1984).Using equation (A1) in equation (A22), we get

_HH 2Z

S

Ds x

Ds Ds x Ds zz ds : A23

The second term in the integral may be transformed according toZS

Ds x Ds zz ds

ZS

Ds Ds x zz ds

Z

S

Ds Ds x zz ds

0 ; A24where Stokess theorem has been applied to get the last line, and then

_HH 2ZS Ds x Ds ds : A25Effecting in the latter relation an integration by parts, applying Gausss theorem, and using equation (A3), we have also

_HH 2Z

S

D2s ds 2

ZS

Bz ds : A26

It thus appears that does not explicitly contribute to changing the magnetic helicity. However, it plays an implicit role, whichis hidden in the time dependence ofBz on S(see eq. [A16]).

Finally, we also note for future reference the following expression, which is obtained by using equation (A10) in equation(A26):

_HH c

ZSS

BzrEsr0 x r r0

jr r0j2dsds0 : A27

APPENDIX B

A METHOD FOR CONSTRUCTING TIME-SEQUENCES OF FORCE-FREE FIELDS

We describe here a general method that allows us to construct a time sequence of force-free fields from an arbitrarily givenforce-free field. This method generalizes the one given in Paper I and used therein to provide an analytical class of fieldsevolving quasi-statically because of horizontal converging motions imposed on the footpoints on S.

Let B0r be an arbitrary force-free field in , and setBr; t lB0r ; B1

where l and are two arbitrary differentiable positive functions of t t0 ! 0 satisfying l0 0 1. With the same



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type of arguments as in Paper I, it is easy to check the following:

1. The field Br; t is also force-free at any time t ! t0, withBr; t0 B0r : B2

It can be computed from the vector potential

Ar; t lt2

t

A0tr : B3

2. Its -function is given by

r; t 0r ; B4which implies the current density

Jr; t c4

r; Br; t lJ0r : B5

3. The energy and the relative helicity ofBchange according to

Wt l2

3 Wt0 ; B6

H

t

l2

4

H

t0

;

B7

respectively.

4. The energies Wt of the potential field and Wt of the open field associated with the changing Bzrs; 0; t given byequation (B1) have the same scaling as Wt, and thus the ratios W=W and W=Wkeep constant values.

5. The fields Band B0 have the same ideal stability properties; i.e., they are either both stable or both unstable (linearly andnonlinearly).

The model of Paper I is recovered by choosingl 2.

APPENDIX C

SCALING SOLUTIONS OF THE TWO-DIMENSIONAL DIFFUSION EQUATION

We discuss here the existence of solutions to equation (8) of the form

qrs; t lq0rs ; C1where and l are two positive differentiable functions of t t0 satisfying 0 l0 1. Using the latter equation inequation (8) leads immediately to

_llq0 l _rs x Dsq0 b2 D2s q0 ; C2where a dot denotes a time derivative and the functions q0,

Dsq0, and

D2s q0 are taken at the point rs. Equivalently, we can write

at the arbitrary point rs

_ll

lq0

_

rs x

Dsq0 b2 D2s q0 : C3

Since there is no longer any time dependence left in the functions q0,Dsq0, and

D2s q0, the three functions ofappearing in front

of them, respectively, need to be proportional to each other for equation (C3) to possibly have a solution. Then there do existtwo numbers kand such that

_ll

l k

_

k

22 C4

(the second constant has been written in the form k=2 for later convenience). This system of equations for and l can beintegrated immediately to give

11 1=2

; C5

l 11 k=2

; C6



20/20

where the initial conditions on and l have been used to fix the values of the two integration constants. It results clearly fromthe form of these solutions that we need to have > 0 [for otherwise would become singular at jj1].

Moreover, equation (C4) allows us to rewrite equation (C3) in the form

2D2s q0 2rs x

Dsq0 2kq0 0 ; C7

where we have set 2 4b=. Of course, we are interested here only in solutions that have zero total flux through S anddecrease sufficiently fast at infinity. It turns out that the latter condition implies k> 1 (to see that, multiply the equation by q0,integrate over S, and apply Gausss theorem). Then let us first assume that k 2 Ds x rs, in which case equation (C7) takesthe form

Ds x 2 Dsq0 2q0rs 0 : C8

It does admit as a particular integral

Dsq0 2q0rs 2q0Ds ln q0

B0 r

2s

2

0 ; C9

where B0 is an arbitrary constant, and integrating once more leads to the particular solution

q00rs B0er2s=2 : C10The function qrs; associated with this initial value is easily obtained by using equations (C1) and (C5)(C6), and it is recog-nized at once as being identical to the well-known self-similar solution to the two-dimensional diffusion equation (with an

initial singularity at 1

; see, e.g., Tikhonov & Samarskii 1963).Unfortunately, the function q00 is not physically satisfying since it has a nonvanishing total flux (just note that it keeps thesame sign on the whole S). We can construct from it, however, an admissible solution by merely noting that

q0rs 2@yq00rs B0y

er

2s=

2 C11

is a solution of equation (C7) with k 3 (this is seen at once by differentiating with respect to y both sides of eq. [C7], in whichwe take k 2), and has zero total flux (it is odd in y).

By taking equations (C1) and (C5)(C6) into account, we thus obtain eventually as an example of satisfying the scalingsolution to the diffusion equation on S

qrs; t B0 y1 2 exp

r2s

21

: C12

To conclude, we note the following formulae that give, for any scaling solution, the values of the unsigned flux (flux throughS), of the energy of the associated potential field, and of that of the open field, respectively:

t 11 k2=2

t0 ; C13

Wt 11 k3=2Wt0 ; C14

Wt 11 k3=2Wt0 : C15

They are obtained by a simple change of variables in the integrals defining these quantities (see eqs. [18][19] for the last two).

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